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 Russell 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 physics?
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.
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.
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.
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.
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.
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More about the philosophy of mathematics: