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First there was Plato who with his Forms introduced a solution before the problem was defined. From his theory of the Forms, Plato advanced the ontological position that mathematical objections, for examples the number 2 or lines, do in fact exist. Although we may not have direct access to mathematical objects, in the concrete corporeal sense, but they do exist, eternally.
Soon afterwards, Aristotle solved the epistomialogical concern about how to best determine mathematical truth. With his introduction of Logic, Aristotle set all of Western thinking on the path of demonstrative proof of knowledge.
Note: In philosophy, knowledge comes in two forms. (1) a prior -- knowledge or truth derived from pure rational thinking and (2) a posterior -- knowledge attained through empirically through sense data and experiences.
Later, Kant introduced two notions of what a true proposition or statement may be. (1) analytic -- wherein a statement or proposition is true by the very definition of the words used versus (2) synthetic -- propositions that are true because they map the real world. According to Kant, mathematical propositions are a prior synthetic.
Leibniz comes along and agrues that mathematics is really just Logic and that all mathematical propositions are necessarily true. When Leibniz claimed 'must be necessarily' true, he unintentionally introduced a modal concept: the difference between what is necessarily true versus what is possibily true. The debate continues in the subject called Modal Logic.
Centuries later, Frege takes up the Leibnizian cause and establishes the Logicism agenda, to reduce all of mathematics to Logic.
Russell after noting an error in Frege's reasoning, which has become known as Russell's Paradox, adopts and continues Logicism almost to the nth degree in his and Whitehead's Principia Mathematica. Unfortunately, to date, the Logicism scheme has been abandoned. (Well almost abandoned, there is the new school of Neo-Logicism.) In his attempt to solve Frege's problem, Russell advanced the field of Set Theory which continues to-day. The Set-theory agenda aims to reduce all of mathematics to a finite number of set-theoretic axioms. In other words, all mathematical propositions can be proven by a finite number of 'self-evidently true' rules of mathematics, currently known as ZFC (Zemelo-Frankle + Axiom of Choice).
Then we have some one like Brouwer who comes along and purposes that Classical logic is flawed in some way and therefore leads mathematics based on Logic astray. Instead of Classical interpretation of Logic, Brouwer purposed Intutitionism in which we alter the once held belief of logic and re-define some logical notions, i.e. re-interpeting the logical principles.
In addition, there was the idea of Constructivism. That is defined as deriving mathematics from what can be rationally thought and applied and not necessarily proven. Its strength depends on proof through application.
Then, Hilbert suggests the Formalist programme. Since mathematics may have different foundations e.g. Logic, Set-Theory, Constructivism, or Intutitism, why not search for the ultimate Foundation of Mathematics and determine the laws that govern this Foundational structure. After all, mathematics is no different from a game of pushing symbols on a page around.
Well, some guy by the name of Gödel put the whole Hilbert programme in the rubbish with his Incompleteness Theorem which states: no system of Mathematics can ever be complete because: (a) it cannot prove the self-consistency of the system (sort of like defining a word with the word), (b) because of (a) there are propositions that are un-decideable to be true or not true, therefore (c) the such a system can never be complete i.e. reduced to a finite set because these un-decideables will always appears.
At this instant, I am having second thoughts about the validity of Hilbert's programme. I am now of the opinion that Hilbert's programme is an improper avenue of investigation and not worth pursuing. Does this mean that I may come around to accepting Gödel's Imcompleteness Theorem? Do not hold your breath waiting for that day.
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