Kengo Okamoto （ Tokyo Metropolitan University ） 2019.6.2

1. On Internal Employment of “Formal Concepts” - A Neglected Aspect of the Logicist Program Revisited KengoOkamoto （Tokyo Metropolitan University） 2019.6.2

2. TheGoalofthe Logicist Program • The Fundamental Questionof theLogicistProgram: What is the reason for our accepting mathematics - more precisely, number theory and (some reasonable portion of) analysis - as we actually do? • Not the question of psychological motivations or inclinations but of some appropriate normative reason. • Thus, in other words: How can we (and should we) rationalize our acceptance of mathematics?

3. Logicist Requirement (1) • Insufficiency of isolated fragmentary formalizations of mathematical theories: E. g. First Order Peano Arithmetic: Here, the notion of natural number and the reason for accepting the induction principle for it are not explicated in the theory (i.e. object-language), but only presupposed to be understood in the meta-language. • What needs to be explicated is such understanding itself. • But then, how about such fairy global formal theories as Second-order Peano Arithmetic and Axiomatic Set Theory (in both of which one can define the set of natural numbers and prove the induction principle)? This is a good question but I think I can point out rather the following:

4. Logicist Requirement (2) • The basic ingredients of mathematics (numbers, functions, structures and so on) and the principles governing them should be specified and explicated in the theory itself so that ordinary rational agents, when confronting the theory, would naturally agree to admit of the existence of the former and accept the validity of the latter. • Hence the appeal to the term “logic”: Mathematical notions and principles should be explicated as something one might call logical in the sense that they are just what are ubiquitous and serve us in our rational activity in general (e.g. reasonings) . • Also, hence the appeal to “application”: The logicist formalization of mathematics should never fail to account for in some appropriate way the applicability of mathematics. I think these requirements are by and large the characteristic features of the logicist conception of formalizations of mathematics that are missing in other formalizations such as second-order arithmetic or axiomatic set theory. One might refer to the distinction made by Michael Dummett between the full-blooded and the modest theory of meaning. What the Logicist Language purports to do is similar to what Dummett’s full-blooded theory of Meaning does. No wonder, since Dummett is strongly influenced by Frege’s ideas.

5. Consequences of the Logicist Conception of Language 1: Need for Internalization • In the Logicist Language, we would need to be able to formulate (and prove the validity of) our ordinary situation-dependent reasonings such as the following: “Let us bring all the chairs in the rooms A and B into the room C. What will be the number of the chairs that actually will have come to be in C?” “Well, if nothing particular happens, the number will be 12, since there are for the moment 5 chairs in A and 7 chairs in B.” Cf. Wittgenstein’s worry. • I believe we need to synthesize an appropriate mathematical language with a certain kind of a situation-sensitive language such as temporal hybrid language, in which one can refer to current state and various past and future states just by means of free temporal variables (so called “nominals”) without using any indexical nor demonstrative devices at all.

6. Consequences of the Logicist Conception of Language2: Need for Universal Type • In the Logicist Language, we also would need to be able to formulate (and prove) various meta-theoretical statements (i.e. statements in which “formal concepts” - “x is an object”, ”x is a function”, x is a concrete object” and so on - are employed) such as the following: “In contrast to the successor function, which is a first level function on the natural numbers, both 2 and Caesar are an object but the latter has never been, is not and will never be a number, since it is not a logical nor a mathematical object at all but a concrete one”. In other words, we should be able to make various cross-type classificatory statements which would be simply inexpressible in e.g. Axiomatic Set Theory.

