truth

Are the Natural Numbers Supernatural?

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.

Related posts:
Atheism, Religion, and Science
The Limits of Science
Three Perspectives on Life: A Parable
Beware of Irrational Atheism
The Creation Model
The Thing about Science
Evolution and Religion
Words of Caution for Scientific Dogmatists
Science, Evolution, Religion, and Liberty
Science, Logic, and God
Is “Nothing” Possible?
Debunking “Scientific Objectivity”
Science’s Anti-Scientific Bent
Science, Axioms, and Economics
The Big Bang and Atheism
The Universe . . . Four Possibilities
Einstein, Science, and God
Atheism, Religion, and Science Redux
Pascal’s Wager, Morality, and the State
Evolution as God?
The Greatest Mystery
What Is Truth?
The Improbability of Us
A Digression about Probability and Existence
More about Probability and Existence
Existence and Creation
Probability, Existence, and Creation
The Atheism of the Gaps
Probability, Existence, and Creation: A Footnote
Scientism, Evolution, and the Meaning of Life

Demystifying Science

“Science” is an unnecessarily daunting concept to the uninitiated, which is to say, almost everyone. Because scientific illiteracy is rampant, advocates of policy positions — scientists and non-scientists alike — often are able to invoke “science” wantonly, thus lending unwarranted authority to their positions.

WHAT IS SCIENCE?

Science is knowledge, but not all knowledge is science. A scientific body of knowledge is systematic; that is, the granular facts or phenomena which comprise the body of knowledge are connected in patterned ways. Moreover, the facts or phenomena represent reality; they are not mere concepts, which may be tools of science but are not science. Beyond that, science — unless it is a purely descriptive body of knowledge — is predictive about the characteristics of as-yet unobserved phenomena. These may be things that exist but have not yet been measured (in terms of the applicable science), or things that are yet to be (as in the effects of new drug on a disease).

Above all, science is not a matter of “consensus” — AGW zealots to the contrary notwithstanding. Science is a matter of rigorously testing theories against facts, and doing it openly. Imagine the state of physics today if Galileo had been unable to question Aristotle’s theory of gravitation, if Newton had been unable to extend and generalize Galileo’s work, and if Einstein had deferred to Newton. The effort to “deny” a prevailing or popular theory is as old as science. There have been “deniers’ in the thousands, each of them responsible for advancing some aspect of knowledge. Not all “deniers” have been as prominent as Einstein (consider Dan Schectman, for example), but each is potentially as important as Einstein.

It is hard for scientists to rise above their human impulses. Einstein, for example, so much wanted quantum physics to be deterministic rather than probabilistic that he said “God does not play dice with the universe.” To which Nils Bohr replied, “Einstein, stop telling God what to do.” But the human urge to be “right” or to be on the “right side” of an issue does not excuse anti-scientific behavior, such as that of so-called scientists who have become invested in AGW.

There are many so-called scientists who subscribe to AGW without having done relevant research. Why? Because AGW is the “in” thing, and they do not wish to be left out. This is the stuff of which “scientific consensus” is made. If you would not buy a make of automobile just because it is endorsed by a celebrity who knows nothing about automotive engineering, why would you “buy” AGW just because it is endorsed by a herd of so-called scientists who have never done research that bears directly on it?

There are two lessons to take from this. The first is  that no theory is ever proven. (A theory may, if it is well and openly tested, be useful guide to action in certain rigorous disciplines, such as engineering and medicine.) Any theory — to be a truly scientific one — must be capable of being tested, even by (and especially by) others who are skeptical of the theory. Those others must be able to verify the facts upon which the theory is predicated, and to replicate the tests and calculations that seem to validate the theory. So-called scientists who restrict access to their data and methods are properly thought of as cultists with a political agenda, not scientists. Their theories are not to be believed — and certainly are not to be taken as guides to action.

The second lesson is that scientists are human and fallible. It is in the best tradition of science to distrust their claims and to dismiss their non-scientific utterances.

THE ROLE OF MATHEMATICS AND STATISTICS IN SCIENCE

Mathematics and statistics are not sciences, despite their vast and organized complexity. They offer ways of thinking about and expressing knowledge, but they are not knowledge. They are languages that enable scientists to converse with each other and outsiders who are fluent in the same languages.

Expressing a theory in mathematical terms may lend the theory a scientific aura. But a theory couched in mathematics (or its verbal equivalent) is not a scientific one unless (a) it can be tested against observable facts by rigorous statistical methods, (b) it is found, consistently, to accord with those facts, and (c) the introduction of new facts does not require adjustment or outright rejection of the theory. If the introduction of new facts requires the adjustment of a theory, then it is a new theory, which must be tested against new facts, and so on.

