I hope you will excuse my butting in here, but I was passing through on
a different mission
and became disturbed by reading some earlier posts of this thread.
My 2 cents worth:
I tend to think that David Nyman has the more sceptically acceptable
slant on this. Mathematics and logic are constructions of the human
brain. They are extremely useful, in appropriate contexts, because they
allow effective, efficient and economical representations of processes
in the world.
There is no particularly good reason however to think that mathematical
objects exist outside of human brains or phenotypic extensions such as
computers. I think it IS fair to say though that, for example, numbers
and formulae written on a page or blackboard are literal extenstions of
the constructs within the active mathematical mind.
That so much of what occurs in 'the world' CAN be represented by
numbers and other mathematical/logical objects and processes, is better
expained by assuming that the great 'IT' of noumenal nature is actually
made up of many simple elements [taken firstly in the general sense].
This underlying simplicity which yet combines and permutates itself
into vast complexity, is something we infer with good reason - it
works! But you cannot DEDUCE from it that numbers and other
mathematical objects exist 'out there', except in those particular
regions of space time that happen now to be mathematically active
[EMAIL PROTECTED] wrote:
> 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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