Is mathematics the language of the Universe?

[Whack-a-Mole]

I this thread (http://boards.straightdope.com/sdmb/showthread.php?t=621064) on Evolution I'm afraid I started a hijack on what is a "fact" by suggesting 2+2=4 is the only "true" facts we can count on (which is to say math...not just that equation).

Derelth responded by saying:

If you let me define the axiom system, I can make 2+2 equal anything you want. I can make it equal 0 quite easily, in fact. I'm not saying the resulting axiom system would be interesting or useful to you or anyone else, but I can do it, and the equality would be perfectly true within that axiom system.

My point is that axiom systems allow us to derive truths, but they are only true within the axiom system from which they are derived; they may be false or meaningless in other axiom systems, and are only relevant to the outside world as descriptions or models of physical relationships and processes. However, something that is true within an axiom system is always and absolutely true within that axiom system. That kind of absoluteness is not available to us in the physical world.

The conversation moves from there and most recently John Mace and I had this exchange:

Always seemed to me that math was the language of the universe.

When prying into the fundamental reality of things the bottom line for doing so is math.

I have been told by physicists (one of my brothers is a physicist) that at the end of the day words fail them when explaining certain concepts and if I really want to understand what is happening I need to understand the math.

Hell, I think that gold record we sent on Voyager to "talk" to aliens if they found it used math to communicate. Obviously an alien will not know English but presumably math is universal and they'd understand that.

John Mace's response to that:

Math is the language we use to describe the universe. The universe doesn't "care" about math. It's a human construct.


I studied physics, and I agree. But does that say anything fundamental about the universe itself, or our ability to understand it?


Only in the sense that sentient beings should understand this. Again, the universe just is. It doesn't "need" math.

Now, I get that the Universe does not "care" about math. I understand that we use human words and concepts to try to describe the universe but they are inexact.

Nevertheless it seems, to me, that the language of the universe is math.

True a moon does not "understand" math and "agree" to follow a certain orbit. That said the orbit the moon follows is thoroughly described by math.

That gold record on Voyager uses math as a universal language because any reasonably intelligent alien will know that 2+2=4

Perhaps we are talking past each other here. I suspect we agree (in the other thread) more than we disagree on this point. Still, figured it was something for the Teeming Millions to have a go at.

[John Mace]

I'm not sure I can say more than I already said on the subject other than to just repeat myself, but let's give it a go.

The question is: did the universe "invent" math in order to exist (pardon the anthropomorphizing), or did humans invent math in order to understand the universe? I'm not sure how you would prove this, but it seems intuitively obvious to me that it's the latter.

[Whack-a-Mole]

The question is: did the universe "invent" math in order to exist (pardon the anthropomorphizing), or did humans invent math in order to understand the universe? I'm not sure how you would prove this, but it seems intuitively obvious to me that it's the latter.

I agree the universe did not "invent" math but I am proposing that math is a fundamental aspect of the universe.

Math is a language and a human invention. The universe does not know what "two" is.

That said in a universe with two particles in it then there is a reality to the numbers. You can use whatever words you want (two or dos or два).

A planet orbits in a very predictable way. We can describe that with math very precisely (in theory to any precision you want till you bump into Heisenberg).

[Buck Godot]

I think I'm going to have to go with the others to say that the universe acts in concordance with the mathematical language we made for ourselves, but is independent of it. The moon orbits in a ellipsoid because its momentum is conserved and it is acting under a force of gravity that is proportional to the product of the masses and the square of the distance. This is because there are certain additive forces and fields at work*. We can use math to manipulate symbols that correspond calculate what those additive forces and fields are, and so predict the future orbit of the moon and other satellites we put up. However all of this symbol manipulation is just an accounting system designed by man so that it matches what we see in the universe.

As another example, if I have 42 8oz cans of tomato sauce and a 5 gallon drum, one might want to know if I have enough sauce to fill the drum. This can be done entirely without math by opening all the cans and trying to fill the drum. That is the physical fact of the situation. But instead I use some math to find that 48x8oz cans = 384oz= 3 gal and so I know the drum won't be filled. I could reduce this problem to a symbol manipulation problem because I designed the symbolic the laws of multiplication to be such that when 48x8oz < 5 gal, it will be the case that 48 8oz cans will fit into a 5 gallon drum.

