FOM: Definition of Platonism

Michael Hardy hardy at
Wed Jan 14 15:58:53 EST 1998

	In the posting numbered 9801.53 in the archive,
Stephen G Simpson <simpson at> wrote:

> (i.e. Platonism, the view that mathematical truth exists independently
> of, and reveals itself to, passive consciousness).

	I think this gets the definition of Platonism wrong, in just
the way mathematicians often get it wrong.  Platonism is the proposition
that the existence and nature of _universals_ is independent of
_particulars_.  The fact that mathematics deals _only_ with universals
is perhaps why mathematicians often fail to realize that Platonism
says something about universals that it doesn't say about particulars.
Note also what Platonism holds things to be independent of: They are
held to be independent, not of consciousness, but of particulars.

	The proposition that things in general exist independently
of consciousness is not at all the same thing as Platonism.  I would
call this, not ``Platonism,'' but, perhaps, ``objectivity.''

	To me this distinction is important, because I believe that
(1) there is a universe outside of our minds, whose existence and
nature does not depend on our minds, and that this universe includes
that which is studied by mathematics; and (2) Platonism is false, i.e.,
the nature and existence of universals _does_ depend on particulars.

	This discussion erupted on sci.math.research 13 months ago.
Michael Huemer of Rutgers posted the comments below, included here
by permission.

	Mike Hardy

Michael Hardy
hardy at

Forwarded message:

> From news Mon Dec  9 10:45:13 1996
> From: owl at (Michael Huemer)
> Newsgroups: sci.math.research
> Subject: Re: UC Davis math research profiles
> Date: 7 Dec 1996 21:29:20 -0500
> Organization: Rutgers University
> Approved: Daniel Grayson <dan at>, moderator for sci.math.research
> greg at (Greg Kuperberg) writes:
> > Platonism: The habit among mathematicians, especially geometers, of
> > pretending that mathematical objects actually exist.
> This doesn't seem like a very fair definition.
> As the lone philosopher here, I feel obliged to add something about this.
> 1. Platonism is a position on the problem of universals, and
> derivatively a position in philosophy of mathematics.  Platonism is
> the view that universals exist, and their existence is independent of
> the existence of particulars.  (Also called "transcendent realism,"
> contrasted with "immanent realism," which holds that universals exist,
> but only insofar as some particulars exemplifying them do.)
> - A PARTICULAR is just an individual entity (or event), like me, or
> Mount Everest, or the Earth.
> - A UNIVERSAL is the kind of thing that (a) can be predicated of
> something, and (b) can be wholly present in multiple different things
> at once.  For example, redness is a universal:  you can predicate
> redness of something (as in "The flower was red"), and redness can be
> present in multiple different things at once (e.g., a flower can be
> red, and also a fire hydrant, and a cup, all at the same time).
> Neither of those things could be said of a particular.  Also, the
> relation 'biggerness' is a universal -- it can be predicated of an
> ordered pair of things (as in "Bob is bigger than Sally"), and
> multiple different pairs of things could exemplify it.
> - For another example:  *war* is a universal, but *the war of 1812* is
> a particular.  The war of 1812 is a single, individual event, but
> *war* is a general class that many different particular events fall
> under.
> - A particular is said to EXEMPLIFY or 'instantiate' or 'fall under'
> whatever universals can be correctly predicated of it.  For instance,
> the particular coffee cup I'm drinking out of now exemplifies the
> universals 'cup', 'black', 'cyllindrical', and indefinitely many more
> universals. 
> 2. The problem of universals in philosophy is roughly this:  Do
> universals exist?  And if they do, in what manner?
> - If you say they don't exist, then you're called a 'nominalist'.
> - The view that universals do exist is called 'realism'.  If you
> adhere to realism, then you can ask the further question:  do
> universals depend for their existence on particulars?
>   - The view that universals exist *independently* of particular
>   things, is Platonic realism.
>   - And the view that they depend on particular things for their
>   existence is immanent realism.  For example, this view would say
>   that the universal 'redness' exists if and only if there is at 
>   least one red object in the universe.  The Platonist would say it
>   doesn't matter whether there are any red things in the world;
>   redness exists anyway.
> 3. What does this have to do with philosophy of mathematics?  Well,
> mathematical objects are universals (at least I think they are).  The
> number 2, for example, is a universal:
> (a) you can predicate it of something, as in, "These are TWO socks"
> (predicating 'twoness' of the pair of socks);
> (b) you can have multiple instances of it.  For instance, 'two-ness'
> can be found in my shoes, in my two hands, in the two tires on my
> motorcycle, and so on.
> Hence, Platonism in philosophy of mathematics is the view that these
> mathematical objects, which are just a subclass of universals, exist
> in a manner independent of particulars.
> 4. Platonism is not *merely* the view that mathematical objects exist,
> full stop.  You could have the view that mathematical objects exist
> *insofar as* there exist particular objects and/or events in the world
> that exemplify them - for instance, numbers 'exist' because there are
> things to be counted.  This view is not called "Platonism" (but it is
> a form of 'realism').
> 5. My final comment (and this is more in the way of opinion than
> explanation):  If some form of realism is not true - that is, if
> mathematical objects do not exist in some manner - then it would seem
> that mathematics is merely a repository of peculiarly widespread
> delusions, no better than astrology or mythology.
> -- 
>                                               ^-----^ 
>  Michael Huemer <owl at>        / O   O \
>             |   V   | 

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