# Why Mathematics works so well with Physics

• FallenApple
In summary: If something that can no longer be described by math (say beyond the standard model) can still affect the physical or has experimental results, then it is definitely physics. The proof for this statement is that mathematics is a product of human thought and that humans can always find equations to describe anything that affects the physical.
FallenApple
Clearly math is a product of human thought. But is there a prior reason for this?

A possibility is that math is a byproduct of basic information that we obtain from nature, where that information come in the form of ideas in which we have to sort out into something more rigorous.

We absorb information from the environment, for example, from an image, we can see a primitive idea of the geometric relation between things, ideas that represent numbers, sets, and perhaps a very basic logical structure of the picture etc. So we take this very basic information obtained from the outside world, and then mix it together to synthesize new stuff with it. But is it really "new" stuff? I mean, when chemists make a non naturally occurring chemicals out of elements, they didn't invent anything new, because the compound must have always been allowed within the laws of nature; that is, it must have always been so. Hence, maybe, somehow, logic itself is absorbed from the outside world. Even if logic is inherent from birth, we could argue that it could be absorbed from the environment via DNA in the learning process of evolution. In that sense logic is, very roughly, like an element, or component if you will, of nature and hence when we create new mathematical structures with it, it has the possibility of being a actual physically valid structure and most certainly, a potential physically valid structure, using this picture. Although this may be stretching it too far, but we see that for geometry, this is the case.

Geometry relies on images of the outside world. We see huge successes between curved geometry and relativity theory even though curved geometry came prior. But this is because differential geometry is built from the basic ingredients obtained from nature in the first place; concepts such as logic, shapes, curved shapes, numbers etc. And we see how algebraic topology can be possibly used to model a mouse brain where it is clear that the mouse brain is just an emergent phenomena of the laws of nature. Algebraic topology may not obviously manifest itself in the basic laws of physics, but in the emergent physical structures that result because of these basic laws. Graph theory shows up in social networks, again another system in nature, and with the reductionist view, a product of physics. The list goes on.

Thoughts?

timeuntotime and ISamson
I think this is a good paper that deals with this topic:
https://arxiv.org/pdf/1111.6560v3.pdf

The purpose of this essay is to bring out the unique role of Mathematics in providing a base to the diverse sciences which conform to its rigid structure. Of these the physical and economic sciences are so intimately linked with mathematics, that they have become almost a part of its structure under the generic title of Applied Mathematics. But with the progress of time, more and more branches of Science are getting quantified and coming under its ambit. And once a branch of science gets articulated into a mathematical structure, the process goes beyond mere classification and arrangement, and becomes eligible as a candidate for enjoying its predictive powers ! Indeed it is this single property of Mathematics which gives it the capacity to predict the nature of evolution in time of the said branch of science. This has been well verified in the domain of physical sciences, but now even biological sciences are slowly feeling its strength, and the list is expanding.

The question itself is so wide spread that an answer can range between my short quotation here and a book about life, the universe and everything.

I need to know something related to this.

Physics is said to be about math and models and experiments.

What would happen if something that can no longer be described by math (say beyond the standard model) can still affect the physical or has experimental results.. does it mean it's no longer physics and physicists will just ignore them?

Blue Scallop said:
What would happen if something that can no longer be described by math (say beyond the standard model) can still affect the physical or has experimental results..
If it affects experimental results, phenomenologists will figure out some equations to describe it -- trust me.

And then theorists will drive themselves mental trying to uncover a deeper principle to account for the phenomenology.

does it mean it's no longer physics and physicists will just ignore them?
Quite the contrary. If it affects experimental results, it would cause a sensation throughout the physics community.

strangerep said:
If it affects experimental results, phenomenologists will figure out some equations to describe it -- trust me.

What if there is just no equation to describe it? Or do you mean anything that can affect the physical always has equation (and can be described by math)? What is the proof for this statement? (I mean in terms of theorem or axiom or whatever that whatever affects the physical can be model mathematically?)

And then theorists will drive themselves mental trying to uncover a deeper principle to account for the phenomenology.

