# Is math real? Is physics math describable?

• Borek
In summary, some mathematicians believe that mathematics is a creation of the mind, while others believe that mathematics is discovered through abstraction.f

#### Borek

Mentor
This question arised somewhere else (https://www.physicsforums.com/showthread.php?t=236919). It started with

Maths IS, essentially, games of our own making.

This made me wonder - is math real? I don't mean real like a hammer - math is competely abstract, there is no doubt about it. However, is math really a game? Once we have made some basic assumptions and added some definitions, world that emerges is not random. We can discover its properties but statements that are true are already true and statements that are false are already false - even if we don't know them yet. So when we work on some branch of math we are not creating it - we are in fact uncovering construction that was there from the very beginning. It was there even before we have selected axioms and definitions.

Now, why is math so efficient tool in describing physics phenomena? Could be the reason is similarity - there is a set of axioms (rules, definitions) underlying all physics, and these axioms define all physics - just like some simple sets of axioms and definitions create huge branches of math. I am not aiming at Equation Of All Equations here, One To Rule Them All, One That Will Answer All Questions, 42. I just wonder if the fact, that now and then we hear that someone have realized that some esoteric math theory perfectly describes fine details of observable physiscs is not some sort a sign that these worlds (math & physics) are in a way parallel? Just like in math some statements are false, in physics some things can't happen - for the same reason. They are prohibited by logic. And just like in math starting point (set of axioms and definitions) generates whole world even before we start to think about possible outcome, whole world of physics is generated by some starting point. (Don't ask me what this starting point is - I have no idea).

If the math analogy is OK, looks like our physics has a good starting point, that leads to many emergent properties.

Disclaimer: English is my second langugae - and I am not sure if I wrote exactly what I mean. Hopefully I did.

Disclaimer: English is my second langugae - and I am not sure if I wrote exactly what I mean. Hopefully I did.
English is my third language. No wait, fourth ! Whatever

Is this a poll ?

Some think we discover math (platonists). Other prefer to think we construct them. This discussion can be very interesting.
Now, why is math so efficient tool in describing physics phenomena?
That's a very good question indeed !

Did you check Wigner ?

The Road to Reality Penrose R. is a mathematical one.

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"Don't confuse the map with the territory."

Some think we discover math (platonists). Other prefer to think we construct them. This discussion can be very interesting.

Try this interesting essay, called http://www.fdavidpeat.com/bibliography/essays/maths.htm" [Broken] by David Peat.

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I hold mathematics to be an unambiguous and precise language for describing relations, quantities and so on. Therefore, there is no surprise in mathematics being effective in the natural sciences. Saying that it is a surprise that mathematics is effective in the natural sciences is like claiming that the fact that English is effective in the natural sciences is surprising. The fact that clear descriptions of relations and quantities and so on are useful in the natural sciences is hardly surprising.

I guess this rules out platonism and intuitionism for me. I think I subscribe to some combination of formalism, mathematical realism, and mathematics as a language.

I gave this link in an earlier thread, and it appears that it may be relevant again in this thread.

Zz.

Wigner link was already posted here by humanino - and it is relevant.

The most important point - for me - being, I am not the only one to think about these things

Wigner link was already posted here by humanino - and it is relevant.

Zz.

Not that it particularly matters for the discussion at hand, but it appear that Wigner is a creationist. Well, at least some character assassinations are valid. In any case, both Howell and Steiner has made a similar argument before. Richard Carrier discussed the argument in detail here.

Not that I think about it, there is a good deal of chance that I first read this with your initial link "in an earlier thread" So, I should thank you

Wigner is a creationist.
I strongly oppose to that statement. Most important of all, by the time Wigner was alive, the mere word "creationism" did not carry the heavy controversial weight it does today. So using this word, purposedly or not, is an anachronism.

A few friends of mine sharing more or less the same point of view as experssed by Wigner would be pretty upset to be called creationists

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Honestly I did not read the entire link. Can you however show me where Wigner is mentionned ?

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I strongly oppose to that statement. Most important of all, by the time Wigner was alive, the mere word "creationism" did not carry the heavy controversial weight it does today. So using this word, purposedly or not, is an anachronism.

