THE success of Pythagoras' program was immediate and devastating. His demand for proof in geometry and arithmetic inspired the invention of the most absorbing game our race has ever played. As intricate as three-dimensional Chinese chess, it had an advantage not shared by any other real game: it could be played without men, counters, or cards, and without putting up any money. A stylus and a pad of wax made the play easier, but were not necessary. Good players could do all that was required by talking. This endless game of deductive reasoning still has its fascinations and its uses, although it is no longer esteemed as a mystic rite for the discovery and worship of eternal verities. The only “truths” which it reveals are the tautologies it grinds out endlessly like a primitive perpetual motion machine with three gears. And these are the gears:

(1) AisA.(2)Everything is eitherAor not-A.(3)Nothing is bothAand not-A.

Since the time of Aristotle (B.C. 384–322) these have been called the Laws of Thought. The first is called the Law of Identity; the second, the Law of Excluded Middle; the third, the Law of Contradiction.

I shall not attempt to elucidate their meaning. Any treatise on formal logic will undertake to explain their meaning, except those treatises (there are such) which declare that the laws have no meaning. Instead of rushing in where metaphysicians tread I shall try to illustrate how these laws have been used in reasoning. The only instance I can recall of the use of the law of identity is the classic assertion “pigs is pigs.” A biologist who says “pigs is mammals” is speaking both ungrammatically and illogically.

The law of excluded middle (everything is either A or not-A) is used at least subconsciously every time we give an “indirect” proof in geometry. Instead of A we shall use the word “true.” To see how the law works, suppose we wish to prove that a certain proposition is true. Law (2) tells us that the proposition is either true or not-true or, as we usually say, either true or false. Let us assume provisionally that the proposition is false. Then, if we can show from this provisional assumption that the proposition is also true, we shall be in conflict with (3), the law of contradiction. But we have agreed to accept (1), (2), (3) as the rules of our game. Therefore our proposition cannot be both true and false. But the error was shown to follow from the provisional assumption that the proposition is false, and it was this step which brought us into conflict with (3). From that we conclude that the provisional assumption was wrong. Therefore the only other possibility under the rules is that the proposition is true.

The foregoing somewhat involved example was concocted
with several ends in view. A careful reading of it, or of any
actual example of “indirect” proof, brings out the Janus-like
character of the abstract “truth” to which appeal is made: a
proposition is *either* true *or* it is false; one or other of these faces
*must* be turned toward us whenever we beseech Truth to reveal
her dazzling countenance to us—the poor creature has no other.
If you imagine her head as thin and flat as a silver dollar, and
squint sideways at her, you will see only a thin vertical rectangle where her shining head should be. This figure however
is incorrect. It contradicts the law of excluded middle. For if
you could see neither Truth's “false” face nor her “true” one,
you could see nothing at all, for those two faces are all she has.
To correct the figure, imagine the dollar to be infinitely thin—if
you can (I can't).

Such is the cardinal *assumption* of the system of deductive
reasoning which Greece bequeathed to its posterity. As already
remarked several times, this assumption is merely an *assumption*; it is *not* a necessary “law” of consistent reasoning.

Before introducing an important technical term, I shall give
another simile to illustrate what the law of excluded middle
“means” in regard to “truth.” Our infinitely thin dollar is so
thin that it cannot stand upright on its edge—we shall assume
this. When it is tossed it must fall *either* heads (“true,” say) *or*
tails (“false”). Every time it falls heads write down a T (for
“true”), and every time tails an F. Keep a record of all the
throws: it might be TTFTFTFF for eight throws. Suppose now
that we are shown a record like this, FTGTF, and are told that
our thin dollar produced it. We must conclude that G is a mistake for either T or F, but we cannot say for which. The simple
point of all this is merely to illustrate what is meant by saying
that “Truth” is *two-valued*, and that its “values” are "true"
and “not-true” (or “false”). They might as well be “blue” and
“not-blue”; it is the *assumed* “two-ness” which is important
here. A “proposition” is a statement which is either true or
false; it has just two “values,” and these values are “true” and
“not-true.” Whether we accept this view of “Truth” or not is
beside the point in considering Aristotle's rules; he evidently
did accept the two-valued idea of truth. Plato also made much
use of it in his speculations. As we have already indicated
several times, the advances in deductive reasoning since 1930
have shown this cardinal assumption to be unnecessary. Thus
much of Greek speculation concerning the natures of truth and
reasoning is vitiated at its source.

Suppose now that instead of tossing a thin dollar we were to toss an ordinary solid die with six faces. In imagination at least we could follow in Aristotle's footsteps and go beyond him, laying down a set of rules in which “Truth” is not two-valued but six-valued. “Absurd!” some will exclaim. Nevertheless it has been done within the past four years. And it works as well as Aristotle's system, possibly better. This slight anticipation may serve to sharpen our suspicions of Plato's perpetual-motion-machine-made conception of “Truth” when we come upon it presently. Let us go on with Aristotle.

The third law, that of contradiction, “nothing is both A and
not-A,” or “no proposition is both true and false,” is the telephone pole which brings us up short just as we begin to shoot
into really brilliant speculations at eighty miles an hour. It
appears to be useful. If a theory contradicts itself in any of its
deductions we usually throw it away as useless. The law seems
to show that if we run forward and backward as hard as we can
go, simultaneously and in one place, we are not likely to get
anywhere, in spite of relativity. Conclusions which violate the
third law are called *inconsistent*. The demand that the conclusions of our deductive reasoning be *consistent* is *retained* in the
new outlook. Thus far we follow the Greeks.

