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XIII 00114 1960 The Unreasonable Effectiveness of Mat hematics in the Natural Sciences Richard Courant Lecture in Mathematical Sciences delivered at New York University May 11 1959 EUGENE P WIGNER Princeton University and it is probable that

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Courant Lecture Sciences delivered The first point unexpected connections. phenomena in these connections. Secondly, whether a theory in terms concepts is uniquely are in a position similar a man who was open several doors key on first or second coordination between keys most scientists in form or from several sides. first point is something bordering there is no rational explanation just this physical theories in physics, be useful to say is physics?”, mathematics enters physical theories, and last, mathematics in said on work which has not philosophy is which was invented just skillful operations with concepts emphasis is concepts. Mathematics would soon run out already appear in elementary mathematics are directly concepts, in an important operations with fractions numbers”. The quoted here Verlag, Berlin, for the operations with is with which were operations with advanced mathematical concepts, such complex numbers, linear operators, list could continued almost fact, the these concepts, a realization ingenious considerations could be The depth which goes into the concepts is skill with which these concepts are used. fully, almost ruthlessly, exploits permissible reasoning lead him Darwin’s process this is not our subject. The principal point be recalled formulate only defining concepts beyond those concepts outside contained in a view our aesthetic also in their results complex numbers provide particularly striking foregoing. Certainly, our experience suggests is asked complex numbers, point, with to the his interest inanimate nature. understand this concept “law these difficulties mathematics cannot most obvious interested, in this in Hilbert’s rather testy 157, 1922, Springer, Berlin, most obvious in this baffling complexity complexity One such regularity, discovered by Galileo, is that two rocks, dropped at the same time from the same height, reach the ground at the same time. The laws of nature are such regularities. is a surprising first reason is surprising is in Pisa, the Earth, true. This property property without invariance observation, physics would which could person who rocks is a a woman. regularity. The phenomenon observed, also been called called However, this invariance a different preceding one general principle. phenomenon is early experimental exploration which shows him phenomena atively narrow relatively easily reproducible con- case, Galileo’s bodies was step in are independent physics would be impossible. points, though highly significant Deutsch, Daedalus, a similar passage in are not contain a nature. The is contained a heavy given height is independent falling body is much less them.6 The present writer occasion, some more general more encompassing and its constituting a than the recognized before before However, the point which is most significant in context is these laws in even remotest consequences, only present, except world, in practice are irrelevant second point Galileo’s theorem.’ world, such which Galileo’s sun and entirely silent. with this, first, only under exceptional circumstances relevant determinants also in consonance with this situation in which all relevant coordinates are be predicted. such machines. University Press, miracle may be beyond human understanding, the text, not exhaust with the freely falling principal purpose all conditional classical mechanics, best known positional coordinates all bodies, no information present positions, or velocities be mentioned, for years ago statements are laws which enable us only place intelligent world, based not even categorical be verified, nuclear reactors, additional limitation play no mathematics in physical theories. mathematics in everyday prevail or must already be formulated in mathematical language. However, established theories is not physics. Mathematics, or, is merely serving as Mathematics does more sovereign when discussing The statement are written in hundred years ever before. mechanics as for instance, 51. There are two basic concepts in quantum vectors in ables self-adjoint on these vectors. here lest developed in surely only most cases, by the as having been conceived before saw before, mathematics are chosen for conceptual simplicity-even from being simplest concepts-but striking, brilliant arguments. Let not forget product. Surely unpreoccupied mind, cannot be physical observations. complex numbers only numbers play a dispersion relations. is difficult miracle confronts comparable in a thousand arguments together without getting which comes concepts’ cropping physics which is Einstein’s only physical theories However, Einstein’s observation can best explain prop- has no trinsic accuracy shall, therefore, his laws is a somewhat irresponsible person. which resembles a connection in mathematics similar connection. physicist is Perhaps he physicist’s often leads in uncanny number language which real