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Honestly I did not read the entire link. Can you however show me where Wigner is mentionned ?

Eugene Wigner appears to be the author of that article. Creationist was probably a too strong word, and the remark he made was just brief.

I'm sure religion is not the most popular thing to discuss here (or any other diverse community), but I do have to point out that you can't use science to back up your disbelief of a creator; if anything, science proves there is a creator. I'm not religious but I'm just saying that science can only explain what the big bang was, not what lit the fuse. :)

edit: btw, my statement about science proving there is a creator is referring to the understanding that things don't spontaneously begin to exist out of nowhere.

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If math is not real like a hammer, it's real like a ____?

I don't know what 'real' is other than what is physical. I think math is real like a hammer.

So these aliens show up one day from Andromeda. And they've got the same hammer! It's the quadradic equation. Well, it's kind of the same hammer, after applying some symbolic conversion and other mathematical gyrations, it's the same hammer, sorta'.

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Like Moridin said, math is a language. It's a purely syntatic language used for describing and manipulating raw data. It may seem like math is intrinsic to the universe but that's only because it's based (originally) on observations of the relationships and constants seen in reality so that it can be used to model it, and with physics and quantum physics it's often only through these models that we can "see" what is going on. It's taken thousands of years to develope the language and it's models to accomidate the ways that it's used to model reality today. No one simply looked at the world and "uncovered" it's hidden math.

Zero: The Biography of a Dangerous Idea is a very good book that can give you an idea of the history behind the development of math.

Like Moridin said, math is a language. It's a purely syntatic language used for describing and manipulating raw data. It may seem like math is intrinsic to the universe but that's only because it's based (originally) on observations of the relationships and constants seen in reality so that it can be used to model it, and with physics and quantum physics it's often only through these models that we can "see" what is going on. It's taken thousands of years to develope the language and it's models to accomidate the ways that it's used to model reality today. No one simply looked at the world and "uncovered" it's hidden math.

Zero: The Biography of a Dangerous Idea is a very good book that can give you an idea of the history behind the development of math.

In addition, I'd like to note that math doesn't perfectly describe reality, but it's very easy to manipulate and "fit" to reality, as it was designed.

If anything, math (as a concept) tells us more about consciousness and intelligence than it does about reality.
The brain is made in such a way that it quantifies and abstracts things from each other, and then in some mysterious way processes the information.
This is the foundation for 'everything' about the mind.
When a human applies morality, it's a way of processing data in a meaningful way with something that symbolizes the essence of something.
In the same manner 1 + 1 = 2, the 1 is a symbol of a quantity, and the 2 is the processing of these two 1's.

But the most important thing to note, is that like someone else said, the map is not the territory itself.
The map can be used to describe a lot of things to great detail, but they are two separate things, and in the end, the map has no connection or link to the territory itself.
The connection is imo once again made in the brain of a conscious being like a human.
We cannot say "this math perfectly describes the form, motion and weight of this hammer, therefore the form, motion and weight exists in the real world."
We can't in any way describe something objectively, there will always be from a viewpoint of sorts, and math is also a viewpoint, it is a way of perception.

In my opinion of course.

In addition, I'd like to note that math doesn't perfectly describe reality, but it's very easy to manipulate and "fit" to reality, as it was designed.
From another point of view, you can consider that Nature does not perfectly fits in the mathematical model
So Nature seems always more complex than a simple finite number of axioms.

Like Moridin said, math is a language.
Most importantly, this is the only unambiguous language.

But you do not address the universal aspect of mathematics. Let me recap the procedure
• (1) Choose a set of axioms
• (2) Derive logically all provable theorems
• (3) Conjecture new axioms, independent from the previous ones, to enrich your formal system and go back to (1) (possibly getting a Field medal in between)
From Godel and Turing's works we know that this cannot be performed by an automatic machine. Mathematicians need creativity to guess new axioms.

