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Why are the physical sciences described perfectly by mathematics?

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There is a famous paper by Wigner: "The unreasonable efficacy of mathematics in the physical sciences" which gives an idea of how deep this question was considered by a great fellow in the recent past. –  Ron Maimon Jul 7 '12 at 2:29
    
Great Question... –  Sachin Shekhar Jul 7 '12 at 6:51
    
it can not be by pure luck.. There are theories that are correct mathematically and demonstrate un-observed physical phenomena. –  Jose Javier Garcia Jul 7 '12 at 14:41
    
What else would you use to describe them? –  Nathaniel Jul 7 '12 at 16:27
    
@Nathaniel Before mathematics we used similar observations compared to eachother. I would imagine on some basic level all you need to describe the natural world is observation and communication –  Argus Jul 7 '12 at 18:47

6 Answers 6

up vote 10 down vote accepted

Why are the physical sciences described perfectly by mathematics?

They are not.

Here's a simple example: QED is arguably one of the most precisely predictive theories in physics, yet its underlying mathematics are unavoidably based on approximation methods, so you can never really calculate an ultimate answer using it. Feynman himself did not believe that QED was indefinitely accurate in any case -- he was if anything surprised at just how far the method ended up working.

Or here's another even simpler point: there are no points in physics! That is, the mathematical concept of a precise point, or of a precise line, has no real analog in the physical world. Creating finer and finer points in physics requires more and more energy, again creating a situation of approximation, this time in the other direction: physics can only approximate one of the most fundamental concepts of mathematics, and rather poorly at that. String theory postulates line-like strings of... something, I've never quite understood what... that might come closer, but I would assume that even in that case it is unlikely that strings are claimed to be "perfect" lines.

Now if on the other hand your question is more along the lines of why are some forms of mathematics so incredibly good at predicting physics when applied at the right level of resolution and context?, then it grows far more interesting... and harder to answer.

My personal suspicion is that we are paralleling our own universe with the ways our brains work. Mathematics is after all just an application of our built-in spacial reasoning to clusters of symbols that, like the features on rocks from a riverbed, we can rotate and turn or disassemble and recombine in more complex patterns. Vertebrate brains seem to be fantastically well tuned to making energy-efficient use of patterns of invariance and constancy in our universe, and mathematics is in that sense a gorgeously refined and precise application of just that ability. So I don't think it's too surprising that our brains are capable of creating constructs that are impressively "in tune" with a universe that is both regular enough and complex enough to contain creatures like us, creatures who can inspect and predict that regularity and then pose interesting questions about it.

(And I'm betting that was likely not the kind of answer you were anticipating... :)

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I don't think this answer is very good. The perturbative expansion of QCD is similar to the perturbative expansion of QED, but QCD can be simulated on a computer to arbitrary accuracy. In certain string theories, the results of any process can be calculated in principle on a computer to any accuracy, for example, a finite number of particles scattering on AdS space, or a finite matrix mechanics system. Even without strings, we observe mathematical regularities, like, say, the ideal gas law, or the repeatable hardening of metals under quenching, that are quantitative and match models. –  Ron Maimon Jul 7 '12 at 2:48
    
Exactly what I expected. Well at least two people have these ideas. –  Argus Jul 7 '12 at 5:59
    
Argus, cool! I feel heretical whenever I carry over some of my "day job" into physics discussions, but there are some quite remarkable things going on these days in the literature on biological intelligence. I personally like to challenge folks to look at the astonishingly high energy efficiency of vertebrate brains; for contrast, look at how many watts the DARPA Grand Challenge vehicles required to navigate even simple roads. Understanding neural biology necessarily tells us something about how physics and mathematics work, since an equation with no brain to interpret it means nothing at all. –  Terry Bollinger Jul 7 '12 at 14:01
    
The push nowadays we'll over the last 8 years seems to be integrated systems. Combining long lasting components encasing an intricate network of pathways to manage the entire system. Ie Self-determination. –  Argus Jul 7 '12 at 19:43
    
-1. Physics is a subset of mathematics. –  Dimensio1n0 Jun 27 '13 at 3:30

This is a very deep question, and one that we cannot (and, quite possibly, will never be able to) answer.

Nevertheless, I think there are really two parts to this question. The first is actually quite easy to answer - the second, not so much.

First: why is it that the language of mathematics is so useful in describing the structure and orderly behaviour that we see around us? To answer this, I cite Wikipedia: "Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the abstract study of subjects encompassing quantity, structure, space, change, and more." In other words, we define mathematics as the study of structures, patterns, well-defined processes, etc. Humans developed math as a language to convey precise ideas about relationships, properties, changes in quantities over time, etc., so it should come as no surprise that math is well-suited to talk about orderly behaviour.