Quite the contrary. If it affects experimental results, it would cause a sensation throughout the physics community.

Blue Scallop said:
What if there is just no equation to describe it? Or do you mean anything that can affect the physical always has equation (and can be described by math)? What is the proof for this statement? (I mean in terms of theorem or axiom or whatever that whatever affects the physical can be model mathematically?)
Take the empirical data. The fact that it's "empirical" means you can express it in terms of well-known quantities -- like mass, momentum, energy, spin, position, temperature, etc -- whatever the apparatus was designed to measure. Generically, let's say the appartus was design to measure well-known property "##A##".

Perform a regression fit on the data. Start with something simple like least-squares. Or try log-linear, or log-log, etc. Maybe a combination of polynomials. Then you have an equation like "##A = ##some function". Various function choices will match the data to varying degrees of accuracy -- which is normal in physics. We go with the best one until someone thinks of something better.

IOW, mathematics "works" in physics because it's a way to organize our perception and thinking into self-consistent structures.

Blue Scallop
strangerep said:
Take the empirical data. The fact that it's "empirical" means you can express it in terms of well-known quantities -- like mass, momentum, energy, spin, position, temperature, etc -- whatever the apparatus was designed to measure. Generically, let's say the appartus was design to measure well-known property "##A##".

Perform a regression fit on the data. Start with something simple like least-squares. Or try log-linear, or log-log, etc. Maybe a combination of polynomials. Then you have an equation like "##A = ##some function". Various function choices will match the data to varying degrees of accuracy -- which is normal in physics. We go with the best one until someone thinks of something better.

IOW, mathematics "works" in physics because it's a way to organize our perception and thinking into self-consistent structures.

Oh I see. But for even QFT.. they can't do a fully interacting QFT but needs perturbation method. Maybe it's possible there is no non-perturbative QFT even in principle? Then some portion of physics can't even be modeled by math... is it not.

Maths can be a convenient and concise way of summarising observation that can be made.

Blue Scallop said:
Oh I see. But for even QFT.. they can't do a fully interacting QFT but needs perturbation method. Maybe it's possible there is no non-perturbative QFT even in principle? Then some portion of physics can't even be modeled by math... is it not.
All theoretical physics is essentially models. In general, different models match the empirical data with different accuracies. The most accurate model becomes the current "preferred" model of that particular aspect of physics, but may well be replaced by another, more accurate, model in the future -- who knows?

Perturbative QED achieves an astonishingly good match between theoretical and empirical -- see Precision tests of QED . There just ain't no pleasing anyone who isn't satisfied with that.

Blue Scallop
My question about something that can't be modeled by math stems from Penrose non-algorithmic thing in his book "The Emperor New Mind". Did Penrose mean by "non-algorithmic" as can't be modeled by math.. or does he mean math can be both algorithmic and non-algorithmic.. what is example of a non-algorithmic math then?

One of my favourite quotes of mine:
There can't be science (or physics in this case) without math, there can't be math without science.

Ivan Samsonov said:
One of my favourite quotes of mine:
There can't be science (or physics in this case) without math, there can't be math without science.

You mean when we dream at night.. it can be broken down into math if we just have enough computation power?

Blue Scallop said:
You mean when we dream at night.. it can be broken down into math if we just have enough computation power?

This is true, but this is not what I meant.
I meant that if we look deeply into it then there can't possibly be science without math (graphs, averages...), if there was no science then there would not be any medicinals and technology, so all the mathematicians would die out.

ISamson said:
One of my favourite quotes of mine:
There can't be science (or physics in this case) without math, there can't be math without science.

The part after the comma is false, if you mean physical sciences (and not just considering math as a science). There's plenty of math that has no science analog or doesn't represent anything physical.

deskswirl and fresh_42
FallenApple said:
Clearly math is a product of human thought.
Thoughts?

