The Platonic Mathematical World

[SerScot]

I finished Anathem by Neal Stephenson recently. It prompted an interest in Platonic thought and modern interpretations of Plato's theory of the Forms. In any event I'm attempting to read Roger Penrose's book The Emperor's New Mind. In it there will be heavy discussions of the idea of Plato's Mathematical World and the idea that when we have a flash of insight (I love the term "Upsight" from Anathem) we are actually perceving, however briefly, the Platonic Mathematical World and seeing the "true" forms and that is from where we derive our Upsight.

Now, I'm Christian so I don't really have a problem with the idea of a reality that I can't prove empirically being the source of this sort of "upsight". It seems like briefly touching the mind of God to me. When I've had this sort of "upsight" that's always what I've thought. This is my first experience in reading scientists who aren't arguing we are touching God's mind but that we are touching some "other reality" where information is more real.

What does everyone think about the idea of the Platonic Mathematical World and the idea that the Forms may be real in some way we don't understand and can't completely empirically explain?

[Vraith]

I leave room...just a little...in my worldview for the possibility of extra-natural things, but what I really believe is that most of the "real" ones...whether religious/mystical, or "good god! Benzene is a RING!" ...are currently unexplainable [which is why we have a hard time proving some of them, and separating the actual ones from those of a really good liar], but they won't always be so. Most of us are intuitive...even visionary...to some extent, and I think it's a natural function of consciousness.
However, while visions of the angels, or pure mathematical ideas can carry meaning, or at least information, that does not mean they are necessarily real in an object/objective sense. And even if they are in some sense/place, their absolute/unchanging nature does not make them in any way superior or more real or more true, it makes them inferior, less real, less true.
What I mean [one part of what I mean] is this: It's almost universal now to believe the world/universe follows mathematical laws...even most of those who believe in God[s] generally believe something like God created those laws, and certain aspects of the universe follow them...they just believe there are additional non-scientific/non-mathematical laws as well.
I think that is purely silly. Mathematical "laws" are merely descriptive, not determinative, if that's a word. They have no function/force in the workings of things. They're only more pragmatically useful than a song of love...ummm...in most cases not an example of "The Truth."
In any stable universe, where beings that think could arise, mathematic laws could [almost certainly would] be invented. But notice, they would NOT have to be the same mathematics, or the same laws, and they WOULD have to be invented. The Universe doesn't follow mathematics, mathematics conforms to the universe, if it is to be useful.
I also speculate...I'll have to think on this a while, it's an unexamined intuition...that if various mathematical assumptions/constructs/objects were able to really, physically exist, OR be absolutely true, one other thing must be true as well: a different universe for every kind of mathematics. And one thing probably true: in those universes beings that resemble us in any way could not exist.

[aliantha]

I think you're spot-on, Vraith, at least in regard to your theory that mathematics describes our universe, as opposed to defining it. I think that's true of all of the sciences, as well. I mean, as soon as you run up against chaos theory and fractals and whatnot, you realize that the current mathematical theories can only carry us so far in understanding how the universe was "built."

I can also see what you're saying about alternate mathematical theories and other types of beings in other universes -- but yeah, it would take some time to wrap my brain around that fully.

[Avatar]

IIRC, Plato thought the Forms preceded the reality, right? I can posit their existence, but I think that they're hypothetical ideals constructed in the mind through an aggregation of perceptions of the actual things. In other words, the reality precedes the Forms rather than vice versa.

--A

[SerScot]

Vraith,

Hypothetically, if a spaceship appeared in orbit around the Earth that could not have been built in our Universe (for reasons established by they materials used in the ship's construction) that had a visual proof of the Pythagorean Theorem on the exterior of the Ship would your opinion about the extra universality of mathematical laws continue to hold true.

In Anathem the Characters speculate that there are other Universes where Mathematics don't "work" but that under "Anthropomorphic principles" those Universes don't usually last very long and don't support intelligent life.

Further, they speculate that the PMW is necessary to the development of intelligent life that quantum connection to the PMW is necessary for the development of life and intelligence.

I will concede that what is proposed here is metaphysical speculation. But, as I read more, I'm hoping to discover more information that gives credence to the idea of the PMW being necessary to our own Universe. Kurt Godel, the man who proved we couldn't prove the validity of a mathematical system without assuming the system is valid and therefore couldn't prove the validity of a given mathematical system, was a believer in the PMW. I've got a book on Godel on it's way.



I'm enjoying The Emperor's New Mind so far. Penrose spends the first chapter discussing the ideas behind "Strong-AI" and starting his criticism of those ideas. It does all tie back to Platonism.

