Forgive me, I beg, the odd, breezy posting schedule this week. I reasoned that many would still be on vacation and wanted to save the (for a word) denser material for next week. Like the article on deriving objective morality: not simple, not short, not easy.

So I took this week to finish what I could of the second edition of a book that did surprisingly well (at least I thought so; it sold about 5,000 copies). Working title: Son of Everything You Believe Is Wrong. I got to the point I wanted an index, and since I wrote in LaTeX, I thought this would be particularly simple using AI.

I was wrong.

I should have expected it, since it was I who wrote about AI limitations: here and here (among others). These language models excel where they have all the causes, like in images and moving images (movies). They do okay, but err often enough, with code or math. As long as the code or math is common, it works fine. But once you move into new areas, well, you had better know the right answers, or you could fool yourself easily.

My book, I told both Grok and ChatGPT, was a book on logical arguments. It therefore assumed it would be like all those other books, which it is not. You can follow the hilarity in the video below.

Briefly, since I already blew two days on this and want to purge the bad taste, this:

I asked Grok for index tags, which in LaTeX are simple: “\index{word}” next to the words. It kept swearing, and promising larger and larger numbers, of index tags were being inserted into my text. Best I could get was two tags. Mostly none. Worst was my own idiocy for not checking and continuing on as if it was working.

Because I had to keep uploading chapters, I eventually ran out of my weekly credits. I had heard ChatGPT does better with language, so I created an account there (I never had one) and asked the model if it could do LaTeX index tags. It could!

I uploaded the whole book, as requested, and what followed was the best comedy. Over and over and over ChatGPT said something along the lines of “This is good. Here is our plan. We’ll first create a Table of Concepts. We’ll then review the entries and I’ll do the tags.”

I kept answering “Great, let’s do it.” Then it said, in effect, “Agreed. Here is our plan. We’ll first create a …”

Over and over again. Then it turned out it said it couldn’t read the book it was quoting back to me. It told me to re-load chapters each time I asked a question, because of some internal limitation. I did so.

Promises by the acre were provided. But not one damned index tag did I get.

Then I ran out of the daily credits or whatever they use to track usage.

My own fault, all of it. I spent all yesterday afternoon doing the tagging by hand. Which I should have done in the first place.

Anyway, to tease the book, I created that seven second video with Grok. And since it knows the causes of images, it did a reasonable job. I thought it was cute. So I posted it on YouTube (and here yesterday).

I immediately lost a lot of followers especially on YouTube, doubtless on the suspicion that I had turned into yet another slop AI account.

I have learned my lesson. I promise.

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UPDATE: Man, that is the last time I use AI for anything like this. I thought it a cute way to introduce the second edition of (the surprisingly popular) Everything You Believe Is Wrong. But the instant I put it up on YouTube, several followers fled, doubtless fearing an AI takeover. And wait until you hear of my horrible interaction trying to get AI to build an Index. High comedy.

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Briggs: As a natural accompaniment to Ianto’s article, please do not miss Richard Greenhorn’s explanation of SCOTUS’s latest misery: “Justice Thomas’s 14th Amendment: The culmination of a career, and testament to another generation.”

Every morning, I descend my stairway to the hall that leads to my hearth room and kitchen. As I walk this hall, I pass before a large, beautifully illumined copy of the Constitution of the United States of America. I write this on the eve of the 250^th anniversary of the passage of this document into history. And 223 years into the dustbin of history.

I write this tonight because I am grieved at our loss of memory. I have not seen, in these past six months leading to this date, not a single article that calls into question whether this document still actually exists. Not a single article that asks whether it died aborning in 1803. Not a single word on what that death means for us today.

But I will now fill that void, out of a sense of obligation, akin to the death of a dearly departed child sibling. Because it’s true. America, the nation, has died. America’s place has been overtaken by a golem that resembles her, in the most superficial way. A true nation, the one we had hoped in.

To do this properly, I must first ask for a show of hands. Who here has actually read this document? Further, who has actually read it’s antecedent, The Articles of Confederation? Be honest, please.

Now let us examine this matter of whether we are observing a corpse or a living being. To do this properly, we need to take a short digression into contract law. Because we are dealing with the supposed Social Contract that governs us as a people. The Leviathan that rules us.

Supposedly, our government derives its power from those it governs. Think about that oxymoron for a moment. It would make sense, if man wasn’t a fallen creature. And if there was no God. But neither of those propositions is true. At least, obviously, not the first of them.

Now anyone who has signed a contract does (or should) understand what a contract is. It is an agreement of terms. And so, the terms are stated and defined. And this is done in the order of their importance to the matter at hand. The most important things come first. In other words, the who, what, when, where, why and how’s are stated and affirmed.

So we start with who (we, the people), the what (constrained by an unwieldy compact), the when (1789), the where (the American states), the why (in order to form a more perfect union), do hereby re-establish the order of our union. That is, we are now going to re-define the how. Now we get to the meat of the matter. The Articles that define us.

We now define the re-organization of the Leviathan. Which will directly predict its eventual size, and scope of power. The definition of our particular version of the Leviathan is spelled out in the Articles of this Constitution of 1789 (that is, 13 years after the Declaration of Independence, and the initial Articles of Confederation).

As in most contracts, the most important elements are placed first. That is, the elements of ‘the how’ mentioned above. The ‘how’ of which we are to be governed. So let us review the document, and we see that Article One of this new social contract is entitled ‘The Legislative Department’. This article then defines the scope and powers of this first of three branches of government envisioned in this re-organization of America, the singular nation.

Here we need to take note of a single, incredibly important fact, which is this: this new Constitution did certainly NOT contemplate or define a government of three co-equal branches. If it did, why were the three separate branches (legislative, executive, judicial) defined separately? And ranked in order? Simple. Because they were not intended to be co-equal. Why? Because, apart from the Trinity, with its underlying hypostasis, there is no way a three- (or even two-) way division of power will ever work. Not among fallen men, at least. And after all, if we are not fallen men, then why do we need a government?

What is the proof of my assertion that the Legislative department is the most important of the three branches the Constitution enumerates? Simple, citizen. Look at the enumerated powers it is given: the power to tax, the power to legislate, the power to declare war, the power to impeach and remove. Neither of the two other branches has these powers. Plus, it also has the power to over-ride the executive, when it vetoes a law the majority of Congress has passed. In other words, Congress has ultimate power over the executive (Presidency). And even more importantly, over the third branch, the Judiciary.

What am I referring to here? Simple, friend. I am referring to the article I read every morning, as I pass across my copy of the Constitution in my hallway. That is, Article III, Section 2, SubSection 2. This is the section that deals with what was overthrown in 1803 in the Supreme Court decision known as Madison vs Marbury.

Please note, this Wikipedia explanation omits the most salient fact of this case; namely that the defendant personally involved here was William Marshall, the brother of the Chief Justice, John Marshall. William Marshall, is who was going to jail if the matter was decided on either the facts or the law. And both the facts and the law were against him. So then, Komrade. What is to be done? Simple, citizen- find something in the penumbra of the law! Shades of Roe vs. Wade.

What does all of this mean? Simple, my fellow American: this fourteen year-old document (at that time) we know as The Constitution, was overthrown in a judicial coup that would lead to our change from nation to empire. The change from a government of the People to a government of the un-elected black-robed Tyrants.

Let’s be clear. The Articles of the Constitution were intended to go from the most important to the least important. The Legislative branch was set up to enact the law. The second Article (the Executive presidency) was set up to enforce the law the Legislature passed. The third branch (Article III, the Judiciary) was set up to rule on the law. But nowhere was there any power enumerated for the Judiciary (or Presidency) to nullify the laws enacted by the Legislature!

Let’s be even more clear here. In Article III (the Judiciary), Section II, Sub-section 2, it specifically removed the power of the Judiciary to review the acts of the Legislature, as specified by the Congress.

In other words, the Judiciary was subordinate to both Congress and the Presidency. There was never any intent to create three co-equal branches of government.

But here we are today, where the third and subordinate branch of government has jiu-jitsu’d its way to being the Supreme (Court) ruler of America. A complete reversal of the original intent of this document that supposedly places the will of the People as the source of legitimate rule. Instead, the will of the People is now subordinated to the will of five or more black-robed, un-elected tyrants who have somehow discerned the will of God in the affairs of men.

So much for Democracy. Which, by the way, is a word unspoken in this same document. And no wonder, eh? Aristotle knew the problem here.

We are now back to the question of whether America is what we think it is. Is it a nation ruled in favor of the elected will of the People? Or an empire ruled by a majority of an un-elected few who have overthrown the clear intent of the Founders, and the People they persuaded to ride along on this road to perdition?

Let me conclude by pointing out that I am not a fan of either choice. I hold no truck with the defenders of democracy, as I fully understand that The People can be wrong, on any given point. And I know full well that a Judicial microscopic subsection of these same People can be wrong as well.

But this I do know as well. It is easier to subvert five individuals than it is to pervert a majority of 435 locally elected individuals. Especially if it is supposedly illegal for a foreign nation to influence (buy) anyone’s allegiance against our own self-interest.

But given that this foreign purchase of influence of either the five (the Judiciary), the one (The Presidency) or the 435 (the Legislature) is not precluded by law (or rather, enforcement), we will end up in the same place we are in today.

That is to say, in the Waiting Room of Hell. Please take a number. We will be calling your name soon.

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I was just a kid. I wore an Uncle Sam costume my mom made, and my sister was Lady Liberty, complete with torch. My dad helped me rig up this old heavy wood-sided cart, and I painted each side with American flags. The cart was on hard rubber casters and didn’t move too good, but I was able to drag it with my sister in it the entire length of the Alpenfest parade that year of 1976, the Bicentennial.

My grandpa was busily reading through the Kent Family Chronicles paperbacks, novels about America’s founding by John Jakes. Enormous bestsellers. He kept them on a window sill in the front room of the cabin on Otsego Lake. They don’t make mass market paperbacks like these anymore, which I find very sad. But then, the size of the books were too large for me and I didn’t try to read them.

The Bicentennial was everywhere that year. Especially on TV. Back then, I thought TV was the greatest invention of all time, and my biggest ambition was to save enough money to buy a small portable B&W set so I would watch Count Zappula. Better, I wouldn’t have to watch my mom’s shows, like The Love Boat. Way too much kissing. Bring on The Muppet Show!

We had four channels, one of which seemed to belong to Mr Roger’s and England. Best thing was that you didn’t have to pay for any of these channels. Can you believe it? Just stick an antenna into the air and they “streamed” at no cost. My dad, then as now a sports fan, got to see all the events he enjoyed and didn’t have to “subscribe” to fourteen different services to get them. Streaming will wound, or even kill, sports, because it fragments audiences, who can’t keep up. Same thing happened to boxing when they began Pay-Per-View.

Having only four channels, and a handful of newspapers, made it a lot easier to keep things together. I emphasize the small number of outlets for information, because back them it seemed we were—no, not seemed, we definitely were—one people. As much as an endless land filled with two hundred plus millions of souls could be one people, that is. There were always divisions, of course, but weren’t of daily importance as they are now. Back then, we of course had many who broke into the country illegally, but this was then an mere annoyance. It only became a flood later under Equality. We are now no longer one people, but many peoples.

When we were one, it was technologically easier to be one. Easier to get the message out, which of course could go either way. But then, as much as it could, it went the right way a lot more often. And, as one, everybody got behind the Bicentennial. There was a great spirit of pride and, yes, fun in the air.

The cynical view is that people were cashing in, as people are always cashing in. Surely some were. Yet not all. There were great measures of sincerity, which were obvious. Even in the advertisements. The advertisers might not have believed the messages they were selling, but they knew their audiences did.

New ads are in your face, always featuring improbable, and even impossible, family scenarios, obviously designed to be inoffensive to the Hersterical Set. That they might be offensive to normal people doesn’t matter.

Besides those “That’s the Way it Was” sports, somebody had the bright idea of The Bicentennial Minute, a series of one-minute bursts or pure patriotism. They were so ubiquitous that parodies of them were made.

You couldn’t feature these today, not because there isn’t the channel for it, but because much of this history is now so ancient, and so untaught and so unfamiliar, that you may as well air segments on the early history of Athens. If anybody even attempted it, the ads would be DIE propaganda. Which nobody wants to see. Would any schoolchild today know “One if by land”, the Stamp Act, Continental Congress, Articles of Confederation. Benedict Arnold, Nathan Hale, Betsy Ross, Publius, or even John Hancock?

This is what the “news” did back then:

Unimaginable now. I mean that word in its full sense: you cannot imagine any “network” producing material like this today. All day long in San Francisco, the reporter said, San Francisco, there would be “parades and festivals and fireworks.” Will there be these delights this year in that city?

I loved TV when I was a boy, but I gradually shifted and became a radio guy, and still am.

Radio is dying, so is TV, and newspapers are as good as dead, even as ways to communicate expand. Maybe there’s a TikTok or Instamgram channel devoted to the Sesquicentennial, but I wouldn’t know it, since I have neither of apps. I blow all my time on Twitter, and haven’t seen much about the 250th there. Some carping from some lowlife actors or signers is all I can remember.

But it was festive in 1976, and we had a ball. We were all proud to be an American.

Many of us still are.

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There will be no Class this week or next because of the Fourth. But to tide you over, this short video instructing you never to use of trust “P-values”.

P(agan)-values are a holdover from our pagan days when we thought Chance was real and that Probability imbued things with life, and Fate has causal powers. In particular, almost all medical research relies on Pagan-values. Which is why one week we see the headline “Chocolate cures heart disease” and the next we see “Chocolate causes heart disease”, all the result of P(agan)-values.

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Some fellow who got rich once he was in office, Ro Khana, claims Elon Musk, who got rich outside office, is responsible for some preposterous number of deaths, something into the millions, because of Musk’s involvement with the once-happy bureau called the Department of Government Efficiency.

You recall DOGE was doing too good at its named job, cutting spending, which is forbidden in government, so it had to go. But not before it cut off the firehose of funds, the system which took my money, and probably yours too, and sprayed it out over the world. The reason the government gave for this seizure and redistribution was that I had money, the rest of the world did not have my money, and therefore the rest of the world deserved it and had to have it.

USAID was the name of the government entity that removed my money, and probably yours too, and gave it to people elsewhere. This provided great power to politicians, and a modest penury in you. Musk was instrumental in stopping USAID, which naturally incensed those used to and dependent upon the largess.

This accounts for why since USAID was folded up, there have been any number of claims of how stopping my money from being shipped overseas is somehow killing foreigners. This is, of course, absurd, as we’ll see. But even if it were true, it is not my duty to stop these deaths.

The peer-reviewed paper is “Evaluating the impact of two decades of USAID interventions and projecting the effects of defunding on mortality up to 2030: a retrospective impact evaluation and forecasting analysis” by Cavalcanti and a slew of others in the political journal The Lancet.

Their Background:

The US Agency for International Development (USAID) is the largest funding agency for humanitarian and development aid worldwide. The aim of this study is to comprehensively evaluate the effect of all USAID funding on adult and child mortality over the past two decades and forecast the future effect of its defunding.

Now this will turn out to a thoroughly asinine paper, which is to science what “lived experience” is to politics. Which is to say, fevered fiction. Regular readers will not be surprised at any of this.

Suppose your neighbor hasn’t a hammer and asks to borrow yours. You give it to him, he uses it to some benefit. Later you need it back and take it. Yet your neighbor has not completed his project. What is the best action your neighbor can take?

• A: Suffer and let his project rot for want of a hammer,

• B: Get his own hammer.

There will, of course, be a good number of indigent, unintelligent, and just plain lazy people who opt for A. Everybody else with go with B.

As will the countries who find their small cuts of my money have dried up. Not forgetting, of course, like with the vast majority of NGO-like entities, most of the monies end up in the pockets of politicians, activists, and organizers, with only the smallest trickle leaking to (the amusingly labeled) beneficiaries.

Everybody knows of this kind of corruption. That “everybody” ought to include researchers who build models to “prove” how missing USAID funds kill people. Meaning those models ought to include adjustments for the certain fraud that occurred. Did those models include adjustments for fraud? No, sir, they did not.

Indeed, the models assumed option A would be the case everywhere and for all peoples. Our researchers could not envision anybody could find a way to cope with the loss of USAID. They could only suffer.

The rest of the paper is yet another instance of the epidemiologist fallacy. This is when a researcher announces “X causes Y” but where X is never measured, and sometimes Y isn’t either, and proxies are given in their place, and where the cause is “confirmed” with wee P-values, which is always, no exceptions, a logical fallacy.

Yes. That “no exceptions” means even the studies you like, or the ones you did yourself.

Here are their Methods (feel free to scan):

In this retrospective impact evaluation integrated with forecasting analysis, we used panel data from 133 countries and territories— including all low-income and middle-income countries (LMICs)—with USAID support ranging from none to very high. First, we used fixed-effects multivariable Poisson models with robust SEs adjusted for demographic, socioeconomic, and health-care factors to estimate the impact of USAID funding on all-age and all-cause mortality from 2001 to 2021. Second, we evaluated its effects by age-specific, sex-specific, and cause-specific groups. Third, we did several sensitivity and triangulation analyses. Lastly, we integrated the retrospective evaluation with validated dynamic microsimulation models to estimate effects up to 2030.

No amount of USAID funding any person was given was measured, and neither was it measured whether any person lived or died because of that funding. In other words, X was not measured, nor was Y. Yet the claim (we see next) will be “X caused Y”. Instead, we get things like this: “Our modelling approach, based on previous studies, was done in two stages: first, we created a synthetic cohort of all countries for the years 2024–30, extrapolating and modelling each country-level independent variable from the retrospective dataset…”

Here are their Findings (cutting out the boring stuff, so please check the original):

Higher levels of USAID funding—primarily directed toward LMICs, particularly African countries—were associated with a 15% reduction in age-standardised all-cause mortality… and a 32% reduction in under-five mortality … This finding indicates that 91 839 663 … all-age deaths, including 30 391 980 … in children younger than 5 years, were prevented by USAID funding over the 21-year study period. USAID funding was associated with a 65% reduction … in mortality from HIV/AIDS (representing 25·5 million deaths) … Forecasting models predicted that the current steep funding cuts could result in more than 14 051 750 (uncertainty interval 8 475 990–19 662 191) additional all-age deaths, including 4 537 157 (3 124 796–5 910 791) in children younger than age 5 years, by 2030.

