Re: (OT) Re: Object identity


Dmitry A. Kazakov wrote:
On 6 Jul 2006 19:55:45 -0700, David Barrett-Lennard wrote:

Dmitry A. Kazakov wrote:
On 5 Jul 2006 17:16:07 -0700, David Barrett-Lennard wrote:

Dmitry A. Kazakov wrote:
On 5 Jul 2006 08:47:20 -0700, David Barrett-Lennard wrote:

Dmitry A. Kazakov wrote:
On 5 Jul 2006 05:39:21 -0700, David Barrett-Lennard wrote:

Dmitry A. Kazakov wrote:
On 4 Jul 2006 19:10:28 -0700, David Barrett-Lennard wrote:

For me the word Platonism refers specifically to belief in the
independent existance of things like integers, and has nothing to do
with absolute truth.

Existence independent on what?

Our universe or multiverse or physical reality or whatever you want to
call it. I don't believe mathematics exists because of physical
reality. I believe it's more likely to be the other way around.
Physics is a branch of mathematics!

But you have no mathematics. There is a many formal systems floating
around. Where do numbers exist?

For me it is obvious that numbers exist. Can you imagine that numbers
don't exist?

Huh, there are tribes which have only numbers 1, 2 and "many."

I was asking you personally!

Yes, I do. Numbers do not exist, in material sense. They don't have taste,
color, mass, spin, charge etc.

I never asked "... exist, in material sense".

There is no other senses. In fact there is no any absolute sense of
existence. That's the point.

That's only the point for you.


On the other side do hyper-inaccessible cardinals exist?

That's a good question. I'm not sure if I can dodge it! It seems to
me that some mathematical things like the integers are obviously
self-consistent,

Integers aren't consistent or not. Consistency is a property of a formal
system.

I'm assuming *any* of the intuitively reasonable formal systems that
contain the axioms that define the integers and allow for the proof
that they are unique up to isomorphism. I don't care exactly which
formal system you pick. Only you seem to care!

I do care, because anything you state is a function of the system within
the system. This includes integers. So talking about self-consistency of
integers is meaningless.

That's simply not true. All the common formal systems that include
axioms for the integers only differ in relation to the fuzzy boundary
of what's provable. The underlying integers are the same in all
cases. You confuse completeness with uniqueness up to isomorphism.


(I presume that "intuitively reasonable" should read "platonic?" (:-))

Therefore I *postulate* that a chosen formal system is consistent.

Nice, I'd like *postulate* myself a millionaire... The only problem is that
my bank has postulated something different.

That's a silly analogy.

It is very reasonable to postulate consistency of ZF (say). That's
the whole point. A good mathematician finds axiomatic systems that are
powerful and intuitive. There is an implicit assumption that the
reader is willing to accept self-consistency.

This is tantamount to postulating that the integers exist. Yes of
course it can't be proven. But we are talking about philosophy not
science.

In any case, can I postulate that numbers don't exist? If you answer YES,
then that closes the discussion, because your postulate is nothing better
than mine.

Answering yes doesn't close the discussion. You can certainly
postulate that numbers don't exist (ie that even the smallest
axiomatic system defining the integers is inconsistent). However, in
practice that is a rather useless postulate, because it is at odds with
most people's innate belief systems.

I normally postulate things that I believe in. Don't you?

Ergo, numbers both don't exist and exist or neither (at your
choice, but with the same end effect). Alternatively, you could say NO, but
then you'll need to present some other postulate, which would support the
first one and refute mine. On that second one we (all, now and future
generations) should agree. [No chance, really]

Remember that you can't express anything at all without some implicit
"belief system" in place. I'm happy to put set theory and
numbers into the core of my belief system. This is a platonic
viewpoint.

No. Plato would say that you have no choice, because there is an absolutely
true set theory.

If you are unwilling to let me use my own characterization of Platonism
then it's difficult to have a reasonable discussion.


I use the rule : if it's self consistent then it exists. That's my
definition of "exists".

And if it exists then? Do inconsistent things exist?

Relevance?

A contradiction. A Platonic must accept existence of inconsistent things.
After all, the very inconsistency is an idea, it must exist. So
inconsistency is a reality. Therefore a Platonic cannot equalize
consistency with existence.

That's an absurd argument. A formal system known to be inconsistent
doesn't define anything because nothing satisfies the axioms. It is
quite simple, really.


Inconsistency exist and existence is inconsistent.

I give you an example: is MS-Windows consistent? Or, let's take German
taxation system. Is it? (:-))

BTW, you can take any inconsistent system and wrap it into a consistent
one. It is trivial, you replace each statement P of the original system
with <P is X> in the new system. For example:

"All Cretans are liars" ---> <"All Cretans are liars" is contradictory>

Yes but it is rather boring.

I don't think the taxation laws are just boring, they are revolting! (:-))

You must be very careful about "obvious" things. Consider a lamp. Let you
turn it on at 11:30. Then you turn it off at 11:45, on then at 11:52:30,
and so on. In a platonic Universe such lamp definitely exists. Now, will it
be on at 12:01?

I just see incompleteness as being orthogonal to inconsistency.

Egh? You do mean that in a larger super platonic system you'd be able to
answer this question? Why didn't you started with that system? Platonic
philosophy is too naive to elude such attacks.

No. You continue assuming that Platonism implies completeness.

It does not imply that, it *is*. Either universals are real or not. Plato
does not accept any incompleteness. So the continuum hypothesis is either
true or not.

You are trying to argue a point with me that I don't even agree with.
Why don't you focus less on "Plato" or "Platonism" and
actually argue with what I'm saying.


Instead it is only about postulating consistency.

It is rather so that you postulate its consistency.

So you don't mind spending a lot of your time thinking and reasoning
about numbers (like all that time as a child learning your
multiplication tables), and yet believing they don't exist?

Sure I don't. Precisely I don't believe in a system where this question
could be asked.

That's a belief! You're very choosy in the things you believe in.
If you were a true atheist you wouldn't be able to speak :)

Materialist don't speak, *it* emits sound waves. It is same as to say (:-))
that a FM receiver gives a speech.

I don't think this is going anywhere!

Cheers,
David Barrett-Lennard

.

Relevant Pages

  • Re: (OT) Re: Object identity
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