For Andy, though of interest to just about everyone:

**Euclid, Reimann, Poincare, and Einstein: Alternative Mathematical Systems**

Taught by Michelle Granger

*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 lifeyime, an alarmingly deep crisis in the foundations of the exact sciences had begun. For years scientific truth had been beyond the possibilty of 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. Th mission of science was now simply to refine the answers to greater and greater accuracy. True, there were still some 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 explaind 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 is one and exactly one straight line parallel to another given straight line that does not pass through that point, 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:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

Yet the fifth postulate is not nearly so obvious or intuitive. Yet you couldn't get rid of it without destroying huge portions of 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 unimanageable, Poincare said.

Finally, in the first quarter of the nineteenth century, and almost at the same time, a Hungarian and a Russian - Bolyai and Lobacheveski - established irrefutably that a proof of Euclid's fifth postulate is impossible. They id 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 Euclidian geometry.

Thus by his failure to find any contradiction he proves tha tthe fifth postultae is impossible to reduce 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 logical contradtictions 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 sustaine by pure 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 fifth postulate, but also the first axiom, 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 Rheory of Relativity, Riemann geometry best describes the world we live in.

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*judgements, 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 seemd to be contrary to the whole nature of geometry itself.

Poincare concluded that the axioms of geometry are

*conventions*, our best choice among all possible convntions is

*guided*by experimental facts, but it remains

*free*and is limited only by the necessity of avoiding all contradictions. 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 false; whether Cartesian coordinates are true and polar coordinates are false. One geometry cannot 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 not 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 completely mechanical explanation it will admit an infinity of others which will account equally well for all peculiarities disclosed by the experiment." If the scientiest had at his disposal infinite time, Poincare said, it would onlybe 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 rule: 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 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 ell 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 resemblences 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 resemblences 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 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 htought 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 couldnt' 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 crystallzation had taken place.

He described how a second wave of crystalizatoin, 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 a 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 ot fathom more deeply what had happened.

Mathematics, he siad, isn't merely a question of applying rules, any more than science. It doesn't merely make the mots combinations possible according to certain fixed laws. The combinations so obtained would be 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 impossibel to state them precisely; they must be felt rather than formulated.

Poincare then hypothesized that this selection is made by what is called the "subliminal self, an entity that corresponds exactly with what others have called, "preintellectual awareness." The subliminal self, Poincare said, looks at a 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 ont he basis of "mathematical beauty," of the harmony of numbers and forms, of geometric elegance. "This is a true aesthetic 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 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 gream 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 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 guaruntees 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 tha tthese reasonable beings have seen the same things as we; thus it is that we know we haven't been dreaming. It is this harmong, this

*quality*, if you will, that is the sole basis for the only reality we can ever know.

Poincare's contemporaries refused to acknowledge that facts are preselected becasue they thought that to do so would destroy the validity of scientific method. They presumed that "preselected facts" meant that truth is "whatever you like" and caleld his ideas conventionalism. They vigorously ignored the truth that their own "principle of objectivity" is not itself an observable fact - and therefore by their own criteria should be put in a state of suspended animation.

They felt they ahd to do this because if they didn't, the entire philosophic underpinning of science would collapse. Poincare didn't offer any resolutions ot htis quandry. He didn't go far enough into the metaphysical implications of what he was saying to arrive at the solution.

*Largely quoted and paraphrased with only slight modifications from Robert Pirsig's

*Zen and the Art of Motorcycle Maintenance*