If you really believe that math is a product of human thought, let me ask you this question. When Evariste Galois wrote down the first description of a group, do you really believe that at that point he invented the monster group, with its 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements? If not, when was this group "invented"? I think the answer is obviously that the monster group existed in some sense before Galois conceived of the idea of a group. I think the idea that mathematics is an invention of the human mind is absurd. There is a fundamental order to the universe which we are discovering as we develop new lines of mathematical thought.

fresh_42 and dkotschessaa
He'd be better tried to prove the existence of a simple group with 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements than to get engaged in a duel.

deskswirl
phyzguy said:
If you really believe that math is a product of human thought, let me ask you this question. When Evariste Galois wrote down the first description of a group, do you really believe that at that point he invented the monster group, with its 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements? If not, when was this group "invented"? I think the answer is obviously that the monster group existed in some sense before Galois conceived of the idea of a group. I think the idea that mathematics is an invention of the human mind is absurd. There is a fundamental order to the universe which we are discovering as we develop new lines of mathematical thought.

I get why we avoid philosophy on PF but it's pretty darn compelling isn't it?

One thing that I find missing in these conversations usually is history. We have the math we do because of the people and cultures who developed it, the order that it was developed, and the notation that was cooked up to communicate it. The equals sign, for example, has only been around since the 1500s. But it seems like a very natural thing to have.

There's also the fact that the very foundations of mathematics have been rocked by philosophical contests. What if the intuitions had "won' over the formalists? Was this the best way to go or was it a popularity contest? (Another thread I suppose).

The skibble-bibble people of Planet Gurzog, who I just made up, probably have something that we'd call math, but most certainly they do not share our notation, and if they have anything like numbers, they might not even use our base 10 system. There might even be some analog of calculus - certainly multiple earthlings seem to have discovered it - conceptually at least, a few times, independently.

I could probably take a physical system of some sort, say something in biology, and model it as a differential equation, or a cellular automaton, or a graph. The skibble bibbles could have something completely different.

-Dave K

dkotschessaa said:
I get why we avoid philosophy on PF but it's pretty darn compelling isn't it?
Mathematics and philosophy was long two sides of the same coin and most mathematicians at early universities studied both, often accompanied by legal science and theology (to achieve the possibility of an income).
dkotschessaa said:
There's also the fact that the very foundations of mathematics have been rocked by philosophical contests. What if the intuitions had "won' over the formalists? Was this the best way to go or was it a popularity contest?
Well, formalism is a completely new phenomenon, which I would say originated in the second half of last century, probably mainly driven by Bourbaki. I have a book on group theory (Kurosh, 1970) which is mainly written in words and formulas only occur if they have to. Also van der Waerden's books on algebra can be read, in contrast to be decoded. Even Mathematical Methods of Physics, Courant, Hilbert 1924, in which one would expect mainly formulas, has surprisingly a lot of text. The big advantage of formulas is, that one doesn't need to learn some dozens of languages to understand them. I've quoted a few German, French or Spanish Wikipedia pages if their English counterpart didn't contain the same, although this isn't allowed here. But they were so completely full with formulas, e.g. proofs about the value of a series, that it simply didn't matter. However, the real truth might be a lot more trivial than this, as it could well be what I like to call Gutenberg effect: If you've ever tried to typeset a formula with a mechanical - or even an electrical - typewriter, then you appreciate text instead. It's not that long ago, that LaTeX and computer writing entered our world (late '80s). So it is simply far easier nowadays to write a formula than it has been formerly.
dkotschessaa said:
The skibble-bibble people of Planet Gurzog ...
I like to think that they will have a right and wrong, a way to count, and an imagination of shapes (spherical stars, e.g.). The rest is only a generalization of these.

dkotschessaa
Blue Scallop said:
My question about something that can't be modeled by math stems from Penrose non-algorithmic thing in his book "The Emperor New Mind". Did Penrose mean by "non-algorithmic" as can't be modeled by math.. or does he mean math can be both algorithmic and non-algorithmic.. what is example of a non-algorithmic math then?
I can't comment on that, beyond saying that I think stuff involving "quantum consciousness" and "wavefunction collapse related to brain function" are a load of crackpottery.

