Modeling Revisited

Posted on July 5, 2017 by Loquitur Veritatem

Arnold Kling comments on a post by John Taylor, who writes about the Macroeconomic Modelling and Model Comparison Network (MMCN), which

is one part of a larger project called the Macroeconomic Model Comparison Initiative (MMCI)…. That initiative includes the Macroeconomic Model Data Base, which already has 82 models that have been developed by researchers at central banks, international institutions, and universities. Key activities of the initiative are comparing solution methods for speed and accuracy, performing robustness studies of policy evaluations, and providing more powerful and user-friendly tools for modelers.

Kling says: “Why limit the comparison to models? Why not compare models with verbal reasoning?” I say: a pox on economic models, whether they are mathematical or verbal.

That said, I do harbor special disdain for mathematical models, including statistical estimates of such models. Reality is nuanced. Verbal descriptions of reality, being more nuanced than mathematics, can more closely represent reality than can be done with mathematics.

Mathematical modelers are quick to point out that a mathematical model can express complex relationships which are difficult to express in words. True, but the words must always precede the mathematics. Long usage may enable a person to grasp the meaning of 2 + 2 = 4 without consciously putting it into words, but only because he already done so and committed the formula to memory.

Do you remember word problems? As I remember them, the words came first:

John is twenty years younger than Amy, and in five years’ time he will be half her age. What is John’s age now?

Then came the math:

Solve for J [John’s age]:

J = A − 20

J + 5 = (A + 5) / 2

[where A = Amy’s age]

What would be the point of presenting the math, then asking for the words?

Mathematics is a man-made tool. It probably started with counting. Sheep? Goats? Bananas? It doesn’t matter what it was. What matters is that the actual thing, which had a spoken name, came before the numbering convention that enabled people to refer to three sheep without having to draw or produce three actual sheep.

But … when it came to bartering sheep for loaves of bread, or whatever, those wily ancestors of ours knew that sheep come in many sizes, ages, fecundity, and states of health, and in two sexes. (Though I suppose that the LGBTQ movement has by now “discovered” homosexual and transgender sheep, and transsexual sheep may be in the offing.) Anyway, there are so many possible combinations of sizes, ages, fecundity, and states of health that it was (and is) impractical to reduce them to numbers. A quick, verbal approximation would have to do in the absence of the real thing. And the real thing would have to be produced before Grog and Grok actually exchanged X sheep for Y loaves of bread, unless they absolutely trusted each other’s honesty and descriptive ability.

Things are somewhat different in this age of mass production and commodification. But even if it’s possible to add sheep that have been bred for near-uniformity or nearly identical loaves of bread or Paper Mate Mirado Woodcase Pencils, HB 2, Yellow Barrel, it’s not possible to add those pencils to the the sheep and the loaves of bread. The best that one could do is to list the components of such a conglomeration by name and number, with the caveat that there’s a lot of variability in the sheep, goats, banana, and bread.

An economist would say that it is possible to add a collection of disparate things: Just take the sales price of each one, multiply it by the quantity sold, and if you do that for every product and service produced in the U.S. during a year you have an estimate of GDP. (I’m being a bit loose with the definition of GDP, but it’s good enough for the point I wish to make.) Further, some economists will tout this or that model which estimates changes in the value of GDP as a function of such things as interest rates, the rate of government spending, and estimates of projected consumer spending.

I don’t disagree that GDP can be computed or that economic models can be concocted. But it is to say that such computations and models, aside from being notoriously inaccurate (even though they deal in dollars, not in quantities of various products and services), are essentially meaningless. Aside from the errors that are inevitable in the use of sampling to estimate the dollar value of billions of transactions, there is the essential meaninglessness of the dollar value. Every transaction represented in an estimate of GDP (or any lesser aggregation) has a different real value to each participant in the transaction. Further, those real values, even if they could be measured and expressed in “utils“, can’t be summed because “utils” are incommensurate — there is no such thing as a social-welfare function.

Quantitative aggregations are not only meaningless, but their existence simply encourages destructive government interference in economic affairs. Mathematical modeling of “aggregate economic activity” (there is no such thing) may serve as an amusing and even lucrative pastime, but it does nothing to advance the lives and fortunes of the vast majority of Americans. In fact, it serves to retard their lives and fortunes.

