What is mathematics about? Is there any mathematical reality, and in that case, where is it located? Such questions, I presume, are seldom discussed in the classroom. I think they should. Integrating them into courses is one of my ideas about how to humanize mathematics education. But wouldn’t that be a complete waste of time? Not if it opens new roads into an understanding of the esoteric formalism and language that mathematics use to capture this very reality, whether it exists or not.
And isn’t it very strange that a subject that deals with abstract objects never discusses what these objects are, or where they are? Is it a wonder that most people have problems with mathematics? Who wouldn’t have problems understanding a subject that is about things you don’t know what they are or where they are?
Sometimes answers to these questions collapse down to a duality: Either mathematics exists ”out there” and is discovered by the mind, or it’s all a mental construction, and is consequently invented by the mind. However there are many nuances and the question has a similarity to the old philosophical question about the existence of universal concepts. I got this notion from Lars Mouwitz’s PhD dissertation. Four distinct directions of thought seems to have crystalized (in my free-swinging interpretation).
- universalia ante res
Universal concepts come first. This is Plato: Concept realism, concepts are not invented, rather they exist before reality and real things are copies of the concepts. Mathematical concepts are real (in this sense) and are discovered by the mind. This is Mathematical Platonism. It is sometimes jokingly said that most mathematicians are Platonists on weekdays due to the very strong feeling they have that they are working with real existing objects. A critique against this view is based on the obvious problem of locating where the concepts actually reside.
- universalia in res
Universal concepts reside in real ”things”. This is Aristotle: Concepts exist in reality and are extracted by the mind, not invented, but discovered. They are sort of built into things. Knowledge is empirical and abstracted. Mathematics becomes the language of nature.
- universalia post res
Universal concepts come after real ”things”. This sounds like a more modern view. There are no concepts where there are no minds. Concepts are invented by the mind based on empirical studies, but they don’t exist in physical reality itself. Concepts are cultural phenomena, propagating through society (space) and history (time). An analogy would be the ”memes” of Dawkins. In pedagogy and philosophy, this is constructivism. Concepts are created by the mind in the mind. A question: How can concepts be private, yet commonly shared, correct and useful? An answer: By social and cultural processes and communication, and the concept-forming minds are parts of reality. The position of David Hersh lies here (more on that below). Mathematical concepts are social constructions.
There are no concepts, only the things themselves. Concepts are just names. In mathematics this corresponds to formalism. Still jokingly, when the weekday Platonism of mathematicians are challenged, they resort to Sunday formalism. Mathematics is about nothing, it is just a play with symbols. Wittgenstein held the view that there is nothing beyond the signs and symbols, it’s all language-games. Perhaps many students end up here. The mathematical formalism doesn’t mean anything, the symbols are disconnected from reality. It becomes a meaningless and largely incomprehensible game.
To me, the first and last position are too extreme. Platonism may be a beautiful idea, but it is not scientifically plausible. Nominalism is too poor. In his book What is mathematics, really? David Hersh argues for mathematics being a social construction. This is an idea with precursors. There is an article arguing that point of view in The World of Mathematics (don’t have the exact reference just now) and I’ve seen more. A very interesting such argument is constructed by Oswald Spengler who wrote a long section on the meaning of numbers in his The demise of the West.
A problem with full constructivism (mathematical concepts as social constructions) is the Wigner problem: ”The unreasonable effectiveness of mathematics …”. There is no denying (I think) that mathematical principles seem to be built into physical reality. So some kind of compromise between the second and third viewpoint may the most viable.
A far out aside: There are even theoretical physicists who claim that underlying what we perceive as physical reality is nothing but mathematics. This is sometimes popularized by saying the world is a simulation run on a super-computer. This is of course hard to check near non-sense and it is open to the obvious infinite regress problem (and that super-computer, what’s the physics underlying that?) Still there is a strange problem in fundamental physics: What is an electron? No-one knows, the only thing we have is a description of such particles as quantum fields obeying the field equations and the rules of quantum field theory. That is applied mathematics.
Discussions like these are of course related to the foundational crisis and debates more than a hundred years ago. The history is nicely summarized in David Hersh’s Some Proposals for Reviving the Philosophy of Mathematics in New Directions in the Philosophy of Mathematics.
Discoveries in analysis such as for instance continuous but nowhere differentiable functions and Fourier series showed that the geometric intuition underlying infinitesimal calculus was insufficient. This lead to arithmetization of analysis and Cantor’s set theory. Then to the Russel paradox and the breakdown of Frege’s system of logic and mathematics. The classical foundational programs of logicism, formalism and intuitionism were all attempts to resurrect certainty of mathematical knowledge. They all failed. Hersh writes that mathematics has no foundations and needs no foundations.
Certainty of knowledge has been a preoccupation of philosophers of all time, in particular since Descartes and onwards. Today, this preoccupation seems very antiquated. Of course, our scientific knowledge of for instance fundamental physical reality is certain to a very high degree, but no-one claims it to be 100% certain or even hope for it ever to be. The modern scientist can live with uncertainty. Indeed, if you can’t stand living with uncertainty, then you’re no scientist at heart. Mathematics is very likely to be even more certain than fundamental physics. But isn’t it more interesting if there is an epsilon risk of error rather than zero risk? And historically, no paradox or inconsistency ever discovered has been able to destroy mathematics. The only consequence of the Russel paradox is: Don’t deal with such silly ideas. Isn’t it obvious from the very beginning that the idea of the set of all sets is ill conceived?
Anyway, my real question is not what is the best philosophy of mathematics, but rather: These kinds of discussions, may they help in teaching and learning mathematics?