BillTre
I think mathematics works so well with physics, because physics needs precision and numbers and mathematics allows physical events to be described exactly.
Physic needs precision ans math gives this to physics.

One explanation is extremely simple, straightforward and classical: there is Consciousness (the Process of awareness) playing the Game.
To organize any game, the mathematical tree of the game must be generated according to certain mathematics. If the Game is the whole physical universe, then the Tree is the Everettian branching worlds.

As the Game unfolds, new subprocesses (within the main Process) of awareness are introduced to become new players of the Game - our human minds are among them.

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Perhaps the harmony between mathematics and physics is a consequence of something as simple as the following. The incoherent configurations and phenomena, if they occur, do not last long enough to study them. Then, we only study coherent configurations and phenomena. Chaos theory ? Study traits of chaos that can be expressed in coherent terms. Mathematics formulates coherent relationships. Thus, the correlation between physics and mathematics should not be surprising.

phyzguy said:
If you really believe that math is a product of human thought, let me ask you this question. When Evariste Galois wrote down the first description of a group, do you really believe that at that point he invented the monster group, with its 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements? If not, when was this group "invented"? I think the answer is obviously that the monster group existed in some sense before Galois conceived of the idea of a group. I think the idea that mathematics is an invention of the human mind is absurd. There is a fundamental order to the universe which we are discovering as we develop new lines of mathematical thought.

It is a product, but it is built from logic which I suspect is isomorphic to the structure of the data we absorb from our surroundings. So math is actually made of ingredients that are fundamental in nature. So it's not really inventing. It always had to be so because it was always possible.

When I say product of the mind, I meant it like an engineer. Engineers build things out of real materials. Hence, whatever they made must be a real material as well. May be new, but it was always possible. Same with math. For lack of a better analogy, it's like building stuff from clay. The resulting sculpture is new. But it's made out of something that is actually real, namely clay. Hence, the sculpture says something deep about the clay in which it is built from. Namely, that is possible to form said sculpture from clay. It tells you about the possibilities of clay. Similarly, math tells about the possibilities of logic. And logic isn't invented. It is fundamental.

FallenApple said:
It is a product, but it is built from logic which I suspect is isomorphic to the structure of the data we absorb from our surroundings. So math is actually made of ingredients that are fundamental in nature. So it's not really inventing. It always had to be so because it was always possible.

When I say product of the mind, I meant it like an engineer. Engineers build things out of real materials. Hence, whatever they made must be a real material as well. May be new, but it was always possible. Same with math.

So I think you've answered your question of , "Why Mathematics works so well with Physics?" Because both are derived from the fundamental structure of the universe.

phyzguy said:
So I think you've answered your question of , "Why Mathematics works so well with Physics?" Because both are derived from the fundamental structure of the universe.
Well some people may disagree with that. It's a strong hunch that I have, but I'm not sure what the pitfalls of that approach is. That's why I posted the thread.

gleem said:
This topic was discussed by Eugene Wigner in 1960 in his widely discussed paper http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

For a brief review of this paper see https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences

There's also the recent Argument from the Applicability of Mathematics in philosophy (which has gained a lot of academic traction and interest lately), which is an argument for God's existence.

It argues that the "happy coincidence" (a phrase coined by philosopher Mary Leng of The University of York) of math applying to the real world (as a language) is best explained by God orchestrating/designing it that way versus the highly improbably coincidence that it does so on its own.

There's no reason why our maths would just so happen to "magically" apply to the real world and physics. Sometimes people confuse the argument with necessary truths in math and logic (such as 2 +2 =4, which is a necessary truth, once you have the foundations for math in place). But the argument isn't saying there's anything special about those necessary truths (as they are necessary, obviously), but rather the entire system of our mathematics that's built up to involve them. Lots of non-sensical abstract "inventions" of math also apply beyond that.

Think of negative numbers, imaginary numbers, infinity, etc., which seem non-sensical outside of the realm of abstraction (like what is -5? in the physical world?), yet are required for our system of mathematics to work. Yet, these map onto the real world in physics (we do use these "numbers"/concepts in the math).

