When professional mathematicians describe the moment a difficult proof finally falls into place, the language they reach for is, on the available evidence, almost always closer to discovery than to invention. They describe the proof as having “come to them.” They describe themselves as “finding” the solution. They describe the result, once it has emerged, as having “always been there.” The vocabulary is the vocabulary of someone uncovering something that existed before they began looking, rather than the vocabulary of someone constructing something that did not previously exist.
The consistency of this framing, across cultures, centuries, and different mathematical traditions, has been remarkable enough that it has driven philosophers of mathematics, for nearly all of the twentieth and twenty-first centuries, to a serious and unresolved question. The question is whether mathematical structures exist independently of the minds that find them. The question is, on close examination, not a question the mathematicians themselves can easily answer from inside their own practice, because the experience of doing mathematics does not, by itself, settle whether the experience is tracking something real or producing an unusually consistent illusion.
The mathematicians’ own language
It is worth being precise about what the mathematicians actually say, because the standard cultural register has not, on the available evidence, taken the consistency of their language as seriously as the philosophy of mathematics has.
The mathematician G. H. Hardy, in his 1940 essay A Mathematician’s Apology, wrote that “I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations,’ are simply our notes of our observations.” Roger Penrose, in his book The Road to Reality, has argued for a similar view, suggesting that the consistency and surprising effectiveness of mathematics in describing the physical world is best explained by the existence of mathematical structures that have an independent reality. Kurt Gödel, whose incompleteness theorems are among the most important results in twentieth-century mathematics, was, by every available account of his views, a committed mathematical realist who believed that mathematicians perceive abstract objects much as ordinary observers perceive physical ones.
This is not, on the available evidence, a fringe view among working mathematicians. Surveys of contemporary mathematicians, including those reported in Jim Holt’s Why Does the World Exist?, place the proportion of working mathematicians who hold some version of mathematical realism at roughly two-thirds. The estimate is disputed in its precise figure, but the general direction is well-established. A majority of professional mathematicians, when pressed, report believing that the mathematical structures they work with exist in some sense independently of the minds that discover them. The minority who do not hold this view tend to hold it less because they are confident the structures are invented than because they are skeptical of the philosophical framework in which the question is usually posed.
What the philosophical position actually claims
The philosophical position that takes the mathematicians’ language seriously is called mathematical platonism. The Stanford Encyclopedia of Philosophy defines mathematical platonism as the view that mathematical objects exist, that they are abstract rather than physical, and that they are independent of intelligent agents and their language, thought, and practices. The claim is structurally similar to the standard scientific claim about physical objects. The electrons exist whether or not anyone is observing them. The number seven, on the platonist account, exists in essentially the same sense, even though it is not physical and cannot be located in space.
This view, when stated plainly, sounds strange to most people who have not previously encountered it. The strangeness is one of the reasons the position has remained philosophically contested for over two thousand years. The strangeness is also, on close examination, partly a function of the fact that the position is making a claim about a category of objects that is genuinely unlike the categories most adults have intuitions about. The strangeness is not, by itself, evidence that the claim is wrong.
The argument for platonism, in its strongest contemporary form, derives from Gottlob Frege and is sometimes called the indispensability argument. The argument runs roughly as follows. Mathematics is indispensable to our best scientific theories. Our best scientific theories make claims that are, on the available evidence, true. The claims of our best scientific theories quantify over mathematical objects. Therefore, our best scientific theories commit us to the existence of mathematical objects. The argument has been refined and contested in various ways since Frege’s original formulation, but the core structure has remained influential.
The opposing positions, and why they are also taken seriously
The opposing positions, broadly classified under labels including formalism, intuitionism, fictionalism, and various forms of nominalism, hold that mathematical objects do not exist independently in the way platonism claims. The various positions differ in their accounts of what mathematics is, instead. Formalism treats mathematics as a game played with symbols according to specified rules, with no claim that the symbols refer to anything beyond themselves. Intuitionism treats mathematics as a construction performed by the human mind, with mathematical objects existing only insofar as they can be constructed. Fictionalism treats mathematical claims as something like literary fiction, useful and meaningful within the fictional framework but not literally true.
Each of these positions has, on close examination, substantial philosophical motivations and substantial philosophical difficulties. The formalist position has trouble explaining why mathematics is so successful in describing the physical world. The intuitionist position has trouble accounting for parts of mathematics that appear to outrun any actual construction performed by any actual mind. The fictionalist position has trouble explaining why mathematicians overwhelmingly report their experience as discovery rather than as fiction-writing.
The wider philosophical debate among these positions has, on the available evidence, not resolved into a clear consensus. Contemporary philosophers of mathematics remain divided, with strong positions held on multiple sides. The lack of resolution is not, in itself, surprising. The question is the kind of philosophical question that may not, by its structure, admit of the kind of resolution that empirical scientific questions can sometimes achieve.
What the experience of doing mathematics is, in fact, like
What does seem to be settled, by the testimony of mathematicians themselves across cultures and centuries, is what the experience of doing serious mathematics actually feels like from the inside. The experience, by overwhelming testimony, has the structural features of discovery. The mathematician is working on a problem. The problem resists. The mathematician tries various approaches, most of which fail. At some point, often after a period of working on something else or sleeping, the solution arrives. The arrival is not, in the mathematician’s report, accompanied by the feeling of having constructed something. The arrival is, more accurately, accompanied by the feeling of having seen something that was already there.
The seen-something-already-there feeling is, in some real way, what has been driving the platonist position for the last two and a half thousand years. The feeling is not, on its own, philosophically decisive. The feeling could, in principle, be a consistent illusion produced by the structure of the human cognitive system rather than by anything corresponding to a real mathematical realm. But the consistency of the feeling, across mathematicians who have never met each other, across cultures that have developed mathematics independently, across centuries that have produced very different views about almost everything else, is, on close examination, a piece of evidence that the philosophers of mathematics have not been able to easily dismiss.
The consistency could be explained, on the platonist account, by the fact that the mathematicians are all looking at the same independently existing realm of mathematical objects, in roughly the way that astronomers from different cultures looking at the night sky all see the same stars. The consistency could be explained, on the non-platonist account, by the fact that the human cognitive apparatus is structurally similar across individuals and produces similar mathematical experiences for that reason. The philosophical debate, on close examination, is partly about which of these explanations is more parsimonious and partly about which one better fits the various other features of mathematical practice that have to be accounted for.
The acknowledgment this article wants to leave
The question of whether mathematical structures exist independently of the minds that find them is, by every available measure, a serious philosophical question that has not been resolved. The question is taken seriously by professional philosophers of mathematics because the working mathematicians, by their overwhelming and consistent testimony across cultures and centuries, report their experience as one of discovery rather than invention, and because the philosophical positions that take this testimony at face value have substantial arguments in their favor that the opposing positions have not been able to decisively defeat.
The wider cultural register has, on the available evidence, mostly absorbed mathematics as a human invention, similar in structure to other forms of human creative production. This absorption may be correct. The absorption is also, on close examination, not what most professional mathematicians, when describing their own experience, would actually claim. The mathematicians’ claim is closer to the claim that they are explorers of a realm that exists independently of their exploration. The question of whether this claim is literally true, or whether it is an unusually consistent and useful illusion, is one of the more interesting unresolved questions in contemporary philosophy. The question is, on close examination, worth more attention than the wider cultural register has been giving it.