[SerScot]

Avatar,

The position you are taking, if I understand it correctly, is that we provide semantic content to the idea of the forms, that it cannot have an existence independent of the existence of our minds. However, consider this, is a Prime number prime without humans or other intelligent beings around to perceive that number as prime? Is its nature as a Prime Number dependent upon our perception of it as prime? Does it have semantic content or is it pure syntax that we provide meaning to by imposing our idea of prime upon it?

[Vraith]

Vraith,

Hypothetically, if a spaceship appeared in orbit around the Earth that could not have been built in our Universe (for reasons established by they materials used in the ship's construction) that had a visual proof of the Pythagorean Theorem on the exterior of the Ship would your opinion about the extra universality of mathematical laws continue to hold true.

In Anathem the Characters speculate that there are other Universes where Mathematics don't "work" but that under "Anthropomorphic principles" those Universes don't usually last very long and don't support intelligent life.

Further, they speculate that the PMW is necessary to the development of intelligent life that quantum connection to the PMW is necessary for the development of life and intelligence.

I will concede that what is proposed here is metaphysical speculation. But, as I read more, I'm hoping to discover more information that gives credence to the idea of the PMW being necessary to our own Universe. Kurt Godel, the man who proved we couldn't prove the validity of a mathematical system without assuming the system is valid and therefore couldn't prove the validity of a given mathematical system, was a believer in the PMW. I've got a book on Godel on it's way.



I'm enjoying The Emperor's New Mind so far. Penrose spends the first chapter discussing the ideas behind "Strong-AI" and starting his criticism of those ideas. It does all tie back to Platonism.

I would think that said ship came from a Universe much like ours. I'm saying something I think a little different from what you think I'm saying. [I think.. ] The existence of PMW, or any ideal "forms" is not necessary for any universe, or for the developement of life/intelligence. However, in any universe that physically resembles ours, contains minds resembling ours, the concepts of PMW will be functional and valuable in describing that universe, and pragmatically useful in various ways [maps, buildings, describing gravity, etc, etc]. However #2, any universe where, for instance, an absolute Euclidean Geometry triangle could physically exist would be unable to sustain anything remotely like our world or life.
I do not think these ideals/forms are metaphysical at all. They are sub-physical. Artifacts of, not necessary for, real things. This universe isn't a mere imperfect reflection of Forms, the forms are reflections of things stripped of particularity and actual existence. Pure mathematics, by definition, excludes real properties of actual things. Our universe isn't what it is because it follows mathematical laws, mathematical laws can describe it because of what it is.
Even if [and I suspect this is possibly so] mathematics can describe all possible universes, it still isn't necessary or even likely that the PMW is "real." Rough analogy: the fact that I can speak, describe, explain a mountain [mathematically, geologically, religiously, poetically, whatever] in no way creates or is necessary for the mountain to exist or be possible.

[Fist and Faith]

Alas, I don't know anything about Plato's Mathematical World. The only thing I know about anything Plato is that a cave is somehow important.

[aliantha]

I'll admit it, every time I look in this thread, my eyes glaze over. Would it be worth reading Anathem to figure out what you guys are talking about? (And btw, SerScot, I got a copy of Dune for my e-reader a couple of weeks ago. It's next on my reading list. )

[Vraith]

I'll admit it, every time I look in this thread, my eyes glaze over. Would it be worth reading Anathem to figure out what you guys are talking about? (And btw, SerScot, I got a copy of Dune for my e-reader a couple of weeks ago. It's next on my reading list. )

It does a decent job of explaining basic concepts of PMW, [and some language/knowledge stuff] though you might need to google occasionally. It does an absolutely crappy job...to the point of dismissiveness...of opposition ideas, even though some characters are oppositional.

It's as if SRD had written the Chronicles and, instead of them being a story that works according to what he thinks fantasy literature is/can/should be, he included in the text some characters to deliver a couple dozen chapters of lecture/essay on what fantasy really is/can/should be. [and they were completely right about everything].

[Fist and Faith]

I'll admit it, every time I look in this thread, my eyes glaze over. Would it be worth reading Anathem to figure out what you guys are talking about? (And btw, SerScot, I got a copy of Dune for my e-reader a couple of weeks ago. It's next on my reading list. )

ARE YOU SAYING YOU'VE NEVER READ DUNE?!?!???

[aliantha]

I'll admit it, every time I look in this thread, my eyes glaze over. Would it be worth reading Anathem to figure out what you guys are talking about? (And btw, SerScot, I got a copy of Dune for my e-reader a couple of weeks ago. It's next on my reading list. )

ARE YOU SAYING YOU'VE NEVER READ DUNE?!?!???

