Calculation, Proving and Rule
Mitsuhiro Okada
Department of Philosophy, Keio University（This paper is related to joint work with Mathieu Marion on the uniqueness rule.）
We discuss rules related to calculation and proving. Although it is often consideredthat Wittgenstein’s view on calculation and rules had been changed by periods(Tractatus, the early transition period, the later transition period, later periods), we
propose a different reading of Wittgenstein’s texts, according to which his views could
be understood coherently through periods, on the base line.
For example, Wittgenstein introduces “numbers” in terms of a version of general
operational rule and “calculation” in means of a version of “proof-as-calculation”
paradigm, in Tractatus; the “propositions” and the “proofs-calculation” are strictly
disconnected; any “proof” cannot be a proof of any “proposition”. We demonstrate
how these claims of him are kept in the transition periods and later., in principle.
Despite the similarity on the base line, there are, of course, differences in his viewson calculation and proofs among the different periods. Wittgenstein “extends” his“Tractatus” view on proofs-calculation in the transition periods; his proofs-asprogram view becomes, in the author’s opinion, clear in the transition periods (thanin the Tractatus period) because of this extended view. His arguments on rulefollowing calculations (such as inductive-proofs) will be discussed in this context.
Such arguments are also related to various issues, such as surveyabiliy, calculability,purity, translatability issues in philosophy of mathematics. We, among others,consider his discussions on commitments to proofs and on accepting proofs and theirrelation to networking of norm, in the later period, eg. RFM.
At a slight look, the scope of Wittgenstein’s discussions on proofs-calculations looks
limited to a naïve mathematics: for example, when he talks about arithmetic, he does
not say any algebraic equations/properties as sayable (through all periods). However,
we note that Wittgenstein suggests further about algebra although algebraic proofs
belong to a language game different from the arithmetic one in which we stay with
our arithmetic. It is important to think how the arithmetic language game is related
to our ordinary life on the one hand, and to the algebra on the other hand. We discuss
how Wittgenstein argues on these issues, coherently from some transition period tothe later.
Changing the way of calculation-proving would be an aspect-switch, and taking adifferent way of calculating-proof could commit to a new norm. Wittgenstein’s
arguments are restricted to the cases of numerical arithmetic rule introductions. We
would like to challenge to extend his idea to much wider scope. We remark that theunderlying arguments of Wittgenstein’s on these are about transformations from aseries (progressing sequence) to another series (progressing sequence), or sometime
from a sequence of sequences to another sequence of sequences. So, it is not just acommitment to a single new calculation-proving, but to a new (progressive) sequenceof calculation-proving (eg. not just 3ⅹ3ｘ3ｘ3 to 34, but, a progressive sequence to a
progressive sequence such as 31,32,33.34,35,…). Taking a transformation corresponds
to a norm-setting; we discuss how committing to transformations forms a new normnetworking. We discuss what this standpoint would offer to the rule-following
controversy.
This standpoint suggests the idea that demonstrating-proving-calculating shows
norms and what are being demonstrated, not vice versa; not that norms-propositions
provide (propositional) demonstrations-proofs-calculations. We would like to place
this idea in some traditional issues of philosophy of mathematics and logic (andfurther), and try to present some hints for fruitfulness of the idea on the issues.