Much (perhaps too much) has been written on the relations of the three laws to one another, and a good deal of what has been written will be found discussed in the stupendous German treatises on the history of logic. The three rules of course are not all that there is to the game. They are, however, the only part of the game which it is necessary for us to remember in our search for truth. The theory of the syllogism and all the rest of the vast development is just so much jam on a cake that is already sweet enough for any normal taste. Although we shall not need any of this it would be interesting to know whether our race has ever invented anything more futile.

The agreement sought by Pythagoras was sealed by Aristotle in his logic in the Fourth Century B.C. It had the effect of paralysing thought in general for a full two thousand years.

We cannot disentangle all the theories of logic invented by
the great Greek thinkers. A considerable part of their effort
seems to have centered round the verb “to be.” What “is” this,
that, or the other? “*Is*” this an apple, or “*is*” it a projection in
space and time of an ideal, extra-spatial, extra-temporal, non-rottable Apple, “existing” in the changeless and eternal realm
of “ideas” as the ethereal ampleness of all the little apples God
ever made? Is it irreverent to suggest that this type of question
originated with the ancient Cambodians and not with Plato?
Again, there was Parmenides (Sixth Century B.C.), who stated
that what can be thought can “be,” and that “truth” and.
“reality” are to be determined by what is “necessary” in
“thought.” From there to a denial of the “existence” of the
“material universe” was but a step, and some of the thinkers
took it. Is it any wonder that the Greeks never made a wheelbarrow? The *necessity* they postulated was by no means
necessary.

There must have been something in what they said; so many of them said it. Many of them were intellects of the first rank. Possibly they said about all that the unaided intellect (verbalizing?) can say that is of any value. A new method for the exploration of the universe, utterly beyond the capacities of the Greek genius, was needed before further progress could be made. Of course it is possible to quibble over “progress” and to maintain that “thought” has not “advanced” since Aristotle and Plato. To avoid an argument, let us say change instead of progress. I presume the most ardent partisan of Greek thought will admit that the great Greeks would have called the conspicuous moderns incurably insane, and that goes in particular for the mathematicians. So there is at least a difference. The important part of that difference is the scientific method of precise experiments directed to definite ends. Our habit of profitless and sometimes self-contradictory speculation in science goes back to Pythagoras, so possibly the Greeks would not consider us so barbarous after all.

While the philosophizing logicians were getting all balled up in the thread which Pythagoras brought from Egypt like a mess of fighting cats in a knitting basket, the sophists rushed in to render first aid and try to prevent a wholesale strangling. They were too late. Aristotle and Plato were presently to prepare the ingenious noose with which the Middle Ages were later to hand themselves, and Euclid was to tie knots around the vital parts of geometry which were to paralyse its creative function for two thousand years. To call a man a sophist today is to accuse him of something akin to sodomy. Nevertheless we must swallow our dislike and have a look at what one of these disputatious dialecticians did, as it is of importance in tracing the thread through the Middle Ages into our own times.

One of the great Greek philosophers was nicknamed “the laughing philosopher.” If it wasn't Democritus (B.C. 460–370), it should have been, for what one of his outstanding achievements did to Aristotelian logic as time went on was enough to make a mummy laugh. That achievement was the proof for the formula which gives the volume of a pyramid. The Greek mathematician and scientist Archimedes attributes this feat to Democritus, so probably the honor is placed where it is due.

We observed some time back that only one way of proving the formula “one-third base times altitude” for the volume of a pyramid standing on a triangular base is possible.¹

Let us see how Democritus did it. If we think of those steps
formed by the stone tiers laid down, one on top of another, by
the Egyptians in building a pyramid, the rest is easy. Democritus imagined his pyramid cut into slices of the same thickness
by planes parallel to the base. The more such slices he imagined, the thinner each became. Finally he imagined them so
thin that the slope of their sides could be neglected, and each
slice could be regarded as a thin triangular sandwich with
vertical sides. It is easy to find the volume of one of these
sandwiches: multiply the area of the base by the thickness.
Pythagoras (or the Egyptians) had shown how to find the
area of a triangle, so there was no difficulty about the base of
the sandwich. He then added up the volumes of all the slices and
thus got an *approximation* to the volume of the triangular
pyramid.

Now, no matter how great a number of slices he took, provided the number was *finite*, he could never get the correct
formula for the volume. There would always be a slight discrepancy introduced by considering the sides (not the top and
bottom) of the sandwiches to be vertical instead of sloped, like
the wall of an embankment, as they really are.

To give a *proof* that the discrepancy vanishes when the number of slices is increased beyond all finite numbers-that is,
when the number of slices is *infinite*—demanded something
more than mere talent of the man who first gave such a proof.
I rather suspect that Democritus took 10 slices first, then 20,
and so on, possibly up to 100, considered the sides of the slices
to be vertical, and calculated the sums of the volumes of the
slices in the respective cases. Seeing that he was getting closer
and closer to “one-third base times altitude” the more slices
he took, he jumped into the infinite and guessed that the
formula of the Egyptians is “true.” Now, *is* it? After all that
has been said, surely it is not necessary to go over the ground
again and point out that the formula is no “truer” than the
consistency of the reasoning by which the formula is “proved,”
and that “proof” itself depends upon the postulates from which
we start.

I am well aware that many take another view. For them
there “exists” in some mystical mathematical heaven an
“ideal” pyramid, and the volume of this pyramid is *necessarily*
what is given by the Egyptian rule. Some of these devout
believers are first rate mathematicians. I am not trying to
destroy anyone's belief in Fairyland; much less would I attempt to rob him of his faith in Santa Claus. All I am trying
to do is to point out, or at least indicate, the kind of evidence
on which some who disbelieve in the *human* origin of abstract
mathematical “truths” base their belief in the mathematical
heaven of Plato. After all, belief was more important to the
men of the Middle Ages than it is to us.

