It always seemed incredible to me, and still does, I guess, that Phædrus should have traveled along a line of thought that had never been traveled before. Someone, somewhere, must have thought of all this before, and Phædrus was such a poor scholar it would have been just like him to have duplicated the commonplaces of some famous system of philosophy he hadn’t taken the trouble to look into.
So I spent more than a year reading the very long and sometimes very tedious history of philosophy in a search for duplicate ideas. It was a fascinating way to read the history of philosophy, however, and a thing occurred of which I still don’t know quite what to make. Philosophical systems that are supposed to be greatly opposed to one another both seem to be saying something very close to what Phædrus thought, with minor variations. Time after time I thought I’d found whom he was duplicating, but each time, because of what appeared to be some slight differences, he took a greatly different direction. Hegel, for example, whom I referred to earlier, rejected Hindu systems of philosophy as no philosophy at all. Phædrus seemed to assimilate them, or be assimilated by them. There was no feeling of contradiction.
Eventually I came to Poincare. Here again there was little duplication but another kind of phenomenon. Phædrus follows a long and tortuous path into the highest abstractions, seems about to come down and then stops. Poincare starts with the most basic scientific verities, works up to the same abstractions and then stops. Both trails stop right at each other’s end! There is perfect continuity between them. When you live in the shadow of insanity, the appearance of another mind that thinks and talks as yours does is something close to a blessed event. Like Robinson Crusoe’s discovery of footprints on the sand.
Poincare lived from 1854 to 1912, a professor at the University of Paris. His beard and pince-nez were reminiscent of Henri Toulouse-Lautrec, who lived in Paris at the same time and was only ten years younger.
During Poincare’s lifetime, an alarmingly deep crisis in the foundations of the exact sciences had begun. For years scientific truth had been beyond the possibility of a doubt; the logic of science was infallible, and if the scientists were sometimes mistaken, this was assumed to be only from their mistaking its rules. The great questions had all been answered. The mission of science was now simply to refine these answers to greater and greater accuracy. True, there were still unexplained phenomena such as radioactivity, transmission of light through the “ether”, and the peculiar relationship of magnetic to electric forces; but these, if past trends were any indication, had eventually to fall. It was hardly guessed by anyone that within a few decades there would be no more absolute space, absolute time, absolute substance or even absolute magnitude; that classical physics, the scientific rock of ages, would become “approximate”; that the soberest and most respected of astronomers would be telling mankind that if it looked long enough through a telescope powerful enough, what it would see was the back of its own head!
The basis of the foundation-shattering Theory of Relativity was as yet understood only by very few, of whom Poincare, as the most eminent mathematician of his time, was one.
In his Foundations of Science Poincare explained that the antecedents of the crisis in the foundations of science were very old. It had long been sought in vain, he said, to demonstrate the axiom known as Euclid’s fifth postulate and this search was the start of the crisis. Euclid’s postulate of parallels, which states that through a given point there’s not more than one parallel line to a given straight line, we usually learn in tenth-grade geometry. It is one of the basic building blocks out of which the entire mathematics of geometry is constructed.
All the other axioms seemed so obvious as to be unquestionable, but this one did not. Yet you couldn’t get rid of it without destroying huge portions of the mathematics, and no one seemed able to reduce it to anything more elementary. What vast effort had been wasted in that chimeric hope was truly unimaginable, Poincare said.
Finally, in the first quarter of the nineteenth century, and almost at the same time, a Hungarian and a Russian… Bolyai and Lobachevski… established irrefutably that a proof of Euclid’s fifth postulate is impossible. They did this by reasoning that if there were any way to reduce Euclid’s postulate to other, surer axioms, another effect would also be noticeable: a reversal of Euclid’s postulate would create logical contradictions in the geometry. So they reversed Euclid’s postulate.
Lobachevski assumes at the start that through a given point can be drawn two parallels to a given straight. And he retains besides all Euclid’s other axioms. From these hypotheses he deduces a series of theorems among which it’s impossible to find any contradiction, and he constructs a geometry whose faultless logic is inferior in nothing to that of the Euclidian geometry.
Thus by his failure to find any contradictions he proves that the fifth postulate is irreducible to simpler axioms.
It wasn’t the proof that was alarming. It was its rational byproduct that soon overshadowed it and almost everything else in the field of mathematics. Mathematics, the cornerstone of scientific certainty, was suddenly uncertain.
We now had two contradictory visions of unshakable scientific truth, true for all men of all ages, regardless of their individual preferences.
This was the basis of the profound crisis that shattered the scientific complacency of the Gilded Age. How do we know which one of these geometries is right? If there is no basis for distinguishing between them, then you have a total mathematics which admits logical contradictions. But a mathematics that admits internal logical contradictions is no mathematics at all. The ultimate effect of the non-Euclidian geometries becomes nothing more than a magician’s mumbo jumbo in which belief is sustained purely by faith!
