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There have arisen three main
philosophies, or schools of thought, concerning the foundations of mathematics
the so-called logistic, intuitionist, and formalist schools.
Russell and Whitehead’s
logistic thesis was that mathematics is a branch of logic. Rather than being
just a tool of mathematics, logic becomes the progenitor of mathematics. All
mathematical concepts are to be formulated in terms of logical concepts, and
all theorems of mathematics are to be developed as theorems of logic; the
distinction between mathematics and logic become merely one of practical convenience.
The Brower intuitionist thesis is that mathematics is to be built solely by finite constructive methods in the intuitively given sequence of natural numbers. According to this view, then, at the very base of mathematics lies a primitive intuition, allied, no doubt, to our temporal sense of before and after, that allows us to conceive a single object, then one more, then one more, and so on endlessly. For the intuitionists, a set cannot be thought of as a ready-made collection, but must be considered as a law by means of which the elements of the set can be constructed in a step-by-step fashion.
The Hilbert formalist thesis
is that mathematics is concerned with formal symbolic systems. In
fact, mathematics is regarded as a collection of such abstract developments, in
which the terms are mere symbols and the statements are formulas involving
these symbols; the ultimate base of mathematics does not lie in logic but only
in a collection of pre-logical marks or symbols and in a set of operations with
these marks. Since, from this point of
view, mathematics is devoid of concrete content and contains only ideal
symbolic elements, the establishment of the consistency of the various branches
of mathematics becomes an important and necessary part of the formalist
program.