Once the axioms are chosen, there is no freedom to departe from the rigourous rules of logical derivations. But new axioms already wait for you to discover them, in the form of true but unprovable statements. Just another point of view

Eugene Wigner appears to be the author of that article. Creationist was probably a too strong word, and the remark he made was just brief.

I did some digging and I retract my original claimed that Wigner was a creationist.

http://www.talkorigins.org/faqs/edwards-v-aguillard/amicus1.html

He and 71 other Noble laureates signed in support of science in the Edwards vs. Aguillard trial in 1987.

Most importantly, this is the only unambiguous language.

Tell that to Kurt Gödel :)

Tell that to Kurt Gödel :)
What is your point ? I indeed mentionned Godel's work to argue in favor of genuine mathematical creativity intrinsic to the process itself, meaning that true unprovable statements are already there before you discover them.

Re: Godel. On Formally Undecidable Propositions

Most importantly, this is the only unambiguous language.
It's more than ambiguous, it's meaningless except in reference to itself until values are defined.

But you do not address the universal aspect of mathematics.
It's "universal" because of it's lack of a semantic quality. You can apply any value to it you wish both meaningful and meaningless to reality. And as I've already pointed out it's taken thousands of years for math to come far enough to be as "universal" as it is. In reference to the book I mentioned It's fairly recent (in the scheme of things) that the idea of zero and it's implications and rules surounding it were developed. It's fairly integral (so to speak) to math as we know it today and didn't even exist in the beginning of math.

What is your point ? I indeed mentionned Godel's work to argue in favor of genuine mathematical creativity intrinsic to the process itself, meaning that true unprovable statements are already there before you discover them.
And what does this mean? You argue in favour of genuine creativity intrinsic to the process, but they are discovering math not creating it?

So, what's it going to be? Is math real like a hammer or is it magic?

So, what's it going to be? Is math real like a hammer or is it magic?

Reality is the magic, especially the more you try to describe it fundamentally. Or the more fundamental you try to describe it...

Math isn't so easy in a philosophical atmosphere. You'd have to tell me your definition of real and your definition of math. They both tend to be arbitrary. At one level, there may be differences in the definitions based on our academic background (my own definition as a physics student may even differ from my definition as a practical person, and is probably a lot more shallow than any of my professor's definitions). On the other level, our more fundamental beliefs could heavily influence the definition of both (I'm a bit of a nihilist, but also an idealist. A statistician might call me agnostic).

It's not completely like a hammer. A hammer has a more specific definition. But, again, in philosophical atmosphere it wouldn't be too complicated too analogize them. They're both tools that do work, but math is growing and developing in a much more complicated manner than the hammer. It's applications probably vastly out number a hammer, too. The hammer strikes high on the tangibility meter, the math does not.

It's more than ambiguous, it's meaningless except in reference to itself until values are defined.
If I tell you "red is colder than blue" this is an ambiguous sentence with which you will probably disagree. If we define the temperature of a color from a blackbody, then the statement becomes well-defined and true. No matter what your representation or conception of a blackbody or temperature, if they are to make sens, they must agree with the mathematical statement.

To come back to the initial issue, I am not trying to convinve anybody, as I myself am not compelled by any agrument

Let me take another example to illustrate the Godel-Turing argument. Consider the Riemann hypothesis. If it is wrong, it will be possible to find a counter-example to it. But it can very well be true and unprovable. And mathematics proceeds by assuming it for decades now. If it turned out to be wrong, many theorems would vanish in thin air. And it is called "hypothesis" and not "conjecture" which is remarquable. One can not disagree that, in this case, mathematicians have taken this statement which was found likely but is still unproven. From Godel-Turing's point of view, the statement can be true but unprovable, and it has been discovered in this sens.

Reality is the magic,

Right on, man--at least, that's the way I see it. The one magic.

especially the more you try to describe it fundamentally. Or the more fundamental you try to describe it...

I don't think you mean to say there's more reality, the more you describe it.

Math isn't so easy in a philosophical atmosphere.

Good grief! I didn't know. I was searching the newly posted threads.