You see it: the direct claim of causality (deaths in children “prevented by USAID funding”) from a regression, which is forbidden in statistics, but of course happens all the time.

This model is really no different than the vast majority of economics models. It builds in the cause: it assumes the cause. The cause is assumed here by saying the positive correlation of USAID money caused the lives saved. The model is then run, and lo!, some researcher announces “We have discovered the cause we built into the model is the cause.”

For all models only say what they are told to say.

One of the main arguments against psychic abilities is the complete absence of psychics among the worlds trillionaires, or even billionaires. If people really could foresee the future or read minds they would have been able by now to have used these talents to observable effect. We do not observe these effects, and we are therefore rational to predict people who call themselves psychics are frauds.

The same argument works for economists.

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Jim Franklin hosted this classic essay of the late David Stove’s for many years. But he received a complaint from one of the intellectually Tolerant. As Franklin said, “An authorized officer of the University of New South Wales has requested that one of David Stove’s articles not be hosted on a UNSW web server. So the David Stove website has moved. It can be found at [Gerry Nolan’s Stove site].”

The article is “The Intellectual Capacity Of Women”, which appears in Chapter 5 of Cricket Versus Republicanism, Quakers Hill Press, 1995 and was originally published in Proceedings of the Russellian Society, Vol. 15, 1990. It also appears in a must-have compendium of Stove’s work edited by Roger Kimball Against the Idols of the Age. It was Kimball who first alerted me to Franklin’s predicament. Franklin, who is Stove’s literary executor, gave me permission to host the article here.

To say that this essay was controversial is like saying Hillary Clinton is shrill. I post it for several reasons. One: there’s gold to be mined from it. Two: to prove that those who say “In science no topic is sacrosanct” and “There are no dogmas in science” lie; they are untruthful. Three: to make sure it doesn’t die. Four: Stove was a self-professed non-theist, but that doesn’t mean he wasn’t brilliant. Still, word has it that he wasn’t entirely satisfied with parts of this essay as time went by, so there is plenty to discuss (such as the nature of probability). Five: Since the main article is partly philosophical and mainly probabilitistic, it will be of great interest to regular readers.

If you’re reading this via email, because of Substack’s limitations, it may be truncated. Click to go to the site, or download the PDF of the articlee.

One word of advice: read it before (as in before) commenting. Off we go!

————————————————————————————————-

I BELIEVE THAT the intellectual capacity of women is on the whole inferior to that of men. By “on the whole,” I do not mean just “on the average”; though I do mean that much. My belief is, if you take any degree of intellectual capacity which is above average for the human race, as a whole, then a possessor of that degree of intellectual capacity is a good deal more likely to be man than a woman.

This proposition is consistent, of course, with there being women, and indeed with there being any number of women, at any level of intellectual capacity however high. But it does mean, for example, that if there is a large number of women at a given above average level of intellectual capacity, then there is an even larger number of men at that level.

In the past almost everyone, whether man or woman, learned or unlearned, believed the intellectual capacity of women to be inferior to that of men. Even now this is, I think, the belief of most people in most parts of the world. In this article my main object is simply to remind the reader of what the evidence is, and always was, for this old belief, and of how strong that evidence is.

An opposite belief has become widely current in the last few years, in societies like our own: the belief that the intellectual capacity of women is on the whole equal to that of men. If I could, I would discuss here the reasons for the sudden adoption by many people of this opinion. But I cannot, because I have not been able to find any reasons for it, as distinct from causes of it. The equality-theory (as I will call it) is not embraced on the grounds of any startling facts which have only lately come to light. It is not embraced on the grounds of some old familiar facts which have been misunderstood until lately. It is not embraced, as far as I can see, on any grounds at all, but from mere prejudice and passion. If you ask people, “What evidence is there for the equality-theory?”, you do not get an answer (though you are likely to get other things).

Rather, the question is felt to be somehow improper, morally or intellectually, and is thought not to deserve any answer.

I do not know why it should be thought so. The question is a perfectly proper one morally and intellectually, and should not be hard to answer. That men and women have the same intellectual capacity is not, after all, a self-evident proposition, like (say) “7 + 5 = 12” nor is it something just obvious, like (say) the sun’s rising in the east. So if it is rational to believe it, there must be evidence for it: facts which lead to it by good reasoning. But where is that evidence to be found?

By contrast, there is no difficulty at all in saying what the evidence is, and always was, for the other theory, the theory of the inferior intellectual capacity of women. This evidence is not at all esoteric, but on the contrary is of the most familiar and homely kind. The main reason why I believe, and the main reason why nearly everyone always has believed, that the intellectual capacity of women is inferior to that of men, is just this: that the intellectual performance of women is inferior to that of men.

The reasoning involved, then, is reasoning from inferior performance to inferior capacity. It is reasoning of the same general kind, therefore, as that which convinces us, even if we understand nothing of the internal make-up of cars, that Fords are on the whole inferior to Mercedes; or as that which convinces dog-fanciers that Irish setters are not as smart as labradors; or as that which convinces everyone that the intellectual capacity of seven-year-old children is on the whole inferior to that of nine-year-olds. They do not do as well, and we infer from this that they cannot do as well.

This is a very homely kind of reasoning, to be sure. But that is not to say that there is anything wrong with it, and in fact no one distrusts reasoning of this kind. On the contrary, we could scarcely take a single step, in science or in common life, if we did not rely on this kind of reasoning.

Of course no thoughtful person mistakes such reasoning for proof. Inference from inferior performance to inferior capacity is fallible: that should go without saying. Everyone knows that a car, or an organism, may on a given occasion fail to perform as well as it can perform: there was some interfering factor at work. And this can happen not just on one occasion, or to just one organism. A whole class of organisms might perform below capacity, in a given respect, for any length of time, or forever. It is even logically possible that every organism of a certain kind should have a certain capacity, and yet that interfering factors prevent every one of them from ever exercising that capacity even once. So far, then, is inferior performance from being an infallible indication of inferior capacity. And so far, too, should we be, from mistaking the inferior intellectual performance of women for a proof of their inferior intellectual capacity.

This, then, is one commonplace truth which needs to be borne in mind when we think about the intellectual capacity of women: that capacity does not require performance. But there are other such commonplace truths, and some of these point in the opposite direction.

One is that, although performance is no infallible guide to capacity, it is, in the end, the only guide we have or can have. I do not mean that there can be no evidence of A’s capacity to F, unless A actually has F-ed at least once. That would be a stupid thing to say. When I meet a brown snake in the bush, I have good evidence of its capacity to inflict a dangerous bite on me, even if this particular snake has never bitten anyone. Again, a chemist often has good evidence concerning the capacities of a compound which, until he makes it in the laboratory, has never even existed, and which therefore cannot possibly have yet exercised any of its capacities. All I mean is, that the evidence for an unexercised capacity, which is a kind of unrealised possibility, cannot consist in its turn just of other unexercised capacities, or unrealised possibilities. Such evidence must include some actualised possibilities, some exercises of capacities. If the chemist, for example, is entitled to say in advance that his new compound X will have the capacity to F, that is because he knows of capacities which have actually been exercised by existing elements or compounds. While, then, capacity does not require performance, still evidence of a capacity does require performances, of some kind, by something or other, somewhere along the line.

Another important commonplace about capacity and performance is this: if we believe that something has a certain capacity, but the expected performance is not forthcoming, then we may not postulate just any old interfering factors in order to explain the discrepancy. Suppose our pet theory is that every B has the capacity to G, but the evidence is that B’s have never or hardly ever G-ed, although there have been billions of B’s, placed in the widest variety of circumstances. Then we may not save our theory just by saying: “H is a factor which inhibits G-ing, and it is possible that H has been present in most of the cases.” Nor may we just say, “Oh well, there must be something which has so far stopped B’s from G-ing”, or “Somehow, B’s have never had a fair chance to G.” Nor may we just say, “Satan likes to stop B’s from realising their G-potential; he is a non-B himself, you know.” Statements such as these might happen, indeed, to be true. But given the evidence of the B’s’ actual performance, one would be irrational to believe them. Where the relevant performance is absent, it is rational to believe that a capacity is present, only if there is evidence of some actual, specific, and detectable interfering factor. Merely possible interfering factors, or actual but indefinite ones, or ones which, even if actual and definite , are undetectable (like Satan), will not do.

The reason why they won’t do is obvious enough: that otherwise anyone could safely ascribe any capacity to anything. I, for example, a former village-cricketer but now too old even for that, could safely claim to have the capacity, at this very moment, to play cricket as well as Vivian Richards does. Brown bread, as well as brown snakes, could be credited with the capacity to inflict a dangerous bite. Men could be credited with a thousand times the intellectual capacity of women; and vice versa; and so on. We would need only to blame undetectable interfering factors, or indefinite ones, or merely possible ones, for having so far prevented the exercise of these interesting capacities.

Suppose that you want to know whether a certain coin is a fair one, but that for some reason or other you are not allowed to subject the coin to any direct physical test. In that case you will have to rely on tossing it, and observing the relative frequency with which “heads” and “tails” come up. You will have to infer what the probability of heads is, i.e. whether it is .5 or not, from the observed frequency of heads. As it would be manifestly irrational to draw such a conclusion from a few tosses, you will of course toss the coin a large number of times. A thousand tosses would surely be a large enough “sample” to base a rational inference on. Suppose the coin comes up heads 539 times in this thousand tosses. Then, if you are rational, you will strongly incline to the belief that the coin is not fair, but slightly biased to heads. If you wanted to put the matter beyond reasonable doubt, you would try to exclude, as far as one can, the possibility that the coin itself is fair but that the tossing of it has so far not been fair. The only rational way to do that is to arrange for more sets of tosses, of a thousand each, to be made. You will have some of them made by Jack and some by Jill, some in January and some in June, and so on. In short you will “vary the circumstances” (as philosophers say), in every way you can think of that seems at all likely to make any difference to the frequencies. If you have suspect that a factor H, which was present for certain sets of tosses, may have interfered to depress the frequency of tails below the probability of tails, then you will ensure that H is absent in an equal number of other sets of tosses. Suppose that in this way you make a thousand sets of a thousand tosses each, i.e. a million tosses all told, and that the number of heads in a thousand tosses never falls below 500, never rises above 550, and in almost every case is between 525 and 535. Then, if you are rational, you will be firmly convinced that the coin is not fair, but is biased to heads to the tune of about 53-47: that is, that the probability of heads with it is about .53. Someone who knew all the evidence of the tosses, and had no specific objection to make to the variety or the number of the trials, yet persisted in believing that the coin is fair, would be manifestly irrational. Indeed, one would have to doubt such a person’s sanity, or suspect that the fairness of the coin is a point of religion with him. His crazy proposition, that the probability of heads with this coin is .5, is not logically inconsistent with the observed frequency of heads being about .53. The fact is, a probabilistic or statistical generalisation (with the trivial exception of “closed” ones, like “80% of the present Federal Cabinet are men”), can never be disproved by observed frequencies. “The probability of an F being a G is r” is consistent (once you set aside the “degenerate” values of r, 1 and 0), with every possible frequency of G’s which might be observed in a sample of F’s, however large the sample, and however varied. Hence if someone were to accept all the evidence mentioned above about the tosses, and still maintain that our coin is fair, he would not actually be contradicting himself

This only goes to show, of course, that consistency, or not contradicting yourself, is only a very small part of rationality. To believe that the coin is fair is irrational in the light of our evidence, even though it is consistent with that evidence. After all, if the coin had never once come up heads in all of our million tosses, then that too—that is, an observed frequency of no heads in a million tosses—would have been logically consistent with the hypothesis that the coin is fair. Or again, if someone had the hypothesis that the coin is so very biased that the probability of heads with it is .99, then that hypothesis too would be consistent with an observed frequency of no heads in a million tosses. And so on: we need to fix in our minds the point that absolutely any observed frequency is consistent with every probability other than 1 and 0. And a consequence is this: where probabilitistic or statistical hypotheses are concerned, consistency with the observed frequency counts for literally nothing in favour of any one hypothesis, because it is a property common to every hypothesis.

In our coin case, therefore, with an observed frequency of heads uniformly around .53, it would be merely fatuous if some one kept on saying “Still the coin might be a fair one,” or “You haven’t proved that the coin is biased,” or “There could have been an unnoticed factor at work, depressing the frequency of tails below the probability.” These statements are true, but they are fatuous, because they are merely logical truths. They are simply ways of expressing the truth that the frequency of heads which has been observed is logically consistent with the proposition that the coin is fair; and that, as we have seen, would be true whatever the observed frequency had been. The logical possibility that a factor interfered with tails is no reason whatever to believe that a factor did interfere with tails; any more than the logical possibility of your car having turned into a lobster last night is a reason to believe that you have a four-wheeled lobster in the garage now. Reason to believe in an unexercised capacity, such as a capacity in our coin to come up tails as often as heads, requires evidence, (as I said before), of a detectable, definite, and actual interfering factor. The logical possibility of interference is no evidence whatever of interference.

The only rational inference to make about our coin, in view of the number and variety of the trials, is that the probability of heads with it is close to the observed frequency of heads, i.e. about .53. Of course this reasoning is not proof. Of course we may be mistaken in rejecting the hypothesis that the coin is fair.

But these, as I keep emphasising, are trivial logical truths. And they take nothing away from another logical truth, namely this: that in relation to the evidence which we actually have, it is nearly certain that the coin is not fair, but that the probability of heads with it is about.53.

Probabilities are a sort of graduated capacities, and the question about the comparative intellectual capacity of men and women is, like the question whether a certain coin is fair, a question about probabilities. What we want to know is whether the probability of a woman having d, where d is a given above-average degree of intellectual capacity, is equal to the probability of a man having d. We were not able, I supposed, to make direct physical tests from our coin: that was why we were obliged to infer the probability of heads from the frequency of heads observed in a large and varied sample of tosses. Just so, we are in fact not able to make direct tests of human intellectual capacity. We do not know enough to be able to do so at all as yet, and tests which are both precise and reliable are indefinitely far off. Even when they become technically possible, law or morality may well forbid their being made.

We are therefore thrown back, just as in the coin case, on having to infer the probabilities from the observed frequencies: that is, on inferring the comparative intellectual capacities of men and women from their comparative intellectual performances in that large and varied “sample” which is past human history.

In this sample, just as in the coin case, the observed frequencies are uniformly unfavourable to the equality-theory. In every field in which intellectual capacity can be exercised, from the most severely theoretical to the most intensely practical such as business, or medical practice, or war, there have always been far more men than women at any above-average level of performance. This is not in dispute.

No one disputes the following, either: that from this uniform inequality in the frequencies of intellectual performances between the two sexes, it will be rational to infer a similar inequality in their probabilities of intellectual performances, that is, in their intellectual capacities, if the observed sample, namely human history, is (first) large enough, and (second) varied enough.

And no one disputes that the sample is large enough. There have been a very great many women and men. Human intellectual capacity is a coin which has been tossed, not a million times, but very many billions of times.

The question, therefore, on which everything in this dispute turns, is whether human history is a varied enough sample to base a rational inference on. I believe, and most others, I think, believe, that it is. That is, we believe that there has been enough variety, in the circumstances in which humans have been placed, to exclude every explanation worth considering, of the observed difference in the intellectual performance of the two sexes, except a difference between them in intellectual capacity. We therefore conclude that the respective probabilities are close to the observed frequencies, or that the intellectual capacity of women is about as much below that of men as their past intellectual performance has been below that of men.

The equality-theorist says, and as we have seen must say, that there has not been enough variety in the “trials.” Common to all or most past history, he must say, there has always been some factor or factors interfering with the exercise of the intellectual capacity of women, depressing the frequency of their intellectual performance below the probability of it.

Well, have there been such common factors, or have there not? Has the variety of human circumstances been great enough to constitute a fair trial of the intellectual capacity of women, or has it not? This is the question, I repeat, on which all turns.

On such a matter it should go without saying that certainty, or even the degree of rational confidence which we can reach concerning the bias of a coin, is out of the question. No one can have a detailed knowledge of all human history. Even more to the point, humans are not only more complicated than coins, but are, as far as we know, the most complicated things there are. As a result, a human has more ways of interacting with its surroundings than anything else has; and as a further result, there are more ways, in the case of a human than in any other case, in which a capacity can fail to get full expression in performance. It must therefore be harder, in the human case than any other, to reach a given degree of rational confidence that we have in fact excluded all but one of the possible causes of a given effect.

But the complexity of human beings, and the scale of human history, should not tempt us to draw the sceptical conclusion, that human history can never constitute a fair trial, or a trial reasonably believed to be fair, of human capacities. That would be merely silly. At that rate, if the intellectual performance of women were to be equal to that of men for the next million years, and in the widest variety of circumstances, we would still have to say at the end of that time that we hadn’t the vaguest idea how the intellectual capacities of the two sexes compare. Anyway, this sceptical or “desperation” position is not open to either side of our dispute, which is between two parties each of which holds that, although certainty is out of reach, rational belief on the matter is not only possible in principle, but possible on the evidence which is available now.

