FOM: Hersh/Rota/Feferman/Lakatos

Reuben Hersh rhersh at
Fri Jan 2 20:18:32 EST 1998

On Fri, 2 Jan 1998, Robert S Tragesser wrote:

> 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), --
>         [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 
>         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.
>         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

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