What Mathematics is not

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.”

---

Seven varieties of metaphysics

I was having a discussion with someone recently on metaphysics, so I thought I would blog about it. Here are seven varieties of metaphysics, describing three “layers” of reality (and yes, I am oversimplifying for brevity).

The first is Platonism. Plato believed that there was a hierarchy of Forms (Ideals), of which the highest was The One (Plato’s version of God). These Forms or Ideals were the true reality, and the physical objects we touched, saw, and tasted were only shadows of that true reality (that is the point of the allegory of the cave). The physical orange which we see and eat reflects Ideals such as “Fruit,” “Sphere,” and “Orange.” Neoplatonism continues and extends this point of view.

Saint Augustine and many later Christians held to a Christianised Platonism, in which the Ideals were thoughts in the mind of God (the Christian God, naturally). The physical objects we touched, saw, and tasted had a greater importance in Christian Platonism than they did for Plato – after all, when God created those objects, “God saw that it was good.” Much as with Platonism, the regularities that people see in the physical universe are explained by the fact that God created the universe in accordance with regularities in the Divine thoughts. However, Christian Platonism does not have the metaphysical hierarchy that Platonism or Neoplatonism have – in Christian Platonism, God makes direct contact with the physical universe.

Aristotle also reacted to Plato by increasing the importance of the bottom layer, and Aristotle’s thought was Christianised by Thomas Aquinas as Thomism. However, in Thomism the all-important bottom layer does very little except to exist, to have identity, and to have properties assigned to it. It is also not observable in any way. This can be seen in the Catholic doctrine of transubstantiation. According to the Tridentine Catechism of 1566, the bread and the wine of the Eucharist lose their bottom (“substance”) layer (“All the accidents of bread and wine we can see, but they inhere in no substance, and exist independently of any; for the substance of the bread and wine is so changed into the body and blood of our Lord that they altogether cease to be the substance of bread and wine”), while the bottom (“substance”) layer of the body and blood of Christ becomes metaphysically present instead.

Idealism denies that the physical universe exists at all. The followers of Mary Baker Eddy take this view, for example, as did George Berkeley. Only thought exists. To quote a famous movie line, “there is no spoon.” These thoughts may be independent of whatever God people believe in or, as in monistic Hinduism, they may be actually be the thoughts of God (in which case, only God exists).

The last three kinds of metaphysics deny the existence of any kind of God. In Platonist Materialism, this denial is combined with a Platonist approach to mathematics, about which I have written before. Mathematics exists independently of the physical universe, and controls the physical universe, in the sense that the physical universe follows mathematical laws. Roger Penrose is one of many scientists holding this view.

In what I am calling Extreme Materialism, the existence of an independent mathematical world is also denied, i.e. there is an empiricist approach to mathematics (mathematics simply describes observed regularities in nature). This view seems to be increasing in popularity among non-religious people, although it causes philosophical problems for mathematics.

Finally, the concept of the Mathematical Universe holds that the so-called “physical universe” is itself composed only of mathematical objects – only mathematics exists (which makes this, in fact, a kind of Idealism).

---

Three Worlds

Roger Penrose, in his book Shadows of the Mind, outlines an idea adapted from Karl Popper – that there are “three worlds.” The physical universe needs no explanation, except perhaps to Bishop Berkeley, while the subjective world of our own conscious perceptions is one we each know well. The third world is the Platonic world of mathematical objects.

Penrose says of the third world: “What right do we have to say that the Platonic world is actually a ‘world,’ that can ‘exist’ in the same kind of sense in which the other two worlds exist? It may well seem to the reader to be just a rag-bag of abstract concepts that mathematicians have come up with from time to time. Yet its existence rests on the profound, timeless, and universal nature of these concepts, and on the fact that their laws are independent of those who discover them. The rag-bag – if indeed that is what it is – was not of our creation. The natural numbers were there before there were human beings, or indeed any other creature here on earth, and they will remain after all life has perished.” (Shadows of the Mind, p. 413)

Edward Everett, whose dedication speech at Gettysburg was so famously upstaged by Abraham Lincoln, put it more poetically: “In the pure mathematics we contemplate absolute truths, which existed in the Divine Mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven.” G. H. Hardy was ambivalent about the Divine, but like most mathematicians he believed “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.”

This trilogy of worlds raises some questions, of course. The first is what Eugene Wigner called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” As William Newton-Smith asks, “if mathematics is about this independently existing reality, how come it is useful for dealing with the world?” Why does the world follow the dictates of eternal Reason? Or, as Einstein put it, “how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”

The second question is the mind-body problem. How are we conscious of the universe, and how do our decisions to act affect it? Does even our perception have strange quantum effects?

Finally, how do we become aware of the Platonic world? Elsewhere, Penrose says “When one ‘sees’ a mathematical truth, one’s consciousness breaks through into this world of ideas, and makes direct contact with it… When mathematicians communicate, this is made possible by each one having a direct route to truth” (The Emperor’s New Mind, p. 554). But what exactly does that mean? Does one’s soul go on some kind of “spirit journey”?

Doubling the square

Plato, in the story of Socrates and Meno’s slave, tells how an uneducated slave is prompted to discover how to double a square. Plato saw this as evidence of memory from a past life, but it provides an example of mathematical intuition that all (successful) students of mathematics will recognise. As Saint Augustine said, “The man who knows them [mathematical lines] does so without any cogitation of physical objects whatever, but intuits them within himself.” Yet Plato’s (and Augustine’s) belief in such an intuitive soul makes the mind-body problem more acute. How do the three worlds tie together? It seems a mystery.

Three Worlds by M. C. Escher