# Why is mathematics so ridiculously effective?

1. Apr 14, 2006

### bobbytkc

Why? Why Why? Is there an answer?

ON MATHEMATICAL IMPLICATIONS IN PHYSICS

It is a topic that has been well discussed and thought over by many well learned individuals. Scientific endeavors are not always necessarily pursued using the mathematical method that we have come to assume in this day and age. In fact, the mathematical method appears to me to have matured only after the introduction of Newton’s mechanical vision of the universe as well as the introduction of calculus to the available tools. In Aristotle’s day, misguided if well meaning scholars rarely make use of mathematical tools at all, nor in fact do they make use of any experimental basis for their conclusions, firm in their belief of the infallibility of their logic and intuition.

These days, the importance of mathematics in the study of science is well documented and taken for granted. Regardless of the fact that some eminent scientists (such as the late Richard Feynman, who had once claimed that physics would have progressed even without mathematics) still remain doubtful of the mathematical methods and their overarching meaning in the overall picture, they remain far in the minority. Most people involved in the study of the physical sciences now freely admit that without mathematical methods, they see no way of gaining any further understanding of the mechanics of the universe. Indeed, there are many examples of physical theories appearing only after certain mathematical methods have matured or have been discovered. One notable example is the geometry of non-Euclidean space, studied in great detail by eminent mathematician Herbert Minkowski, which precedes the appearance of the General Theory of Relativity by his student, Albert Einstein.

Of course, the great usefulness of mathematics in physics does not give any clear indication of why mathematics has any implications in the real world at all. In what has been called the ‘ridiculous effectiveness of mathematics’, mathematicians and physicist alike cannot supply convincing explanations for the inextricable link between the two fields. Why is it in fact that the Universe follows mathematical laws at all? Perhaps this has something to do with why we have developed mathematical reasoning in the first place, and mathematical reasoning exists only because the physical universe behaves mathematically. But of course, this answer is not the least bit satisfying since it employs circular reasoning. In this way, it is similar to saying the Universe has the properties it possesses only because we exist to observe it. That is in no way close to the rigorous explanation that physicists seek and expect.

Of course, given the axiomatic nature of physics (as well as mathematics, and indeed, any precise science in general) could mean that the mathematical nature of the Universe could be a fundamental axiom, one which forms the basis for all of physics, and need not, or could not, be explained in any manner. Unfortunately, there is no way for us to prove that the mathematical nature of physics is a fundamental axiom, nor is there a way for us to show that there is no way to explain this mathematical nature, similar to Gödel’s proof that in any axiomatic mathematical system undecidable propositions exists (see ON FORMALLY UNDECIDABLE PROPOSITIONS OF PRINCIPIA MATHEMATICA AND RELATED SYSTEMS by Kurt Gödel). Therefore it appears for now that we remain stuck in a conundrum, looking for a proof when no proof can be found, even if the proposition is correct.

This in some ways parallels the search for the proof of Fermat’s Last Theorem, long suspect to be one of the undecidable propositions in Gödel’s Theorem, resisting nearly 300 years of mathematical research, including contributions by giants such as Euler. However, this theorem was finally cracked recently by the number theorist Andrew Wiles, after a new mathematical technique was developed.

Perhaps in this case, what we need is just another mathematical revolution.

2. Apr 15, 2006

### Jimmy Snyder

I don't believe that it does. What laws did you have in mind?

3. Apr 15, 2006

### HallsofIvy

Staff Emeritus
The universe doesn't follow mathematical laws. People make up laws based on what they see the universe doing.

Mathematics is effective because because a mathematical structure is a "template"- to model a physical situation mathematically, we choose a mathematical structure (out of the many that have been defined) that seems to fit, then tailor the definitions to the situation.

4. Apr 15, 2006

### arildno

Since, presumably, ANY law that the Universe follows allows a mathematical representation of it (in some sense of the term "mathematical"),
one can, indeed (rather facetiously), say that the Universe follows some mathematical laws.

5. Apr 15, 2006

### Jimmy Snyder

Such as?

This text added to satisfy a curious criterion.

6. Apr 15, 2006

### bobbytkc

Well, in specific, i refer to Paul Dirac's discovery of anti-matter.

Anti matter wasn't even expected or hinted at prior to its actual detection, but through mathematical tinkering (the additional negative sign resulting from a square root) Paul Dirac hypothesized on the existence of anti-matter through only a mathematical existence (the negative sign was previously only ignored as non-representative of the physical world).

