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.”

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Answering Gracie Cunningham

A 16-year-old TikTok user called Gracie Cunningham recently went viral with two short videos (second video here) asking questions about mathematics. Like a few other people, I thought that they were sufficiently interesting to answer.

1. How did people know what they were looking for when they started theorising about formulas? Because I wouldn’t know what to look for if I’m making up math.

Well, first, contrary to your comment “I don’t think math is real,” mathematics is indeed real. Even if the universe was completely different from the way it is, mathematics would still be true. Edward Everett, whose dedication speech at Gettysburg was so famously upstaged by Abraham Lincoln, put it like this: “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.” (OK, not everybody has this view of mathematics, which is called “Platonism,” but in my opinion, it’s the only view that explains why mathematics works).

Second, mathematics is discovered, not invented. The great mathematician G. H. Hardy pointed out “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.” When you embark on a journey of discovery, you don’t know where you’ll end up. That’s what makes it exciting. When the early Polynesians set off in canoes across the Pacific, thousands of years ago, they didn’t know that they would discover Hawaiʻi, Samoa, and New Zealand. They just headed off into the wild blue yonder because that was the kind of people they were.

Now the Babylonians and others developed mathematics primarily because it was useful – for astronomy (which you need to decide when to plant crops) and engineering and business. But the Greek started to do mathematics just for fun. They discovered mathematical truths not because they were useful, but simply because, as Joel Spencer, put it, “Mathematics is there. It’s beautiful. It’s this jewel we uncover” (quoted in The Man Who Loved Only Numbers, p 27).

It is also worth pointing out a subtle kind of prejudice (almost a kind of racism) that is widespread – many people think that the ancients were primitive. They weren’t. They had plumbing, and architecture, and astronomy. They even knew about zero. Some of them were very smart people. There were Babylonian versions of Albert Einstein, whose names are now long forgotten. There were many Greek versions of Albert Einstein (including Archimedes, among others). I kind of wish that schools would teach young people more about what the past was actually like. If people learn anything about the past at all, it’s usually from popular literature:

2. Once they did find these formulas, how did they know that they were right? Because, how?

Short answer: Euclid. Around about his time, the Greeks started to ask themselves exactly that question, and developed the concept of rigorous mathematical proof as an answer. The fact that typical mathematics classes don’t introduce simple proof is yet another indication of how badly broken modern education is. Even this simple visual proof (uploaded by William B. Faulk; click to view animation) gets the idea across:

3. Why is everyone being really mean to me on Twitter? Why are the only people who are disagreeing with me the ones who are dumb, and the physicists and mathematicians are agreeing with me?

Well, that’s also an interesting question. First, for reasons that I don’t fully understand, Twitter just makes people mean.

Second, as Dunning and Kruger famously pointed out, it is the people who know the least that are the most confident.

Third, the original video was in teen-girl English, with multiple uses of the word “like” (I have a sneaking suspicion that this was deliberate). Using teen-girl English for a “serious” subject like mathematics makes people’s heads explode (this will be useful to know when you have your first job interview).

But thank you, Gracie, for asking some really good questions.

Some follow-up remarks on what mathematics is not are here.

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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).

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Alternatives to mathematical Platonism (2)

In my previous two posts, I outlined the Platonist view of mathematics, and the empiricist alternative. There are also two other alternatives:

Formalism

The truths of mathematics appear to be different in nature from the truths of physics. Formalism accepts this, but suggests that the nature of mathematics is inherently cultural. Different branches of mathematics are essentially just games with symbols and arbitrary rules – games that don’t have any particular meaning. Mathematicians simply work within the chosen rules. However, apart from the problem of the “unreasonable effectiveness of mathematics,” the idea that these rules are chosen arbitrarily runs counter to the experience of most mathematicians. When the concept of “number” was extended to include the imaginary numbers, for example, consistency with the existing rules meant that there was very little choice about how imaginary numbers behaved. In the words of mathematician 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” (from the introduction to The Psychology of Invention in the Mathematical Field).

Many officially formalist mathematicians are Platonists at heart. Jean Dieudonné once wrote with refreshing honesty: “On foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hide behind formalism and say: ‘Mathematics is just a combination of meaningless symbols,’ and then we bring out Chapters 1 and 2 on set theory. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working on something real. This sensation is probably an illusion, but is very convenient.” (from “The Work of Nicholas Bourbaki,” American Mathematical Monthly, 77(2), Feb 1970, p. 134–145).

