Sometimes I find myself to be the crossroads of the universe. Related things arrive at me independently from different directions at much the same time.
Arriving from one direction came George Spencer-Brown’s Laws of Form containing, in the introduction, the following incisive paragraph:
In arriving at proofs, I have often been struck by the apparent alignment of mathematics with psycho-analytic theory. In each discipline we attempt to find out, by a mixture of contemplation, symbolic representation, communion and communication, what it is we already know. In mathematics, as in other forms of self-analysis, we do not have to go exploring the physical world to find what we are looking for. Any child of ten, who can multiply and divide, already knows, for example, that the sequence of prime numbers is endless. But if he is not shown Euclid’s proof, it is unlikely that he will ever find out, before he dies, that he knows.
At the centre of this crossroads lies a roundabout (what a fruitful metaphor this is turning out to be!). Chasing themselves endlessly around the roundabout are:
- what I call the concept of just in time mathematics: the phenomenon of pure mathematical concepts turning up just in time for scientific theories to be built on them. As Corfield argues, this happens too often to attribute it to coincidence.
- Mach’s argument that scientific progress is not a continual refinement of better measurements leading to better theories, it is a progression reflecting what scientists can think.
- Spencer-Brown’s argument that mathematical progression is not an outward exploration but an inner discovery of what we already know.
This loop supports my view (expressed in various blogs below—see the waterfall at Wakefield) that mathematicians are constrained by the universe, of which they are a part, in what they discover and scientists are thereby enabled by those mathematical advances to uncover the mathematics which is the universe.