7. Need for Some Kind of Universal Type Is the Language in Frege’sBasic Laws of Arithmetic a Name Language? As is well-known, Frege himself explicitly and vehemently claims that it is. But I think this opinion of his is not sufficiently reliable. Here, by Name Language is meant a language, unlike ordinary languages, not with two base types (the type of objects and that of propositions) , but with only one basic type, i.e. that of objects. Of course, one can derive other types (types of n-ary functions of various levels) by applying the abstraction operation. Still, all the non-functional (i.e. saturated) well-formed expressions uniformly have as its type the type of objects. In particular, formulae in ordinary sense of the word count as (are assimilated to) a kind of name in the Name Language. In view of the examples referred to in the former slide, I think Frege’s type of objects could best be regarded as a kind of Universal Type. At the same tme, we should note that various internal indefinability results are obtained. Cf. Frege Structure developed by Peter Aczel.

8. Appendix: “Post-Logicist” Conception of Mathematics in later Wittgenstein Mathematical Proposition not as Justification-Condition but Possibility of Application : • Wittgenstein,L., ed. by Wright, G. et al and tr by Anscombe: Remarks on the Foundations of Mathematics (rev. ed.), MIT Press, 1978 . Henceforth [RFM] [RFM]283 Now, when does one understand it [= a mathematical proposition]? － I believe: when one can apply it. Wannversteht man ihn nun? － Ich glaube: wenn man ihnanwendenkann.

9. Two Roles of “Facts” in Calculation • [RFM]382 [The interlocuter says that ]“...surely our calculating [= Rechnen] must be founded on empirical facts!” Certainly. But what empirical facts are you now thinking of? [1] The psychological and physiological ones that make it [= calculating] possible, ＝　the cognitive process, in particular, its characteristic visual features, that normally accompany and support our activity of calculating [2] or those that make it a useful activity? ＝ facts that are seemingly exhibited and discovered in some sort of experiment, but in reality playing the role of models/paradigms intimating to us various possible applications of the calculation

10. Two Roles of “Facts” in Calculation • [RFM]382 The connexion with the latter [=[2]] consists in the fact that the calculation is the picture of an experiment as it practically always turns out. Der Zusammenhangmitdiesenbestehtdarin, dass die Rechnungdas Bild eines Experiments ist, wie es naemlich, so gut wieimmer, ablaeuft. ＝ picture as a model From the former [=[1]] it gets its point, its physiognomy; Von den anderenerhaelt es seine Point, seine Physiognomie; ＝ syntax has “a face” but that is certainly not to say that the proposition of mathematics have the functions of empirical propositions. aber das sagtduchausnicht, dass die Saetze der Mathematik die Funktionen der empirischenSaetzehaben. ＝　Calculation is never to be confused with experiment.

11. Mathematics creates the forms of what we call facts: [RFM]383 Calculating, if it is to be practical, must be grounded in empirical facts’ － Why should it not rather determine what empirical facts are? >Das Rechnen, um praktisch sein zukoennen, muss auf empirischenTatsachenberuhen.< － Warumsoll es nichtlieberbestimmen, was empirischeTatsachensind? ＝ Mathematics determines what empirical facts are. [RFM]381 ’To be practical, mathematics must tell us facts.’ － But do these facts have to be the mathematical facts? － But why should not mathematics, instead of ‘teaching us facts’, create the forms of what we call facts? >Die Mathematik, um praktischzu sein, muss unsTatsachenlehren.< － Aber muessendieseTatsachen die mathematischenTatsachen sein? － Aber warumsollsienicht, stattuns >Tatsachenzulehren< die Formendessenshaffe, was wirTatsachennennen. ＝ Mathematics creates the forms of empirical facts. ・　form＝　possibility?

12. Literature Frege[GLA]: Grundlagen der Arithmetik. Frege[GGA]: Grundgesaetze der Arithmetik. Frege[BLA]: Basic Laws of Arithmetic tr. by A. Ebert & M. Rossberg with C. Wright, Oxford U.P.. Aczel, P. [1978]: Frege Structures and the Notions of Proposition, Truth and Set. In Kleene Symposium, Elsevier. Cantini, A.[1996]: Logical Frameworks for Truth and Abstraction – AnAxiomatic Study, Elsevier.