The real question here is: why does the universe exhibit structure and orderly behaviour? Why is it that the amazing complexity we see around us is often explainable by very simple rules? This is an intriguing question, and I don't think we'll ever be able to answer it properly.

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Regarding your last paragraph. Consider the following: to exist, to be, is to be something specific, to have an identity, a nature, and to act according to that nature. A thing is what it is and not something else. This comes directly from a rule of thought; the law of identity: A is A. A disorderly universe? Full of ... what? Certainly not existents because, to exist is to be something specific. Could something exist that isn't what it is? –  Alfred Centauri Jul 7 '12 at 12:03

IMHO, I think that it should not be very surprising that Mathematics perfectly describes tha natural sciences. Firstly, mathematics was developed initially as a tool for natural sciences. ( I am talking about modern mathematics, starting with say calculus). So, i think of maths as not another subject in its own right, but just a tool to study natural sciences, which was later on made more rigorous and more powerful by the 'mathematicians'. In other words, what I feel is that we have voluntaily chosen maths as the language for physical sciences, and shouldn't be surprised. Its as if saying 'Wow, I can communicate (or get my thoughts across) using English (or any another language), how mysterious'. The second point, which I am entirely sure of is that the world looks orderly only from a distance. We all like to think that why the world obeys laws, but it is the human tendency to see order in everything. If we probe deeper maybe, there many be some underlying disorder ( like QM proved the underlying uncertainty in everything). So maybe the universe is just apparently mathematical. These are just conjectures.

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mathematics was developed initially as a tool for natural sciences fractals were originally designed as a rebellion to find something that doesn't occur in nature. It's only later they found that their rebellion is in vain when it turns out that there is fractals everywhere in nature. –  Lie Ryan Jul 7 '12 at 9:52
    
thats just one example. –  user7757 Jul 7 '12 at 11:10
    
Mathematics is a bit older than the time calculus was developed. –  MBN Jul 7 '12 at 11:17
    
Your answer sounds a lot like this. While I don't really like Arnolds pov, I think yours is the most reasonable answer so far. –  NikolajK Jul 7 '12 at 12:32
    
I think of math as useing numbers for anything other than counting. New math is no more significant in its respective time frame. As a whole the concept of the numbers themselves and the process of minipulation. How does this describe sciences? –  Argus Aug 10 '12 at 20:42

Mathematics cannot perfectly describe physical phenomena. However, mathematics, as a language, provides us an easier way of describing and predicting what will most likely happen (given a situation). These predictions are based on well known laws, principles and theorems.

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Physics
For an arbitrary example I take Coulomb's law. The interaction between two charged objects is proportional to the electrostatic charge of each object, and inversely proportional to the square of the distance between them.

As far as we know the Coulomb law has been valid as long as the Universe has existed, throughout all of the Universe.

By contrast, imagine a situation where all things related to electrostatic charge are entirely random. Two objects may randomly attract or repulse, and/or the distance depencency of the interaction changes randomly. Then there is no law whatshowever.

In our actual universe, there are to our knowledge no exceptions to Coulombs law. Of course, that's why it has the status of 'law' in the first place. It gets to that status because we're never confronted with an exception to it.

This is the case for our entire corpus of physics laws: we find many properties of the universe that are perfectly consistent throughout time and space.

My best guess is that this is necessary for a Universe to exist at all. Atoms can exist because of the existence of the Coulomb attraction between the positively charged nucleus and the negatively charged electrons.

My guess: to have a Universe at all its properties must be consistent throughout time and space. An existing Universe is a self-consistent universe.


Mathematics
In doing mathematics, in creating mathematics, there is just a single criterium: self-consistence. That is the only rule.
Example: the three self-consistent geometries: euclidean geometry, spherical geometry, hyperbolic geometry. It doesn't matter that those three are incompatible with each other, the point is that each of them is self-consistent. Any thought system that meets the criterium of being self-consistent is a form of mathematics.



The Universe and mathematics have that one property in common: perfect self-consistency. Hence the efficacy of mathematics in the physical sciences

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The axioms of math are observations of the real world. Math is a particular study of the real world. Logic, the other tool of math, is a codification of how the brain deals with reality. If it did not work we would not study it. The question is a bit like asking why the hammer is so effective at driving nails. The answer is that Math works because that is what we developed it for and if it did not work we would have kept looking until we found something that did. The fact that it is so simple at its core is a reflection of the fact that the universe is fundamentally simple at its core. Another restatement of the question might be why is the reflection in the mirror so accurate. The question has an anthropic principle feeling about it.

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