[FOM] Simpson on Tymoczkoism

[P.T.M.Rood@ph.vu.nl]

On Sun Sep 28 16:01:53 2003,
Neil Tennant <neilt at mercutio.cohums.ohio-state.edu> wrote:
>Thanks to Ron Rood for an enlightening message.
>
>There is another philosopher's distinction that might be relevant to
>Steve Simpson's exchange with David Corfield. This is Reichenbach's
>distinction between the context of discovery and the context of
>justification. (Apologies to anyone who might have made this point
>already.)
>
>In the 23-page sample of his book which I downloaded---David,
>why can't CUP cough up the whole of the first chapter for us?!---Corfield
>emphasizes the relevance of real/core mathematical practice to the
>philosophy of mathematics, and complains that philosophers pay too little
>attention to it. Perhaps his emphasis here is really on the context of
>discovery---involving the Lakatosian process of conjecture, failed
>attempts at proof, subsequent conceptual tinkering, and eventual success
>with proof---rather than the context of justification.
>
>In response to this, the philosopher of mathematics who emphasizes instead
>the context of justification will be waiting on the sidelines, as it were,
>for the mathematician (or mathematical community) to "clean up its act",
>and present us with the finished product. Then, and only then, does the
>mathematical foundationalist of the Simpsonian kind roll up his or her
>sleeves and get down to the business of anatomizing the conceptual
>structures and deductive resources underlying the results proved.
>

My impression is that "mathematical practice"--a term used by Tennant and
many others on this list--is often taken to be a kind of catch-all term in 
order
to cover everything that cannot be captured, or cannot be readily captured,
by means of formal logic (and axiomatic set theory). (One often encounters a
quite similar phenomenon in the case of semantics: everything that cannot be
captured by means of the techniques of formal logical semantics is considered
part of pragmatics, i.e., is "thrown into the pragmatic waste basket".) I 
think
that making a few distinctions here helps us begin to to come to closer grips
with this thing called "mathematical practice." Also, it may help us to begin 
to
understand the relation between "mathematical practice" and the 
foundationalist
enterprise as advocated by Simpson and others. Finally, it will throw some 
light
on the discovery/justifcation distinction as brought up by Tennant. In 
particular,
it puts into question whether, and if so, in what sense, there really is such 
a
distinction. (Philosophers of science have already put this into question for 
years,
and for good reason.--See, for example, my reference to W.C. Salmon below.)

Tennant suggests--rightly, I think--that "mathematical practice" (whatever it 
is)
is constituted by some kind of processes or acts. Now, processes are temporal
somethings-- processes take place in time. However, processes can take place
at different time scales. It is relevant to sort these out, for they relate to 
different
"kinds of mathematical practice".
 
First, solving a mathematical problem typically takes place within a 
relatively
short time interval (0-60 minutes, say). Furthermore, solving a mathematical
problem is, I take it, typically an individual task. Of course, there are 
mathematical
problems that mathematicians (plural) were working on for decades or even
centuries. Indeed, there are mathematical problems that are still open since 
centuries ago (e.g. the Goldbach conjecture). Without underestimating the
importance or of many such problems, I do not wish to call them "typical
mathematical problems" (see the third point below).

Second, learning mathematics typically takes place one a relatively larger 
time scale.
What I have in mind is learning a "part" of mathematics such as some part of
topology (e.g. elementary topology, dimension theory), algebra (e.g. group
theory, representation theory), or analysis (e.g. elementary analysis, measure
theory). Let us say that learning mathematical typically takes place within a 
time
interval of 60 minutes - 1 year. (The exact boundaries are not important.)

Third, the development of mathematics as a historical phenomenon takes place
at an even larger time scale. Let us say within an interval of 1 year - 
several centuries.
Mathematical practice as taking place on such a time scale seems to form the 
subject
matter of Lakatos' work. (See e.g. his Conjectures and Refutations from 1976.)
Furthermore, I also think we should put the famous mathematical problems (e.g. 
Fermat,
Goldbach, the Poincare conjecture, Riemann hypothesis etc.) within 
mathematical
practice thus understood. These are typically problems that several 
mathematicians
have worked on for years. (I have not yet read Corfield's new book that is 
often
mentioned on this list, except for the (part of) the sample chapter that can 
be read from
CUP's website. From Corfield's references to Lakatos I get the impression that 
he
thinks of mathematical practice primarily in the sense of large scale 
historical
developments. It would be interesting to hear from him whether this is 
correct.)