There are many branches of mathematics which so far have no relevance to the universe, and similarly there are many things going on in the universe that as yet we have no mathematics to model. (ie quantum gravity)


*I'm no physicist so I'm leaving this intentionally ambiguous so as to not get in trouble

[RTFirefly]

The question is: did the universe "invent" math in order to exist (pardon the anthropomorphizing), or did humans invent math in order to understand the universe? I'm not sure how you would prove this, but it seems intuitively obvious to me that it's the latter.Depends on what you mean. To the extent that any sentient race, in any physical environment anywhere in this Universe, would come up with the same mathematics, I think you'd have to regard that math as inherent in the nature of things in this universe, rather than an invention.

For instance, having things to count, and counting them, gets you the counting numbers. Subtraction gets you zero and the negative numbers; division gets you the rationals; roots get you the real and complex numbers. If a sentient race has occasion to count, subtract, divide, and take roots, they're going to come up with all the sets of numbers that are the basis of contemporary mathematics.

[bump]

Math's a way to represent what we see in the physical world, a language if you will.

The ratio of a circle's circumference to it's diameter is constant, whether it's denoted as 3.14159, 22/7, 3.11037 or 3.243F6. (pi, if you haven't figured that out)


What people are trying to say when they say that "math's the language of the universe" is that instead of saying that an alien would look at 3.14159 and recognize pi, that there are abundant examples of mathematical "rosetta stones" (pi, etc...) out there, so that we and any other species could relatively easily reconcile our respective mathematical systems.

[John Mace]

Depends on what you mean. To the extent that any sentient race, in any physical environment anywhere in this Universe, would come up with the same mathematics, I think you'd have to regard that math as inherent in the nature of things in this universe, rather than an invention.

For instance, having things to count, and counting them, gets you the counting numbers. Subtraction gets you zero and the negative numbers; division gets you the rationals; roots get you the real and complex numbers. If a sentient race has occasion to count, subtract, divide, and take roots, they're going to come up with all the sets of numbers that are the basis of contemporary mathematics.
I agree that math is "universal" in the sense that sentient beings will come up with the same rules. I said that in the other thread.

I'm just not sure what the OP means when he says that the language of the universe is math. The universe doesn't have a language.

[Whack-a-Mole]

I'm just not sure what the OP means when he says that the language of the universe is math. The universe doesn't have a language.

Language is: One, two, three or uno, dos, tres.

As mentioned if you have two particles in a universe then you have two particles. They behave in accordance with mathematical rules.

We are not inventing math. We are discovering math.

[John Mace]

Language is: One, two, three or uno, dos, tres.

As mentioned if you have two particles in a universe then you have two particles. They behave in accordance with mathematical rules.

We are not inventing math. We are discovering math.

I think you are making a mistake by confusing math with reality. The particles, as far as we know, obey physical laws that we represent using math. But the physical laws are not math. Not to mention that there isn't any such thing as a particle in the first place. That, too, is a human construct.

[Odesio]

Is math something that is invented or discovered? I ask because, in philosophy at least, you don't invent anything you just discover it.

[Whack-a-Mole]

I think you are making a mistake by confusing math with reality. The particles, as far as we know, obey physical laws that we represent using math. But the physical laws are not math. Not to mention that there isn't any such thing as a particle in the first place. That, too, is a human construct.

We are stuck using language to describe things.

In English we say two plus two equals four.

Those words may be incomprehensible to someone else but no matter where you go in this universe the underlying premise is true.

Hence why we used math on that gold record on Voyager. As said above it is a Rosetta Stone (like that analogy). No matter where you go everyone will understand it (assuming some moderate intelligence).

Why? Because it is universal.

[Whack-a-Mole]

Is math something that is invented or discovered? I ask because, in philosophy at least, you don't invent anything you just discover it.

Maybe both.

2+2=4 is discovered.

As mentioned in the OP there apparently is modulo math which makes 2+2=0.