Secondly, you also could imagine a different set of mathematics that might be internally consistent that could be the language of the universe, but our math seems to fit perfectly as a "happy coincidence."

So, there's the contemporary philosophical argument too.

kyphysics said:
It argues that the "happy coincidence" (a phrase coined by philosopher Mary Leng of The University of York) of math applying to the real world (as a language) is best explained by God orchestrating/designing it that way versus the highly improbably coincidence that it does so on its own.
I certainly do not want to discuss anything religious here, nor anything philosophical, but I have to claim that this is right away nonsense. The term "happy coincidence" is already an inappropriate assessment and does by no means require an orchestration. The argument goes along the lines: "Because it is as it is and it makes me happy, there must be a reason for my happiness." That's even below the anthropological reasoning and completely deliberate.
There's no reason why our maths would just so happen to "magically" apply to the real world and physics.
There is a reason. It is generally called evolution and on a more practical level, the need to improve. We developed mathematics in order to make life in a work sharing society easier and did so by naturally given means. And although mathematics is a form of abstraction, it did arise from the perception and needs of our environment. Combined with a thumb, a language, fantasy and curiosity mathematics became the logical consequence of our capabilities as part of the universe. It doesn't need an orchestration to distinguish true and false, count and recognize geometric shapes. The entire rest is based on that.

kyphysics said:
There's also the recent Argument from the Applicability of Mathematics in philosophy (which has gained a lot of academic traction and interest lately), which is an argument for God's existence.

It argues that the "happy coincidence" (a phrase coined by philosopher Mary Leng of The University of York) of math applying to the real world (as a language) is best explained by God orchestrating/designing it that way versus the highly improbably coincidence that it does so on its own.

There's no reason why our maths would just so happen to "magically" apply to the real world and physics. Sometimes people confuse the argument with necessary truths in math and logic (such as 2 +2 =4, which is a necessary truth, once you have the foundations for math in place). But the argument isn't saying there's anything special about those necessary truths (as they are necessary, obviously), but rather the entire system of our mathematics that's built up to involve them. Lots of non-sensical abstract "inventions" of math also apply beyond that.

Think of negative numbers, imaginary numbers, infinity, etc., which seem non-sensical outside of the realm of abstraction (like what is -5? in the physical world?), yet are required for our system of mathematics to work. Yet, these map onto the real world in physics (we do use these "numbers"/concepts in the math).

Secondly, you also could imagine a different set of mathematics that might be internally consistent that could be the language of the universe, but our math seems to fit perfectly as a "happy coincidence."

So, there's the contemporary philosophical argument too.

I think the problem with her argument is that probability is defined in terms of measure theory, which is mathematical. The math has to be prior. So any argument using probability has to be Bayesean since math has to exist beforehand.

It's hard to argue the origins of probability using probability. I think that requires determining the consistency of a system from within a system, which shouldn't be possible according to Godel.

It seems that the proof building process of using logic, the foundation of mathematics, has a temporal ordering to it. That is, it uses an ordered sequence of steps to arrive at propositions using previous propositions, all the way down to the beginning, the basic assumed axioms. This process is highly reminiscent of human psychological/empirical perception of the flow of time. Very uncanny.

Also, we have geometry, which borrows it's ideas from spatial properties.

kyphysics said:
It argues that the "happy coincidence" (a phrase coined by philosopher Mary Leng of The University of York) of math applying to the real world (as a language) is best explained by God orchestrating/designing it that way versus the highly improbably coincidence that it does so on its own.

I don't consider this a coincident at all.
The math we understand resides in our brains.
Physical reality is separate from that.
The brain structure/functioning underlying mathematical understanding must be in some way provide a basis for those ideas to occur.
To me, this is the question here. Why is the brain's math abilities so well adapted to this purpose?

