Is there a known limit to relationship between physics and mathematics?

Question

I am much interested in discussions such as Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". It's quite amazing that mathematics so well applies to our universe, and this raises many interesting questions which have been discussed here already, but I am interested in the converse as well: "There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world." (Lobachevsky)

So, then, my question is thus: what is known of the limits to the relationship between mathematics and physics. It seems that no matter the subject, it finds its way into a description of our reality. Is this so because we, as denizens of this universe, cannot imagine anything due to our mental structure being at some level composed of the very physical laws we wish to comprehend? Might alternative constructions of mathematics be more applicable in other universes, and we merely neglect their study, not seeing their worth? Or is there so deep, arcane tie-in between mathematics and nature, such that nature must effect each mathematically formalizable notion in some form or another. The last point is to say, how accurate was the second quote I posted? Is there a 'use' so-to-speak, a physical or social or cognitive implementation of any given arbitrary mathematical system? And what might I read on other's thoughts of why mathematics seems to hold such power on our minds and our world?

There's the idea that mathematics merely exists because it's a useful tool, though there is a certain grand structure in many areas of mathematics which start with merely few axioms, and the complexity just falls out of it naturally. Does our reality play catch-up with mathematics, or vice versa?

Answer 1

I had asked a similar question and the insight I got is this:

Our hardwired perception is root of our logic. And anything built upon such logic will be natural for us.

The grandness/beauty/natural-feel that falls out of our mathematical endeavors is nothing but a revelation, that its that same perception.

Colors for example:

we can see colors , so every thing visible to us in universe will have a color.

Did one simple ability of perception just forced a universal rule that all visible objects must have color ? Did we just painted the whole universe colorful, with a single mind trick ?

How wonderful is that, to us.

Now, You can have an abstract painting, but not an abstract color. Also, no perfect color, no universal color.

Is color a property of everything visible in universe ?

should 'Red+Blue+Green=White' be considered a Mathematical theorem ?

Mathematics faces similar questions here, when one assumes that it can define any phenomenon in universe.

Answer 2

Alex, excellent question, here.

I think Wigner's paper gives little defense to the Platonist case. It seems to me that he's merely saying "applying mathematics to physics is useful to us," but this doesn't quite lead us to believe that "mathematical entities are real," which is what the Platonists would want to say. This is why I think that: consider the possibility of stripping the theories of science of their mathematics. If we were able to do this, would we have any reason to think of numbers as still being real? I'm not sure that we would.

Note that I'm not saying our theory would be better, or anything. It'd probably be much worse! You could definitely argue that keeping the numbers in the theory has value, that it's good; perhaps it's easier for humans to solve, or better yet, computers. Or maybe the theories are more compact with numbers. But if the theories don't actually depend on the mathematics, who cares, right?

It's useful for me to think that I'm on just on a bus when I'm 35,000 feet in an airplane, because that helps me get through the flight without freaking out. But of course I have no reason to think that this useful thought has any bearing on what's real.

This idea that I've introduced is called the indispensability argument, or the Quine-Putnam thesis. It basically says that we absolutely rely on mathematical entities to explain scientific entities, so we have just as much reason to believe in those mathematical entities as our scientific entities. A good thing about this theory that you don't even need to think that scientific entities are real to accept it. It just says that we should have the same confidence in mathematical entities as we have in scientific ones, whatever that amount of confidence might be (even if it's none).

There are a vast number of criticisms about the original Quine-Putnam thesis, some of which you can read in that article or on the Wikipedia. One criticism was that some philosophers doubted that any scientific theory depended on math at all!

However, just a few years ago, a paper was published where a brilliant philosopher, Alan Baker, found such a theory. You can read it in his paper here.

So where do we go from here? If Baker is right, then we do rely on a mathematical property for a scientific explanation. Should we only place confidence in those mathematical entities (prime numbers?) Or all mathematical entities? Is that one example enough?

Anyway, I just thought these sorts of thoughts and questions were relevant, so I decided to share them. I hope you find it interesting!

Answer 3

Your question presupposes in some ways that it is possible to perceive (in the broadest sense possible, not just through our senses, but also through our senses + tools) a universe that is not compliant (in some way) with our logic or mathematics. Not all philosophers would grant you this premise. In fact, most have a very difficult time in showing that there is an 'external' world that we are modeling. For a very clear example of this see Hume's radical empirical skepticism which reduced Locke and Berkley's empiricism to a completely subjective (and rather solipsistic) philosophy. If you want this taken even further, consider Kripke's presentation of Wittgenstein.

As an example of a noted philosopher (unfortunately one that was not very knowledgeable of the math of his day) who would not grant you this premise is Kant. For Kant, the world in itself is not organized in any particular spatial or temporal manner. Geometry and time-sense are synthetic a priori to our mind, and thus all perceptions and sensations we could ever have are inherently forced to comply to this structure of our mind. In that setting, the applicability of math is no surprise since we can only perceive things that comply or can be expressed in our a priori math-sense.

Of course, this aspect of Kant is easy to argue with (because he had pretty big misconceptions about the math of his day, and would definitely be behind in our day), but it has been refined by contemporary thinkers among the same line. The basic premise is that if it doesn't comply with logic/something-math-like then we cannot possible perceive it. For these philosophers, the effectiveness of mathematics would not be surprising.

For Plato in particular, the effectiveness of mathematics would be absolutely no surprise. Math for Plato is in the world of perfect forms, and the sensible world is just shadows and imperfect copies of these forms. Thus, the fact that we remember the mathematical forms necessary to describe idealizations of the world is to be expected.

