This is a conversation I seem to having a lot lately with math students, particularly seniors anxious to enter the fabled “real world” and questioning the value of their major. I figured I might as well organize my thoughts, and here is as good a place as any to do so and to, who knows, carry on the conversation.
It’s about one of the oldest philosophical issues of mathematics, debate as old as mathematics itself. It’s one of the few issues that can completely polarize mathematicians, worse that the usual turf wars of pure vs. applied or analysis vs. algebra. It’s something so fundamental to the discipline that Barry Mazur just calls it The Question. In it’s simplest form, it’s this:
Is mathematics something that one discovers or that one creates?
To a supporter of the discovery position, all mathematical truths and concepts actually “exist” in our universe in some real, albeit ideal and possibly nonphysical form. For example, twoness, the common thing shared by all pairs of objects, is a real, eternal, concept. Even in a universe devoid of life — devoid of anything, even — twoness would still exist. Under this view, mathematics is true in a fundamental, essential, universal way…. human beings merely reveal it with the tools of logic and proof in the same way that a paleontologist unearths the bones of long dead dinosaurs with a pick and brush. Because this viewpoint shares a lot with Plato’s theory of forms, and because mathematicians are prone to hero worship, this position is is often called mathematical Platonism.
On the flip side, to a supporter of the creation position, mathematics is nothing more than a specialized sub-genre of art, splashes of paint on the canvas of humanity’s collective mind that have no more permanence outside human existence than a faint fragrance carried on a breeze. For these folks, twoness is just a construct of the human mind to describe collections that can be put into bijective correspondence with the set {1,2}, so a universe devoid of life would also be devoid of twoness, since there would be no one to think it. Under this view, mathematics is true only in the sense that it’s logically sound: it is a silly game played by the minds of Men (and Women!) rather than a profound insight into the mind of God. Consequently this is often called mathematical Formalism.
Questioning The Question
The question about The Question, then, is who is right? Does mathematics live out there somewhere, eternal and unchanging; or does it live only in here, a fleeting dance of synapses in the gray stuff in your cranium?
The experience of doing mathematics suggests a little of both: devising new definitions or ideas or isomorphisms sure feels like an act of creativity – flashes of insight or playful what-if’s — but investigating the consequences of these constructs sure feels like an act of discovery. Leopold Kronecker seemed to toe this diplomatic line when he famously quipped “God created the natural numbers; all the rest are the works of Man.”
There are compelling arguments in favor of either side, which I can summarize by sticking them in the mouths of Plato and Imannuel Kant.
Plato: Obviously mathematical ideas exists independently of humanity. Mathematics is the language that describes these inevitable, basic truths of the universe around us.
Kant: English is another language also describes the world around us, but no one would think it is universally inevitable. Mathematical concepts need not be bound to anything observed or connected to the world around us. Simply choose your axioms and let your mind wander.
Plato: But that proves my point! Even the bits of mathematics purposely designed to have no real utility or existence — negative numbers and imaginary numbers and non-Euclidean geometries — nevertheless routinely end up describing the universe around us. Physicist Eugene Wigner called it the unreasonable effectiveness of mathematics in the natural sciences, and argued that it is the strongest case for the “universality” of mathematics.
Kant: But the counterargument is obvious: if basic mathematical language was initially devised to describe the real world, why should it be surprising that ever more refined versions of this language describe the real world as well?
Plato: Even granting the more abstract or abstruse bits of mathematics are inventions of the mind, the bedrock language you speak of, which I take to include such basic concepts as number and order, are discovered over and over again by different cultures at different times, and even in various animal species. Does not their unfailing, inevitable consistency across all peoples and all times prove that these things must exist objectively in the universe, rather than subjectively in the mind?
Kant: Perhaps, but it seems more reasonable this common mathematics speaks less of a universality of common concepts than a common humanity of perception. Perhaps “twoness,” to take an example, is so obvious to humans precisely because we’re ingrained to see the universe in terms of I and YOU. Would not, say, a sentient ant colony or an intelligent magnetic cloud be so profoundly different from us that, to it, twoness would be an unrecognizable or irrelevant or even flawed idea? Would we even recognize their kind of mathematics?
Plato: Just because I may not understand a description of the Grand Canyon written in a foreign tongue doesn’t mean the Grand Canyon does not exist. Perhaps these other mathematics, if they exist at all, are a part of the grand collection of “universal” truths
… and round and round it goes.
So how do I assess The Question? Whose arguments do I find most compelling or most flawed? Let me summarize my thoughts by viewing The Question from three perspectives.
1. The mathematical perspective
Let’s consider The Question mathematically. Mathematical Platonism is one of those unfalsifiable propositions, since it postulates the existence of things beyond the physical or provable. I can’t, for example, disprove the objective existence of 2 by not finding its “ideal” form hidden away on some distant moon.
From a mathematician’s perspective, then, it seems more like an axiom. The best axioms are the ones that bear a minimum of additional hypotheses while generating several new, useful results.