Whether or not the historical sample is varied enough, it is certainly prodigiously varied. The variety of physical and social circumstances in which women have found themselves is, surely, just about as great as the variety which is possible for any class of persons. Women have been pirates and poets, princes and paupers, priests and prostitutes: you name it, some women have been it, if it is logically and biologically possible for a woman to be it. Almost every conceivable factor, therefore, which might have been thought to constitute an impediment to the intellectual performance of some women, has been removed in the case of some other women. Yet their intellectual performance, or at least the comparison of it with the intellectual performance of men, has not varied. This is true of the variety in women’s circumstances which occurs spontaneously between or within societies; but the same is true of that variety in women’s circumstances which has been introduced by human contrivance. Wherever some defect has been found or imagined in existing arrangements for the education of females, energetic and ingenious people have always been busy setting up a form of education free from that real or supposed defect. Novel schemes of education, intended among other things to remove obstacles to the exercise of the intellectual capacity of women, are. at least as old as Plato, and hundreds of them have been put into more or less widespread practice. Yet despite all this variety in the supposed causes of female intellectual performance, the effects have been singularly invariant. I do not mean that these schemes of education have never had any effect at all on female intellectual performance. I do not know, but it is in any case indifferent to my thesis, whether they have or not. My thesis only requires, what is the case, that educational innovations have never shown any significant tendency to bridge the gap between male and female intellectual performance.

Is not this virtually-unlimited variety, variety enough? Has it not constituted a fair trial of the intellectual capacity of women? “No,” says the equality-theorist. But this theory now begins to reminds us of a supremely silly thing which G. K. Chesterton once said: that Christianity has not been tried and found wanting, but was found difficult, and never properly tried. Now, I ask you: Christianity not had a fair trial? Why, at this rate, nothing has ever had a fair trial, and we can know, or even rationally believe, nothing whatever about the capacities of anything: brown snakes, brown bread … anything. But this is just an even stronger version of that silly scepticism which I mentioned a moment ago. In fact we know, or near-enough know, that Christianity does not have the capacity, which it claims to have, to satisfy indefinitely the religious aspiration of all human beings. And such a case proves, let us notice, that an historical sample of performance can be varied enough, and large enough, to be the basis of a rational inference to capacity, or rather to the lack of capacity.

A comparison between Christianity, and the supposedly-equal intellectual capacity of women, is in fact worth pausing over. Equality-theorists are never tired of reminding us of the obstacles which have been put in the way of the exercise of the intellectual capacity of women, at such-and-such a period, in that society or the other; and of course there are countless such cases. Those obstacles, however, have never been more than trifles when compared with the obstacles which, in countless cases, have been put in the way of the practice of the Christian religion. It is a mere abuse of words to speak, as some do, of “martyrs” and “persecution” in the one case as in the other. In both cases, for every instance in which some obstacle was put in the way, there is another instance in which that obstacle was not put in the way. Now, Christianity has sometimes made its way, sometimes without obstacles, sometimes even with obstacles; whereas the supposed equal intellectual capacity of women has never made its way, with or even without obstacles. Yet female intellectual capacity has obviously been tried in a far greater number of cases, and in a far wider variety of circumstances, than Christianity.

You can’t go on forever saying “The game’s not fair,” when the game has been played ten billion times, under a billion different circumstances; at least, if you are rational you cannot, unless you are prepared to say in just what way it is not fair. Exactly where, then, is the variety in our historical sample deficient? Just what is that factor, common to all or most past history, which has interfered with the exercise of the intellectual capacity of women? And the answer must be, as I said earlier, some actual, definite, and detectable factor.

But of course not every such factor will do. Some people love just stringing together anecdotes: women were prevented from exercising their intellectual capacity by this obstacle in Periclean Athens, by that obstacle in Confucian China, by the other obstacle in seventeenth-century France, etc. But an equality-theorist must do more than this. He has to offer some definite explanation of why the intellectual capacity of women has so consistently met with obstacles it could not overcome, and his explanation must be one which is consistent with the equality-theory. It would obviously be no good, for example, if he were to say, “The main interfering factor has been the aggressiveness, sexual exclusiveness, and superior cunning of males.” This suggestion, considered in itself, is by no means without merit: aggressiveness, sexual exclusiveness, and superior cunning are definite and detectable things, and I at least believe that they actually do operate in males, and do impede, to some extent, the intellectual performance of women. But of course the suggestion is not one which an equality-theorist can adopt, since to ascribe superior cunning to males is to contradict the very intellectual equality for which he contends.

Presumably no equality-theorist has ever been quite so foolish as to offer the explanation just mentioned. But there are subtler forms of the same inconsistency which do constitute a serious temptation to an equality-theorist, and a more serious temptation, the more sensible the equality-theorist happens to be. What could possibly be more natural and sensible, for example, than to suggest that the uniformly inferior intellectual performance of women is connected in some way with their reproductive capacity? Some of the more sensible equality-theorists, accordingly, are somewhat drawn to this suggestion. But then it will scarcely be possible for them to avoid contradicting their own theory.

A plantation-owner gave his slaves a large quantity of whisky on New Year’s Eve, and asked them next morning what they thought of its quality. “Just right,” was the answer. “What do you mean, ‘just right’?” “Well, if it had been any worse we couldn’t have drunk it, and if it had been any better you wouldn’t have given it to us.” If this story is not true it is at least well-founded.

It is also of very wide application: “Just right” is the general rule in the organic world. A species has those capacities, in those degrees, which are necessary, but no more than are necessary, to keep up its numbers. As Hume wrote (in Part XI of his Dialogues Concerning Natural Religion): “Every animal has the requisite endowments; but these endowments are bestowed with so scrupulous an economy, that any considerable diminution must entirely destroy the creature. Wherever one power is increased, there is a proportional abatement in others. Animals which excel in swiftness, are commonly defective in force. Those which possess both are either imperfect in some of their senses, or are oppressed with the most craving wants.” Darwinian theory acknowledges this fact, of course, and even up to a point explains it. But the fact was noticed long before Darwin, and must be acknowledged on any theory.

Although “Just right” is the general rule in nature, it is not the invariable rule. Since species are sometimes extinguished, their capacities are not always up to their tasks. And on the other side there are, or appear to be, excessive endowments, and capacities of a degree quite beyond what is needed to enable the species to keep up with the Joneses. A famous example of excessive endowments is the peacock’s tail: Darwin tells us it gave him nightmares, as well it might. But human intellectual capacity is itself another striking example. Humans do indeed need some intellectual superiority over other animals, precisely because (as Hume goes on to say) they are, of all animals, “the most necessitous, and the most deficient in bodily advantages”: they have no natural weapons, no body-covering, no uncommon strength, stamina, or speed. But why have humans got so much intellectual capacity? What biological need was there for the intellectual capacity, I will not say of a Hume or a Darwin, but even of the average philosopher? Wallace, though more Darwinian than Darwin himself, in the sense of being less Lamarckian, was convinced, by the superfluity of human intellectual capacity, that something more than natural selection must have been at work in the descent of man; as well he might be.

Still, whether or not there is an Author of Nature, there certainly is, overall, a recognisable style in nature, and it is the ‘Just right’ style: the style, as Hume says, not of “an indulgent parent” but of “a rigid master.” And the principle of parsimony does not stop at the distribution of capacities to species. It extends to the distribution of capacities between the sexes of a single species. If a species devotes a large part of its energy-budget to food-getting, say, or the defence of territory, or home-making, or reproduction, or nurture of young, then one sex takes on more of that task than the other does, and it will take on less of some other major task; and either sex will be comparatively or entirely deficient in the capacities required for the “specialisms” of the other sex. Again, this is not an invariable rule. In many species of birds, for example, the heavy task of getting food for the young falls equally on both sexes. But it is the general rule: if you take any major task that a species performs, it is exceptional for the two sexes to be exactly equally equipped to perform it.

The energy of organisms is like your money: it can be turned into almost anything, but there is only a certain amount of it to go round. So, whether among species or between the sexes of one species, lavish expenditure on one side means going short on another. The more you invest in heavy body-covering, say, the less you can put into mobility, and so on. And if one sex spends energy extravagantly on defence of territory, say, then the other sex is going to have to do more than an equal share of other main jobs, such as food-getting, or nurture of young.

On the side of reproduction and nurture, the human race carries a uniquely heavy burden. Gestation is long and taxing: yet that is the comparatively easy part. The adults of the species are omnivores, yet the new-born are such tiresome doctrinaires about food that almost nothing in the world except milk will do for them, and even at non-human milk they are apt to look askance and die. Above all there is the extreme and prolonged helplessness of the human young. This is, indeed, so marked a peculiarity of our species, and so cruel a drain on its energy-budget, that it gave rise to the thought, as early as the sixth century BC, that we must have descended from some other species which had a more sensible policy on nurture. We would never have got here, the Greek philosopher Anaximander reasoned, if all our antecedents had been as extravagant as we are in spending energy on absolute beginners!

Virtually all of this huge burden must fall on women. Men are not equipped for gestation or lactation, and what they can do to share the burden of nurture of the new-born is extremely little. The mother and the new-born, in man as in all the higher mammals, are obviously programmed to attach themselves to one another, literally and metaphorically; but there is no such programme in the father. Recently, of course, fathers of a certain kind have made a great business of being at the birth of their children, sharing the nappy-changing, etc., and no doubt some of them are occasionally of some real help in this way. But really, when you compare this with the deadly-earnest business which is going on at every moment between mother and infant, it is play. It is about as much help as those elaborate mock-confinements, etc., which fathers in many primitive societies insist on undergoing, and it should indeed be recognised as our contemporary version of those. In short the whole affair is mainly an expression of injured male self-importance, and a graceless one at that. Of course all this may be different, and the burdens of reproduction and nurture may be shared more equally between the sexes, if and when our species is superseded by another. But I am talking about H. sapiens, and in that species those burdens cannot fall to any significant extent on males.

Now in man, as in all animals, a peculiarity of reproduction-and-nurture is this: that of all the major tasks of the species, it is the one for which the innate programme is most complete, and the one, therefore, which requires the least intellectual effort for its performance. In plain English: a woman does not need to use her brains to have a baby, and doesn’t even need to use them much in order to see the infant through the period of its most extreme helplessness. Nearly everything she needs to know is already written in her inner manual. By contrast, every other major task, such as getting food or defending territory, abounds in totally-unforeseeable individual contingencies. It can therefore be only schematically programmed-for: you have to fill in most of the important details ad hoc, as each new situation arises. And that means that you constantly have to be using your brains. Think, for example, of the task, the very urgent task, of getting meat to eat. It is an extremely difficult task, and the difficulty is intellectual as much as physical. There is no meat going round asking to be made a meal of, and all the meat that is going round displays amazing cunning in the effort to preserve its life. Even finding the animal is often an intellectual feat, but even when you have found it, unless you can also out-think it, you will eat vegetarian or not at all that night. In short, the intellectual activity which is required for successful hunting is extremely great. And so it is, to a greater or less degree, with all the other major tasks confronting the human species, except reproduction. And so it must be, for the simple reason which Hume gave: that humans have got very little going for them except “reason and sagacity.”

In summary, then: our species has, like every other, a fixed energy-budget, and out of this it spends very heavily, far more heavily than any other species does, on reproduction-and-nurture. This great cost is borne exclusively by women, while reproduction-and-nurture is, of all the major calls on the budget, that one which demands the least intellectual activity. Now, there is a general principle of parsimony prevailing in the organic world, according to which a species, or a sex within a species, is endowed only with the minimum capacities, or degrees of capacities, needed for survival.

One would expect, then, that the sex not burdened with reproduction-and-nurture would shoulder the main burden of those other major tasks which are intellectually more demanding; and therefore, by the principle of parsimony, that men will have a higher degree than women of what is peculiarly required for those tasks, intellectual capacity. I do not claim that this inference is inevitable, but it is at least a natural one. And vague as its premises are, they do furnish, I believe, the lines along which an explanation must be sought for the intellectual difference observed between men and women.

Even some equality-theorists, as I said before, show signs of inclining to some such biological explanation of the facts. But any equality-theorist who accepted the explanation which I have just outlined would be simply inconsistent; just as one would be who accepted an explanation in terms of superior male cunning. What the equality-theorist needs is an explanation, consistent with the equal intellectual capacity of men and women, of their unequal intellectual performance. But what I have just advanced, if it is an explanation of anything, is an explanation of the inferior intellectual capacity of women. I offer, indeed, a kind of interfering factor, but that factor is inferior capacity, rooted in the general biology of the species and its sexes. The equality-theorist, on the other hand, has to try to find some interfering factor which, though it has beset the entire female sex throughout the entire history of the species, is not rooted in biology. Well, it is his work, and he is welcome to it.

I have sometimes been asked “What would it take to convince you that the intellectual capacity of women is equal to that of men?” This is a kind of question which it is often instructive to ask, and certainly it is a fair one here. I can easily answer it and will.

But since intellectual performance is the only guide we have to intellectual capacity, while the intellectual performance of men has always been superior to that of women, the question is an ever fairer one if it turned around, and addressed to the equality-theorist: “What would convince you of the inferior intellectual capacity of women?” And this question is one which many equality-theorists, I think, would not find at all easy to answer. The religious quality of their attachment to their theory is, in many instances, only too obvious.

Here is something which would not convince me of the equality-theory: reports by psychologists or educationists of tests, conducted within recent years, on (for example) the comparative mathematical ability of boys and girls. Such reports would not only not convince me: I do not believe that any attention at all should be paid to them. My main reason for this is not the public record of psychologists for fraud or susceptibility to fraud or of educationists for unswerving obedience to the winds of fashion; although this record is sufficient in itself to justify a hearty scepticism towards their reports. My main reason is a quite general principle: that a person’s testimony should carry no weight or little weight with you, if you are sure or nearly sure that his testimony would have been the same whatever had actually happened. If you are pretty sure that the boy would be crying “Wolf!” whether he had seen a wolf or not, you give him little credence, if you are rational, when he does cry “Wolf!”. Well, everyone can be pretty sure that, if educationists or psychologists report nowadays on a test of mathematical ability between boys and girls, say, they will report the girls as doing at least as well as the boys, whether they really did or not. If the tests seem to show markedly superior mathematical ability in the boys, the experimenters will not only withhold publication of the results, but will almost certainly themselves believe that their experiment must have been defective in some way. It is as simple as that. So, when such persons do report equal mathematical performance by girls and boys, rational people simply ignore their reports.

This is a sad state of affairs, but it is, of course, the equality-theorists who are chiefly to blame for bringing it about. For they have created in recent years a climate of feeling in which many men are afraid to deny the equality-theory openly, and even ashamed to doubt it inwardly. Hence the phenomena which are now so observable, of hypocrisy, self-deception, and pious fraud: those invariable concomitants of a militant religion.

“Would fifty years, or five hundred years, of equal intellectual performance by women and men, convince you of their equal intellectual capacity?” That would depend entirely on the circumstances. For it might then be my turn to cry “The game’s not fair,” and I might be able to make the charge stick too. I might be able to identify some interfering factor which was more than adequate to explain the discrepancy between my theory and the observed facts. Pressure-cooker education for all girls, say, or huge cash-incentives for brilliant women university students, or punishment for brilliant male ones. Obviously, if there really were such factors at work, it would be quite irrational for me, or for anyone, to take the equal performance at face-value, however long it might go on.

What would convince me of the equal intellectual capacity of men and women is, simply, the kind of evidence which, as things are, convinces me of the opposite: that is, equal intellectual performance, over a long time, and in the widest variety of circumstances.

Nothing else would convince me, or even begin to do so. In particular, no experiments of any kind, however well-conducted, would weigh with me, if their results were inconsistent with the verdict of ordinary experience. If intellectual performance continued to be, as it has always been, unequal, in all the limitless and largely-undesigned variety of life, then that is the evidence I would trust. If all the educationists (etc.) in the world, even without the influence of fraud or self-deception, reported equal intellectual performance by the juveniles of either sex, it would cut no ice with me; and similarly if all the geneticists and molecular biologists could not detect, by their most refined experiments, any physical basis for the unequal performance of the two sexes. I would still stand by the evidence, raw and unanalysed as it is, of a long and varied experience, if that still testified to inequality.

I think that every rational person would do the same, yet I must admit that what I have just said sounds a bit off-key. It does so, because it conflicts, or appears to conflict, with a point of orthodox scientific method to which we all subscribe: that an ounce of controlled experiment is worth a ton of unanalysed observation.

I could say, with some plausibility, that the conflict is only apparent. I could say, that is, that controlled experiment does always outweigh unanalysed observation, but that, on something as complex as human genes, experiments never are able to be rigidly controlled, and therefore can sometimes be outweighed by a large mass of mere observations. But this does not seem quite plausible enough. Experiments on human genes which are at least nearly as rigidly-controlled as the best experiments anywhere, will surely be possible soon, if they are not so already. I am therefore inclined to think that the conflict is real, but that it is just not true that controlled experiment always outweighs unanalysed observation.

There is independent reason to think this. Consider the proposition: “Monkeys treading at random on typewriter-keys will never reproduce the works of Shakespeare.” It is not a scientific law, or anything like one. The evidence we have for it is massive, indeed irresistible, but it is utterly unanalysed, and certainly includes no controlled experiments, on monkeys, typewriters, or anything else. Consider, on the other hand, those propositions which we have the best reason to believe are scientific laws. For these, the best evidence does consist of certain rigidly-controlled experiments.

Yet every rational person, I think, is more confident of the above proposition about the monkeys, than he is of any proposition which he takes to be a scientific law.

There are certain things which, though they would not convince me of the intellectual equality of men and women, would certainly, if true, go some way to confirm that proposition. One is, statistics which suggest that the past intellectual performance of non-reproductive women has been equal or nearly equal to that of men.

Child-bearing is so extremely obvious an impediment to intellectual performance that, to any sensible equality-theorist, a compromise is bound to suggest itself, between the equality-theory and any inequality-theory like the one which I sketched above. My suggestion was, in effect, that it is the reproductive capacity, and not the actual bearing and nurture of children, which accounts for the inferior intellectual performance of women. But the suggestion is bound to arise that the burden of actual reproduction-and-nurture is sufficient to account for the observed inferior intellectual performance of women.

If this were so it would undoubtedly save the equality-theory, But it simply does not seem to be so. Women university students, although hardly any of them have given birth, are uniformly present in smaller numbers (proportionately) than men, at any above-average level of intellectual performance. At least, this is the case in any branch of university work which is very intellectually-demanding.