7. Apr 15, 2006

### bobbytkc

Also, I think the most omnipresent effect of the mathmatical nature of physics is that no matter the nature of any physical theory, we cannot test every single possible scenario in that theory, and we assume that the theory works for other cases only by mathematical extension (for example, we can only test newton's laws of motion using only a definite range of masses and speeds, but we assume that the theory works as well outside our experimental range through mathematical extension, and thus far, this assumption has not failed us)

8. Apr 15, 2006

### bobbytkc

and indeed my question can simply be rephrased to why does the universe have laws that allow mathematical representations in the first place? And why is it that if we apply mathematical methods on these physical laws (with the mathematical method having no true physical meaning) the results are almost invariably represented in the real world?

9. Apr 15, 2006

### arildno

It might just be that your question is ill-posed in the first place, in the sense that it is meaningless to assume that the universe cannot be tackled with applied logic (i.e, maths).

Anyway, that's my view.

10. Apr 15, 2006

### bobbytkc

That is somewhat of a circular reasoning there. My question is why it is possible that the universe has a mathematical representation at all, and by your reasoning, you say that it is meaningless to assume a universe that does not apply to mathematics, since the way we can understand universes is mathematically.

But just as well, though the universe may be non-mathematical, that does not mean there can be no ordered system of rules. Informal logic (e.g. logic of languages) employ systems of rules and yet are non-mathematical logic. It is not necessary to have a universe construct observers to understand it like ourselves, or indeed be intuitively understandable from our frame of reference. What i am suggesting is whether there is any possible explanation at all as to why our universe employs mathematical logic, which in this case created us that understands the universe mathematically.

11. Apr 16, 2006

### arildno

Not really. You are the one assuming that there might be some intrinsic limitation inherent in applying logic in some form to the Universe and hence, that the Universe might be ordered in such a way that it is impossible to model it according to a set of logical rules of inference and a quantification method (i.e, maths).
Yet, you haven't given any reason why such an assertion cannot simply be dismissed as nonsense.

12. Apr 16, 2006

### bobbytkc

I see I see, so you are saying that I am assuming without evidence that the universe is limited to mathematical logic in some way.

However, I am not trying to imply that the universe only works mathematically. Obviously, the existence of the conception of the other forms of logic means that these other logic take some form within our universe.

However, what i have in mind (as given in the articles above's title) is not so much that the universe abhors other forms of logic, but that Physics in particular is representable in mathematics so effectively. Perhaps other forms of logic or emplyed in other capacities in other areas, but certainly, the physics world view is extremely mathematical. The most solid evidence of this is empirical, that all our tests show that it is true, and indeed, it remains still an assumption, but one that is not explainable yet, but also this assumption has not failed us.

This is somewhat similar to the heydays of the Law of Conservation of Energy, only recently derived from supersymmetry, but previously an assumption, because it holds true for all our empirical test. So it is extremely probable that there are no deviation from this rule.

So my question remains: why is this so? or can it even be explainable in the first place? Is there an effect that 'crystallizes' the Physical world into a mathematically consistent system?

In the first part, you said i assumed that the universe is limited to employing only certain kinds of logic

and yet in the 2nd part, you seem to be implying that I am suggesting that other forms of logic (other than the mathematical) are possible and viable.

That seemed to me a little contradictory, on the one hand you say i assumed a narrow view of the kinds of applicable logic, but because of that I throw the door open to all other kinds of non-mathematical logic?

I guess, to make it clear again, I am trying to say that the universe (from the point of view of physics) is very mathematical, and it is supported by evidence (every experiment, test is a validation of it), but there appears no reasons why other kinds of logic and rules are also not equally viable systems for physics. So why is it NOT possible for physics to employ other systems of logic?

13. Apr 17, 2006

### matt grime

I think your premise is flawed. The mathematics used to describe parts of the physics of the universe are not consistent. No successful synthesis of quantum effects and relativity exists yet there are mathematical models of both phenomenon. We know there is effectively a minimal workable distance, planck length, yet no workable model of space-time as a discrete system exists. The exact nature of the universe actually is mathematically inexplicable right now, but then, arguably, so is the exact nature of the universe, at least to such an extent that allows us to model it and explain it mathematically, which is the standard you have chosen to adopt. Is string theory the true explanation, even?

Last edited: Apr 17, 2006
14. Apr 17, 2006

### bobbytkc

Ah yes, but string theory is a more complex mathematical system that encompasses both GR and Quantum theory AND provides explanation of the discrete behavior of particles. In this case, it is not the mathematics that limits the unification, but rather, choosing among the more encompassing forms of mathematics available.

15. Apr 18, 2006

### matt grime

The versions of string theory that are mathematically understood do not even begin to pass any criteria for being a model of the physical world. There are many competing versions. None of them works. Some physicists can't see it ever working, but that is their problem. And it doesn't address the discrete nature of space-time (I didn't mention anything about the discrete nature of particles) as far as I am aware; I only know about the mathematics not the physics.

