It is not at all hard to enter a mathematician’s mind without being a psychologist. This is because all that individual does in front of a piece of paper – observing, finding patterns and regularities, abstracting, describing, inducing and deducing, generalizing, applying and looking for consistency, but also for symmetry, elegance and beauty (call these latter aesthetics) – we regularly do in our daily life. Whatever mathematics is (science, method, formal language or logical symbolism), we do mathematics without being mathematicians. Of course, our involvement depends to a certain degree on our ability and willingness, which relates to education and experience, but the mental processes underlying this activity are the same for all of us. Then, to find where to draw a kind of “psychological” line, what essentially differentiates the mathematician from the non-mathematician?
Contemporary pioneering studies in what is called perceptual mathematics [see, for instance, that of (Teissier, 2005), (Ye, 2010), and (Mujundar & Singh, 2015)] came to shape an interdisciplinary cognitive theory that claims that all mathematics is human, resides in the mind, and is not an external product of the mind. The human mind is endowed with innate primordial perceptions such as spatial (metric, linearity), numerical, and topological (proximity, relational structures), reflected by the common empirical concepts such as distance, motion, change, flow of time, and matter. It is further hypothesized that animals also hold such perceptions. Thus, the concepts of mathematics are not platonic, but are built in the brain from these primordial perceptions, and brain neurophysiology gives rise to the extremely precise and logical language of mathematics.
Giving credit to this theory, the question then arises: If all of the vastness of mathematics, with its beautiful theorems and interconnected concepts from various separate theories and branches, is innate and resides in the mind, how is it that it is not an at-hand resource to which we can appeal whenever we need whether we are mathematicians or not? How is it that mathematicians struggled for dozens of years to prove such-and-such a conjecture? One answer comes from evolutionary biology, which says that the energy- or material-saving feature characterizing evolution in general also applies to the brain. Moreover, the human brain has developed a special energy-saving strategy for its more than 86 billion neurons, more intense and targeted that any other part of the human organism or other species. In 2010, some Yale researchers claimed to have deciphered the way brain conducts its energy-efficiency plan. In brief, the work is carried out by the inhibitory neurons, yielding an effect that has been called the “iceberg phenomenon”, where the type of the iceberg is the only essential information that needs to be processed because the submerged iceberg is information that can be suppressed by the inhibitory neurons. These neurons dictate how much of the iceberg we actually see, and thus they act like a regulator that keeps the brain running efficiently (Haider et al., 2010).
This answer is supported – surprisingly – in the philosophy of mathematics, where nominalist Jody Azzouni (2000) argued for what he called ‘implicational opacity’, namely, our inability to recognize implications and consequences of various mathematical statements before a proof is offered. (He also gives a thought-provoking definition of mathematics as being the discipline that deals with “interesting” non-obvious things). He invoked this psychological fact to argue that explaining the so-called ‘miraculous’ success of applied mathematics (what in philosophy of science and mathematics is known as Wigner’s puzzle) is not a genuine problem; that is, if we had such ability, the so-called surprise of the success of applied mathematics would vanish. (Knowing the brain is equipped by evolution with that ‘iceberg regulator’, Azzouni’s ‘implicational opacity’ seems more an ability than an inability.)
Bringing applied mathematics into this philosophical setup will shed some light on our discussion of “non-mathematician mathematicians”. In its earliest stage – from the ancient Greeks to the Babylonian “empirical” mathematics, which are the earliest available writings – pure mathematics assumed a mathematical creation of concepts that reflected the surrounding reality. This is what Euclidian geometry, arithmetic and early algebra are about – our perceived physical universe, described in concepts and a language that allow first- and second-order logic to operate with the mathematical propositions in order to reach unseen deductions. The results of such mathematical deductions were and are all empirically confirmable. We can say that early pure mathematics is implicitly an applied mathematics, as the empirical concepts are constitutive for the axioms of the former.