All of that because pointy-headed academics, power-lusting politicians, and bamboozled bureaucrats believe that economic aggregates and quantitative economic models are meaningful. If they spent more than a few minutes thinking about what those models are supposed to represent — and don’t and can’t represent — they would at least use them with a slight pang of conscience. (I hold little hope that they would abandon them. The allure of power and the urge to “do something” are just too strong.)

Economic aggregates and models gain meaning and precision only as their compass shrinks to discrete markets for closely similar products and services. But even in the quantification of such markets there will always be some kind of misrepresentation by aggregation, if only because tastes, preferences, materials, processes, and relative prices change constantly. Only a fool believes that a quantitative economic model (of any kind) is more than a rough approximation of past reality — an approximation that will fade quickly as time marches on.

Economist Tony Lawson puts it this way:

Given the modern emphasis on mathematical modelling it is important to determine the conditions in which such tools are appropriate or useful. In other words we need to uncover the ontological presuppositions involved in the insistence that mathematical methods of a certain sort be everywhere employed. The first thing to note is that all these mathematical methods that economists use presuppose event regularities or correlations. This makes modern economics a form of deductivism. A closed system in this context just means any situation in which an event regularity occurs. Deductivism is a form of explanation that requires event regularities. Now event regularities can just be assumed to hold, even if they cannot be theorised, and some econometricians do just that and dedicate their time to trying to uncover them. But most economists want to theorise in economic terms as well. But clearly they must do so in terms that guarantee event regularity results. The way to do this is to formulate theories in terms of isolated atoms. By an atom I just mean a factor that has the same independent effect whatever the context. Typically human individuals are portrayed as the atoms in question, though there is nothing essential about this. Notice too that most debates about the nature of rationality are beside the point. Mainstream modellers just need to fix the actions of the individual of their analyses to render them atomistic, i.e., to fix their responses to given conditions. It is this implausible fixing of actions that tends to be expressed though, or is the task of, any rationality axiom. But in truth any old specification will do, including fixed rule or algorithm following as in, say, agent based modelling; the precise assumption used to achieve this matters little. Once some such axiom or assumption-fixing behaviour is made economists can predict/deduce what the factor in question will do if stimulated. Finally the specification in this way of what any such atom does in given conditions allows the prediction activities of economists ONLY if nothing is allowed to counteract the actions of the atoms of analysis. Hence these atoms must additionally be assumed to act in isolation. It is easy to show that this ontology of closed systems of isolated atoms characterises all of the substantive theorising of mainstream economists.

It is also easy enough to show that the real world, the social reality in which we actually live, is of a nature that is anything but a set of closed systems of isolated atoms (see Lawson, [Economics and Reality, London and New York: Routledge] 1997, [Reorienting Economics, London and New York: Routledge] 2003).

Mathematical-statistical descriptions of economic phenomena are either faithful (if selective) depictions of one-off events (which are unlikely to recur) or highly stylized renditions of complex chains of events (which almost certainly won’t recur). As Arnold Kling says in his review of Richard Bookstaber’s The End of Theory,

people are assumed to know, now and for the indefinite future, the entire range of possibilities, and the likelihood of each. The alternative assumption, that the future has aspects that are not foreseeable today, goes by the name of “radical uncertainty.” But we might just call it the human condition. Bookstaber writes that radical uncertainty “leads the world to go in directions we had never imagined…. The world could be changing right now in ways that will blindside you down the road.”

I’m picking on economics because it’s an easy target. But the “hard sciences” have their problems, too. See, for example, my work in progress about Einstein’s special theory of relativity.

Related reading:

John Cochrane, “Mallaby, the Fed, and Technocratic Illusions“, The Grumpy Economist, July 5, 2017

Vincent Randall: “The Uncertainty Monster: Lessons from Non-Orthodox Economics“, Climate Etc., July 5, 2017

Spooky Numbers, Evolution, and Intelligent Design

Posted on August 13, 2013 by Loquitur Veritatem

“Spooky numbers” refers to Steven Landsburg’s position — expressed here in commenting on a post by Bob Murphy about intelligent design — that natural numbers just are. This encapsulates Landsburg’s thesis:

The natural numbers are irreducibly complex, moreso (by any reasonable definition) than anything in biology. But the natural numbers were not designed and did not evolve….

