Leopold Kronecker (1823-1891) rejected Georg Cantor's (1845-1918) transfinite numbers and sets; he also rejected infinitesimals as well as other mathematical entities - such as transcendental numbers. What was his philosophical view from which he derived his rejection of various mathematical entities? (Or rather, did he reject the mentioned entities based on mathematical arguments?)
- Anne Troelstra & Dirk van Dalen, Constructivism in mathematics: An Introduction. Volume 1 (1988), page 17.
There is no "explicit" philosophy of mathematics in Kronecker's works.
He may be regarded as a "constructivist", or perhaps as a precursor of the finitist approach.
In his essay "Uber den Zahlbegriff" (1887) he outlined the project of "arithmetizing" Algebra and Analysis; that is, to found these disciplines on the fundamental notion of number, avoiding thus geometrical intuition.
In his arithmetization project he considered a mathematical definition acceptable only if it could be checked in a finite number of steps, criticizing the "pure" existence proofs. He stated that an existence proof for a number could be considered correct only if it contained a method to find the number whose existence was proven.
Some of his remarks belong to mathematical folklore, like his widely reported statement that:
"the Lord made the natural numbers (ganze Zahlen), everything else is the work of men";
the same idea is reiterated in the following statement:
"I consider mathematics only as an abstraction of the arithmetical reality".
Here are some philosophical statements of Kronecker from his Public lecture in summer semester 1891 at Berlin – Kronecker's last lecture. "Sur le concept de nombre en mathematique":
The whole mathematics is there to be applied.
Mathematics is a natural science – not better, not more complete, and not simpler the phenomena can be described than mathematically.
All considerations and speculations about the magnitudes which are defined as limits fail, as soon as there are multiple limits.
And here is his Credo:
I believe that we will succeed one day to "arithmetize" the whole contents of all these mathematical theories, that is to base it solely on the concept of number in its stricter sense, i.e., to get rid of the added modifications and extensions (namely the irrational and continuous magnitudes) which mainly have been caused by the applications on geometry and mechanics. [Leopold Kronecker in K. Hensel (ed.): "Leopold Kroneckers Werke" III, Teubner, Leipzig (1895-1931) p. 253]