The role of mathematics in physics (and science)

In summary, the conversation discusses the idea that mathematics is the foundation of the universe and whether it exists as a human construct or a Platonic ideal. The argument is made that the regular behaviors in the universe are due to its underlying mathematical basis, and that without mathematical constraints, the universe would be chaotic and inconsistent with the existence of life. However, others argue that mathematics is just a way of describing quantities and relationships, and not the actual source of these behaviors. The conversation also delves into the relationship between mathematics and physics, with some arguing that the behaviors of objects are due to underlying physics rather than mathematics. Overall, the conversation raises interesting questions about the role of mathematics in the universe and its relationship to the physical world.
  • #1
William Nelso
21
1
Hi guys,

In my blog www.letstalkphysics.com I present an argument (pretty simple)
that mathematics must be the foundation of any possible universe (as
it certainly seems to be in ours). I wrote it because it seems that so many
people, both mathematicians and physicists, not to mention laypeople,
seem to think that it is a curious
accident that our universe runs on math. I doubt it, but I am curious what
some of the math aficionados here think!
 
Physics news on Phys.org
  • #2
I think a fallacy in your argument is that the universe must have some set of rules. Why must this be so ? Why can't the universe just be a big set of rather arbitrary happenings (events), say, determined by the fancy will of some or other deity ? What excludes "magic" a priori ? Why can't the universe just be a dream ?
 
  • #3
Note that you say "rather arbitrary", instead of "arbitrary". There is a big difference between "rather" arbitrary, which implies rules, and completely arbitrary, which is the case with no rules.
If you put yourself in the position of the creator of a universe, and attempt to further specify a set of "rather arbitrary" constraints, you will quickly find that it is impossible except by being completely precise (or at least, precisely probabilistic, as in quantum mechanics).

And as for a "supernatural deity", it is the same thing. If such a being exists, it just means that our universe contains things we haven't seen yet. It doesn't change the need for rules, since without some constraints the deity could not even influence its own actions. How could it, since with no rules there can be no connection between the past and the present?

Indeed, this may be a better way to make my argument (and it is also David Hume's argument). The present has to be constrained by the past, otherwise there can be no coherent sequence of events and no real universe. However, there is no middle ground between complete, mathematical constraints, and no constraints at all. It is simply impossible to write down any kind of "partial" constraints like the "rather arbitrary" rules you suggest above. The closest you can come is exact probabilities, as in quantum mech.
 
  • #4
I think you can argue that even though completely chaotic universes might exist, with no physical laws ordering events within them, these would be inconsistent with the existence of life within them.

The reason that "our universe runs on math" would therefore seem to be anthropic in nature.

Of course this begs the question about whether mathematics exists as some Platonic ideal, discovered by mathematicians, or whether it is a human construct.

Garth
 
  • #5
It does not run on math; it runs on a few behaviours (such as electrons like to orbit protons and atoms like to absorb and emit photons) which, when repeated countless times, result in larger, emergent behaviours. Because the small behaviours and the large behaviours are so consistent across untold light years, we can use numbers to represent them.

i.e. the universe is entirely dervied from simple processes, and these processes are predictable ( = math ).
 
  • #6
It's difficult for me at least to picture the concept of "no physical laws". For example, is there a space with fixed dimension in which things happen? That is a law...
Can truly any event happen, e.g. universe turns into a potato, then Bart Simpson, then
War and Peace wrapped on a torus? If not, then something is constraining the events,
and those constraints are law of physics. What I claim is that by the time you are
done specifying what events can occur and which cannot, you will have to define
them in terms of mathematical constructs, for the simple reason
that nothing else really can be defined.



Garth said:
I think you can argue that even though completely chaotic universes might exist, with no physical laws ordering events within them, these would be inconsistent with the existence of life within them.

The reason that "our universe runs on math" would therefore seem to be anthropic in nature.

Of course this begs the question about whether mathematics exists as some Platonic ideal, discovered by mathematicians, or whether it is a human construct.

