Mathematics and the Universe

[CoxRox]

Would you say that 'patterns' like the Pythagorean triplet are just coincidences?

[Purple Rabbit]

No they are inherent to Euclidean Geometry. EG however is not the way the universe is built according to General Relativity. The Pythagorean triplets aren't valid in the neighboorhood of mass.

[CoxRox]

I only mentioned them as they seem to attract 'importance' to some people , as do prime numbers??

[Purple Rabbit]

So are you saying that mathematics is no different to a language, it is purely human made? We choose symbols to depict values etc. 'Language' needs a giver and a receiver in order for it to have a purpose or be understood. We decide what the values are and the meanings, but surely this is not the case with mathematics. Are you saying that pi for example or Einstein's famous equation isn't really out there waiting to be found, rather we have 'created' these equations? Sorry, I'm struggling to convey what I'm thinking. I hope you see where I'm leading...

By the way, have you read much of Kurt Gödel?

Rereading your post triggered some other thoughts on this. Your question is not only about mathematics but about it's application to the real world. Assuming that there is a real world out there, it does not necessarily follow from it that mathematical concepts are relevant in the real world. Always the application to the real world needs validation. Reality does not follow from mathematics. Euclid's Geometry shows this. Assuming Euclid's axioms as true thus has two meanings that should be distinguished sharply: 1) are Euclid's axioms true in mathematices, and 2) are Euclid's axioms true in the real world. It seems essential to me that these are separate questions.

But what does it mean to apply mathematics to nature? For instance we have no way on deciding if perfect circles in nature exist, because that would mean that physical objects can be arranged with random precision to form a perfect circle. But validating this requires measurement with random precision. This is not possible, not practically (we would need perfect tools), and not theoretically (due to Heisenberg's Uncertainty Principle). So it seems to me that anyway you look at it, by saying that a certain geometry (whatever geometry!) is applicable to reality is already an assumption on the nature of reality. It's a model of reality and again it follows that mathematics has no necessary implications on reality.

[CoxRox]

Purple Rabbit, you said:
''it does not necessarily follow from it that mathematical concepts are relevant in the real world.''

Would you not agree that mathematics is still somewhat of a mystery? Its 'nature' for want of a better word, allows for possible 'worlds' or applications 'foreign' to our experience e.g infinity is used in mathematics. Having noted what you are saying, and they are interesting points, how do they tie in with the assertion that 'mathematical objects and rules enjoy an independant existence: they transcend the physical reality that confronts our senses'. (The Mind of God- Paul Davies)?
An example of this 'independant existence' given by Penrose is the Mandelbrot set. Do you have any thoughts on this? Sorry if I've 'missed' your intended points.

[Purple Rabbit]

Purple Rabbit, you said:
''it does not necessarily follow from it that mathematical concepts are relevant in the real world.''

Would you not agree that mathematics is still somewhat of a mystery?

For me the nature of mathematics in itself is not the biggest mystery. What is it more than a structured sort of fantasy? But that we are able to describe with high precision the real world with it, is intriguing to say the least.

Having noted what you are saying, and they are interesting points, how do they tie in with the assertion that 'mathematical objects and rules enjoy an independant existence: they transcend the physical reality that confronts our senses'. (The Mind of God- Paul Davies)?
An example of this 'independant existence' given by Penrose is the Mandelbrot set. Do you have any thoughts on this? Sorry if I've 'missed' your intended points.

If I can fantasize, into finest detail if you will, about a unicorn, what it eats how it flies through the sky, what it's horn is made of and what magical powers it has, then does that in itself make the uncorn into an existing being? Of course not, it all plays out in the realm of mental manipulation. Mathematics also is completely in the mental realm. All mathematics is based on axioma's, assumptions that are not justified by something other than what I would call their "duh"-ness, all too obvious truths that humans are eager to accept. What makes mathematics into a special kind of fantasy is the intricate web of unrelenting logical rules in which it is weaven. Paul Davies is too much a romantic who equals the beauty of the web that is weaven with divine purpose and mathematical proof with existence. What would he think of unicorns if their horn could be described as mathematical projections of a Mandelbrot fractal?

[CoxRox]

Well, I've just removed the Mandelbrot fractals from up the arses of Davies and Penrose from where you seemed to send them. They weren't too pleased and henceforth repay the complement with the said unicorn horn.

I'm reading Penrose (and recently Davies) and he mentions Gödel. Why do you think the greatest mathematical genius of the twentieth century was a mathematical platonist? Surely not another romantic?

[Purple Rabbit]

Well, I've just removed the Mandelbrot fractals from up the arses of Davies and Penrose from where you seemed to send them. They weren't too pleased and henceforth repay the complement with the said unicorn horn.

I reckon a Mandelbrot fractal perceived as an existing reality up in the arse is a rather spiky experience, while a virtual unicorn horn leaves no traces.

I'm reading Penrose (and recently Davies) and he mentions Gödel. Why do you think the greatest mathematical genius of the twentieth century was a mathematical platonist? Surely not another romantic?

Sometimes a one track mind is needed to achieve breakthrougs in thinking. What precisely is your question regarding Gödel?

In "The Road To Reality" on page 12 to be exact, Roger Penrose is not very convincing imo in defending the stance that mathematical forms have independent existence. He writes the following:

"Yet, there is something important to be gained in regarding mathematical structures as having a reality of their own. For our individual minds are notoriously imprecise, unreliable, and inconsistent in their judgements. The precision, reliability and consistency that are required by our scientific theories demand something beyond any one of our individual (untrustworthy) minds. In mathematics we find a far greater robustness than can be located in any particular mind. Does this not point to something outside ourselveswith a reality that lies beyond what each individual can achieve?"

With that kind of logic any idea that is a product of human collaboration gains immediate independent existence. Peace would be an independent existing entity. So would slavery, war, socialism, torture, etc, etc. Roger and out.

[CoxRox]

Why do you think the greatest mathematical genius of the twentieth century: Gödel, was a mathematical platonist? Is his reasoning also 'suspect' like Penrose's?

As for Penrose's logic you have quoted, I will ponder this and come back to you tomorrow.

[CoxRox]

Purple Rabbit, my dad has replied to you:

'Peace', 'slavery', 'war', 'socialism', 'torture', etc, are abstract concepts, relating specifically to human conditions. (They are in a word, relative.) Mathematical truths are Absolute, and therefore, universal - ie non relative. (Gödel's theorum demonstrates that mathematics cannot be reduced to a positivistic, formal system, because truth is absolute, proof merely relative).


Regards Michael