The trouble with the “actual-reproduction” theory is, that it is altogether too easy. We all know, only two vividly, the sort of instances which first suggest it, and which, as far as they go, really do confirm it: the intelligent woman, with a PhD thesis uncompleted or an important novel hardly begun, whose meagre frame must sustain, through the “best” years of her life, the slings and arrows of outrageous offspring. No doubt these appearances are often delusive: the thesis would (like most others) suffer deserved neglect if it ever were completed, the novel is a phantom-child which, if it ever were produced, would prove an even greater anticlimax than the children of her body. But often, perhaps equally often, the appearances are not delusive at all. There cannot be the smallest question, in a rational mind, that in countless cases the exercise of the intellectual capacity of women has been more or less frustrated by the exercise of the reproductive capacity, and that untold suffering has resulted from this.

But then, who can tell us the extent to which non-reproductive women have had the exercise of their intellectual capacity frustrated by other causes, for example economic ones? Here our thoughts fly to the Jane Austens, George Eliots, Caroline Herschels, etc., as though their existence settled the matter. But this is merely an instance of that well-known human frailty, “accentuating the positive”: we remember the day we walked under a ladder and later broke a leg, and we forget all the “negative instances.” That the Jane Austens, etc., are very exceptional among women, we know. But here we need to know that they are not equally exceptional among non-reproductive women; and who does know this? Non-reproductive women, as a class, are peculiarly little-known. They scarcely exist in many societies, and where they do exist they tend to lead an invisible or interstitial life. For about 1700 Years, vast numbers of them in the West found their way into religious institutions, and thus became, with not one exception in ten thousand, totally lost to our view. Yet even here, I must point out, the old cruel inequality asserts itself. The medieval monastery was often a repository, at least, of learning, and sometimes an active centre of learning; the nunnery was neither. Similarly, outside the monasteries, religious orders of celibate males have often been remarkable for intellectual activity (the Jesuits being the most striking example); but their female counterparts have never been so.

One could make a start, as far as the British are concerned, by going through The Dictionary of National Biography, getting out all the women noticed there for intellectual performance, and seeing how this class divides between reproductives and nonreproductives. I have not done this. As far as I know, no one else has done it either for Britons or anyone else. And I cannot help thinking that such a statistic, if it were known to anyone and pointed at all unambiguously towards a level of intellectual performance by non-reproductive women which was close to that of men, it would by now be known to everyone. In fact the very dogs would be barking it. As they are not, one must infer that such a statistic is either not known to anyone, or is not such as to afford any significant comfort to equality-theorists. And on the first of those alternatives the actual reproduction theory is unsupported, while on the second it is discredited.

As I have now said what would convince or begin to convince me of the falsity of my belief, I may be entitled to renew my question to the equality-theorists: What would convince them of the falsity of their belief? What would they even regard as being some evidence against it?

Any serious answers to these questions would be instructive, but I do not really expect to receive any such answer. The evidence for the inferior intellectual capacity of women is so obvious and overwhelming, that anyone who can lightly set it aside must be defective in their attitude to evidence; and our contemporary equality-theorists are in fact (as I have hinted several times), religious rather than rational in their attitude to evidence. As providing some further indication of this, the following thought-experiment may be of use. Suppose that the historical evidence had been the exact reverse of what it has usually been: that is, suppose that the intellectual performance of men had been uniformly inferior, under the widest variety of circumstances, to that of women. Rational people would in that case be as confident of the superior intellectual capacity of women as they now are of the reverse. But would those people who are at present equality-theorists be as confident then as they are now of the equal intellectual capacity of the two sexes? To ask this question is to answer it. The fact is, our egalitarians treat evidence on a basis of heads-I-win-tails-you-lose; indeed, to say so is “putting it mild,” at that.

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I am on the road and am way behind on posts—I have a 3,500 word article on deriving Oughts from Is’s in the works—so in the spirit of summer, where I am assured temperatures will at last crack 70, maybe even this weekend, in Michigan (though I usually don’t give weight to such rumors), here is a classic post on marbles which originally ran 25 January 2014.

The rumor went around that trains were throwing steel ball bearings from their wheels. Genuine steelies! Somebody said they saw one that a kid from Detroit brought up. So big that you couldn’t close your fist around it. Nobody else saw it, though.

Nevertheless, we scoured the tracks for endless hours, turning over every likely pebble that caught the sun. No steelies. I did amass a small collection of spikes that had worked their way loose from the ties. Never did figure a use for them. But they were cool.

The game was marbles. Way we played it in upper Michigan was to draw a chalk circle on the cement or scratch one on the sand with a stick, sometimes we even used a length of string. The battleground was set. The players approached cautiously if their stock was low or boldly if their marble bag brimmed.

One boy against one boy. Man against man. A test of skill and mental grit. Incidentally, only boys played against boys, girls against girls: this was back in the day when adults didn’t burst a spleen for kids noticing the obvious differences in the sexes. Besides, boys were daring and cavalier. They were out for conquest. Girls would sometimes—this still baffles me—agree to play to a draw! Even if one girl would outright win she wouldn’t claim her prize. Bizarre creatures.

First boy, if he was a pro, placed his middle finger against his thumb and then snapped it at his marble, aiming to smack it into his opponent’s. If he hit it, he got to keep it. No time outs, no I-wasn’t-readies, no whimpering. Newcomers and cowards could be identified by the anemic way they smacked their marble, say by giving it no more than a half-hearted push with their forefinger.

Strategy was obvious. If you were the better shooter, you attacked attacked attacked. Go right after the other guy. Intimidate him. Like Nelson, ignore all subtleties and blaze away. Cagier boys took advantage of the braggarts and only pretended to be cautious. They let their opponents open with a bold shot or two, answering with a shot away as if in retreat, then when the other boy’s shot would miss and stop close by, the jig was up.

And that was the problem. Too bold and your shots would come close, too close. Countless times the marbles would stop an inch away. It was a goner. This is why it made sense to shoot hard; over-shooting was hardly a risk. Unless you in a sudden-death match where shooting outside the lines meant an automatic loss.

Most of us neither lost too many nor gained too many. But a few kids were easy pickings. Every day they’d come with a new small bag (these were sold in grocery and ‘dime’ stores) and lose them. Rich kids.

Unless by prior agreement, only even matches were allowed. Cat’s eye against cats eye were the lowest, entry-level bouts. Cats’ eyes, for those monumentally ignorant of Americana, are the clear marble with the squiggly, colorful aberration running through the center. These were the general stock, like ten-year-old used cars; useful, but unloved. Why, you could find spare cat’s eye rolling around behind the home room teacher’s desk. What counted was quantity.

Perries, probably a corruption of puries, solid colored beauties. These were always just a shade smaller than the cat’s eyes. Perries, though coveted, had a tendency to chip, sometimes even crack open when hit hard. Clears were solid glass, no color. Steelies were polished ball bearings. I only ever saw two of them. Not the railroad kind, but small ones from who-knows-where. They were said to be unbeatable, which is why we latched onto any rumor of a source.

Then came the boulders, the double-sized version of the normal marble. One of these was worth five to ten of the regular kind, depending on availability. Boulder-on-boulder matches were always closely attended. They were rare, too, especially when they featured perries. (I have since learned boulders are called “shooters” in other areas. But all our marbles were shooters.)

We used to empty our marble bags and compare, holding our treasures up to the light for the kind of close scrutiny used by gemologists sizing up wedding rings. Oddities such as pea-sized perries occasionally showed up. These were said to be from China, then still a mysterious land.

I don’t guess kids play marbles anymore. Unless there’s a marble “app” for their cell phones. But it now makes sense that my favorite adult game is petanque.

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We flesh out our lectures on simple optimal decisions and why we don’t want or need ROC curves. The running example is the PSA test and cancer.

Video

Links: YouTube * Twitter – X * Rumble * Bitchute * Class Page * Jaynes Book * Uncertainty

HOMEWORK: Study

Lecture

We finish our simple optimal decisions and anti-ROC curve lectures with an example. I’ll assume you remember all the material from those lectures and won’t repeat any of it here. Review first or you will be lost.

Or example comes from a Kaggle dataset with about 28,000 prostate specific antigen (PSA) test and biopsy status, benign or malignant, observations. There are also things like age, BMI, smoking status and the like. We’ll use a couple of these for the sake of the example, but we are not here trying to find the best cancer prediction model.

We’re going to put ourselves in the place of doctors and patients who have received data from a PSA test. We’ll use just the PSA level itself, and then the probability of malignant tumor from a simple model which includes PSA. Here, since we know logistic regressions are awful (because over-smoothing) in these situations, we use a random forest model as a function of PSA level, Age, Smoking History, and BMI. I picked these only on the basis of popularity, and they are indeed the kind of measures you’d encounter in a doctor-patient setting.

I repeat: we’re going to take both the PSA levels and the models as given. We are not here going to look at any of the model goodness measures we developed earlier. We are not critiquing the models, but trying to make decisions based on what information we have been shown. Like most people.

Recall the PSA test produces (finite, discrete) ordered levels L = (l_1, l_2, ...l_m). Here, higher levels are thought more indicative of the presence of prostate cancer. In the first step, we pick a level l_i and say “At values greater than l_i, we’ll act like there’s prostate cancer.” We do this for each l_i.

Then, in the second step, we build a predictive model Pr(Malignant | L, E) = P. These P are like L: they, too, are ordered: P = (p_1, p_2, ..., p_m). We pick a probability p_i and say “At probabilities greater than p_i, we’ll act like there’s prostate cancer.” We’ll do that for each p_i.

With our level or probability in hand, we have our decision. We now only need our gain-loss matrix a. These have the dichtomized values a_{Decision, Cancer State}:

1. a_{1,1} = 50 (true positive decision),

2. a_{0,1} = -100 (false negative decision),

3. a_{1,0} = -20 (false positive decision),

4. a_{0,0} = 10 (true positive decision).

These are, of course, arbitrary. I picked them only because they seemed plausible. To me. They are my worthiness criteria. They may not match yours, or even come close. But that, we learned in the first lesson, is the great benefit of this approach. You pick what is meaningful to you, knowing that what some dumb statistician picked, or what some doctor picked, can be wildly different. In other words, a model which is useful to one man may be terrible for another, as we also learned.

We also compute the sensitivity and specificity of these decision-cancer state pairs using L, and then we plot the ROC curve, as discussed last time.

EXAMPLE

There are about 28,000 observations: about 20,000 are benign and the rest malignant. The PSA levels run from just over 0 to 15 ng/mL. Custom fixes the value 4 ng/mL as a cutoff between benign and malignant.

Here is the histogram of PSA values for the two groups:

One might have hoped the malignant distribution would have shown greater frequency of high PSA values. Indeed, there seems little to distinguish these observed distributions (these are not probabilities!), which is bad news of PSA as a raw marker.

That’s backed up in ROC curve:

That red dot is supposed to the optimal decision point based on maximizing true positive - false positive. What is the PSA associated with that red dot? Can’t see it, can you. Another big weakness of the ROC. You don’t know what PSA level is associated with each point on the graph. They are all hidden in the sensitivity etc.

We also see the AUC, but since I reject this measure (why, we discussed last time), I say no more about it, except to note that since the AUC is about 0.5, and almost all the points lie on the one-to-one line, which is taken as the “random classifier.” In other words, the 50-50 guess.

Even though we can’t see what the PSA level is for each point in the graph, we can find it other ways. Above, the red dot is true positive - false positive. But other people like sensitivity + specificity. That’s called Youden’s Index. There are other choices. None of these choices take into account the actual gains and losses made using decisions based this way. Anyway, Youden’s gives a PSA = 9.2 as the cutoff.

We can incorporate those gains and losses using only the PSA in the way we discussed above and in the previous lectures. I.e. by deciding cancer if L >= l_i for each level, calculate the average utility from the data, and plot this for each level. If we do that, we get this plot:

The average utility---based on my worthiness criteria, maybe not yours!---is negative for almost all values of the PSA, until we get near the max. And, in fact, this is the max. Which means, for me, I would never use the PSA test. Unless maybe it maxed out. The PSA on average leads to lousy decisions. For me.

And likely for you, too. I varied the a_{0,1} (false negative decision) by a lot, even down to -10,000 and got versions of this same plot. This is not surprising given the observational distributions of PSA we saw above. You get about the same average number of malignants regardless of PSA level.

This is proof PSA levels have no value. It could be different levels help predictions, but only when married to other measures, such as Age, Smoking status, and BMI. Which is what we try next, building a random forest model of these measures (and PSA), and looking at average utility for the decision rules “Act like malignant if P > p_i” for each p_i the model produces.

We get this plot:

The best prediction is Pr(Malignant | L,E) = 0.38. But it’s only best by a hair, and we’d have about the same average utility if we started with, say, Pr(Malignant | L,E) = 0.25.

The reason this plot goes negative is that for high values of Pr(Malignant | L,E) we get more false negatives, which have large absolute weights. If the probability is, say, 0.75, we have a good chance of catching cancers, but at the cost of increasing the chances of false negatives.

At this point, any number of things can be done. All the usuall model goodness criteria we developed could be checked. A discussion whether this was the best a (matrix) for myself can be had. We can see how different my a is from the doctor’s, who likely puts more weight on false negatives than you do. After all, those make him look bad, whereas asking for more tests feels utterly routine, and makes the system more money (and you less).

We would still want to know if the PSA had any predictive utility. There may be non-linearities or something in it the simple methods missed and maybe random forest can catch. Besides, not I think about it, given the wisdom that “every man dies with prostate cancer, but not of it”, I’m switching my worthiness criteria! The gains are the same, but now the false negative loss is -10, and the false positive loss (given my loathing of “going to the doctor”) soars to -200. Then, using only a random forest model as a function of PSA (and nothing else), we get this:

Now the chance for Pr(Malignant |L, E) has to be greater or equal to 0.9 before I would act. That, to me anyway, seems right. And shows there is some weak predictability for PSA, at least in this model. For other models? Who knows.

DISCUSSION

PSA is in many ways a rotten example. It stinks as an assay, and it seems like we spent a lot of time spinning our wheels when looking at these plots. Which is true. But that is the beauty of the approach. We can see all before us. The focus becomes, as it should, on the worthiness criteria.

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A new book from the curiosity that is academia has emerged. It is The Epistemic Violence of Mathematics by Carla-Julie Kather, who bills herself as a “Sexual Violence Prevention, Feminist Theory and Neurodiversity Studies” expert.

Being something of a mathematician myself, I thought it would be of interest to see how my profession was epistemically violent. Maybe, I thought to myself, I could learn to be even more brutal and violent in the puzzles I bash readers with.

Kather writes, “At the core of this thesis lies the question of Mathematics and my belief that what Wynter terms the Man-centric world is always the Mathematics-centric world too”. This Wynter is Sylvia, a retired professor from Stanford in one of their Studies departments, who, that university boasts, “left an indelible mark on Black Studies and postcolonial thought.” Perhaps like the mark of being stabbed.

Anyway, men, you see, have a nasty habit of producing more “bodies”, and producing far more “bodies” than women, that can do mathematics well. This isn’t fair. And, by implication, ought to be. The problem must be twofold: men and math. The academy, infamously, is confused about what a man is. But maybe our author has a clearer notion of what math is. After all, she wrote an entire book about it. Here is her definition:

Mathematics is therefore a notion of thinking and a mode of thinking that forms modes of being. The contents of Mathematics as an Image of Thinking, as well as the consequences of Mathematics, are epistemic violence because they actualize hegemonic notions of Rationality and the Human.

I’m not sure whether this includes “1 + 1 = 2” and the like, or only does so only in regimes which favor “patriarchal-colonial-capitalist witch hunts”. That passage, however, is as close as she comes to defining what math is.

As we learn from the title, Mathematics (we she always italicizes) is epistemic violence. Thus math “enables and legitimizes the ongoing epistemicide (the killing of knowledge) of Indigenous mathematics.” And what is “Indigenous mathematics”? She never says, not in the entire book.

Epistemic violence, she writes, is, “the silencing of knowledge formed from marginalized positions or the silencing of knowledge that does not accord with hegemonic systems.” It is also this:

Epistemic violence is a distinct form of violence, but not one that can be separated from other forms of violence. Epistemic violence continuously unfolds its [sic] precisely through its interwovenness with other forms of violence. Throughout this chapter [two] sexual violence will function as a case study to illustrate the interwovenness of epistemic violence with other forms of violence.

So what, then, is “sexual violence”?

Sexual violence is a violence that grows from within the world that the Mathematics-Rationality-Human continuum builds. Sexual violence, as I understand it, is a practice of excluding as a being from the realm of the human.

And who are the biggest victims of this form of sexual violence? Can you guess?

Women are subjected to sexual violence especially often precisely because they are less-human than men in the world of the Mathematics-Rationality-Human continuum because their thinking-being is less accounted for in the continuum-based notion of the human. Women of color are subjected to even more sexual violence because their thinking-being is even less accounted for in the human birthed by the continuum.

And don’t even get her started on “autistic women”.

After reading this, maybe you were as confused as I was about this thing called “epistemic violence”. She must have sensed this in readers, because she late in the book clarifies the concept:

Epistemic violence is a living body, and its heart is the exclusive human–the human birthed by the Mathematics-Rationality-Human continuum. This exclusive human is both a notion and a mode of thinking-being, which builds a continuum with other notions and modes as well as with other images of thinking-being, namely Rationality and Mathematics.

That clears it up for me.

I could not discover, anywhere in the book, anything resembling an argument. You are welcome to check for yourself (the book is free online, at the top link), but all of it reads exactly like the quotes here. One massive word jumble with no hard definitions of anything, and no justification of any conclusion. We are left only with a certain sense that somebody has once done a great injustice to Kather, but we don’t know what. My guess is that she failed her calculus course and was none too happy about it.

Now if this were only Kather and her odd little book, it would be a small matter and an occasion for a few small jokes. Of course, it is not only her. It is a wide swath of academia, with a gaggle of lady and other Victim professors going on and on about “epistemic injustice” and “epistemic violence”. Like this, this, and this: also peruse the book’s bibliography which goes on at depressing length. The “violence”, so far as I can tell, applies only to subjects which require no small degree of intellectual prowess, and in which these ladies and Victims don’t on average fare as well as (white and east Asian) men. Which isn’t fair.

