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 theoretic 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 Ph?drus 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
