FOM: Hersh/Rota/Feferman/Lakatos

[Robert S Tragesser]

ABSTRACT: A sketch of the origins of mathematics that 
explains what is distinctive about mathematical knowledge 
but does not essentially depend on a reference to 
traditional mathematicals (such as numbers).   This 
explanation explains why mathematical experience 
supports Platonism but also why Platonism can likely be 
never more than a regulative idea(l).
        Feferman's questions about the distinctiveness of 
mathematical thought are put to Hersh/Lakatos,  and 
answered via Rota's phenomenological thinking,  and 
answered in a way without undercutting the phenomena 
which fund Platonism,  but which at the same time reveal 
the extreme difficulties in sustaining Platonism as anything 
more than a regulative idea(l). [End of Abstract.]

        I've been enjoying Hersh's _What is Mathematics,  
Really?_  I think that Reuben Hersh is importantly 
correcting the course of the philosophy of mathematics,  
but at the same time he _over-corrects_ it.
        How should the over-correction be itself corrected?
        Hersh and Lakatos stand in need of such correction,  
and in need of the very same correction -- they both overly 
collapse -- in a phenomenologically undiscerning way -- 
uses of 'certainty' (and its "cognates").
        I've been writing a long review essay on the 
phenomenological studies of mathematical thought in 
Gian-Carlo Rota's _Indiscrete Thoughts_.   It is worth 
observing that,  although Rota and Hersh,  through 
forewards and dust-jacket copy,  strongly applaud one 
another's (philosophical) work,  Rota's phenomenological 
appreciation of mathematical proof and mathematical 
understanding provides the needed correction to Hersh's 
course.
        I want to try to explain this here,  and do so in a way 
that points to some answers to questions raised by Sol 
Feferman.
        In a recent FOM posting,  Cook asked Hersh how 
mathematics is distinguished from other "academic 
subjects".   Hersh answered that mathematics was one of 
the "humanities" (because it is entirely a human product) 
and is distinguished among the humanities by being about 
mathematicals.
        In his review of Lakatos,  Sol Feferman asked a series 
of questions -- one close to that of Cook's -- which we,  
qua phenomenological Rota,  put to Hersh (but don't wait 
around for his answer), --

FEFERMAN'S QUESTIONS:
        [1] What is distinctive about mathematics?
_i.e._,
        [2] What is distinctive about its verification structure?
        [3] What is distinctive about its conceptual content?
The absolutely important _necessary condition_ for giving 
a good answer to these questions:
        Necessary Condition:  They must allow us to 
characterize mathematics in a way that is independent of 
the kind of appeal Hersh makes,  viz.,  that mathematics 
studies the mathematicals (studies what mathematicians 
study).
        It is exactly by evading this issue that Hersh misses 
the distinctive features of the verification structure of 
mathematics (or:  is able to play a bit too much of the old 
fast and loose with it).
                                       ****
I'll make this short,  though details can be supplied (some 
will be supplied in a posting giving an account of 
Lebesgue's conception of arithmetic).
        First,  I'll describe the two stages in the origin of 
mathematics,  and then I'll give an example.
[1] First stage of the origin of mathematics:  Witty persons 
become aware of (practical) problems which have 
solutions which (a) can be framed or represented in 
thought and (b) can be seen by thought alone to be 
definitive solutions (seen by a peculiar sort of light;  
mathematical proofs will be architectures in that light).  
The recognition and cultivation of such problems is the 
first stage in the development of mathematical thought.   
Their cultivation may go beyond practical interests (e.g.,  
as in play or poetry. . .riddles. . .problems to be solved in 
contests,  etc. . . .N.B.,  Ian Hacking's phantasy of 
language issuing more from play rather than work).
[2] Second stage of the origin of mathematics (Berkeleyian 
abstraction): it is observed that the light by which one sees 
that such and such is THE solution to the problem "Q?" 
has an authority that is not bound to the concrete 
particulars of the problem.   In successfully reframing such 
problems,  their solutions,  and the 
exhibition/demonstration of such solutions as _the_ 
solutions,  mathematics proper begins.

EXAMPLE:  [This example is meant to illustrate the ideas,  
so it is ideal;  but it is exemplary,  too,  in that more 
realistic examples having to do with the "real" origins of 
arithmetic,  algebra,  geometry,  combinatorics. . . will be 
patterned after it.]

