rhersh at math.unm.edu
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
> I want to try to explain this here, and do so in a way
> that points to some answers to questions raised by Sol
> 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
NO, NOT AT ALL, NOT IN THE LEAST. I EXPLAINED ALL
ABOUT THIS IN ANSWERING PROF. TRAGESSER'S LAST POSTING.
> 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:
>  What is distinctive about mathematics?
>  What is distinctive about its verification structure?
>  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
NO NO ONCE AGAIN NO!
> 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).
WHERE? WHEN? HOW? DETAILS, PLEASE.
> 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.
>  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).
>  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.
> 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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