definition of "platonism"
Michael Hardy
Greg Kuperberg wrote:
[ snip ]
> http://www.math.ucdavis.edu/profiles/
[ snip ]
> However, the pages also have some
> serious mathematical content, both in the research descriptions and the
> glossary of mathematical terms, which you might find interesting. Some
> excerpts from the glossary:
>
> Platonism: The habit among mathematicians, especially geometers,
> of pretending that mathematical objects actually exist.
I'm not sure how seriously this definition was supposed to be
taken, but I've run into a number of mathematicians who insist on defining
"platonism" either as the belief that the existence and nature of _things_
_generally_ does not depend on anyone's awareness of them (for this I would
rather use "objectivity" than "platonism") or that the existence of
mathematical objects does not depend on anyone's awareness of them.
I think "platonism" should be defined as the tenet that _abstract_
things (including, but not limited to, mathematical objects) exist as
_concrete_ things in a "world of platonic forms" (or "platonic heaven").
I confess to a suspicion that the only reason why the unfortunately
prevalent definition is used at all, is that the people who are using it
are not subtle enough to see any difference between (1) saying that abstract
things exist as concrete things in a world of forms, and (2) saying that the
existence and nature of things generally (including ordinary concrete
objects like the table in front of me) does not depend on anyone's awareness
of them. (I have not named any names in this paragraph . . . .)
Mike Hardy
Michael Hardy
ha...@stat.umn.edu
Nancy McGough
ha...@umnstat.stat.umn.edu (Michael Hardy) writes:
>> Platonism: The habit among mathematicians, especially geometers,
>> of pretending that mathematical objects actually exist.
>
>
> I'm not sure how seriously this definition was supposed to be
>taken, but I've run into a number of mathematicians who insist on defining
>"platonism" either as the belief that the existence and nature of _things_
>_generally_ does not depend on anyone's awareness of them (for this I would
>rather use "objectivity" than "platonism") or that the existence of
>mathematical objects does not depend on anyone's awareness of them.
>
> I think "platonism" should be defined as the tenet that _abstract_
>things (including, but not limited to, mathematical objects) exist as
>_concrete_ things in a "world of platonic forms" (or "platonic heaven").
>
> I confess to a suspicion that the only reason why the unfortunately
>prevalent definition is used at all, is that the people who are using it
>are not subtle enough to see any difference between (1) saying that abstract
>things exist as concrete things in a world of forms, and (2) saying that the
>existence and nature of things generally (including ordinary concrete
>objects like the table in front of me) does not depend on anyone's awareness
>of them. (I have not named any names in this paragraph . . . .)
What do you mean by "exist as concrete things"? For example, what does it
mean to say that the power set of the reals exists "as a concrete thing"?
And then, using your definition of platonism (call it Hardy-platonism),
how do you state the Hardy-platonic view of the continuum hypothesis? And
how does the Hardy-platonic view of CH (version 1) differ from the version
2 view of CH.
Thanks,
Nancy
--
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@..@ Nancy McGough /\_/\
(----) Infinite Ink ( o.o )
( >__< ) http://www.jazzie.com/ii/ > ~ <
john baez
In article <589dh9$g...@epx.cis.umn.edu>,
Michael Hardy <ha...@umnstat.stat.umn.edu> wrote:
> I think "platonism" should be defined as the tenet that _abstract_
>things (including, but not limited to, mathematical objects) exist as
>_concrete_ things in a "world of platonic forms" (or "platonic heaven").
That's better, thought I wouldn't quite say the platonic forms are "concrete
things" --- it's hard to know quite they'd be "concrete", and in any event,
Plato actually regards the forms as *more* real than the objects we encounter
in daily life!
Actually there is both an ontological and an epistemological side to
Platonism, i.e., a theory both of what exists and how we know it.
The following definition from the WWWebster Dictionary sums them both up
reasonably well:
"the philosophy of Plato stressing especially that actual things are
copies of transcendent ideas and that these ideas are the objects of
true knowledge apprehended by reminiscence"
though I would prefer to say that things *participate* in the forms.
If the term "reminiscence" seems odd, recall how in the "Meno" Socrates
teaches a boy that the diagonal of a square is sqrt(2) times as long as
the side. He considers the following sort of picture:
_________
| /|\ |
| / | \ |
| / | \ |
|/ | \|
---------
|\ | /|
| \ | / |
| \ | / |
| \|/ |
---------
and asks the boy lots of leading questions about the ratio of the
area of the big square to the smaller diamond-shaped one. Eventually
the boy gets it and then the Socrates character says:
Socrates: Without anyone having taught him, and only through
questions put to him, he will understand, recovering the
knowledge out of himself?
Meno: Yes.
Socrates: And is not this recovery of knowledge, in himself and
by himself, recollection?
Meno: Certainly.
Socrates: And must he not have either once acquired or always had the
knowledge he now has?
Meno: Yes.
Socrates: Now if he always had it, he was always in a state of
knowing; and if he acquired it all some time, he could not have
acquired it in this life. Or has someone taught him geometry?
You see, he can do the same as this with all geometry and every
branch of knowledge. Now, can anyone have taught him all this? You
ought surely to know, especially as he was born and bred in your
house.
Meno: Well, I know that no one has ever taught him.
[from http://www.perseus.tufts.edu/cgi-bin/text?lookup=plat.+meno+82b]
It may seem ludicrous to suggest that "every branch of knowledge"
can be taught in this way, but remember, for Plato true knowledge is
primarily knowledge of the forms.
> I confess to a suspicion that the only reason why the unfortunately
>prevalent definition is used at all, is that the people who are using it
>are not subtle enough to see any difference between (1) saying that abstract
>things exist as concrete things in a world of forms, and (2) saying that the
>existence and nature of things generally (including ordinary concrete
>objects like the table in front of me) does not depend on anyone's awareness
>of them. (I have not named any names in this paragraph . . . .)
It's probably just my overly cynical disposition, but a further reason
springs to mind, namely that these unfortunates have not in fact read
their Plato before venturing to summarize his philosophy.