Garth
 
  • #7
The source of these regular behaviors is their underlying mathematical basis.
Indeed, it is not even possible to define an atom or a proton on its own
(consider all the intermediate
states of partially-destroyed atoms and protons).
Only the underlying constituents, electrons and quarks and photons, have a
definition, because they are fundamental mathematical objects
within quantum field theory.


DaveC426913 said:
It does not run on math; it runs on a few behaviours (such as electrons like to orbit protons and atoms like to absorb and emit photons) which, when repeated countless times, result in larger, emergent behaviours. Because the small behaviours and the large behaviours are so consistent across untold light years, we can use numbers to represent them.

i.e. the universe is entirely dervied from simple processes, and these processes are predictable ( = math ).
 
  • #8
William Nelso said:
The source of these regular behaviors is their underlying mathematical basis.
...
Only the underlying constituents, electrons and quarks and photons, have a
definition, because they are fundamental mathematical objects
within quantum field theory.
No. You've got it backwards.

The behaviours are due to the underlying physics of the objects. They operated just fine before humans came along and quantified them with mathematics.
 
  • #9
In my view, mathematics is just a way of describing quantities and relationships. OP should not confuse the map with the territory.
 
  • #10
DaveC426913 said:
No. You've got it backwards.

The behaviours are due to the underlying physics of the objects. They operated just fine before humans came along and quantified them with mathematics.


Can you express this "underlying physics" in some form other than mathematical? I would be interested to see it...and so would the Nobel Prize committee :-)
 
  • #11
Moridin said:
OP should not confuse the map with the territory.
This sentence repeated over and over again... Georges Perec, one of my favorite author, wrote a lot among other things on this subject. One can argue how the territory is not real until you have a map. One can argue about a 1 to 1 scale map. One can argue about a map, the knowledge of part of it allowing you necessarilly to be able to extend it, by pure thought, without end, and being able to check that it always matches with the territory... which one is the most real ? Please don't over-simplify.

Platonists vs (?) constructivists : as far as I'm concerned, I was disappointed to face the impossibility of a discussion. Most people are convinced either way, and do not care about pure arguments in favor of one or the other way. I myself do not claim to be convinced either way, at least I am much more interested in hearing other people's arguments than trying to prove them wrong.
 
  • #12
William Nelso said:
Can you express this "underlying physics" in some form other than mathematical?
Certainly*. But even if I couldn't, so what? The math is just the description. The description is merely for the benefit of humans.

As before, the universe got along just fine before math came along.

* let's see:
1] For thousands of years, describing the underlying physics as the actions of gods or demons worked quite well. Granted, it is no longer sufficient for some of us (though not all of us), but you can't deny that gods and demons did explain why the sun chased the moon across the sky.
2] Aliens could easily describe our universe using a completely different structure. Granted, they wil have their own form of mathematics, but the point is, the language is just the tool for describing the underlying physics, and it doesn't much matter the details of the tool so long as it's complex enough.


Moridin said:
OP should not confuse the map with the territory.
Hear hear!
 
  • #13
By "description" I mean of course a description in terms that you can define.
One can define an electron, using mathematics, but good luck defining a god or
demon.
As for "explanation", that is not the role of science. That is the role of fairy tales.




DaveC426913 said:
Certainly*. But even if I couldn't, so what? The math is just the description. The description is merely for the benefit of humans.

As before, the universe got along just fine before math came along.

* let's see:
1] For thousands of years, describing the underlying physics as the actions of gods or demons worked quite well. Granted, it is no longer sufficient for some of us (though not all of us), but you can't deny that gods and demons did explain why the sun chased the moon across the sky.
2] Aliens could easily describe our universe using a completely different structure. Granted, they wil have their own form of mathematics, but the point is, the language is just the tool for describing the underlying physics, and it doesn't much matter the details of the tool so long as it's complex enough.



Hear hear!
 
  • #14
There is a big assumption here: you're assuming that the Universe behaves rationally. There is no guarantee that the Universe follows the laws that govern the reason of men. The Universe is only describable mathematically if is indeed a model of a logical system; it may not be - we don't know.
 
