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Hilbert's Programme Notes

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This is an extract of our Hilbert's Programme document, which we sell as part of our Philosophy of Mathematics Notes collection written by the top tier of University Of Oxford students.

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Formalism (Game-) Formalism is the view that mathematics has no content. Mathematics is mere symbol manipulation according to certain rules, akin to following the rules of a game. Problem: game formalism cannot make sense of applications of mathematics. The game formalist is limited to the shallowest form of explanation: mathematics applies to the physical world because it does. For the formalist, the fact that bridges do not fall down, men land on the moon and so on is near-miraculous. Problem: the theory of formal symbols is, in itself, a mathematical theory. Therefore one cannot consistently be a formalist about all mathematics. Consider a 'string theory' in which we have an alphabet of string variables, with an empty string and the operation of concatenation. Then, if we introduce a number of plausible-sounding principles into the theory - for example induction - the string theory becomes equivalent to a theory of arithmetic: we can interpret PA2 in the theory. Problem: are the strings taken to be actual physical symbols - tokens - or as types? If they are to be taken as tokens it is hard to see how there can be enough of them, and how theorems do not 'break' when they are no longer physically expressed. On the other hand, if they are taken as types they seem to become the kind of abstract objects the formalist wishes to avoid.

Semi-formalism Semi-formalism attempts to avoid the problems of formalism by taking a fragment of mathematics - generally taken to approximate arithmetic - to be contentful, while taking a formalist approach to the rest of mathematics.

Hilbert's programme Hilbert wanted to establish the 'security' of mathematics - that it would never "fall prey" to paradox again - without a substantial revision of mathematical practice (in contrast to, say, Brouwer). Hilbert's programme: to establish that mathematics will never fall prey to paradoxes (like in set theory and the infinitesimal calculus again). By: showing infinitary mathematics is a reliable and adequate 'instrument' for deriving finitary truths.

Finitary and infinitary mathematics Hilbert introduced a distinction between finitary and infinitary mathematics, intended to mirror the observation/theory distinction in the empirical sciences. Infinitary mathematics was then taken to be an instrument for deriving finitary truths, without any intrinsic value of its own.

1 The subject matter of finitary mathematics was supposed to be sequences of concrete symbols of strokes. Then infinitary mathematics was introduced as 'higher' or 'ideal' elements - akin to the introduction of complex numbers or a point at infinity - in order to shorten finitary proofs. Infinitary mathematics, unlike finitary mathematics, was composed of formulas with no content, manipulated according to formal rules. Kant thought of finitary mathematics on broadly Kantian lines. The certainty attached to finitary statements derives from the representability of signs to the intuition. No contradictions can arise from finitary mathematics because there is no logical structure in the propositions of contentual number theory. Finitary mathematics is generally taken to be identified with Primitive Recursive Arithmetic (PRA). By contrast, infinitary mathematics was taken to be potentially unreliable - Hilbert pointed to the paradoxes of the infinitesimal calculus and set theory as examples. Hilbert aimed to show that mathematics could never fall prey to the paradoxes again. Hilbert distinguished between

1. Elementary finite statements. Example: '2+3=5'. These contain numerical terms whose denotation is directly graspable or finitely computable, function symbols for finitely computable operations, and predicates for decidable relations. Bounded quantifiers are placed here.

2. Finitary generalisations. These are distinguished from elementary finite statements since they include free variables(s). Substituting constant terms for each of the variables then yields an elementary finite statement.

3. Infinitary statements. Example ' [?] x ( x+ 2=3) . These include the negations of finitary generalisations - so finitary mathematics is not closed under negation - and unbounded generalisations. The scope of finitary mathematics Parsons has emphasised the characterisation of finitism as primarily to do with intuition and intuitive knowledge. Parsons argues that what can count as finitary on this understanding is restricted to the arithmetical operations that can be defined from addition and multiplication using bounded recursion. For Parsons, exponentiation and general primitive recursion are not finitarily acceptable. In contrast, Tait has argued that finitism coincides with primitive recursive arithmetic (PRA). For Tait, the hallmark of finitary reasoning is that it is a kind of minimal reasoning supposed by all non-trivial mathematical reasoning about numbers. He takes the notion of Number to be the form of finite sequences. Particular numbers are then specific subforms of the same structure. Tait then analyses finitary operations and methods of proof as those implicit in this notion. For Tait, the fact that any nontrivial reasoning about number will presuppose finitist methods lends it a kind of 'Cartesian indubitability': it would be simply pointless to doubt it.

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