With this in mind, note the following two observations. First:
- Mathematical Platonism has one helluva hypothesis, namely, that every mathematical concept, definition, and truth that every was or ever will be articulated already exists in some timeless, nonphysical, ideal state.
Where do these things exist? Is there a cosmic storehouse in which every conceivable number system, arithmetic operation, geometric construct, epsilon-delta proof, and whatever else is stored? Perhaps like Kronecker, you suppose there to be only a small number of these eternal, untouchable, unsullied objects, but this still entails appending some deeply theist assumptions to your maths.
Second:
- Mathematical Platonism is completely irrelevant to how these mathematical truths are proved.
That is, the set of discoverable, logically provable mathematical statements are the same, whether we accept this axiom or not. Hence, whether or not the unfalsifiable position of Platonism is true, it has aboslutely no effect on what kinds of mathematics we can prove.
That would suggest that Platonism is a poor axiom to accept.
2. The practical perspective
What, then, is the harm in accepting Platonism, if it has no effect on the kinds of mathematics we can prove? The harm is that Platonism affect what kinds of mathematics we should prove.
Look at it this way: you will only argue that there’s no such idea as “less than o” or “infinitely many lines parallel to a given line through a given point” unless you actually think those things exist in some pristine form that can be damaged or corrupted. If mathematics is supposed to be a search for truths, Platonism undermines that by blindly asserting what is true.
Worse, even if Platonism is true, devout Platonists have demonstrated a spectacularly poor track record of sussing the “true” bits from flotsam of failed ideas. Every new breakthrough in mathematics — be it the discovery of negative numbers, or imaginary numbers, or noneuclidean geometry, or the differential calculus, or transfinite arithmetic — has always been resisted only by those who subscribe to a Platonist view. Jesuit preist Giovanni Saccheri discounted non-Euclidean geometries as being “repugnant to the nature of straight lines.” Papal celebrity Gerolamo Cardano dismissed the imaginary number as being “as subtle as it is useless.” Even our accommodating Platonist Kronecker, whom I quoted above, actively campaigned against Georg Cantor’s formalizing the concept of infinity, to the point of name-calling in journal articles, heckling Cantor as a “scientific charlatan” and a “corrupter of youth.” If those objections had been heeded, mathematics would be far less beautiful and profound and, paradoxically enough, less useful as a model for reality.
Said more succinctly, the discipline of mathematics, the progress of mathematics, and the practitioners of mathematics have only ever suffered under mathematical Platonism.
3. The scientific perspective
The scientific utility argument — “is not the utility of mathematics in fleshing out the science of the universe around us is evidence that mathematics exists independent of us?” —seems easy enough to answer. The basic building blocks of mathematics — arithmetic and geometry — were built to model the most basic aspects of the observable world. As Kant alluded to in his second counter-attack above, it’s almost a tautology that a subject designed to model the universe around us might, in fact, model the universe around us.
But as Plato countered, what of the unreasonable utility of mathematics? Why, for example, does something as esoteric as the imaginary number, originally thought to have no real-world applicability, and now scribbled on a dusty slate board in a university lecture hall, do so well in describing the processes of electric engineering in a pristine lab in another university? Is that not evidence that imaginary numbers are, in fact, real things that govern science?
This is a tough argument to counter, but science itself suggests that we might be asking the wrong question. Is it that the universe is rife with mathematical patterns awaiting for us to unearth… or is it that humans are very good at creating and connecting patterns of all sorts?
In fact, human beings are amazing, rather hyperactive pattern recognizers. It’s a survival trait honed over millions of year of evolution. We see faces in clouds, hear rhythms in sounds, and numbers in the world, whether or not they’re there. The human brain is wired to find patterns and create connections. Moreover, evolutionary biologists have given a number of explanations for why our species also evolved a natural tendency towards the intentional stance: the predisposition to assign intent to things we experience, regardless of whether or not intent is implied (or even possible). If we see a pattern, we’re compelled to assume there is a reason for it. Couple the apparent utility of mathematics to solve problems bigger than the ones first posed with our species’ biologically-wired need to assign meaning to it, and you’ve got the ingredients for belief in mathematical Platonism that has nothing to do with whether or not Platonism is true.
So consider, which is more likely? Is humanity evolving to a better understanding of the eternally unchanging nature of mathematics, or is mathematics is evolving to better fit the ever-changing, increasingly complicated needs of humanity? Both of these say similar things, but the first requires a leap of faith to believe an unseeing pantheon of eternal forms, unseen and unquantifiable in any way. The second requires only belief in the cleverness and gullibility of humans, and that’s supported by reams of evolutionary, biological, and psychological scientific data.
The Answer…
No argument can conclusively disprove the preternatural existence of “mathematics,” but the three perspectives offered above show
- there is no logical reason to assume mathematical Platonism is true,
- there are practical reasons against assuming mathematical Platonism is true, and
- there are a number of scientific alternatives to the apparent truth of Platonism that don’t require it at all.
Taken all together, I see no reason to believe in mathematical Platonism, and so I don’t.
