Everything about Foundation Of Mathematics totally explained
Foundations of mathematics is a term sometimes used for certain fields of
mathematics, such as
mathematical logic,
axiomatic set theory,
proof theory,
model theory, and
recursion theory. The search for foundations of mathematics is also a central question of the
philosophy of mathematics: On what ultimate basis can
mathematical statements be called
true?
Philosophical foundations of mathematics
The foundational philosophy of
Platonist mathematical realism, as exemplified by mathematician
Kurt Gödel, proposes the existence of a world of mathematical objects independent of humans; the truths about these objects are
discovered by humans. In this view, the laws of nature and the laws of mathematics have a similar status, and the
effectiveness ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation. The obvious question, then, is: how do we access this world? (cf Anglin 1991 p. 218)
The foundational philosophy of
formalism, as exemplified by
David Hilbert, is based on
axiomatic set theory and
formal logic. Virtually all mathematical
theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is then nothing but the claim that the statement can be derived from the
axioms of set theory using the rules of formal logic (cf Anglin 1991 p. 218).
Merely the use of formalism alone doesn't explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why do "true" mathematical statements (for example, the
laws of arithmetic) appear to be true, and so on. In some cases these may be sufficiently answered through the study of formal theories, in disciplines such as
reverse mathematics and
computational complexity theory.
Formal logical systems also run the risk of
inconsistency; in
Peano arithmetic, this arguably has already been settled with several proofs of
consistency, but there's debate over whether or not they're sufficiently
finitary to be meaningful.
Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own
consistency. What Hilbert wanted to do was prove a logical system
S was consistent, based on principles
P that only made up a small part of
S. But Gödel proved that the principles
P couldn't even prove
P to be consistent, let alone
S!
The foundational philosophy of
intuitionism or
constructivism, as exemplified in the extreme by
Brouwer and more coherently by
Stephen Kleene, requires proofs to be "constructive" in nature – the existence of an object must be demonstrated rather than inferred from a demonstration of non-existence. For example, as a consequence of this the form of proof known as
reductio ad absurdum is suspect (cf Anglin 1991 p. 218).
Some modern
theories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on
mathematical practice, and aim to describe and analyze the actual working of mathematicians as a
social group. Others try to create a
cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world. These theories would propose to find foundations only in human thought, not in any objective outside construct. The matter remains controversial.
Foundational crisis
The
foundational crisis of mathematics (in
German:
Grundlagenkrise der Mathematik) was the early
20th century's term for the search for proper foundations of mathematics.
After several schools of the
philosophy of mathematics ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within
mathematics itself began to be heavily challenged.
One attempt after another to provide unassailable foundations for mathematics was found to suffer from various
paradoxes (such as
Russell's paradox) and to be
inconsistent: an undesirable situation in which every mathematical statement that can be
formulated in a proposed system (such as 2 + 2 = 5) can also be
proved in the system.
Various schools of thought on the right approach to the foundations of mathematics were fiercely opposing each other. The leading school was that of the
formalist approach, of which
David Hilbert was the foremost proponent, culminating in what is known as
Hilbert's program, which thought to ground mathematics on a small basis of a logical system proved sound by
metamathematical finitistic means. The main opponent was the
intuitionist school, led by
L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols (van Dalen, 2008). The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of
Mathematische Annalen, the leading mathematical journal of the time.
Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program couldn't be attained. In
Gödel's first result he showed how to construct, for any sufficiently powerful and consistent finitely axiomatizable system – such as necessary to axiomatize the elementary theory of
arithmetic – a statement that can be shown to be true, but that doesn't follow from the rules of the system. It thus became clear that the notion of mathematical truth can not be reduced to a purely
formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system wasn't powerful enough for proving its own consistency, let alone that a simpler system could do the job. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means. Meanwhile, the intuitionistic school had failed to attract adherents among working mathematicians, and floundered due to the difficulties of doing mathematics under the constraint of
constructivism.
In a sense, the crisis hasn't been resolved, but faded away: most mathematicians either don't work from axiomatic systems, or if they do, don't doubt the consistency of
ZFC, generally their preferred axiomatic system. In most of mathematics as it's practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as
logic and
category theory), they may be avoided.
A working perspective
To give an example, in number theory there's a huge body of doctrine, a tiny fraction of which has been developed in a particular axiomatic system, say Peano arithmetic (PA). Most of this work could be developed in PA; as a famous example, the
prime number theorem is provable in
PRA (Sudac (2001)), a much weaker theory than PA. But the working number theorist is concerned with proving theorems from initial assumptions which are obviously true using proof methods which are obviously correct, not with any particular logical system. In fact, the "crisis"-causing assertions discovered by Gödel
are assertions about
Diophantine equations, one of the main avenues in number theory. It may or may not be the case that there's a fundamental limit to what humans can understand about numbers (for example, there may be true number-theoretical principles which can't be perceived as being true by any human), but Gödel's theorem doesn't tell us which of these is the case, and we've no way of knowing. It may or may not be that we're required to introduce principles which are not expressible in the language of first order arithmetic in order to decide questions which are (for example the consistency of PA), but Gödel's theorem doesn't tell us which of these is the case, and again we've no way of knowing. It is often asserted that in light of Gödel's theorem one must introduce set-theoretical principles in order to decide certain number theoretical questions, but this assertion is unjustified. Gödel's theorem doesn't put any such constraints on the nature of the principles involved (for example the language in which they must be expressed). The attitude of the working number theorist is thus a reasonable one: one doesn't spend time thinking about such things, as there's simply no way to know. Instead one continues to prove theorems, and true principles which may be outside this or that logical system will be appealed to as required. Such principles will be introduced by people thinking about and solving actual problems, on the frontline. The problems (assuming there's no limit to what humans can understand about numbers) will be solved by people carrying on in the same way as they did before.
Summary of the three philosophies
» Platonism
» “Platonists, such as Kurt Gödel, hold that numbers are abstract, necessarily existing objects, independent of the human mind” (Anglin (1994) p. 218)
» Formalism
» “Formalists, such as David Hilbert (1862-1943), hold that mathematics is no more or less than mathematical language. It is simply a series of games...” (Anglin (1994) p. 218)
» Intuitionism
» “Intuitionists, such as L. E. J. Brouwer (1882-1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which wouldn't exist if there were never any human minds to think about them.” (Anglin (1994) p. 218)
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