Author Topic: Questions on mathematical platonism from Imperishable Star v. I  (Read 313 times)

Xepera maSet

For @merytseth

How is it concluded that the empty set is an axiomatic fact? Related to your article, why does the ability to construct X prove X exists?
AKA: Three Scarabs, 1137

"You look up into the night sky - whether as a child or an adult - and if you open yourself honestly, then it is a gateway to mystery, to the unknown."

merytseth

  • Guest
Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #1 on: June 03, 2017, 05:51:38 am »
The empty set is a set with no elements.  It contains nothing.  If you can construct a set of no elements, then the empty set exists (in the platonic sense).  Surely I can construct a set with nothing in it.  I just did.  So did you.  It's an axiom because we can't "prove" it, but it doesn't really make sense to think that it may not be true.

In your second question, existence is also in the platonic sense.  If the form of, say, a triangle did not exist, would one be able to construct a triangle?  The natural numbers can be constructed set-theoretically, and the rationals from those, and a good number of real and complex numbers can be algebraically constructed.  If we can construct a number algebraically, then clearly that number exists in the platonic sense.  

The more interesting thing, at least as far as algebra goes, is the vast array of transcendental numbers which indisputably exist, but cannot be constructed The most famous transcendental is pi, but really most of the real number line is transcendental.  If you were to throw a dart at the real number line, the probability of that dart landing on an algebraic number (that is, not transcendental) is zero.  In fact, most of the real number line is even weirder - they are indescribable.  Even beyond transcendental, most real numbers can't even be described.  Here is a layman's-speak version of the proof the indescribable numbers exist, that also goes into some of the different 'kinds' of infinity which I've mentioned to you before.  http://blog.ram.rachum.com/post/54747783932/indescribable-numbers-the-theorem-that-made-me

EDITED TO ADD:  In an even more "first principles" kind of way, here is the more complete set theoretic construction of the natural numbers.  It goes into something called the Peano axioms, which is pretty abstract stuff about the properties of the natural numbers.  http://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_proofs_f2014/ccs_proofs_f2014_lecture4.pdf
« Last Edit: June 03, 2017, 05:59:24 am by merytseth »

merytseth

  • Guest
Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #2 on: June 03, 2017, 06:08:10 am »
If you enjoy the idea of reducing maths to first principles and formal logic (I know I do), you may be interested in Bertrand Russell's Principiahttps://plato.stanford.edu/entries/principia-mathematica/

It was a 1925 attempt to construct all of math from first principles, and has a lot of really interesting stuff in it.

Unfortunately, Godel's incompleteness theorems show that no single logical system can describe all of math, but it's still a neat thing to study.

Xepera maSet

Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #3 on: June 03, 2017, 12:15:00 pm »
Thanks so much! It's weird coming at this from philosophy to math rather than the other way around. I don't think I ever liked math before!
AKA: Three Scarabs, 1137

"You look up into the night sky - whether as a child or an adult - and if you open yourself honestly, then it is a gateway to mystery, to the unknown."

merytseth

  • Guest
Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #4 on: June 03, 2017, 01:28:30 pm »
Thanks so much! It's weird coming at this from philosophy to math rather than the other way around. I don't think I ever liked math before!
The two aren't entirely unrelated. ;)  Unfortunately, no one tells you that until after you've learned more calculus than anyone will ever use.  Happy to help!  

pi_ramesses

Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #5 on: June 21, 2017, 01:19:59 pm »
Great paper, @merytseth . I was delighted to have read it. It took me back to a time when all I read pertained to predicate calculus. 

There is a playlist that I saw a couple years back that still makes my heart leap when I watch it over again:


Pro omnis dominos viae sinistra, sic itur ad astra
Nylfmedli14

Xepera maSet

Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #6 on: June 21, 2017, 01:55:54 pm »
I am reading about Godel right now and it is quite fascinating. The discrediting of the axiom of parallel lines literally blew my mind, I had to stop reading for the day. 
AKA: Three Scarabs, 1137

"You look up into the night sky - whether as a child or an adult - and if you open yourself honestly, then it is a gateway to mystery, to the unknown."

pi_ramesses

Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #7 on: June 21, 2017, 01:58:07 pm »
Non-Euclidean geometry is definitely mind blowing.
Pro omnis dominos viae sinistra, sic itur ad astra
Nylfmedli14

Xepera maSet

Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #8 on: July 11, 2017, 08:59:50 am »
Would the following statement be true?