I previously addressed Landsburg’s claim about natural numbers, here; for example:

Why have humans, widely separated in time and space, agreed about numbers and the manipulation of numbers (mathematics)? Specifically, with respect to the natural numbers, why is there agreement that something called “one” or “un” or “ein” (and so on) is followed by something called “two” or “deux” or “zwei,” and so on? And why is there agreement that those numbers, when added, equal something called “three” or “trois” or “drei,” and so on? Is that evidence for the transcendent timelessness of numbers and mathematics, or is it nothing more than descriptive necessity?By descriptive necessity, I mean that numbering things is just another way of describing them. If there are some oranges on a table, I can say many things about them; for example, they are spheroids, they are orange-colored, they contain juice and (usually) seeds, and their skins are bitter-tasting.

Another thing that I can say about the oranges is that there are a certain number of them — let us say three, in this case. But I can say that only because, by convention, I can count them: one, two, three. And if someone adds an orange to the aggregation, I can count again: one, two, three, four. And, by convention, I can avoid counting a second time by simply adding one (the additional orange) to three (the number originally on the table). Arithmetic is simply a kind of counting, and other mathematical manipulations are, in one way or another, extensions of arithmetic. And they all have their roots in numbering and the manipulation of numbers, which are descriptive processes.

But my ability to count oranges and perform mathematical operations based on counting does not mean that numbers and mathematics are timeless and transcendent. It simply means that I have used some conventions — devised and perfected by other humans over the eons — which enable me to describe certain facets of physical reality.

Mathematics is merely a tool that can be useful in describing some aspects of the real world. Evolution and intelligent design, on the other hand, are theories about the real world. Though evolution and intelligent design are not complete theories of the real world, they are far more than mere mathematical descriptions of it.

To understand the distinction that I’m making, consider this: Some of the differences between apples and oranges can be described by resorting to the mathematics of color, taste, shape, and so on. But an apple or an orange — as an entity — is more than the sum of its various, partial descriptors. So, too, is the real world more than the sum of any number of mathematics or descriptors (physics, chemistry, biology, etc.) that have mathematical components. The real world encompasses love, hate, social customs, and religion — among many things that defy complete (or even partial) mathematical description.

Now, what about evolution and intelligent design? Are they reconcilable theories? Murphy implies that they are. He says that

Michael Behe–[a leading proponent of intelligent design] who (in)famously said that the bacterial flagellum exhibited too much design to have arisen through unguided evolution in the modern neo-Darwinian sense–does not have a problem with the idea that all of today’s cells share a common ancestor….

So yes, Behe is fine with the proposition that if we had a camera and a time machine, we could go observe the first cell on earth as it reproduced and yielded offspring. There would be nothing magical in these operations; they would obey the laws of physics, chemistry, and biology. The cells would further divide and so on, and then over billions of years there would be mutations and the environment would favor some of the mutants over their kin, such that natural selection over time would yield the bacterial flagellum and the human nervous system.

Yet Behe’s point is that when you look at what this process spits out at the end, you can’t deny that a guiding intelligence must be involved somehow.

The question-begging of that last sentence is what frustrates scientists. It says, in effect, that there must be a guiding intelligence, and the complexity of the products of evolution proves it.

No, it doesn’t prove it. God — as an entity apart from the material universe — cannot be shown to exist by pointing to particular aspects of the material universe, be they evolution or the Big Bang (to offer but two examples). God is a logical necessity, beyond empirical proof or disproof.

I greatly respect the sincerity of theists and the credence they give to sacred texts and accounts of visions and miracles. Their credence may be well-placed. But I am just too much of a doubting Thomas to rely on unfalsifiable, second-hand evidence about the nature of God and His role in the workings of the universe.

I will say this: Given the logical necessity of God, it follows that the universe operates in accordance with the “laws” that are inherent in His creation. Intelligent design, as an explanation for the forms taken by living creatures, is therefore something of a truism. But intelligent design cannot be proved by reference to products of evolution.

