FOM: History and f.o.m. wtait at
Tue Mar 31 11:06:12 EST 1998

The following is a posting that I composed some time ago in response to one of Harvey F's postings on what I found to be the significance of historical studies. For various reasons (concerning real relevance) I hesitated posting it; but it also partly answers Steel's (3/25) loft-level question:

> On a more lofty level, I wonder what people on the list think of the
>"Gentzen Program" of giving "as-constructive-as-possible" consistency
>proofs for strong systems like 2nd order arithmetic and set theory.

What follows is my original draft.

 Harvey Friedman (23\1\98) wrote in resonse to my reservations about discussing history on FOM

>I do think that a discussion of why history is interesting and/or important
>for f.o.m. is appropriate for fom. The question I have is this: 

>Are you
>interested in the history of logic and/or philosophy because you find it
>intrinsically interesting, or intrinsically important, or because it sheds
>light on important curerently unresolved issues in f.o.m.? >If the latter is
>an important consideration for you,

 >then can you tell us which history 

>historical figures are most relevant?

>Have you, or do you expect to make
>advances in f.o.m. with the help of historical studies?

Let me reply to 1 now, and think about how, or if, I should reply to 2-4. 

The short answer is: both. But I'll give the long answer. (The question often comes up in professional philosophy: unlike most fields, history is taught as an intrinsic part of the curriculum in philosophy.) I came to be intrinsically interested in the history of the philosophy of math; but my original interest was in trying to find the itch I wanted to scratch. I think that a lot of philosophy is like that, including some of the issues we discuss on FOM. We inheret the words and phrases and questions: they have a history and pick up an aura that persists through time, though their meaning, which can depend upon a particular historical context, may radically change---indeed, they may become meaningless or at least pointless. It helps to free oneself from the aura surrounding them to trace back to see where they touched ground.

A good example of the latter is a question that comes up on FOM (the `unreasonable effectiveness Š' question): Why is mathematics such a good tool for describing, controlling, predicting, whatever, nature?   Or, Why does reason work? (It has been brought up as a difficulty for the view that mathematics is a social institutions or construct.) Before the theory of evolution by natural selection, this surely presented a profound puzzle, that seemed to demand for its solution a grand designer---e.g. in Plato and Leibniz, the view that there is a best order of things and a creator who designed the cosmos with this order and humans with at least a limited propensity to understand it. But surely evolution by natural selection allows us to believe that the explanation for the usefulness of mathematics and for our ability to develop and apply it can be accounted for in the same way as the usefulness of other social institutions and individual physical and mental organs and powers. One may well feel that we are very far from actually having such an account; but the profound sense of mystery that had to prevail prior to Darwin no longer seems (to me) forced on us. But for many people the question (about the effectiveness of math, but not of the eyeball!) still seems to demand some kind of extra-natural answer, not because (or so I think) they can specify just what an empirical answer would necessarily lack, but because the question came down to us labelled `non-empirical'. They want a _justification_ for how we do science, rather than just a, so to speak, _causal_ account of why it works.

Incidently, this example bears on Steel's remarks (Wigner quote, 1/27/980)
>There is an old parallel in
>the medieval Argument from Design. Theologians "explained" the order and
>harmony of our universe by postulating a superior version of our own
>intelligence as its creator. The "explanation" goes nowhere, because it
>rests on concepts so much less clear than the laws being explained.
It seems to me that the medievals---and the ancients and the early moderns---had a good reason to postulate divine designs. They were asking a question to which we can perhaps give an empirical answer, if only in principle, but to which they could not even conceive of one.  

This solved a puzzle for me---an historical one, not in f.o.m.: Leibniz, for example, seems a very rational fellow and I follow his arguments reasonably well---though there are some problems of interpretation. But then he steps over the line into the domain of theology and, as I for a long time thought, he abandons reason and is an alien. Of course one could always say that he was a person of his times and/or that he was reacting to the menace of the church. But I now think that a plausible explanation is the one I have just given: it is more reasonable to believe in a Grand Designer than to think that it is just a coincidence that we happen to have so many appropriate tools (including, for L, our innate ideas as well as our eyeballs) that just happen to work for us. 

It is interesting to think of Kant from this point of view, too, who is unique in the pre-Darwinian treatment of this problem. He wanted to explain the unreasonable effectiveness of reason by conceiving the worf and woof of the empirical world as our construct, as built into the way we necessarily experience: the mountain comes to Mohammed, so to speak; because he couldn't understand how Mohammed could come to the mountain. But the theory of natural selection allows us to understand how _the species of_ Mohammeds could have come to the mountain.   
A more personalized answer: In the 60's I worked on a kind of extended Hilbert's program in proof theory---to find a constructive interpretation (in some loose sense, an extension of finitistic mathematics) of parts of classical mathematics. (I learned the project from Kreisel, but it goes back to Bernays and Godel,in which order I don't know. But in 1933 Godel had clearly already conceived the programm as I understood it in 1958.) In the late 60's, I realized i) (a technical point) that there was an absolute barrier to this project, in any reasonable sense of a constructive extension of finitism, namely Phi^{1}_{2} analysis (second order arithmetic with the comprehension axiom restricted to Pi^1_2 formulas)[see the note * below.], and ii) (a philosophical point) that I could no longer credit constructive methods with any favored epistemological status. So I had started with a fairly concrete project, which simply went flat. I still felt that I did not understand mathematical meaningfulness and truth; but I no longer had a clear direction to take to gain understanding. For me, going back to the history of the origin of basic mathematical concepts was an attempt to gain some bearings. 

I should say, since I am responding to Harvey, that, even leaving aside ii), reverse mathematics wouldn't have satisfied me. I wanted a constructive interpretation of e.g. Phi^1_n analysis in general. What strikes me about the vertical point of view of reverse mathematics is that it responds to historical accidents---what theorems we have happened to proved. (I don't mean to imply contempt for our human interests; only that, at least as I conceived it, foundations shouldn't take them into account.) I wanted to understand the quantifiers over sets of numbers and, more generally, the levels of logical complexity, across the board. Since we can't do that constructively, then so much for constructivism (even, as I said, leaving aside ii)). 

My attitude towards this is surely debatable; but it is not something I want to (or will) spend time debating (in part because of ii))---though I would be very happy (I hope!) to hear the response of Harvey and Steve S. and others to what I have said.

* I have some recollection that Harvey fed into my conviction that Pi^1_2 analysis was not attainable by constructive methods; but I don't remember how (that's the way it is with pensioners)---maybe a conversation at the Buffalo conference in 1968?  Final conviction for me was the connection between the proof-theoretic ordinal of a subsystem of second order arithmetic and the least admissible ordinal or limit-of-admissibles alpha such that the second order version of L_alpha is a model of the system. (So it was connected with work of Kripke in the mid-60's, that I knew but just didn't take in soon enough.) Namely, the former is a collapse of the latter; and constructive conviction that induction on the proof-theoretic ordinal is valid depended, as it seemed to me, on a constructive understanding of alpha.[See note **] But, in the case of Pi^1_2 analysis, alpha had to be the least non-projectible ordinal; and there seemed to be no reasonable sense of constructivity under which one could understand that ordinal.

**So for me, it wasn't the well-orderedness of the ordering of the natural numbers which carried constructive conviction; it was the constructive theory of iterated inductive definition and, in particular, the constructive higher number classes. 

Bill Tait

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