The Efficacity and Identity of Mathematics

Eugen Wigner left us with a formulation that continues to impress: in a brief article, he talks about the “unreasonable efficacity of mathematics in the natural sciences”.

Wigner is amazed that since Galileo, we have so often seen that mathematical knowledge is useful in physics, and that the most improbable mathematical conceptions find application in so many aspects of the dominant science.

But does it then go without saying that the success of mathematics is “unreasonable”? There is at least one perspective that seems to preclude such a notion: the view numbers – the object of mathematics – are the essence of all reality. If this is the case, then indeed we should not be surprised. To master mathematical objects is to master the stuff of which everything real is made. It follows, then, that mathematical theories speak the truth about the logic of the world.

Now, this idea that mathematics is by rights the purified theory of the essence of being, that mathematical harmony is what binds together the complexity of reality, is consistent with a strong tradition known to philosophers.

Perhaps what speaks to us most in this sense today is the Pythagorean concept of whole numbers, their distribution and structure, the secret of the world’s harmony and its musical expression. But this conception appears in philosophy in different guises.

A significant moment in this regard is Plato’s Timaeus dialogue, which derives the world from elementary solids in which the right-angled triangles that form half of an equilateral triangle seem to be privileged.

From there, we can move to the theological translation of these ideas: that the mathematical entities structuring the world are the mark of its sublime creator. The Pythagorean vision is thus explained as follows: if reality is mathematical, it is because the divine power to whom we owe it is mathematically inspired. Even Leibniz’s God, in the end, seems to have chosen our world from among all possible worlds in the name of mathematical considerations of optimality.

There is, then, a strong and traditional way not to be surprised by the efficacity of mathematics in the natural sciences: it is consistent with a divine quality of being that we recognise in our science insofar as in doing maths we are raising our lives up to the level of the divine. From this perspective, Wigner’s observation invites us to move to a metaphysical or theological level and to leave behind our surprise in order to ward off the incomprehensible.

Everything depends on our definition of mathematics, its world, and its objects. To share Wigner’s surprise, and to see the efficacity of mathematics in the natural sciences as “unreasonable”, we must, I think, feel mathematics and its objects as typically “ours”, on “our side”, expressing our human subjectivity, characteristic of our mental and symbolic habits.

So, we arrive at the question what exactly is mathematics?

The answer is not simple, and the difficulty has several aspects. A quasi-political aspect is that mathematics never stops reinterpreting its past so as to reattach itself to it, by retaining the traditional assertions. What is mathematical is what mathematics itself sees as its experience in the aftermath. We might think here of the rewriting of history by a triumphant Stalinism: once Beria had been identified as a spy and a traitor, his whole history can be rewritten teleologically, as one that was leading from the start to that destiny. Similarly, mathematics reappropriates the old truth and makes of it a special case for its new assertions. This practice confers an exceptional degree of unity and strength of identity. We can offer the simple and telling example of Pythagoras’ theorum, which we read today as the classic formulation of a truth that is articulated in any Euclidian space (roughly speaking, <x ;y>=0 if and only if ||x+y||²=||x||²+||y||² ).

This example draws our attention to the question of the mathematical object. If mathematics maintains an identity throughout history, it does so nevertheless by forever changing the face of the object: the vectors of a Euclidean space were not known to Euclid (despite the apparent paradox in that statement).

The most ordinary vision links the identity of mathematics with the theme of the quantitative: mathematics is essentially the discipline of counting and calculating. From the start though, the geometry half protests this view: mathematics is also thinking about figures and demonstrating truths about them.

Since Cantor and Dedekind, we have been used to seeing the whole as the object par excellence of mathematics: the discipline is interested in how collections of entities are arranged and combined, without making any assumptions about the entities in question. We are quite naturally led to the Bourbaki group’s view, which describes mathematics as the study of the most general structures and the classifying of them into three main types (the algebraic type, the topological type, and the ordered structures type).

In the end, this vision imposes a qualitative conception: mathematics brings to light the specific qualities of various structures. It also suggests the conceptual dimension of mathematics: according to the Bourbakist programme, we often understand that the truths that for us emerged from an opaque calculation, are in fact properties that simple reasoning shows belong necessarily to concepts embedded within a network of concepts.

To see mathematics as a science of structures encourages us, it seems to me, to understand it as ours. Building structures upon the structures that precede us, guide us and accompany us in our lives – is that not precisely our fundamental practice as symbolic animals? Structuralism has enhanced this view by describing the human plane as one on which we share symbols which enter into relationships and constitute for us a normative framework. It also sees language, the milieu of human endeavour, as the greatest proof of the symbolic and systematic character of all that concerns us. But doesn’t understanding humanity as originally embedded in the symbolic fabric and the systematicity of that fabric imply interpreting mathematics – as a science of structures – as the science par excellence of our symbolic powers, the a priori science of our symbolic games? One that is interested in their various ways of giving rise to complexity?