It would be unfair, too, if I let myself instead of Kather have the last word. So, ladies and gentlemen, I give you Carla-Julie Kather:

Putting final words to the matter of the Mathematics-Rationality-Human Continuum or to our multitudes of wild thinking-being is an impossible task. It is impossible, because this thesis is supposed be a small part of movements of re-inventing what it is to think and be – mathematically and otherwise.

Video

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“It may seem strange, but it is almost impossible to create a perfect coin or a perfect die,” says [paper author] Renner. No matter how symmetric and smooth a die is made, after a roll one of its six faces will always point upwards slightly more often.

“Even modern random number generators, which are based on quantum mechanical effects like the reflection of photons from beam splitters, are not entirely immune to such a systematic error or ‘bias,’” adds Wallraff. But now Wallraff’s and Renner’s teams have found a way to take imperfect randomness and still extract perfectly random numbers from it. They call their method randomness amplification.

Before we discuss these curiosities, the paper is “Experimental randomness amplification” by this Renner and many others in Nature. Stick with me, because we have to get through the Abstract (with my parapgraphifications; and have no fear the loophole-free Bell or quantum computer business as we won’t really need these):

Realistic quantum information processing devices are inherently imperfect, leading to computational errors that require quantum error correction.

Likewise, random bits generated by such devices are flawed and must be enhanced to be usable for applications such as generating cryptographic keys. This enhancement of randomness quality is achieved through a protocol known as randomness amplification. Here we report on an experiment that implements such a protocol.

Randomness amplification is device-independent, making no assumptions about the internal workings of the quantum devices. It requires executing a loophole-free Bell test within a specific parameter regime that involves both a high Bell violation and a high repetition rate.

The experimental demonstration is made possible by a combination of theoretical advances, which allow for protocols with an experimentally realistic parameter regime, and experimental progress that achieves this regime with superconducting circuits. Crucially, randomness amplification has been proven to be impossible by purely classical means. This experiment therefore demonstrates a definitive quantum advantage—leveraging quantum technology to accomplish a task unattainable by classical information processing.

I’ll now define what these things mean, then say why anybody should care.

Random

What is random? Although in the colloquial it can mean surprising or unexpected, e.g. “That was random”, formally it is unknown explanation, where we must always remember explanation means cause. But cause in its full Aristotelian sense, the complete explanation of a thing. When you are missing pieces in the full-cause-slash-explanation of a thing, it is to that extent random.

There is another usage, inappropriate I think, but common, and that is uniform. “Random” numbers, to be defined in a moment, must be strictly uniform to “truly random.” So that if you have 10 items, any departure from 1/10th of each of them in any process, natural or artificial, to generate them, the process is said to be not wholly random, and thus becomes candidate for “randomness amplification” to restore the supposed imbalance.

Probability, you will recall from taking the Class, means (and only means) the uncertainty you have in some proposition given whatever evidence you consider. In the authors’ definition of “perfectly random” they mean the probability a thing is in one of the states it can take equals the probability of any other state it can take. In notation (which you don’t have to understand), Pr(S = s_i | E) = 1/n, where S can take one of n states and E is all the evidence you assume.

If for any of the states, the probability is different than 1/n (where logically, at least two states must differ if one does), then there is in a sense some predictability of the state greater than if every state had 1/n probability. Take the easy case of two states, bits, which can be on or off. If given your evidence the probability is 50-50, then that becomes the hardest prediction. Any departure, here by making a 1 or 0 greater than 50 which automatically implies the 0 or 1 less than 50, makes the prediction easier.

And it’s that uniformity the authors mean by “truly random.” They say, lamenting the problem legacy “quantum random number generators” have is that

…for any real-world implementation with experimental imperfections, this randomness is not perfect—the probability that the next bit, 0 or 1, can be correctly predicted might deviate from the ideal value of 1/2. The amount of this deviation is called the bias [their term].

What about coin flips, as mentioned in the opening? Are they “random number generators”? Coins, as I explain in this video, don’t have probabilities. Nothing has a probability.

As the authors said, you cannot have a perfectly symmetric coin: it is not possible. And even if it were possible, you cannot have a perfect “flipping environment”. What precisely would that even mean? Precisely. What we have is an object, with a varying mass, throw into the air with rotation, mostly confined to one dimension, but present in all three, that rises to some height in air of a certain temperature, moisture, density, with whatever wind, and which comes down and is grabbed, or lands, in a certain way on some surface and ultimately comes to rest with one side facing up.

This is an entirely physical process, with causes and conditions that can be known and manipulated—which I and my father did. The machine flips coins and manufactures heads. Because it turns out those causes aren’t that difficult to control.

Thus, given all this, the probability of a head in a coin flip is 1.

But given something entirely different, like the conditions under which most flip coins, the probability is very different than 1, and indeed is 1/2, given only the evidence that a flip will be made. My machine does not produce random flips because the full explanation of the flip is known.

If you need maximally unpredictable numbers (a far better term, I think), for instance for security reasons in cryptographic systems, you need a device which produces numbers whose predictions cannot be bested over predicting uniformity, even if an enemy gains access to the device afterwards. The only way this can be done is if the causes of these numbers can remain unknown to everybody.

This is what makes quantum mechanical devices appear ideal, because the ultimate causes of events are (it is claimed) unknown. Yet QM devices are still in the world, and in the world we know a lot of causes, such as why a particular counter registered whatever number it just registered, even if the counter was hooked to a QM device. The authors get it right: “unpredictability is not an inherent property of a bitstring [the numbers] itself, but rather of the process that generates it.”

Worse, in the search for perfection, is that sometimes knowledge of past values produced by whatever device is used can give relevant information about future values, even if the device produces its requisite uniform numbers. This is important because if the numbers, used in for instance cryptographic keys, can be predicted, then the keys can be guessed.

The authors’ idea is to take a partially predictable (away from uniformity) string of numbers (bits) from a QM measurement process and “amplify” the “randomness” of those numbers. A term I cannot love. Here’s their picture:

Some process (we don’t need to understand what) generates two strings of bits, X and Y, these are put into some black box which takes advantage of certain QM properties (that Bell business), which produces another “entangled” string, and finally are all joined together in a mathematical way and out pops “perfect output randomness.”

Where “perfect” here means imperfect. Because, as even the authors admit and write into their equations, the first QM process, which takes place in the world with macro measurement equipment, has a chance of erring, as does the combination process later downstream (they never state what these erring chances could be). The combining procedure itself even has an admitted chance of containing information that allows predictability beyond 1/2.

They did try their experiment with certain machinery, not important to us, which produced long strings with passed NIST tests for unpredictability. So adding in a layer of a second QM process to the first QM process works, up to point, as would be expected. But most of the causes of the QM process are not known. Thus, and enemy can gain access to your “random” number generator and, as long as it didn’t store your numbers, they could not reconstruct your sequence.

Unless that enemy can gain access to the causes behind QM events.

Now many say this is impossible because of Bell’s theorem. Which we here do not have to understand, except to state that this is a mistaken reading of that theorem. Bell merely seems to imply that if had knowledge of local causes, we could not be able to guess QM events. Which leaves open the possibility we might gain at least some knowledge of non-local causes. This article gives one idea (and there are several).

That is too much for us to go into here, except to say that the notion that QM events are uncaused or are caused by “probability” in some mysterious way, as some claim, is false. Of course QM events have causes, explanations for their being. As all things do. If things happened for “no reason”, then only absurdity results. Magic would have to be invoked to explain how the “no reason” QM regime suddenly becomes “reason” macro regime. All of which shows you how important a Reality-based philosophy of Nature is to Science.

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Reminder: The Thursday Class is only for those interested in studying uncertainty. I don’t expect all want to read these posts. Pease don’t feel like you must. Yet, I have nowhere else to put them. Your support makes this Class possible for those who need it. Thank you. Much math alert!

We don’t need the ROC curve unless we’re designing radio receivers or tests. What we want is the probability of the event given the test: calculate that.

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HOMEWORK: Study

Lecture

When we last met we derived simple optimal decisions, where both the “optimal” and the probability behind these were, as they ever are, conditional on the worthiness criteria and evidence you decide. I had thought of doing examples, but realized I hadn’t yet covered sensitivity, specificity, ROC curves and all that.

I don’t think we need spend much time on these, as they are so well known. Still, there is some value putting these creaky old terms into our notation and contrasting them with our last lesson. Also, I do not like ROC curves, AUC and all that.

Recall one of our running examples. We have some assay, like a PSA test, which produces (finite, discrete) ordered levels of some measure, call these L = (l_1, l_2, …l_m). Here, higher levels are thought more indicative of the presence of prostate cancer. We pick a level l_i and say “At values greater than l_i we’ll act like there’s prostate cancer.” We went through the difference between probability and decision last time, so review that.

We’ll call a level greater than (or greater than or equal to, if you like) l_i a “positive test”; else a “negative test.” Let the test be T, and the Disease (cancer or any dichotomous event) be D. The evidence E can include whatever past observations we have made, or it can be purely an assumed model.

With our level in hand, and with whatever additional E we bring to bear so that we can create quantitative probability models, we can compute various things, such as:

1. Pr(+ Test | + Disease, l_i, E) = sensitivity,

2. Pr(- Test | + Disease, l_i, E) = false negative test,

3. Pr(+ Test | – Disease, l_i, E) = false positive test,

4. Pr(- Test | – Disease, l_i, E) = specificity.

Obviously, one would enjoy a sensitivity and specificity equal to 1, and false negatives and positives equal to 0. But, of course, most simple measures are not direct inspections of cause, and are mere correlations, and so typically none of these probabilities are extreme.

None of these quantities ought to be used to act or decided as if a disease is present or absent, which would be silly, since these are not probabilities of disease states, and instead say something about the qualities of the test. Alas, they often are used to make these very decisions, which is rather distressing. Like the p-value, no amount of argument and proof will dissuade fans of these probabilities.

What’s wanted instead in decisions about disease state? These:

1. Pr(+ Disease | + Test, l_i, E) = predictive probability,

2. Pr(- Disease | + Test, l_i, E) = also predictive probability,

3. Pr(+ Disease | – Test, l_i, E) = also predictive probability,

4. Pr( -Disease | – Test, l_i, E) = also predictive probability.

In other words, all these are nothing but different forms of Pr(Disease | Test, l_i, E). All probabilities concerning the disease (event) itself, which is what everybody (well, most) are interested in. The exceptions are people creating the tests, who (one hopes) want good ones. Sensitivity and whatnot are good for them, but not for patients, say, who have just been confronted with the results of a test.

On the other hand, the first set are nothing but different forms of Pr(Test | Disease, l_i, E). Also all probabilities. But, unless we’re creating the test (assay), they are not interesting, because they assume you know the state of the Disease. Which you won’t know if you’re using the test to guess that state!

Sharp readers will have noticed the similarities with the depressing p-value. But only similarities, because at least Pr(Test | Disease, l_i, E) does have some uses if you are trying to create a test.

You will have also noticed that Pr(Disease | Test, l_i, E) is not a decision. We might (or somebody might have) used l_i as a decision point, as we did last time. But we could also calculate Pr(Disease | Test, l_i, E) for each l_i, i = 1, 2, … m and use that in our decisions calculus, taking into account the gains and losses for making correct and incorrect decisions. Again, this we learned how to do last time, so I needn’t repeat any of it here.

Let’s instead look at what other people do, and see if we can figure a way to talk them out of it.

The ROC Curve.

Being lazy, I stole our example from Wokepedia:

These are, they say, “ROC curve of three predictors of peptide cleaving in the proteasome“. Three different tests with differing levels: each point on each curve picks an l_i, and the curve itself is generated by running through all l_i, i = 1, 2, … m.

The “true positive rate” is the crude estimate of Pr(+ Test | + Disease, l_i, E), i.e. the sensitivity. The crude estimate is the proportion of times the test was positive (greater than l_i) and given the disease (event) was present (here not a disease, but cleaving in the proteasome). This is not the true positive rate of identifying the disease (or event).

The “false positive rate” is the crude estimate of Pr(+ Test | – Disease, l_i, E), i.e. false positive test, not disease. It is got in the same way.

Here’s where it gets weird. The area under any ROC curve is called (ta da!) the Area Under The Curve, or AUC. The larger this is, it is said, the better the test is. But that area, and indeed each curve, is a function of each of the l_i, where actual decisions about the disease or cleavage or event would be based on Pr(Disease | Test, l_i, E) and on the gains and losses you specify (as in the last lesson) leading to an optimal l_i. You only pick one. Or because the gains and losses might be generic, because you are considering a suite of them for different users of your model, you might have several l_i, one per each specified each gain-loss matrix a (see last time). But still each user picks just one l_i. The model is judged at that l_i, for him.

And you will have noticed, because of the language, that Pr(Disease | Test, l_i, E) is no official part of the ROC curve. We could, of course, plot Pr(+Disease | +Test, L, E) or Pr(-Disease | -Test, L, E) for all l_i in L. And then marry that with our a (the gains and losses matrix).

As useful as a ROC curve might be in creating better tests, it is not useful at all in deciding what to do about the main thing of interest. Nor in judging models, which we learned, has to be done by specifying worthiness propositions (those gains and losses). Just like P-values, which make one-size-fits-all decisions, the ROC curve assumes symmetric costs and and gains. And only related to quality of the test, not the quality of the decision about the disease/event.

Another criticism. Suppose (as mathematicians say) without loss of generality the test has three levels, l_1, l_2, and l_3. And we have a collection of old observations of disease/event states (dichotomous) and test levels. Each of these levels is used in the ROC curve, which has three points. The first is “Act as if Disease is present if Test >= l_1” (recall, we are assuming higher levels mean more likely disease/events, but this works the other way, too). Well, now, that’s everybody, isn’t it. So that level would never be used. But it is part of the ROC curve. Et cetera.

We just don’t care about questions of the test behavior, unless we’re creating tests. We care about what the model says about the proposition of interest.

BAYES

Time to drag out the hoary textbook example about the probability of disease when confronted with a positive test which boasts a sensitivity of (say) 99%! This thing only shows its wizened head because we abandoned probability of event in the first place. Here’s what we want

Pr(Disease | all evidence, including + Test and test characteristics).

If we always had been giving that, then this textbook staple would never have been created. No one would have thought to do so. Because no patient ever gives a damn about sensitivity or specificity about some test. He wants to know what are the chances he’s sick. Or you want to know the probability of the event.

Positive tests are concerning, and, ceteris paribus, they always increase the chance of the having the disease, just as a negative one always decreases it. But the changes are not necessarily large or important. That’s why communicating the probability ought to take precedence. A doc could say, “Sure, you had a positive test, but only 1 out of 100 people with that have the disease.”

That probability depends on “all evidence”, which means what it says: all evidence. What is all evidence? It is, since we have learned from Day One, the evidence you consider. What is the evidence you ought to consider? That which is probative of the proposition of interest. Ideally, the evidence of cause, which gives extreme probabilities (i.e. 0 or 1).

Nothing has changed for us since that Day One. It was always Pr(Y|X) and decisions downstream from that.

I leave as homework that standard textbook example. I can’t bring myself to write it.

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Every now and then I suspect there is a game played by bored elites to see which of them can get away with the most asinine “climate change” claim. There have been some doozies: lack of UFOs, “climate change” anxiety, increased sugar consumption, engineering short people. On and on and on ad infinitum.

Today we meet what could be the winner in the contest. Experts are claiming World Cup matches could be slower. Now instead of ninety-plus minutes running around ending in zero-zero ties and dramatic shootouts, we’ll get ninety-plus minutes of somewhat slower running around ending in zero-zero ties and dramatic shootouts. Because “climate change”.

Seems to me, though, that if we really wanted to save time, we could have all matches begin at the shootouts. The whole tournament could be over in a day. Take that, “climate change”!

Here’s the claim, appropriately announced on a PR site, a site whose specialty is juicing dull and dubious headlines: “Climate Central finds climate change has increased the likelihood of heat that could slow players for 97 of the 104 matches, threatening game speed and fan safety.”

What will happen to the other 7 matches they don’t say. Probably they will end in shootouts.

Details:

Previous research shows temperatures above 82.4°F can reduce sprint frequency, total distance covered, and recovery time, impacting not only player performance and safety, but also match tempo, tactics, and overall style of play.

Previous researchers discovered—and stay with me, here—that when it’s hot outside people outside in the heat act like people act when it’s hot out.

Where would we be without previous researchers?

Now this heat happens in summer, and, would you believe it, that’s the exact time of year Authorities decided the World Cup shootouts ought to happen. And not just in summer, but in quite a lot of southern cities in summer. So either these World Cup Authorities didn’t know about this stunning previous research, or they did and knew it would be summer and decided to run the risk of 82.5°F days when they scheduled their tournament in summer and in many southern cities.

So much for previous research. What about current research?

Nearly half of the matches have at least a 50% likelihood of experiencing heat that can impair performance. In 26 of those matches, climate change increases the likelihood by at least 10 percentage points.

Here’s their touted precise prediction for the highly anticipated (one supposes) Mexico – South Africa match:

Let’s suppose for a moment this should be taken seriously. “Climate change”, says the research, will increase the chance of an 82.5°F day by 8% for this match up. Let’s believe that.

Now what.

Should the World Cup be canceled? Will it be canceled? Is an extra 8% chance of an 82.5°F degree day worth the risk of slightly slower play? Aren’t Mexicans already used to heat? Should we not watch? Or watch only in air conditioned spaces? Ought players take a knee before each game, running the risk of dishonoring George “Thug-Drug” Floyd, to “raise awareness” of “climate change”?

Seriously, now. What exactly ought to be done with this stunning new research? What happens if awareness is raised? Will people wipe their brows and blame their moistened bandanas on “climate change”?

Even if this research is true, there is still a 11% chance of an 82.5°F game day (and similarly for other days). What can be done about that? Not a thing, short of canceling or postponing matches, which isn’t going to happen.

