Class 2: Uncertainty & Probability Theory: The Logic of Science: Quid Est Veritas

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(Hagfish: I was in such a hurry, I forgot my jacket.)

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It still feels like I’m rushing things, but the videos are already a half hour. I can’t see your faces so it’s difficult to tell. Let me know about the tempo.

I also said I could not think of a homework in the video. Well, I have.

HOMEWORK (which I said there was none in the video, but here it is anyway): Find in the “media” what you think is a local/conditional truth and demonstrate why. Then find in your life what you think is a universal/necessary truth and demonstrate why.

Lecture

This is an excerpt from Uncertainty. All the references have been removed.

“Quid est veritas?”

The answer to the above, perhaps the most infamous of all questions, was so obvious that Pilate’s interlocutor did not bother to state it. Truth was there, in the flesh, as it were, and utterly undeniable. Everyone knows the sequel. Since that occasion, at which the answer was painfully obvious, the question has been re-asked many times, with answers becoming increasingly skeptical, tortured, and incredulous. The reasons for this are many, not the least of which is that denial of truth leads to interesting, intellectually pleasing, unsolvable but publishable puzzles.

Skepticism about truth is seen as sophistication; works transgressive to truth are rewarded, so much so that finding an audience accepting of truth is increasingly difficult. More than sixty years ago Donald Williams, exasperated over the pretended academic puzzlement over the certainty of truth, said the academy

in its dread of superstition and dogmatic reaction, has been oriented purposely toward skepticism: that a conclusion is admired in proportion as it is skeptical; that a jejune argument for skepticism will be admitted where a scrupulous defense of knowledge is derided or ignored; that an affirmative theory is a mere annoyance to be stamped down as quickly as possible to a normal level of denial and defeat.

Yet truth is our goal, the only destination worth seeking. So we must understand it. There are two kinds of truth: ontological and epistemological, comprising existence and our understanding. Tremendous disservice has been done by ignoring this distinction. There are two modes of truth: necessary and local or conditional. From this seemingly trivial observation, everything flows.

Truth

Truth exists, and so does uncertainty. Uncertainty acknowledges the existence of an underlying truth: you cannot be uncertain of nothing: nothing is the complete absence of anything. You are uncertain of something, and if there is some thing, there must be truth. At the very least, it is that this thing exists. Probability, which is the language of uncertainty, therefore aims at truth. Probability presupposes truth; it is a measure or characterization of truth. Probability is not necessarily the quantification of the uncertainty of truth, because not all uncertainty is quantifiable. Probability explains the limitations of our knowledge of truth, it never denies it. Probability is purely epistemological, a matter solely of individual understanding. Probability does not exist in things; it is not a substance. Without truth, there could be no probability.

Why a discussion of truth in a book devoted to probability? Since probability is the language of uncertainty, before we can learn what it means we need to understand what it is that probability aims at. Hempel understood this, but couldn’t help himself from writing the word without scare quotes, as if “truth” might not exist. What is the nature of probability’s target? What does it mean to be uncertain? How do we move from uncertainty to certainty? How certain is certain? It will turn out that statements of probability (assuming they are made without error, an assumption we make of all arguments unless otherwise specified) are true. When we say things like “Given such-and-such evidence, the probability of X is p“, we mean to say either that (the proposition) X is true, or that not-X is. So truth must be our foundation. What follows is not a disquisition on the subject of truth, merely an introduction sufficient to launch us into probability. This chapter is also a necessity because the majority of Western readers have grown up in a culture saturated in relativism. There is ample reason Pilate’s question is so well remembered.

Our eventual goal is to grasp models, and models of all kinds, probabilistic or otherwise, are ways of arguing, of getting at the truth. All arguments, probabilistic or not, have the same form: a list of premises, supposeds, accepteds, evidence, observations, data, facts, presumptions, and the like, and some conclusion or proposition which is thought related to the list. Related how and in what way is a discussion that comes later, but for now it loosely is associated with what {\it causes} the proposition to be true. Arguments can be well or badly structured, formally valid or invalid, and sound or unsound. Unlike most logical, mathematical, and moral arguments, which often end in truth, probabilistic arguments do not lead to certainty. Whenever a probabilistic argument is used, it is an attempt to convince someone how certain a proposition is in relation to a given body of evidence, and {\it only that} body of evidence.