I assume there are real world uses for that kind of math but I do not think that kind of math describes the universe we live in.'

[John Mace]

First of all, we assume it's universal. We don't really know.

We also assume that diagrams are universal, since they are also used on that plaque that went with Voyager. There was some sort of symbol that was supposed to represent a Hydrogen atom. And there was a diagram of our solar system.

[Asympotically fat]

Eugene Wigner, a highly influential figure in quantum mechanics (though at the same time he did have a few controversial views, i.e. he believed conciousness played an importnat role in quantum physics) wrote the famous essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences (http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html).

As I said on the other thread, maths is about abstraction. If yous tart requiring maths to have some sort of roots planted firmly in the empircial then your in danger of losing the full power of abstraction.

I would say mathematics is the language of physics. ratehr than of the universe.

[Asympotically fat]

There are many branches of mathematics which so far have no relevance to the universe, and similarly there are many things going on in the universe that as yet we have no mathematics to model. (ie quantum gravity)


*I'm no physicist so I'm leaving this intentionally ambiguous so as to not get in trouble

Actually I sort of disagree, nearly all areas of maths find practical application. Catergory theory which is a an area of maths about catergorizing abstract mathematical structures such as sets, fields, vector spaces, topological spaces, etc, etc was the first thing that sprangt o mind for me when thinking about areas of maths that are very abstract and don't have applications that leap out at you. Howvwer it does seem from a quick google that compuetr programmers have used it for practical purposes.

The mathematics do actually exist to model quantum gravity (though it's empircally untested for reasons that I'm just about to mention), you can derive the maths of quantum gravity using canocial quantization (i.e. the standard way of creating a quantum field theory from a classical field theory), the problem is that it has an in-built resistance to peturbative solutions (a method for obtainign approximate solutions) which makes it totally impractical as you can't even obtain approximate solutions (by normal methods at least).

[John Mace]

It can take centuries before a branch of mathematics finds an application in the real world. Sure, Newton invented calculus to solve a real-world problem, but that may be the exception rather than the rule.

[Boyo Jim]

I believe that the OP's question cannot yet be answered. We know there are real physical phenomena that cannot presently be described by mathematics. We have some faith that this will not always be the case, but we can't say this for certain.

If we ultimately find the math to describe all physical phenomena, then I think the OP's conjecture can be said to be correct for all practical purposes. Even if it turns out that mathematics has uses beyond the description of reality, that doesn't preclude it from being the "language" of it.

But if we fail to find the math to describe all physical phenomena, then the OP's question cannot be answered yes or no. I'm not sure that one could prove a physical phenomenon to be not reducible to math, so we may never know for certain whether this is the case or whether we have simply failed to discover that correct math to describe it.

[Superfluous Parentheses]

I don't really know what the OP means by "[math is] the language of the universe". Mathematics itself is probably "universal", but that doesn't seem to mean the same thing at all.

AFAIK, the truth is, we don't really know what the relationship between logical and mathematical systems and "physical reality" is, or even if there really is a relationship other than that we can use one to make mental models of the other. We don't even really know what exactly the relationship between either of those and our mental models is. These are old questions, going back at least to Plato.

Roger Penrose's book "The Road to Reality" touches on these issues in the introduction, but it mostly makes it pretty clear that our mathematical descriptions of reality are if anything much more complicated than you'd expect (interesting, but complicated).

[Smurfie]

This is another face of the old Math debate...Do Mathematicians 'discover' new things or 'invent' new things.

[Mijin]

I don't think the Universe is mathematical any more than it is based on the English language.

How many points are there along a 1 metre line; I mean in the real, physical universe?
1 / planck length? Infinity? Then how come I can do useful mathematics, and get correct answers, with a much lower granularity?

The answer is because mathematics is just a tool for deriving non-obvious facts from obvious facts. It's a means of manipulating information in a way we would find difficult without the intermediate step of representing it symbolically.

That said, the fact that maths can be applied to the universe tells us something: that the universe is self-consistent and behaves in the same way as far as we can currently see.

