As a mathematician, I have given a certain amount of thought to the question “What is mathematics?”, and I confess myself somewhat dissatisfied with the “silly clever” answer that mathematics is what mathematicians do. Mathematicians in general seem to like this answer, perhaps because “silly clever” answers are a speciality of mathematicians. But mathematicians, above everybody else, should appreciate that this is a circular definition and therefore is not a definition at all.
In mathematics, the purpose of a definition is to distinguish between things that are (A) and things that are (not A). But in fact this is impossible, and all one can actually hope to do is to distinguish between things that are (A and not B) and things that are (B and not A). That is what all the early 20th century work on logic established beyond all reasonable doubt.
In other words, mathematicians have proved that it is not possible to answer the question “what is mathematics?” The kinds of questions one can hope to answer are (a) “what is mathematics but not physics?” and (b) “what is physics but not mathematics?” When framed in these terms, the questions and their answers are a good deal more illuminating. Thus, we may take a particular subject which we assume is part of mathematics and/or physics, and ask ourselves which it is.
But in order to do this, we need a definition to distinguish mathematics from physics. We don’t need to define physics, and we don’t need to define mathematics, but we do need to define the distinction between the two. Now I have no doubt that you have your own definition, but I would like to propose my own: physics is the study of our universe, and mathematics is the study of all possible universes.
You may not agree with my definition, but I think you have to agree it is succinct, it is clear, and it can be applied in practice. Let’s try it on an example: is string theory physics or mathematics? The answer is obvious: string theory is the study of all possible universes, and therefore it is mathematics, not physics. There are two famous books, by Peter Woit and Lee Smolin, published in 2006, which argue this thesis in depth. I am prepared to admit that string theory is mathematics, and these two books argue conclusively that it is not physics, so I don’t think there is really much room for any further discussion.
This entry was posted on May 29, 2021 at 6:13 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.