Anybody who engages in any argument thus accepts that certainty and truth exist. We should have no patience for philosophical skepticism, which is always self-defeating. If you are certain there is no certainty, you are certain. If it is true that there is no truth, it is false there is no truth. If you are certain that “Every proposition is subject to uncertainty” then you speak with forked tongue. Certainty and truth therefore exist. But we must understand that truth resides in our intellects and not in objects themselves, except in the sense of existence. That being so, probability also does not exist physically; it also resides in our intellects and not in things themselves.

All arguments have stated and tacit premises, with those tacit usually about the meaning of the words and grammar used to state the argument, but also about how arguments themselves are to be interpreted, about how we move from premise to conclusion. Confusion usually enters when there are misunderstandings or disagreements formed about the tacit premises. Badly structured arguments are incautious in their use of tacit premises, containing too many or those which are prone to dispute. Ajdukiewicz confirms this in his lost classic Pragmatic Logic, an excellent book for students to understand the nature of arguments. There is also a burgeoning field called argumentation theory which can be looked up.

Realism

No definition of truth is better or more succinct than Aristotle’s: “To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true.” St Thomas Aquinas, following Aristotle’s Metaphysics, in his Summa Theologica (First Part, Q. 16, articles 1 and 8) said “the true denotes that towards which the intellect tends.” “Truth, properly speaking, resides only in the intellect, as said before ; but things are called true in virtue of the truth residing in an intellect.”

This view encapsulate what is called correspondence and reflect the metaphysics of (moderate) realism; see below in the Chapter on Causality. When we later say of a proposition “It is necessarily true”, this is never meant to imply that the proposition is true in or because of some theory. The proposition is necessarily true for reasons in the proposition itself and the evidence which supports it; the proposition is not true “in” or because of a theory. It is true because it is true.

Moderate realism is the common-sense position that there exist real things, that there is an existence independent of our minds, that an external world is “out there” and that we can know it, that we can “know things as they are in themselves”, to coin a phrase. Moderate realism holds that greenness exists apart from or in addition to individual green things; exists as an intellectual idea, that is. Realism says the idea of color exists independent of individual colored things. Mathematicians are realists when they insist all triangles have three straight sides and an interior sum of angles of $180^o$. Individual approximations to or implementations of triangles also exist, but given the way the world is, all are imperfect representations of the universal ideal. Try drawing one. Catness exists and so do individual cats. We can tell cats from dogs because we know the nature or essence of both. Knifeness exists as do individual knives, even though it’s not always clear if a given object is a knife or only acts like one.

These natures (or ideas) are universals. They don’t exist as physical objects in some ethereal realm, à la Plato; instead they exist in the objects which instantiate them—redness exists in red apples, knifeness exists in cleavers—or they exists as idea in intellects, as immaterial concepts. This is scholastic realism, a modified form of Aristotle’s. Excellent introductions to moderate realism are given by Feser.

Contrary to realism is nominalism, which denies universals exist. Under this view, individual triangles exist but there is no concept of an ideal, perfect triangle. This appears to leave out mathematical definitions and, it would seem to follow, all of mathematics, since this field is founded on universal truths (see below; also see Franklin’s Aristotelian conception of mathematics. Under nominalism, two drawings of triangles are not two drawings of triangles, just two drawings which might have vague similarities, the similarities bespeaking of no central thing in common. How, then, if nominalism is true, could we even have the word triangle or even similarity? Man is also therefore a meaningless term: there are individual bipedal creatures which might coincidentally look somewhat alike and share some DNA (but is all DNA actually DNA?), just as they are more dissimilar to quadrupedal creatures. The higher concept of man or human being holds no higher meaning. Things do not instantiate natures. Things just {are, never mind how. Most working scientists are not nominalists, for obvious reasons.

Nominalism comes in various forms and subtleties, but no branch holds any interest for probability and statistics. If there were no universals, there would be little point in conducting experiments or grouping data, which admits of universals or essences. The acts of grouping and collating say, do they not?, “All these data represent the same underlying essential thing.” Even those dismal objects p-values admit of universal “null” and “alternate” hypotheses; these surely bespeak of universal essences and do not point to physical substances (p-values, God rot them, are discussed later). And neither is probability, as de Finetti taught us in a loud voice, a tangible physical quantity, something that can be measured with a physical apparatus. Probability, like logic, as we’ll see, assumes universals.

The opposite of nominalism (if such a thing could have an opposite) is idealism, the concept that reality does not exist, rather that individual physical objects do not exist, but that only universals do. Our thoughts are capital-I It, our thoughts are everything, our thoughts define existence. If so, how do we know when you and I are thinking of the same thing? We cannot. I don’t consider idealism to be on any interest. The best overview and refutation of idealism is found in David Stove’s essay “Idealism: A Victorian horror story”.

There are many other ways for thought to go wrong, and those which have a bearing on probability will be outlined later. For now, I’ll boldly state all scientists are realists, or ought to be. There’s no use for a scientist who subscribes to some form of idealism. After all, if the universe is only in his mind, there’s no guarantee that the universe which is my mind is in any way the same thing as the universe in his. If idealism is true, why not make up how the universe is? Saves research time. If nominalism is true, what is true here might not be true there, and it is of little to no use to speak of “laws” or causes.

Epistemology

Can we know any truths? Yes. And if you disagree you necessarily agree. In disagreeing you’d at least know that you can’t know anything, which would be a truth, and then you’d realize you bit yourself in the tail. Any attempt to deny there are truths is self-contradictory. Roger Scruton said that the people who tout theories which deny truth or our knowledge of it are inviting us to disbelieve them, an invitation which we eagerly accept.

That there are truths and we can know them is traditionally called rationalism. A prime example of a known truth is Aristotle’s principal of non-contradiction. The epistemic version states that a proposition cannot be both true and false simultaneously (given the same evidence). It is impossible, and not just unlikely, for somebody to doubt this principle. It is possible, and unfortunately not uncommon, for some to claim to doubt it. But claiming and doing are not identical as everybody knows, and that is why we have the words like deception, mistaken, and lying—words, incidentally, which admit the existence of truth and knowledge. Claiming to doubt the principle of non-contradiction is like the man who boasts of disbelieving the reality of gravity. No matter the degree of his earnestness or the number of his scholarly credentials, if he takes a long walk off a short dock he is going to end up wet.

A ontological version of the non-contradiction principle is that something cannot be and not-be at the same time, that something cannot exist and not-exist simultaneously. Existence is an ontological truth. You cannot exist and not-exist at the same time; further, it is impossible, and not just unlikely, to believe that you exist and that at the same time don’t exist. This is not the same as saying, for example with respect to certain very small objects in physics, that you do not know if or where a thing exists or not. A thing’s existence and our knowledge of it are different. Indeed, the mixing up of epistemological and ontological claims is a routine problem in probability.

Everyone, regardless of what they might claim, knows that an external world exists. And all scientists ought to admit it, else they’re in the wrong business. This is another way to state realism. Anybody asking the question of another, “Does an external world exist?” has answered it affirmatively, since to ask it requires a person to ask and another to answer it, hence an external world in which the other person exists to answer it, hence we can know it exists, hence we know there are other people, too (the traditional way to phrase it is that we know there are “other minds”).

Another truth known to everybody is that solipsism is impossible. Again, if you disagree with me, you agree with me and acknowledge the complete fallaciousness of your position because, of course, to disagree with me implies someone other than yourself exists, hence solipsism is false.