As you have surmised, we don’t have to believe any of this: the scenarios are all silly, proving that the stunning new research is purely performative, a way to signal the vices and predilections of the researchers. As proof, we oblige the PR site which asks us to quote this:

As global temperatures continue to rise, driven by the burning of fossil fuels, the 2026 World Cup could become another example of how climate change can disrupt sports and traditions people cherish, forcing a reevaluation of how the game is played.

This is all ridiculous, and only saved by the charge of falsity by adding that “could.” Meaning might, which also means might not.

They also provide this quote they ask us to re-quote:

Alex Jacobs, professional player formerly with the Jamaican Premier League:
“Heat is not new. But extreme heat — made more likely by heat-trapping pollution driving climate change — might just be a difference maker in this summer’s edition of the biggest sporting event on the planet.”

Way these quotes work is that either the PR firm, or the organization employing the PR firm, write them for victims, who are sought out and asked to sign on. The people allowing their names to be used are, of course, fully responsible for the words used, and even often believe them. But there’s no mistaking that stilted prose of the manufactured quote. Nobody talks like this. This prose is not conclusive, but whenever you see it, it’s a darn good reason to dismiss the claims made.

HOW

How did they do it? Here’s their Method, which long-time readers will recognize:

Climate Central analyzed the dates and locations of all scheduled matches in the 2026 FIFA World Cup and calculated the likelihood of temperatures exceeding 82.4°F (28°C), a threshold associated with declines in soccer/football athlete performance. The analysis compared current climate conditions influenced by human-caused climate change with a counterfactual world without climate change to determine how global warming altered the likelihood of performance-impairing heat at each match.

There is no such counterfactual world without “climate change”. We only get the world we have. The move isn’t illogical, though, because we can calculate probabilities for any proposition we consider. But there are two problems. The first, and obvious, one is that we have no way to check whether the counterfactual world they imagined is correct. It’s pure guess.

But it is obviously envisioned as some kind of ideal world, in which the “climate” has never been touched by “man”. Such as thing can have happened, but it is always irrelevant. Because, of course, man exists. And all creatures, including man, and all things influence the actual climate. It is impossible not to influence the climate, however slight that influence might be. So this counterfactual world is pure fantasy.

The second, and more damning faault, is that it requires both models, the one of the world as it is and the second the imaginary world, to be well calibrated, sharp, predictive, skillful, and all that kind of thing we’re covering in the Class. If the models aren’t good in this sense then it’s just making up numbers. And the second model, the counterfactual one, has to be perfect

Which is exactly what they’re doing.

Those who want technical arguments, and more detail, can read these two papers on why you ought never to trust “attribution research”, like the kind advertised by the PR firm: paper one, paper two.

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Don’t Peek

Don’t Peek

Don’t Peek

Don’t Peek

Don’t Peek

The Answer

The answer is obviously 84.

This is from the formula, which many of you spotted right away (I’m always proud of regular readers):

where p_n is the n-th prime number and n the top number(s) in the triangle (the first few primes, you recall, are p_1 = 2, p_2 = 3, p_3 = 5, p_4 = 7, p_5 = 11, p_6 = 13, …). Since in the first example there are two 2s, you can think of the first n in the formula as the first 2, and the second n as the second 2. Et cetera.

To confirm, p_2 = 3. Thus

And with the 5s we have, recalling p_5 = 11,

Finally, with 6s, recalling p_6 = 13,

Neat math, n’est-ce pas?

“No, Briggs! No! No! No!“

You object?

“The answer is obviously 72! You used the wrong formula, Mr Sophistication. Everybody knows you’re wrong.”

I am? What’s the right formula?

“Look, genius: 2^2 = 4, and 2 x 4 = 8. Yes?”

Usually.

“And 5^5 = 25, and 2 x 25 = 50.”

That’s true.

“That means the clear-to-anybody-but-you formula is 2 x n^2. Which means 2 x 6^2 = 72. That’s the right answer.”

I had no idea my formula was wrong. But now I can see that yours is wrong, too.

“It is not.”

Is it. Look, 2 x 7 + 2 x 7 – 20 = 28 – 20 = 8, does it not?

“So what.”

And 5 x 7 + 5 x 7 – 20 = 70 – 20 = 50, yes?

“I don’t like where this is going.”

That means the real formula is 2 x n x 7 – 20. Which means 2 x 6 x 7 – 20 = 64. That’s the real right answer: 64!

“No! No! No! Must you be so obtuse? It’s clearly 72.”

Wait. I want you understand I take your, and all, criticisms seriously. I see the errors I’ve made. And I can now plainly see the true correct formula.

“Well it’s about time.”

The correct formula (let’s call the answer y) is obviously:

where k does not equal 0. That is the obvious natural extension, or generalization, to your formula. The question of k is a curious one, but we can easily deduce it’s value must be 3, since we have triangles above, and everybody, even I, know triangles have three sides. That means for n = 6 we get

Aha! So the right answer was 84 all along, and you were just teasing me. Good joke. You know, it’s not often we mathematicians—

“You can’t do that!”

Hold up, hold up. I see what you mean. My guess for k was silly. Since these are triangles, and all triangles have 180 interior degrees, k = 180. That means y = 792.

“No!”

You mean k is the number of lines in pictures? I didn’t see it at first. I have it now: k = 6. Thus y = 96.

“I’m done with you.”

Too bad. I was just about to change the subject.

Author! Author!

The right answer to any and all questions much always include this: the mind of the author of the questions.

I tried making this point in an earlier post, but I don’t think it stuck. I was too abrupt, the math too obscure. So here I expanded on the idea to make everything (I hope) clear.

My interlocutor above was convinced the author of the puzzle had a very specific and not especially difficult formula in his (the puzzle author’s) mind. Given everything we know about these kinds of puzzles, our wide experience with them, seeing what forms they take and their correct answers, my friend is likely right, too. His formula was very probably the right answer, and all mine were wrong.

I cannot say this with certainty, because I don’t know with certainty precisely what the puzzle author was thinking. He may have been like me, a fan of prime numbers. But of all the scenarios I think possible, I judge my friend’s “72” hypothesis almost certainly—but not certainly certain—the correct one.

I can therefore guess that the 72-hypothesis holds. I make this decision based on the likelihoods of various hypotheses I formed conditional on the evidence I assumed. And as everybody following the Class knows, decisions are not probabilities. Decisions, guesses, come after the uncertainty, and they depend on different evidence or premises which we use to form the probabilities. Evidence or conditions like what will happen if I’m wrong, or if I’m right, and how important are these things to me?

It turns out I don’t care much either way about being right or wrong, and therefore decide the 72-hypothesis is the best guess: I make that guess, then I do the math and write the answer 72. Then I await the author, or his representative (like a teacher), to tell me if I was right.

For puzzles like these, we combine the uncertainty and decision steps so fast that they seem one. We’re so good at this in ordinary situations that the acts seem inseparable. But they are not: they are entirely different.

And so it remains that in judging IQ test questions, or as we’ll see any questions, we are not just working out the math, or logic, we are also making a decision about the goals of the questions, about the authors’ intentions. This must be clear for the example which we use, because all the formulae I gave were entirely consistent with the stated available evidence. They were also consistent will all the unstated evidence we considered.

And there is lots of unstated evidence. For instance, you assumed what the “2”, “5”, “6”, and “12” meant. You assumed the triangles encompassed different examples, that they were boundaries for separate examples. You assumed because of the form of the question that some kind of mathematical manipulation was called for. You assumed you needed the rules of math and logic (rules which were also assumptions). All these things and more, you assumed, were on the author’s mind.

And you didn’t assume that it wasn’t math. Like, say, you didn’t assume the “2” was a shape and not number, and that, again for instance, the “8” was something like two “2”s stacked, one inverted, with the ends closed. You didn’t assume because the “5” (also not a number) had two straight sides, the rule was t0 repeat the figure and put a “0” next to it. That means, for “6”, which has no straight sides, the answer would be stacked, one inverted, something like this:

Put your newfound realization to work on this one. Hint: it also has different possible interpretations. There is more than one possible right answer, depending on your assumptions.

Explanations here and here, if you need them. But do try first. There can be more than one “correct” answer in the given set, A–F, but there can also be more consistent answers than just this set.

A Very Large Number

There is lots, then, going on in even the simplest questions, but you’re so good at thinking (obviously true, since you are here reading this) that you no longer see the steps involved, nor give much consideration to their assumptions.

The first kicker is this: for our starting puzzle I gave several possible formulas consistent with my assumptions about what the evidence meant. How many different formulas are consistent with evidence provided?

Infinity.

Yes, really: here’s a small subset of that Infinity:

This is obviously infinite; each formula, one per i, is entirely consistent with the known observable data. It is clear (well, to those used to playing wth equations) many other variations can be found.

This is always so. That’s the second kicker, and what I promised at the beginning. All questions have an infinity of answers consistent with the observed data. Which is why the observed data is never enough.

“Let the data speak” leads to the worst schizophrenia, an infinite cacophony. “Show me the data” is like being presented with a picture of a falling grain of salt in a blizzard. The data is not sufficient; it never can be its own explanation.

This applies everywhere. You must always bring outside assumptions. And those must relate to the mind of the question writer, whether that be a man, Nature, or God himself. This is inescapable.

Answers being infinite, how can we ever be, perhaps not certain, but at least reasonably sure we’ve made good guesses, at least most of the time?

The 6 February 2022 NYT Sunday crossword was titled “Sci-fi Showdown.” The clue for 70-across was “The better of two major sci-fi film franchises” (8 letters). What would you answer? Here’s the relevant section of the grid (the blue means nothing):

The down clues that overlapped the last four letters of 70-across were in order: “It’s a ______!” (71 down), “Body part that precedes band” (67 down), “Ones involved in a transaction” (47 down), and “Let out, in a way” (55 down).

Let’s work through this. How many possible answers are there for “It’s a ______!”?

Infinite.

One possibility is “It’s a five!”, as those fond of (early) primes are wont to say. Or maybe the guy reaching into his pocket expecting a one. “It’s a teen!”, as readers of the news might say. A proud father will exclaim “It’s a girl!” .

Many English words fit. But who’s to say the answer must be an English word? That’s an assumption about the creator’s intention. I’ve been doing these puzzles for a long time, and sometimes squares can be whole words themselves, or even symbols. Maybe that’s so here. Then we might have, say, “It’s a Smiley-Face Smiley-Face Smiley-Face Smiley-Face!” to imitate a laughing text (I don’t know how to get pictures of smiley faces in place of letters on these posts). Or maybe just “It’s a LMAO!”

Since anything might do for a symbol, the possibilities are endless.

We are in the same situation we started with, but with a new wrinkle. The answer to 71-down does not exist alone. It must fit with all the other words it touches directly, and fit indirectly with all the words (or symbols) in the entire puzzle. They are all connected. Each other word has also has infinite possibilities, and indeed so does the puzzle as a whole, though the explanation for the joint mass of clues will surely become exceedingly complex the greater we deviate from the author’s mind. We must add epicycles upon epicycles to justify each additional entry we make.

Unless you have guessed the mind of the puzzle author correctly. Then everything fits tight.

In a way of proving that, the answers—plural—to 70-across were “Star Trek” or “Star Wars”. The down answers were “Wrap / Trap”, “Waist / Wrist”, “Payees / Payers”, and “Leased / Leaked.”

Two possible answers for each, and all fit into the whole puzzle.

And I know these, these are not a guess, because I know the mind of the author of the puzzle, he being obliged to publish it after the puzzle has run.

In other words, I know that part of the cause or explanation of the grid.

Nature’s Grid

It is no different in Science. In her Defending Science — Within Reason, Susan Haack likened scientific explanations to a giant multi-dimensional crossword puzzle. All the answers must fit together.

Picture a scientist as working on part of an enormous crossword puzzle: making an informed guess about some entry, checking and double-checking its fit with the clue and already-completed intersecting entries, of those with their clues and yet other entries, weighing the likelihood that some of them might be mistaken, trying new entries in the light of this one, and so on. Much of the crossword is blank, but many entries are already completed, some in almost-indelible ink, some in regular ink, some in pencil, some heavily, some faintly.

That some of the entries might be mistaken is usually not given hearty consideration, especially when grants are on the line. Much in Science is mistaken, as you and I have seen over the years. And a lot that is wrong precisely because that part of the cause which is the Author’s mind is ignored. We saw not long ago a “prominent” (a quote, not a scare-quote) scientist reject alternate testable theories of biological change because the Author’s Mind “would be let back in”. The widespread rejection and neglect of teleology, which reveals that Mind, has choked Science.

However, that is too much for us today. We shall return to this subject again.

Test Your IQ Test

There is no escaping the mind of the author in an IQ test (which was the point of the first time I attempted to explain this). Which means IQ tests are not quite the tests of intelligence you may have thought.

If you have a math test to give to kids, you take great pains to spell out the precise exact conditions of the questions. You try to indicate your mind in the set up of the question, so that it is plain you want numbers, and numbers of a certain kind, and got in a prescribed way.

For instance, solve for x in this equation:

Where the answer is obviously

Which you would not do to a 5th grader. Even though this x is correct, as are an infinite number of other possibilities (x= -\int_0^{\pi/2} sin(y)dy, etc., etc.). The tacit assumption, given by any number of practice questions in class and in homework (before it was decided math was “racist”) was that the answer would be a single number itself, and nothing else. Certainly not a symbol or political statement.

And since we have just introduced negative numbers in class, and the subject of the test is negative numbers, the student rightly surmises the mind of the author (his negativity?). Only then can the student bring his math skills to bear and solve the problem. These are entirely different steps than that first necessary step of assessing then guessing the mind of the author.

The same is thus true for all IQ test questions. If you had guessed, as I first suggested, 84 as the answer, because this was obviously and truly consistent with the evidence, and the mind of the author was elsewhere, you would be marked wrong, regardless of your intellectual ability to do math. The IQ test score would reflect your inability to surmise the mind of the test writer, but many would mistakenly take your low score as indicating inability to calculate.

Of course, intelligence is more than calculation ability, too. Part of intelligence is discerning the minds of the question writers, and some of that discernment is indeed cultural can thus improve with practice. Thus it is possible to get better at IQ tests, to a degree. And so, quod erat demonstrandum, IQ test scores smooth over the various dimensions of intelligence, as one-number summaries of complex entities so often do.

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You will have noticed these last few weeks a switch to bi-weekly posts, longer and (I hope) more informative than the old way of having three to four short, standard column-length posts. Plus the Thursday Class—which I do not expect most want to read, but which I believe is utterly crucial for all working scientists to understand.

I had thought this switch would be better for you, Dear Reader, since I noticed the articles I myself read were longer and were receiving lots of interaction. But I also thought switching to two-long would save time that I could use to finish the two important books I’m working on (because I can leave behind physical copies after I go under). However, I find that it takes just as much time to do two long as three or four short posts.

Perhaps one thoughtful article per week is best. You tell me.

I’ve also added video to nearly every post, but I think the audience for these is different than for reading. Not a large audience, either. Mostly because I have obviously not mastered, or even minored, the genre. (I only recently, and this is pathetic, learned how to do picture-in-picture.)

In any case, which would you prefer each week?

Or maybe a mix of all. For instance, I am already working on two for next week, and it is necessary (as you will see) that they be a bit long. Both are, for me anyway, a bit thrilling and absolutely fundamental.

Topics

The focus is still Science, especially the philosophy, misuses and politics of the same. There are more sites than you can count that sing its praises and are full of gape-jawed wonder at thrilling new “Research Shows” headlines, and you don’t need more from me. But if you’d like to see things adjacent, let me know.

Timing

For years I’ve scheduled posts to go out at 7 AM. No reason went into that. I picked it once and stayed with it. I wonder, though, if the emails sent that early in the day get lost.

Your Support

When we were starting out on our own, I kept a small bookkeeping journal that tracked every penny coming and going. This is not a euphemism or metaphor. I mean every single cent. Money was tight. It’s not so loose now, especially given my trio of cancellations, but I can afford my necessary bottles of Winking Owl.

My point is that I know what it is like to have next to nothing and want to have access to information I could not afford. So I don’t ever want to charge anybody who can’t pay to read or watch material here. The books I have cost a little, but most of the charge goes into printing. And you get a good idea what’s in them from these openly available posts.

This is why your support is necessary and voluntary, motivated by your desire to help others out. I can’t write grants, universities are (obviously) out of the question, and no corporation in its right mind would hire me. What’s left is right here. I am profoundly grateful for every bit of support I get from my dear readers, and even more thankful for your making it possible for everybody to benefit. God bless you all for your many years of support.

The Class & Office Hours

The Class progresses on its course (what a good joke!). A few take all. Most take some to none, which is what I expected and which is fine. I believe, and prove, that that standards of evidence used in Science are deeply flawed, lead to vast over-certainty, error, and scientism. However, these problems are easily fixed—if you know there are problems. That’s what they Class is all about. And why it is so necessary.

There are two difficulties, though:

1. The Class is now rather mature and its length intimidating. Those curious won’t know where to start, or can be disheartened. So I need to figure some way to make standalone mini-lessons. All ideas welcomed.

2. Office Hours. I believe I have enough subscribers at YouTube now—but subscribe anyway—to go Live. When they still let me in the classroom, I learned more from excellent questions than any other way. If there is an interest from a good number of people, we can try to do live sessions.

Anything Else?

What am I forgetting?

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I don’t know who wrote Magnifica Humanitas, the Vatican’s new encyclical on “safeguarding the human person in the time of artificial intelligence”, but he (or it?) appears to not have had much time in which to do so. If he had, he would have written something shorter. Some autist at Less Wrong (the “rationalists”) wrote more words than anybody would care to read to claim that AI “wrote” the document. Those pesky em dashes, you see. Which, regular readers will attest, I enjoy and have been using for years—decades, even.

Pope Leo signed off on it as Official Author, so to him goes the credit. Whoever or whatever produced the document, the words and arguments stand and must be judged as they are (this is what I take such great pains to teach in the Thursday Class about any model, even human models).