16. Apr 18, 2006

### Nancarrow

My thought-about-just-now view is that mathematics is the systematic study of patterns and structure (thank you Wikipedia!). So the 'unreasonable effectiveness of mathematics in the natural sciences' is due to the presence of structure in nature. IOW the question can be rephrased as 'why does the universe follow laws at all rather than being a chaotic random mess'? Cue Einstein quote here.

17. Apr 18, 2006

### Gokul43201

Staff Emeritus
The "unreasonable effectiveness" of mathematics becomes suudenly all too reasonable if you think of mathematics as a method of identifying things that are equal to (or the same as) each other, and applying the logic that two things that are both equal to a third thing must be equal to each other.

For instance, if we have a*b = c [1], then we find b = c/a [2]. Given a relationship [1], we have identified, using math, another pair of quantities that are equal to each other [2]. So, if we also know that b = d*e, we can deduce that d*e = c/a.

Physics involves the process of :

(i)Starting off with some empirically observed "truths" which state in essense that two well-defined things are equal to each other.

For instance, one might consider as an empirical truth that the flux of the electric field emerging from a closed surface is equal to the charge enclosed within it. ~~ Gauss' Law

(ii) Applying the tricks of math to determine quantities that are equal to some of the things in our empirical relation.

For instance, one such trick tells us that by adding up the area of very thin rings, one can find the surface area of a sphere, and this quantity is equal to $4 \pi r^2$.

Using the mathematical equality of the surface area of the sphere and $4 \pi r^2$ in our empirical statement and identifying the equality of b and c/a as in the first example, one finds that the field due to a point charge obeys and inverse square law behavior. ~~ Coulomb's Law

Should we be amazed that the math we used led us from one physical truth to another ?

18. Apr 18, 2006

### -Job-

One way to approach this question would be by proposing that numerical values have equivalent physical quantities and that mathematical operations have equivalent physical interactions. This would generate a "closed set" perhaps provable by mathematical induction.
NOTE: This is a purposedly ambiguous approach, i'm more interested in conveying the idea than actually proving it.
Such a proof by induction might state that if indeed numbers are associated with physical quantities and operations are associated with physical interactions between quantities, producing, in turn, a different set (distribution) of quantities, then physical quantities, before and after physical interactions, have equivalent numerical values. If, on top of this, basic mathematical operations are equivalent to basic physical interactions then:
. any set of physical quantities can be numerically described
. any set of basic physical interactions can be described with numbers and basic mathematical operations
Furthermore, if complex physical interactions can be broken down into basic physical interactions then:
. any non-basic (complex) physical interactions can be described with numbers and basic mathematical operations.

In this manner there would be a "one-to-one relationship", or equivalence, between mathematics and the physical world. Hence, wherever one leads, the other one follows.

This is mostly common sense but it's interesting to study some repercussions of such an identity. For example what are we doing when we attempt to explain gravity? We are basically generating a physical model from which we can extrapolate the relevant mathematics. It's especially useful regarding gravity because whatever produces this force isn't clearly visible (i.e might require dealing with a 4th dimension) so we are more confortable with a model such as that of a sphere bending some surface. This is a model we can analyze, observe, and interact with, so it's more natural and simplifies the process of finding the mathematical model that is representative of it.
If physical model A guarantees similar behavior to gravity, then, by our 1-to-1 correspondence above between math and the physical world, model A and the physical process responsible for gravity are describable by similar mathematics. Clearly, it is irrelevant whether a given model is "believable" or not, if it behaves the same, then it is mathematically similar and will help us understand the physical process we wanted to understand in the first place.

There is one additional point i want to make and that pertains to the describability of the basic physical interactions of the universe. If basic physical interactions have associated basic mathematical operations, then an attempt at explaining a basic physical behavior/quantity (such as movement or matter/energy) is equivalent to trying to explain numbers and basic mathematical operations using mathematics. How would this be possible? In trying to explain basic math operations with mathematics you'd find yourself using the same basic operations in your explanation as the ones you are trying to explain, a circular process. What is curious is that this happens in the real world as well. For example try to create a physical model which explains why things move without using movement. I can't do it, and my failure might indicate that movement is a basic physical process, not explainable through the use of our current mathematics.

Last edited: Apr 18, 2006
19. Apr 19, 2006

### Thrice

As Nancarrow said, any law allows mathematical representation. I would add that even "a chaotic random mess" allows mathematical representation.

20. May 3, 2006

### nameta9

The reason why mathematics is so effective is because MATTER KNOWS ITSELF. Why wouldn't matter be able to decode itself ? After all we are simply matter talking to itself, decoding itself and ultimately manipulating itself. So matter can discover its own laws and regularities, so why be surprised ? We are made up of matter, we can discover ourselves by following certain pathways, so matter can ultimately decode and discover how it works all by itself.

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