From: http://home.mira.net/~andy/works/maths.htm

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.