Democritus at any rate took a long step ahead of the Egyptians. Following Pythagoras, he saw clearly that a *proof* of the
formula was necessary. More significant yet, he realized (according to tradition) that it is futile to seek such a proof by
*finite* means; the *infinite* must be brought into play.

If the problem of the pyramid is hard, the problem of finding the area of the surface of a sphere of any given radius seems doubly hard. The Egyptians gave the correct answer to that also (the surface equals the area of four great circles of the sphere) some time before 1800 B.C. This demands the same kind of infinite summations (the integral calculus we call it today, invented by Newton in the Seventeenth Century) as for the pyramid. Until the Moscow papyrus was deciphered this achievement has usually been rated as one of Archimedes' (B.C. 287–212) greatest. The Egyptians are coming up.

All this geometry is introduced for a definite purpose: there is no other way of viewing the first assaults of skepticism on the supposed inviolability of what finally crystallized in to Aristotle's logic with its three “laws of thought”—that impregnable citadel of absolute truth. The first serious assault came from the sophist Zeno (B.C. 495–435). Some will no doubt be delighted to hear that Zeno lost his head for plotting against the government, even if it did happen more than 2300 years ago.

Zeno objected to taking an infinite number of slices of anything, from ham to pyramids to space or time, even in imagination. To put a point on his objections he constructed several paradoxes to show that reasoning about infinities is on a different footing from reasoning about tight little collections that can be counted by human fingers. He also objected to “motion” as an abstract, logical “possibility.” As his paradox about motion is the simplest of all, we may state it first.

“A body is either in the place in which it is or in the place
in which it is not.” Even Aristotle would have to admit that;
it is an application of his own law of excluded middle. “The
body cannot move where it *is*,” Zeno next asserts; “for if it *is*
in a place, it *is* there.” This is not so clear, but we can take his
word for it; Plato would have understood it perfectly. “And,”
Zeno concludes, “the body certainly cannot move where it is
*not*; for it is not there to move or to do anything at all. Hence
it cannot move, and therefore motion is *logically* impossible.”
Since we find it convenient in science and everyday life to think
of bodies moving, it would seem to follow that logic is invades
quate for at least some human purposes.

Many “solutions” of this paradox have been given; too many, in fact. As some contradict others, it seems unlikely that all are right, and we must choose according to our individual tastes. Having no taste in the matter I give one more of Zeno's paradoxes, the famous “Achilles and the tortoise,” so often mentioned in school. My excuse for repeating it is that it seems the most “reasonable” of the lot. It also has been resolved in too many ways.

Achilles gave the tortoise a start of 1000 yards in a race. He could run ten times as fast as the tortoise, and was rash enough to wager that he could overtake the tortoise. The tortoise covered the ground at the rate of a yard a minute. After he had run the 1000 yards, Achilles of course found that the tortoise had gone one-tenth of 1000 yards. So it was now only 100 yards ahead. When Achilles had covered this 100 yards, he was surprised to find the tortoise still going strong, 10 yards ahead. Achilles began to hold his breath. It was no use; when he had done the 10 yards, the plugging tortoise had done its one-tenth of ten, and was still ahead, but only by one yard. Achilles covered the yard; the tortoise was still a tenth of the yard ahead; he covered the tenth; the tortoise was still ahead, by a hundredth of a yard, And so it went, forever and forever and forever. When last seen the tortoise was still a tenth of an atom ahead, and Achilles' tongue was hanging out half a yard.

One interpretation of Zeno's purpose in manufacturing this paradox states that he was denying the infinite divisibility of “time” or of “space,” or of both. Whatever his purpose, it seems fairly obvious that too much faith must not be placed in an argument or “proof” merely because its logical pattern is unobjectionable. Some attention must be paid to our hypotheses and to what it is that we think we are talking about, also to brute facts of common, non-verbalized observation. If the reader has never tried to give an operational solution of this paradox, he may find it interesting to analyse the situation in terms of operations which Achilles could actually perform. The so-called solutions in the advanced textbooks of mathematics only verbalize the difficulties into others as irritating.

Zeno was only one of many. His attacks and those of others were partly responsible for the final cast-iron form of Aristotle's rules. In those simple “truths” consistency surely must abide, if anywhere. But how do we know that those prolific rules will not produce a slower Achilles and a swifter tortoise, or that they have not already done so? We do not know, except by an act of faith (which, by the way, is not at all like knowledge), that Plato's mathematical heaven is as right as a trivet. In that faith we may rest assured that no Achilles is chasing any obstinate tortoise round and round the nebulous walls of the expanding universe provided, of course, that we accept the faith. But it is not necessary to do so.

Before dropping in on Plato's heaven let us put a more modern specimen of infinities beside Zeno's for comparison. This one essentially was manufactured by Galileo in the Seventeenth Century, and again by some of the mathematicians of the Nineteenth. We have already alluded to it; here we may consider it in some detail.

Consider all the whole numbers 1, 2, 3 . . . . . There is no end to the sequence, at least in imagination, for if there were a last number we could add 1 to it and get another. (This is made more respectable in a mathematical treatment by a definition and two postulates instead of the loose intuitive statement. But no number of definitions and postulates will of themselves ensure consistency, or freedom from self-contradiction, in what is deduced from them. The laying down of postulates is not an act of God, but of man; and whatever truth or consistency there may be in deductions from the postulates is not to be sought for profitably in Plato's heaven.) Now, it is “obvious” that in the sequence 1, 2, 3, 4, 5, . . . . there are “more” numbers than there are in the sequence 2, 4, 6, 8, . . . . , each being continued indefinitely; for the first contains all the evens 2, 4, 6, 8, . . . . that make up the second, and in addition all the odds 1, 3, 5, 7, . . . . , none of which occur in the second.

