David Hilbert (1862-1943)
-With his formalist project he tried to create a ‘theory of everything’ (now called Hilbert’s program) of mathematics. Hence he rejected self-defeating thinkers: in a speech (1930) to the Society of German Scientists and Physicians he stated that “We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion none whatever in natural science. In opposition to the foolish ignorabimus our slogan shall be:We must know — we will know!”.
Hilbert was probably referring to a speech (188) of the German physiologist Emil du Bois-Reymond. In that speech Bois-Reymond claimed that there are seven ‘world riddles’ that science has to explain, but that three of them are impossible to solve due to our limitations. The three riddles being: 1) the nature of matter and force; 2) the origin of motion; 3) the origin of sensations (regarding consciousness)- Bois-Reymond believed that the answers to this question transcend our intellectual capabilities. This is why the phrase “ignoramus et ignorabimus” was used.
The biologist Ernst Heckel, like Hilbert, rejected the ‘ignorabimus’ concept by pointing out instead that science would ultimately answer to those riddles- hence Heckel suggested that the philosophy of science should be ‘impavidi progrediamur’ (‘we must proceed without fear’)
-In 1899 he proposed new axioms (twenty) for the foundations of Euclidean geometry. Hilbert was particularly focused on the independence, completeness and consistency of the axioms.
-In 1900 Hilbert presented the famous 23 outstanding mathematical problems of his time.
-In the 1900 speech at the International Congress of Mathematicians in Paris he describes Fermat’s last theorem as “a free invention of pure reason, belonging to the region of abstract number theory”. My impression is that it is difficult to understand whether Hilbert’s believed mathematical objects are created or discovered, since verbs suggesting the act of creation or the act of discovering are used interchangeably- though considering Hilbert’s formalist program, one would be more inclined to believe that he would support the idea that mathematical objects are created.
Hilbert explains that rigor in maths is achieved when a proof is developed through a finite number of steps that are logically deduced. The simpler the steps the better the quality of the proof.
He discusses the fact that mathematicians need to develop independent, complete and non-contradicting axioms for arithmetic.
He believes (just like Poincaré) that mathematical ‘existence’ arises when a mathematical object does not create contradiction. As an example, Hilbert cites the square root of -1.
Hilbert defends Cantor’s continuum hypothesis.
Hilbert describes mathematics as “an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts”
Hilbert’s goal is to find a general law (axiom of solvability) through which every mathematical problem can be either solved or proved to be impossible to prove.
“This conviction of the solvability of every mathematical problem is a powerful incentive…. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.”
–Hilbert’s infinite dimensional space will find useful applications in quantum physics
-In a 1925 speech at the Westphalian Mathematical Society he expresses his support for Weierstrass’ foundation of analysis.
-In 1928, at the Bologna congress, Hilbert discussed the problem of the completeness of first-order-logic (this problem and others concerning logic will be solved soon later by Gödel).
 1900 speech: http://aleph0.clarku.edu/~djoyce/hilbert/problems.html