But what if I were an illusion? What if, that is, you were hallucinating my obstreperousness? From David Stove’s masterful essay “I only am alone escaped to tell thee: Epistemology and the Ishmael Effect” (pp. 61–82):

[I]t is true, and also contingent, that some of us sometimes hallucinate. But it does not follow from that, (even if Descartes thought it did), that it is logically possible that all of us are always hallucinating. Some children in a school-class may happen to be below the average level of ability of children in that class, but it not logically possible that all of them are. Neither is it logically possible that we are all always hallucinating. For we—that is, all human beings—are perceived by (unless indeed we are hallucinations of) at least one human being: ourselves if no other. Whence, on the supposition that we—that is, all human beings—are always hallucinating, it follows that all human beings are hallucinations of at least one human being. And that is not logically possible.

Empiricism insists on the observational verifiability of all propositions, in contrast with realism, which does not. But not all propositions can be verified; think, for example, of truths reached by induction or mathematical deduction, especially statements about various infinite sets and so forth. The realism-rationalism view says all knowledge begins in sense impressions, and then moves from those particulars to grasp universals, which are entities which cannot be checked or verified empirically. Since most of our reasoning involves these universals, we’d be in a world of intellectual hurt if empiricism were true. Only the experiments we have seen are those we can say are true. We could never extrapolate from them. Those results which are merely similar or the same at other times and places might, under strict empiricism, be different. Mathematical axioms cannot be seen, touched, tasted, heard, or smelt, yet we insist on their truth. That logic in particular cannot be wholly empirical is dealt with in the next chapter, a useful exercise because probability follows directly from logic.

Necessary & Conditional Truth

Given “x,y,z are natural numbers and x>y and y>z the proposition x>z is true (I am assuming logical knowledge here, which I don’t discuss until Chapter 2). But it would be false in general to claim, “It is true that ‘x>z‘.” After all, it might be that “x = 17 and z = 32″; if so, “x>z” is false. Or it might be that “x = 17 and z = 17″, then again “x>z” is false. Or maybe “x =$ a boatload and z = a humongous amount”, then “x>z” is undefined or unknown unless there is tacit and complete knowledge of precisely how much is a boatload and how much is a humongous amount (which is doubtful). We cannot dismiss this last example, because a great portion of human discussions of uncertainty are pitched in this way.

Included in the premise “x,y,z are natural numbers and x>y and y>z” are not just the raw information of the proposition about numbers, but the tacit knowledge we have of the symbol >, of what “natural numbers” are, and even what “and” and “are” mean. This is so for any argument which we wish to make. Language, in whatever form, must be used. There must therefore be an understanding of and about definitions, language and grammar, in any argument if any progress is to be made. These understandings may be more or less obvious depending on the argument. It is well to point out that many fallacies (and the best jokes) are founded on equivocation, which is the intentional or not misunderstanding double- or multiple-meanings of words or phrases. This must be kept in mind because we often talk about how the mathematical symbols of our formulae translate to real objects, how they matter to real-life decisions. A caution not heard frequently enough: just because a statement is mathematically true does not mean that the statement has any bearing on reality. Later we talk about how the deadly sin of reification occurs when this warning is ignored.

We have an idea what it means to say of a proposition that it is true or false. This needs to be firmed up considerably. Take the proposition “a proposition cannot be both true and false simultaneously”. This proposition, as I said above, is true. That means, to our state of mind, there exists evidence which allows us to conclude this proposition is true. This evidence is in the form of thought, which is to say, other propositions, all of which include our understanding of the words and English grammar, and of phrases like “we cannot believe its contrary.” There are also present tacit (not formal) rules of logic about how we must treat and manipulate propositions. Each of these conditioning propositions or premises can in turn be true or false (i.e. known to be true or false) conditional on still other propositions, or on inductions drawn upon sense impressions and intellections. That is, we eventually must reach a point at which a proposition in front of us just is true. There is no other evidence for this kind of truth other than intellection. Observations and sense impressions will give partial support to most propositions, but they are never enough by themselves except for the direct impressions. I explore this later in the Chapter on Induction.