But look at this:

1, 2, 3, 4, 5, 6, 7, . . . .2, 4, 6, 8, 10, 12, 14, . . . . ;

the numbers in the two rows are paired off, one-to-one, no matter how far out we go. Therefore, if we keep on going, and never
stop pairing numbers, each number in the bottom row will have
a unique mate in the top, for the numbers in the bottom row
are got by doubling those in the top. But these rows are the
sequences 1, 2, 3, 4, . . . . and 2, 4, 6, 8, . . . . with which we
started. The argument about the paired rows shows that there
are just as many numbers in the bottom infinite row as in the
top. Therefore there are just as many even numbers as there
are numbers altogether, odds *and* evens. But we saw first how
obvious it was that there are fewer evens than numbers altogether. *We have landed in a flat contradiction:* Aristotle's Law
of Contradiction is violated by a couple of sequences of numbers which defy it. The first sequence both *has*, and *has not*
more numbers in it than the second. If we examine the argumentmore carefully, we see that “pairing” and “counting”
have not been explained. If we now *define* two collections to
contain the *same number* of things only when the things in the
respective collections *can* be paired one-to-one, the difficulty
shifts definitely to “pairing.” There we leave it. Incidentally
also the postulate or axiom of elementary geometry that “the
whole is greater than any of its parts” is exploded as a “universal truth.” Again the reader will find it interesting to
criticize the foregoing argument in detail by the operational
method.

So axioms or postulates are not necessarily either “self-evident” or “true.” They are mere assumptions accepted by a common, temporary agreement.

It may be stated here that this difficulty is overcome in an orthodox treatment by enlarging the domain of mathematics by the annexation of infinities of all denominations, and then inflating the mathematical code by the printing of postulates enough to satisfy the infinities and make them behave. But it is yet to be shown that the postulates are strong enough to hold down any revolutions that may start (several are in progress at this moment) and prevent the whole kingdom of mathematics from going to pieces in anarchy, a martyr to its own imperial generosity. If the tight little finite system was hard enough to govern, the difficulty of keeping the vaster domain in order can be easily imagined. We have no concern with these troubles here, except to point out that most of the really spectacular speculations and prophecies of physical science are reached by flights of the unscientific imagination from deductions proceeding by mathematical analysis. The last in turn stands upon the shakiest spots in all the quaking realm of mathematics. What if the ground were to go from under it? Would that destroy the speculations? I believe not; nothing can destroy ungrounded speculations so long as prosperous congregations with mediaeval minds can be found to listen to honest quacks. The chief importance of these riddles of the infinite for our present purpose is that they were the first difficulties which roused suspicion regarding the validity of logical arguments proceeding in the classical pattern. They first wakened the critical faculties of mathematicians.

The astonishing success of the deductive method in geometry, rather than any of Socrates' ethical “demonstrations,” seems to have been the real inspiration which filled Plato's (B.C. 429–348) mind and lungs and caused him to discourse endlessly on the mystical universe of “Ideas.” Of that universe our own grossly sensual material world is but a shabby and shopworn reflection.

Anyone who has followed the mechanical ruthlessness of a
long chain of geometrical deductions can easily see how the apparent inevitability of conclusion after conclusion might deceive an overimaginative man into believing that the theorems
had an existence and life of their own, independent of the efforts
of the mere human being who first linked the chain together.
From that to a belief in the independent existence of the postulates from which the whole chain rolls, as “eternal verities”
abiding forever in the insubstantial ether of pure, disembodied thought, is but a short step, and perhaps a natural one
for human beings to take in the childhood of their race. The
cow, we now believe, did not jump over the moon; but there
is nothing to prevent us from believing in an ideal cow ideally
skipping over an ideal moon. That is, if we wish to believe.
Similarly for the independent, superhuman “truth” of things
which *appear* to be *self-evident and necessary*.

Parmenides appealed to Plato as an eminently practical thinker. According to Parmenides, we have on the one hand the opinions of mortals, and on the other, “divine truths.” There is also an eternal and unchangeable thing called “Being,” which is identical with “Thought,” and a perpetually disintegrating thing, called “non-Being,” for which the human senses are largely if not wholly responsible. Just as in geometry, so in a number of other arguments designed on the deductive pattern “if A, then B,” there appear to be whole swarms of “concepts” which are identical with themselves (Aristotle's “A is A”?) and which, everlastingly unchangeable, beget like on like, or even on unlike, by a mystical union of their genes and chromosomes with those of the unravished and eternal virgin Truth. Here was a gorgeous opportunity for the three-geared engine of formal logic to grind out eternal verities without end. The “concepts” have a self-perpetuating “existence”; it is the high purpose of formal logic and epistemology to discover the “divine laws” by which “ideas” move and live and have their “being.”

The senses, according to Plato, have nothing to do with the generation of “ideas,” although he appears to believe that these same senses do play a considerable part in the corruption of ideas. Certainly some of his own ideas seem to be void of anything that a modern would call meaning. Nor, as Aristotle would have us believe, are these “ideas” mere “abstractions” evolved (by verbalization?) from “generalities.” Far from it: they owe no part of their existence to any activity of the human mind or of the human vocal organs; they are woven into the warp and woof of the “soul,” which itself has “being,” and these “ideas” were from the beginning of eternity the indestructible essence of the “being” of the soul. Mere sense impressions impinging on our eyes, ears, noses, tongues, and skins had nothing whatever to do with this mysterious process called “filling yourself up with all sorts of ideas, some crazy, some not so crazy.”