Mysteries: Sacred and Profane

Posted on April 19, 2012 by Loquitur Veritatem

A philosopher named Jamie Whyte, about whom I have written before (“Invoking Hitler“), is the author of Bad Thoughts – A Guide to Clear Thinking. According to the publisher, it is a

book for people who like argument. Witty, contentious, and passionate, it exposes the methods with which we avoid reasoned debate…. His writing is both laugh-out-loud funny and a serious comment on the ways in which people with power and influence avoid truth in steering public opinion.

Bad Thoughts is witty — though “laugh-out-loud funny” is a stretch — and, for the most part, correct in its criticisms of the kinds of sloppy logic that are found routinely in politics, journalism, blogdom, and everyday conversation.

But Whyte is not infallible, as I point out in “Invoking Hitler.” This post focuses on another of Whyte’s miscues, which is found under “Mystery” (pp. 23-26). Here are some relevant samples:

…Consider … the orthodox Christian doctrine of the Unity of the Holy Trinity. The Father, the Son and the Holy Ghost are three distinct entities — as suggested by ‘Trinity’. Yet each is God, a sinle entity — as suggested by ‘Unity’. The doctrine is not that each is part of God, in the way that the FM tuner is part of your three-in-one home stereo. Each is wholly God.

And there’s the problem. It takes only the most basic arithmetic to see that three things cannot be one thing. The doctrine of the Unity of the Trinity is inconsistent with the fact that three does not equal one.

Whyte goes on and on, but the quoted material is the essence of his “case” that the Blessed Trinity (Catholic usage) is impossible because it defies mathematical logic. What is worse, to Whyte, is the fact that this bit of illogic is “explained away” (as he would put it) by calling it a “mystery.”

I am surprised that a philosopher cannot accept the idea of “mystery.” Anyone who thinks for more than a few minutes about the nature of the universe, as Whyte must have done, concludes that its essence is beyond human comprehension. And, yet, the universe exists. The universe — a real thing — is, at bottom, a mystery. Somehow, the mysteriousness of the universe does not negate its existence.

And there are scientific mysteries piled on that mysteriousness. Two of those mysteries have a common feature: They posit the simultaneous existence of one thing in more than one form — not unlike the Blessed Trinity:

Wave–particle duality postulates that all particles exhibit both wave and particle properties. A central concept of quantum mechanics, this duality addresses the inability of classical concepts like “particle” and “wave” to fully describe the behavior of quantum-scale objects.

* * *

The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction, but denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternative histories and futures are real, each representing an actual “world” (or “universe”).

As Shakespeare puts it (Hamlet, Act I, Scene V), “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” Or in your physics.

If Whyte wants to disprove the Blessed Trinity, he must first try to disprove the existence of God — a fool’s errand that I have addressed in other posts; for example:

“A Digression about Probability and Existence”
“More about Probability and Existence”
“Existence and Creation”
“Probability, Existence, and Creation”
“The Atheism of the Gaps”
“Not-So-Random Thoughts (II)” (see the first section, “Atheism,” which inter alia addresses Lawrence M. Krauss’s A Universe from Nothing, which is summarized in this article by Krauss)

Are the Natural Numbers Supernatural?

Posted on February 15, 2012 by Loquitur Veritatem

Steven Landsburg writes:

…It is not true that all complex things emerge by gradual degrees from simpler beginnings. In fact, the most complex thing I’m aware of is the system of natural numbers (0,1,2,3, and all the rest of them) together with the laws of arithmetic. That system did not emerge, by gradual degrees, from simpler beginnings….

…God is unnecessary not because complex things require simple antecedents but because they don’t. That allows the natural numbers to exist with no antecedents at all—and once they exist, all hell (or more precisely all existence) breaks loose: In The Big Questions I’ve explained why I believe the entire Universe is, in a sense, made of mathematics. (“There He Goes Again,” The Big Questions Blog, October 29, 2009)

* * *

The existence of the natural numbers explains the existence of everything else. Once you’ve got that degree of complexity, you’ve got structures within structures within structures, and one of those structures is our physical Universe. (If that sounds like gibberish, I hope it’s only because you’re not yet read The Big Questions, that you will rush out and buy a copy, and that all will then be clear.) (“Rock On,” The Big Questions Blog, February 8, 2012)

With regard to the first quotation, I said (on October 29, 2009) that

Landsburg’s assertion about natural numbers (and the laws of arithmetic) is true only if numbers exist independently of human thought, that is, if they are ideal Platonic forms. But where do ideal Platonic forms come from? And if some complex things don’t require antecedents, how does that rule out the existence of God … ?