And of course once that door was opened one could hardly expect the number of contradictory systems of unshakable scientific truth to be limited to two. A German named Riemann appeared with another unshakable system of geometry which throws overboard not only Euclid’s postulate, but also the first axiom, which states that only one straight line can pass through two points. Again there is no internal contradiction, only an inconsistency with both Lobachevskian and Euclidian geometries.
According to the Theory of Relativity, Riemann geometry best describes the world we live in.
At Three Forks the road cuts into a narrow canyon of whitish-tan rock, past some Lewis and Clark caves. East of Butte we go up a long hard grade, cross the Continental Divide, then go down into a valley. Later we pass the great stack of the Anaconda smelter, turn into the town of Anaconda and find a good restaurant with steak and coffee. We go up a long grade that leads to a lake surrounded by pine forests and past some fishermen who push a small boat into the water. Then the road winds down again through the pine forest, and I see by the angle of the sun that the morning is almost ended.
We pass through Phillipsburg and are off into valley meadows. The head wind becomes more gusty here, so I slow down to fifty-five to lessen it a little. We go through Maxville and by the time we reach Hall are badly in need of a rest.
We find a churchyard by the side of the road and stop. The wind is blowing hard now and is chilly, but the sun is warm and we lay out our jackets and helmets on the grass on the leeward side of the church for a rest. It’s very lonely and open here, but beautiful. When you have mountains in the distance or even hills, you have space. Chris turns his face into his jacket and tries to sleep.
Everything is so different now without the Sutherlands… so lonely. If you’ll excuse me I’ll just talk Chautauqua now, until the loneliness goes away.
To solve the problem of what is mathematical truth, Poincare said, we should first ask ourselves what is the nature of geometric axioms. Are they synthetic a priori judgments, as Kant said? That is, do they exist as a fixed part of man’s consciousness, independently of experience and uncreated by experience? Poincare thought not. They would then impose themselves upon us with such force that we couldn’t conceive the contrary proposition, or build upon it a theo
retic edifice. There would be no non-Euclidian geometry.
Should we therefore conclude that the axioms of geometry are experimental verities? Poincare didn’t think that was so either. If they were, they would be subject to continual change and revision as new laboratory data came in. This seemed to be contrary to the whole nature of geometry itself.
Poincare concluded that the axioms of geometry are conventions, our choice among all possible conventions is guided by experimental facts, but it remains free and is limited only by the necessity of avoiding all contradiction. Thus it is that the postulates can remain rigorously true even though the experimental laws that have determined their adoption are only approximative. The axioms of geometry, in other words, are merely disguised definitions.
Then, having identified the nature of geometric axioms, he turned to the question, Is Euclidian geometry true or is Riemann geometry true?
He answered, The question has no meaning.
As well ask whether the metric system is true and the avoirdupois system is false; whether Cartesian coordinates are true and polar coordinates are false. One geometry can not be more true than another; it can only be more convenient. Geometry is not true, it is advantageous.
Poincare then went on to demonstrate the conventional nature of other concepts of science, such as space and time, showing that there isn’t one way of measuring these entities that is more true than another; that which is generally adopted is only more convenient.
Our concepts of space and time are also definitions, selected on the basis of their convenience in handling the facts.
This radical understanding of our most basic scientific concepts is not yet complete, however. The mystery of what is space and time may be made more understandable by this explanation, but now the burden of sustaining the order of the universe rests on “facts.” What are facts?
Poincare proceeded to examine these critically. Which facts are you going to observe? he asked. There is an infinity of them. There is no more chance that an unselective observation of facts will produce science than there is that a monkey at a typewriter will produce the Lord’s Prayer.
The same is true of hypotheses. Which hypotheses? Poincare wrote, “If a phenomenon admits of a complete mechanical explanation it will admit of an infinity of others which will account equally well for all the peculiarities disclosed by experiment.” This was the statement made by Phædrus in the laboratory; it raised the question that failed him out of school.
If the scientist had at his disposal infinite time, Poincare said, it would only be necessary to say to him, “Look and notice well”; but as there isn’t time to see everything, and as it’s better not to see than to see wrongly, it’s necessary for him to make a choice.
Poincare laid down some rules: There is a hierarchy of facts.
The more general a fact, the more precious it is. Those which serve many times are better than those which have little chance of coming up again. Biologists, for example, would be at a loss to construct a science if only individuals and no species existed, and if heredity didn’t make children like parents.
Which facts are likely to reappear? The simple facts. How to recognize them? Choose those that seem simple. Either this simplicity is real or the complex elements are indistinguishable. In the first case we’re likely to meet this simple fact again either alone or as an element in a complex fact. The second case too has a good chance of recurring since nature doesn’t randomly construct such cases.
Where is the simple fact? Scientists have been seeking it in the two extremes, in the infinitely great and in the infinitely small. Biologists, for example, have been instinctively led to regard the cell as more interesting than the whole animal; and, since Poincare’s time, the protein molecule as more interesting than the cell. The outcome has shown the wisdom of this, since cells and molecules belonging to different organisms have been found to be more alike than the organisms themselves.