Answer 4

Our own existence is evidence that the Universe is predictable. Evolution could not occur in an unpredictable universe. In fact all life depends on predictability - from bacteria, to mammals, to humans, to nervous systems and brains. The main goal a brain has is to predict what actions it can perform to achieve it's goals. During human evolution, human brains have created language as an enhanced way to model the universe and to achieve better predictions about the universe. Science and mathematics are simply extended versions of language that we find to be useful when it comes to predicting the universe. So as we have tried to develop laws of physics we have found that we can get better predictions by using mathematics to state the laws of physics. However, the real meaning of physical laws is simply that they enhance our ability to predict the universe - I don't believe they have any reality beyond allowing enhanced predictability.

As to whether all sub-fields of mathematics will find some applicability in the laws of physics is something that I think cannot be known. What is known is that many fields of mathematics that at one time were thought to have no applicability to physics have been found to be applicable and to be useful in developing physical laws. A recent example is that advanced mathematical topology seemed to only be a mathematical field with no applicability to physics, but from what I hear some very deep topological theorems are allowing some breakthroughs in the area of string theories and quantum theories in general.

Answer 5

Is it equally unreasonable to assume language should apply to physics?

After all, didn't we evolve language exactly to apply to our perceived word? From the point of view of symbolic logic, Mathematics is just a very deep, systematic probing of language. And from a point of view of many philosophers of science, physics is just a very deep, systematic probing of our perceptions.

So, of course we would expect extremely refined language to fit extremely refined perception as well (and perhaps only as well, note quantum indeterminacy) as ordinary language fits ordinary perception.

Answer 6

There is an extremely popular notion that mathematical platonism (every mathematical object has a platonic existence) is well-defined. But that's false. Due to the incompleteness theorem, we know that any reasonable foundation for mathematics is either inconsistent (useless) or incomplete (does not prove every true sentence about ℕ), and so cannot pin down a unique mathematical universe. Thus the popular notion of mathematical platonism is ill-defined, because "mathematical object" is ill-defined without reference to a specific foundational system. One can amend it to "there is some real-world interpretation of ZFC", but nobody can justify such a statement (so far). So what then? So far all the empirical evidence that mathematics is relevant to the real world only justifies that there seems to be a real-world interpretation (maybe even merely approximately) of some very tiny fragment of ZFC, probably between ACA and ATR0 (see Reverse Mathematics), but almost surely no more than HOA (higher-order arithmetic). The success of applied mathematics cannot justify all of modern mathematics at all.

People who claim that every possible axiomatic system for mathematics have a platonic model, are even more wrong. Again by the incompleteness theorem, if PA is consistent then PA+¬Con(PA) is also consistent, but it proves itself inconsistent! So obviously some FOL theories are simply utterly false (with respect to the standard model of PA on which FOL syntax and deduction itself is based)! Worse still, there are people who claim that every mathematical structure exists. This claim is not even wrong because it is ill-defined in itself. There is no way anyone can define what mathematical structure means without reference to some formal system or some class of formal systems, and the mere definition of "formal system" needs to be based on a rudimentary amount of assumptions about finite strings.

The bottom line is that you need to define "mathematical entity" or "mathematically formalizable notion" otherwise your question is ill-defined. For instance, the real numbers as constructed within a formal system do not have concrete existence in reality any more than a checkmate strategy from a given winning chess position. The question you should ask is whether there is an embedding of the mathematical structure of the reals into reality. Unfortunately, that is ill-defined, because different formal systems prove different things about real numbers!

For example, ACA0 suffices for all applied real analysis till today, because one can encode any real number as a subset of ℕ, and can encode any real functions with countably many discontinuities as a subset of ℕ, and ACA0 is strong enough to facilitate manipulation of such encodings. But ACA0 has a model M comprising ℕ and the arithmetical subsets of ℕ (equivalently the subsets of ℕ whose membership can be computed using some finite Turing jump. Note that in this model M there are only countably many subsets of ℕ!!

Think about it. For applied real analysis we do not seem to need anything beyond ACA0, but there is a model M of ACA0 that has countably many reals (per the encoding). Where did all the other 'reals' go? Observe that ACA0 does prove (an encoding of) Cantor's theorem, namely "there is no surjection from ℕ to ℝ". There is no contradiction here; in the model M this just means there is no finite-jump-computable surjection from ℕ onto arithmetical sets, which is correct.

Now Z set theory proves that ℝ is uncountable, but that is just a statement encoded in the language of Z. By Lowenheim-Skolem you know that if Z is consistent then Z has a countable model, so that should let you realize that ℝ is not an absolute concept. Unless you come up with a privileged or real-world model of Z, you simply do not have any objective notion of "real numbers", and hence asking whether "the reals" exist would not be a well-defined question.

Regarding the relationship between human mathematics and reality, I think it is quite obviously due to the fact that the only ways we currently know how to interpret mathematical theorems in reality apply only to arithmetical sentences, and it is intrinsically difficult to come up with an arithmetical sentence that cannot be decided (i.e. proven or disproven) by PA (without using the incompleteness theorem). And if one knows about PA and the incompleteness theorem, but does not believe in existence of uncountable sets, then it is still intrinsically difficult to come up with an arithmetical sentence that cannot be decided by ATR0 (unless one employs transfinite induction) or Π[1,1]-CA0 (even if one employs transfinite induction, since one lacks uncountable well-orderings).

ATR0 is roughly the limit of predicativity. Some (but not the majority of) logicians doubt Z2 (full 2nd-order arithmetic) due to its impredicativity. While Z2 proves lots of interesting mathematical theorems, if one day it is found to be inconsistent then all those theorems would simply be useless for the real world, contrary to Lobachevsky. It may not happen for Z2, but what about ZFC or ZFC plus some sufficiently large large cardinal? Some people have studied Berkeley cardinals within ZF, which were known to be inconsistent with AC before they were invented. But it is possible that one day it is found inconsistent with just ZF, which would render this "mathematical notion" useless.