[FOM] Indispensability of the natural numbers
isles@kingcon.com
isles at kingcon.com
Tue May 18 20:25:45 EDT 2004
A few comments on T. Chow's FOM email of May 17, 2004 05:16:42.
1. I agree with the opinions expressed in paragraphs 4 and 7 of that email. As I understand it, he is saying that any mathematical "entity" which is defined inductively, e.g. wffs or derivations in formal systems, depends for its intelligibility on a prior understanding of counting (i.e. the enumeration of NN, the natural numbers I, II, III, etc.) But "the buck stops somewhere": the natural numbers are themselves "given" by inductive rules e.g. R1: I is in NN; R2: if x is in NN, xI is in NN; R3: continue. Since to understand "continue" we need an understanding of counting i.e. of NN, the natural numbers seem to be an impredicative notion.
It does not seem to me that Andrew Boucher's response (FOM May 17 20:32:38) adequately addresses this point of Chow. For to say, as Boucher does, that one can avoid the prior knowledge of the natural numbers required by, for example, an inductive definition of wffs by saying "..instead:'X is a wff if: it is atomic, or if it is of the form (A->B) or...' " seems no advance. For you have to add to such a definition "and A and B are wffs" and this again commits one to a sequential understanding of this definition.
2. I do not agree with Chow's claim in paragraph 3: " ..if you try to create a 'small perturbation' of the concept [of the natural numbers] (e.g. by taking a nonstandard model of PA), you typically get something that is clearly 'more' complicated than the natural numbers." Just the contrary. The standard notion of the natural numbers includes the belief that "they" are closed under, for example, all primitive recursive functions. But the original understanding of the natural numbers- upon which the justification of mathematical induction rests- is given by the inductive rules R1, R2, and R3 given above and these clearly do not include this feature. The argument that the natural numbers are so closed depends on the uncritical use of mathematical induction. But there are arguments which question this use and point out where it might not be appropriate (for examples see some of E. Nelson's or my own work.) Roughly speaking, as I understand it, the usual picture of the "standard" natural numbers (closed under all primitive recursive functions) is to the "true" natural numbers (given by rules R1, R2, and R3) as the "nonstandard" model theoretic natural numbers are to the "standard" natural numbers.
3. Finally, I disagree with Chow's comments in paragraph 5: " Formal languages are interesting mostly because they capture some features of an entity that we're studying, and if the entity vanishes, then why bother with the formal language anymore?" This argument has a Platonic quality: there is an "entity" (for example, the natural numbers), and we try to describe it linguistically via a formal language (for example, PA.)
It seems to me that the following alternative picture is a more accurate description of what happens: We have a practice of which we conceive a vague notion (e.g. counting.) This "entity" we discuss linguistically and/or with pictures. The use of the linguistic and pictorial representations suggests new aspects of the notion AND WITHOUT THE USE OF THESE REPRESENTATIONS WE WOULD NOT HAVE CONCEIVED OF THESE ASPECTS. These new aspects are then incorporated as part of our notion of the "entity". But this enriched understanding requires a richer linguistic framework with which to grasp and discuss it. Etc. Thus the notion of an "entity" such as the natural numbers is not a static one: it grows and is redefined as our experience with it and discussions about it continue. In this interactive preocess, notations and formalisms play a creative, not merely a descriptive, part.
Comments on the above opinions would be appreciated.
David Isles
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