With the development of sciences, especially physics, pure mathematics extended its mission and roles. Once mathematicians and physicists ascertained the descriptive power of mathematics, they used it effectively in the constitution and development of the sciences. Starting with Newton’s and Galileo’s physics, science used the concepts, tools and language of mathematics to formulate the laws of nature and infer its properties and behavior. All of this methodology was legitimated through three major principles: first, the descriptive role and empirical success of early classical mathematics; second, the rigorousness of mathematical language; and third, the epistemic status of the necessary truths of mathematics. This practice has led us to the current physics, in which the content of some theories is entirely mathematical, so that it has become difficult to say whether those theories belong to physics, applied mathematics or pure mathematics.
The history of application of mathematics in the sciences also has a “mysterious” element. Being driven by their natural impulses of inquiring and generalizing, but also following some special criteria of beauty, symmetry and elegance specific to the mathematical creation, mathematicians began to develop new, more complex concepts starting from the classical ones. They did this for several possible reasons: for pragmatic needs (to make the concepts consistent with other theories, or to improve the way concepts reflected reality), for unification of theories, for implementing the concepts into a particular scientific theory, for solving a particular scientific or applied-mathematics problem, or simply as an intellectual game. Having at hand a set of axioms that define a concept reflecting reality and “playing” with those axioms to create a more complex concept leads eventually to a new concept which loses contact with reality in the sense that it has no empirical interpretation or immediate application. Well, the history of science has proved that whatever complex concepts mathematicians created, they finally came to be applicable in the mathematics of physics or even to directly describe an empirical context. Take for instance the classical example of complex numbers. Created in the 16th century just to contribute to the fundamental theorem of algebra by allowing every polynomial equation to have a solution (say, through criteria of symmetry and elegance), complex numbers did not show any empirical interpretation and did not express any magnitude. However, they came to be essential for various theories in physics and technology, such as fluid dynamics, quantum mechanics, relativity, electromagnetism, control theory, signal analysis and many others. Take also the classical example of Minkowski geometry, which has nothing to do with our immediate spatial perception, on the basis of which General Relativity was developed and provided us with the beautiful relativistic cosmology that was empirically confirmed. Many such mathematical concepts were not immediately applied. History of science is replete with examples of mathematical concepts or theories which had to wait several decades before coming to be applied effectively.
Then, philosophers of science asked: Why have mathematics and its abstract entities proved so effective in empirical applications? How it is that mathematical concepts created with no empirical influence come to be applicable in physics or discoverable in nature? Or how is it possible that concepts created from aesthetic reasons are applicable in exact sciences? Or, as mathematician-philosopher Mark Steiner rhetorically asked: “How does the mathematician – closer to the artist than to the explorer – by turning away from nature, arrive at its most appropriate descriptions?” (Steiner, 1995).
These questions are the object of what is called Wigner’s puzzle (1960); he labeled their premises as “the unreasonable effectiveness of mathematics”. In other words, we as philosophers still doubt a bit that Galileo was right when saying that the great book of nature is written in the language of mathematics, and now we are told that we can recreate the unseen content of this book by ourselves as mathematicians, from our table, merely with our minds and not from observing out there in nature! This requires an explanation (even though some philosophers of science, especially nominalists, say it doesn’t – that is, Wigner’s puzzle is not a genuine problem), at least for the reason that the mathematical method should be entirely legitimated as the main method of scientific investigation.
Despite its “mysterious” applicability in science, mathematics became more and more applicative in both its results and the motivations for its development. Indeed, mathematics has perpetually fed from the problems of sciences, has created new concepts and theories just for solving these problems, and thus has contributed to the advances of those sciences; mathematics has also developed itself and strengthened its ability to solve future problems. There is a mutual exchange of a metabolic type between mathematics and sciences that has been acknowledged even before Wigner’s philosophical concern. In 1950, Nicholas Bourbaki, who shaped the concept of mathematical structure and types of structures from a set-theoretic perpective, asked in his influential The Architecture of Mathematics whether the unity of mathematics is the outcome of formal logic or simply this scientific fertility. In other words, the unity of mathematics is not one of an inert structural skeleton, but one of a more complex organism in evolution with the scientific environment that influences the organism through mutual exchanges. Are mathematical structures abstract, inert forms, or do they have a certain “life” consigned by their applicability? Can we somehow have the certitude of this applicability in the future course of the evolution of science? Nowadays, these questions have been reformulated, deconstructed, and refined within the problems of philosophy of applicability of mathematics; however, mathematics has continued to play its roles better and better regardless of the answers of philosophers.