No no, I read it years ago. I just don't remember much about it, other than that I came away with the impression that Herbert was a misogynist. There's a thread over in the Tank where I mentioned that, and SerScot and a couple of other folks asked me what specifically gave me that impression, and I had to admit I couldn't remember. So I promised to give it another go and report back.

Vraith, you're not making Anathem sound very enticing.

[SerScot]

Aliantha,

I loved Anathem. I'm reading other books mentioned in Neal Stephenson's acknowledgements I enjoyed it so much.

[Vraith]

Vraith, you're not making Anathem sound very enticing.

Heh...not my job, besides I knew SS would stand up for it...you don't have to believe me.
Ol' Neal...I thought Snowcrash freakin amazing, Cryptonomicon smart but hit and miss, Anathem some of Cryptonomicon's smarts, few of its hits and all of its misses. [heard good things about his series that distantly connects to crypto...but haven't read any.]

[Avatar]

...is a Prime number prime without humans or other intelligent beings around to perceive that number as prime? Is its nature as a Prime Number dependent upon our perception of it as prime?

What makes a number prime? It's conformity with human derived "rules" for what constitutes a prime number. Hell, the concept of numbers themselves are human derived. "Numbers" per se don't have a physical or material existence. All they are is a way of describing some material aspect of the world.

A prime number is meaningless...it's only prime because that is what we decided to call numbers that fit certain criteria. It can't exist without those rules that define it. In other words, we made them up.

--A

[SerScot]

Avatar,

So, could another species that has had no contact with Humans determine "prime" numbers are not "prime"? Is the mathematical concept of "prime numbers" dependent upon human intellect and culture?

[Vraith]

...is a Prime number prime without humans or other intelligent beings around to perceive that number as prime? Is its nature as a Prime Number dependent upon our perception of it as prime?

What makes a number prime? It's conformity with human derived "rules" for what constitutes a prime number. Hell, the concept of numbers themselves are human derived. "Numbers" per se don't have a physical or material existence. All they are is a way of describing some material aspect of the world.

A prime number is meaningless...it's only prime because that is what we decided to call numbers that fit certain criteria. It can't exist without those rules that define it. In other words, we made them up.

--A

In a lot of ways, this is exactly what the argument is about: part of the Platonic/neo-Platonic view is that such do exist even without anyone to think about them...that there exists such a thing as "two-ness" independent of other objects or viewers.
I don't think it's so.

[SerScot]

Vraith,

So is it possible for another species or an unusual culture to find a way to make 1, 2, 3 and all other prime numbers evenly divisiable by numbers other than themselves and one? If not then aren't primes unique in a way independent of culture? If Primes exist completely independently of human culture how can we say they are a facite of human culture or a creation of the human mind?

Additionally, doesn't the fact that mathmatical concepts are universal, that 1+1=2 is not limited to the Western Culture or that the null set is univerally accepted, suggest a concreteness to mathmatics that means math is something discovered not created? After all there haven't been any alternate methods of counting created by different cultures. They may use different symbols but the meaning of those symbols are all precisely the same. 5=V=*****. There is no varition among these symbols based upon the cultural differences.

[Vraith]

I'll try it a different way...though I'm still thinking about what I'm about to say, I'm not sure it works or not if followed through/examined.

Forms/Mathematics, according to Plato actually are things, are real. They are the content of a superior/truer/absolute universe. I say that is junk:
Mathematics is purely a context, with no actual content, no actual substance, only definitions and operations and consistency. The universality is irrelevant to the reality.

[SerScot]

Vraith,

I'll try it a different way...though I'm still thinking about what I'm about to say, I'm not sure it works or not if followed through/examined.

Forms/Mathematics, according to Plato actually are things, are real. They are the content of a superior/truer/absolute universe. I say that is junk:
Mathematics is purely a context, with no actual content, no actual substance, only definitions and operations and consistency. The universality is irrelevant to the reality.

I absolutely disagree.

The very universality of mathematics suggests they are absolutely real as they have an existence outside of the context of the human mind and Culture. If, as you suggest, mathematics is "pure context, with no atual content, no actual substance" it wouldn't be universal. It would have some cultural variation. It wouldn't be a means of communcation with something that has no context with us as humans.

To claim (as you appear to), "there's no there, there", is to suggest mathematics should change given context. Context is irrelevant in all irreduceable Mathematical concepts. As such they have a reality independent of the human mind.