This theory of “ideas” being inherent in the structure of the
soul is responsible, among other disasters, for Kant's (1724–1804) elaboration of the “*a priori*” in his Critique of Pure Reason in connection with “space.” Although the *a priori* theory
of geometrical “truths” was exploded once and for all in 1826,
it still lingers on tenaciously in philosophy. So Plato is not yet
as dead in the world of science as he might be. We shall examine
the explosion of 1826 when we come to it in its proper chronological order.

Going on with Plato, let us take a step or two into his
celestial bower. Material things—bricks, pigeons, potatoes—
owe their “existence” to ideas. This disposes of the old query,
“Which came first, the hen or the egg?” Neither; the *idea* of a
hen preceded both her and the egg. Similarly for the egg. The
senses, however, do have some use in this topsy-turvy world:
they are “occasions” through which the gestating “ideas” are
quickened into life in the—wherever it may be that ideas do
pass that first, long, dark period of their mundane “existence.”
To avoid indelicacy I shall simply retail what Plato says—the
“consciousness.” I believe it was our old friend William James
who first of all mortals was bold enough to doubt whether consciousness exists.

The varying individual sense impressions give no perception of material things; these bricks and pigeons of which we mistakenly think we are aware never “are,” they are always on the way, though whence or whither is a mystery. Bricks and pigeons are “images” of the respective immutable Brick and Pigeon, as “ideas.” All knowledge is a “recognition” of “ideas” by the senses. From the few features of the eternal ideas glimpsed by the Senses, we ascend through “thought” to the one and only “true reality.” All this is according to Plato.

Next he tells us in some detail how the ascent from the world of the senses to the heaven of ideas is made: through the sciences, particularly mathematics. The last he pays a very high compliment: mathematics is a sort of fluoroscope through which the human mind may View the insubstantial bones of the ideal unobscured by the dense clay of the senses. “God,” he declares, “ever geometrizes,” thus starting the current fashionable superstition that God is a great mathematician. Plato however did not go as far as some of the great mathematical physicists of today who confuse themselves with God in their attempts to create the universe out of nothing.

On another occasion Plato has something more to say about the mathematical method, which, when stripped of its verbose mysticism, might seem sensible to the least godly mathematician living. “The soul is compelled to use hypotheses in its search, . . . , as though unable to ascend higher than hypotheses. . . . This is so in geometry. . . . . What reason itself attains is this: by its dialectic (talkative?) faculty it forms hypotheses from which it starts on its ascent to the real principle of the universe (of discourse P). Intersecting that universe, reason fuses with it, thus attaining its end without making use of anything given by the senses. In this way thought, starting from ‘ideas,’ and threading its way from one ‘idea’ to another, begins and ends in ‘ideas.’”

He might have said it all more tersely in one modern definition: “Mathematics is the set of all propositions of the form ‘P implies Q,’ where P, Q are any propositions whatever.” But perhaps this is not what he meant, and anyway it will probably not bear analysis on its own account.

From all that has been said of Plato's heaven, it should be clear that he believed exactly what some of the professional scientific speculators would have us believe. If the universe is not exactly one vast and incomprehensible Great Thought, nevertheless our ideas about it, and especially our mathematical ideas about it, are more real than the universe itself. Perhaps “more real” is incorrect, since there is nothing else but our ideas, or “ideas” of which we form ideas. It would be more conservative to say that our theories about the universe are more important to us than the universe is to itself or to us. If not to all of us, then to the theorizer himself.

Before inspecting one or two extremely curious samples of Platonic science, let us note a characteristic feature of the Greek approach to nature, inherent in Plato's philosophy, which also has turned up again, obstinate as ever, in our own day.

The early (pre-Socratic) Greek scientific philosophers took it for granted that the whole range of natural phenomena can be verbalized into propositions deducible from a few fundamental assumptions. Here we see deductive reasoning elevated almost to the status of a creative principle. As was facetiously said of the philosopher Kant, these early speculators undertook to evolve the universe out of their inner consciousness. In Kant's case tobacco smoke is sometimes included as a substantial aid to cogitative creation. I do not know Whether Kant smoked; remembering his notoriously frail health, I trust that he indulged in tobacco, if at all, less stout than his philosophy. But leaving the question of tobacco smoke aside, there are those who declare the project of evolving the universe by deductive reasoning out of one's private assumptions to be the supreme instance of man's conceit and the unsurpassable acme of infernal gall. This project, these dissenters assert, is the original and incurable form of the Jehovah complex, which manifests itself in milder cases in the will to rule the world. The incurables would not only prescribe the laws the entire universe must obey; they would create the universe first so that it could not possibly disobey. Without being as harshly unsympathetic as this to the early Greek program, we need not believe that it will work, and in fact we shall see that it did not always do what was hopefully expected of it. I must state first, however, that nothing definite is known of what the early Greek speculators actually said. All we know of their theories is second or third hand. A good deal of this “they said” sort of evidence is retailed by Aristotle, and a lot more by Plato. Modern classical scholars, perhaps taking a tip from their scientific colleagues, have extrapolated whole philosophies back onto the defenceless Greek pioneers; and for practically any assertion of what the pioneers taught or believed, it is possible to find an equally strong assertion of the exact opposite. What follows appears to be one of the points on which expert disagreement is a minimum.