"If X can potentially be constructed, the Form of X exists, even if X is never constructed."
AKA: Three Scarabs, 1137

"You look up into the night sky - whether as a child or an adult - and if you open yourself honestly, then it is a gateway to mystery, to the unknown."

merytseth

  • Guest
Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #9 on: July 11, 2017, 10:48:54 am »
Would the following statement be true?

"If X can potentially be constructed, the Form of X exists, even if X is never constructed."
Yes, I would say so, and would go even farther, because indescribable numbers can't be constructed at all (if they could, they wouldn't be indescribable), but it isn't hard to prove that they exist.  The forms of these numbers would also be indescribable.

Xepera maSet

Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #10 on: July 11, 2017, 10:55:53 am »
Would the following statement be true?

"If X can potentially be constructed, the Form of X exists, even if X is never constructed."
Yes, I would say so, and would go even farther, because indescribable numbers can't be constructed at all (if they could, they wouldn't be indescribable), but it isn't hard to prove that they exist.  The forms of these numbers would also be indescribable.

This has given me an interesting idea, I'll start a thread hopefully later on it.
AKA: Three Scarabs, 1137

"You look up into the night sky - whether as a child or an adult - and if you open yourself honestly, then it is a gateway to mystery, to the unknown."

Xepera maSet

Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #11 on: July 12, 2017, 02:19:53 pm »
Would the following statement be true?

"If X can potentially be constructed, the Form of X exists, even if X is never constructed."
Yes, I would say so, and would go even farther, because indescribable numbers can't be constructed at all (if they could, they wouldn't be indescribable), but it isn't hard to prove that they exist.  The forms of these numbers would also be indescribable.

This has given me an interesting idea, I'll start a thread hopefully later on it.

Uggggghhhhh I have no idea what my train of thought was haha. Hopefully it comes back!
AKA: Three Scarabs, 1137

"You look up into the night sky - whether as a child or an adult - and if you open yourself honestly, then it is a gateway to mystery, to the unknown."

pi_ramesses

Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #12 on: July 13, 2017, 05:43:55 am »
Would the following statement be true?

"If X can potentially be constructed, the Form of X exists, even if X is never constructed."
Yes, I would say so, and would go even farther, because indescribable numbers can't be constructed at all (if they could, they wouldn't be indescribable), but it isn't hard to prove that they exist.  The forms of these numbers would also be indescribable.

This has given me an interesting idea, I'll start a thread hopefully later on it.

Uggggghhhhh I have no idea what my train of thought was haha. Hopefully it comes back!

Something to do with Setian metaphysics perhaps? Suppose that X was your future Self. Who can describe that? Also @merytseth and @Xepera maSet what are your thoughts on Kant's argument that existence is not a predicate? Logically, I think he had a point and that he was right about that. And when you say X exists, is it intended to mean that it just exists or some other state of existence as with Meinong's Jungle?
« Last Edit: July 13, 2017, 06:00:58 am by Nylfmedli14 »
Pro omnis dominos viae sinistra, sic itur ad astra
Nylfmedli14

Xepera maSet

Re: Questions on mathematical platonism from Imperishable Star v. I
« Reply #13 on: July 14, 2017, 06:50:16 pm »
Would the following statement be true?

"If X can potentially be constructed, the Form of X exists, even if X is never constructed."
Yes, I would say so, and would go even farther, because indescribable numbers can't be constructed at all (if they could, they wouldn't be indescribable), but it isn't hard to prove that they exist.  The forms of these numbers would also be indescribable.

This has given me an interesting idea, I'll start a thread hopefully later on it.

Uggggghhhhh I have no idea what my train of thought was haha. Hopefully it comes back!

Something to do with Setian metaphysics perhaps? Suppose that X was your future Self. Who can describe that? Also @merytseth and @Xepera maSet what are your thoughts on Kant's argument that existence is not a predicate? Logically, I think he had a point and that he was right about that. And when you say X exists, is it intended to mean that it just exists or some other state of existence as with Meinong's Jungle?

It was the Zeena thread about self creation. Her belief fits with the idea that if X can be constructed its Form exists.
AKA: Three Scarabs, 1137

"You look up into the night sky - whether as a child or an adult - and if you open yourself honestly, then it is a gateway to mystery, to the unknown."


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