I admit to having said that without the benefit of reading The Big Questions. I do not plan to buy or borrow the book because I doubt its soundness, given Landsburg’s penchant for wrongheadedness. But (as of today) a relevant portion of the book is available for viewing at Amazon.com. (Click here and scroll to chapter 1, “On What There Is.”) I quote from pages 4 and 6:

…I assume — at the risk of grave error — that the Universe is no mere accident. There must be some reason for it. And if it’s a compelling reason, it should explain not only why the Universe does exist, but why it must.

A good starting point, then, is to ask whether we know of anything — let alone the entire Universe — that not only does exist, but must exist. I think I know one clear answer: Numbers must exist. The laws of arithmetic must exist. Two plus two equals four in any possible universe, and two plus two would equal four even if there were no universe at all….

…Numbers exist, and they exist because they must. Admittedly, I’m being a little vague about what I mean by existence. Clearly numbers don’t exist in exactly the same sense that, say, my dining-room table exists; for one thing, my dining-room table is made of atoms, and numbers are surely not. But not everything that exists is made of atoms. I am quite sure that my hopes and dreams exist, but they’re not made of atoms. The color blue, the theory of relativity, and the idea of a unicorn exist, but none of them is made of atoms.

I am confident that mathematics exists for the same reason that I am confident my hopes and dreams exist: I experience it directly. I believe my dining-room table exists because I can feel it with my hands. I believe numbers, the laws of arithmetic, and (for that matter) the ideal triangles of Euclidean geometry exits because I can “feel” them with my thoughts.

Here is the essence of Landsburg’s case for the existence of numbers and mathematics as ideal forms:

Number are not made of atoms.

But numbers are real because Landsburg “feels” them with his thoughts.

Therefore, numbers are (supernatural) essences which transcend and precede the existence of the physical universe; they exist without God or in lieu of God.

It is unclear to me why Landsburg assumes that numbers do not exist because of God. Nor is it clear to my why his “feeling” about numbers is superior to other persons’ “feelings” about God.

In any event, Landsburg’s “logic,” though superficially plausible, is based on false premises. It is true, but irrelevant, that numbers are not made of atoms. Landsburg’s thoughts, however, are made of atoms. His thoughts are not disembodied essences but chemical excitations of certain neurons in his brain.

It is well known that thoughts do not have to represent external reality. Landsburg mentions unicorns, for example, though he inappropriately lumps them with things that do represent external reality: blue (a manifestation of light waves of a certain frequency range) and the theory of relativity (a construct based on observation of certain aspects of the physical universe). What Landsburg has shown, if he has shown anything, is that numbers and mathematics are — like unicorns — concoctions of the human mind, the workings of which are explicable physical processes.

Why have humans, widely separated in time and space, agreed about numbers and the manipulation of numbers (mathematics)? Specifically, with respect to the natural numbers, why is there agreement that something called “one” or “un” or “ein” (and so on) is followed by something called “two” or “deux” or “zwei,” and so on? And why is there agreement that those numbers, when added, equal something called “three” or “trois” or “drei,” and so on? Is that evidence for the transcendent timelessness of numbers and mathematics, or is it nothing more than descriptive necessity?

By descriptive necessity, I mean that numbering things is just another way of describing them. If there are some oranges on a table, I can say many things about them; for example, they are spheroids, they are orange-colored, they contain juice and (usually) seeds, and their skins are bitter-tasting.

Another thing that I can say about the oranges is that there are a certain number of them — let us say three, in this case. But I can say that only because, by convention, I can count them: one, two, three. And if someone adds an orange to the aggregation, I can count again: one, two, three, four. And, by convention, I can avoid counting a second time by simply adding one (the additional orange) to three (the number originally on the table). Arithmetic is simply a kind of counting, and other mathematical manipulations are, in one way or another, extensions of arithmetic. And they all have their roots in numbering and the manipulation of numbers, which are descriptive processes.