The sciences have become mathematized not only in their methodology, but also in their conceptual framework and theoretical content. Mathematics has reached domains where we have never imagined that it could be applied – for instance, in biological sciences, where cell life, tumor growth, body fluid dynamics, neurophysiology and many other processes and phenomena from narrow fields are described and investigated through precise mathematical models. These achievements are the result of that perpetual applicability and fertility of mathematics.
This is the briefest possible story of how pure mathematics became applied mathematics and implicitly how pure mathematicians became applied ones. Coming back to our initial question of what essentially differentiates the mathematician from the non-mathematician (given the mental-biological nature of mathematics), let us bring again into discussion Wigner’s puzzle and observe meanwhile that ordinary people are applied mathematicians rather than pure mathematicians. Indeed, we are educated from early school that mathematics is applicable and confirmable empirically. The fact that two apples plus two apples make four apples and that we were given examples of physical objects was enough for us to establish the relation between pure mathematics and the physical universe without entering the subtleties of this relation. Then, mathematical thinking and simple successful applications from daily life made us applied mathematicians without our even being aware of Wigner’s puzzle. Don’t worry, most professional, experienced applied mathematicians are not aware of it, either. And even if they were, nothing would change in their behavior and work. Their work is both creative and technical, following a precise applicative goal under the conviction that whatever the final results, their mathematical truth will be enough for legitimating them and ensuring their empirical success. There is no speculation in their work; they rely on mathematics and don’t waste time with its philosophy. Things change a bit with pure mathematicians, who are more inclined toward its philosophy. The aesthetic criteria of the mathematical creation raise philosophical questions in their minds, as well as whether their concepts will find applicability somewhere. They somehow recreate Wigner’s puzzle even if they are not aware of it, and they have a special philosophical-mathematical sense of “why is this and not that, why this way and not that way, what would be if…” – call this the sense of the “meta”; they will always think in terms of a metatheory when dealing with a theory. Neither ordinary people nor many applied mathematicians have such a sense. In brief, I think philosophy is what differentiates mathematicians from non-mathematicians and applied mathematicians. The former are implicitly philosophers, even if they do not practice philosophy as a profession. There are numerous examples of mathematician-philosophers, starting with ancients Plato and Aristotle, then Descartes, Leibniz and Laplace in the middle age, Gödel and Russell in the first half of the 20th century, just to mention the best known. In contemporary times, several mathematicians are also philosophers. Some of them have dealt with Wigner’s puzzle and some of them have actually made a professional shift from mathematics to philosophy.
A great example is René Thom, a recipient of the Fields Medal in 1958, the author of theory of catastrophes, which has a wide range of applications in Earth physics, ray optics, biology, genetics and many other fields. He also investigated Wigner’s puzzle from the standpoint of the applicability of its own theory. In a lecture held in 1991 called Leaving Mathematics for Philosophy, Thom described how, by creating, as a pure mathematician, the catastrophe theory, his focus was moved unavoidably to the philosophical aspects of this theory and how this experience self-revealed him as a philosopher of science. In detailing a comparison between the profiles of a mathematician and a philosopher, he says:
“This might seem strange to mathematicians, but I will say that if I had the choice between an error which has an organizing power of reality (this could exist) and a truth which is isolated and meaningless in itself, I would choose the error and not the truth.” (Thom, 1991, p.11)
This is just a spark of the description of what I called earlier the “meta” sense of the mathematician-philosopher, which may be able to either organize or sacrifice a given epistemic coherence with the power and authority of mathematical thinking.
