The Tricycle Down The Rabbit Hole

In my previous blogpost (What Is Mathematics?) we saw that platonists believe that what mathematicians do is discovering things about a world which exist independently of themselves. Intuitionists, on the other hand, believe that mathematicians do not discover but create mathematical theorems*. For many mathematicians Platonism is the obvious choice here. Doesn’t nature just show us how mathematics works? We need only look at triangles drawn in the sand to see that Pythagoras’ theorem is true, don’t we? Isn’t that discovery? Let’s take a look at an example of a mathematical theorem to see whether Platonism is really that obvious.

Suppose that some brilliant mathematician, burning the midnight-oil in her lonely attic, finds some very complex geometrical theorem (say, about the volume of an exotic geometrical shape). It may be that this theorem is not instantiated; it may be that there is nothing in nature which actually has this shape (maybe the shape has a large number of dimensions). It is not so obvious that the geometrical theorem that our lonely mathematician has come up with exists anywhere outside her mind (and the paper she wrote it down on). Why should we believe that the mathematician has discovered anything? That would imply that it was already there before she came along. Might we not say that she just ‘made it up’? Such questions show that it is far from obvious that Platonism is self-evident while Intuitionism is not.

Why should we care?

‘Ok, fine.’ I hear you think, ‘discovering and creating are two different things. Platonists and intuitionists have different ideas about what mathematics is. They disagree on whether mathematics necessarily exists outside the mind of the mathematician. But why is that important? Does it matter for how we do mathematics?‘

Yes, it does. One of the rules that is often used in mathematics is ‘the exclusion of the middle ground’. It is the simple rule that any mathematical proposition is either true or it is false – there is no middle ground. Mathematicians use this rule whenever they prove a proposition by contradiction. That works as follows. Suppose we want to prove a certain proposition. If we can show that the negation of our proposition (the assumption that it is false) leads to a contradiction, then ‘the exclusion of the middle ground’ tells us “assuming the falseness of the proposition leads to a contradiction, so the proposition must be true.”

Intuitionists argue that proof by contradiction isn’t really proof at all. They believe that real proof is constructive. The only thing that is constructed in a case of proof by contradiction is that the negated proposition is not true, not that it is false. Mathematical proof, according to the intuitionist, shouldn’t be based on the metaphysical assumption that the truth is in a sense ‘binary’ (that propositions must be either true or false). In cases where ‘the exclusion of the middle ground’ seems like a useful rule, it must be possible to find real, constructive proofs. Intuitionists, therefore, believe that much of what we call mathematical knowledge is actually based on conjecture (because it is based on ‘the exclusion of the middle ground’-rule).

Can you think of a reason why the intuitionists are skeptical of the ‘exclusion of the middle ground’-rule? (please leave your answer as a comment to this blogpost).

Intuitionism & P vs NP

Some computer scientists believe that the skepticism of the intuitionists regarding ‘the exclusion of the middle ground’ is connected to the infamous P vs NP problem. In one sentence, the P vs NP problem is finding an answer to the question whether any mathematical problem whose solution can be quickly verified (to be an actual solution), can also be quickly solved. Some say that the difference between constructive and non-constructive proofs as well as the difference between solving problems and verifying solutions can be understood as a difference between bottom-up and top-down mathematical reasoning.

If there is indeed this connection (between the intuitionism debate and the P vs NP problem) then the interpretation of mathematics may be of great importance to, for example, cryptography. Much of the security of financial transactions today relies on certain mathematical problems being very difficult to solve (such as breaking very large numbers down into prime factors). What if whole swathes of mathematical knowledge turn out to be based on conjecture? Of course, these conjectures are not necessarily false, but intuitionist philosophy casts doubt on them. But how will the world’s economy fare, if so much is based on trust in financial systems? Trust which is often based on mathematical proof?

What is a Mathematician?

Back to the philosophy of mathematics. Can we answer the question from my earlier blogpost ‘What is a Mathematician’? We might say that the platonist believes that all physical things can be a mathematician, as long as they follow (or embody?) the rules of mathematics (whether it be some conscious being, a lifeless calculating machine, or even a falling stone). The intuitionist, on the other hand, believes that a mathematician must be able to apply the rules of mathematics.

I’ll illustrate this difference by comparing mathematics to a game of chess. Let’s say that every possible state of affairs on the chessboard represents a mathematical proposition and that the moves to get to that state of affairs are limited by the rules of mathematics.

The platonist would say that any state of affairs on the chessboard that can be reached by following the rules of mathematics is a mathematical proposition – whether it actually has been reached or not. They just exist. Or they don’t. The intuitionist, on the other hand, would say that mathematical propositions are constructed by the mathematician by following the rules of mathematics in moving about the pieces on the chessboard. Before the mathematician moved the pieces, the proposition simply didn’t exist yet. 

