FOM: Hersh's tripartition of reality; Simpson's definition of intuitionnism

[Patrick Peccatte]

fom member digest:
Name: Patrick Peccatte
Position: Software engineer and philosopher
Institution: Independant private company and Paris 7 University
Research interest: History and philosophy of sciences and mathematics
(specially quasi-empiricism, experimental mathematics, ontology)
More information: http://peccatte.rever.fr
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1. The Hersh's partition of reality (see: Thu, 26 Feb 1998 14:54:09)
seems based upon the abrupt cartesian distinction between body -
physical reality - and mind - mental reality - to wich a third category
has been added. This third category, namely social-cultural-historical
reality, is very heteroclit and operates like a saturation of the hard
split mentionned above; it is like a big bag, a jumble which could
capture almost all the reality as in certain versions of this way of
thinking (cf. for exemple Paul R. Gross and Norman Levitt's "Higher
Superstition"). The argument used here is the following: take the very
popular distinction between mind and matter and saturate it with a third
class, so you cannot distinguish other kinds of realities (this is the
first oversimplification). Since "mathematical objects are not material,
and they are not mental" (Hersh words and the second
oversimplification), mathematics obviously belongs to the third
category. Like all the taxonomies, and specially the very holistic
classifications, it is a succession of oversimplifications. It is useful
of course, but we cannot consider that it is a complete description of
the real essence of what is so classified. Take another example: natural
species taxonomy is much more sophisticated than the Hersh's
tripartition, but from the point of view of this classification, the
fact that humans are primates like apes does not explain all the
humankind features (and particularly, the importance of the
social-cultural-historical reality for the humans...). It could be more
plausible to consider that mathematics have features rooted in the three
realites, and by that, wins its own and specific reality like other
similar "social-cultural-historical objects" - like computer programs
for example. I don't want extract too much from analogies and I don't
want to appear *obsessed* by computer programs (see my mail: Wed, 25 Feb
1998 21:39:54) but let me try to explain briefly this point. I hope it
appears obvious that both mathematics and computers programs are
effectively social-cultural-historical objects; but they are also the
expression of mental activity. And they are also anchored in the
physical reality. Not only in the sense of the empirical reality (the
"five fingers"), but in the mathematical notation (think to matrices),
in quasi-empirical facts (the zeroes of the Riemann zeta function -
comments are welcome here...) and so on. Maybe the more physical aspect
common to mathematics and computer programs is the importance of time.
Time is obviously important in programs because a program only lives
running, executing its instructions. Despite the fact that proof in
mathematics is habitually recognized independent of time, it seems to
have same characteristics: a beginning, an end, a progression step by
step, the search for shortest demonstrations, etc. Somebody knows
studies or references about this analogy ?

2. The characterization of constructivism given by Steve Simpson
("mathematics consists of mental constructions ", in different mails) is
an approximation which is roughly true only for the historical
intuitionnism (Brouwer). Intuitionnism is multiform and a its
characterization is less solipsistic.
I quote here four theses from "Intuitionnisme 84" by Jacques Harthong
and Georges Reeb:
"Nous considerons:
a) que la mathematique formelle (le theorie axiomatique des ensembles de
Zermelo-Fraenkel, qu'on nous a toujours presentee, lorsque nous etions
etudiants et naifs, comme le fondement absolu et eternel de la
mathematique) n'est qu'une theorie scientifique eventuellement sujette a
la refutation  [...], une simple interpretation d'une realite qui lui
est exterieure, et non la realite mathematique elle-meme;
b) que ce qu'il y a de scientifique dans la mathematique, c'est
l'articulation entre la theorie (quelle qu'elle soit) et cette realite,
et non le caractere formel et abstrait de la theorie qui n'est qu'un
instrument;
c) que cette articulation entre la theorie et la realite repose sur la
methode experimentale dans les sciences de la nature, mais sur la
methode constructive dans la mathematique;
d) que l'ideologie formaliste ayant nie cette realite exterieure a la
theorie en pretendant que la seule realite methematique est la theorie
formelle elle-meme, sa domination ecrasante pendant un demi-siecle a eu
pour consequence de prendre cette theorie beaucoup trop au serieux, d'en
oublier toute la relativite historique et semantique, d'en faire un
dogme pesant et tyrannique, une langue de bois qui interdit toute
expression personnelle, qui change le travail du mathematicien en
publications standardisees et insipides, et les chercheurs en robots."
(in "La Mathematique non-standard", Paris, Editions du CNRS, 1989, p.
214).
As a realist in philosophy of mathematics, I don't agree completly with
these four theses (specially with the too strong faillibilism expressed
in it and with the confusion between *formalism* as an ideology and
*formalism* as a constitutive feature of mathematics). But my aim is to
claim that intuitionnism and constructivism philosophical backgrounds
are more subtle than the radical subjectivism in the Steve's definition.
The main characteristic seems to me the insistance in all kind of
intuitionnism on a mathematical reality external to mathematical
theories themselves.

Regards
[I apologize for my poor english]
Patrick Peccatte