The deductive scientific method of the early Greeks did less
than its inventors expected of it because the gratuitous assumptions from which each of the two main rival schools started
were too simple. These two schools of speculative deductive
reasoning may be called the *continuous* and *discrete*. The
simplest instance of something that is continuous is a segment
of a straight line: between *any* two distinct points on the line
we can always find (or imagine) another point on the line;
there are no breaks as we pass from one end of the segment to
the other. Contrast this with the sequence 1, 2, 3 . . . . of the
natural numbers. Between 1 and 2 there is *not* another number
of the sequence, since there is no whole number which is both
greater than 1 and less than 2. Similarly for 2 and 3, 3 and 4,
and so on. As we pass along the sequence we must step over
definite gaps between consecutive numbers. The “continuous”
school held that the universe is built on the model of the
straight line; the “discrete” school postulated that the sequence
1, 2, 3, . . . . is the frame-work of the universe. These assumptions are mutually contradictory—so long as we retain the law
of contradiction (Aristotle's third) in our reasoning, as all ages
of reason have agreed to do. The possibility of a *partly* discrete
and *partly* continuous universe is not precluded, but compromises of this sort do not seem to have appealed to the early
extrapolators. The following analogy may bring out the radical
distinction between discrete and continuous universes.

Imagine a motion picture to be gradually slowed down. Presently individual pictures will flicker onto the screen, and what
we saw as *continuous* action will now appear as a succession of
sharply defined individual pictures, each distinct from those
immediately preceding and following it. Roughly this is what a
*discrete* universe would look like if we could “slow it down,” or
sharpen our senses, to the point Where we could observe the
“ultimate particles” of matter and radiation at their work. Following a particular “atom” we should see it move forward in a
succession of jerks or jumps, or rather we should see it in one
position, lose sight of it for instant, and then see it in the next
position. Extrapolating yet farther, some have declared that
“space” and “time,” the simplest of all the frameworks on
which we hang our experiences of “reality,” are also discrete.
(If so, Achilles and the tortoise present new puzzles.) The other
kind of universe, the continuous, would be like a motion picture in which no amount of slowing down would separate the
action on the screen into individual pictures that could be
counted of? 1, 2, 3, . . . . “Space” and “time” in a continuous
universe are also assumed to be continuous.

Between these extreme hypotheses many others are of course conceivable, but these two, either singly or in compromise, have dominated physical speculation for over 2000 years. Like two boys playing leap frog, this active pair have tumbled down the centuries from 600 B.C. to the present day, now one uppermost, now the other, but neither one for very long ahead of its sprightly companion. The “discrete” assumption can claim all “atomic” theories, including those of chemistry, physics and, more recently, radiation (part of modern physics) as its share of the game; the “continuous” assumption has run off with all theories of the ether and, until quite recently, of the electromagnetic field. So neither has been exactly idle. Being mutually contradictory in their strict, classical forms, these prolific assumptions have made it possible for later generations of scholars to attribute each and every scientific theory of modern times to the Greeks of the Sixth Century before Christ; for what the “discrete” assumption contradicts, the “continuous,” must, by Aristotelian logical necessity, confirm. With such a pair behind them, it is logically impossible for modern speculators to create a new heaven or even a new earth. Even the revolutionized logics of 1930 offer no hope of escape from the old man of Greece, for they also insist that reasoning be not self contradictory. So we may expect the game of leap frog to continue.

The “discrete” hypothesis was that first favored by Pythagoras. When he found that the side and diagonal of a square have no common whole number measure, he more or less modified his position. If the “discrete” assumption was contradicted by One of the simplest figures a savage or child might draw (a square with one of its diagonals-the figure Pythagoras asked us to “measure” earlier), it seemed unlikely that the numbers 1, 2, 3, . . . . could suffice for a full mathematical description of the universe.²

The “continuous” theory has also had its spectacular ups
and downs. In pursuing the tortoise with Achilles we witnessed
the first serious setback suffered by the theory. Strict deductive
reasoning there produced a conclusion at variance with everyday experience. Achilles *does* overtake the tortoise; we know
that. But if “space” and “time” are “infinitely divisible,” or
“continuous,” Achilles has a hard time overtaking the tortoise
by unobjectionable deductive reasoning in the orthodox Aristotelian pattern. Here, as with Pythagoras' decisive defeat by the
diagonal of a square, the paradoxical upset was caused by
stumbling over the very beginnings of mathematical reasoning
or, as an early Greek might have said, geometry. Yet Plato had
no qualms in affirming that “God ever geometries,” while at
the same time elevating the “laws of thought” to the supernatural status of a “fate” or “necessary” form in which
“Truth” reveals itself. Thus “geometry,” which God uses in
his construction of the universe, and which Achilles was unable
to make the obstinate tortoise obey, dwells in celestial harmony in the realm of Platonic ideas with the logic which demands a consistency it fails to obtain. It is difficult to see that
this theory of truth made any very radical advance beyond the
sheer guessing of the ancient Egyptian who hit upon a consistently usable formula for the volume of a truncated pyramid.

In spite of these and many similar setbacks, the Greek speculators stuck to their cardinal hypothesis that *it is possible to
construct the universe by deductive reasoning from a few* (prefix
erably one) *“self-evident” assumptions*. Modern speculation is
less ambitious; the number of initial assumptions is greater
than it was some years ago. But again, in spite of temporary
setbacks, the speculative mind still appears to believe that the
Cardinal hypothesis of the pre-Socratic Greeks is both fruitful
and true. As to the first, it has been argued by scholarly mathematicians that the Greek hypothesis is the most helpful for
science ever imagined. An operationalist, however, might disagree. As to the second, it does not have much significance till
we agree what “true” means, and when we have done that, we
shall agree, I think that the “truth” of the extrapolators and
speculators is non-existent. They also, like some of the Greeks,
seem to believe in thesuperhuman necessity of the classical
laws of logic and the uniqueness of the machinery of deductive
reasoning.

