With all my fancy book-learning and abstract axiomatic thinking, there are a lot of classical results that I’ve never learned. I just ran into an interesting classical theorem by Sylvester about symmetric matrices. Let A_n be the submatrix of the first n rows and columns. When you complete the square on the associated quadratic form, the coeffients of each square term is of the form

(det A_n)/(det A_n-1).

From this it follows that the matrix is positive definite if and only if each (det A_n) is positive.

The criterion is not practical for large matrices, but it does imply one interesting theoretical result: the set of positive definite matrices is a real semialgebraic set.

Warning: if your relatives ever find out about this website, you’ve sentenced yourself to a lifetime of receives its wares for birthdays and holidays.

I just came across a reference to the work of the philosopher of mathematics Hartry Field. This is what Wikipedia has to say:

Fictionalism was introduced in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine’s indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as true, Field suggested that mathematics was dispensable, and therefore should be rejected as false. He did this by giving a complete axiomatization of Newtonian mechanics that didn’t reference numbers or functions at all. He started with the “betweenness” axioms of Hilbert geometry to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.

Does anyone know if that is an accurate summary of Field’s argument? It seems obviously wrong to me.

Via Crooked Timber.