The Platonic Mathematical World

[SerScot]

Roger Penrose interview about Godel and the nature of conciousness:

simplycharly.com/godel/roger_penrose_godel_interview.html

[Hashi Lebwohl]

What a great article, SerScot--thank you for the find. I especially liked this part:

To take the argument further, I take the view that the quality of consciousness is something that is potentially out there in the physical world, and is not necessarily something unique to human beings. But I regard the Gödel argument as showing that conscious understanding is something that cannot be properly imitated by a computer. So I argue that if consciousness is part of physics—-describable by the “true” laws of physics—-then the true laws of physics must be non-computable. It is known (using Gödel-Turing-type arguments) that there are many areas of mathematics which are actually non-computable, so I am claiming that the true laws of physics (not yet fully known to us) must also be non-computable. But the known laws of physics are (more-or-less) computable, so we must look outside the known laws. I argue, further, that the only plausible loophole in the laws that we know lies in the issue of quantum measurement, and that the “measurement paradox” (basically “Schrödinger’s cat”) points to where we need to make further progress in our understanding of the laws of physics in order to uncover what is actually non-computable in the true laws).

because I had a similar idea about 15 years ago. The way I phrased it was this: 1) the universe is a formal system, 2) the laws of physics are the axioms, therefore 3) there are things in the universe that are true that cannot be proven by science (physics). However, I didn't think about the "loophole" of quantum mechanics--that must be how those true-yet-unprovable things are true.

[SerScot]

Hashi,

The prompted this though (in me): what if conciousness is thee ability to think about things that are otherwise "non-computable" as Turing defined the term.

[Hashi Lebwohl]

hrm...is the thought of a unicorn a real thought?

He addresses a possibility for consciousness in the interview by touching on quantum mechanics in neurons. We don't know all the rules of quantum physics so we are still making discoveries there. He seems to be making the argument that artificial intelligence won't be possible until computers begin to incorporate quantum effects in their processing, so the Singularity is a little farther off than its proponents claim...if Mr. Penrose's ideas are accurate.

[Zarathustra]

Nice interview. As someone who has read and recommended Shadows of the Mind for years on this board, it's nothing new to me. Not bragging, just pointing out that this discussion would f-ing rock if some of you would take me up on the recommendation. You guys seems interested in it. Screw dissections of Donaldson's work .... we should be dissecting some of this stuff!

[Just teasing. I really don't have time for it. My pace own my own work has recently picked up quite a bit and I'm actually starting to see the end of the tunnel.]

[Morning]

Aliantha,

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

Snow Crash is superb.

Penrose's Road to Reality should serve for the advanced starter

No time on my hands now for more, but good topic. I'd say the platonic forms precede, literally in the temporal sense, the shaping of our reality, and being static they can precede time as well.

[Morning]

What a great article, SerScot--thank you for the find. I especially liked this part:

To take the argument further, I take the view that the quality of consciousness is something that is potentially out there in the physical world, and is not necessarily something unique to human beings. But I regard the Gödel argument as showing that conscious understanding is something that cannot be properly imitated by a computer. So I argue that if consciousness is part of physics—-describable by the “true” laws of physics—-then the true laws of physics must be non-computable. It is known (using Gödel-Turing-type arguments) that there are many areas of mathematics which are actually non-computable, so I am claiming that the true laws of physics (not yet fully known to us) must also be non-computable. But the known laws of physics are (more-or-less) computable, so we must look outside the known laws. I argue, further, that the only plausible loophole in the laws that we know lies in the issue of quantum measurement, and that the “measurement paradox” (basically “Schrödinger’s cat”) points to where we need to make further progress in our understanding of the laws of physics in order to uncover what is actually non-computable in the true laws).

because I had a similar idea about 15 years ago. The way I phrased it was this: 1) the universe is a formal system, 2) the laws of physics are the axioms, therefore 3) there are things in the universe that are true that cannot be proven by science (physics). However, I didn't think about the "loophole" of quantum mechanics--that must be how those true-yet-unprovable things are true.

I think conscience is the exact opposite of what Damasio has been defending. Damn, no time now.

[SerScot]

If math has no objective reality why does it translate across all human cultures? Why are there not cultures that don't have the idea of 3 or, more interestingly, for whom the idea of 3 cannot be described within their language because it will not translate.

Yes, Indian's developed "Zero" but "Zero" then easily translated into every human culture. If mathematics is not objective why can all mathematical concepts be understood by all cultures?

[Avatar]

It's not the numbers that have no objective reality, it's the results. You can say math is universal because everybody gets 6 when they add 3 and 3, but that's not what we mean by math. That's arithmetic at best.

Do Javanese islanders share our understanding of the mathematics of M-Theory?

--A

[SerScot]

Avatar,

If they can understand the mathematics, yes. There is not question. The mathematics are objectively true.

