Mathematical Knowledge

Mathematical Knowledge

We can all agree that human beings have mathematical knowledge. We understand mathematical concepts. But how extensive is mathematical knowledge—what subjects are mathematical? The OED defines mathematics as “the branch of science concerned with number, quantity, and space, either as abstract concepts or as applied to physics, engineering, and other subjects”. But the question is which other subjects—psychology, sociology, geology, geography, history, ethics? It is tempting to divide all learning (academic and non-academic) into two groups, mathematical and non-mathematical; but is this right? One subject you study in math class, the other subjects in other classes (music, literature, Spanish). I am going to argue that this distinction is artificial, misguided, and simple-minded; mathematics is everywhere. All (or nearly all) knowledge is mathematical to one degree or another. In fact, mathematical knowledge is cognitively fundamental. Elsewhere[1] I have argued that logic and ethics are (partly) mathematical; now I will extend that thesis more broadly. It turns out that this is not that difficult to do once we have shed certain prejudices and curricular conventions. I will quickly survey the whole field of knowledge to establish this claim; then I will turn to theoretical conclusions.

Psychology has a mathematical side; mathematics is applied in it. Psychophysics, statistical methods, computational models, laws of memory (recency and frequency laws), theories of learning, IQ tests, and so on. You can’t be a psychologist and be mathematically illiterate. Sociology is much the same. Geology is concerned with questions of depth, hardness, age, weight, and molecular composition. Geography deals with land masses, distance, size, height, and so on. Biology involves morphology, genetic structure, population dynamics, cellular forms, and energy consumption. History records dates and times, movements of populations, rates of social change, distribution of political power. Economics states laws of supply and demand, discusses money, tracks financial markets. Poetry is concerned with meter. Literature may include word counts. Music is mathematical. Space and time are mathematically described and they condition everything. Both in class and in the street the knowing mind is working mathematically. The case of language is of special interest, because it has taken recent linguistics to recognize the mathematical character of linguistic competence: a natural language is a combinatorial capacity to operate on discrete symbols to produce infinitely many sentences; Chomsky’s Merge operation is set-theoretic and recursive. Grammar is basically mathematical (“computational”). There isn’t much that doesn’t have a mathematical side, whether salient or submerged. It isn’t just math proper along with physics and engineering; mathematics seeps into everything (astronomy, botany, cookery, art). Mathematical knowledge is ubiquitous.

This has a bearing on certain philosophical questions. First, mathematical and non-mathematical knowledge are not discrete states of mind but interwoven. The a posteriori is infected by the a priori—what we call applied math. Second, the problem of mathematical knowledge is not limited to pure math, or school math; it applies to all knowledge, give or take a bit. The farmer has this problem as much as the professional mathematician. Third, Quine’s indispensability thesis applies not just to physics but to every branch of science and beyond. Fourth, philosophy of mind (including cognitive science) must take account of the prevalence of mathematics in the human (and animal) mind. The mind is not just computational but also numerical. The brain too—neuroscientists must reckon with the mathematical brain. The mind-brain not only processes information; it calculates, does sums. This doesn’t fit classic empiricism in which perceiving is the fundamental operation of mind, unsullied by such rational faculties as mathematics. If mathematics in innately known, then all knowledge has elements of the innate in it. Epistemology also must acknowledge that the foundations of so-called empirical knowledge cannot be innocent of non-sensory contents. Math is foundational too. Empirical knowledge is a mixture of the purely sensory and the abstractly mathematical. Such knowledge is a fusion of the qualitative and the quantitative. Even colors need numbers (they have extent and intensity).

We are edging towards the doctrine that all knowledge is mathematical, necessarily so. This is a strong doctrine, but it has its appeal. Consider intentionality: all mental states are directed at some object or other—some discrete, distinguishable, countable object. Typically, we have multiple intentional objects before our minds simultaneously, each comparable with the others—larger, smaller, heavier, lighter, louder, quieter, closer, further away. These are all quantifiable relations. The visual field alone is replete with mathematical content (mainly geometry). Some of this mathematical complexity is imposed by the mind, which is mathematically pre-established. Thus, where there is intentionality there is mathematical content (quantity, number); and all knowledge presupposes intentionality. The idea of a spatiotemporal world consisting of discrete particulars builds in mathematical structure, and it is omnipresent. Plus, the very notion of plurality is implicitly mathematical: it is the idea of a set. Not for nothing did Frege liken predication to the function-argument structure in mathematics; it involves the idea of one thing being a member of a set of things. Set-theoretic thinking is basic to human cognition (perhaps all cognition). We are beginning to seem like natural-born mathematicians, at home with abstractions. Born to be mathematical (and wild). Baby, I was born this way (and in other ways too).