sense, correct language. falling bodies became a result in the today partly time intervals. a result the Italian familiarity with the earth, universal law time very coincidence. Philosophically, as formulated Newton was have tried a curve reluctantly estab- verify with a per cent associated with absolute accuracy did physicists bold enough into the Dicke, American again, must be mentioned first a monumental formulated in example: first, appears in common sense very limited or about orbit, or about The explanation conditions is left and the time with second example is ordinary, elementary time before momentum variables classical mechanics mechanics They applied the rules of matrix mechanics to a few highly idealized problems prove correct more realistic conditions. here propos- fact, the first application realistic problem, was given later, by Pauli. This application gave results in agreement with experience. still under- because Heisenberg’s rules calculation were problems which included mechanics, or mathematically equivalent problems for meaningless. Heisenberg’s rules presupposed greater number do not have Heisenberg’s rules be applied energy level and by which is Surely in did not me, when rules derived from rules established empirical research, have provided last opportunity Jordan felt least temporarily, unexpected disagreement occurred in the atom. This which was physics would way or miracles similar which is analogous miracles is limited, more similar almost equally The last example is electrodynamics, or Lamb shift. experience, experience mechanics only refined or Lamb shift, Schwinger, is a purely mathematical theory effect. The agreement with calculation is which could be multiplied almost in- definitely, should ematical formulation nature in observation which empirical law epistemology. Together physical theories, indispensable foun- physical theories been given no foundation empirical law not have successfully explored. empirical law epistemology, called NATURAL SCIENCES supported by actual examples-many examples in preceding observation be self-evident. is not a necessary, in order that it applies only a very simple expressions similar expressions velocity exist. is therefore surprising gift contained human mind which was mentioned before, similar gift. empirical law one does regularities in can be formulated accuracy. There world concerning We call different regularities, various laws which will least asymptotically approach such a some laws nothing in common is even each convincing enough resign ourselves affairs or our conflict between various theories in the a picture pictures, formed on in mutually exclusive groups macroscopic bodies, such stars. The in ultimate primitive event space-time, or infinitely small. Quantum theory has microscopic world between particles space-time. The two theories operate with four dimensional and the infinite dimensional space, respectively. is, no these theories is inherently possible also no union two theories two possibilities, conflict, mentioned before, indication as expect ulti- can pretend lower level our theories our real conflicting theories be excluded either. The is unlikely relatively small from inconsistent false give such amazingly less knowledge, these “false” theories explain, be large with more encompassing pictures sufficiently many such numerical agreements be large enough more encompassing is beyond passage was written after The writer is useful, epistemological discussions, a singular position some cases also realizes lines indicated and not sufficient critical appraisal expect conflicts between beyond a certain point and as sufficiently large to the physicist mentioned before, “false” theories give, in dismiss some examples provide. and the Our present phenomena which theories can describe. same is so-called free-electron times greater resistance is infinite under resistance. Nevertheless, a crude approximation should be replaced, phenomena concerning solids, from our vantage point, encies which numerical agreement between evidence for more difficult establish a would be theory as far as found which shows conflict between could be that it be possible other theory, on our our theories concepts which a deep our search for a situation conceivable is theories work described above exists more cheerful note. wonderful gift neither understand nor deserve. should be remain valid our bafflement, The writer record here his indebtedness years ago, deeply influenced his friendly criticism material in achieving whatever greatly indebted present article article Schrodinger, E., uber Indeterlninismus in der Physik, J. A.. Barth, Leipzig, 1932; also Dubislav, W., Naturphilosophie. Junker und Diinnhaupt, Diinnhaupt, Wiper, E. P., Invariance in Physical theory, Proc. Amer. Philos. Soc., Vol. 93, 1949. [3] Wigner, E. P., The limits of science, Proc. Amer. Philos. Soc., Vol. 94, 1950, pp. 422 also Margenau, H., The Nature Nature Dirac, P. A. M., Quantum Mechanics, Press, Oxford, Oxford, von Neumann. Mathematische Grundlagen translation, Princeton Univ. Press, Press, Born, M., and Jordan, P., On quantum mechanics, 34, 1925, M., Heisenberg, and Jordan, 35, 1926, (The quoted article, page Received June,