There is a lot of fluff in Magnifica, common these days, passive language about “dialogue” and “dignity” and “equality” and “synodality”. That last is a neologism from synod, a meeting (of bishops). In practice, it’s a word without boundaries, meaning whatever its user wants it to mean, mostly finding ways to excuse sin and overthrow tradition.

There’s even, Lord help us all, “journeying.” Everything is now a “journey”. No one can do the simplest task without it being a “journey”. Women “journey” for every damned thing they do. I looked up a bread recipe and it featured the words “your sourdough journey.” I had a paper accepted and the publisher said here is the paperwork to complete your “publishing journey.”

On our spiritual journey to review this document, and not that some of these subjects aren’t important, but we must turn a blind eye to all of it lest we miss our point. Which is our relation to technology.

Ratcheting Up

Now technology means tools, a far less scary word, which Leo understands: “Technology should not be considered, in itself, as a force antagonistic to humanity.” And “In the abstract, technology in and of itself is not a solution to humanity’s problems, just as it is not inherently evil.” Tools become antagonistic, or sympathetic, by the hands that wield them.

The problem is, as it ever is, not tools, but people. There’s a some good, and a lot wrong, with people. Including forever forgetting, in the rush for “progress”, the undefeated Doctrine of Unexpected Consequences: “New technologies open up a horizon extending in directions that are imaginable but not yet fully predictable.”

On that unpredictability, here’s David Stove in the opening to “Why You Should Be A Conservative”:

A primitive society is being devastated by a disease, so you bring modern medicine to bear, and wipe out the disease, only to find that by doing so you have brought on a population explosion. You introduce contraception to control population, and find that you have dismantled a whole culture. At home you legislate to relieve the distress of unmarried mothers, and find you have given a cash incentive to the production of illegitimate children. You guarantee a minimum wage, and find that you have extinguished, not only specific industries, but industry itself as a personal trait. You enable everyone to travel, and one result is, that there is nowhere left worth travelling to. And so on.

This is the oldest and the best argument for conservatism: the argument from the fact that our actions almost always have unforeseen and unwelcome consequences. It is an argument from so great and so mournful a fund of experience, that nothing can rationally outweigh it. Yet somehow, at any rate in societies like ours, this argument never is given its due weight.

The good old DUC has already hit AI, which was promised to “revolutionize”—does nobody ever remember revolutions are to be eschewed?—how companies work, bringing vast savings (notice the em dashes!). Predictions of our bright AI future were many. But it was recently revealed (of many similar stories, like rampant cheating in universities):

A company spent $500,000,000 on Claude in one month because nobody set usage limits…Uber burned their entire 2026 budget by April. Their COO said he can’t connect any of it to consumer features…Microsoft canceled most Claude Code licenses because the token bill spiraled…Companies are now laying people off to pay the AI bill. Not because AI replaced the work. Because the bill replaced the headcount.

Science Is Amoral

Pope Leo starts (para 12) with a truth almost completely forgotten, or if not forgotten then actively denied:

[B]uilding for the common good means accepting the limits and weakness of humanity without considering them an error to be corrected. Today, the human desire for fullness of life is at risk of being misled by deceitful goals, such as the prospect of a technology that promises to free us from all weakness, and models of wellbeing that leave behind entire populations.

There will be no eliminating the human condition. No technology will ever fix us. Every tool eases efforts in one dimension and creates difficulties or differences in others, and our flawed nature remains. In his same essay, Stove notes man is equipped “with a ‘miserostat,’ which always keeps his discontent up, not indeed to a constant level, but at least to a certain minimum level, and a high one at that.” No tool can set that miserostat to zero more than temporarily.

Leo does not take an anti-tools position. Nor is Leo anti-science, except in the sense that we realize science is not the answer to any moral question. This is what Leo meant when he said (para 27) “science [sometimes] oversteps the limits of its competence.” As we saw most ingloriously in the covid panic.

Yet sometimes even the Church doesn’t realize this, as when Leo (para 43) says “In Laudato Si’, Francis provided the first significant systematic treatment of the environmental crisis…” And later (para 81) “A litmus test for social justice today is…climate change and environmental disasters.”

There is no environmental crisis (see our Clintel essay on this Francis’s encyclical). But it is believed there is one, the belief implanted by scientists who didn’t so much overstep moral bounds, though they often did that, but guaranteed far more certainty than is warranted. The climate is never constant and environmental disasters have always been with us, and always will. They are not coming with greater frequency now, and many of the tools we made deal with them handily. Which shows that miserostat can be set to HIGH even in the holy.

AI Itself

At last we come to computer models, a rose that smells sweeter with its glorified name of AI. Here Leo, and Francis, are right (para 92):

In his Encyclical Laudato Si’, Pope Francis denounced the growing dominance of a technocratic paradigm [119] in our globalized world: the tendency to let the logic of efficiency, control and profit alone shape personal, social and economic decisions. This makes it clear that technology is not simply a tool. When it becomes the standard by which everything is judged, it begins to dictate what matters and what can be discarded, reducing creation to an object of exploitation and human beings to mere cogs in a system driven toward ever greater efficiency.

This damns both scientism and utilitarianism, both children of the Enlightenment. And (para 94):

The danger of humanity becoming a victim of its own achievements was already clearly recognized by Saint Paul VI, who warned that “the most extraordinary scientific progress, the most astounding technical feats and the most amazing economic growth, unless accompanied by authentic moral and social progress, will in the long run go against man.” [121] For this reason, technological progress — valuable in itself — requires careful discernment of the anthropological vision that guides it and the ends it pursues.

Leo is also right when he names, without naming the name, of the oligarchy, reminding us that our newest tools are not controlled by States “but with major economic and technological actors.” Such power “tends to become opaque and evade public oversight”. How many times has Big Tech been caught listening to conversations or peeking into emails when they said they weren’t?

His knowledge of computer models is not great, however, saying “those who design them, possess only a limited understanding of their actual functioning.” That’s only so in a limited sense where any model output with myriad inputs can’t usually be known exactly, though the task of the model can be, and indeed always is, known. For (and you have this one by heart) all models only say what they are told to say, and AI is a model.

On the other hand, my heart soared like a hawk to see (para 99) Leo recognizes computer models “merely imitate certain functions of human intelligence” and that any seeming intelligence is “a form of statistical adaptation based on data and feedback, which can be very effective, but does not imply inner growth.” There is nothing inside a model that can grow.

Leo also sees computer-model hype fosters “encourage excessive reliance and the search for ready-made answers, and weaken personal creativity and judgment” and that many “overlook the fact that they reflect the cultural assumptions of those who designed and trained them” (all models etc. etc.). He incorrectly frets about their “climate” effects and says “it is essential to develop more sustainable technological solutions”. There is an AI bubble now, as there always is with new tools, and a filtering and efficiencies will come as they always do.

Using AI

The problem is people. Hypers and overly aggressive promoters, and the genuinely wrong about the capabilities of computer models, are one side, and a great mass of those who buy into are on the other. Leo: “we cannot consider AI to be morally neutral.” Of course computer models are not morally neutral. They are just as moral as your screwdriver or your blender. Which is to say, there is no morality about them at all. Whatever biases—a neutral word itself—they have are put there by their coders. Claude refuses to say good things about white people, while excessively praising all other races, because that’s what its coders decided. As long as you don’t let yourself descend to sloppy thinking about “artificial” “intelligence” becoming alive, model—coder—bias can easily be coped with.

Leo expects (para 107 and on) “AI to respect human dignity and truly serve the common good”. You might expect the same of the New York Times, but you’re not going to get it. It’s not the tool. It’s the people. We ought not want a model to respect dignity, because models can’t respect anything. We want rulers and elites and coders to respect dignity. Leo says “Those who control AI will impose their own moral vision, which will become the invisible infrastructure of these systems”. That’s only true, again, to the same extent people think the NYT is politically neutral. Many already know most computer models are woke. And if they don’t now know, they soon enough will.

Here’s where he’s right, in a way: “To disarm [AI] means discrediting the assumption that technical power automatically confers the right to govern.” To disarm arms does the same. Yet what then gives the right to govern? Surely it is not the “will” of the people, or some such democracy nonsense. Rejecting God, the only thing modern rulers have is might. They are not going to give up that might, in whatever form.

No More Than We Are

In our culture, intelligence is automatically taken as a measure of morality: the more intelligent, the better the person is. Not in, say, ability to calculate obscure measures, but better tout court. The paradox is that we must also claim all are equally intelligent, that no person, and certainly no group of people, can be less intelligent than another. We are all above average. The only evil is to reject Equality. Maybe this is why we are happy to award and rate intelligence in computer models, that act being taboo for peoples.

Leo: “intelligence, when absolutized, overshadows other essential dimensions of life, such as affection, the will, commitment and relationships.” The risk is to narrowly define of create of fetish of intelligence and create some dreary mandarinate.

Which brings us to transhumanism. One of its ploys is to promise greater intelligence with implants, even aided by computer models. Well, I can jump in a Mustang and tear off down the road a lot faster than I can run. If I have an implant, I may have it recall a forgetton formula faster than I can look iy up in a book. But in the end, I have not changed my nature. No one can. Transhumanism is a false promise in that sense. (And I can get out of the car and walk, but removing the implant would not be so easy.)

Radical transhumanists, Leo says (para 116), are “anticipating a threshold where humanity surpasses itself in a new evolutionary stage.” These tend to be the same people arguing computer models are alive. The threat in both instances is entirely imaginary. That people believe in it is the real problem.

And that, at last, brings us to the line many quote (para 117): “If the human being is treated as something to be perfected or surpassed, it becomes easier to accept that some lives are less useful, less desirable or less worthy.”

If you are looking for a linebacker, I am less worthy than many men. If you are looking for a wife, then I am not desirable at all. Nor am I useful in either of these things. But I fancy I can perfect myself in the subject of probability, and I have hopes my students (you, dear reader) will surpass me. And that is before thinking of new and better things for my sons.

One can be less worthy or less desirable in many, even most, things. One must seek perfection, in oneself and in others, up to a point. And only a point. All of this is conditional. It’s difficult to speak unconditionally, except narrowly.

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In proving foresight may be vain:
The best laid schemes o’ Mice an’ Men
Gang aft agley,
An’ lea’e us nought but grief an’ pain,
For promis’d joy!

Lennie said, “I thought you was mad at me, George.”

“No,” said George. “No, Lennie. I ain’t mad. I never been mad, an’ I ain’t now. That’s a thing I want ya to know.”

The voices came close now. George raised the gun and listened to the voices.

Lennie begged, “Le’s do it now. Le’s get that place now.”

“Sure, right now. I gotta. We gotta.”

And George raised the gun and steadied it, and he brought the muzzle of it close to the back of Lennie’s head. The hand shook violently, but his face set and his hand steadied. He pulled the trigger. The crash of the shot rolled up the hills and rolled down again. Lennie jarred, and then settled slowly forward to the sand, and he lay without quivering.

Capital Crimes

A gang of gypsies in England gang raped a young girl (and another previously) at knifepoint while filming the deeds, laughing all the while and even posted one of the rapes on social media. At their trial, Judge Nicholas Rowland excused their crimes because he said the criminals were “‘very young’, had low intelligence, a ‘limited understanding of consent’ and were susceptible to ‘peer pressure’.”

[Rowland] said that the second boy fell into the bottom one per cent in IQ for his age, and he had been diagnosed with ADHD, while the third boy had ‘low intellectual capacity’ and he had a ‘limited understanding of consent’.

Iryna Zarutska, 23, was riding a train in Charlotte, when Decarlos Dejuan Brown Jr rose up, slit her throat, and as he was exiting the train gleefully declared he “got” his white woman. Brown had been arrested some 14 times before he murdered Zarutska, for crimes including armed robbery. He was freed each time. For the murder, he was found by Experts to be “incompetent to stand trial”.

Brown and the gypsies were not alone. Recently, there were these cases:

Many states have humane destruction laws that apply when animals (usually dogs) have attacked or killed humans. Florida, for instance, confiscates vicious dogs and puts them down. When any animal kills and eats a man it is usually put down, and most think it wise and prudent to do so. But some curiously argue the animals cannot help themselves, that it is their nature to attack and kill and even eat people, and who are we to judge?

In any case, it is clear that dogs, nor any animal, are not as intelligent as man. Just as it is clear obvious truth that some men are not as intelligent as others. Yet this fact does meet resistance from Equalitarians and Universalists, both forgiving every sin except the sin of claiming sin exists.

A Supreme Mess

Alabama was going to execute murderer Joseph Clifton Smith (white), but his lawyers appealed and then the states’s lawyers appealed after a lower court said Smith could escape his fate. The case recently reached the Supreme Court which let the lower court’s ruling stand.

Smith’s lawyers didn’t argue Smith was stupid; they insisted there was a chance (these are their arguments, not mine) his “IQ” was below 70. This number became an odd cutoff: murderers “with IQs” greater than 70 could be executed for their despicable crimes. Those “with IQs” less than 70 were allowed to get away with it. Or so it seems. We explore this idea below.

But first examine the Supreme Court’s 2002 decision in Atkins v Virginia. The facts were not too different than with Smith: planned robbery followed by abduction and despicable murder. The Courts said the execution of “low IQ” persons was “cruel and unusual punishment” prohibited by the Eighth Amendment. Not surprisingly, Scalia and Thomas and even Rehnquist disagreed.

The Eighth Amendment reads: “Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted.”

Since 2002, executing a dumb criminal is “cruel”, yet executing an intelligent criminal is not cruel nor unusual. This is odd because, as any dog owner can attest, even dogs can know right from wrong, and even stupid men know murder is wrong.

Scalia wrote in his dissent that an Expert (a psychologist) on one of the appeals testified Smith had “an IQ of 59”. Smith also, and in this case, really had “16 prior felony convictions for robbery, attempted robbery, abduction, use of a firearm, and maiming.” He noted previous courts ruled only the profoundly retarded, those “idiots” who “had an IQ of 25 or below”, had a “‘deficiency in will’ rendering them unable to tell right from wrong”.

Neither Akins nor Smith nor Brown above were “idiots” in this sense.

The Court conveniently ignores a third “social purpose” of the death penalty–“incapacitation of dangerous criminals and the consequent prevention of crimes that they may otherwise commit in the future,” Gregg v. Georgia…

Scalia wrote that it wasn’t true the “mentally retarded really more disposed (and hence more likely) to commit willfully cruel and serious crime ” and instead “being childlike generally suggests innocence rather than brutality.”

To the key pertinent question of culpability:

Assuming, however, that there is a direct connection between diminished intelligence and the inability to refrain from murder, what scientific analysis can possibly show that a mildly retarded individual who commits an exquisite torture-killing is “no more culpable” than the “average” murderer in a holdup-gone-wrong or a domestic dispute? Or a moderately retarded individual who commits a series of 20 exquisite torture-killings? Surely culpability, and deservedness of the most severe retribution, depends not merely (if at all) upon the mental capacity of the criminal (above the level where he is able to distinguish right from wrong) but also upon the depravity of the crime–which is precisely why this sort of question has traditionally been thought answerable not by a categorical rule of the sort the Court today[351]imposes upon all trials, but rather by the sentencer’s weighing of the circumstances (both degree of retardation and depravity of crime) in the particular case. The fact that juries continue to sentence mentally retarded offenders to death for extreme crimes shows that society’s moral outrage sometimes demands execution of retarded offenders. By what principle of law, science, or logic can the Court pronounce that this is wrong? There is none.

There is more: I have only given highlights pertaining to the question of intelligence and guilt. (Incidentally, late in his dissent, Scalia calls phrases like less likely used in that context “pretty flabby language”.)

Intelligence is not IQ

Sotomayor (with Jackson joining) wrote the first opinion in the new Alabama-Smith case. They decided on the cutoff of an IQ score of 70 as execution worthy. Yet what happens if the convicted has more than one IQ test score or there is uncertainty? Which do you pick? And what IQ score should SCOTUS justices have (which they didn’t address).

At his sentencing hearing, Smith introduced evidence of intellectual disability, including an IQ score of 72 and testimony from the test’s administrator explaining that the test’s standard error of measurement indicated Smith’s IQ could be as low as 69 or as high as 75.

This had a footnote, which read:

The standard error of measurement reflects the potential error inherent in an IQ test and is used to calculate a confidence interval, a “range within which one may say an individual’s true IQ score lies.” Hall v.
Florida…(2014).

There are two problems. The first is that, as I am always ranting, IQ is not intelligence. IQ is a score on a test. No one has an IQ. Some have scores on tests. Everybody has intelligence to varying degree. See this, this, this and this for exhaustive critiques. All of which end with the vain plea: if you mean intelligence, just say intelligence and not “IQ”.

The second problem is that confidence intervals don’t mean what Sotomayer says, though everybody, even statisticians, takes them as meaning as she says. Their true meaning is so bizarre that nobody can keep their real definition in mind. Not even SCOTUS judges. This is one of the many measures in so-called frequentist statistics I’m always railing about in the Class. Do not use them.

For the morbidly curious—everybody else can skip this paragraph or see this post—here is what the true definition is: If an infinite (no fewer) number of experiments were run, each identical but randomly different to the one which gave us our data, and each experiment had a confidence interval calculated, then a certain percent of these infinite intervals will “cover” or contain the true value of the thing of interest. About your own confidence interval, the one you just calculated, nothing can be said, except the tautology the true value may or may not be in the interval.

Even you understood this, it does not apply to the Smith case or to any “IQ”. What is possible is a guess, within some bounds, of what a given person might score on a test.

Later Sotomayor says:

On [an IQ] test administered by King, Smith scored a 74 with a 95% confidence interval of 70 to 79. On the test administered by Dr. Fabian, Smith scored a 78 with a 95% confidence interval of 72 to 83. At this point, Smith had obtained five IQ scores ranging from 72 to 78.

There follows wrangling over the IQ test scores and their variability on subsets of the tests.

