By Alexander Fields

In 1960 physicist Eugene Wigner published a paper entitled, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Citing such timeless examples as Newton’s universal law of gravitation and Maxwell’s equations, he reflects on the mystery of how formally constructed mathematical models, when applied to certain observed experiences, often link up surprisingly well with a host of interconnected phenomena—in some cases even well beyond the scope of the original experiments! Unexpectedly, physical explanations end up falling quite naturally out of elegant mathematical schema. He writes, “The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious, and there is no rational explanation for it.” It does seem almost unfair how often we find confirmation for what Wigner calls the empirical law of epistemology, which he describes as: “the appropriateness and accuracy of mathematical formulation for the laws of nature…” Galileo, more than three hundred years prior, expressed similar sentiments in his 1623 publication, Il Saggiatore:



Although taken somewhat for granted by Galileo, this beautiful articulation describes exactly that which Eugene Wigner finds so inexplicable. While the physicist is ultimately unable to account for “unreasonable effectiveness,” he argues that we may nonetheless accept his empirical law of epistemology as an article of faith. The paper concludes,



Wigner despaired over ever having anything in the way of a satisfying answer to his question—such a pursuit can only find its resolution by becoming relegated to the domain of faith. We could never hope to discover the true origins and causes of our most mysterious and pristine science. Or… has it already been done?

In fact, Immanuel Kant, in his Critique of Pure Reason, boasts that he has solved this very mystery; and this of course preceded Wigner by almost 200 years. That formally constructed mathematical models, apart from any experience, can then be applied to nature is a rather bizarre fact of our experience, but for Kant, none of this “happenstance” is at all surprising, and on the contrary: it would be entirely expected! In the Prolegomena to any future Metaphysics, he comments:



That appearances are brought under our intuitions of space and time (the very source of mathematics) seems much more likely than positing a Platonic account—something in the way of an eternal and unchanging realm of pure mathematical objects, for example. And it would be strange to say that we read mathematics out of reality (since it would not then be an a priori endeavor). Furthermore: although there is a real sense in which mathematical truths seem to have their existence before we come across them, such an observation is perhaps better explained in Kant’s view than in the doctrine that they enjoy an existence independent of our thought. There is simply no getting away from this fact: mathematics, “must first exhibit its concept in intuition, and do so a priori … Without this mathematics cannot take a single step”. Just as when I look to the world to synthesize some new fact or predicate into my understanding of a subject, so also do I synthesize extra information from the mental object (coming from intuition) which I fix my gaze upon in any mathematical proposition. Set theory would not be possible without some a priori conceptualization of a collection of individuals, for example. And how could geometry proceed without its intuitions of space or extension?

Kant believed, as I do, that mathematical propositions do indeed contain synthetic information. Thus for a Kantian, it should be no surprise that, when we confront experience, we find mathematics cropping up in all kinds of natural phenomena. Nor should it be surprising that we judge these results beautiful. This effect is owed to the categories which structure our sensations; by this view we are not reading mathematics out of reality, but into it! In a sense, it is not the universe which is written in the language of mathematics, but our minds. There are, of course, valid objections to his argument. Perhaps the hardest blows dealt to Kant’s view of mathematics come from a group of philosophers known as the logical positivists. A.J. Ayer, in his book Language, Truth and Logic, is severely critical of Kant’s position on several points.

Most of Ayer’s concerns are defeated in light of Gödel’s incompleteness theorem (for brevity’s sake we will pass over them in silence), but there does remain one, which is the elephant-in-the-room, and springs from some surprising developments in mathematics during the 19th Century—here even the least judicious gaze must remain skeptical. Since the advent of non-Euclidean geometries, Kant’s description of space and time as sensible forms of intuition has been deemed an unlikely hypothesis:



However, ever since Bolyai and Lobachevski, it has become evident that geometry need not be confined to a study of physical space. In fact, once Riemann enters the picture, we can even define n-dimensional space (as many dimensions as you would like)! This is the principle objection to be dealt with if we want to maintain a defense of Kant’s idealism. Ayer argues:



This is a rather clever way of defending the position which denies the synthetic content of mathematical propositions. However, in turning to their explanations of how these purely analytical statements then become useful in physical application, I find the positivists unconvincing. I have motivation for raising doubts on two grounds. First, they have implied that certain mathematics only arbitrarily link up with natural phenomena, that is, “it just happens.” Second, the reason they judge this practice, where it occurs, to be so surprisingly effective, is because apparently, like mathematics, the book of the universe is written in the language of logic! And so the mystery essentially remains the same; it has only shifted shapes.

Furthermore, at least in my own reading of Kant, it seems that many of these objections have already been answered! Most importantly, I wish to draw attention to the fact that this last objection turns out to be more of a support than a stumbling block. This is not to say that Kant anticipated non-Euclidean geometry, but his transcendental idealism certainly does seem to allow for the possibility of mathematical constructions which make no appearance in our phenomenal representations. He writes, “Sensibility, the form of which is the basis of geometry, is that upon which the possibility of external appearance depends. Therefore these appearances can never contain anything but what geometry prescribes to them.” And so, our intuition sets forth the conditions which make up all possible experience; he does not here say it necessitates experience. That we have Lobachevski or Riemann describing wholly new worlds of geometry based on dissimilar intuitions does not mean we will necessarily find these surfaces or spaces in nature, any more so than we may ever come across a chiliagon. However, we do find applications for non-Euclidean geometries in the natural sciences—case in point: geometry done on the surface of a globe for example, or even better: General Relativity. Again, Kant would be delighted at this discovery! He explicitly says that geometry in fact prescribes the theory to experience, and not the other way around. So that we can conceptualize a 4-dimensional Riemann curved space means: it is possible we will encounter such a space in experience. And we do, it seems. In fact, we could never encounter a physical space which would not admit of some geometrical conceptualization—if we can experience the space then we can define it, and no sensible object could possibly present itself to us outside of a geometrically determined context.

So perhaps we should not be so quick to dismiss Kant’s ideality of space and time in lieu of non-Euclidean geometries after all. To s/he who wishes to present an alternative, Kant quite frankly offers only the following challenge in his Prolegomena to any future Metaphysics:



Good luck! Oh and before you attempt to answer him, I would like to remind you: writing the first Critique swallowed 10 years of Kant’s life.