Are mathematical systems absolute, real things?

[bascule]

I'd really, really like to meet Max Tegmark and hope one day I have the opportunity to.

 

Max Tegmark proposes we live in a mathematical universe where all mathematical systems are absolute, physical things, at least in some context:

 

http://arxiv.org/abs/gr-qc/9704009

 

I perhaps have a fuzzy interpretation of what Tegmark is suggesting, but two of the systems which stand out in my mind as having an absolute physical presence are:

 

• cellular automata
  
• fractals
  

 

The distinguishing characteristic of both of these systems is that they provide for "unlimited novelty" despite being described by relatively simplistic rules.

 

The underlying concept is that from simple rules we can derive extraordinary complexity.

 

What do you think? Does a "theory of everything" encompass all mathematical descriptions of potential universe? Do similarly simple rules lie at the heart of our "universe"?

[Edtharan]

I don't fully agree with the proposition that: "All structures that exist mathematically also exist physically."

 

However, it is noteworthy that any mathematics your create must occur in a physical system because it is a physical being that thinks of it (even if it was a being made of pure energy, it still has a physical reality in the universe in the form of force carrying particles/fields). So in some way Max Tegmark is sort of right because if you can think of the mathematics, then it does "exist physically".

 

Why I don't agree with him is that in mathematics there exist Infinities, and Infinitesimals. and all evidence we can see indicates that the universe is finite. With Infinitesimals, we have even more evidence that they can't exist because of the Plank limits (size and time).

 

These finite limits to sizes (big and small) seem to make any such structures (infinities and infinitesimals) impossible to "exist physically".

[ajb]

Tegmark's postulate is a very interesting one. It is something I have thought about, somewhat casually.

 

It seems an indisputable fact that mathematics and physics go hand in hand. Both can stimulate the other.

 

One should try to make a distinction in physical theories between mathematical aspects and physical aspects. For example, what is really a feature of differential geometry and what is really a feature of general relativity. As general relativity is so tied in with differential geometry can such a separation be made?

 

Similar questions can be asked of all physical theories and it can be come a blur as to what is mathematics and what is physics.

 

Back to the opening points. It seems true, that just about all areas of mathematics have been applied to theoretical physics and applied mathematics. Is this a coincidence? Well, probably not.

 

First we have areas of mathematics that were developed with applications in physics in mind or at least are inspired by physics. Here we can include calculus, differential geometry and operator algebras to name a few.

 

What is interesting are the areas of mathematics originally seeming devoid of physics are just what are needed in theoretical physics. Why should this be?

[bascule]

However, it is noteworthy that any mathematics your create must occur in a physical system because it is a physical being that thinks of it (even if it was a being made of pure energy, it still has a physical reality in the universe in the form of force carrying particles/fields). So in some way Max Tegmark is sort of right because if you can think of the mathematics, then it does "exist physically".

 

Does that mean that when people aren't looking at the Mandelbrot Set or the Rule 110 cellular automaton they go away?

 

These guys seem pretty convinced fractals are absolute, real things:

 

[Edtharan]

These guys seem pretty convinced fractals are absolute, real things:

I never said that these things weren't real. I said I disagreed with the Tegmark's postulate that: "All structures that exist mathematically also exist physically".

 

The key word here is "physically". These are regular structures that can be found in mathematics, and we can arrange matter and energy to represent them. But, until we arrange the matter and energy correctly, then the structures don't exist physically.

 

Information, if it is to exist physically, must have some arrangement of matter and/or energy to represent it. However, the Mandelbrot set is infinitely complex: that is it has an infinite amount of information. But, if this information is to be "physically" existent, then it need an infinite amount of matter and energy to represent it.

 

This is a problem. First of all, if enough matter and energy is concentrated into a small enough space, it forms a black hole and no information can come out from that. Second, if we have an infinite amount of matter.energy to represent an infinite amount of information, then no amount of space could contain such a physical construct without collapsing into a black hole. Even an infinite amount of space would succumb to this collapse (when you realise that even in an infinite space, there has to be an infinite density of matter/energy to handle the amount of information).

 

Therefore, if the Mandelbrot set were to be represented physically, then the universe would collapse into a black hole. But we don't have an infinite density of Matter/Energy do we, so this also rule out the possibility of the Mandelbrot set existing physically, as there is not enough "stuff" to represent it completely.

 

Does that mean that when people aren't looking at the Mandelbrot Set or the Rule 110 cellular automaton they go away?

In a way I do agree with this because it is only when we arrange matter and energy to represent part of these mathematical constructs do they have any "physical" existence. Even if we are just thinking about them, the matter and energy (electrical activity) in our brains are arranged to represent these structures.

 

When we stop representing these structures, that is when we stop "looking" at them, then there is no physical structure representing them and they "go away". But this does not make them any less "real", in that they are a regularity in the "structure" of mathematics.

 

So I agree that they exist, but I disagree that the "exist physically".