In mathematics, logic, and philosophy popular kinds of propositions which are known to be true because induction tells us so are called axioms. Axioms are indubitable—when considered. Arguments for an axiom’s truth are made like this: given these specific instances, thus this general principle or axiom. I do not claim, and it is not true, that everybody knows every axiom. The arguments for axioms must first be considered before they are believed. A good example is the principal of non-contradiction, a proposition which we cannot know is false (though, given we are human, we can always claim it is false). As said, for every argument we need an understanding of its words and grammar, and, for non-contradiction specifically, maybe the plain observation of a necessarily finite number of instance of propositions that are only true or only false, observations which are consonant with the axiom, but which are none of them the full proof of the proposition: there comes a point at which we just believe and, indeed, cannot do other than know the truth. Another example is one of Peano’s axioms. For every natural number, if x = y then y = x. We check this through specific examples, and then move via induction to the knowledge that it is true for every number, even those we have not and, given our finiteness, cannot consider. Axioms are known to be true based on the evidence and faith that our intellects are correctly guiding us.

This leads to the concept of the truly true, really true, just-plain true, universally, absolutely, or the necessarily true. These are propositions, like those in mathematics, that are known to be true given a valid and sound chain of argument which leads back to indubitable axioms. It is not possible to doubt axioms or necessary truths, unless there be a misunderstanding of the words or terms or chain of proof or argument involved (and this is, of course, possible, as any teacher will affirm). Necessary truths are true even if you don’t want them to be, even if they provoke discomfort, which (again of course) they sometimes do. Peter Kreeft said: “As Aristotle showed, [all] ‘backward doubt’ terminates in two places: psychologically indubitable immediate sense experience and logically indubitable first principles such as ‘X is not non-X’ in theoretical thinking and `Good is to be done and evil to be avoided’ in practical thinking,” Part VI.

A man in the street might look at the scratchings of a mathematical truth and doubt the theorem, but this is only because he doesn’t comprehend what all those strange symbols mean. He may even say that he “knows” the theorem is false—think of the brave soul who claims to have squared the circle. It must be stressed that this man’s error arises from his not comprehending the whole of the argument. Which of the premises of the theorem he is rejecting, and this includes tacit premises of logic and other mathematical results, is not known to us (unless the man makes this clear). The point is that if it were made plain to him what every step in the argument was, he must consent. If he does not, he has not comprehended at least one thing or he has rejected at least one premise, or perhaps substituted his own unaware. This is no small point, and the failure to appreciate it has given rise to the mistaken subjective theory of probability. Understanding the whole of an argument is a requirement to our admitting a necessary truth (our understanding is obviously not required of the necessary truth itself!).

From this it follows that when a mathematician or physicist says something akin to, “We now know Flippenberger’s theorem is true”, his “we” does not, it most certainly does not, encompass all of humanity; it applies only to those who can and have followed the line of reason which appears in the proof. That another mathematician or physicist (or man in the street) who hears this statement, but whose specialty is not Flippenbergerology, conditional on trusting the first mathematician’s word, also believes Flippenberger’s theorem is true, is not making (to himself) the same argument as the theory’s proponent. He instead makes a conditional truth statement: to him, Flippenberger’s theorem is conditionally true, given the premise of accepting the word of the first mathematician or physicist. Of course, necessary truths are also conditional as I have just described, so the phrase “conditional truth” is imperfect, but I have not been able to discover one better to my satisfaction. Local or relative truth have their merits, but their use could encourage relativists to believe they have a point, which they do not.

Besides mathematical propositions, there are plenty other of necessary truths that we know. “I exist” is popular, and only claimed to be doubted by the insane or (paradoxically) by attention seekers. “God exists” is another: those who doubt it are like circle-squarers who have misunderstood or have not (yet) comprehended the arguments which lead to this proposition. “There are true propositions” always delights and which also has its doubters who claim it is true that it is false. In Chapter 2 we meet more.

There are an infinite number and an enormous variety of conditional truths that we do and can know. I don’t mean to say that there are not an infinite number of necessary truths, because I have no idea, though I believe it; I mean only that conditional truths form a vaster class of truths in everyday and scientific discourse. We met one conditional truth above in “x>z“. Another is, given “All Martians wear hats and George is a Martian” then it is conditionally true that “George wears a hat.” The difference in how we express this “truth is conditional” is plain enough in cases like hat-wearing Martians. Nobody would say, in a general setting, “It’s true that Martians wear hats.” Or if he did, nobody would believe him. This disbelief would be deduced conditional on the observationally true proposition, “There are no Martians”.

