Why is mathematics so ridiculously effective?

[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.

 

[Jimmy Snyder]

Why is it in fact that the Universe follows mathematical laws at all?

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

 

[HallsofIvy]

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.

 

[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.

 

[Jimmy Snyder]

Since, presumably, ANY law that the Universe follows ...

Such as?

This text added to satisfy a curious criterion.

 

[bobbytkc]

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

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).

 

[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)

 

[bobbytkc]

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.

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?

 

[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.

 

[bobbytkc]

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.

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.

 

[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.

 

[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?

You are the one assuming that there might be some intrinsic limitation inherent in applying logic in some form to the Universe

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

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).

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?

 

[matt grime]

Is there an effect that 'crystallizes' the Physical world into a mathematically consistent system?

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?

 

[bobbytkc]

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?

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.

 

[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.

 

[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.

 

[Gokul43201]

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 ?

 

[-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.

 

[Thrice]

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?

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

 

[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.

 

[nameta9]

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?

Aside from the idea that MATTER KNOWS ITSELF, HENCE IT IS ACTUALLY SIMPLY TALKING TO ITSELF SO CAN UNDERSTAND AND DECODE ITSELF, there are two other important points to keep in mind:

1) Natural language reduces to being purely mathematical when all is said and done. The decomposing of a Natural language into its mathematical-logical components may be complex, but in the end we THINK AND TALK IN PURE MATHEMATICS-LOGIC. I don't think there is any difference between logic and mathematics as a computer demonstrates by its own operation and how it can simulate anything.

2) Our sense organs are simply furnishing a set of numbers being transferred to our brains. We can view our sense organs as furnishing signals as analog signals, with our brain producing something like an ANALOG TO DIGITAL conversion of all the information coming to us and then consider the brain a processing device, that processes all this MATHEMATICALLY-LOGICALLY.

So mathematics-logic is intrinsic in all possible ways to decoding reality through our brain and with our language. There is no way to NOT USE MATHEMATICS-LOGIC. Even the most irrational, is ultimately based on mathematical laws of physics. MATTER IS MATHEMATICS, MATHEMATICS IS MATTER.

Physics is just a bunch of equations ued to predict how certain experimental configurations will behave. It is simply a tool.
Maybe the ultimate elementary particle is the mind which cannot be decomposed into anything else, it is a monolithic slab of consciousness that cannot be reduced to anything, the scientific method of reductionism cannot be applied to the mind.

 

[rewebster]

I agree with posts 3 , 18 , and 19

Math is just representational. Math tried (probably) to represent the pre-Copernican view of the solar system. Did math make that view right just because 'math' described it?


Math is an idea, a concept. Without going into "the whole world is an illusion" thing, 1=1 or x=x has no meaning---except that it 'could' represent 'something'. And then, those two objects in the 'real' world aren't 'equal' anyway (apples)---they are similar.

Math generalizes the world so that we, as humans, can communicate about its similarities and differences.


(Did you know math spelled backwards is htam ?)

 

[matt grime]

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.

This is called a pathetic fallacy (and no, that is not rude). Matter does not 'know' anything because matter is not sentient: a photon does not know it is a photon, it does not know anything, it does not have knowledge, it has not learned that it needs to obey wave-particle duality by taking a good look in the mirror and deciding to pull its socks up and get on with figuring out its purpose in the world.

 

[selfAdjoint]

This is called a pathetic fallacy (and no, that is not rude). Matter does not 'know' anything because matter is not sentient: a photon does not know it is a photon, it does not know anything, it does not have knowledge, it has not learned that it needs to obey wave-particle duality by taking a good look in the mirror and deciding to pull its socks up and get on with figuring out its purpose in the world.

Matt, of course I quite agree with you, but I couldn't help noticing that your post was long on assertion and short on evidence. I can't think of any evidence to falsify the self-aware photon either.

 

[nameta9]

This is called a pathetic fallacy (and no, that is not rude). Matter does not 'know' anything because matter is not sentient: a photon does not know it is a photon, it does not know anything, it does not have knowledge, it has not learned that it needs to obey wave-particle duality by taking a good look in the mirror and deciding to pull its socks up and get on with figuring out its purpose in the world.

So then we are not made up of matter ? what are we made up of, pure spirit ? are we looking at the universe from OUTSIDE the universe ? We are matter looking at itself and talking to itself. Maybe a large complex assembly of matter organized in a certain way, but WE ARE MATTER THAT IS LOOKING AT ITSELF AND TALKING TO ITSELF AND DECODING AND MANIPULATIONG ITSELF. WE CANNOT GET OUTSIDE OF MATTER, WE ARE MATTER.

 

[matt grime]

Matt, of course I quite agree with you, but I couldn't help noticing that your post was long on assertion and short on evidence. I can't think of any evidence to falsify the self-aware photon either.

Ockham's razor. If someone wants to assert that a photon is self aware, which has no extra explicative power, then they have to prove it. I don't need to offer any evidence to disprove them, which sounds like a get out but is perfectly reasonable.