Against the cardinal hypothesis of the Greeks let us put a more conservative way of trying to get a grip on the universe, which commits us to no dogma concerning our assumptions, and which never even raises the question of the “truth” of our conclusions. This way is followed by many first rate scientists, to one of whom I am indebted for the following simple illustration which admirably brings out the points to be observed.

Suppose two large boxes of matches are before us, and suppose the matches in one box are blue, those in the other, red. From one box we take a handful of matches and toss it on the floor; a handful from the other box is also tossed onto the floor so that the matches in the two handfuls do not mix. The matches have fallen in two roughly similar patterns, the one all. of blue matches, the other of red. We now detect certain approximate similarities between parts of the two patterns: here, for instance, three red matches almost close up into a triangle; there, three blue matches also, but more crudely, approximate to a triangle; and so on for more complicated rough correspondences. The correspondences, note, will never be exact (if we accept the mathematics of probability) unless some sort of miracle interfered with the tossing of the second handful.

We now read into this situation an analogy with what the unspeculating kind of scientist does with his postulates,ihis theories, and his experimentally ascertained facts. The pattern of blue matches corresponds to the postulates (assumptions) which the scientist has made regarding the particular range of natural phenomena in which he is interested, together with all theories and predictions which he has reached by deductive reasoning from the postulates. The pattern of red matches corresponds to all the experimental data (pointer readings, etc.) which the scientist has collected for comparison with his postulates and theories. These data cover the same range of phenomena as before, and they have been expressed in language (usually mathematical) which will make possible a comparison of the two patterns, the blue and the red. Unless the postulates are almost miraculously bad, the scientist will usually observe a few rough similarities between his two patterns. If his postulates are as good as those of general relativity, the blue pattern, that of assumptions and deductions, will suggest symmetries and similarities to be looked for in the red pattern, that of experimentally ascertained data. Sometimes these suggested similarities will not be found at once, but do appear—roughly—when new experiments are devised to secure further data with which to fill out the red pattern.

Notice that not a word has been said about the “truth” of
the blue pattern or the “reality” of the red pattern. It is neither
necessary nor helpful to attribute either quality to either pattern. If the blue and red patterns are hopelessly dissimilar, the
usual remedy is to call in the maid—who is a severe critic of
unnecessary rubbish—and ask her kindly to remove the blue
pattern with the vacuum cleaner. The red is left for further inspection, and another handful of blue matches is tossed onto
the carpet in the sanguine hope that *this* time it will not be
quite so unlike the other.

Disasters frequently happen, however, before the maid can clean up. The scientist's cat, who is hazy about colors like most cats, fancies himself as a philosopher of science. At the critical moment he clashes in, rolls in the matches, and succeeds in thoroughly mixing the blue and red into an indescribable litter that is neither fact nor theory. Being kicked out by his exasperated master, the cat gets even by fleeing to the roof, where he sits all night declaiming in a cacophonous metaphysical falsetto that truth is reality and reality is truth. This noncommittal approach to nature has much to commend it to a reasonably critical taste, but it was not the way followed by Greek scientists.³ Let us return to them for a moment and see to what the deductive method led when carried to its conclusion by a master.

What would a confirmed idealist like Plato make of a simple scientific observation or mathematical theorem? I shall not trust myself to say. Instead I shall quote from an impartial historian of Greek science (G. H. Lewes, in his Aristotle, 1864, pp. 105–6). “81. The mathematicians having discovered the five regular solids, Plato naturally made great use of them in his cosmology. Four of them were represented by the four elements—the Earth was a Cube, Fire a Tetrahedron, Water an Octahedron, and Air an Icosahedron. This left the fifth, the Dodecahedron, without a representative; accordingly it was assigned to the universe as a whole.

“The Creator, having thus shaped the visible universe, and
distributed souls over the earth, the moon and other unnamed
places—-and having commissioned the younger gods (*dii minores*) to construct man,—retired to his repose.

“It is needless to add that Plato never thinks of offering any better reason for these propositions than that they are by him judged sufficient. If one of his hearers had asked him why water might not be a cube, and air an octahedron—or what proof there was of either being one or the other—he would have replied: ‘It is thus that I conceive it. This is best.’

“82. Let us proceed. The universe, we learn, has a soul which moves in perpetual circles. Man also has a soul which is but a portion thereof, consequently it also moves in circles. To make the resemblance more complete, man's soul is also enclosed in a spherical body-namely the head. But the gods foresaw that this head, being spherical, would roll down the hills and could not ascend steep places; to prevent this, a body with limbs was added, that it might be a locomotive for the head. As the fore-parts are more honourable and regal than the hind parts, the gods made man's locomotion chiefly progressive.”

Omitting Plato's beautiful theories of the liver and the intestines, I shall merely quote what Lewes says about them.

“§84. In a modern such ideas would not appear profound. (A fat lot Lewes knew about it!) I have not cited them for the poor pleasure of holding up a great name in the light of ridicule; but to show how even a great intellect may unsuspectingly wander into absurdities, when it quits the firm though laborious path of inductive inquiry. . . . . The same confidence in deduction from unverified premises vitiates his teaching in every other department of inquiry, moral and political; but in Science his errors are more patent, because his statements admit of a readier, and less equivocal, confrontation with fact.”

That, it seems to me, is the proper attitude to take. But if a man or an age has made himself or itself ridiculous, who shall unmake him or it?