We sometimes hear people claim conditional truths are necessary truths, especially in moral or political contexts. A man might say, “College professors are intolerant of dissent” and believe he is stating a necessary truth. Yet this cannot be a necessary truth, because no sound valid chain of argument anchored to axioms can support it. But it may be an extrapolation from “All the many college professors I have observed have been intolerant of dissent”, in which case the proposition is still not a necessary truth, because (as we’ll see) observational statements like this are fallible. Hint: The man’s audience, if it be typical, might not believe the “All” in the argument means all, but only “many”. But that substitution does not make the proposition “Many college professors are intolerant of dissent” necessarily true, either.

Another interesting possibility is in the proposition “Some college professors are intolerant of dissent,” where some is defined as at least one and potentially all (I keep this definition throughout the book unless otherwise specified). Now if a man hears that and recalls, “I have met X, who is a college professor, and she was intolerant of dissent”, then conditional on that evidence the proposition of interest is conditionally true. Why isn’t it necessarily true? Understand first that the proposition is true for you, too, dear reader, if we take as evidence “I have met X, etc.” Just as “George wears a hat” was conditionally true on the other explicit evidence. It may be that you yourself have not met X, nor any other intolerant-of-dissent professor, but that means nothing for the epistemological status of these two propositions. But it now becomes obvious why the proposition of interest is not necessarily true: because the supporting evidence “I have met X, etc.” cannot be held up as necessarily true itself: there is no chain of sound argument leading to indubitable axioms which guarantees it is a logically necessity that college professors must be intolerant of dissent. (Even if it sometimes seems that way.)

We only have to be careful because when people speak or write of truths they are usually not careful to tell us whether they have in mind a necessary or only a conditional truth. Much grief is caused because of this.

One point which may not be obvious. A necessary truth is just true. It is not true be because we have a proof of it’s truth. Any necessary truth is true be because of something, but it makes no sense to ask why this is so for any necessary truth. Why is the principle of non-contradiction true? What is it that makes it true? Answer: we do not know. It is just is true. How do we know it is true? Via a proof, by strings of deductions from accepted premises and using induction, the same way we know if any proposition is true. We must ever keep separate the epistemological from the ontological. There is a constant danger of mistaking the two. Logic and probability are epistemological, and only sometimes speak or aim at the ontological. Probability is always a state of the mind and not a state of the universe.

Science & Scientism

The example of the intolerant college professor is like most propositions in science. Examples, “Radium has the atomic weight of w“, “The speed of light is c“, “The earth is warmed by the sun’s rays”, “Creatures evolve by natural selection”, and on and on. These statements are all contingent, meaning there is (so far) no known route to proving their necessary truth (though that is the goal in physics). They are all conditionally true, given various facts and evidence. In any of these propositions none of the conditioned-on facts or evidence meets the test of a sound chain of valid argument leading to indubitable axioms. In other words, none of these propositions are logically necessary. It is a logical possibility that any of them might be (necessarily or observationally) false. That radium does not have the atomic weight of w might be false if the equipment, no matter how sophisticated or fine, erred in its measurements. That the speed of light is some number might also be false for the same or some other reason. Many physical formulas (and this is obviously theory dependent) rely on “constants”, such as the speed of light in a vacuum or Planck’s constant, which are productions of the result of measurements. They are not themselves deduced from earlier truths; i.e. there is nothing which we know of that states Planck’s constant must of logically necessity take the precise value it does (though, as I said, this is the goal of physics). The same is true for all statistical use of parameters. This lack spoken of is what makes these creatures parameters, about which much more later. That means any theory which relies on contingent premises might be false. It might be incredibly improbable, given the evidence we have, for our best scientific theories to turn out false, but we cannot claim any are necessarily true.

