Árpád Kurusa mathematician, associate professor Department of Geometry Bolyai Institute Faculty of Science University of Szeged

## The relationship between reality and geometry

A quotation from the book "Space, time and gravitation" by Eddington.

(The somewhat shorter hungarian translation is from the book "A fizika kultúrtörténete" (cultural history of physics) by Károly Simonyi.)

A conversation between
An experimental Physicist.
A pure Mathematician.
A Relativist, who advocates the newer conceptions of time and space in physics.

Rel. There is a well-known proposition of Euclid which states that Any two sides of a triangle are together greater than the third side.'' Can either of you tell me whether nowadays there is good reason to believe that this proposition is true?

Math. For my part, I am quite unable to say whether the proposition is true or not. I can deduce it by trustworthy reasoning from certain other propositions or axioms, which are supposed to be still more elementary. If these axioms are true, the proposition is true; if the axioms are not true, the proposition is not true universally. Whether the axioms are true or not I cannot say, and it is outside my province to consider.

Phys. But is it not claimed that the truth of these axioms is self-evident?

Math. They are by no means self-evident to me; and I think the claim has been generally abandoned.

Phys. Yet since on these axioms you have been able to found a logical and self-consistent system of geometry, is not this indirect evidence that they are true?

Math. No. Euclid's geometry is not the only self-consistent system of geometry. By choosing a different set of axioms I can, for example, arrive at Lobatchewsky's geometry, in which many of the propositions of Euclid are not in general true. From my point of view there is nothing to choose between these different geometries.

Rel. How is it then that Euclid's geometry is so much the most important system?

Math. I am scarcely prepared to admit that it is the most important. But for reasons which I do not profess to understand, my friend the Physicist is more interested in Euclidean geometry than in any other, and is continually setting us problems in it. Consequently we have tended to give an undue share of attention to the Euclidean system. There have, however, been great geometers like Riemann who have done something to restore a proper perspective.

Rel. (to Physicist). Why are you specially interested in Euclidean geometry? Do you believe it to be the true geometry?

Phys. Yes. Our experimental work proves it true.

Rel. How, for example, do you prove that any two sides of a triangle are together greater than the third side?

Phys. I can, of course, only prove it by taking a very large number of typical cases, and

Rel. I need only trouble you with a special case. Here is a triangle $ABC$; how will you prove that $AB + BC$ is greater than $AC$?

Phys. I shall take a scale and measure the three sides.

Rel. But we seem to be talking about different things. I was speaking of a proposition of geometry --- properties of space, not of matter. Your experimental proof only shows how a material scale behaves when you turn it into different positions.

Phys. I might arrange to make the measures with an optical device.

Rel. That is worse and worse. Now you are speaking of properties of light.

Phys. I really cannot tell you anything about it, if you will not let me make measurements of any kind. Measurement is my only means of finding out about nature. I am not a metaphysicist.

Rel. Let us then agree that by length and distance you always mean a quantity arrived at by measurements with material or optical appliances. You have studied experimentally the laws obeyed by these measured lengths, and have found the geometry to which they conform.

Rel. Let us proceed to examine the laws of natural geometry. I have a tape-measure, and here is the triangle. $AB = 39\frac{1}{2}$ in., $BC = \frac{1}{8}$ in., $CA = 39\frac{7}{8}$ in. Why, your proposition does not hold!

Phys. You know very well what is wrong. You gave the tape-measure a big stretch when you measured $AB$.

Rel. Why shouldn't I?

Phys. Of course, a length must be measured with a rigid scale.

Rel. That is an important addition to our definition of length. But what is a rigid scale?

Phys. A scale which always keeps the same length.

Rel. But we have just defined length as the quantity arrived at by measures with a rigid scale; so you will want another rigid scale to test whether the first one changes length; and a third to test the second; and so ad infinitum.

Math. The view has been widely held that space is neither physical nor metaphysical, but conventional. Here is a passage from Poincaré's Science and Hypothesis, which describes this alternative idea of space:  If Lobatchewsky's geometry is true, the parallax of a very distant star will be finite. If Riemann's is true, it will be negative. These are the results which seem within the reach of experiment, and it is hoped that astronomical observations may enable us to decide between the two geometries. But what we call a straight line in astronomy is simply the path of a ray of light. If, therefore, we were to discover negative parallaxes, or to prove that all parallaxes are higher than a certain limit, we should have a choice between two conclusions: we could give up Euclidean geometry, or modify the laws of optics, and suppose that light is not rigorously propagated in a straight line. It is needless to add that everyone would look upon this solution as the more advantageous. Euclidean geometry, therefore, has nothing to fear from fresh experiments. ''

Rel. Poincaré's brilliant exposition is a great help in understanding the problem now confronting us. He brings out the interdependence between geometrical laws and physical laws, which we have to bear in mind continually. We can add on to one set of laws that which we subtract from the other set. Moreover, we have actually arrived at the parting of the ways imagined by Poincaré, though the crucial experiment is not precisely the one he mentions.

References