[Avatar]

You mean that if somebody works it out the exact way you did, they'll come to the same answer.

I'm not convinced that that is the same thing.

--A

[SerScot]

Avatar,

How could it be different?

[Avatar]

Well, if somebody works it out differently, they may get a different answer.

You're not saying it is universal, you're just saying that anybody who sees/understands/can use math exactly like you do will get the same answer you do. That's obvious. But then you're assuming that everybody/thing that uses math does so in the same way you do. That surely is logically by no means given?

--A

[SerScot]

Avatar,

I'm saying that if the same conditions apply the same answer should result because mathematics is objective.

[Avatar]

But your assumption seems to be that because it is "objective," the same conditions will always apply.

I'm saying that since we cannot know if the same conditions will always apply, we cannot determine it's objectivity.

If indeed objectivity is the right word. My problem with math has never been in claiming 1+1 might equal 2.5 some day. It's in the fact that it is constructed in order to support an answer already decided on. (You can say theorised or hypothesised if you like it better.) Since you're working backward from your conclusion, all that matters is that it works, not whether it was/is real.

This is exemplified in M-Theory, where an additional (11th?) dimension was postulated in order to make the math work.

Is there an 11th dimension? We have no idea. But we're basing the "proof" of the theory on there being one. (Is there a 10th? A 9th? No idea. IIRC though, they were postulated for string theory.)

Do you see what I'm getting at here? Math isn't the real world. It's a system for describing what we perceive the real world to be. As such, it is flawed both by the limitations of our perceptions, and our assumptions (or hopes) about it.

Don't get me wrong. It's amazing stuff. Awe-inspiring sometimes. Essential to our development. But we made it up, which means it can't be "objective" in the sense you seem to mean it. The things it tries to describe are objective. Math is just a language for that description. And language is subjective.

--A

[Fist and Faith]

How about this idea, Av. From Wikipedia:

Einstein included the cosmological constant as a term in his field equations for general relativity because he was dissatisfied that otherwise his equations did not allow, apparently, for a static universe: gravity would cause a universe which was initially at dynamic equilibrium to contract. To counteract this possibility, Einstein added the cosmological constant.

It was assumed that the universe was static. Yet it couldn't be. Gravity would make everything contract. The math said so. In this case, the actual answer was staring him in the face: The universe isn't static. That's what the math was telling him. But that just couldn't be. I guess it was "common knowledge" that it was static. So obvious that even Einstein didn't imagine otherwise. Yet the math said something was wrong, so he came up with his cosmological constant. Wrong solution, but the math was wrong. He knew, because of the math, that the universe was not what he thought it was.

Now, the math says something is wrong with this M stuff. Does the math actually point to an 11th dimension? Or is that just the first idea anyone came up with to make the math come out right? I have not the slightest idea. But if the math says something is wrong with our assumptions, then it's a good idea to reexamine our assumptions. Because, even though math is far beyond 1+1=2, that is what it's all based on. From what I've heard, math is scrutinized to the highest possible degree. Information is shared between mathematicians to an amazing degree, because they all know that they all have to build on each other. They all pick it apart as best they can, trying to find flaws, to make sure it's all correct. Because they'll need it for their own work tomorrow, and they don't want their work to be based on wrong stuff. And we can follow the whole thing backwards, ending back at 1+1=2.

There's a really fun book called Fermat's Enigma, by Simon Singh. Briefly (heh)... Most of us have heard of the Pythagorean Theorem. a^2 + b^2 = c^2. Basic right-triangle stuff, right? And there are an infinite number of these "Pythagorean Triples".

But what if the exponent was something other than 2? Nobody had been able to find an example of such a triple. In 1637, a guy named Fermat wrote a note in the margin of a math book he was studying. The note said that he had a proof that there are no such triples. It only works when the exponent is 2. But the proof was too big to write there in the margin, so he'd get to it later.

Then he died. Lol

In 1995, Andrew Wiles figured it out. He spent many years on it. At one point, he thought he had it. He gave three lectures to a group of his peers, to reveal what he had accomplished, and make sure it was correct. But an error was found. He finally got it right.

The book is really a history of math. It starts in Greece, and works its way up to Fermat. Then it talks about all the effort that Fermat's enigma brought about. Huge advancements in math came about because people were so interested in solving his puzzle. And we follow one person picking up where another left off, on and on, until Wiles solved it. What's fun about the book is not the math you learn. I learned no math from it. But it's written in a way that makes you understand how it all came about. Possibly very geeky to say, but it's an exciting thing! heh

[Vraith]

Possibly very geeky to say, but it's an exciting thing! heh

Possibly. But I admit to being certain to check regularly on news/updates the Millennium Prize Problems

Only one has been solved so far.
I suspect the Fermat would have been included on the list, except it was solved just a few years too soon. It's fascinating stuff...and some of it is important, potentially great/scary for almost everyone in very practical ways.
For instance the P vs NP...if it is solved the way folk EXPECT it will be then security/encryption is safe now for a reasonable amount of time, and perfect, unbreakable encryption is possible, probably soon-ish.
But if it is solved a different way...apparently it makes all current known/theoretical forms of encryption vulnerable, the 'net and all its communications [including the entirety of commercial transactions] wide open. [which it already is...it is just that cracking it requires too much speed and power to do it in a reasonable amount of time].

