*“But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.”*

*“As far as the laws of mathematics refer to reality, they are not certain;*

and as far as they are certain, they do not refer to reality.”

The two Albert Einstein quotations above frequently appear without citation. For example, the first quotation is printed in the “They Say, What They Say, Let Them Say” pages at the beginning of E. T. Bell’s *Men of Mathematics*, and the second can be found in the third volume of James R. Newman’s *The World of Mathematics*. The quotations, placed side-by-side, might look somewhat contradictory at first glance, since one seems to indicate that mathematics is a boon to natural science, while the other appears to say that mathematics is of no use in describing the real world; however, both of these quotations actually originate from the same source and occur only a few sentences apart.

Photograph of Albert Einstein. *Convergence* Portrait Gallery.

On January 27, 1921, Albert Einstein presented a lecture to the Prussian Academy of Sciences entitled “Geometry and Experience.” His complete paper was printed later that year and appeared in English translation a year later. In his paper, Einstein discussed the geometric model of the universe and how to best determine what that model must be. He also considered ways to visualize non-Euclidean concepts such as sets that are finite, but unbounded. Both of the quotations above, however, appeared at the beginning of the paper, when Einstein sets out to distinguish between “axiomatic” and “practical” mathematics.

The paper began with separating mathematics from science by means of certainty: mathematical propositions are not subject to debate, whereas scientific propositions, even when supported by all current facts, may be falsified by later data. Einstein pointed out that, while the natural sciences may not be completely certain, their use of mathematics introduces at least some level of certainty to the mix. It was at this point that the first quotation appeared:

Die Mathematik genießt vor allen anderen Wissenschaften aus einem Grunde ein besonderes Ansehen; ihre Sätze sind absolut sicher und unbestreitbar, während die aller andern Wissenschaften bis zu einem gewissen Grad umstritten und stets in Gefahr sind, durch neu entdeckte Tatsachen umgestoßen zu werden. Trotzdem brauchte der auf einem anderen Gebiete Forschende den Mathematiker noch nicht zu beneiden, wenn sich seine Sätze nicht auf Gegenstände der Wirklichkeit, sondern nur auf solche unserer bloßen Einbildung bezögen. Denn es kann nicht wundernehmen, wenn man zu übereinstimmenden logischen Folgerungen kommt, nachdem man sich über die fundamentalen Sätze (Axiome) sowie über die Methoden geeinigt hat, vermittels welcher aus diesen fundamentalen Sätzen andere Sätze abgeleitet werden sollen. Aber jenes große Ansehen der Mathematik ruht andererseits darauf, daß die Mathematik es auch ist, die den exakten Naturwissenschaften ein gewisses Maß von Sicherheit gibt, das sie ohne Mathematik nicht erreichen könnten [Einstein 1921, p. 3].

One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of all other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts. In spite of this, the investigator in another department of science would not need to envy the mathematician if the laws of mathematics referred to objects of our mere imagination, and not to objects of reality. For it cannot occasion surprise that different persons should arrive at the same logical conclusions when they have already agreed upon the fundamental laws (axioms), as well as the methods by which other laws are to be deduced therefrom. But there is another reason for the high repute of mathematics, in that it is mathematics which affords the exact sciences a certain measure of security, to which without mathematics they could not attain [Einstein 1922, pp. 27–28].

Einstein then asked a familiar question: why should mathematics, based on axioms and independent of experience, be so useful as a tool for modeling reality? It is in response to the idea that human beings might be able to determine the properties of the world by thought and reason alone that the second quotation appears.

An dieser Stelle nun taucht ein Rätsel auf, das Forscher aller Zeiten so viel beunruhigt hat. Wie ist es möglich, daß die Mathematik, die doch ein von aller Erfahrung unabhängiges Produkt des menschlichen Denkens ist, auf die Gegenstände der Wirklichkeit so vortrefflich paßt? Kann denn die menschliche Vernunft ohne Erfahrung durch bloßes Denken Eigenschaften der wirklichen Dinge ergründen?

Hierauf ist nach meiner Ansicht kurz zu antworten: Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nicht sicher, und insofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit [Einstein 1921, p. 3].

At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?

In my opinion the answer to this question is, briefly, this:—As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality [Einstein 1922, p. 28].

Although the quotation itself only says “mathematics,” the following sentences make clear that, in this context, the mathematics in question is “axiomatic” mathematics. Using geometry as an example, Einstein proceeded through the familiar argument that, in its purely axiomatic form, the expressions “straight line” and “point” have no connections to real objects or properties of the real world. Any properties they have proceed from the axioms and only from the axioms.

Einstein then defined “practical geometry,” which takes axiomatic geometry and adds a proposition about how it relates to reality; for example, that solid bodies in the real world behave just like solid bodies in Euclidean Geometry. He pointed out that this proposition is not accepted as an axiom, but instead as a scientific premise, to be tested through experience and experimentation. This is clearly the intended context of the first quotation: that while science benefits from a degree of certainty through the use of “practical” mathematics, it cannot provide full certainty, because future experience may demonstrate that a proposed correspondence between a mathematical object and a property of the universe will be found to be invalid.

##### References

Einstein, Albert. 1921. *Geometrie und Erfahrung*. Berlin: Julius Springer.

Einstein, Albert. 1922. *Sidelights on Relativity*. Translated by G. B. Jeffery and W. Perrett. London: Methuen & Co., Ltd.

“Quotations in Context” is a regular column written by Michael Molinsky that has appeared in the *CSHPM/SCHPM Bulletin* of the Canadian Society for History and Philosophy of Mathematics since 2006 (this installment was first published in May 2007). In the modern world, quotations by mathematicians or about mathematics frequently appear in works written for a general audience, but often these quotations are provided without listing a primary source or providing any information about the surrounding context in which the quotation appeared. These columns provide interesting information on selected statements related to mathematics, but more importantly, the columns highlight the fact that students today can do the same legwork, using online databases of original sources to track down and examine quotations in their original context.