  • #15
Astrophysics provides plenty of evidence that the laws of physics discovered on Earth apply equally well across the observable universe, take as an example the precise pattern of spectral lines that identify specific elements in the spectra of distant stars.

These patterns are described mathematically, there is a numerical relationship between the wavelengths of separate lines in the absorption spectrum.

These examples provide conclusive evidence that the observable universe behaves rationally.

If it didn't it would not be possible to 'do' science at all.

Garth
 
  • #16
William Nelso said:
In my blog www.letstalkphysics.com I present an argument (pretty simple) that mathematics must be the foundation of any possible universe (as it certainly seems to be in ours).

The issue here is that...nobody yet fully understands the universe. And then you say 'it certainly seems to be in OURS'...which means that you're taking just a single example and then you're assuming this single example applies to everything else.

Also, it doesn't make sense to say mathematics must be the foundation of any possible universe...and this is because we don't know what is the origin of the universe...and therefore we don't know what the foundation of the universe is. And...does maths tell where energy and matter actually came from? It doesn't.
 
  • #17
humanino said:
This sentence repeated over and over again... Georges Perec, one of my favorite author, wrote a lot among other things on this subject. One can argue how the territory is not real until you have a map. One can argue about a 1 to 1 scale map. One can argue about a map, the knowledge of part of it allowing you necessarilly to be able to extend it, by pure thought, without end, and being able to check that it always matches with the territory... which one is the most real ? Please don't over-simplify.

Platonists vs (?) constructivists : as far as I'm concerned, I was disappointed to face the impossibility of a discussion. Most people are convinced either way, and do not care about pure arguments in favor of one or the other way. I myself do not claim to be convinced either way, at least I am much more interested in hearing other people's arguments than trying to prove them wrong.

Now, I'm not necessarily a constructivist in regards to mathematics, but have more of a naturalistic / physicalistic view. I think that the quantities and relations mathematics describe is very much real and that mathematics is superior to other 'languages' in its unambiguity and component simplicity, but that the specifics symbols and their semantics is mostly a social construct. Similarily, the term "chair" is a social construct, but chairs are not. I'm not sure my elaboration helped, or if I just repeated what I stated earlier.
 
  • #18
Werg22 said:
There is a big assumption here: you're assuming that the Universe behaves rationally. There is no guarantee that the Universe follows the laws that govern the reason of men. The Universe is only describable mathematically if is indeed a model of a logical system; it may not be - we don't know.

In objectivist metaphysics, the uniformity of nature can be seen as the principle of identity (A is A) applied to action over time. Furthermore, if the uniformity of nature does not obtain, there is no reason for you to think that a rational discussion on the uniformity of nature is possible, since valid perception presupposes the uniformity of nature.
 
  • #19
William Nelso said:
By "description" I mean of course a description in terms that you can define.
One can define an electron, using mathematics, but good luck defining a god or
demon.
As for "explanation", that is not the role of science. That is the role of fairy tales.
You're still confusing the descriptor with the described.

By that logic I could claim that all Earthly existence has, at its fundamental, language, since it requires language to describe it.


I'm still waiting to hear you address the issue as to how the universe managed for 13.69999999 billion years before mathematics came along.
 
  • #20
The language of mathematics describes the mathematical landscape of logical relationships between mathematical objects.

These 'objects': geometrical shapes, real numbers, complex numbers, vectors, tensors, scalars etc. are defined by their properties, i.e. what they do, not what they are.

The language was invented by humans, the mathematical relationships were there all along.

2 + 2 = 4 was true even when there were no humans to think so.

The language of that expression can be translated into other languages such as binary:
10 + 10 = 100

or Roman:
II + II = IV

but in each case it means the same, a numerical property that first was attached to objects such as:
:smile::smile: + :smile::smile: = :smile::smile::smile::smile:

but which exists as an abstract concept that can be logicaly and consistently constructed without any reference to objects at all.

The mathematics was there all along, it was the language of mathematics that humans have invented.

Garth
 
  • #21
Garth said:
Astrophysics provides plenty of evidence that the laws of physics discovered on Earth apply equally well across the observable universe, take as an example the precise pattern of spectral lines that identify specific elements in the spectra of distant stars.