(I should parenthetically add that a quite similar tripartite distinction of 
time scales is
discussed in A. Newell and H. A. Simon's famous book Human Problem Solving
from 1972. My distinction leans much on theirs. I cannot give the FOM readers 
an
exact page reference right now since Newell and Simon's book is in a library 
in a
different building than the one I am currently working in. But if I remember 
correctly,
the relevant discussion takes place in the early pages of their book.)

When Tennant brings up the discovery/justification distinction he seems to be 
looking
at mathematical practice primarily in terms of its historical development (cf 
the third
point above). At least, this seems to be what is suggested when he speaks of a 
"Lakatosian process of conjecture, failed attempts at proof, subsequent 
conceptual
tinkering, and eventual success with proof". Tennant considers these 
Lakatosian
processes as processes of discovery, which may be fair enough. Furthermore, he
sees the "Simpsonian" foundationalist project that follows up a Lakatosian 
process
of discovery (to "roll up his or her sleeves and get down to the business of 
anatomizing
the conceptual structures and deductive resources underlying the results 
proved")
as a process of justifcation. Thus we have, in Tennant's views, the 
discovery/justifcation
distinction.

Contrary to what Tennant seems to do, I hesitate to view the Simpsonian 
foundationalist
project as following up a Lakatosian process of discovery in terms of of 
justification.
Rather, the Simpsonian project, thus understood,  seems to me primarily as 
being
concerned with the *organization* of a reasonably completed body of 
mathematics
in terms of a hierarchical body of propositions. And, as I tried to make clear 
in my
previous message (FOM, Sunday, September 28, 2003), the "structure of 
justifcation"
need not be the same as the "structure of propositions." (This, by the way, 
leads to the
following question: how is the logical organization of the propositions of 
mathematics
(i.e., Simpsonian foundations) related to the epistemic foundations of 
mathematics?
Note that this question presupposes *that*  they are related--a presupposition 
that
definitely needs closer investigation.)

I close off this message by considering mathematical practice in terms of 
problem solving/
theorem proving (cf the first point above). (As I've not thought this over 
very much. I will
not talk about mathematical practice in terms of learning (cf. the second 
point above.)
How does the discovery/justifcation distinction apply there? Consider an 
example. Problem:
show that there are infinitely many primes. Solution: you suppose not; 
specifically, you
suppose p is the largest prime. Now you consider the number (1 * 2 * ... * p) 
+ 1, etc. etc.
(I suppose that FOM readers know how the solution goes.) This amounts to the 
description
of a *process* to the end of solving a certain problem. How does the 
discovery/justification
distinction apply here? Personally, I am not even sure whether it applies. 
What I just described
was, I take it, a (correct) proof procedure [sic], not *just* a process of 
discovery. Discovery
and justification seem to go hand in hand here.

(This point derives from a suggestion made by the philosopher of science 
Wesley Salmon.
He has suggested that whenever you (correctly) apply a certain procedure or 
method  in
order to discover something, justification often comes hand in hand. See his  
paper "Bayes's
theorem and the history of science." In: Minnesota studies in the philosophy 
of science 5:
historical and philosophical perspectives of science, R. H. Stuewer (ed.) 
Minneapolis: University
of Minnesota Press (1970).)

Note that above I did not logically deduce (i.e., I did not apply logical 
rules of deduction)
a proposition from certain axioms within some system of number theory (let 
alone set theory!).
Although, of course, a proposition expressing that there exist infinitely many 
primes must be a
*logical consequence* of the axioms of number theory. But that is a different 
issue. Finally,
what would be the point of presenting a ("Simpsonian") reconstuction in terms 
of a formal
logical deduction in some logical system (of set theory, say) when it is 
epistemic justifcation
that is at stake? Must we really say that what I've just described was, 
contrary to what I
suggested, indeed nothing but a process of discovery? Must we really say that 
justification
--proof--can, in the end, only be gained by means of formal logical deductions 
within
some system of set theory? But that would mean that virtually no mathematician 
ever
has ever really *proved* something. (Not even set theorists!) This I find hard 
to swallow.

Ron Rood
-- 

*****************************
Ron Rood
Department of Philosophy
Vrije Universiteit Amsterdam
De Boelelaan 1105
1081 HV Amsterdam
The Netherlands
e-mail: p.t.m.rood at ph.vu.nl
FAX: +31-20-4446620
phone: +31-20-4446614
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