Real and Free
In the world of open source software there is a need to differentiate between the two uses of the word free. In general, the open source movement supports the concept of software being “free as in speech” as well as “free as in beer“. I am reminded of this by the name of David Corfield’s blog, The Philosophy of Real Mathematics, and his associated book Towards a Philosophy of Real Mathematics. What’re those reals doing in there?
Again this week I met a person with a first degree in mathematics who apparently not only knew no mathematical philosophy but had never actually heard of the subject. I stumbled across it myself as an undergraduate not because of the enthusiasms of my lecturers but because I found a copy of Nagel and Newman’s wonderful book on Gödel’s theorems in my local library. This introduced me to the idea that mathematics was actually alive and Littlewood’s essay The Dilemma of Probability Theory in his Miscellany made me realise that mathematics could actually be disputed. As Littlewood says in that essay: “Mathematics (by which I shall mean pure mathematics) has no grip on the real world; if probability is to deal with the real world it must contain elements outside mathematics…”.
After reading the Nagel book I searched around for courses and attended some of Imre Lakatos’ seminars at the LSE. At the time I had no idea of the reputation of the person to whom I was listening and, in retrospect, would probably never have dared attend had I known. These seminars must have been some of the last he gave before his early death.
But the issue of real is still there. Mathematical philosophy to me has traditionally been about the foundations of mathematics, in particular problems in arithmetic, not about real mathematics and the way mathematicians work: perhaps that is the reason for the disconnexion between mathematics degrees and mathematical philosophy. For this reason I have always believed mathematical philosophy to be particularly accessible to young people, bright teenagers wanting to find the excitement in mathematics and break the common misconceptions of “all mathematicians are dead”, “mathematics was finished some hundreds of years ago, probably by an ancient Greek”, “all of the answers are in the back of the book”, “my teacher knows everything about mathematics”. These teenagers also enjoy being able to push back on their teachers: I suspect that “You’re using Reductio ad Absurdum there. I can’t accept that because I’m an intuitionist” would be a delight for any disruptive high-school mathematics student to use.
Corfield moves from basic arithmetic and logic to real mathematics. What reminded me of the open source software is that real is used here as in real beer (actually ale but the analogy is better with beer). In Britain there has been for many years a Campaign for Real Ale. Real mathematics includes topological quantum field theory: something that, I suspect, may be beyond the resources of most teenagers. And even of those omniscient teachers.