But my ability to count oranges and perform mathematical operations based on counting does not mean that numbers and mathematics are timeless and transcendent. It simply means that I have used some conventions — devised and perfected by other humans over the eons — which enable me to describe certain facets of physical reality. Numbers and mathematics are no more mysterious than other ways of describing things and manipulating information about them. But the information — color, hardness, temperature, number, etc. — simply arises from the nature of the things being described.

Numbers and mathematics — in the hands of persons who are skilled at working with them — can be used to “describe” things that have no known physical counterparts. But that does not privilege numbers and mathematics any more than it does unicorns or God.

Probability, Existence, and Creation: A Footnote

Posted on August 29, 2011 by Loquitur Veritatem

Mario Livio writes:

[H]umans invent mathematical concepts by way of abstracting elements from the world around them—shapes, lines, sets, groups, and so forth—either for some specific purpose or simply for fun. They then go on to discover the connections among those concepts. Because this process of inventing and discovering is man-made—unlike the kind of discovery to which the Platonists subscribe—our mathematics is ultimately based on our perceptions and the mental pictures we can conjure….

[W]e adopt mathematical tools that apply to our world—a fact that has undoubtedly contributed to the perceived effectiveness of mathematics. Scientists do not choose analytical methods arbitrarily but rather on the basis of how well they predict the results of their experiments….

Not only do scientists cherry-pick solutions, they also tend to select problems that are amenable to mathematical treatment. There exists, however, a whole host of phenomena for which no accurate mathematical predictions are possible, sometimes not even in principle. In economics, for example, many variables—the detailed psychology of the masses, to name one—do not easily lend themselves to quantitative analysis. The predictive value of any theory relies on the constancy of the underlying relations among variables. Our analyses also fail to fully capture systems that develop chaos, in which the tiniest change in the initial conditions may produce entirely different end results, prohibiting any long-term predictions. Mathematicians have developed statistics and probability to deal with such shortcomings, but mathematics itself is limited, as Austrian logician Gödel famously proved….

This careful selection of problems and solutions only partially accounts for mathematics’s success in describing the laws of nature. Such laws must exist in the first place! Luckily for mathematicians and physicists alike, universal laws appear to govern our cosmos: an atom 12 billion light-years away behaves just like an atom on Earth; light in the distant past and light today share the same traits; and the same gravitational forces that shaped the universe’s initial structures hold sway over present-day galaxies. Mathematicians and physicists have invented the concept of symmetry to describe this kind of immunity to change….

I started with two basic, interrelated questions: Is mathematics invented or discovered? And what gives mathematics its explanatory and predictive powers? I believe that we know the answer to the first question. Mathematics is an intricate fusion of inventions and discoveries. Concepts are generally invented, and even though all the correct relations among them existed before their discovery, humans still chose which ones to study. The second question turns out to be even more complex. There is no doubt that the selection of topics we address mathematically has played an important role in math’s perceived effectiveness. But mathematics would not work at all were there no universal features to be discovered. You may now ask: Why are there universal laws of nature at all? Or equivalently: Why is our universe governed by certain symmetries and by locality? I truly do not know the answers, except to note that perhaps in a universe without these properties, complexity and life would have never emerged, and we would not be here to ask the question. (“Why Math Works,” Scientific American, August 2, 2011)

What Is Truth?

Posted on September 12, 2010 by Loquitur Veritatem

There are four kinds of truth: physical, logical-mathematical, psychological-emotional, and judgmental. The first two are closely related, as are the last two. After considering each of the two closely related pairs, I will link all four kinds of truth.

PHYSICAL AND LOGICAL-MATHEMATICAL TRUTH

Physical truth is, seemingly, the most straightforward of the lot. Physical truth seems to consist of that which humans are able to apprehend with their senses, aided sometimes by instruments. And yet, widely accepted notions of physical truth have changed drastically over the eons, not only because of improvements in the instruments of observation but also because of changes in the interpretation of data obtained with the aid of those instruments.

