As is often the case with philosophical investigations of any kind, one is at risk of straying off of the original track of thought. An outgrowth of logicism, the formalist movement returns to the notion that the only purpose of an overarching mathematical system of philosophy is to govern what is done within that system. At its core, formalism states that math is little more than a game one plays through the manipulation of symbols, and that the only necessary rules are those which help direct unified “playing” of sorts. To take this reasoning even further, it does not matter if a stated rule (or axiom or theory etc) is false, as long as it is consistent throughout the entire mathematical community.

An example of this reasoning in practice is the definition of 0! as having a value of 1, only since it is a necessary prerequisite for the Taylor Series. Even though the product of all integers leading up to 0 is most definitely not 1, the community as a whole needs to overlook this true fact in order to further future study.

Falling under the general category of foundationalism, formalism attempts to explain not so much why certain things are foundations of classical mathematics, but rather how they are to be used. Hilbert, the leading proponent of formalism, sets up the following prerequisites (which must be met before any tests of validity) under his theory of proof:

1.) To list all mathematical symbols, including primitive symbols: ~ and -> (negation and implication respectively)

2.) To classify, specifically, all of the manners in which the above symbols can be combined to produce meaningful results (even if not true) i.e. no ‘2+~=---2

3.) To devise a procedure whereby formulas can be arranged successively so as to correspond to the provable theses of classical mathematics

4.) To demonstrate (in a finite manner, combinatorially) that the statements which are checked by arithmetical methods can be fit into a provable order (3) if and only if the check of the corresponding classical statement shows it to be true

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