[njtt]

Math is an amazingly useful tool in physics (although even there it is not the whole story: you cannot infer physical knowledge just from mathematics). It is a lot less useful (I am not saying it is no use at all) when we are trying to understand other aspects of reality, for instance, history. Yet history happens in the universe too. Math maybe fairly plausibly be called the language of physics, but it is not teh language of history. Other aspects of reality, such as biology, fall somewhere in between: math has considerable utility in biology, but it is very far from being its "language".

(I am aware that some people may think that biology and history, and everything else, ought to be reducible, in principle, to physics, to the motions of particles or the evolution of waveforms, or whatever. However, this is an unproven, and almost certainly unprovable metaphysical hypothesis, not a fact. Even if it is true, we can't even begin to make the reduction)

[njtt]

This is another face of the old Math debate...Do Mathematicians 'discover' new things or 'invent' new things.

Actually I don't think it is. I can agree that mathematicians discover new things without conceding that it is "the language of the universe."

Mijin makes a very good point, I think. I wish I'd said that.

[Whack-a-Mole]

I don't think the Universe is mathematical any more than it is based on the English language.

How many points are there along a 1 metre line; I mean in the real, physical universe?
1 / planck length? Infinity? Then how come I can do useful mathematics, and get correct answers, with a much lower granularity?

The answer is because mathematics is just a tool for deriving non-obvious facts from obvious facts. It's a means of manipulating information in a way we would find difficult without the intermediate step of representing it symbolically.

That said, the fact that maths can be applied to the universe tells us something: that the universe is self-consistent and behaves in the same way as far as we can currently see.

Aren't your issues really just human inability to do the math?

I mean, in theory, if we knew the starting conditions with perfect precision (ignoring Heisenberg for the moment) isn't every future event predictable?

Heisenberg certainly throws a spanner in those works but hypothetically, if you had the calculational power, you could do that.

[Indistinguishable]

Heisenberg certainly throws a spanner in those works but hypothetically, if you had the calculational power, you could do that.
Lots of things can be reasoned about. Anything which follows rules can have its rules analyzed. For example, chess. If I had the calculational power, given the option to choose whether to be black or white, I could ensure that I never lost a game of chess.

Does that mean math is the language of chess? (Genuine question here; I have no idea what it means to say that "X is the language of Y"). If that's the sort of thing you're talking about, then math is the language of everything. [Anything you can reason about you can, uh, reason about. Anything with rules can have its rules analyzed. Anything you can talk about, you can talk about abstractly. That's all math is...]

[Whack-a-Mole]

Does that mean math is the language of chess? (Genuine question here; I have no idea what it means to say that "X is the language of Y"). If that's the sort of thing you're talking about, then math is the language of everything.


Reminds me of this XKCD comic (http://xkcd.com/435/).

In theory I think everything is reducible to math.

Gets us into free will and all that jazz though.

Perhaps Heisenberg's theory allows for genuine unpredictability. Literally God rolling dice with the universe.

[Indistinguishable]

What does unpredictability have to do with whether math is "the language of the universe"?

If one analyzes situations such as "There is a die which unpredictably produces a value from 1 to 6; based on the value of the die, [something something]" and all their consequences, is that not just as much math as the analysis of deterministic situations? A non-deterministic function is just as much a mathematical concept as a deterministic function.

[Half Man Half Wit]

I think the question resolves itself if one asks why people wonder about the unreasonable effectiveness of mathematics, but not about the unreasonable effectiveness of computers -- since with computers, it is well known that any one (universal) computer can emulate any other (see Turing completeness (http://en.wikipedia.org/wiki/Computational_universality)), so if reality is not 'more complex' than any computer, it's trivial that computers can be used to simulate every aspect of the physical world.

But actually, I think it's not different with mathematics. Certainly, math has similar capacities to 'emulate' other systems as computers do -- it must, since computers are mathematically describable. A famous example of this is Gödel numbering: you can mirror every statement you can make in a formal mathematical system within number theory, as simple (well, actually kind of complicated) arithmetical statements. This establishes the universality of mathematics.

So, the reason mathematics can be used to model the physical universe is the same reason that computers can be used thus -- universal systems are able to emulate, i.e. mirror the behaviour of, all other universal systems, or systems not more complex than that.