This “inconvenient fact” — that an adjusted theory is a new theory –  is ignored routinely, especially in the application of regression analysis to a data set for the purpose of quantifying relationships among variables. If a “model” thus derived does a poor job when applied to data outside the original set, it is not an uncommon practice to combine the original and new data and derive a new “model” based on the combined set. This practice (sometimes called data-mining) does not yield scientific theories with predictive power; it yields information (of dubious value) about the the data employed in the regression analysis. As a critic of regression models once put it: Regression is a way of predicting the past with great certainty.

A science may be descriptive rather than mathematical. In a descriptive science (e.g., plant taxonomy), particular phenomena sometimes are described numerically (e.g., the number of leaves on the stem of a species), but the relations among various phenomena are not reducible to mathematics. Nevertheless, a predominantly descriptive discipline will be scientific if the phenomena within its compass are connected in patterned ways.

NON-SCIENCE, SCIENCE, AND PSEUDO-SCIENCE

Non-scientific disciplines can be useful, whereas some purportedly scientific disciplines verge on charlatanism. Thus, for example:

  • History, by my reckoning, is not a science. But a knowledge of history is valuable, nevertheless, for the insights it offers into the influence of human nature on the outcomes of economic and political processes. I call the lessons of history “insights,” not scientific relationships, because history is influenced by so many factors that it does not allow for the rigorous testing of hypotheses.
  • Physics is a science in most of its sub-disciplines, but there are some (e.g., cosmology and certain interpretations of quantum mechanics) where it descends into the realm of speculation. Informed, fascinating speculation to be sure, but speculation all the same. It avoids being pseudo-scientific only because it might give rise to testable hypotheses.
  • Economics is a science only to the extent that it yields valid, statistical insights about specific microeconomic issues (e.g., the effects of laws and regulations on the prices and outputs of goods and services). The postulates of macroeconomics, except to the extent that they are truisms, have no demonstrable validity. (See, for example, my treatment of the Keynesian multiplier.) Macroeconomics is a pseudo-science.

CONCLUSION

There is no such thing as “science,” writ large; that is, no one may appeal, legitimately, to “science” in the abstract. A particular discipline may be a science, but it is a science only to the extent that it comprises a factual body of knowledge and testable theories. Further, its data and methods must be open to verification and testing. And only a particular theory — one that has been put to the proper tests — can be called a scientific one.

For the reasons adduced in this post, scientists who claim to “know” that there is no God are not practicing science when they make that claim. They are practicing the religion that is known as atheism. The existence or non-existence of God is beyond testing, at least by any means yet known to man.

Related posts:
About Economic Forecasting
Is Economics a Science?
Economics as Science
Hemibel Thinking
Climatology
Physics Envy
Global Warming: Realities and Benefits
Words of Caution for the Cautious
Scientists in a Snit
Another Blow to Climatology?
A Telling Truth
Proof That “Smart” Economists Can Be Stupid
Bad News for Politically Correct Science
Another Blow to Chicken-Little Science
Same Old Story, Same Old Song and Dance
Atheism, Religion, and Science
The Limits of Science
Three Perspectives on Life: A Parable
Beware of Irrational Atheism
The Hockey Stick Is Broken
Talk about Brainwaves!
The Creation Model
The Thing about Science
Science in Politics, Politics in Science
Global Warming and Life
Evolution and Religion
Speaking of Religion…
Words of Caution for Scientific Dogmatists
Science, Evolution, Religion, and Liberty
Global Warming and the Liberal Agenda
Science, Logic, and God
Debunking “Scientific Objectivity”
Pseudo-Science in the Service of Political Correctness
This Is Objectivism?
Objectivism: Tautologies in Search of Reality
Science’s Anti-Scientific Bent
Science, Axioms, and Economics
Global Warming in Perspective
Mathematical Economics
Economics: The Dismal (Non) Science
The Big Bang and Atheism
More Bad News for Global Warming Zealots

The Universe . . . Four Possibilities
Einstein, Science, and God
Atheism, Religion, and Science Redux
Warming, Anyone?
“Warmism”: The Myth of Anthropogenic Global Warming
Re: Climate “Science”
More Evidence against Anthropogenic Global Warming
Yet More Evidence against Anthropogenic Global Warming
A Non-Believer Defends Religion
Evolution as God?
Modeling Is Not Science
Anthropogenic Global Warming Is Dead, Just Not Buried Yet
Beware the Rare Event
Landsburg Is Half-Right
Physics Envy
The Unreality of Objectivism
What Is Truth?
Evolution, Human Nature, and “Natural Rights”
More Thoughts about Evolutionary Teleology
A Digression about Probability and Existence
More about Probability and Existence
Existence and Creation
We, the Children of the Enlightenment
Probability, Existence, and Creation
The Atheism of the Gaps
Probability, Existence, and Creation: A Footnote

What Is Truth?

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.