## Archive for August, 2019

### Science as a literary pursuit

August 24, 2019

I found something in a footnote in the book of Oliver Sacks I mentioned in the previous post which attracted my attention. There is a citation from a letter of Jonathan Miller to Sacks with the idea of a love of science which is purely literary. Sacks suggests that his own love of science was of this type and that that is the reason that he had no success as a laboratory scientist. I feel that my own love for science has a strong literary component, or at least a strong component which is under the control of language. In molecular biology there are many things which have to be named and people have demonstrated a lot of originality in inventing those names. I find the language of molecular biology very attractive in a way which has a considerable independence from the actual meaning of the words. I expect that there are other people for whom this jungle of terminology acts as a barrier to entering a certain subject. In my case it draws me in. In my basic field, mathematics, the terminology and language is also a source of pleasure for me. I find it stimulating that everyday words are often used with a quite different meaning in mathematics. This bane of many starting students is a charm of the subject for me. Personal taste plays a strong role in these things. String theory is another area where there is a considerable need for inventing names. There too a lot of originality has been invested but in that case the result is not at at all to my taste. I emphasize that when I say that I am not talking about the content, but about the form.

The idea of using the same words with different meanings has a systematic development in mathematics in context of topos theory. I learned about this through a lecture of Ioan James which I heard many years ago with the title ‘topology over a base’. What is the idea? For topological spaces $X$ there are many definitions and many statements which can be formulated using them, true or false. Suppose now we have two topological spaces $X$ and $B$ and a suitable continuous mapping from $X$ to $B$. Given a definition for a topological space $X$ (a topological space is called (A) if it has the property (1)) we may think of a corresponding property for topological spaces over a base. A topological space $X$ over a base $B$ is called (A) if it has property (2). Suppose now that I formulate a true sentence for topological spaces and suppose that each property which is used in the sentence has an analogue for topological spaces over a base. If I now interpret the sentence as relating to topological spaces over a base under what circumstances is it still true? If we have a large supply of statements where the truth of the statement is preserved then this provides a powerful machine for proving new theorems with no extra effort. A similar example which is better known and where it is easier (at least for me) to guess good definitions is where each property is replaced by one including equivariance under the action of a certain group.

Different mathematicians have different channels by which they make contact with their subject. There is an algebraic channel which means starting to calculate, to manipulate symbols, as a route to understanding. There is a geometric channel which means using schematic pictures to aid understanding. There is a combinatoric channel which means arranging the mathematical objects to be studied in a certain way. There is a linguistic channel, where the names of the objects play an important role. There is a logical channel, where formal implications are the centre of the process. There may be many more possibilities. For me the linguistic channel is very important. The intriguing name of a mathematical object can be enough to provide me with a strong motivation to understand what it means. The geometric channel is also very important. In my work schematic pictures which may be purely mental are of key importance for formulating conjectures or carrying out proofs. By contrast the other channels are less accessible to me. The algebraic channel is problematic because I tend to make many mistakes when calculating. I find it difficult enough just to transfer a formula correctly from one piece of paper to another. As a child I was good in mental arithmetic but somehow that and related abilities got lost quite early. The combinatoric channel is one where I have a psychological problem. Sometimes I see myself surrounded by a large number of mathematical objects which should be arranged in a clever way and this leads to a feeling of helplessness. Of course I use the logical channel but that is usually on a relatively concrete level and not the level of building abstract constructs.

Does all this lead to any conclusion? It would make sense for me to think more about my motivations in doing (and teaching) mathematics in one way or another. This might allow me to do better mathematics on the one hand and to have more pleasure in doing so on the other hand.