The latter point brings me to logical-mathematical truth. It is logic and mathematics that translates specific physical truths — or what are taken to be truths — into constructs (theories) such as quantum mechanics, general relativity, the Big Bang, and evolution. Of the relationship between specific physical truth and logical-mathematical truth, G.K. Chesterton said:

Logic and truth, as a matter of fact, have very little to do with each other. Logic is concerned merely with the fidelity and accuracy with which a certain process is performed, a process which can be performed with any materials, with any assumption. You can be as logical about griffins and basilisks as about sheep and pigs. On the assumption that a man has two ears, it is good logic that three men have six ears, but on the assumption that a man has four ears, it is equally good logic that three men have twelve. And the power of seeing how many ears the average man, as a fact, possesses, the power of counting a gentleman’s ears accurately and without mathematical confusion, is not a logical thing but a primary and direct experience, like a physical sense, like a religious vision. The power of counting ears may be limited by a blow on the head; it may be disturbed and even augmented by two bottles of champagne; but it cannot be affected by argument. Logic has again and again been expended, and expended most brilliantly and effectively, on things that do not exist at all. There is far more logic, more sustained consistency of the mind, in the science of heraldry than in the science of biology. There is more logic in Alice in Wonderland than in the Statute Book or the Blue Books. The relations of logic to truth depend, then, not upon its perfection as logic, but upon certain pre-logical faculties and certain pre-logical discoveries, upon the possession of those faculties, upon the power of making those discoveries. If a man starts with certain assumptions, he may be a good logician and a good citizen, a wise man, a successful figure. If he starts with certain other assumptions, he may be an equally good logician and a bankrupt, a criminal, a raving lunatic. Logic, then, is not necessarily an instrument for finding truth; on the contrary, truth is necessarily an instrument for using logic—for using it, that is, for the discovery of further truth and for the profit of humanity. Briefly, you can only find truth with logic if you have already found truth without it. [Thanks to The Fourth Checkraise for making me aware of Chesterton’s aperçu.]

To put it another way, logical-mathematical truth is only as valid as the axioms (principles) from which it is derived. Given an axiom, or a set of them, one can deduce “true” statements (assuming that one’s logical-mathematical processes are sound). But axioms are not pre-existing truths with independent existence (like Platonic ideals). They are products, in one way or another, of observation and reckoning. The truth of statements derived from axioms depends, first and foremost, on the truth of the axioms, which is the thrust of Chesterton’s aperçu.

It is usual to divide reasoning into two types of logical process:

Induction is “The process of deriving general principles from particular facts or instances.” That is how scientific theories are developed, in principle. A scientist begins with observations and devises a theory from them. Or a scientist may begin with an existing theory, note that new observations do not comport with the theory, and devise a new theory to fit all the observations, old and new.

Deduction is “The process of reasoning in which a conclusion follows necessarily from the stated premises; inference by reasoning from the general to the specific.” That is how scientific theories are tested, in principle. A theory (a “stated premise”) should lead to certain conclusions (“observations”). If it does not, the theory is falsified. If it does, the theory lives for another day.

But the stated premises (axioms) of a scientific theory (or exercise in logic or mathematical operation) do not arise out of nothing. In one way or another, directly or indirectly, they are the result of observation and reckoning (induction). Get the observation and reckoning wrong, and what follows is wrong; get them right and what follows is right. Chesterton, again.

PSYCHOLOGICAL-EMOTIONAL AND JUDGMENTAL TRUTH

A psychological-emotional truth is one that depends on more than physical observations. A judgmental truth is one that arises from a psychological-emotional truth and results in a consequential judgment about its subject.

A common psychological-emotional truth, one that finds its way into judgmental truth, is an individual’s conception of beauty. The emotional aspect of beauty is evident in the tendency, especially among young persons, to consider their lovers and spouses beautiful, even as persons outside the intimate relationship would find their judgments risible.

A more serious psychological-emotional truth — or one that has public-policy implications — has to do with race. There are persons who simply have negative views about races other than their own, for reasons that are irrelevant here. What is relevant is the close link between the psychological-emotional views about persons of other races — that they are untrustworthy, stupid, lazy, violent, etc. — and judgments that adversely affect those persons. Those judgments range from refusal to hire a person of a different race (still quite common, if well disguised to avoid legal problems) to the unjust convictions and executions because of prejudices held by victims, witnesses, police officers, prosecutors, judges, and jurors. (My examples point to anti-black prejudices on the part of whites, but there are plenty of others to go around: anti-white, anti-Latino, anti-Asian, etc. Nor do I mean to impugn prudential judgments that implicate race, as in the avoidance by whites of certain parts of a city.)

