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Author Topic:   Math: Eternal? If so Who Created It?
Sarah Bellum
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Posts: 224
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Message 16 of 30 (856779)
07-02-2019 8:00 PM
Reply to: Message 15 by Son Goku
07-02-2019 6:04 PM


But how can a set (in any kind of set theory) have the same cardinality as the set of all its subsets?
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GDR
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From: Sidney, BC, Canada
Joined: 05-22-2005
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Message 17 of 30 (856781)
07-02-2019 8:04 PM
Reply to: Message 13 by Son Goku
07-02-2019 2:18 AM


Son Goku writes:

Model Theory can basically show you that a single mathematical statement can have several models with completely different properties and a single formal statement can have multiple realisations.
So let's say the statement "The Reals are not countable". This is a provable statement in the ZFC formalisation of mathematics and basically refers to the fact that there isn't a bijection from the Natural numbers to the Reals.

However there are several universes of sets that satisfy the ZFC axioms, these are models of the ZFC axioms. In some models of ZFC the statement "Reals are uncountable" is true because in that model the set that obeys the axioms of the Reals genuinely have a higher cardinality than the set that obeys the axioms of the Naturals. In other models the set filling the role of the Reals actually happens to be of the same cardinality as the set that is the Naturals but a bijection between them doesn't exist.

So even a simple statement like "The Reals are uncountable", which seems to say something concrete about the Real numbers, is ambiguous because it's not totally fixed what "Reals" or "Uncountable" refer to. It's a purely formal/linguistic statement in ZFC.


Exactly what I was going to say. Couldn't have put it better myself.

He has told you, O man, what is good ; And what does the LORD require of you But to do justice, to love kindness, And to walk humbly with your God.

Micah 6:8


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AZPaul3
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Posts: 4258
From: Phoenix
Joined: 11-06-2006
Member Rating: 3.7


Message 18 of 30 (856792)
07-02-2019 10:40 PM
Reply to: Message 17 by GDR
07-02-2019 8:04 PM


Exactly what I was going to say.

I was waiting to see the English translation before I comment.

Edited by AZPaul3, : No reason given.


Eschew obfuscation. Habituate elucidation.

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Son Goku
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From: Ireland
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Message 19 of 30 (856809)
07-03-2019 2:59 AM
Reply to: Message 16 by Sarah Bellum
07-02-2019 8:00 PM


I mentioned this in my last post, by the construction of the power set being limited within the model they may be of the same size. The lack of appropriate bijections would make them have differing cardinalities within the model, despite being of the same cardinality when viewed externally.

Also one must distinguish a model of one set theory (i.e. different models of ZFC) from different kinds of set theory (i.e. ZFC, Tarski–Grothendieck, Kripke–Platek). The former are different realisations of the same syntactic statements, the others have different syntactic statements.

Edited by Son Goku, : No reason given.


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Son Goku
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From: Ireland
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(4)
Message 20 of 30 (856810)
07-03-2019 3:41 AM
Reply to: Message 18 by AZPaul3
07-02-2019 10:40 PM


The specifics of the case Sarah Bellum asked about aren't too important, those only really have a technical exposition since they're a technical question.

The basic point is that what a mathematical proof refers to is a bit ambiguous and mathematics can throw up problems similar to language like contradiction from over self-referencing. Which leads people to think that it is in fact a language not some externally existing "thing" fully independent of humanity.

For example there are axioms that both the Whole Numbers (Naturals being their technical name) and the Real Numbers obey. These axioms are things like:
For any two numbers a,b if a,b > 0 then a + b > 0 and a.b > 0

If you then try to prove the statement:
There is a number between 0 and 1
Using only those axioms you can't prove it true or false. That's because two different models (i.e. two different systems obeying them) exist. In one of them (Reals) the statement is true, in the other (Naturals) it is false. So from the axioms themselves you can't prove it true or false, because the axioms can only deal with statements common to both.

Work after Gödel's incompleteness theorem showed that this is always true, no matter how precise you are with your axioms there will always be multiple models and so there will be statements you cannot prove. Also statements can be true in the different models for different reasons.

Due to all these models it's ambiguous what a mathematical statement means. And this combined with paradoxes that result if you allow too much self-reference make many people think mathematics is just a language and a creation of human thought.

In addition to this, for myself, the fact that quantum theory suggests a layer of reality that cannot be described with mathematics (i.e. is non-algorithmic or non-mechanical) makes me inclined even more strongly toward mathematics being a human creation.

Edited by Son Goku, : No reason given.

Edited by Son Goku, : No reason given.


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Son Goku
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Posts: 1148
From: Ireland
Joined: 07-16-2005
Member Rating: 6.8


(2)
Message 21 of 30 (856912)
07-04-2019 4:59 AM
Reply to: Message 4 by Sarah Bellum
05-08-2019 10:37 PM


I meant to reply to this.

I wouldn't say the complexity of the Monster Group comes from the group axioms. It is a very complicated mathematical object that is a group, however the complexity doesn't arise in any sense from the group axioms.

