In the period very roughly from the beginnings of modern physics (1905) up to Alan Turing's description of the Turing machine in 1938, one of the focal points of dispute in the theory of knowledge was the foundations of mathematics.
The main players in this struggle are:
Gottlob Frege : the founder of Logicism , the position that the whole of mathematics can be reduced to a set of relations derived one from the other solely by means of logic, without reference to specifically mathematical concepts such as number. Wittgenstein attempted to carry Frege's concepts of mathematics over to the natural language, with predictably inane results. Frege was also the inspiration for Rudolph Carnap and the various schools of Logical Positivism which continued to wrestle with the problems generated by the new physics. Frege took no part in the struggle after 1903, and died in bitterness and isolation in 1925 having failed to complete a system based on his concept without the appearance of contradictions or logical flaws. His project was later continued by Bertrand Russell and Alan Whitehead.
David Hilbert : the founder of Formalism , the position that mathematics consists solely in the generation of combinations of symbols according to arbitrary rules and the application of logic. His first important work in 1899 was to produce a definitive set of axioms for Euclidean geometry without any appeal to spatial references or intuition. In 1905 (and again from 1918) Hilbert attempted to lay a firm foundation for mathematics by proving consistency - that is, that finite steps of reasoning in logic could not lead to a contradiction. But in 1931, Kurt Gödel showed this goal to be unattainable: propositions may be formulated that are undecidable; thus, it cannot be known with certainty that mathematical axioms do not lead to contradictions.
Luitzen Brouwer : the founder of Intuitionism , that views the nature of mathematics as mental constructions governed by self-evident laws. Brouwer is considered the founder of Topology. In his doctoral thesis in 1907, On the Foundations of Mathematics , Brouwer attacked the logical foundations of mathematics and in 1908, in On the Untrustworthiness of the Logical Principles , he rejected the use in mathematical proofs of the principle of the excluded middle , which asserts that every mathematical statement is either true or false and no other possibility is allowed. In 1918 he published a set theory, the following year a theory of measure, and by 1923 a theory of functions, all developed without using the principle of the excluded middle. Brouwer was the first to build a mathematical theory using Logic other than that normally accepted, a method of research since applied to quantum mechanics and more widely.
Kurt Gödel : in 1931, author of the epoch-making Gödel's theorem, which states that within any consistent mathematical system there are propositions that cannot be proved or disproved on the basis of the axioms within that system and that, therefore, it is uncertain that the basic axioms of arithmetic will not give rise to contradictions. The proof was specifically aimed against Russell & Whitehead's Principia Mathematica - an attempt to complete Frege's project. This article ended nearly a century of attempts to establish axioms that would provide a rigorous basis for all mathematics. Gödel was an avowed Kantian and expresses support for Husserl 's Phenomenology (http://www.marxists.org/reference/subject/philosophy/works/husserl.htm).
Alan Turing ; founder of computer science and research in artificial intelligence. Motivated by Gödel's work to seek an algorithmic method of determining whether any given proposition was undecidable, with the ultimate goal of eliminating them from mathematics, he proved instead, in 1936, that there cannot exist any such universal method of determination and, hence, that mathematics will always contain undecidable propositions. To illustrate this, Turing posited a simple device that possessed the minimal properties of a modern computing system: a finite program, a large data-storage capacity, and a step-by-step mode of mathematical operation - the "Turing machine". Using Hilbert's own methods, Turing and Gödel put to rest the hopes of David Hilbert & Co. that all mathematical propositions could be expressed as a set of axioms and derived theorems.
Turing championed the theory that computers could be constructed that would be capable of human thought and his writing on this subject show considerable affinity with behaviourist psychology.
The following extended quote in which Gödel summarises his position is worth considering: "... it turns out that in the systematic establishment of the axioms of mathematics, new axioms, which do not follow by formal logic from those previously established, again and again become evident. It is not at all excluded by the negative results mentioned earlier that nevertheless every clearly posed mathematical yes-or-no question is solvable in this way. For it is just this becoming evident of more and more new axioms on the basis of the meaning of the primitive notions that a machine cannot imitate. "I would like to point out that this intuitive grasping of ever newer axioms that are logically independent from the earlier ones, which is necessary for the solvability of all problems even within a very limited domain, agrees in principle with the Kantian conception of mathematics. The relevant utterances by Kant are, it is true, incorrect if taken literally, since Kant asserts that in the derivation of geometrical theorems we always need new geometrical intuitions, and that therefore a purely logical derivation from a finite number of axioms is impossible. That is demonstrably false. However, if in this proposition we replace the term "geometrical" - by "mathematical" or "set-theoretical", then it becomes a demonstrably true proposition. I believe it to be a general feature of many of Kant's assertions that literally understood they are false but in a broader sense contain deep truths. In particular, the whole phenomenological method, as I sketched it above, goes back in its central idea to Kant, and what Husserl did was merely that he first formulated it more precisely, made it fully conscious and actually carried it out for particular domains. Indeed, just from the terminology used by Husserl, one sees how positively he himself values his relation to Kant. "I believe that precisely because in the last analysis the Kantian philosophy rests on the idea of phenomenology, albeit in a not entirely clear way, and has just thereby introduced into our thought something completely new, and indeed characteristic of every genuine philosophy - it is precisely on that, I believe, that the enormous influence which Kant has exercised over the entire subsequent development of philosophy rests. Indeed, there is hardly any later direction that is not somehow related to Kant's ideas. On the other hand, however, just because of the lack of clarity and the literal incorrectness of many of Kant's formulations, quite divergent directions have developed out of Kant's thought - none of which, however, really did justice to the core of Kant's thought. This requirement seems to me to be met for the first time by phenomenology, which, entirely as intended by Kant, avoids both the death-defying leaps of idealism into a new metaphysics as well as the positivistic rejection of all metaphysics. But now, if the misunderstood Kant has already led to so much that is interesting in philosophy, and also indirectly in science, how much more can we expect it from Kant understood correctly?" [The modern development of the foundations of mathematics in the light of philosophy, Gödel 1961]
Gödel has done a great service here in drawing the very precise and formal development of the foundations of mathematics back to the fundamental questions which drove classical epistemology. The real question is not the building of ever more elaborate logical edifices, but understanding the nature and source of these "more and more new axioms on the basis of the meaning of the primitive notions".
With the more or less decisive defeat of the Formalist and Logicist schools, and Turing's reduction of the problems to questions of programming, controversy in the foundations of mathematics died down after World War Two. Turing's work introduced new concepts of complexity in language which have provided the basis for Noam Chomsky's Kantian structural psychology and the foundations of complexity theory. Gödel's theorem indicates that the behaviour of even purely formal systems cannot be completely described by formal logic, and this is at the root of the inherent complexity, unpredictability and richness of the world of Nature and society.
None of this controversy bore on the issue of how it comes that mathematics finds application in the sciences. Attempts to reduce mathematics to logic failed, so it must be accepted that mathematics is a science which studies an aspect of Nature, viz., Quantity , it is not just rules for manipulating symbols. Nevertheless, the "Third Positivism", which climbed out of the ashes of the positivism of Mach & Co., took inspiration from the Logicist School and remain an important trend to this day. The way in which mathematics found application in the New Physics was central to the development of positivistic philosophy in the period from 1905 up to recent times.