Just recently, I responded to a pair of viral videos by a 16-year-old TikTok user called Gracie Cunningham (her second video is here). Along with a lot of nasty abuse (Twitter is a cesspool), Gracie received many friendly, but in my view quite wrong, replies. So I thought I would say something about what mathematics is not.
1. Mathematics is not a language
Mathematics obviously includes a language, but mathematics is not just a language.
If you think about it, the “notation” section of mathematics textbook is kind of like a dictionary. But the bulk of a mathematics textbook makes (and proves) assertions. In this sense, a mathematics textbook is like a botany textbook or a physics textbook – it has content. It discusses mathematical objects, and makes statements about them.
2. Mathematics is not a cultural artefact that we invent
This should also be obvious. First, if mathematics was simply cultural, it would not be so enormously useful in science. Indeed, as Eugene Wigner famously pointed out, parts of mathematics are often useful in areas quite different from the area where they first arose. In particular, the number π = 3.14159… is useful all over the place.
Second, mathematicians are acutely aware that we can’t just “make things up.” In the words of Jacques Hadamard: “We speak of invention: it would be more correct to speak of discovery… Although the truth is not yet known to us, it pre-exists and inescapably imposes on us the path we must follow under penalty of going astray.”
Third, if mathematics was simply cultural, incompatible versions of mathematics would arise in different cultures, and this is not the case. For example, the picture below shows a sum in modern Western numerals, and the same sum in Devanagari numerals (from India), Chinese numerals, Mayan numerals (in base 20), and Babylonian numerals (in the base 60 that still survives in our hours, minutes, and seconds). The same truth (39 + 47 = 86) is expressed in all five systems, even though the notation for expressing that truth may differ.
3. Mathematics is not a set of empirical truths
This one is less obvious. Pure mathematicians tend to believe, with G. H. Hardy, “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.” Many physicists believe that mathematical truths are discovered from the physical universe, but pure mathematicians tend to believe that mathematical truths lie beyond the physical universe. As Saint Augustine put it, “The man who knows them [mathematical lines] does so without any cogitation of physical objects whatever, but intuits them within himself. I have perceived with all the senses of my body the numbers we use in counting; but the numbers by which we count are far different from these. They are not the images of these; they simply are. Let the man who does not see these things mock me for saying them; and I will pity him while he laughs at me.”
One reason why Hardy and Augustine have to be right is that we can imagine a different universe, with different physical laws. But no matter how different the universe, 39 + 47 = 86 would still be true. It is, somehow, true at a deeper level than the laws of physics.
Another reason is that the empiricist point of view isn’t true to the history of mathematics. Galileo used parabolas to describe the motion of falling objects, but the ancient Greeks had originally described parabolas in a quite different context, that of conic sections. Similarly, imaginary numbers were originally discussed without the slightest idea that centuries later they would become a fundamental part of quantum theory.
A third reason is that a great deal of mathematics has no connection to the physical universe at all. Mathematicians study the properties of numbers far larger than the number of particles in the universe, and the properties of algebraic structures with no (known) relationship to the physical universe. Usefulness of the mathematical truths they discover is the furthest thing from their minds. As Joel Spencer, put it, “Mathematics is there. It’s beautiful. It’s this jewel we uncover.”