I have always had an interest in mathematics. This is despite, or perhaps because of, never being very good at the subject at school, and avoiding it to the maximum extent compatible with getting a science degree at university. Not that I have any fondness for what is called ‘recreational mathematics’, which has always seemed to me to be a contradiction in terms, nor any desire to do substantive academic work involving mathematics, for which I am ill-equipped. My interest rather has been in the qualitative ‘meta’ of the subject: what is mathematics, and how does it relate to the real world. In particular, I always wondered about two points. Why are the mathematical concepts developed by humanity so useful in accounting for the behaviour of the physical universe; what the physicist Eugene Wigner called the “unreasonable effectiveness” of mathematics. And whether mathematics was something independent of people, waiting somewhere to be discovered, or whether it was a human creation.
Quite a long while ago, I was given, for a Christmas present, a copy of John Barrow’s ‘Pi in the Sky‘, and account of the history and nature of mathematics, addressing exactly these issues. It also introduced me to the three main philosophical approaches to mathematics (of which I give only the simplest summary here). The Platonist approach holds that mathematical entities and relationships have an eternal existence, independent of humanity, in some kind of world of abstract ideas. The formalist approach believes that maths is created by humans, and is simply the manipulation of symbols according to arbitrary rules, without any associated meaning. And then there are a number of constructivist approaches, including inventionism and intuitionism, which hold that mathematics is created by humans under the influence of psychological and cultural factors. Barrow carefully pointed out the flaws in each of these, which mean that there is no consensus among mathematicians as to the very basis of what they are doing, and how, and why. Although Barrow was writing over twenty years ago, as I understand it the same is true today.
As I read Barrow, I was struck by what seemed, and still seems, to be a particular contradiction. When mathematicians describe what they do in creating new mathematics, they almost always do so in terms of discovery rather than invention. They think they are finding out things that exist independently of them, rather than creating them; and indeed the many instances when the same new mathematics is derived by several people in ignorance of one another seems to confirm this. And yet it seems self-evident that mathematics is a human creation, since we observe mathematicians creating it.
It occurred to me that one solution to this conundrum might be given by Karl Popper’s idea of World 3 of objective, communicable information. If we can agree that mathematical entities are inhabitants of World 3, as Popper said they were, then we can see how mathematics might be both invented and discovered. World 3 objects are created by humanity, but once created they take on, in a sense, a life of their own, with implications and consequences not necessarily understood by their creators. Mathematics is then indeed discovered, and mathematicians’ sense of what they are doing is not false, but it is discovered in the human-created World 3.
This struck me a particularly nice idea, and I was surprised that no-one else seemed to have thought of it.
Well, of course, someone had, though I only found out years later. Back in 1981, more than ten years before Barrow’s book appeared, the American mathematician and philosopher Reuben Hersh had presented the idea in a book co-authored with Philip J David, ‘The mathematical experience‘.
“Start from two facts”, Hersh wrote. “(1) mathematics is a human creation, about ideas in human minds; (2) mathematics is an objective reality, in the sense that mathematical objects objects have definite properties, which we may or may not be able to discover. Platonism is incompatible with Fact 1, since it asserts that maths is independent of humans; constructionism is incompatible with Fact 2, since there no properties until they are proved constructively; formalism with both, since it denies the existence of mathematical objects… Mathematics is an objective reality that is neither subjective nor physical. It is an ideal (ie non-physical) reality that is objective (external to the consciousness of any one person). In fact, the example of mathematics is the strongest, most convincing proof of such an ideal reality.
This is our conclusion, not to truncate mathematics to fit a philosophy too small to accommodate it – rather, to demand that the philosophical categories be enlarged to accept the reality of our mathematical experience…. The recent work of Karl Popper provides a context in which mathematical experience fits without distortion. He has introduced the terms World 1, 2, and 3, to distinguish three major levels of distinct reality. World 1 is the physical world, the world of mass and energy, of stars and rocks, blood and bone. The world of consciousness emerges from the material world in the course of biological evolution. Thoughts, emotions, awareness are nonphysical realities. Their existence is inseparable from that of the living organism, but they are different in kind from the phenomena of physiology and anatomy; they have to be understood on a different level. They belong to World 2.
In the further course of evolution, there appear social consciousness, traditions, language, theories, social institutions, all the nonmaterial culture of mankind. Their existence is inseparable from the individual consciousness of the members of the society. But they are different from in kind from the phenomena of individual consciousness. They have to be understood on a different level. They belong to World 3. Of course, this is the world where mathematics is located.”
“Mathematical statements”, Hersh concluded, ” are meaningful, and their meaning is found in the shared understanding of human beings, not in an external nonhuman reality. So mathematics deals with human meanings in the context of culture, it is one of the humanities; but it has a science-like quality, since its findings are conclusive, not matters of opinion…. As mathematicians, we know that we invent ideal objects, and then try to discover the facts about them ”
Hersh had stated the idea that I had, formulated more clearly than I could have done, fifteen years beforehand. Ah well.
Hersh has gone on to develop his ideas, though his more recent writings don’t reference Popper, but describe mathematics as being discovered in a human-created socio-conceptual world (which still sounds pretty much like Popper’s World 3 to me). His recent thoughts are nicely presented in a 2014 book, ‘Experiencing mathematics: what we do, when we do mathematics‘, in which he presents the socio-conceptual idea as an explicit alternative to the formalist, Platonist, and intuitionist approaches.
So, although I didn’t think of it first, it seems to me that the mathematical World 3 is a more realistic basis for describing what mathematics is, and what mathematicians do, than any of the alternatives. And also, another vindication that Popper’s World 3, though an idea which has fallen out of fashion to an extent, is still a fruitful concept for any disciplines which deal with abstract concepts at their heart.