Of course, reality might turn out to be 'more complex', i.e. non-computable, though I personally doubt it; but similarly, it might turn out that there are aspects of reality that can't be modelled with mathematics, that we just haven't discovered yet (though if we ever could discover such aspects, our deliberate thinking and modes of communication appearing quite computational in make-up, is an interesting question in itself).

For more on this point of view, see here (http://ratioc1nat0r.blogspot.com/2011/07/universal-universe-part-iii-answer-to.html).

[Derleth]

We must have cross-posted. See my last post in the other thread.

[Hyperelastic]

There is a school of thought (http://en.wikipedia.org/wiki/Digital_physics) that the universe is just a big computer, solving the equations of mathematical physics. It's a fringe view and not one I share.

[Jragon]

Maybe both.

2+2=4 is discovered.

As mentioned in the OP there apparently is modulo math which makes 2+2=0.

I assume there are real world uses for that kind of math but I do not think that kind of math describes the universe we live in.'

Modulo is incredibly useful in Computer Science, where you deal with constraining large amounts of input to finite spaces. To use a VERY simple example, if you have a table with 8 slots, and want to store the number "27", you do 27 mod 8, getting 3 (the remainder of 27/8), of course there are countless problems with this simple example, but that's a simple example of how its useful -- for enforcing bounds constraint.

That said, as far as "the universe" goes, I hear modular arithmetic is very important in chemistry, but I don't know anything as to HOW.

[XT]

Mathematics is the language that humans use to describe the universe. It's a human construct that allows us to describe how we think the universe works and to make predictions about the universe.

-XT

[Indistinguishable]

2+2=4 is discovered.

As mentioned in the OP there apparently is modulo math which makes 2+2=0.

I assume there are real world uses for that kind of math but I do not think that kind of math describes the universe we live in.'
Of course it does. It just describes different aspects of that universe... [Just like "1 + x is guaranteed to be >= 1" is true on some useful interpretations of what addition is ("If you add one item to a pile of items, you'll definitely have at least one item overall"), while "1 + x is sometimes equal to 0" is true on some other useful interpretations of the language of addition ("If you go one mile to the right, and then follow up with a random movement, you might well end up 0 miles from where you started"]

For example, suppose you're keeping track of your car's orientation by counting the number of left turns you make. 2 + 2 left turns is just as good as 0 left turns. In this context, 2 + 2 = 0 is an accurate and useful description of the universe. You wouldn't want a driver who refused to accept 2 + 2 = 0 in this context.

[Indistinguishable]

It's nothing like that 2 + 2 = 4 is "discovered" and 2 + 2 = 0 is "invented", if that's what you're proposing. That would be a ridiculous distinction... They're both equally rules in some abstract games we've chosen to make up and spend time playing. Our interest in these particular games, of course, is often motivated by the desire to model patterns we've noticed in the universe or in other abstract games.

[Jragon]

I don't think it's fair to say math is the language of the universe any more than you can any other language is. Any given language is a more or less powerful way to describe the universe. Just because I'm not quite as PRECISE when I say "The moon orbits around the Earth" doesn't mean I'm not describing the universe the same way math does. I can describe pretty much any physical property I want, it just may not be as succinct as math at describing the particulars. I can probably describe the exact meaning of a given input to a complicated function, even though depending on the language it may take three or four pages to do so.

Math was invented, it just so happens to be a more powerful tool for writing down observations and predicting future outcomes than many other human languages in many cases.


Math's a way to represent what we see in the physical world, a language if you will.

The ratio of a circle's circumference to it's diameter is constant, whether it's denoted as 3.14159, 22/7, 3.11037 or 3.243F6. (pi, if you haven't figured that out)


What people are trying to say when they say that "math's the language of the universe" is that instead of saying that an alien would look at 3.14159 and recognize pi, that there are abundant examples of mathematical "rosetta stones" (pi, etc...) out there, so that we and any other species could relatively easily reconcile our respective mathematical systems.

