I’ll never forget the first math class I took in grad school. The professor, a middle-aged French emigre with long and lanky salt-and-pepper hair and the subtle hints of an accent and the passionate, didactic, decisive (if not at times abrasively impatient) manner of a man who “knew his stuff,” pronounced to the class, after the desultorily predictable reading of the syllabus, without any trace of hesitation in his voice, that:
“After all, mathematics is only a language: and as such we must begin with vocabulary; that is, we must define our terms.”
I have no quarrel with beginning a lesson, any lesson, with a definition of terminology: indeed, I think we live in an “over-informed” age, where the “data” we gather, in admittedly unimaginably larger quantities than ever before, is of such poor semantic quality that we can’t defend any choice we make. Nor is it my desire to critique a teacher who was, in his own way, trying to “bring out treasures old and new” from the storehouses of knowledge to the initiates in his care, and needed to get on with the lesson in an overpopulated classroom mostly full of students who couldn’t care less about the philosophical credentials of his position. Nor do I pretend to even be literate enough in mathematical philosophy to contribute any definitive answer to the question “what is mathematics?” as I haven’t been trained to it.
However, I couldn’t then (and still can’t now) let his remark stand without wondering if he is right or wrong. The debate about what mathematics is and isn’t runs, arguably, back to Greece and to Plato: but for we post-moderns, the debate can effectively be accelerated to Hilbert’s initiation of the “logicist problem,” i.e. the attempt to develop, starting from pure logic alone, a system of axioms and relational operators from which one could develop all of arithmetic. This had its earlier roots in the work of Frege and Dedekind, but the most impressive work that arose out of this is undoubtedly Whitehead and Russell’s Principia Mathematica. This massive tome (three volumes in length, which takes several hundred pages to prove that 1+1=2) was perhaps the best and most rigorous attempt to “axiomatize” mathematics as per Hilbert’s program. If anything was going to solve Hilbert’s “problem” it was going to be PM.
Then along came the seemingly fatal blow, in the form of a humble, unassuming logician named Godel, whose “incompleteness,” theorems, for most mathematicians, proved that Hilbert’s program was simply and strictly impossible. To put it briefly, Godel was able to prove that any system of “first-order” logic (which can essentially be taken to be equivocal with “simple” logic consisting only of nouns and verbs) could not be syntactically complete. This is distinguished from being semantically complete: which means that the system can prove any of its tautologies based purely on the application of the axioms. Syntactical completeness would mean that any statement whatsoever could be proved or disproved by the application of the axioms: and essentially what Godel was able to show was that in any such first-order system, there would always be statements that were true, but could not be derived from the axioms (this is his first “incompleteness theorem,”). Simply put:
In first-order systems that can express basic arithmetic, a complete and consistent finite list of axioms can never be created: and, if a putative axiom were ever added that could made the complete, it does so at the cost of making the system inconsistent. You can have completeness or consistency, but never both.
For those of us who’ve had some mathematical education, this should explode across our horizon like a bombshell, because we are used to the idea that math, almost unlike any other subject, builds up linearly: we start with definitions (just as my professor did), proceed to connect them via inference and argument to form provable theorems, from which we can build up techniques that allow us to solve problems. Pick up any book on any mathematical subject, be it probability or topology, and they all follow this pattern. Essentially what the logicist programme was attempting to do was to do this same process for mathematics itself: that is, they were trying to derive primitive enough definitions or notions or axioms from which it would be possible to derive all of mathematics, including things like the natural and real numbers, thus forever under-girding mathematics as (despite its appearance to the contrary) an extension of logic. And it was to this that Godel said:
So what does this have to do with the statement that “mathematics is only a language?” Well, once the logicist position had been so clearly demolished, most mathematicians were left to choose between becoming one of two kinds of philosophers: Realists (who, like Godel himself, assert that numbers are real, objective things existing independently of the human mind) and Formalists (who hold that all mathematical statements can be boiled down to consequences of arbitrary but ultimately meaningless rules that are nevertheless consistent). In other words, the formalists deny that mathematics has any kind of meaning beyond and outside of itself at all: and that all the work of building up theorems was no more than a “game” (they themselves use the word) with certain (arbitrary) rules (called axioms) through which one proceeds to produce what are called “proofs,” but which in reality are only useful inside the “game” and ultimately hold no significance to the real world of corporals and kangaroos, measuring tapes and tea cozies. It’s most extreme form is found in the founder of logicism itself who, one has to imagine, was more than a little disappointed by the failure of PM: Hilbert eventually regarded anything beyond “finitary arithmetic,” (what we’d call integer arithmetic) as ultimately meaningless.
