David Nyman wrote:

> I fail to see any 'knock-down' character in this argument. Peter says
> that mathematical concepts don't refer to anything 'external', and on
> one level I agree with him. But they are surely derived from the
> contingent characteristics of experience, and AFAICS experience in this
> context reduces to the contents of our brains. So 'infinite sets' is
> just a model (brain material at another level of description) which IMO
> counts as a 'physical notion' unless you start off as an idealist. Put
> simply, you can't think mathematical thoughts without using your brain
> to instantiate them - and you don't literally have to instantiate an
> 'infinite set' in the extended sense in order to manipulate a model
> with the formal characteristics you impute to this concept. In fact,
> the inability to convert infinite and transfinite sets into physical
> notions is excellent empirical evidence that they *don't* exist in any
> literal sense -  they don't need to, as their usefulness is as limit
> cases within models, not as literal existents (nobody has ever
> literally deployed an infinite set).

A particular concrete (brain) instantiation of a mathematical concept
can't be equivalent to the math concept itself.  I pointed out that
many different physical processes can implement the *same* algorithm -
this shows that the mathematical concept of the algorithm can't be
identified with any particular physical instantiation of it.  Read up
on the failure of simple Identity theories of mind.  Surely you
understand the difference between a *Class* (an abstract actegory) and
an *Object* (a particular instance of the concept).?  The Class is not
the object

That's the first part of the argument for platonism. (1)   The second
part of the argument is the argument from indispensibility - you can't
remove mathematical concepts from theories of reality because some
concepts (like inifnite sets for example) can't be converted into
physical notions. (2)  It's the combination of (1) and (2) that
clinches it.


>
> This is a thoroughgoing contingentist position, and I don't see that it
> can be refuted except by rejecting contingentism and starting from
> idealism. But then you've begged what you're trying to prove.

Aren't you guilty of the same thing?  You're simply assuming that
materialism is the ultimate metaphysics and trying to reduce everything
to that.  You do this because the human brain is only capable of
representing *physical* things in conscious experience.

But what is a *physical* thing really?  For instance is the *length* of
the computer screen in front of you an objective value?  Someone moving
close to light speed perpendicular to your computer screen would record
a quite different value for the length of your computer screen than you
would.  This suggests that the physical form is not objectively real.
What *is* objectively out, is a 4-dimensional world-time for your
computer screen as described by general relativity.... but this 4-d
world-time is a *mathematical* concept.

One could imagine an alien race or a super-intelligence which had no
consciousness of physical things, but *sensed* everything purely in
*mathematical* terms.  For instance imagine if they a way to *directly
sense* 4-d world-lines.  Then it might be 'obvious' to alien
philosophers that mathematical things were objevtively real.




>
> >  'If according to the simplest explanation, an entity is complex and
> > autonomous, then that entity is real.' ('The Fabric Of Reality', Pg 91)
>
> Autonomous of what precisely? In what sense is a mathematical concept
> autonomous of your brain, or the collection of brains and other
> recording devices that instantiate it? Remember that we're talking
> about mathematical *concepts* - i.e. things we can grasp - it's merely
> a metaphor to claim that these models *refer* to autonomously existing
> platonic realities. Either a metaphor, or the presumption of such
> platonic reality, not its proof.


See (1) and (2) above.  If the postulation of some entity *simplifies*
our explanations of reality, then this provides (probabilistic)
evidence that the postulated eneity exists.  (Occams razor).  The
evidence for the existence of platonic entities is that they simplfiy
our models of reality.

>
> > As Detusch points out, mathematical entities do appear to match the
> > criteria for reality: 'Abstract entities that are complex and
> > autonomous exist objectively and are part of the fabric of reality.
> > There exist logically necessary truths about these entities, and these
> > comprise the subject-matter of mathematics.'
>
> Truths are only equivalent to 'existents' for an idealist. Fair enough,
> but then this has to be accepted axiomatically, or not at all. I can't
> honestly see why this is so hard to grasp.
>
> David

I certainly wouldn't equate Platonism with Idealism!   We don't seem to
accept anything 'axiomatically'.  Instead we look to see which
postulated entities simplify our explanations of reality the best.


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