Logicism

Most mathematicians feel that the truths of mathematics are indeed in a different category from the truths of physics – that the truths of mathematics in a sense come first. Logicism is a way of rescuing this aspect of Platonism while avoiding the more mystical aspects. The basis for logicism is that logic also comes before physics – all sciences assume logical thought as a starting point. Logical truths “exist” in some sense, and logicists assume that there are no philosophical difficulties about this kind of existence. In other words, a mystical Platonic world is not needed to explain logic. Consequently, if we can provide a foundation for mathematics in terms of pure logic, we can retain all the benefits of Platonism without any of the problems.

In logicism, numbers are defined as being particular kinds of sets. Logicism began with Gottlob Frege, who published two volumes of his Die Grundgesetze der Arithmetik in 1893 and 1903. Sadly for Frege, his fellow mathematician and philosopher Bertrand Russell found a major flaw – now known as “Russell’s paradox” – in the work, just before the second volume was published. With Alfred North Whitehead, Russell was able to repair the flaw, in a three-volume work called Principia Mathematica (published in 1910, 1912, and 1913).

After 379 pages, Whitehead and Russell are well on the way to proving that 1 + 1 = 2.

The logicist programme, however, is not free of problems. First, it is extremely complex. It took Whitehead and Russell hundreds of pages of complicated logic to prove that 1 + 1 = 2. Normally, we try to explain complex things in terms of simple ones. It seems a little perverse to give such a complicated explanation of numerical facts that we understood in kindergarten. And, by including set theory as part of the basis, it isn’t really “pure logic” any more.

Second, there is more than one way of defining numbers as sets, and none of them is obviously “right.” This has led to the suggestion that sets are “what numbers could not be” (the title of an article by Paul Benacerraf in The Philosophical Review, 74, Jan 1965, pp. 47–73), and that numbers must be fundamentally different in nature from sets – if not Platonic objects satisfying certain axioms, then something else which exists in a non-contingent way.

Third, it is unclear whether logicism has actually gained anything. The starting assumption was that logic was simple and obvious, raising no philosophical problems. But if all of mathematics is hidden deep inside the structure of logic, then perhaps logic is not as simple as it first seemed. Mathematics and logic may in fact be different aspects of the same thing, but this may not make the fundamental questions about mathematical existence go away.

Personally, I still see Platonism as the best answer. How about you?

Alternatives to mathematical Platonism (1)

In my last post, I outlined the view of mathematical Platonism taken by Roger Penrose and other mathematicians. Briefly, in the words of Joel Spencer, “Mathematics is there. It’s beautiful. It’s this jewel we uncover” (quoted in The Man Who Loved Only Numbers, p 27).

Uncovering hidden jewels (James Tissot).

However, some modern mathematicians feel that the time for such ideas is past. In the words of Brian Davies, “It is about time that we … ditched the last remnant of this ancient religion” (in his article “Let Platonism die,” European Mathematical Society Newsletter, June 2007).

One alternative commonly presented is Empiricism. Santa Claus and the Tooth Fairy do not exist, and neither do infinite decimals, or perfect circles, or the set of all natural numbers. Only the physical universe exists. In the words of astronomer Carl Sagan, “The Cosmos is all that is or ever was or ever will be” (the opening sentence of his book Cosmos). Our only truly certain knowledge is physics, and mathematics is in fact a branch of physics. When we say that 2 + 2 = 4, we are not talking about a relationship between Platonic number-objects. Instead, what we mean – and all that we mean – is the empirical truth that two atoms plus two atoms gives four atoms, and likewise for stars, rocks, or people. “Four” is not a noun, it’s an adjective.

The empiricist point of view seems to solve the mystery of the “unreasonable effectiveness of mathematics.” Mathematics is just part of physics, and it isn’t surprising that different branches of physics agree with each other. Empiricism also avoids the need to postulate a “soul” or some other mechanism for peering into an ethereal Platonic world.

Conic sections in theory and practice.

There are two problems with the empiricist point of view, however. First, it 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. There’s still a mystery there: why should mathematics from one context work so well in another?

The second problem is that scientific truths are contingent. We could, for example, live in a world where plants were purple. The fundamental forces of physics could be different from the way they are (although that could imply a lifeless universe). We can even imagine a universe containing no matter at all, only empty space. The laws of mathematics, however, could not have been different – they are necessary. It is impossible to imagine – at least, to imagine consistently – a universe in which 2 + 2 = 5. Even God cannot make 2 + 2 = 5. This indicates that there must be more to mathematical truth than just physics. Statements about mathematics, such as “2 + 2 = 4” are in a different category to statements about the universe, such as “light travels at 299,792,458 metres per second.” But if that is the case, then mathematical truth must in some sense lie outside the universe – which brings us back to Platonism.

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