While this may be true, this is more of a sign that we happened to study the same things. I'm sure they have plain Glorkinese for "running" too, which almost all languages share a word for. It's just the chances of them writing a document describing running similar enough to ours to decode are much smaller than the chances of them computing the area of a circle in a similar way to how we do it.

[Indistinguishable]

It's funny that π = 3.14... is ubiquitously held up as an example of the one universal mathematical constant that every sophisticated alien culture can be expected to have canonized as profoundly important, since any alien mathematical culture is more likely to have hit upon τ = 6.28... as the more natural, fundamentally significant "circle constant". Our primary veneration of π (the number of radians in a half revolution) is a bizarre historical anomaly, rather like making a big hullabaloo over sqrt(e) or 1 + the Euler-Mascheroni constant or some such thing. Sending it out into space is begging to be mocked (though if we do it in binary, we can hope the aliens don't quite notice the factor of 2...).

[Measure for Measure]

This is another face of the old Math debate...Do Mathematicians 'discover' new things or 'invent' new things. Precisely. The question is: did the universe "invent" math in order to exist (pardon the anthropomorphizing), or did humans invent math in order to understand the universe? I'm not sure how you would prove this, but it seems intuitively obvious to me that it's the latter. Intuition says "Invention". But how do we account for fractals such as the Mandelbrot set (http://en.wikipedia.org/wiki/Mandelbrot_set)? The set continues on into infinity, so no human can hold it in his head at once. … no one, not even Benoit Mandelbrot himself [...] had any real preconception of the set’s extraordinary richness. The Mandelbrot set was certainly no invention of any human mind. The set is just objectively there in the mathematics itself. If it has meaning to assign an actual existence to the Mandelbrot set, then that existence is not within our mind, for no one can fully comprehend the set’s endless variety and unlimited complication. An alien civilization would come up with the same Mandelbrot set that we do, yet they probably wouldn't create the plays of Shakespeare. [1] So the latter was created and the former discovered.

Unfortunately, I find the idea of a discovered mathematics to be imponderable as well. If math is discovered, then it has an existence independent of the human mind. Math has no mass: is it measurable? I find this topic rather confusing. Penrose tentatively posits a Platonic world of mathematics, which is a reasonable model though it's metaphysically unsettling. Reverend Berkeley (http://en.wikipedia.org/wiki/George_Berkeley) would have probably said that the Mandelbrot set is situated in the mind of God.


[1] Idea lifted from here: http://blog.hvidtfeldts.net/index.php/2010/04/the-reality-of-fractals/

[Jragon]

It's funny that π = 3.14... is ubiquitously held up as an example of the one universal mathematical constant that every sophisticated alien culture can be expected to have canonized as profoundly important, since any alien mathematical culture is more likely to have hit upon τ = 6.28... as the more natural, fundamentally significant "circle constant". Our primary veneration of π (the number of radians in a half revolution) is a bizarre historical anomaly, rather like making a big hullabaloo over sqrt(e) or 1 + the Euler-Mascheroni constant or some such thing. Sending it out into space is begging to be mocked (though if we do it in binary, we can hope the aliens don't quite notice the factor of 2...).

Nah, the universe will be filled with quirks, specifically built around the circle constant, just to confound everyone. One alien with have found pi-squared, yet another will have 2*tau, another still will SOMEHOW have their base circle constant be the pitau. One day we'll eventually figure it all out and have a good laugh... which the aliens will misinterpret for a battle cry and throw us in the salt mines for threatening their peaceful existence.

Edit:

Even if we can't hold the entirety of the Mandlebrot set in our minds at once, that doesn't really matter, we can still hold the rules for constructing it. In language terms, we basically just invented a really witty turn of phrase for the math world that happens to be a good comeback to damn near everything.

[Indistinguishable]

An alien civilization would come up with the same Mandelbrot set that we do, yet they probably wouldn't create the plays of Shakespeare.
How about, say, tic-tac-toe? The rules are so simple and semi-natural that I can well imagine large numbers of alien civilizations independently coming up with it... Yet, in each case (including our own), I am also perfectly happy to say it was "invented" or "made up"* by whoever, well, came up with it. I don't feel that we should deny that person X invented Y for reasons Z just because some other person also independently invented essentially the same thing for similar reasons.