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More about the philosophy of mathematics:

https://plato.stanford.edu/entries/philosophy-mathematics/#For

*) Wait a minute! What happened to Formalism? Why aren’t we discussing that anymore? To see how Formalism differs from Platonism and Intuitionism it is important to know that formalists believe that mathematics in itself is ’empty’ – consisting in tautologies whose truth is independent of anything material. Understanding this is at the basis of what I wrote about numbers at the end of the post What Is Mathematics?. I’ll discuss the emptiness of mathematics in a later blogpost and here focus on the difference between Platonism and Intuitionism.

Mathematics is a language. More specifically, mathematics is the language that scientists use to organise and order observations.

For example, physicists may describe falling stones in terms of mathematical concepts like parabolas and perfect spheres and sociologists describe their observations of large numbers of people in terms of normal distributions and differential equations.

However, stones are not perfect spheres and groups of people don’t behave exactly as described by normal distributions. It follows that talking about falling stones is not the same as talking about spheres that follow parabolas. Considerations like these prompted Bertrand Russel to say that

“mathematics may be defined as the subject in which we don’t know what we are talking about, nor whether what we are saying is true.”

So what is the relation between the work of mathematicians and the observations that physicists make? What is the relation between mathematics and physicis?

Mathematicians do not agree on this issue. In this blogpost I want to show you that there are different views on the nature of mathematics. The view that most mathematicians have (and, perhaps without being aware of the discussion, most physicists as well) is called Platonism. Platonism is named after the ancient Greek philosopher. And that’s not a coincidence.

Platonism

The Plato of old is known for many things, but perhaps most for his ‘theory of forms’. According to the theory of forms the reality that we humans see is made up of imperfect copies of some other ideal reality: the reality of forms. According to Plato, our situation is comparable with that of prisoners that are chained inside a cave. The prisoners don’t see the real world outside the cave, but only the shadows cast by things in this real world on the walls of the cave. It is the task of the philosopher to try to escape the cave and see the real world – the world of perfect forms. The way to do so, according to Plato, is by studying mathematics. Plato’s reality of forms is a reality described by mathematics (in terms of perfect spheres and ideal straight lines).

I know of no present-day scientists who take Plato’s view seriously in the way Plato formulated it. There is, however, a sense in which the theory of forms is still a part of modern philosophy of science. In the modern philosophy of science, Platonism is the name associated with a view on the nature of mathematics. Not all platonists agree on what Platonism exactly is, but for our present discussion we may characterise it as follows:

Platonists believe that what mathematicians do is discovering things about a world that exist independently of themselves. The theorems that mathematicians discover exist independently of the mathematician, and un-proven yet provable theorems exist just as proven theorems do.

Intuitionism

Although Platonism is probably the most common view on the nature of mathematics, it is not the only possible view. Another way of looking at these matters is that of the intuitionist. The intuitionist regards mathematics as a creative process in which mathematical theorems are created as intuitions in the mathematician’s mind. Mathematical theorems do not exist independently of the mathematician. In other words: mathematical theorems do not exist before they are proved.

Formalism

Yet another way to look at these matters is that of proponents of Formalism.

Formalists believe that mathematics is merely a formal game in which a fixed set of rules (the rules of logic) are followed to go from axioms to theorems and from those theorems to yet more complex theorems.  The axioms and theorems are about numbers. It may be the case that these numbers refer to things we can observe (tables or chairs or what have you) but numbers themselves are meaningless symbols, and the theorems in higher mathematics are strings of such symbols within the game of mathematics – nothing more.

Difference

According to the formalist, we saw, mathematical axioms and theorems are about numbers. Neither the intuitionist nor the platonist would disagree. The difference between Formalism, Intuitionism and Platonism lies in the belief about what numbers are. The platonist believes that numbers in some sense exist independently of the mathematician; the intuitionist believes that numbers exist only in the mind (or intuition) of the mathematician; and the formalist believes that numbers are meaningless abstract symbols that don’t exist outside the game of mathematics.

It seems to me that much of the debate introduced above depends on our idea of what a mathematician is. Does it have to be a living creature? …with a PhD perhaps?

The most general answer possible would be that any physical process can be a ‘mathematician’. Read my next blogpost to find out what the consequences are of such a ‘most general answer’. In my next blogpost I will also give you a concrete example in which the different interpretations are illustrated.

Don’t forget to subscribe to the updates on this blog so that you receive the next blogpost automatically! 😉

[fill in your email-address in the box at the upper-right corner of the page (below ‘categories’)]

More about the philosophy of mathematics:

https://plato.stanford.edu/entries/philosophy-mathematics/#For