Nameta is now asserting that because we are made of matter that matter must be self aware, which is one huge leap. Last time I checked I am not a photon. My table is also made of matter, does that mean that it is 'talking to itself' too? What has the fact that Human beings are selfaware got to do with the fundamental laws of particle physics or relativity? Nothing, this line of argument does not deserve time in a scientific debate any more than an assertion that when matter collides with a solid impediment at 2am it swears because that's what I do when I stub my toe on the bed.

 

[nameta9]

Nameta is now asserting that because we are made of matter that matter must be self aware, which is one huge leap. Last time I checked I am not a photon. My table is also made of matter, does that mean that it is 'talking to itself' too? What has the fact that Human beings are selfaware got to do with the fundamental laws of particle physics or relativity?

OK. Yes the idea is more in the arena of philosophy or metaphysics. But if we look at a material system composed of two items, one item is a "scientist" thinking, experimenting and interacting with another item that is an experimental apparatus configured in a certain way, well the total of these two items is a material system that seen from outside is TALKING TO ITSELF.

Of course the subject - object relationship within the system remains, one item thinks the other responds, a table doesn't think etc. But the total of both seen from outside or from afar is one system, one material system made up of matter that is talking to itself and decoding itself and manipulating itself. Interesting is that there could be many other combinations of matter-energy that could be in an analogous state (beings living in computer simulations etc). So matter talks to itself, can know itself, and mathematics demonstrates that matter can find out all by itself how to decode, understand and manipulate itself.

 

[matt grime]

Putting it in capital letters doens't make it any more reasonable, you know (emphasis is one thing but this is another). You must be using a different definition of 'talk' and 'know' from the norm. In particular 'talk' appears to you to mean any kind of interaction.

 

[nameta9]

Matter is infinite, our universe contains within itself all the universes that are not part of it, that are outside our universe. It also contains all the possible universes.

If matter is self interacting, then maybe it is simply a self invention, it creates itself by observing itself, it may have no material substrate at all. It could simply be a pure state of information, much like the IT FROM BIT theories of reality. The mind may simply be a monolithic slab of pure consciousness that cannot be reduced to anything, since the scientific method of "reductionism" cannot be applied to the mind.

Any explanation is simply a painting, an arbitrary assignment of causes and effects that have no deeper necessity or meaning than that they are arbitrarily assigned. So anything is equivalent to our universe, even a contradictory one, even one with no physical laws or any quirky physical laws, even one with an elvis presley as god and atoms as small guitars. That god could magically assign effects and it would in the end be equivalent and just as necessary as ours .

 

[selfAdjoint]

Matter is infinite, our universe contains within itself all the universes that are not part of it, that are outside our universe. It also contains all the possible universes.

This is just deliberate mistification, spinning words without meaning.

If matter is self interacting, then maybe it is simply a self invention, it creates itself by observing itself, it may have no material substrate at all. It could simply be a pure state of information, much like the IT FROM BIT theories of reality. The mind may simply be a monolithic slab of pure consciousness that cannot be reduced to anything, since the scientific method of "reductionism" cannot be applied to the mind.

Still raises the question, what is the basis for this information? Whece this consciousness?

Any explanation is simply a painting, an arbitrary assignment of causes and effects that have no deeper necessity or meaning than that they are arbitrarily assigned. So anything is equivalent to our universe, even a contradictory one, even one with no physical laws or any quirky physical laws, even one with an elvis presley as god and atoms as small guitars. That god could magically assign effects and it would in the end be equivalent and just as necessary as ours .

This is just wrong. Science is not arbitrary; its theories have to work out in practice. Yes they may be falsified and replaced by better theories, but that is not arbitrary.

 

[nameta9]

This is just deliberate mistification, spinning words without meaning.

This is just wrong. Science is not arbitrary; its theories have to work out in practice. Yes they may be falsified and replaced by better theories, but that is not arbitrary.

I agree somewhat with you, when you reach the edge of knowledge it is hard to express thoughts.

If you look at science and the laws of physics from a far enough point of view, they are just quirk assignments, they have no intrinsic necessity. This does not mean they do not work or that they are incorrect, yes they work and are correct especially for us humans as they pertain to our subjective experience, our sense organs, the way our mind is hardwired, the way we carve out our knowledge by following paths through pain/pleasure measurements, etc.

But seen for what they are, the laws of physics are just quirks, they could have been anything else, they could be anything else and even a complete lawless universe could be. They are equivalent, they are like paintings, arbitrary designs. If a god assigns miracles magically, it would be completely equivalent to a sequence of causes and effects, magical assignments make just as much sense as physical laws and mathematical laws. They are bizarre quirks.

 

[matt grime]

What utter nonsense this has degenerated into.

 

[kmarinas86]

What utter nonsense this has degenerated into.

Oh I see, that's very scary isn't it?

 

[selfAdjoint]

Oh I see, that's very scary isn't it?

Not scary, silly and boring. Philosophy does not mean a teen-age bull session.