In an address to the Prussian Acadamy of Science in Berlin on Jan 27, 1921, Einstein was discussing the significance of mathematics in the history of scientific thought, and remarked "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 breifly 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." This same issue was later addressed by Eugene Wigner in his famous essay "The Unreasonable Effectiveness of Mathematics", which appeared in the "Communications in Pure and Applied Mathematics", vol 13, 1960, and is probably the most often cited reference for this notion, although it's somewhat odd that the idea became so closely associated with the physicist Wigner, considering that Einstein could say in 1921 that this enigma "in all ages has agitated inquiring minds". It isn't as if Wigner provided a compelling explanation for the enigma, since his conclusion was basically identical to his premise: The miracle of appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it, and hope that it will remain valid for future research, and that it will extend, for better or for worse, to our pleasure even though perhaps also to our bafflement, to wide branches of learning. I would say Einstein's consideration of this enigma (as quoted above) was far more insightful, so perhaps all those Wigner references don't do justice to Einstein (to say nothing of all those other inquiring minds through the ages). On the other hand, Einstein gets quite a few undeserved citations for Kant's epigram about comprehensibility, because he happenned to quote it in his famous 1936 essay "Physics and Reality" The very fact that the totality of our sense experiences is such that by means of thinking...it can be put in order, this fact is one which leaves us in awe, but which we shall never understand. One may say "the eternal mystery of the world is its comprehensibility". It is one of the great realizations of Immanual Kant that this setting up of a real external world would be senseless without this comprehensibility. Of course, Kant was famously discredited among physicists by his pronouncements on "final categories" and necessary modes of thought, among which he unluckily listed the framework of Euclidean space. With the advent of non-Euclidean geometry Kant fell into disrepute among physicists and mathematicians, so it was perhaps inevitable that they would find another source for that epigram, which is simply too good to discard just because you've decided the author was an idiot. (Of course, notwithstanding this opinion popular among mathematicians, Kant was far from being an idiot, and Einstein had a healthy respect for him and his ideas.) One of the earlier inquiring mind to be agitated by the link between math and nature was Pythagoras, circa 550 BC. Certainly the Pythagorean doctrine that "all things are numbers" is a fairly explicit assertion of correspondence between the concepts of mathematics and the elements of the physical world and experience. On the other hand, this connection didn't originate with the Pythagoreans, but was carried over from various traditions in the earlier Eastern cultures, particularly Babylonia. Of course, with ancient peoples it's not always easy to distinguish between mathematics and numerology, nor between philosophy and mysticism. Then again, the mystical "hermetic" tradition survived in the scientific world at least through the time of Isaac Newton.

Return to MathPages Main Menu