Ramble, Part 70: Function Follows Form
May. 14th, 2009 02:15 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowAt the heart of the Logicist program was a desire to prove that mathematical statements were true, by resting them on absolutely indisputable bedrock - on logic itself. The second of the major schools to arise, Formalism, took a radically different tack.
David Hilbert, the founder of the Formalist school, proposed that, at bottom, mathematics deals with the manipulation of symbols according to specified formal rules. Axioms are fundamental strings of symbols; theorems are strings derivable from the axioms by use of the specified rules. The symbols themselves have no inherent meaning, and thus "truth" is an irrelevancy. (It is possible that the symbols can be assigned meanings, in such a way that the axioms become true statements; but this possibility is not, on the Formalist view, the concern of the mathematician.)
The Formalist school aims to encompass all of classical mathematics under its aegis; therefore, it is necessary to produce a relatively small list of axioms and, likewise, a small list of manipulation techniques, from which algebra, geometry, and analysis can be in principle constructed. This system should be complete; that is, for every mathematical statement - every statement writeable in terms of the provided symbols - it should be possible to prove either that statement or its negation. On the other hand, it should also be consistent; it should not be possible, for any statement, to prove it and its negation. In wariness over the possible misbehavior of infinite collections, the system should allow a proof that, if a statement about ordinary mathematical objects can be proven using extraordinary methods - such as uncountable sets - then it can also be proved without those methods. (Perhaps paradoxically, this frees the mathematician to use those extraordinary methods wherever they are convenient, in the serene assurance that they could be avoided if it were absolutely necessary.)
In the pursuit of this goal, Hilbert proposed a new branch of mathematics, metamathematics - now more commonly called proof theory - which would study systems of axioms, methods of proof, and so on, with the particular aim of proving the consistency and completeness of whatever axiom system was ultimately chosen to be fundamental. Hilbert's proposal, in its original form, ultimately was found to be unachievable, as will be discussed later, but scaled-down versions are still being pursued, with some success. That last, though, lies beyond the reach of my knowledge, so I'll say no more about it.
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David Hilbert, the founder of the Formalist school, proposed that, at bottom, mathematics deals with the manipulation of symbols according to specified formal rules. Axioms are fundamental strings of symbols; theorems are strings derivable from the axioms by use of the specified rules. The symbols themselves have no inherent meaning, and thus "truth" is an irrelevancy. (It is possible that the symbols can be assigned meanings, in such a way that the axioms become true statements; but this possibility is not, on the Formalist view, the concern of the mathematician.)
The Formalist school aims to encompass all of classical mathematics under its aegis; therefore, it is necessary to produce a relatively small list of axioms and, likewise, a small list of manipulation techniques, from which algebra, geometry, and analysis can be in principle constructed. This system should be complete; that is, for every mathematical statement - every statement writeable in terms of the provided symbols - it should be possible to prove either that statement or its negation. On the other hand, it should also be consistent; it should not be possible, for any statement, to prove it and its negation. In wariness over the possible misbehavior of infinite collections, the system should allow a proof that, if a statement about ordinary mathematical objects can be proven using extraordinary methods - such as uncountable sets - then it can also be proved without those methods. (Perhaps paradoxically, this frees the mathematician to use those extraordinary methods wherever they are convenient, in the serene assurance that they could be avoided if it were absolutely necessary.)
In the pursuit of this goal, Hilbert proposed a new branch of mathematics, metamathematics - now more commonly called proof theory - which would study systems of axioms, methods of proof, and so on, with the particular aim of proving the consistency and completeness of whatever axiom system was ultimately chosen to be fundamental. Hilbert's proposal, in its original form, ultimately was found to be unachievable, as will be discussed later, but scaled-down versions are still being pursued, with some success. That last, though, lies beyond the reach of my knowledge, so I'll say no more about it.
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