Distinguishing in this way between the pure mathematician and the applied one by no means exhibits any hierarchy between them, nor gives higher merits to one category with respect to contemporary achievements of science. In fact, I think that as a species, we should be more grateful to the applied mathematicians. Being problem-solving and application addicted, they have contributed to the rapid technological progress that makes our lives better. The unexpected negative effects of technological progress, were not, of course, their responsibility, nor was it that of scientists or engineers. However, applied mathematicians are present with mathematical models and solutions in the front line of any battle concerning issues of Earth or humanity, and I am sure they do this out of duty rather than pleasure. Take for instance ice melting due to global warming. Mathematicians recently created a partial differential equations-based model with tuned stochastic spatial heterogeneity to estimate the sea ice concentration in the polar regions unobserved by satellites (Strong & Golden, 2017). There is ongoing interdisciplinary research to improve this model and to develop other methods to fill the critical data gap that will impact our understanding of Earth’s changing climate. I take these kinds of applications to be the most beautiful mathematical applications ever, more beautiful than those performed to take the man to the Moon!
As Mark Steiner (1998, p. 55-58) argued, both mathematics and applied mathematics have a strong anthropocentric character, so it is not surprising at all that applied mathematics might be goal- and user-dependent, as some philosophers have claimed. In the light of this anthropocentrism, this dependence will, of course, relate to the survival of our species. Applied mathematicians will always be near the scientists whenever a global issue threatens us; they will help in a broad range of applications, from predicting the spread of an epidemic or computing the trajectory of an asteroid to planning the voyages for terraforming other planets, if necessary. In fact, even if we eventually live on another planet, we will continue to ask ourselves: If humans are all applied mathematicians, and God is a pure mathematician (to paraphrase a former bestseller’s title), why is God’s mathematics human? Does this question qualify us as applied-mathematician-philosophers? Of course it does, and this is a good thing. After all, the new scientific era we are entering is interdisciplinary.
Azzouni, J., 2000. Applying Mathematics: An Attempt to Design a Philosophical Problem. The Monist, 83, pp. 209-227.
Haider B., Krause M. R., Duque A., Yu Y., Touryan J., Mazer J. A., Cormick, D. A., 2010. Synaptic and Network Mechanisms of Sparse and Reliable Visual Cortical Activity during Nonclassical Receptive Field Stimulation. Neuron, 65(1), pp. 107-121 DOI: 10.1016/j.neuron.2009.12.005
Bourbaki, N., 1950. The Architecture of Mathematics. The American Mathematical Monthly, 57(4), pp. 221-232.
Mujumdar, A.G. and Singh, T., 2016. Cognitive science and the connection between physics and mathematics. In: Trick or Truth?, Springer, Cham, pp. 201-217.
Steiner, M., 1995. The Applicabilities of Mathematics. Philosophia Mathematica, 3(3), pp. 129-156.
Steiner, M., 1998. The Applicability of Mathematics as a Philosophical Problem. Cambridge, MA: Harvard University Press.
Strong, C. and Golden, K. M., 2017. Filling the Sea Ice Data Gap with Harmonic Functions: A Mathematical Model for the Sea Ice Concentration Field in Regions Unobserved by Satellites. SIAM News, 50(3).
Teissier, B., 2005. Protomathematics, perception and the meaning of mathematical objects. In: Images and Reasoning, P. Grialou, G. Longo andM. Okada (Eds.), Tokyo: Keio University.
Thom, R., 1992. Leaving mathematics for philosophy. In: Mathematical Research Today and Tomorrow. Springer, Berlin, Heidelberg, pp. 1-12.
Wigner, E. P., 1960. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications on Pure and Applied Mathematics, 13(1), pp. 1-14.
Ye, F., 2010. The Applicability of Mathematics as a Scientific and a Logical Problem. Philosophia Mathematica, 18(2), pp. 144-165.