These patterns are described mathematically, there is a numerical relationship between the wavelengths of separate lines in the absorption spectrum.

These examples provide conclusive evidence that the observable universe behaves rationally.

If it didn't it would not be possible to 'do' science at all.

Garth

Actually, I find this very debatable. If anything, astrophysics provides evidence that our understanding of gravity and matter still has a long way to go. Also it doesn't matter if all of our observations seem to confirm that Universe behaves rationally. For the philosophical argument to be impeccable, we would need to argue purely on deductive grounds that the universe is indeed rational; this is clearly impossible - we cannot ascertain the universe behaves rationally everywhere and at all times.
 
  • #22
Werg22 said:
Actually, I find this very debatable. If anything, astrophysics provides evidence that our understanding of gravity and matter still has a long way to go. Also it doesn't matter if all of our observations seem to confirm that Universe behaves rationally. For the philosophical argument to be impeccable, we would need to argue purely on deductive grounds that the universe is indeed rational; this is clearly impossible - we cannot ascertain the universe behaves rationally everywhere and at all times.

In objectivist metaphysics, the uniformity of nature can be seen as the principle of identity (A is A) applied to action over time. Furthermore, if the uniformity of nature does not obtain, there is no reason for you to think that a rational discussion on the uniformity of nature is possible, since valid perception presupposes the uniformity of nature.
 

1. What is the relationship between mathematics and physics?

The relationship between mathematics and physics is very close and symbiotic. Mathematics is the language of physics, providing a precise and quantitative way to describe and understand the laws and phenomena of the physical world. Many areas of physics, such as mechanics, thermodynamics, and electromagnetism, rely heavily on mathematical concepts and equations to explain and predict the behavior of physical systems. In turn, physics often inspires the development of new mathematical tools and theories.

2. How does mathematics help in the study of physics?

Mathematics plays a crucial role in the study of physics by providing a universal and rigorous framework for understanding and solving complex physical problems. It allows scientists to make precise and quantitative predictions about the behavior of physical systems, test these predictions through experimentation, and develop new theories and models to explain observed phenomena. Mathematics also provides a way to express and visualize abstract concepts and relationships that may be difficult to grasp intuitively.

3. Can physics be done without mathematics?

While some basic concepts of physics, such as motion and force, can be understood without mathematics, the vast majority of modern physics relies heavily on mathematical tools and techniques. Without mathematics, it would be challenging, if not impossible, to make precise and quantitative predictions about the behavior of physical systems, understand the fundamental laws and principles of nature, or develop new theories to explain complex phenomena. Therefore, it is safe to say that physics cannot be done without mathematics.

4. What are some examples of the use of mathematics in physics?

There are countless examples of the use of mathematics in physics, as virtually every area of physics relies on mathematical concepts and equations. Some notable examples include Newton's laws of motion, which are described by differential equations, Maxwell's equations, which describe the behavior of electromagnetic fields, and Einstein's famous equation E=mc^2, which relates mass and energy. Other examples include the use of calculus in understanding motion and change, linear algebra in quantum mechanics, and statistics in thermodynamics.

5. Is a strong background in mathematics necessary for studying physics?

While a strong background in mathematics is not a strict requirement for studying physics, it is highly beneficial and recommended. As mentioned before, mathematics is the language of physics, and having a solid understanding of mathematical concepts and techniques will greatly enhance one's ability to understand and apply the principles of physics. In addition, many advanced topics in physics, such as quantum mechanics and relativity, require a deep understanding of complex mathematical concepts. Therefore, a strong foundation in mathematics is essential for success in the study of physics.

Similar threads

Replies
2
Views
680
Replies
5
Views
844
Replies
14
Views
847
Replies
5
Views
665
Replies
29
Views
2K
  • Science and Math Textbooks
Replies
2
Views
206
  • STEM Academic Advising
Replies
15
Views
1K
Replies
12
Views
1K
Replies
3
Views
2K
  • Science and Math Textbooks
Replies
28
Views
1K
Back
Top