You'd have to tell me your definition of real and your definition of math. They both tend to be arbitrary. At one level, there may be differences in the definitions based on our academic background (my own definition as a physics student may even differ from my definition as a practical person, and is probably a lot more shallow than any of my professor's definitions). On the other level, our more fundamental beliefs could heavily influence the definition of both (I'm a bit of a nihilist, but also an idealist. A statistician might call me agnostic).

It's not completely like a hammer. A hammer has a more specific definition. But, again, in philosophical atmosphere it wouldn't be too complicated too analogize them. They're both tools that do work, but math is growing and developing in a much more complicated manner than the hammer. It's applications probably vastly out number a hammer, too.

If there were funding available to come up with a lot of wildly different kinds of hammers, intricately ornimented and detailed, where the maker of such hammer is not required to make a knowably useful hammer, there would be a large variety of hammers.

It's applications probably vastly out number a hammer, too. The hammer strikes high on the tangibility meter, the math does not.

I've been trying to provoke an argument from those who seem a bit worshipful of math, so here goes.

Math is a lot of marks on paper and in books, stored in electronic media, and most importantly, to tie all these pen scratches together, in our minds. It's not pre-existing and "out there" somewhere looking to be discovered. As a lot of ideas and pen marks, it's invented.

I don't think you mean to say there's more reality, the more you describe it.

I just mean it's more "magical" at the fundamental level. Another way to say this is that it's less intuitive.

If there were funding available to come up with a lot of wildly different kinds of hammers, intricately ornimented and detailed, where the maker of such hammer is not required to make a knowably useful hammer, there would be a large variety of hammers.

But the funding isn't available because the diversity of the hammer is much more limited than math. There are, in deed a large variety of hammers (from machine to hand held to slide-pull hammers) but mathematic is vastly more versatile. It's sort of the language of fundamental understanding (like atoms are the fundamental in terms of matter) so it ought to be versatile.

I've been trying to provoke an argument from those who seem a bit worshipful of math, so here goes.

Math is a lot of marks on paper and in books, stored in electronic media, and most importantly, to tie all these pen scratches together, in our minds. It's not pre-existing and "out there" somewhere looking to be discovered. As a lot of ideas and pen marks, it's invented.

This is the basis of my argument too. There was a user on here previously, Captain Quasar, who stood on the other side of the argument. The thread is around here somewhere (with the same theme as this one) and a lot of interesting side arguments came up between him and I.

So... one could make the argument that we're a product of nature and so math is a product of nature since it comes from us (a product of nature) but I don't think that's the question. I think the question is does math exist in the universe independent of us, and I don't think it does, I think it's just the only we've found to fit our most accurate, meaningful understanding to physical reality (which is perhaps the only reality).

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The brain is made in such a way that it quantifies and abstracts things from each other, and then in some mysterious way processes the information.
This is the foundation for 'everything' about the mind.

My little brain wants to take the vague notion of abstraction and make something more managable out of it. Would you call this abstraction: "A:B :: C:?" ?

I notice a relationship between A and B. Comparing A to C, I consider that C implys some candidate D.

I think we do more catagorizing and story telling than abstracting; categorizing being the process of grouping similar things. Categorizing seems near at the synaptic level, where weighted sums saturate to either of two choices; membership or nonmembership. This is the common attribute with digital logic, nevermind Von Neumann revesibility. But I want to know what it is you call abstraction.

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"Don't confuse the map with the territory."
Deep..!

Question : is mathematics discovered or invented?

Many inventions come from discoveries/accidents/refinement/improvements etc. And the methods that we have in mathematics didn't come from just one person, and didn't come overnight... it is like linux ... somebody started off with an idea and then it started to get built up/evolve ... like open source.

This is my understanding of mathematics:
We state a list of axoims, use deductive logic and the variety of theorems pop out.
This collection is then called a theory.

Now, in choosing the axioms, have we not fixed any and all theorems that are derived from them? Hence it is a discovery, since those theorems exist as a result of the axioms and formal logic that operates on those axioms, even though we don't yet know what they are.

Related question: Where did formal logic come from, and how is it valid? <- This may in fact be a meaningless question since being valid is a logical state.

I would just like to know to say to someone who asks why we bother with being logical at all.