An MIT guy...I've lost the name, I just have the text quoted in a doc...said something interesting about it in non-math terms/application:

If P=NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in “creative leaps,” no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss; everyone who could recognize a good investment strategy would be Warren Buffett. It’s possible to put the point in Darwinian terms: if this is the sort of universe we inhabited, why wouldn’t we already have evolved to take advantage of it?

[Avatar]

But the solution that Wiles found could not have been Fermat's solution. Because Wiles relied on heavy computing power to find his, and IIRC, Fermat noted he had an "elegant" solution.

--A

[Fist and Faith]

Wiles' solution was most certainly not Fermat's. Wiles' used a few centuries' worth of mathematical ideas that hadn't been thought of until after Fermat died. I'm just saying it's a fun book.

What about the math not working out with Einstein? Math is a good tool for understanding the universe, and in much more complicated ways than simple arithmetic. Because it is all built up from simple arithmetic, in a path that can be clearly seen. So when the math says things cannot be the way we think they are, we should pay attention. As Einstein did. Now, it's true that Einstein postulated the cosmological constant to make the math work. But the problem isn't that he was wrong to try to make the math work. The problem is that his first solution was not the right one. Which often happens, and may be the case in M-Theory, for all I know. But the right answer will be found eventually. As it was with Einstein's problem. Something will suggest the correct answer, as the Hubble redshift did for Einstein. Or the math won't work out in the step that follows plugging the wrong solution into the equations, and a new solution will have to be tried. But the math was arrived at very carefully, over hundreds of years starting at 1+1=2, with the entire mathematical world looking for flaws, so we can be reasonably sure it's doing the job we think it's doing.

[Zarathustra]

But your assumption seems to be that because it is "objective," the same conditions will always apply.

I'm saying that since we cannot know if the same conditions will always apply, we cannot determine it's objectivity.

If indeed objectivity is the right word. My problem with math has never been in claiming 1+1 might equal 2.5 some day. It's in the fact that it is constructed in order to support an answer already decided on. (You can say theorised or hypothesised if you like it better.) Since you're working backward from your conclusion, all that matters is that it works, not whether it was/is real.

This is exemplified in M-Theory, where an additional (11th?) dimension was postulated in order to make the math work.

Is there an 11th dimension? We have no idea. But we're basing the "proof" of the theory on there being one. (Is there a 10th? A 9th? No idea. IIRC though, they were postulated for string theory.)

Do you see what I'm getting at here? Math isn't the real world. It's a system for describing what we perceive the real world to be. As such, it is flawed both by the limitations of our perceptions, and our assumptions (or hopes) about it.

Don't get me wrong. It's amazing stuff. Awe-inspiring sometimes. Essential to our development. But we made it up, which means it can't be "objective" in the sense you seem to mean it. The things it tries to describe are objective. Math is just a language for that description. And language is subjective.

--A

You're mixing math and physics here. Yes, we do "make it up" to some extent when we're seeking a mathematical model for our observations. Yes, we do sometimes works backwards from a hypothesis, to get a mathematical description that fits. Sometimes we settle upon a mathematical model that's just an approximation (like Newtonian mechanics), and then later revise it to a more accurate model.

But all this is separate from the question of the objectivity of math. It does not get its objectivity from the physical world. Speaking of "pure" mathematics, not applied math, there is a sense of it being objective that has nothing to do with physics or "making it up." It's only subjective in the sense that humans can "hold" parts of this structure "in their heads." We can force (seriously, it takes effort) our consciousness along these paths. But we don't create those paths in thinking them. We follow those paths. '1 + 1' doesn't equal '2' by convention or preference. It's a necessary conclusion. There is no other answer.

These are pure formal structures. They have no content, only form. But that form is meaningful. It's not ambiguous. It's clearly, rigorously defined. And not defined like words are defined, but defined by its own logical necessity. There is no conceivable universe in which math could be different (though there might be endless paths to the same results). It's entirely independent of physics, entirely independent of matters of fact or states of affairs.

These structures are real. They exist "above" physical reality. They are ideal objects. And not only can we hold them in our consciousness, or shape our thoughts according to their structure, but they also shape physical reality in some mysterious way.