Those who profess to understand Platonic idealism, or who are forced, by the hard circumstance of having to make a living, into professing to understand the theory, affect a scornful superiority over those who can get nothing but words—beautiful words, perhaps, if you admire pompous mysticism—out of Plato's heaven. The fault, these superior beings tell us, is our own, not Plato's. Admitting that the fault, if it is one, is our own, why should any human being fuddle his thinking today by trying to understand that which can be seen through by anyone who will take the trouble to use his eyes and hishands.P

“Ah, but it is a beautiful and inspiring thing, this heaven of
Plato's. You *don't understand*.”

Possibly, and indeed probably.

The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid (B.C. 330–275) did to geometry. His Elements were so nearly perfect a performance for the age in which he lived that it took some countries of Europe till the year 1900, or shortly after, to get elementary geometry untied.

In one form or another Euclid's geometry has gone through
more editions than any other printed book except the Christian
Bible, and more human beings have attempted to master a few
propositions of it than ever mastered our Bible. All civilized
western nations, including the Mohammedan Arabs, struggled
to understand at least the first book of the *Elements*. Its effect
on their mentality may have been beneficial; we have no way of
judging whether it was or not. But the reverence and respect
in which Euclid's allegedly rigorous reasoning was heldby all
educated men for well over two thousand years cannot possibly
have done their own reasoning faculties any good.

A diluted sort of Euclid, looser than his loosest, is one of the mainstays of American education today. It is supposed to quicken the reason, and there is no doubt that it does in the hands of a thoroughly competent and modernized teacher, who lets the children use their heads and see for themselves exactly how nonsensical some of the stuff presented as “proof” really is. But, as nothing else even half so good is offered in the way of deductive reasoning in school, we must not be too hard on what is actually handed out. The great miracle is that there are not a hundred million gullible boobies in America, eager to swallow all the latest and craziest speculations, instead of the negligible few there are—their number can be approximated by a statistical analysis of the “sucker lists” of any of the larger business houses that go in for that sort of fishing.

Who would think of teaching boys who want to tinker with
automobiles and radios their physics out of Aristotle's classic
“Physics”? It is admitted by all that the world has moved
*scientifically* since the Third Century B.C. But it seems to be
less generally admitted that the world has also moved in mathematics and in the technique of straight thinking. The best that
could be done now in the way of elementary school geometry
would be no harder for a normal intelligence than the sorry
third-best that still passes as a training in deductive reasoning.
In fact it would be simpler, because less muddled, less specious,
and less confusing. The rules of the game are only two: State *all*
of your assumptions; see that *no other* assumption slips in during the course of a proof. On these two essentials Euclid—and
his modern diluters—fell down so badly that there is absolutely
no hope of ever getting him on his feet again. Uncritical reverence for the supposed rigidity of Euclid's geometry (he himself excelled mostly as a compiler and logical arranger of other
men's work) had much to do with the retardation of progress in
close reasoning, so we should try to keep him in mind till we
meet his shade in the Nineteenth Century, as we cannot follow
him in detail through the Middle Ages.

I realize that these may sound to some like pretty strong statements, so I shall quote an opinion on the last point by a judge whom all mathematicians agree is competent, Bertrand Russell. Russell is perhaps best known to the general reader as a writer on philosophical and social subjects; his earlier work (the second book he ever wrote, in 1897, was on the foundations of geometry) marked a new epoch in our attempts to understand the foundations of mathematics and what mathematics is all about. This is what he has to say about Euclid.

“It has been customary when Euclid, considered as a textbook, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, and his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid's earlier proofs fail before this test.”

Russell proceeds then to analyse in detail the first seven propositions in Euclid's geometry&mdashwhich Euclid thought he had proved—and some of the later propositions. Euclid had not proved one of them. Here are a few of Russell's comments: “The fourth proposition is a tissue of nonsense.” (Reasons backing this assertion follow.) “The seventh proposition is so thoroughly fallacious that Euclid would have done better not to attempt a proof.”

“Many more general criticisms might be passed on Euclid's methods and on his conception of Geometry; but the above definite fallacies (which Russell points out, but which are omitted here) seem sufficient to show that the value of his work as a masterpiece of logic has been very grossly exaggerated.”

If school children fail to get some conception of geometry and close reasoning out of their course in “geometry” they get nothing, except possibly a permanent inability to think straight and a propensity to jump to conclusions which nothing in reason or sanity warrants.

Our debt to Greece is indeed great. It is long past time that we settle up.

The stage is now set for the first dramatic impact of religious intolerance on the problem of truth. That the intolerance happened to appear first from the particular quarter that it did may be more or less of a historical accident. Any pretext in the hands of a rapidly degenerating race would have served equally well as an excuse for assaulting the perfect thing the Greeks left the world—their specific system of deductive reasoning. What follows is only the green bud of bigotry and intolerance. We shall not see the perfected flower till we reach the Middle Ages.

¹ I am not sure that the reasoning by which this truly remarkable theorem on the unique possibility is established would be considered free from serious objection today. The proof was given by Dehn in 1900. Anyone interested may consult Dehn's paper in the Göttingen Nachrichten for 1900. The point is of no importance for the statement in the text; it is inconceivable that Democritus could have proceeded by some method other than that described; such a method would have been mentioned by his contemporaries and immediate successors. But it should be remembered that the most important phases of the modern revolution in logic did not begin until many years after Dehn's work was published.

² In my Numerology (Williams & Wilkins, 1933), I have given a fuller account of the Pythagorean assumption and its consequences in the history of pseudo-scientific (as well as scientific) speculation.

³To offset my account, the reader may like to consult Professor W. A. Heidel's recent monograph, The Heroic Age of Science: the Conception, Ideals and Methods of Science among the Ancient Greeks (Williams & Wilkins, 1933). This is delightfully sympathetic and illuminating.

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