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.
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.
[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)]
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, and John Baez and collaborators’ work on higher gauge theory etc.; apparently, there’s already enough material to write a 129-page paper on just the pre-history of categorical physics…
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.
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 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.
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).
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.
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.
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.
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.
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.