Torah and Geometry

By the Grace of G‑d
25 Sivan, 5712 [June 18, 1952]
Brooklyn, N.Y.

Greeting and Blessing:

... From your letter I surmise that you are an engineer, though it is unclear to me whether your work involves the construction of buildings or is in the field of measuring distances or areas or the like. In any case, fundamental to all of the above is the science of geometry. And what is one of the things that the nature of this science can teach us?

Geometry has the characteristics of an exact science as well as of an applied science. The same is true of our holy Torah (lehavdil ad infinitum). For though the Torah is the wisdom of G-d--the ultimate in truth and exactitude--and "no man can know its worth... and it is hidden from the eyes of all living,"1 nevertheless, as its name, Torah (from the word hora'ah, "instruction"), implies, its purpose is to instruct our daily lives in this physical and material world. Thus, the difference between these two disciplines (Torah and geometry) can enlighten us as to the fundamental and infinite difference between the Torah, of which it is said "for it is your wisdom and understanding before all nations,"2 and the wisdom and understanding of the nations and also of the intellect of the "animal soul"3 of the Jew.4

All human sciences, including the "exact sciences," are founded upon axioms that are wholly unscientific. For science, especially exact science, accepts only proven facts, while the axioms of all sciences, mathematics and geometry included, are not proven in any way, so that a person is free to accept them or reject them. This is most apparent in the science of geometry, which has three different systems, each of which is founded upon a number of axioms, and the axioms of one system are contradictory to the axioms of the others.

In other words, no science can present to a person anything definite, only a series of contingencies. It can only say: if you accept a number of axioms as true, and you accept a particular method of deduction from them, the results will be such and such.

There are two points here: a) It is the person's prerogative whether to accept the axioms or not. b) Also in the case that he does accept them, there is nothing to compel him to act in a manner that is consistent with the results of the particular method. For all the method says is, "If you act this way, the result will be such and such." But if the person is willing to accept the adverse consequences, there is nothing that compels him not to act in any way he desires. In other words, science does not instruct life, but only narrates--as a sort of fortune-teller--a sequence of events, maintaining that according to past experience, and based upon certain axioms which we fancy to accept as true, things will unfold in such and such a manner.

Utterly different is our holy Torah. As the wisdom of the Absolute Existence--the Almighty--it is absolute. Its axioms, as well as the rules that dictate the manner in which laws are to be derived from these axioms, are utterly true. And since this is the wisdom of the Creator of the entire world, man included, it is self-understood that these laws obligate a person to act in concurrence with them, and in no other way.

This is one of the points that you, as a scientist, ought to engrave in your mind: there can be no refutation of Torah by science, since the Torah is absolute truth, while science itself attests that it is not absolute, but dependent upon the person's willful acceptance of certain givens; that the science has full license to establish several contradictory systems, all legitimized by the person's choice to accept their axioms, as is the case with the three systems in geometry--the Euclidean, Lobachevskian and Riemannian....

(A freely-translated excerpt from a letter)5