The brain's structure/function is an evolved thing, based on at least millions of cycles (generations) of selection, where brain functioning was tested for its ability to match up with the real world around, which affected its ability to survive and reproduce.
This argument should work for counting, logic, and geometry, all things an organism would need to navigate the complex world in which we find ourselves.
Since brain function analyses (mostly) macro (not micro) scale events and is graded on that by evolution, the brain abilities evolved to deal with those scales most naturally (as opposed to say, quantum mechanics, which seems a less natural (more abstract) fit to many people).

Its a kind of circular argument, but to me it makes sense.
Its does not involve invoking God and is not anthropocentric, but one might says its brain-pocentric.

ISamson
FallenApple said:
Well some people may disagree with that. It's a strong hunch that I have, but I'm not sure what the pitfalls of that approach is. That's why I posted the thread.

The difficulty for me is that there are mathematical concepts that are, as far as I can tell, pure abstraction. My favorite example is large cardinals: https://en.wikipedia.org/wiki/Large_cardinal

It's probably one of my favorite areas of mathematics, perhaps because it is so entirely divorced from any sort of reality I know. If someone were to find an application for it I'd probably be disappointed!

-Dave K

Mathematics is all about axioms and definitions followed by logic, at least in my opinion. Within that axioms, any logical derivation is universally true. So anything that applies mathematics, by adding conditions and restrictions, is also universally true under that axioms as long as it is a logical derivation. And of course, science is essentially about finding something that is universally true. So they work well.

dkotschessaa said:
The difficulty for me is that there are mathematical concepts that are, as far as I can tell, pure abstraction. My favorite example is large cardinals: https://en.wikipedia.org/wiki/Large_cardinal

It's probably one of my favorite areas of mathematics, perhaps because it is so entirely divorced from any sort of reality I know. If someone were to find an application for it I'd probably be disappointed!

-Dave K
But the mathematician thinking about a large cardinal corresponds to specific firing pattern of neurons which takes place in the real physical world. One can even say that a specific large cardinal is isomorphic to a specific neuronal electrical pattern.

For those wishing to avoid forays into philosophy or religion, I think the best one can conclude is that "it does." My personal view is in agreement with Eugene Wigner, but he was a man of faith and seemed to be appealing to faith in his assessment, "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. "

But for a strict naturalist or instrumentalist, all one has is the empirical fact that so many natural phenomena are accurately modeled with mathematics. This trend gives one hope and expectation that the next phenomenon we stumble upon with also be amenable to being accurately modeled with mathematics, but there is no guarantee until it happens.

But for me, it's never a matter of whether a problem is amenable to mathematics, it's a question of which mathematical tool is best for the job. But I admit that is an article of faith, and I trust problems that are yet unsolved are unsolved because the right mathematical tool is not yet found, not because math is not applicable.

HAYAO

## 1. Why is mathematics considered the language of physics?

Mathematics is considered the language of physics because it provides a precise and universal way to describe and quantify physical phenomena. The laws and principles of physics can be expressed and understood through mathematical equations, making it a powerful tool for studying and predicting the behavior of the physical world.

## 2. How does mathematics help to explain physical phenomena?

Mathematics helps to explain physical phenomena by providing a framework for understanding and quantifying the relationships and patterns in nature. By using mathematical models, physicists can make predictions and test their theories, leading to a deeper understanding of the underlying principles governing the physical world.

## 3. Why do mathematical concepts often have real-world applications in physics?

Mathematical concepts often have real-world applications in physics because mathematics is based on abstract concepts and principles that can be applied to various physical systems. This allows physicists to use mathematical tools and techniques to analyze and solve problems in the physical world.

## 4. What is the role of mathematics in the development of new theories in physics?

Mathematics plays a crucial role in the development of new theories in physics by providing a language and framework for formulating and testing hypotheses. Through mathematical analysis, physicists can make predictions and determine the validity of their theories, leading to new discoveries and advancements in the field.

## 5. How does the use of mathematics in physics contribute to technological advancements?

The use of mathematics in physics contributes to technological advancements by providing a way to understand and manipulate the physical world. Many modern technologies, such as computers, rely on mathematical principles and equations to function. Additionally, advancements in physics, made possible through the use of mathematics, have led to groundbreaking technologies such as nuclear power and lasers.

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