[E1]  It becomes practically important for me to know the 
minimum number of fruitfly I have to capture in order to 
be certain that I have two which are the same sex.
        I proceed experimentally.
        First,  I form collections each containing one fruitfly. 
I notce that none of them contains two fruitfly which have 
the same sex (but dull empiricist that I am,  I do not notice 
that each collection fails to contain two fruitfly of the same 
sex _because_ each collection contains only one fruitfly).
        Second,  I form collections each containing _three_ 
fruitfly (because dull empiricist that I am I don't see that 
the next logical step would be to try collections of two).   I 
find that each collection contains at least two fruitfly of the 
same sex.   But now I worry whether this is the least 
number.   And dull empiricst that I am,  I first make a lot 
of collections containing _two_ fruitfly and a lot of 
collections containing _five_ fruitfly,  comparing them to 
the collections containing three fruitfly to see whether the 
collections containing five fruitfly or the collections 
containing two fruitfly contain fewer fruitfly than the 
collections containing three fruitfly.   I learn that the 
collections containing two fruitfly contain fewer fruitfly;  
so I will not look through those collections to see if two 
fruitfly of the same sex invariably occur.   I find that they 
do not.   So I conclude: _3_ is the answer to the initial 
question.   Of course,  it might have happened that male 
fruitfly were very rare,  so that in fact all the pairs I 
collected were pairs of females,  and I was led to answer: 
2,  instead of 3.   But in any case,  someone will cast 
doubt on my answer _3_ because I overlooked so many 
other possibilities,  such as 4,  23,  197,  272. . .
        Observe that it is rather unlikely that there should be 
such very dull empiricists. . .who never let in any as it 
were _apriori_ thinking,  for whom it could never be 
decisive that collections containing two things have fewer 
things than those containing three or four things.
        But here is the example of _apriori_ thinking I want 
to dwell on:
        A collection of three firefly must contain at least two 
which have the same sex,  for here are the only possible 
combinations of three firefly identified up to sex (M, F):
        MMM,  MMF,  MFF,  FFF.
"Check that this lists all possible combinations."
        (There are of course a number of ways that the check 
can be made _apriori_ . . .)
        Here one has solved the problem,  definitively,  and 
_apriori_.   Any skeptic must be either witless or fail to 
understand the terms of the problem (such as its being 
assumed that all firefly are either M or F -- see below ON 
HIDDEN ASSUMPTIONS).
        This problem illustrates the first stage.   The second 
stage,  mathematics proper,  begins with the observation 
that in dmeonstrating to onesself that the solution to the 
problem is three,  one actually has proved a more general 
proposition (which in the framing becomes more abstract):
        The demonstration of the correctness of the solution 
to the problem does not essentially depend on fruitflies or 
on sex.
        Ginning up concepts to state and prove,  and then 
stating and proving,  the more general/abstract proposition 
is the very essence of logico-mathematical activity.   It is 
one thing to notice that the "proof" proves more than the 
particular proposition (about collections of sexed fruitfly) 
at issue.   That's the first phase of the second stage on the 
way to mathematics.   The second phase is to find the 
concepts in which to frame the more general/abstract 
proposition.   For examples,  a language containing 
'individual',  'property',  'collection',  'has the property',  
'does not have the property',  and so on,  and more or less 
explicit rules for using these terms.
        
        Some conjectures about the characteristic trait of 
mathematical concepts:
        What enable emergence of the definitive solution to 
the firefly problem?
        Being able to canvas all the possibilities in advance.
        This suggests then a tentative characterization of 
mathematical concepts:  that they enable us to canvass 
(directly as above,  or indirectly,  as is more typical) in 
advance all the relevant possibilities of what they frame (of 
the problems which can be framed through them).
        This suggests why Platonism might seem supported 
but in actuality false:
        Concepts arising in mathematics (through the process 
of Berkleyian "abstraction/generalizatiuon" sketched 
above) may sustain _apriori_ solutions to a wide range of 
problems posed through them (the concepts),  but the 
concepts may also be rough around the edges -- we in fact 
cannot canvass all the relevant possibilities relating to all 
problems framed in those concepts because the concepts 
are not decisive on all such ranges.

REMARKS:
        Rota calls Evidenz (which he translates "insightful; 
understanding") the kind of demonstration/light by which 
we decisively (and _apriori_) find proposed solutions to 
mathematical problems to be solutions to problems.   Yes,  
one can always be "skeptical" in the sense of demanding 
greater explicitness,  etc.   But there comes a point at 
which one either has understood the problem or one has 
not. . .whereafter we have to say that,  as far as the 
problem at issue is concerned,  the skeptic is not cooking 
on all four burners.   (For those who can read Descartes' 
First meditation with great care,  it can be seen that 
Descartes -- like Wittgenstein and Cavell -- makes 
essentially this point. . .that doubt uninhibited by authentic 
understanding is madness,  skeptical terror only.)

        I had best break off here.
                rbrt tragesser