(*: While also, at the same time, I would not be surprised to discovered the existence of phenomena in the universe (even apart from the phenomenon of people playing tic-tac-toe) which were usefully modelled in some fashion or another by the abstract system of tic-tac-toe. Such that someone studying such phenomena might be motivated to codify and analyze tic-tac-toe even if it wasn't a game which previously existed among their culture)

[Indistinguishable]

I'm also happy to describe ideas as both discovered and invented, I suppose. To invent the telephone is to discover a way of transmitting sound over long distances. To invent the wheel is to discover that a rolling, round object experiences mitigated friction. One can use either piece of language to describe the same accomplishment.

[JKellyMap]

I'm also happy to describe ideas as both discovered and invented, I suppose. To invent the telephone is to discover a way of transmitting sound over long distances. To invent the wheel is to discover that a rolling, round object experiences mitigated friction. One can use either piece of language to describe the same accomplishment.

Good point. Etymologically, the first means to "remove what was hiding something", while the second means "to come upon something". Neither, etymologically speaking, implies pure creative genius on the part of the finder.

[Buck Godot]

A Catergory theory which is a an area of maths about catergorizing abstract mathematical structures such as sets, fields, vector spaces, topological spaces, etc, etc was the first thing that sprangt o mind for me when thinking about areas of maths that are very abstract and don't have applications that leap out at you. Howvwer it does seem from a quick google that compuetr programmers have used it for practical purposes.

However computer science isn't physics. I would argue that computer science is just another branch of mathematics. One that can be very useful in terms of information manipulation, but not one that describes the universe outside of the symbolic constructs of humanity.

[Indistinguishable]

For what it's worth, lots of people use category theory for practical purposes, just without knowing that they are doing category theory (just as many people frequently carry out calculations in abelian group theory without having any idea what abelian groups are or the full multitude of contexts in which their calculations can be applied).

A "category" is just any system of nodes, routes between nodes, and equivalences between routes (between the same nodes). If you've ever asked yourself "How can I get from here to there? Is this way just as good as that way?", you were thinking about a category.

So there are plenty of things in the universe that category theory describes well. But, yes, the reason it's able to describe them isn't anything more mystical than that categories are a very natural, ubiquitous pattern of information organization.

[Indistinguishable]

[The point I'm trying to make in that semi-hijack is that, even though category theory is so often presented in an incredibly esoteric way (with a focus on such obscure examples as the category of homomorphisms between fields or the category of continuous functions between topological spaces), such that one gets the picture its applications are sparse and quite removed from ordinary life, that's not really the case at all. Category theory is actually a very basic idea that totally does have applications that leap out at you. It just also has all these esoteric applications as well (as it happens, historically, its name and explicit study arose from these esoteric applications. But that's just history)]

[Half Man Half Wit]

Just parenthetically, category theory is actually somewhat of an up-and-coming subject in physics, with people like Bob Coecke using it in the foundations of quantum mechanics (http://arxiv.org/abs/0808.1023), and John Baez and collaborators' work on higher gauge theory (http://arxiv.org/abs/math/0511710) etc.; apparently, there's already enough material to write a 129-page paper (http://arxiv.org/abs/0908.2469) on just the pre-history of categorical physics...

[Boyo Jim]

Very interesting Nova program tonight on fractal geometry. It ended with a statement relevant to this thread by a scientist whose name I unfortunately didn't catch. Here is an inexact quote, "Fractals have given us a much larger vocabulary with which to read the book of nature."

I like the metaphors of nature as a book to be read, and mathematics as the language of the writing. I think it addresses the objections many here have been raising -- that nature exists and would exist whether or not we humans had the math to describe it.

[Hellestal]

I'm a little surprised at the... hostility? reluctance? toward Platonism.

I don't have any strong opinions one way or another, but on those rare occasions when I'm feeling ontological, I find myself slipping into Platonic descriptions of the universe out of sheer intuitiveness and convenience. And I think Max Tegmark's Mathematical Universe Hypothesis (http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis) is nifty, even though I can't claim to properly understand it.



