FOM: what are the most basic mathematical concepts?
Stephen G Simpson
simpson at math.psu.edu
Mon Jan 12 20:00:18 EST 1998
Charles Silver writes:
>
> On Sun, 11 Jan 1998, Stephen G Simpson wrote:
>
> > 2. As a tentative list of the most basic mathematical concepts, I
> > offered the following:
> >
> > list 1:
> >
> > number
> > shape
> > set
> > function
> > algorithm
> > mathematical axiom
> > mathematical proof
> > mathematical definition
> >
>
> To my mind "set" is utterly distinct from the other members of
> this list. I don't wish to argue that "set" is *not* basic, but if it is,
> it is basic in an entirely different sense than the others. I won't go
> through all the items in the list, but it seems to me that each concept,
> with the exception of "set", can be argued to be "epistemically basic".
> That is, each concept (except "set") is arguably rooted in certain
> fundamental conceptions in our minds that arise naturally. I don't think
> the concept of "set" comes to our minds in such an epistemically primitive
> way. Rather, it seems to me, the concept of "set" reflects a later
> mathematical development. Using the concept of "set" (plus some axioms
> *about* sets), one can unify other mathematical concepts in a desirable
> fashion. In this way and in many others, "set" can be considered "basic".
> But, I don't see it as being basic in the same way as "shape" is, for
> example. Steve, could you please explain your reasons for including "set"
> in the above list?
Yes, I'll gladly explain.
First of all, note that I said "tentative". In other words, I am not
claiming that the above list is immutable for all time. Clearly there
is room for FOM discussion about which concepts are the most basic.
And this could result in some important conceptual clarification and
research programs.
There has already been some discussion of "shape" and foundations of
geometry; see for instance my posting
FOM: foundations of geometry; set-theoretic foundations; Chow
on November 6, 1997. (Some people may have overlooked this, because
it came just before Lou's massive posting in which he gave a fully
detailed high-school algebra formulation of Faltings' theorem, just to
show that it could be done.)
The main reason I included "set" on my list is because of
set-theoretic foundations. According to the doctrine of set-theoretic
foundations, mathematical concepts such as the the natural numbers,
real numbers, manifolds, various topological and algebraic structures,
etc. etc. are to be built up from the notion of "set" as formulated in
systems such as ZFC. This doctrine has been fruitful for mathematics
and for f.o.m. research. Moreover, the set-theoretic foundational
scheme is generally accepted as orthodoxy by the mathematical
community. See for instance Bourbaki's horrible paper "Foundations
for the Working Mathematician", JSL, 1948, which has been discussed
here on the FOM list. So clearly "set" is regarded as basic by the
mathematical community here and now.
It's fun to speculate on whether or how long Bourbaki/ZFC-style
foundations will continue as the accepted orthodoxy. I remember
Nerode saying that it would probably disappear within at most 50 or
100 years.
A subsidiary reason for including "set" on my list is because it does
have at least a little bit of pedigree as being epistemologically
basic, in that it is at least vaguely similar to epistemologically
basic notions such as "concept", "predicate" and "property". But I
admit that this is a very weak argument. I'm well aware that
ZFC-style set theory is a far cry from a good theory of concepts or
properties. For one thing, it's not philosophically sound to say that
if A is a concept and B is a concept then A minus B is a concept.
(There is a concept "car", and there is a concept "red", but there is
no such concept as "cars that are not red".)
There is room for more discussion about which mathematical concepts
are the most basic. I would think that the most controversial item on
my list would be "function". Has there ever been a decent
foundational scheme based on functions rather than sets? I know that
some category theorists want to claim that topos theory does this, but
that seems pretty indefensible. For one thing, there doesn't seem to
be any way to do real analysis in a topos.
-- Steve
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