FOM: Re: Hersh/Rota/Feferman/Lakatos
rhersh at math.unm.edu
Fri Jan 2 20:04:49 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. NO! Please let me repeat my answer here (though in somewhat
different words because I am thinking, not copying.)
Here is my answer: mathematics is both one of the humanities
and one of the sciences. It is that humanity which is
scientific, and it is that science which is humanistic.
Its subject matter, mathematical concepts and objects, are
like novels, symphonies, Constitutions, and ideologies--human
creations, part of human culture. But they are different from
the other humanities in being science-like. Science is said to be
distinguished from non-science by possessing reproducible results,
which means high degree
of consensus. Mathematics, even more than any other science,
possesses reproducible results, so in this repect it is the most
science-like of sciences. We prove the Pythagorean theorem,
no only by rereading it in Euclid, but remarkably by finding other,
unrelated proofs. And of course mathematical calculations
get the same result by means of independdent calculations. When,
rarely a discrepancy appears, it is always straightened out quickly.
This answer does not refer to "mathematicals", or "what mathematicians
do." The traditional definitions of math said it was about number
and geometric figure. Such definitions are obviously no longer
tenable. Mathematical logic is an excellent example of
mathematics which is not about any of the traditional "mathematicals!"
Mathematics can be about anything, as long as its
subject matter is immaterial (not a natural science) and as long as
it attains an extremely high degree of reproducibility and consesus.
If a new field appears tomorrow which satisfies those
` two critieria, everybody will recognize it as mathematical.
> 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
FALSE! MISQUOTATION! WRONG! I DO NOTHING OF THE KIND!
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).
WHAT FAST AND LOOSE? YOU CAN'T MAKE THAT KIND
OF ACCUSATION (OR IS JUST AN INSINUATION?) WITHOUT
BACKING IT UP WITH CHAPTER AND VERSE. IF I'M
REALLY FAST AND LOOSE, SHOW ME AND I'LL TIGHTEN UP
AND SLOW DOWN. BUT DON'T JUST SAY IT WITHOUT ANY
BACK UP AT ALL.
> 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
v> 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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