Scientific statements are therefore contingent statements which can only ever be conditionally true and not necessarily true (the math used in science is an exception, of course). All scientific propositions are therefore subject to doubt. Not always reasonable doubt. Here is a scientific proposition, “If I walk off the edge of the twenty story building I will fall.” There is no chain of argument which proves this is universally true, therefore the proposition is contingent. It is not logically necessary that falling must occur. But I will not be walking off the edge of any twenty-story buildings. I’m also happy with the atomic weight of Radium, even though I’ve taken none of the pertinent measurements myself. The premise of trust is ever present, though as the business of science expands, this premise is weakened.

All science is an attempt to remove as much of the contingency as possible from the supporting evidence for propositions (theories) of interest. The ultimate Theory of Everything would be one which is necessarily true, which begins at indisputable axioms and progressed toward a complete explanation for how everything works, including complete deduced explanations of why the speed of light and Planck’s constant (if they should turn out to remain finally important) take the values they do. Those who have read in physics know how distant, and perhaps even unattainable, this goal is.

People before Newton knew apples fell, and would say so. The reasons they gave for this produced conditional truths—“Apples fall because they have an affinity for the ground”, maybe—which allowed for good predictions: sure enough, apple always fell. Nobody not delusional walked off a mountain cliff in anno Domini 1600 in expectation of not falling because they didn’t understand Newton’s theory of gravitation. Newton’s great trick was to replace highly contingent and more-or-less dubious premises with better evidence which had less contingency. He never remove the contingency completely, of course. But then neither did Einstein when he refined Newton’s premises further. And still nobody has supplied a universally true argument which shows the logical necessity of gravity behaving the way it does. Scientists labor still to remove the remaining contingencies (and there are plenty). Whether they can eliminate them entirely and arrive at scientific statements with all the rigor of mathematical proofs is not known. There is plenty of reason to doubt it, however; but that discussion would take us too far afield. Suffice to say that no known scientific theory is necessarily true. All are at best conditionally true, many are only probably true, and still others are probably or certainly false (examples of these will follow).

Many scientists, perhaps heeding too closely to their citizen cheering section, have the bad habit of insisting that their conditional truths are necessary truths. Some have the even worse habit of insisting probable truths are not only not conditionally but are universally true. Bad habits lead to iniquity, which in this case is the sin of scientism. This is the false belief that the only truths we have are scientific truths. Since scientific truths are only conditional at best, and likely only probable and sometimes false in fact—a truth captured in the slogan “science is self-correcting”, which implies it errs—it is not possible that it is a necessary truth that conditional or probable truths are necessary truths. Tongue twisting? It is not from science we learn “I exist”. Though, if it can be credited, some scientists would say that consciousness of our existence is an “illusion”, an obviously self-contradictory proposition. Who is having the illusion? But that’s not a problem for us to solve in a book on probability. Science is also mute on all mathematical (necessary) truths, which is amusing because scidolators (those who inveterately practice scientism) often wield mathematical truths to show how scientific they are.

Jacques Barzun said this about scientism: “Scientism is the fallacy of believing that the method of science must be used on all forms of experience and, given time, will settle every issue.” And Pascal in his Pensees had this to say, an observation which could be the motto of this book:

The world is a good judge of things, for it is in natural ignorance, which is man’s true state. The sciences have two extremes which meet. The first is the pure natural ignorance in which all men find themselves at birth. The other extreme is that reached by great intellects, who, having run through all that men can know, find they know nothing, and come back again to that same ignorance from which they set out; but this is a learned ignorance which is conscious of itself. Those between the two, who have departed from natural ignorance and not been able to reach the other, have some smattering of this vain knowledge, and pretend to be wise. These trouble the world, and are bad judges of everything. The people and the wise constitute the world; these despise it, and are despised. They judge badly of everything, and the world judges rightly of them.

The increasing politicization of science is also distressing. This is found whenever is heard somebody (almost never himself a scientist) screeching (this is never spoken politely) about some contingent proposition, “The debate is over!”, as if their frenzy or level of ardency removed the obvious contingencies from the proposition in dispute. This tactic is always an obvious fallacy (unless it is applied to a demonstrable necessary truth). But this subject is too depressing to continue, so let it pass. We later meet many examples of scientism.

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