Background
I am writing a paper about David Hilbert's impact for my history of mathematics class. The MacTutor Mathematics history website says that,
As we saw above, Hilbert's first work was on invariant theory and, in 1888, he proved his famous Basis Theorem.
and elaborating,
He discovered a completely new approach which proved the finite basis theorem for any number of variables but in an entirely abstract way. Although he proved that a finite basis existed his methods did not construct such a basis.
As I understand, some mathematicians of the time were uncomfortable with Hilbert's novel methods. He proved that a "basis" existed without ever showing any example of such a basis. Felix Klein later wrote to Hilbert,
I do not doubt that this is the most important work on general algebra that the 'Annalen' has ever published.
Questions
I have the following questions:
- What is the significance of invariance theory on mathematics? What is the basic idea of invariance theory, and why is the Basis theorem important?
- How impactful was Hilbert's new "abstract" approach to mathematics?
If you have an answer for (1) but not (2) or (2) but not (1), that is fine! I would appreciate any thoughs on the impact of Hilbert on mathematics.
The intuitive meaning of invariant theory is it looks for properties invariant under linear coordinate changes. For example take a quadratic form $$ax^2+bxy+cy^2$$ this factors as a product of two linear factors. Now these two factors could be equal or distinct. That is a property invariant under a linear change in the variables $x,y$. This is also determined by whether the discriminant $$b^2-4ac$$ is equal to zero or not. The discriminant (and its powers) are in fact the only invariant of a quadratic. Invariant of other higher order expressions would describe further more subtle properties, again properties not depending on the coordinate system.
One word describes Hilbert's influence: immense.