Math is nothing more than one of many languages we use to describe our existence as we see it.

Nothing more than a language? Do you really believe that math is on the same footing as English? I think you're being disingenuous for dramatic effect.

Math indeed has a language: we agree to certain conventions of symbols and syntax, which must be learned in order to communicate mathematical ideas. But the language is arbitrary and has nothing to do with math itself. We could (and have) changed the language and still retain all the meaning. I point at Fluffy and say "dog"; my German friend says "Hund", but we're both talking about the same thing, independent of the language chosen. Math is fundamentally about that "thing in itself", irrespective of the language used.

It's no wonder that the math we have developed to describe the universe, well, describes it.

The "thing in itself" that is the subject of math is pure logical thought, with all the irrelevant details abstracted away. As such, math comes equipped with an enormous amount of machinery for describing, predicting, and verifying the relationships between anything and everything. In other words, math can not only describe this universe, but any other universe. That's pretty wonderful to me. Now, you may say "so what?" -- if math is so wonderfully generic that it can describe any and all universes, then of course it can describe this particular universe. Duh, right? Indeed, but that's missing the point that there even exists such a universal (multiversal?) system of logic, which we somehow have access to. Out of all human inventions, none come even remotely close to mathematics; I mean, try to visualize where we'd be without it! Given its unimaginably vast richness (of which we've so far explored only an infinitesimal fraction), it seems more than plausible that we didn't actually invent math, but discovered it.

It's axiomatic.

You say that like it's a bad thing, but that's how formal systems work -- use axioms and rules of inference to derive theorems. Remember, you're free to choose any axioms you like; math is not about the axioms, it's about the interesting things that happen after you've chosen a set of axioms.

 

In other words, math can not only describe this universe, but any other universe.

Prove it.

 

Prove it.

There are quite literally infinite examples. One useful example of a completely different (though possible) universe is anti-de Sitter space:
https://en.wikipedia.org/wiki/Anti-de_Sitter_space

 

There are quite literally infinite examples. One useful example of a completely different (though possible) universe is anti-de Sitter space:
https://en.wikipedia.org/wiki/Anti-de_Sitter_space

Premise: one can describe any number of theoretical universes.
Inference: one can describe "any other universe".

This is a non sequitur. The inference does not follow the premise.

 

Premise: one can describe any number of theoretical universes.
Inference: one can describe "any other universe".

This is a non sequitur. The inference does not follow the premise.

What are you actually arguing here? That there exist universes which cannot be described mathematically? Prove it.

 

What are you actually arguing here? That there exist universes which cannot be described mathematically? Prove it.

Negatives cannot be proven. You must prove your assertion that math can describe "any other universe". The burden is on you.

I'm going to edit this:

You made an unsubstantiated assertion. I did not. Therefore, as I said above, the burden is on you.

 

Negatives cannot be proven. You must prove your assertion that math can describe "any other universe". The burden is on you.

I'm going to edit this:

You made an unsubstantiated assertion. I did not. Therefore, as I said above, the burden is on you.

Lol ok. Suppose A is a universe with cosmological constant Λ. If we change the value of Λ, we get an entirely different universe A'.

Let κ = Λ + ε parametrize the cosmological constant, for all ε in ℝ. Then the set defined by each κ has the cardinality of the continuum. This represents an uncountably infinite number of possible universes.

In like manner, parametrize each constant and initial condition that characterizes a universe. This represents a power set of an uncountably infinite number of possible universes. However, all these universes obey our laws of physics, so we're not even close to accounting for all possible universes.

Now, let the set of functions S = { φ1(x1, x2, x3, ...), φ2(x1, x2, x3, ...), ... } represent a particular model of physics, e.g., the differential equations of GR combined with the Maxwell equations, or the Lagrangians of the Standard Model, etc. This set represents the "laws of physics", the equations of motion that describe the dynamics of objects in some particular universe (presumably ours). Let T be the set of real-valued functions, all of them, both continuous and discontinuous. Clearly, S ⊆ T. In fact, the cardinality of T is so unimaginably much larger than the cardinality of the continuum that it belongs to a different category of transfinite numbers. Take the powerset of T, replacing each φ in S with a subset of functions in T. That takes care of all of the universes with physics that can be described by real-valued functions.