Also, I joined the SDMB in 2002. I have now reached post number 1000, nine years later. The stat sheet tells me this is around 0.30 posts a day. Nice to see the ole odometer turn over. At the same pace, it'll only take me another 81 years to see 10000 posts.

[Whack-a-Mole]

How is 2+2=4 not universal?

Nevermind the words which will differ. If I have two particles and you give me two more we have four particles.

This is unavoidable no matter where you live in our universe.

Again, there is a reason (as mentioned above) that math is a sort of universal Rosetta Stone. Any reasonably advanced alien culture will understand that 2+2=4.

As such why can't one say this is a feature of the universe? It is true everywhere.

Granted math, as we know it, remains imprecise. Pi will never get us a perfect circle. We can come arbitrarily close to one but never achieve one. But isn't that also a fundamental feature of the universe? We can calculate Pi to insane levels of precision but the Universe, as Heisenberg noted, places hard limits on us drawing an ever more precise circle. There we leave reality for theory (as in we aught to be able to draw a more precise circle but reality limits us).

[Boyo Jim]

I don't think the concepts of pi and out inability to calculate the circumference of a perfect circle is relevant -- the perfect circle is itself a mathematical concept, Does it exist in physical reality?

But whether or not it does, it brings to mind another objection to the OP -- math is not a perfect expression of nature, it is an approximation, or a simulation of it.

Does this mean math cannot be the actual expression (or 'language' if the OP insists) of nature, or just that we have not yet developed (or discovered) the math that allows a perfect expression? I dunno.

[Mijin]

How is 2+2=4 not universal?

Nevermind the words which will differ. If I have two particles and you give me two more we have four particles.

This is unavoidable no matter where you live in our universe.

Again, there is a reason (as mentioned above) that math is a sort of universal Rosetta Stone. Any reasonably advanced alien culture will understand that 2+2=4.

As such why can't one say this is a feature of the universe? It is true everywhere.


Consider the logic of inference:

All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.

This is a verbal language being used to derive potentially new information (to us).
And we find that such reasoning works, everywhere, because it's just a rearranging of abstract information.

Now you might say that inference is part of mathematics too; it's formal logic. But then we see why mathematics appears special; because any useful or non-trivial representation of information acquires the mathematics tag.

[Half Man Half Wit]

How is 2+2=4 not universal?
Well, it's certainly universal in the sense that there exists a set of axioms from which it follows unequivocally, no matter the conditions under which it is derived. But there are aspects of reality to which it doesn't apply -- 'addition' of fluids might be one example: add two fluids and two other fluids, and what you get isn't four fluids, but rather, one mixed fluid. Indeed, the argument has been made (though I forget by whom) that it's entirely conceivable that, say, a sole jellyfish alone in a giant ocean would never come up with the concept of integers, but rather, use concepts more suited to fluid dynamics, and nevertheless come to a description of his world every bit as accurate as ours. If the jellyfish thus has no need in his description of the world of '2 + 2 = 4' (or an equivalent statement), and nevertheless is able to create a picture that pertains to the relevant facts as well as ours does, in what way is '2 + 2 = 4' universal? To the jellyfish, it is not a fact that's either part of the world, nor part of the description of it -- he can do entirely without.

[Naxos]

...
Nevertheless it seems, to me, that the language of the universe is math.

Not exactly, even though defining the term "language" on a universal scale would be a separate unanswered question on its own.

Mathematics is the ability of the human brain to be aware, observe and find patterns in nature, or the universe.

Math is an abstract mental function that depends on symbols and the ability of an "intelligence" (another undefined term) to use those symbols as placeholders for natural phenomena.

2 + 2 = 4 is actually meaningless on it's own. It doesn't mean anything. But it helps humans figure out and even predict some patterns we have been observing around us.


That gold record on Voyager uses math as a universal language because any reasonably intelligent alien will know that 2+2=4

Not true. Human algebra may or may not have common axioms with the "language" invented by other alien intelligence. We already know that 2+2=4 is a subjective statement based on some axioms we have to assume as true.

Math can deal with this issue too in a satisfactory and comprehensive manner.