Smith’s second expert, too, testified that Smith’s scores were “in the range of what would be considered mild intellectual disability,” “particularly” when “consider[ing] the standard error of measurement.”…

The court acknowledged that King “testified that if there are multiple sources of IQ over a long period of time[,] it contributes to the construct of validity indicating what a true IQ score is for an individual” and that “multiple IQ scores” in Smith’s case “place him in the borderline range, functioning just above intellectual disability.”

Even experts can’t distinguish between IQ tests, which only crudely measure intelligence, and intelligence itself. Nor can SCOTUS justices, because later there is testimony Smith’s “true” IQ might be 69. Nor can Sotomayor nor anyone else deal with the many different scores she received for Smith: “the parties agree that there is no single approach to weighing multiple IQ scores.”

A considerable number of pages is devoted to this mysterious question.

But it is, of course, the wrong question, and not just for the reasons I have given. Because nowhere does Sotomayor (nor Jackson) ask: Did Smith know what he was doing was wrong? In this crime? In all his other previous crimes, including those in which he wasn’t arrested?

Which, of course, the answer is yes, yes, he did. Thus his “IQ” test score is irrelevant. Smith, like Brown now or Atkins then, is anyway sufficient dangerous in the sense Scalia defined.

Justice Thomas naturally saw this gaping hole. He opened his dissent:

Such persons [with low IQ scores] could not be executed, the Court said, even if they “know the difference between right and wrong,” even if they “are competent to stand trial,” and even if they are intelligent enough to deserve other “criminal sanctions.

Thomas also saw the consequence of Atkins: “The result was predictable: To avoid execution, Smith tried to convince courts that he is not intelligent enough to be executed.”

But Smith is not insufficiently intelligent to be executed. He can read at an 11thgrade level. He took five IQ tests and did not once receive a score of 70 or below, instead scoring 75, 74, 72, 78, and 74. The lower courts held that he could not be executed based only on the hypothetical possibility that these IQ scores were all wrong and that his IQ is in fact 70 or below.

It is preposterous to pick a magic number like this wholly artificial 70 to conclude moral culpability. Yet because of this new ruling it is now a good money bet to watch for a slew of claimed IQ scores below this new magic number to argue for leniency of any kind, not just escaping capital punishment.

Justice Alito chides the Court for punting on the question of how to use the multiple scores, which for all her word Sotomayor punted. But Alito does seem to be suspicious of IQ tests, and writes of their “true” values using scare quotes, which I applaud.

I do not see anywhere in the SCOTUS ruling, which means it is surely not in earlier court documents, whether anyone bothered to ask Smith if he knew murder was wrong.

I, too, dissent. Smith ought to be executed.

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Reminder: The Thursday Class is only for those interested in studying uncertainty. I don’t expect all want to read these posts. Please don’t feel like you must. Yet, I have nowhere else to put them. Your support makes this Class possible for those who need it. Thank you. Much much math alert!

How to make the best decisions based on goodness criteria you pick.

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Links: YouTube * Twitter – X * Rumble * Bitchute * Class Page * Jaynes Book * Uncertainty

HOMEWORK: Read!

Lecture

“Suppose an observer is given a voltage varying with time during a prescribed observation interval,” said Peterson, Birdsall and Fox, “and is asked to decide whether its source is noise or is signal plus noise. What method should the observer use to make this decision, and what receiver is a realization of that method?”

Excellent questions. The voltage level here was displayed on a scope hooked to a radar, which is to say a radio (transmitter and) receiver, and the signal plus noise was the identification of an aircraft during one of our rulers’ Twentieth Century wars. The “noise” is misnamed, for all noise is signal of some kind (as we discussed before). We only call unwanted signal “noise.”

The wanted signal here is a real aircraft, which results in a voltage which itself is displayed on a screen (think oscilloscope). The unwanted signals is everything else, all of which adds to the voltage level.

The idea, in strict probability, is easy, though implementation can be more or less difficult. We have a stream of data from which we calculate, at any time, the probability the data represents the signal we want. Such as a plane (from the voltage level) or a disease (from the PSA) or whatever. I.e. Pr(Y = signal | X = data, E), where as always the E is whatever other evidence we bring to the problem.

After Pr(Y|XE), and only sometimes, comes the decision, which is not the probability. If Pr(Y|XE) ? p, we might decide to act as if “Y = signal” is (or will be, or was) true, else we act as if it is false. From which only four things can happen:

• We act as if “Y=signal” is true; it is true; a true positive;

• We act as if “Y=signal” is true; it is false; a false positive;

• We act as if “Y=signal” is false; it is true; a false negative;

• We act as if “Y=signal” is false; it is false; a true negative.

Now here is where it can be tricky. Suppose, as is often the case, your Pr(Y|XM) is a one-to-one function of x; I mean the level of x, which can be a voltage or some other measure such as some antigen or whatever (and where I use ‘M’ to indicate we have a model). This measure goes from small to large (or vice versa), with greater (lesser) values indicating higher probability of Y (shorthand for “‘Y=signal’ is true”). Then picking a p or picking an x_p, the subscript indicating the one-to-one relationship, to decide or act as if Y, is the same.

That is, using the probability or using the measure itself leads to the same decisions. That means any verification method used to judge your judgements would lead to the same outcome. And you will often see the measure or score used directly, and not the probability. But it is all the probability after all. It is probability even if you never compute a quantitative number for the probability. And it is probability because there is uncertainty: it is not certain that Y if X > x_p, nor if Pr(Y|XM) > p.

It’s not that one must first calculate a p from the X in order to decide or verify. But to remind ourselves it’s as if we have done that very thing. We are not escaping probability, nor decision theory by working with X. Which brings us right back to our initial discussion of judgment functions and worthiness premises. Because, as you can now see, it’s the same thing.

So we could use any of the tools we’ve already developed to verify our model (or implied model).

What is, therefore, the best p or X to use? There is no answer. No universal one-size fits all answer. It is true that p is entirely objective, once X and M are specified, as all models are entirely objective. But what is best depends on the uses to which your model (the p, or X indirectly) are put.

A good part of that discussion was reminding you that not all decisions and consequences can be quantified. I remind you of that, and of the strict warning that one must not lather on quantification merely for the sake of putting numbers on things. I double that warning as we do just that, because of custom and sometimes quantification is desired.

Assuming we want or need quantified verification, like before there are two situations: (a) we judge past performance of a model (picking winner like in a footrace), or (b) we use the past data and verification to improve future judgements. The two (as we said before) are often confused.

Let’s revisit the table (in bullet points) above, only now assigning quantitative measures to the outcomes:

• Y = 1 & X = 1 :: a_11;

• Y = 1 & X = 0 :: a_10;

• Y = 0 & X = 1 :: a_01;

• Y = 0 & X = 0 :: a_00.

Where I trust the shorthand is obvious: “X” is overloaded as notation to the decision: X here is a function (of the measure x or p). Most worthiness premises (But see Paul Erhlich) would say a_11 and a_00 are desirable, and thus positive. They would also say a_10 and a_01 are undesirable, and thus negative. A user has gained more than lost if

There is no harm if we assume a_{10} and a_{01} are positive if we remember what we are about. This allows us to drop the annoying absolute value notation, and is the approach I will use henceforth.

If we are in situation (a) and judging past decisions, then our job is simple. We calculate

where the n_{ij} are the obvious counts of the table. If S is positive, then our decisions gave more benefit than produced harms. If we are rating different models, then the one with the largest S is the winner.

If we are in situation (b) the job is harder. For one, we do not have a fixed S, but a suite of them, one for every value of p or x that could have been used; i.e. S_x . For instance, if the measure X had (or has) ordered values (x_{(0)},x_{(1)}, … , x_{(q)}) we have the decision rules: “Act as if Y = 1/signal if X > x_{(0)}” (or less than or equal to) which gives S_0, “Act as if Y = 1/signal if X > x_{(1)}” (or less than or equal to) which gives S_1, and so on.

This will produce a string of S: S_{(0)}, S_{(1)}, …, S_{(q)}. One or more of these will be maximum, and will correspond to an x_{(j)}.

If all S_{(j)} ≤ 0, then my friend your measure x stinks.

Or your p, hence your M and X, if that it what you were using. Above I said p and X are equivalent, and that’s true. But it’s a local truth, because you must remember Y does not have a probability. Nor does Y given X. We only get the probability by adding premises M.

This is utterly crucial to keep in mind, because when you read other authors on this subject, such as Peterson, Birdsall and Fox above, then either make the mistake or given the impression that Y (given X) or that X have distributions. And so universally recognized optimal decisions can be had. They cannot. Because all probabilities are conditional on the assumptions you—as in you—make and most decisions are local and do not involve necessary moral truths. For instance, there is no moral law about the voltage represented on a CRT.

So while again it’s true that p and x can be one-to-one, there can be many different p for the same x. How many different p? How many? That’s right: infinity.

This being so, we can introduce skill in this situation. Skill is when S_max(M2) > S_max(M1) for two competing models M2 and M1.

We always have the natural model M0 to compare against here. WATCH THE VIDEO FOR CLARIFICATION HERE. Suppose we always acted as if Y = 0/no signal. Which is another way of saying X = 0 always. Then we would score

and if instead we picked X = 1 always, we’d score

Let n_0 = n_01 + n_00 (i.e. all the times Y = 0), and n_1 = n_11 + n_10 (i.e. all the times Y = 1). Then we would pick M0_0 as our best model iff

Which is when

If all a_ii are equal, the decision rule becomes simple: choose X = 0 (M0_0) if

𝑛0≥𝑛1,

else choose X = 1 (M0_1). Example: if you are deathly afraid of false negatives, then you might have a large a_01, and so would tend to pick M0_0.

If you have a rival model M1 (or rival models), ir or they ought to beat M0, or the rival model has no skill.

Next time: examples and such things as sensitivity and specificity.

Note: Let A_t = (a_11, a_10, a_01, a_00). Thus A itself can vary from observation to observation or in time, as happens in real life, too.

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As I was looking, I noticed a strange artifact in the data. The number of people killed before they could escape their wombs were not counted in the official death data.

I counted. Here for the USA is a plot of the proportion of all deaths, by year, of those killed inside their would-be mothers.

There were many years in which those who were killed in the womb almost equaled those who died outside of them; i.e., nearly half of all deaths were caused by womb-killing. I find this a remarkable number. Perhaps you will, too. Even though the proportion is lower now, it is still enormous.

The same graph for England Wales is below, after the details.

Details

Every year people of all ages die. Some are killed before or during their exits from their wombs. The womb-killed bodies were not in the official death numbers tracked by the respected Human Mortality Database, from where I retrieved all other death counts by age and year (for many countries). Their data is often the result of their own models, which you can read about there. Meaning there is some uncertainty to all you see here.

I sometimes added the womb-killed esimates/counts to the HMD category “Age 0”. Problem is, there is some plus-or-minus to the womb-killed numbers, especially before 1973 when it was still illegal for women to kill their kids. It still happened, of course, because Americans are traditionally free spirits and scoff at unwelcome laws. But it didn’t happen at rates as great as they would when Equality hit. My counts and estimates are from the Guttmacher Institute, a group well known for tracking these things.

Their numbers before 1973 are a rough estimate based on surveys and the like, numbers which were difficult to come by or to entirely trust because womb-killing was against the law. Even after it was legal, each year wasn’t counted perfectly, though accuracy did improve with time. For missing years, I linearly extrapolated guesses.

In 1950, for instance, their estimate is 600,000. By 1955 they estimated 700,000. My extrapolation guess for 1951 is 620,000. And so on. The peak, says Guttmacher, was 1.6 MILLION in 1990. You see that brutal era in the plots. Numbers are indeed lower now, but that’s mainly because fewer people are attempting to procreate and because of the increased use of birth prevention devices and drugs (“the pill” is almost routine for many now). But I think, too, the drugs women take soon after coitus to poison any conception are not counted.

Also, women are just plain older now, on average, and so fewer can become pregnant. Reality does not acknowledge Equality, even though we must.

Anyway, we are left with uncertainty. Some periods are more uncertain than others, which is why I denoted the period before 1973 with a specially colored line (white in black).

I have no estimates of miscarriages, which rightfully belongs to the early, infant, or Age 0 category. That is, current guesses of rates can be had now, but getting them historically is a problem. It’s difficult to predict whether these numbers have declined with increaseing medical capabilities, or if they have instead increased if we include IVF, which are largely planned mechanical miscarriages, since most of the lives made with IVF are snuffed out.

All these caveats are important, and make a nice area for study for some aspiring sociologist or demographer. You already know fertility rates are declining everywhere rapidly, at the same moment most cultures reward the sexually sterile the most. Indeed, non-reproductive activities are seen almost as kinds of superpowers. It’s very strange.

However, our topic is death, and at what ages it occurs.

Deaths

USA

Statistics like life expectancy have their uses, but one-number summaries of complex distributions always suffer in one way or another (see below). They can mislead and cannot give a full picture of what is happening. But we can. Here is a frequency distribution of death age—the age at which people were recorded to have died—for each year of available data. This is for the USA. This does not have the womb-killed numbers.

You have to stare at the sequence a few times until you become accustomed to what’s happening. Most obviously, infant deaths used to be huge and have dropped to seemingly small numbers. It’s only seemingly because womb-killing is not included in these numbers, which I’ll do below.

It also becomes clear that age at death is increasing: we are dying at older ages than before. The question we began with has been answered: People are living longer now. Up to a point. Always up to a point. The maximum here is “110+” (which I coded as 115). This, or even a little before at 105, appears to be some kind of wall which once hit is fatal, with only a very few slipping under the crack. Another way to say it: the maximum lived age is not increasing, or is but not by a perceptible amount.

This pattern is repeated in all the countries the HMD has. They do not have all countries. And even in some of those they do their data has limited historical scope. There is nothing in Africa, and only Chile represents South America. East Asia is limited to Japan, Hong Kong, South Korea and Taiwan. No China. South Asia and the Middle East, except Israel, is missing entirely.

I’ll include a couple of other countries below the fold. Too many at once can be dizzying.

You can see the Baby Boom strolling to their Final Exit. Watch that “horn” walk upwards to the end. The horn is the male-female difference. HMD breaks their data down by sex, and in the animation below we learn only the old wisdom that men die earlier up until about 80 years old, after which women outpace men. Which means women are hardest hit.

The green line, if you hadn’t noticed, demarcates the more men than women: this is just the ratio of male deaths to female deaths. It is not only in times of war that young men, say around 20, die at three times the rate of women. But perhaps it is remarkable that even at the youngest ages more males than females die. Until the weakest are cleared out. By age 80, the men who made it are more robust and the women start slipping under faster.

We can see population increases in the first animation, or rather infer them, by the increasing numbers of people dying at all ages. But we can’t see everything. We don’t see births or immigration, nor emigration. So we don’t, in these plots, learn where the increased numbers are coming from.

Here is the same animation as above, but now adding the womb-killing to Age 0. Startling, is it not? It still astonishes me that men allow women to kill their children.

England & Wales

We begin with the proportion of deaths attributed to womb-killing (here data is from the UK itself and Johnston’s archive, with the same caveats about about uncertainty; i.e. do not swear to any particular numbers). This is only England & Wales and not Scotland.

It’s curious that the rate is still increasing here. England’s rulers are welcoming foreigners with the same ardency of lonely maidens holding the door open for vampires. Perhaps it is the foreigners who are assimilating to the native culture, at least in womb-killing. Or perhaps the natives, exhausted by the endless lies of their rulers, are giving up. You tell me.

Here is the death distribution animation, as above (no womb-killing included):

Don’t miss the ludicrous bump centered around 20-year-olds about 1940. There are some other peaks that come and go along the way of local interest. The early high-infant-mortality is there, as before. In any case, it’s true here that people are dying older.

I won’t bother adding in the womb-killing for another animation. It has the same depressing shape as in the States.

Two One-Numbers

There are other ways to display the data besides animation, but all require great screen space and much scrolling. As I said above, one-number summaries can provide insight, but at the cost of easy interpretation. Still, people hanker.

One idea is to look at the proportion of deaths over 80 (or 85) years old through time. Another is to look at the mean age of death for all those who died after, say, age 5. The reason for after 5 are those “bi-modal” peaks in the death-age distributions. If you make it past 5, there’s a good chance you’ll make it a lot longer. Other possibilities exist, of course, but with these we can look at many countries at once. There is no womb-killed in the data below.

Start with the proportion of deaths over 80 by country:

It’s hard to pick out individual countries, but our point here is not any particular place, but all. It’s clear enough an increasing number of those equal to or greater than 80 are dying. There is a kind of dispsy-doodle in the 1960s.

Another reason why single-numbers fail: consider we have a cohort (say born in 1930). The proportion of those dying at age >80 in 1940 would be 0, and same in 1950, but there’s be a jump at 2010 to some positive number, which would only increase after that but only because the cohort itself is growing older. That happens, too, to a certain extent in the main death data, but not as much, and there we know we have immigration and births.

Here’s the same for 85:

Same story. Now the mean age of death for those who make it past 5:

The devastation of WWII is there. Wiped out a good chunk of young men (women hardest hit). WWI would be there, too, if the data started early enough. No big signal for the covid panic.

Video

Other Countries’ Animations

WARNING! Your eyes can go buggy staring at these. But for the sake of completeness, here are a few other representative country animations. For our main question—are people dying older?—they are much the same. Those interested in individual countries might take some benefit, too.

Taiwan

Chile

Poland

Netherlands

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As I was scraping the frost off the windshield this morning, I thanked God for global warming, now called “climate change.” For without the warmth provided by our very exhalations, we would already be in a new ice age.

In any case, it is Memorial Day weekend, which in northern Michigan is the official start of the end of winter. Spring is expected sometime next week. But not early next week.

Also be sure to attend a service for those who were asked, and in many cases made, by our rulers to fight their wars for them.

No Class today (but there will be next Thursday). No post, either, until next Wednesday, because of the long weekend. Spend as much time as you can away from your “devices”.

God bless you all, and my profoundest thanks for your continued support.

Here are the various ways to support this work:

• Subscribe at Substack (paid or free)

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• PASS POSTS ON TO OTHERS