How then choose the interesting fact, the one that begins again and again? Method is precisely this choice of facts; it is needful then to be occupied first with creating a method; and many have been imagined, since none imposes itself. It’s proper to begin with the regular facts, but after a rule is established beyond all doubt, the facts in conformity with it become dull because they no longer teach us anything new. Then it’s the exception that becomes important. We seek not resemblances but differences, choose the most accentuated differences because they’re the most striking and also the most instructive.
We first seek the cases in which this rule has the greatest chance of failing; by going very far away in space or very far away in time, we may find our usual rules entirely overturned, and these grand overturnings enable us the better to see the little changes that may happen nearer to us. But what we ought to aim at is less the ascertainment of resemblances and differences than the recognition of likenesses hidden under apparent divergences. Particular rules seem at first discordant, but looking more closely we see in general that they resemble each other; different as to matter, they are alike as to form, as to the order of their parts. When we look at them with this bias we shall see them enlarge and tend to embrace everything. And this it is that makes the value of certain facts that come to complete an assemblage and to show that it is the faithful image of other known assemblages.
No, Poincare concluded, a scientist does not choose at random the facts he observes. He seeks to condense much experience and much thought into a slender volume; and that’s why a little book on physics contains so many past experiences and a thousand times as many possible experiences whose result is known beforehand.
Then Poincare illustrated how a fact is discovered. He had described generally how scientists arrive at facts and theories but now he penetrated narrowly into his own personal experience with the mathematical functions that established his early fame.
For fifteen days, he said, he strove to prove that there couldn’t be any such functions. Every day he seated himself at his work-table, stayed an hour or two, tried a great number of combinations and reached no results.
Then one evening, contrary to his custom, he drank black coffee and couldn’t sleep. Ideas arose in crowds. He felt them collide until pairs interlocked, so to speak, making a stable combination.
The next morning he had only to write out the results. A wave of crystallization had taken place.
He described how a second wave of crystallization, guided by analogies to established mathematics, produced what he later named the “Theta-Fuchsian Series.” He left Caen, where he was living, to go on a geologic excursion. The changes of travel made him forget mathematics. He was about to enter a bus, and at the moment when he put his foot on the step, the idea came to him, without anything in his former thoughts having paved the way for it, that the transformations he had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. He didn’t verify the idea, he said, he just went on with a conversation on the bus; but he felt a perfect certainty. Later he verified the result at his leisure.
A later discovery occurred while he was walking by a seaside bluff. It came to him with just the same characteristics of brevity, suddenness and immediate certainty. Another major discovery occurred while he was walking down a street. Others eulogized this process as the mysterious workings of genius, but Poincare was not content with such a shallow explanation. He tried to fathom more deeply what had happened.
Mathematics, he said, isn’t merely a question of applying rules, any more than science. It doesn’t merely make the most combinations possible according to certain fixed laws. The combinations so obtained would he exceedingly numerous, useless and cumbersome. The true work of the inventor consists in choosing among these combinations so as to eliminate the useless ones, or rather, to avoid the trouble of making them, and the rules that must guide the choice are extremely fine and delicate. It’s almost impossible to state them precisely; they must be felt rather than formulated.
Poincar
e then hypothesized that this selection is made by what he called the “subliminal self”, an entity that corresponds exactly with what Phædrus called preintellectual awareness. The subliminal self, Poincare said, looks at a large number of solutions to a problem, but only the interesting ones break into the domain of consciousness. Mathematical solutions are selected by the subliminal self on the basis of “mathematical beauty”, of the harmony of numbers and forms, of geometric elegance. “This is a true esthetic feeling which all mathematicians know”, Poincare said, “but of which the profane are so ignorant as often to be tempted to smile.” But it is this harmony, this beauty, that is at the center of it all.
Poincare made it clear that he was not speaking of romantic beauty, the beauty of appearances which strikes the senses. He meant classic beauty, which comes from the harmonious order of the parts, and which a pure intelligence can grasp, which gives structure to romantic beauty and without which life would be only vague and fleeting, a dream from which one could not distinguish one’s dreams because there would be no basis for making the distinction. It is the quest of this special classic beauty, the sense of harmony of the cosmos, which makes us choose the facts most fitting to contribute to this harmony. It is not the facts but the relation of things that results in the universal harmony that is the sole objective reality.
What guarantees the objectivity of the world in which we live is that this world is common to us with other thinking beings. Through the communications that we have with other men we receive from them ready-made harmonious reasonings. We know that these reasonings do not come from us and at the same time we recognize in them, because of their harmony, the work of reasonable beings like ourselves. And as these reasonings appear to fit the world of our sensations, we think we may infer that these reasonable beings have seen the same thing as we; thus it is that we know we haven’t been dreaming. It is this harmony, this quality if you will, that is the sole basis for the only reality we can ever know.