"Group" is a type of object like "Vehicle". You could have a very simply designed vehicle like a bob sleigh and an incredibly complex one like a plane, but the complexity of a plane doesn't arise from the basic properties that make it a vehicle.


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Sarah Bellum
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Posts: 224
Joined: 05-04-2019
Member Rating: 2.4


Message 22 of 30 (856934)
07-04-2019 10:19 AM
Reply to: Message 21 by Son Goku
07-04-2019 4:59 AM


Oh, but in this case it does! The definition of a vehicle doesn't include plans for a Model A, so if you ask the question, what are the possible configurations of a "vehicle" based solely on the definition you don't get much of an answer. But for groups you get very specific blueprints!
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Son Goku
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Posts: 1148
From: Ireland
Joined: 07-16-2005
Member Rating: 6.8


(1)
Message 23 of 30 (856940)
07-04-2019 10:56 AM
Reply to: Message 22 by Sarah Bellum
07-04-2019 10:19 AM


Oh, but in this case it does! The definition of a vehicle doesn't include plans for a Model A, so if you ask the question, what are the possible configurations of a "vehicle" based solely on the definition you don't get much of an answer. But for groups you get very specific blueprints!

You get the generic properties of all groups not specific blueprints. You cannot derive the structure of the Monster Group from the Group axioms hence its complexity does not arise from them.
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Sarah Bellum
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Posts: 224
Joined: 05-04-2019
Member Rating: 2.4


Message 24 of 30 (856978)
07-04-2019 1:23 PM
Reply to: Message 23 by Son Goku
07-04-2019 10:56 AM


How was the structure derived, if not from the axioms?
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Son Goku
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Posts: 1148
From: Ireland
Joined: 07-16-2005
Member Rating: 6.8


Message 25 of 30 (856983)
07-04-2019 1:39 PM
Reply to: Message 24 by Sarah Bellum
07-04-2019 1:23 PM


This gets very complex, but at its most "basic" it's the set of symmetries of the Griess algebra. This is how it was first constructed. None of its particular structure follows directly from the Group axioms.
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Theodoric
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From: Northwest, WI, USA
Joined: 08-15-2005
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(4)
Message 26 of 30 (857005)
07-04-2019 5:18 PM
Reply to: Message 24 by Sarah Bellum
07-04-2019 1:23 PM


This is one the great things about this forum. If you are not careful you can get a graduate level seminar on a subject. Sit back and enjoy.

Facts don't lie or have an agenda. Facts are just facts

"God did it" is not an argument. It is an excuse for intellectual laziness.

If your viewpoint has merits and facts to back it up why would you have to lie?


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AZPaul3
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Posts: 4258
From: Phoenix
Joined: 11-06-2006
Member Rating: 3.7


Message 27 of 30 (857011)
07-04-2019 6:44 PM
Reply to: Message 26 by Theodoric
07-04-2019 5:18 PM


It's good when people teach.

Eschew obfuscation. Habituate elucidation.

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Sarah Bellum
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Posts: 224
Joined: 05-04-2019
Member Rating: 2.4


(1)
Message 28 of 30 (857014)
07-04-2019 7:21 PM
Reply to: Message 25 by Son Goku
07-04-2019 1:39 PM


You are merely describing how the structure was determined by algebraists, not the reason for the structure itself.

Here's an example. Michelangelo carved his David out of marble. The structure of the young man was not inherent in the marble itself.

The Venus de Milo was discovered by a Greek peasant two thousand years after it had originally been sculpted.

This is not to denigrate the work of the algebraists. Their work is incomparably more difficult then digging up a field on an island in the Aegean, but they did not shape the group whose structure they described.

Another sculptor with another block of marble could produce a different sculpture than David, but anybody else would have discovered the same Venus de Milo. And even if they hadn't discovered it, it still would have existed!


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Son Goku
Member
Posts: 1148
From: Ireland
Joined: 07-16-2005
Member Rating: 6.8


(1)
Message 29 of 30 (857035)
07-05-2019 2:55 AM
Reply to: Message 28 by Sarah Bellum
07-04-2019 7:21 PM


You are merely describing how the structure was determined by algebraists, not the reason for the structure itself.

I'm not certain what "the reason for the structure" means.

However all the complexity of the Monster group is related to it being the set of symmetries of the Griess algebra. It does not come from the Group axioms.

It is literally impossible to derive the details of the Monster Group from the group axioms because they are generic conditions for all groups and hence say nothing about the specific structure of any group.

The rest of your post appears to be an analogy for Platonism. Whether one is a Platonist and thinks the Monster Group "already existed" in some sort of eternal sense is not relevant to this point. Even if you are a Platonist you still cannot derive the structure of the Monster group from the group axioms.


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Chiroptera
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Posts: 6692
From: Oklahoma
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Member Rating: 5.8


(1)
Message 30 of 30 (857079)
07-05-2019 10:03 AM
Reply to: Message 29 by Son Goku
07-05-2019 2:55 AM


When both the Monster and obey the group axioms, then clearly the complications of the Monster are due to something else.

Edited by Chiroptera, : Typo.


It says something about the qualities of our current president that the best argument anyone has made in his defense is that he didn’t know what he was talking about. -- Paul Krugman

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