Thus, mathematics would be the a priori universal doctrine of articulation, but we do not know whether it is an articulation of objects, language signs, or concepts. The formalisation of mathematics, which apparently reduces them to a writing game and where we produce theorems according to the rule of symbolic manipulation of the formal derivative, seems in any case to confirm such a view.

A similar vision of mathematics comes to us from the philosophical tradition, in a Kantian critique. Kant in effect associates mathematics with the intuitive dimension, but at the same time revolutionises the concept of the intuitive in two ways.

On the one hand, he invents the notion of pure intuition: intuition in a form to which all phenomenal intuition must conform. We “live” a pure intuition of space, meaning that we put all phenomena of an external object into space, which we carry as a framework. We also conceive of that space as the object of a priori necessary mathematical knowledge (that is, geometry). Thus mathematics is our way of controlling the intuited strangeness – preceding any particular intuition – that is space.

On the other hand, he proposes a kind of imaginative intuition, which produces “schemes”. Here, he interprets numbers in arithmetic as our imaginative production of a translation of logical relations expressing quantity in our judgements. According to this view, we invent the erasure of the unit that constitutes the general integer, as a temporal scenario (like the famous pom pom pom pom in Beethoven’s Fifth Symphony) “transposing” our logical thought about unity, particularity, and the totality.

Hence, at one and the same time, geometry and arithmetic are linked to an original power of our intuition or our own faculty of intuitive configuration. Mathematics is thus ours, rather than theological or metaphysical.

The Kantian concept does not really answer Wigner’s question about the unreasonableness of the efficacity of mathematics. Its key contribution is, rather, to justify the question itself, through making a subjective reading of mathematics plausible by presenting it as our symptom, our typical expression; “human, all too human”, we might say.

Indeed, the Kantian philosophy of knowledge says no more than that what we call knowledge – or more precisely, science – cannot consist of anything but the a priori control of the phenomena that make up our world (science must be predictive). As such knowledge necessarily entails inscribing phenomena in space and time, subject a priori to our mathematical reading: always already underpinned by mathematical knowledge already proven to be necessary. Science is necessarily infiltrated by mathematics simply because we would not call science any discourse that does not try to reconstruct the world mathematically. The Kantian approach does not explain the success of knowledge but rather tries to establish that even before any success is registered, we need the mathematical form of science because it expresses our prejudices about being, and our way of anticipating it.

What might an “answer” to Wigner’s question be? To explain “why” mathematics are efficacious for deciphering the real, we need to hypothesise about the ultimate structure of reality and its relationship to the mathematical object indicated by said structure. That is, we must venture into the realm of the metaphysical. Here, the theological reference has the virtue of putting an end to further debate by nominating an original instance that has no need of explanation: at this moment, we assign the genesis of the real at the same time as the development of mathematical ideals.

The Kantian position is a refusal to respond to the question associated with Wigner’s remark: from a Kantian viewpoint, any response takes us beyond all real knowledge, to where we are seduced by metaphysics or the powerful charm of theology.

If it is true, as we said at the beginning, that we must presuppose a “subjective” view of mathematics in order to find, as Wigner does, the success of mathematics in the natural sciences unreasonable, then the crucial point seems to be the accurate perception of mathematics as a traditional human activity.

One way of looking at mathematics is to take into consideration the kind of organisational form that it has taken since the beginning: that of the school. Mathematics is something that we pass to successive generations, through the institution of the school. Already in Plato, we find evocations of mathematical pedagogy, for example in The Meno, Socrates teaches a young slave the effect of doubling the side of a square. David Lachterman, in his historic The Ethics of Geometry, insists on the importance of the fundamental structure of the master and the slave in geometric thought among the ancient Greeks1 . It suffices, in fact, to have experienced the pedagogic situation in order to feel how fundamental it is, how the essence of mathematics passes through it. Husserl said that in the case of the transmission of geometric content from the master to the pupil, the latter comes to imagine the thought process of the teacher and reproduces it in himself2 .