Finally, for any universes that somehow cannot be described by real-valued functions, replace ℝ by every other mathematical structure, one at a time, and repeat the above procedure. I contend that the complement of the resulting set is empty. Why should we be able to describe an astounding number of universes but not some non-empty subset of them? The only reasonable retort is that math can't even describe one universe, but experience doesn't bear this out.

 

Lol ok. Suppose A is a universe with cosmological constant Λ. If we change the value of Λ, we get an entirely different universe A'.

Let κ = Λ + ε parametrize the cosmological constant, for all ε in ℝ. Then the set defined by each κ has the cardinality of the continuum. This represents an uncountably infinite number of possible universes.

In like manner, parametrize each constant and initial condition that characterizes a universe. This represents a power set of an uncountably infinite number of possible universes. However, all these universes obey our laws of physics, so we're not even close to accounting for all possible universes.

Now, let the set of functions S = { φ1(x1, x2, x3, ...), φ2(x1, x2, x3, ...), ... } represent a particular model of physics, e.g., the differential equations of GR combined with the Maxwell equations, or the Lagrangians of the Standard Model, etc. This set represents the "laws of physics", the equations of motion that describe the dynamics of objects in some particular universe (presumably ours). Let T be the set of real-valued functions, all of them, both continuous and discontinuous. Clearly, S ⊆ T. In fact, the cardinality of T is so unimaginably much larger than the cardinality of the continuum that it belongs to a different category of transfinite numbers. Take the powerset of T, replacing each φ in S with a subset of functions in T. That takes care of all of the universes with physics that can be described by real-valued functions.

Finally, for any universes that somehow cannot be described by real-valued functions, replace ℝ by every other mathematical structure, one at a time, and repeat the above procedure. I contend that the complement of the resulting set is empty. Why should we be able to describe an astounding number of universes but not some non-empty subset of them? The only reasonable retort is that math can't even describe one universe, but experience doesn't bear this out.

Sorry, I had to pop out for a bit...daughter's band concert (she's a violinist). She did great!

First, excellent dissertation! Are you a mathematician? That would explain your zealous defense of math as something more than just a mere language.

I will admit, you've put much more work into your thoughts than I would have into mine. The nice thing is that I understand them. I have a couple of choices here:

1) Graciously accept defeat
2) Nod thoughtfully and change the subject
3) Think about it. Are there bases you haven't covered?

I'll choose 3 for now. This "argument" has become more interesting than I imagined it would.

 

Sorry, I had to pop out for a bit...daughter's band concert (she's a violinist). She did great!

Nice! I'm a musician and have been trying (perhaps too gently) to get my daughter to pick up an instrument for years. So far, she'd rather play a violin on a video game than pick up the real thing (and I have many instrument choices available).

First, excellent dissertation! Are you a mathematician?

Thanks! It was fun to think about. Not a mathematician, just a former math-phobe who saw the light. Learning the basics as an engineering undergrad gave me the tools to study further. Still have a long way to go!

I'll choose 3 for now. This "argument" has become more interesting than I imagined it would.

Looking forward to your thoughts.

 

Nice! I'm a musician and have been trying (perhaps too gently) to get my daughter to pick up an instrument for years. So far, she'd rather play a violin on a video game than pick up the real thing (and I have many instrument choices available).

Actually, my daughter is a much better pianist than violinist. She wrote a piano composition at 8 y.o. that won 2nd place in the U.S. for her age group. She's 11 now and starting her rebellious phase. She doesn't see the value (yet -- hopefully) of continuing to perfect her musical skills and so practices grudgingly. I told her she can quit playing piano when she is 18 and moves out of the house.

 

BTW, it looks like I just nodded thoughtfully and changed the subject.

I'll give you this: I think you are correct that any math we have developed that describes our universe to the extent that it describes it likely describes any other possible universe to the same extent.

