What is the relationship between mathematical structures and real existence?

[PhanthomJay]

I haven't got a copy of Hawking's new book yet ('The Grand Design'), hoping I'll get it as a Christmas present . But i was quickly thumbing through it (mosly the side flaps) at the bookstore yesterday, and Hawking seems to subscribe to the 'Sum of Histories' theory (?), where the Universe must take on every possible outcome (as in the quantum photon taking on every possible path to reach its 'destination', as in the double split experiment), implying the existence of an infinite number of universes (in reality, or mathematically speaking?). If this is correct, how does this theory differ from M-Theory with its multiple (infinite?) number of 'multiverses' ?

 

[Haelfix]

The sum over histories with regards to cosmology is typically referring to a Wheeler-DeWitt style approach to quantum cosmology.

This is conceptually different than the multiverse due to say eternal Inflation, or to the myriad metastable vacua in string theory or to the multiple worlds interpretation of quantum mechanics.

Obviously, some authors will try to draw parralels or analogies in specific instances but its important to note that they appear in different calculations and contexts entirely.

I suggest this review (see eg pg 10 onwards)

arXiv:gr-qc/0101003

 

[Chalnoth]

I rather like Tegmark's delineation of the types of multiverses (which I'll modify with my own wording, hopefully keeping the essence):

Level 1: Far away. Different regions of the universe separated by a cosmological event horizon. That is, parts of the universe that are far enough away that we can never interact.

Level 2: Different laws. If you go far enough away, different regions are likely to have undergone different spontaneous symmetry breaking events, and thus experience different low-energy physical laws (e.g. different force strengths, different masses). This type of multiverse is what is normally talked about in the string theory/M-theory context.

Level 3: Many worlds. From the many worlds interpretation of quantum mechanics, the full wavefunction of the universe includes many approximately-classical, approximately-noninteracting worlds. What we observe comprises one of these worlds. This is the multiverse that Hawking was talking about.

Level 4: Different math. This is Tegmark's mathiverse, the idea that different universes can stem from entirely different mathematical structures, and that every mathematical structure may well exist as a universe in and of itself.

These four different multiverse levels are basically independent of one another, though there is some overlap between the Level 1 and Level 2 multiverses. I don't really see how there is any significant overlap between the Level 2 and Level 3 multiverses, however.

 

[Chronos]

None of these hypotheses appear to promise any observational evidence. On that basis, I deem them irrelevant.

 

[Chalnoth]

None of these hypotheses appear to promise any observational evidence. On that basis, I deem them irrelevant.

Well, only the Level 4 multiverse is a hypothesis. The others are conclusions based upon other features of physical law, features which are very much testable.

 

[Max™]

Level 4 is based upon the "unreasonable effectiveness of math in science" as Wigner put it, I recall.

 

[Chalnoth]

Level 4 is based upon the "unreasonable effectiveness of math in science" as Wigner put it, I recall.

That's one way to think it might be reasonable. Tegmark's argument, as near as I can tell, can be boiled down to the following:

1. We can be pretty sure that at least one mathematical structure has real existence (ours).
2. It is simpler to propose an entire class of things rather than just one thing.
3. Therefore, it is simpler to propose that all mathematical structures exist than just ours does.

 

[Chronos]

Not everything mathematically possible necessarily exists in nature. This is true because of our finite understanding of mathematics, and imperfect assumptions. I do, however, share your certainty that mathematics ultimately explains all things that occur in nature.

 

[Chalnoth]

Not everything mathematically possible necessarily exists in nature. This is true because of our finite understanding of mathematics, and imperfect assumptions.

I don't see how that follows. If we define a mathematical structure as a self-consistent symbolic system, then it is true that many things that we consider to be mathematical structures aren't in actuality. It is also true that there will be many mathematical structures that we will never discover.

But I don't see how this can in any way be related to the hypothesis that all mathematical structures have real existence. We just have to be careful to state that things we think are mathematical structures may not be so.

P.S. To be a bit pedantic, the reality is that many of our current mathematical structures have inconsistencies, which may mean that they cannot be "true" mathematical structures, but they remain useful because we they are closely related to a "true" mathematical structure that doesn't have the same inconsistencies. This was the case with early calculus, for instance, which was inconsistent. We have since discovered how to make calculus fully-consistent by being more careful about our definitions for differentials.