It should be quite plain what side of the debate my graduate school professor landed on: he was espousing unadulterated Formalism when he told us that mathematics was “just” a language. As to the question of whether he ought to have told us before beginning his class, then and there, about the metaphysical argument concerning the roots of the subject he was about to teach us and his own answer I will have very little to say: he was in a hurry, and in his admirable expedience had to leave the question for later. I only bring it up at all because it struck me as odd at the time that the students he was teaching were made up almost exclusively by students of the applied sciences: that is, engineers, physicists, biologists etc. The only reason we were taking the class at all was to eventually apply the methods contained therein directly to real, physical phenomena, from which we hoped to derive real insights into what was really going on: and real insight is precisely what an arbitrary game can’t give you. This would be like signing up for a class in car maintenance and the instructor telling you on the first day that he’d be teaching us about how elephants would work in Elfland. Interesting, maybe, but not very pertinent.
So why did I and my fellow “applied” students stick around to learn the things anyway (apart from the fact that we had to take the class, I mean)? The answer is perhaps plain: because we knew that, whatever else our professor meant about it being “just” a language, we knew that mathematics could do more. We knew this not through any theory, but by actual experience: we had done experiments, which prompted us to develop (or, as is the case more often, modify) models in software or in pure math, and then test them. And as we refined the models we noticed that they were able to be refined: that is, math was the right tool to describe the physical universe; if it erred it only wanted improvement, not wholesale abandonment. In other words, we stuck around because we were tacitly familiar with a really astounding property of math (and science, which is a discipline of applied math) that our professor didn’t: what Wigner called “the unreasonable effectiveness of mathematics.”
Math, put simply, works: it seems to describe the universe most aptly. As Galileo put it, “the great book of Nature is written in the language of mathematics,” or as Einstein put it, “the most incomprehensible thing about the universe is that it is comprehensible.” If math didn’t “work,” didn’t give us any “true” insight into the nature of things, then it’s very likely you wouldn’t be reading this on a screen designed by assuming that electrons behave according to the equations that Maxwell derived in the 1860s.
In other words, the formalists, so long as they stay away from the sciences, might be able to get away with saying that their subject is “only” or “merely” (always a dangerous word) a string manipulation game. Much of it, perhaps, is: don’t you remember when you were first taught about negative numbers in school? Don’t you remember a certain reluctance, a certain hesitation in following your teachers argument? Maybe our first instincts were more right than we thought: the great mathematician immediately mentioned above still managed to be a great scientist though he referred to complex numbers as “absurd.” But the instant they start to apply their little game to describe the real universe they hit a (wonderful) snag: the indisputably “unreasonable” effectiveness of mathematics to describe the real universe.
There’s no getting away from this obvious power: we can’t shut our eyes to it or turn aside: Math works. It is down to the formalists to explain away the fact that when we treat all concrete physical things, from comets to kangaroos, as if they were pure quantities (that is, as if they really were merely the masses and inertias that make them up), that we can predict how they will behave with any accuracy at all? All the burden of proof is on their side.
Not so for the Realists. They find this burden so formidable that they take the following solution to the problem, “The reason that math works is because the property we call ‘quantity’ is no mere arbitrary abstraction, but a real thing: if matter didn’t have any kind of quantitative nature whatsoever, then we could never quantify it at all: there’d be nothing to work on. No scientist would pretend that the objects he’s studying are the formulas that describe them: the total reality of the object of his study in all it’s nitty-gritty detail is replaced by a model which is mathematically manipulable. Of course this model isn’t the actual thing: it has a symbolic relationship to the truth, the real thing we’re studying.”
For the Realist, then, science is a work of modeling: of taking reality as it is and translating or transcribing it into a form which is mathematically manipulable, that, as Milton put it, “saves the appearances,” which is to say that it captures as much of the reality as we can. Generating these models is a work because he has something (the reality) to work on: it is not a mere working in a void, but more like the work of a poet in turning a feeling into words. For what could be more unlike: the feeling of falling in love and the ink on the page? And yet the words can give rise to the feeling nonetheless: there exists some invisible accord between these two unlike things. It is the same with the scientist: only that the feeling is the physical object (the comet or kangaroo) and the ink his mathematical model.
The model is, admittedly, only a model, like the reality in some ways and not like it in others, as contour lines can only at best be “like” the real mountain they describe. But despite this deficiency, the model is “true” insofar as the mountain actually has contours and elevations and actual features of which the lines are the projection or translation of them onto the page. It is “less” than the real mountain, but it is not unlike or different: there is continuity between the two of them. If the model is an accurate projection of the quantitative nature of the reality of the concrete thing under study, then it is a “true” model and we can use it (as such); provided we keep the essential limitations of models (as such) in mind as we use it (the scientists are perhaps a little too prone to conflate their models with the indescribably real thing they’re describing).
One last word and I am done. I began this essay by declining to answer the question, “what is mathematics,” and to that I will hold. I wished only to make a small contribution to the debate as to what mathematics is, and it’s this: whatever else it might be, mathematics is certainly a language, but it’s also certainly not only a language.