As you are well aware, our math is not yet a complete description of everything. Gravity is a glaring example -- as well as a model that adequately predicts the mass of an electron or the values of e0 or µ0 (and therefore c).

Are these constants the same everywhere? Who knows. But if they are not, would the same (hypothetical) tools used to predict them here work as well elsewhere?

 

I'll give you this: I think you are correct that any math we have developed that describes our universe to the extent that it describes it likely describes any other possible universe to the same extent.

Given the massive amount of unknowns, that seems eminently reasonable. We can call it the weak version of the all-universes conjecture.

As you are well aware, our math is not yet a complete description of everything. Gravity is a glaring example -- as well as a model that adequately predicts the mass of an electron or the values of e0 or µ0 (and therefore c).

I 100% agree that, given our current level of knowledge and understanding, we cannot describe this particular universe mathematically. No doubt about that. What I'm saying is that this (and every other) universe does have a mathematical description that perfectly characterizes it, we just don't know what it looks like yet. Maybe we'll never know -- maybe the structure of the universe has 100% information entropy, requiring as many bits to describe as there are degrees of freedom. Clearly if the universe is incompressible (in the information sense), then we have no hope of being able to describe it from within.

I suppose the strong all-universe conjecture is that mathematics has greater extent than any particular universe and so could, in theory, describe all universes. That leads to some weird thoughts, such as: if math is part of the universe, then how can the part be greater than the whole? But such things do occur. A simple example: the decimal digits of an irrational number, such as sqrt(2), go on forever. But the digits are not unpredictable -- we can easily write a small program that will produce as many digits as we want. Let's say that the smallest such program requires 1,000 bits; then the information content of sqrt(2) is 1,000 bits. But if we focus on a particular part of the sequence of digits -- say, the 10 digits after the first hundred-- we need to increase the size of our program to specify the desired offset. So, in a very real sense, the part can contain more information than the whole.

This kind of thing happens naturally when the thing being investigated has some kind of pattern or symmetry. The repeating or invariant nature of the thing allows us to compress the information required to describe it. A random trillion-digit integer requires a full trillion digits to describe, but we can easily describe every integer in a few bits (we even have a single character to do it: ℤ). It turns out that our universe seems to have a lot of pattern and symmetry, which strongly suggests that the information required to describe it is compressible. And if this is so, then I believe there is a mathematical description of it. We just have to find it.

Are these constants the same everywhere? Who knows. But if they are not, would the same (hypothetical) tools used to predict them here work as well elsewhere?

Man, this really is the fly in the soup. If the laws of physics aren't even universal -- if they change depending on where in the universe you are -- then we really are just spinning our wheels. But I'm hopeful and optimistic that this isn't the case. What we've seen of the universe so far hasn't given us any indication that it's batshit crazy out there. It really does seem that the more we learn in the lab here on Earth, the better we understand the stuff happening in the cosmos at large (the CMB radiation is a great example).

 

Man, this really is the fly in the soup. If the laws of physics aren't even universal -- if they change depending on where in the universe you are -- then we really are just spinning our wheels. But I'm hopeful and optimistic that this isn't the case. What we've seen of the universe so far hasn't given us any indication that it's batshit crazy out there. It really does seem that the more we learn in the lab here on Earth, the better we understand the stuff happening in the cosmos at large

This is something I've argued here a few times. And was told not to even think about it being true. To be so "Earthcentric" to believe that physics is the same everywhere in the Universe to me isn't even being scientific. Until we go we can't know.

 

To be so "Earthcentric" to believe that physics is the same everywhere in the Universe to me isn't even being scientific

That's an educated assumption based on the behavior of the celestial bodies that we can observe from earth... but you're right, a trip to the far places of the universe is needed to confirm it ... maybe there are entire worlds out there that are made from anti-mater and that's something that cannot be confirmed by mere observation?

 

That's an educated assumption based on the behavior of the celestial bodies that we can observe from earth

But that is still only in our solar system. We only a few years ago found actual evidence of other planets around other stars. Even with the little exploration on the moon or Mars know that their physics aren't exactly like Earth's, close but not exactly.