This prompts two further thoughts: is it the same with other animals, and does it also apply to perception. I think yes in both cases, though human knowledge no doubt has additional properties. Animals size each other up, travel great distances, compute the time from the position of the sun, estimate the reproductive fitness of potential mates, and so on. In these activities they demonstrate basic mathematical skills, make quantitative comparisons, count objects (up to a point).  They are not mathematical ignoramuses. I suspect bees are pretty math-savvy. Survival depends on getting your numbers right. Even worms need to estimate how deeply to bury themselves (see Darwin on worms). Number is part of Life on Earth, part of our biological heritage, embedded in our genes (the mathematical gene). The lioness must count the number of her cubs and estimate how much food they need. In the case of perception, it is eminently plausible to suppose that the eyes perform calculations to gauge distance and produce impressions of depth. People write books called things like “The Intelligent Eye”; someone could write one called “The Mathematical Eye”. The senses are calculating prodigies in the production of perceptual constancies, computing size (say) from perceived distance. So, mathematics is biologically primitive and no doubt ancient; number sense may first have evolved with the most primitive of organisms—but then went on to perform spectacular feats of mathematical cognition, culminating in Isaac Newton and the great mathematicians of history. Mathematics (the faculty) evolves. It has a natural history.

Philosophers are fond of the question of what is ontologically basic and universal. Some say sense-data, others material objects in space and time. You don’t have to be Plato or Pythagoras to see the universality of number: everything real is countable and hence subject to arithmetic. Kant located material objects in space and time, and these are quantifiable things; mathematics is part of the spatiotemporal manifold. And sense impressions can also be counted and measured (e.g., for vividness). Yet people seem reluctant to make numbers ontologically basic (the opposite in fact), probably because they are not “concrete”. In our conceptual scheme mathematics occupies a unique place as a kind of universal language or framework or background. It needs more respect; more room at the epistemological table. In any “descriptive metaphysics” it should be accorded a central place, because it is all-pervasive, an aspect of the lens through which we perceive reality. Sense, reference, and number: for reference depends on singling out one thing from other things, and that involves the deployment of concepts of identity, difference, and plurality—the very basis of number science. We refer to what we can count. If you imagine a mind transitioning from a blurred world to a differentiated world, thus achieving individuated reference, then you will see that mathematical concepts come into play when reference comes on the scene. We might speak of the “mathematical theory” of reference—the embedding of reference in a matrix of mathematical ideas. A space of points, a plurality of particulars. Referential competence presupposes mathematical competence (as linguistic competence in general does). Not school arithmetic, to be sure, but its cognitive underpinnings (quantifiers, sortal concepts, etc.).

It strikes me as not surprising that mathematics was the first science to develop: it is the most salient characteristic of the scientific mind, and the most rigorous. A species bad at arithmetic will not last long. It’s a pity, then, that in its modern academic incarnation it is so forbidding and inaccessible as a field of study (it should be easy!). We are born to be mathematical, as we are born to be linguistic, but the knowledge is not easily articulated or systematized. It is more of a practical skill than conscious theoretical knowledge. Yet everything mathematical seems to be in us from the start (as Plato taught). Of course, this knowledge is philosophically puzzling, even mysterious, but there it is, bubbling beneath the surface. Moreover, it is good knowledge to have (as Plato also taught); it isn’t epistemic junk (a type of gossip). You can be proud of your mathematical knowledge. Fortunately, the human mind has evolved to be in tune with mathematics. Evolution has produced something both useful and admirable. If God existed, he would be a mathematician. In the beginning was the number.[2]

[1] See my “Mathematical Ethics” and “Mathematical Logic”.

[2] Not the deed, not the sense-datum, not the material object, not the process, not the thought—but the number. For all the above are numerable. We have the finite and the infinite, the natural, the negative, the prime, the real, the imaginary, the number zero. You can’t get away from them. You can’t eliminate them from your conceptual scheme. Counting is part of human nature.