Such an event is always associated with the notion of ex-cathedra teaching. When knowledge is dispensed by the teacher, the pupil is likely to receive it perfectly (especially if the teacher has managed to give translucid form to her explanation): in this hypothesis, he instantly becomes the equal of the teacher and is able to correct her if she deviates from what he, the pupil, has understood fully. All who believe in ex-cathedra teaching know that such an experience is just waiting around the corner for them. The masterly dispensation of knowledge can be equalising, even in principle. Moreover, this equalisation is dynamic, it spreads through educational channels: having understood, the pupil becomes the teacher of other pupils, whom he makes his equals, and so on. School, originally conceived to remedy a situation of relegation due to ignorance, is virtually equalising without limits in the ex-cathedra mode.

This does not mean that mathematics does not sometimes constitute a fortress that many see as threatening. Earlier, we described mathematical development as a kind of Stalinism, constantly reinterpreting its past so that it matches what is called the truth in the present. Beyond this, and in line with the fact that mathematics develops by self-confirmation rather than by being surpassed, as physics develops (e.g. Einstein and Dirac refuting Newton), it is hard to deny that in some way, mathematics is difficult to access: one must play its game in order to enter its discourse and hear its message, turn towards objects completely different from ordinary objects, and adopt a language that is not the common language (an aspect that the formalist revolution highlighted). Briefly, to access mathematics, one must go to its school, there is no alternative. Maybe, as I have just said, the school is, in principle and in intention, the egalitarian ex-cathedra school. Nevertheless, at the first level, it repels: it is enough, from the outside, to see how demanding it is to feel repelled.

And yet, we must not use this view of the separate/separating construction of mathematics to associate it with an image of mastery, to see mathematics as the discourse par excellence in which control and power are displayed and exercised. In the end, we can hear Wigner’s “unreasonable efficacity” in this way: mathematics would be the tool of mastery par excellence, and its instrumentalisation in the service of physics would be the secret of its phenomenal material power. In reality, some philosophies – ceding to an “infinite thrill” – offer us such an image of mathematics: making itself the master of infinity, mathematics has already taken over the world. A bit like, in Lovecraft’s writings, he who possesses the secret of a 40^th degree equation is virtually the master of everything.

Yet there are two possible objections.

First, we can raise again the Kantian view: for critical philosophy, when mathematics is involved in the sciences, it is to provide its structures as interpretations of space that host phenomena. That is, mathematics is linked to the “reduction to phenomena”, insofar as we accept knowing the real not as something in-itself (having a substance beyond and independent of us, subsistence), but as signalling itself to us by its phenomena. Thus the scientific penetration of being is limited in principle: no matter what its profundity and systematicity, it is linked to a mathematical reinvention of the world, inspired by the mathematical establishment of the parameters of this world. It never reaches the limits of the in-itself, said by critical philosophy to be radically out of reach.

On the other hand, there is an opening within the internal workings of mathematics, created by the discourse itself. In effect, mathematics has its ways of forever broadening the horizon of the problem, of inventing anew the field of what is to be known or interrogated. It does this by redefining the object that we are addressing (the number, the figure, the totality, the category), by reconfiguring the branches that form it (arithmetic versus geometry, algebra versus analysis, and more recently, topological structures versus algebraic structures versus ordered structures), by redefining the problematic places (algebraic geometry, algebraic topology, arithmetic geometry, dynamic systems). It also does so by interrogating all the regularities and irregularities that appear in the overall architecture of the mathematical object.

Mathematics is not content with the perfect coverage of anything made by a calculation or a dictionary: it goes looking for the object and the landscape that lie beyond our grasp and continues to create problems for us. A major and far from anodyne example is that in the 1930s, mathematics questioned the old notion of calculation, revealing that we never knew exactly what we meant by such a word. Several and varied answers to the calculation question were proposed, from which was established, nevertheless, that the convergence on a certain sense (I am thinking here of the three classic notions of calculability: the one that involves the notion of recursive function, the one that refers to the lambda calculus, and the one that is formulated using the Turing machine concept). We might say that in this case, mathematics set in motion a mass of certainty, whereas common, albeit philosophical, thinking was revealed to be incapable of identifying an issue.

Mathematics is thus an eminent part of what we might call, following Hegel, the Unruhigkeit of thought, its faculty of never resting on itself (its worry, if we translate from the German). That is to say that we must look at mathematics as an integral part of culture, as one of the places where humanity accumulates treasures in whose folds lie the wherewithal to revive our curiosity and exaltation indefinitely.

This text is the written version of my contribution to the programme on mathematics organised by France Culture, in which I was greatly stimulated by my colleague Jean-Jacques Szczeciniarz.

• 1Cf. Lachterman, D.R., The Ethics of Geometry, London, Routledge, 1989
• 2Cf. L’origine de la géométrie, trad. Jacques Derrida, Paris, PUF, 1974, p. 185