Will math continue to be unreasonably effective?

[bostonnew]

Hi all,

I was just reading Wigner's 1960 paper 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences'.

For all of you familiar with his reasoning who understand mathematics better than I do:

Do you think mathematics will continue to be as successful as it has been so far? What are some good examples of natural phenonema that we don't yet have an explanation for so far? Do you believe that mathematical explanations do ultimately exists for everything that happens in our cosmos and it's just a question of whether we're smart enough to figure it out? Or is it conceivable that there are things that math is of no use in trying to understand?

Could it be that math becomes just reasonably effective and not unreasonable anymore?

Thanks!

 

[HallsofIvy]

Well, personally, I disagree with Wigner that math is or has been "unreasonably" effective. There exist an infinite number of "mathematical models" that we can use for any given physics (or other discipline) problem. The question is NOT whether "mathematical explanations" exist for physics phenomena but whether we can have physical theories for those phenomena. Once we have that, we can create the mathematics to fit.

 

[Stephen Tashi]

Existing math seems unreasonably ineffective for guiding us to do things like creating computer programs that can analyze images as well as human beings or make forecasts of the weather and the economy. It may be that there are no elegant solutions to such problems.

I think its interesting to sample peoples opinions on what it takes to solve such complicated problems. Phrased as a multiple choice question, we can offer answers like:

1. The math to solve the problem already exists. The fact that computer programs can already do specialized tasks like recognizing faces or short term weather predicition is evidence for this. It's merely a matter of handling a myriad of details. It could be done with a effor on the scale fo the Manhattan Project.

2. The math to solve the problem hasn't been discovered yet. It will be impractical to solve the problem without new mathematical discoveries.

3. The problem is unsolvable.

 

[Jamma]

Human endeavors based upon faith tend to have poor outcomes with little application in the real world and when they do they often make disappointingly innacurate predictions. Human endeavors based on logic tend to be unreasonably effective since the world often seems to work in a somewhat logical way- it is not totally random and many patterns about it can be ascertained.

In that respect, it is not surprising that a subject built totally on the application of mere logic is extremely effective.

 

[Functor97]

Well if this question pertains to physics, then yes i think that Mathematics will always be our "tool". I agree with HallsofIvy, mathematics describes an infinite number of possible universes, the trick is using experimental data to figure out which one.
I do not like how people try and make mathematics out to be something mystical, given to us by the gods, nor do i like the postmodern view that math is just numbers, created by bored mathematicians. We have evolved with a certain construct of logic, and creating axioms from these logics and drawing out the conclusions is the process of mathematics. The same thing could be said about imagination, is imagination unreasonably effective? I could imagine a unified field theory, that in 40 years turns out to match experimental data, does that mean that our imagination is a component of platonic reality? Is Harry Potter walking around twenty seven realities away? It really is a silly thing, this neo platonism.

 

[bostonnew]

Existing math seems unreasonably ineffective for guiding us to do things like creating computer programs that can analyze images as well as human beings or make forecasts of the weather and the economy. It may be that there are no elegant solutions to such problems.

I think its interesting to sample peoples opinions on what it takes to solve such complicated problems. Phrased as a multiple choice question, we can offer answers like:

1. The math to solve the problem already exists. The fact that computer programs can already do specialized tasks like recognizing faces or short term weather predicition is evidence for this. It's merely a matter of handling a myriad of details. It could be done with a effor on the scale fo the Manhattan Project.

2. The math to solve the problem hasn't been discovered yet. It will be impractical to solve the problem without new mathematical discoveries.

3. The problem is unsolvable.

I really like the examples of climatology and economics. And would add psychology and social science more generally as areas where the mathematical models today are quite far from being useful as explanatory devices.

Do you have a sense of how mathematicians and physicists would vote in your multiple choice quiz?

 

[Stephen Tashi]

Do you have a sense of how mathematicians and physicists would vote in your multiple choice quiz?

From casual conversations, most people with a technical specialty (be it computer science, math or whatever) have the attitude that technical aspects of the image recognition problem and weather are already solved and that working out a solution is simply a matter of detail, although the amount of detail needed may make a solution impossible. The same people tend to dismiss economic and social theory as nonsense, so they don't admit there are any well posed problems to solve.

I, myself, take the attitude that the appropriate mathematics for these problems hasn't been discovered yet. My only justification for that outlook is that it's boring to think about the situation in any other light.

 

[Jamma]

I really like the examples of climatology and economics. And would add psychology and social science more generally as areas where the mathematical models today are quite far from being useful as explanatory devices.

Do you have a sense of how mathematicians and physicists would vote in your multiple choice quiz?

Yes, but you surely admit that any eventual future model which is of any use at all (and all current models) would be mathematical in nature? I suppose you could make the following lemma:

Lemma: Any postulate which accurately describes a sufficiently complex system in the real world is, at least somewhat, mathematical in nature.

And, as far as I can see, mathematics is trying hard to keep producing the volume of work it has to guide and keep up with theoretical physicists. Just look at many of the modern theories in physics, and you will find that there was either some mathematical framework in which the theory sat nicely already (the most stunning example is probably quantum theory- eigenvectors, Hilbert spaces, Fourier transforms...), the mathematics directly encouraged an idea in physics, or the physics inspired a new line in mathematics. People are often lead by the aesthetic value in mathematics and try and make them fit into some physical theory (there are plenty of theories where physicists are trying to incorporate E_8 into their theory, and there is hope for the monster group to be used at some point). And this isn't all crazy hogwash, the standard model, for example, predicted plenty of particles that hadn't been found at the time.

In terms of the OP's question- well I think that it is unlikely that there will ever be a time that mathematics ceases to be unreasonably effective, although I can imagine a future where science is so advanced that mathematics has to become exponentially more advanced to have any use, so that in relative terms, it gets to the point of being impossibly slow to advance usefully for real life applications. Of course, this is just speculation. But there will always be mathematics researchers, no question!

 

[jmoorhea]

I think you might be interested to read up on Max Tegmark (of MIT) and his theory that the entire universe is at its most fundamental level just maths. I really like this theory. Simple and beautiful. Probably will never be proven or disproven. If he is wrong and his theory is rubbish then nature is indeed incredible.

Roger Penrose takes a different view, which is that the mathematical universe exists seperately to the physical universe. He believes that only a small section of the mathematical universe is needed to descibe the physical universe completely. Personally I believe that to be pants. Although being someone that's laughs at people who have unfounded religious beliefs, I suppose I am two faced for holding such a view

My basis for rubbishing Penrose is that, if you take the physical universe we live in and describe it perfectly by a finite but still only a fraction B of the entire mathematical universe A then a human would have to essentially know that there is another section C of the mathematical universe A not contained in B. By merely knowing about this part C of the mathematical universe, you have essentially made C part of the physical universe described by B...So C doesn't really exist or can never be found??
Sorry for my very very loose argument.

 

[Jamma]

I don't understand your argument, at all.

Do you suspect, for example, that there are only finitely many laws that govern the universe (even if they are very complex, and many of them)? Then, of course, the amount of mathematics needed to describe the world is negligible compared to the total amount of possible mathematics, which seems can be made arbitrarily complex. Even if you don't expect the universe to be governed by only finitely many laws, it seems to be pushing it to say that there won't be one piece of redundant mathematics which isn't needed in describing it.

Actually, rereading what you've written, I think where I would object is that simply coming up with some mathematics doesn't mean that that mathematics needs to manifest itself in the physical world. You are basically denying the possibility of conceptual mathematics that doesn't have any basis in the physical world, I think.