# The role of mathematics in physics (and science)

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!

vanesch
Staff Emeritus
Gold Member
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 ?

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.

Garth
Gold Member
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

DaveC426913
Gold Member
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 ).

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.

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

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.

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 ).

DaveC426913
Gold Member
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.

In my view, mathematics is just a way of describing quantities and relationships. OP should not confuse the map with the territory.

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 :-)

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.

DaveC426913
Gold Member
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.

OP should not confuse the map with the territory.
Hear hear!

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.

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!

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.

Garth
Gold Member
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

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.

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.

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.

DaveC426913
Gold Member
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.

Garth
Gold Member
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:
+ =

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

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.

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.