A close parallel is found in the linkage between the psychological-emotional truth that underlies a jury’s verdict and the legal truth of a judge’s sentence. There is an even tighter linkage between psychological-emotional truth and legal truth in the deliberations and rulings of higher courts, which operated without juries.

PUTTING TRUTH AND TRUTH TOGETHER

Psychological-emotional proclivities, and the judgmental truths that arise from them, impinge on physical and mathematical-logical truth. Because humans are limited (by time, ability, and inclination), they often accept as axiomatic statements about the world that are tenuous, if not downright false. Scientists, mathematicians, and logicians are not exempt from the tendency to credit dubious statements. And that tendency can arise not just from expediency and ignorance but also from psychological-emotional proclivities.

Albert Einstein, for example, refused to believe that very small particles of matter-energy (quanta) behave probabilistically, as described by the branch of physics known as quantum mechanics. Put simply, sub-atomic particles do not seem to behave according to the same physical laws that describe the actions of the visible universe; their behavior is discontinuous (“jumpy”) and described probabilistically, not by the kinds of continuous (“smooth”) mathematical formulae that apply to the macroscopic world.

Einstein refused to believe that different parts of the same universe could operate according to different physical laws. Thus he saw quantum mechanics as incomplete and in need of reconciliation with the rest of physics. At one point in his long-running debate with the defenders of quantum mechanics, Einstein wrote: “I, at any rate, am convinced that He [God] does not throw dice.” And yet, quantum mechanics — albeit refined and elaborated from the version Einstein knew — survives and continues to describe the sub-atomic world with accuracy.

Ironically, Einstein’s two greatest contributions to physics — special and general relativity — were met with initial skepticism by other physicists. Special relativity rejects absolute space-time; general relativity depicts a universe whose “shape” depends on the masses and motions of the bodies within it. These are not intuitive concepts, given man’s instinctive preference for certainty.

The point of the vignettes about Einstein is that science is not a sterile occupation; it can be (and often is) fraught with psychological-emotional visions of truth. What scientists believe to be true depends, to some degree, on what they want to believe is true. Scientists are simply human beings who happen to be more capable than the average person when it comes to the manipulation of abstract concepts. And yet, scientists are like most of their fellow beings in their need for acceptance and approval. They are fully capable of subscribing to a “truth” if to do otherwise would subject them to the scorn of their peers. Einstein was willing and able to question quantum mechanics because he had long since established himself as a premier physicist, and because he was among that rare breed of humans who are (visibly) unaffected by the opinions of their peers.

Such are the scientists who, today, question their peers’ psychological-emotional attachment to the hypothesis of anthropogenic global warming (AGW). The questioners are not “deniers” or “skeptics”; they are scientists who are willing to look deeper than the facile hypothesis that, more than two decades ago, gave rise to the AGW craze.

It was then that a scientist noted the coincidence of an apparent rise in global temperatures since the late 1800s (or is it since 1975?) and an apparent increase in the atmospheric concentration of CO2. And thus a hypothesis was formed. It was embraced and elaborated by scientists (and others) eager to be au courant, to obtain government grants (conveniently aimed at research “proving” AGW), to be “right” by being in the majority, and — let it be said — to curtail or stamp out human activities which they find unaesthetic. Evidence to the contrary be damned.

Where else have we seen this kind of behavior, albeit in a more murderous guise? At the risk of invoking Hitler, I must answer with this link: Nazi Eugenics. Again, science is not a sterile occupation, exempt from human flaws and foibles.

CONCLUSION

What is truth? Is it an absolute reality that lies beyond human perception? Is it those “answers” that flow logically or mathematically from unproven assumptions? Is it the “answers” that, in some way, please us? Or is it the ways in which we reshape the world to conform it with those “answers”?

Truth, as we are able to know it, is like the human condition: fragile and prone to error.

Politics and Prosperity

“Government is a disease masquerading as its own cure.”

Tag Archives: mathematics

Modeling Revisited

Spooky Numbers, Evolution, and Intelligent Design

Mysteries: Sacred and Profane

Are the Natural Numbers Supernatural?

Probability, Existence, and Creation: A Footnote

What Is Truth?

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