Is Mathematics invented or discovered?

This is more of a philosophical discussion. But since you posted it in Linguistics we should honor the spirit of your request and look at it from a linguistic perspective, rather than adding to the omphaloskeptic responses you've been getting so far from the philosophers.

I think the root of the problem here is in fact linguistic. What we call "mathematics" is not a single discipline. I'm not enough of a mathematician to pursue this point as far as it needs to go, but even I can divide "mathematics" into two broad and quite different categories of study.

First there is "arithmetic." The theories of arithmetic are derived from pure abstract reasoning and are not dependent on empirical observation, experimentation, testing, peer review, and the whole scientific method. One plus one equals two, and in fact that is the definition of "two," both in a mathematics class and a language class. Each of us has the ability to close his eyes and envision one object, then a second object being added to the image, and the result of that image now containing two objects. We do not need to observe and manipulate actual physical objects. This is an abstract truth.

We're not sure whether any other animal's brain has the capability of performing this abstract exercise, but ours is. The same holds true, to a greater or lesser extent, for the operation on larger numbers, as well as the other arithmetic operations such as subtraction and multiplication. Even the operations of algebra such as exponentiation and factorials fall into this category, since they are built upon the four basic operations. If x^2 - x = 6, then x = 3 or x = -2 and no observations or experiments are required to prove this. Even the operations of calculus fall into this category, since they are built up from infinite series which are built from algebraic operations.

But the other branch of mathematics (there are only two in my simple model) is geometry. Geometry was derived by measuring objects in the physical universe. The area of a circle is pi * the radius squared. Trigonometry is an offshoot of geometry, and solid geometry is an expansion on plane geometry. All of these disciplines were discovered, but unlike arithmetic they were not discovered from abstract reasoning, but from empirical observation.

Sure, now with more than two millennia of hindsight we can say that geometry is also an abstraction, that the area of a circle is always pi * r^2... or can we?

It turns out that what we discovered by empirical observation of the natural universe was only "Euclidean" geometry, named after its primary discoverer. This geometry ONLY applies to the natural universe in which we live. We have subsequently "invented" other types of geometry that apply to universes which do not exist. The two that I know of are Riemannian geometry and Lobachevskian geometry. The essence of the difference is that in Euclidean geometry, for any straight line on a plane there is one and only one "parallel" line that extends to infinity in both directions without ever intersecting it. On a Lobachevskian surface there are an infinite number of such lines, and on a Riemannian surface there are none at all: all lines eventually meet.

In a two-dimensional model Riemannian geometry can be illustrated on a sphere: two straight lines will always cross twice. A Lobechevskian surface is more complicated but it looks something like a saddle that extends to infinity. This seems like a sophomoric exercise until we contemplate that it's possible to hypothesize a three-dimensional Riemannian or Lobachevskian space which may not be so easy to imagine or draw an image of.

The point is: We have "invented" two geometries that have no relationship to the natural universe. (I'll leave it to the next mathematician who logs in to explain the use of these "non-Euclidean geometries," but I don't know whether they have any practical value.)

Therefore, part of mathematics--arithmetic--was discovered, whereas another part--geometry or at least some of geometry--was invented. I'll leave it to that same math whiz to tell us which other fields of mathematics were invented and which were discovered. I crept out of that academy with a headache when we got to partial differential equations, and I never really did understand set theory.

 

It wouldn't be "one of those other geometries" that I've read about, anyway. They are not compatible with Euclidean geometry. Perhaps they are local approximations in other parts of the natural universe. The geometry of which these are all local approximations would have to be something much more complex than any of them.

I don't know about that. So far Euclidean geometry has explained the increasingly large portion of the universe that we've been able to observe fairly well.

What troubles me is some of this wacky new cosmology. The outer edge of the visible universe is farther away than those stars could have traveled at lightspeed since the Big Bang. So, to explain this the cosmologists tell us that the empty space in between the stars is expanding. Uh, "space" is another word for "nothing," so it sounds to me like they're saying "nothing is expanding," which is a rather meaningless statement and hardly explains the problem. Or if it isn't empty, and it's full of photons and gravitons and other subatomic particles that are the plumbing infrastructure of the universe, what happens to those particles, that plumbing, when the "space" they occupy is stretched? If they retain their original size, then the space between them is expanding in a really complex way. But then, so is the space between the stars, if the stars themselves are not expanding proportionally.

I think this theory has a long way to go. Perhaps it requires a trans-Euclidean geometry to really make sense.

No, it's a scientific question. Real numbers measure distances and other physical attributes of the universe. Imaginary numbers (and wouldn't every scientist like to shoot Descartes for being so famous that his dismissive name for a concept he initially rejected stuck, so we have to constantly explain it) are intermediate values that appear in calculations whose end purpose is to predict those measurements.

In electrical circuit theory, for example, voltage, current, resistance, inductance, frequency, and every other dimension of a circuit is measured in real numbers. There's no such thing as a device with an impedance of 6+j ohms. (Electrical engineers use j for the square root of -1 since i already stands for current.) But the calculations for the way complex circuits work are full of j's, they just all resolve into real numbers by the time you get to the end of the calculation. They're just intermediate values with no analog in the natural universe and no analog is necessary.

It's like infinity. Infinity is a "number" we have to have because it's what we get when we divide something by zero, and it serves a purpose in many calculations, e.g. "infinite" series. But it doesn't exactly represent anything in the natural universe.

The reason nobody has actually shot Descartes for naming them "imaginary" (besides the fact that the dude is already dead) is that the term is not entirely inappropriate. He of course called them that because he didn't believe they would prove necessary in the mathematics of his future. But in a context like this it's useful to remember that "imaginary" numbers are not representations of any measurements in the natural universe.

We may wind up making wholesale revisions to relativity in order to explain the expanding universe, discovering an amazingly complicated relationship between gravity and electromagnetism, or finding that particle physics and uncertainty fall neatly in place when the Next Big Thing After String Theory comes along... but none of these steps forward in science will change the fact that the dimensions of the "real" universe are measured in "real" numbers.

Just look at the way complex numbers are graphed: there's a real axis and an imaginary access. They are not dimensions because they have two dimensions by definition!

Remember that mathematics is a tool of science, not a science itself, so all the rules of science don't apply to it. Mathematical theories are derived from pure abstractions rather than empirical observation. Therefore unlike scientific theories they are proven true. No subsequent observation in the natural universe will prove them false.

 

Intelligence is hardly remarkable given life, and I just don't find life all that remarkable given chemistry and physics. Sure, we don't know how abiogenesis works but that doesn't mean we won't. A hundred years ago we didn't know how earthquakes worked. With all the solar systems that are likely to exist in a universe with one sextillion stars, intelligent life was bound to happen. And the intelligent creatures were bound to start wondering about it. That's us.

Okay, but no one has developed the theories of this all-inclusive geometry of which what we see is a microcosm. I stated that the universe is not going to turn out to be a Riemannian space or a Lobachevskian space--the only two well-developed non-Euclidean geometries I know about--because a Euclidean space could not exist inside either one of them.

Sure, we once thought the earth's surface was a Euclidean plane and it turns out to be a Riemannian "plane" because we were only seeing a microcosm of it, but as soon as we could see a decent sized microcosm we began calculating its diameter. We already can see a much larger portion of our universe than the Ancients could see of theirs when they made that discovery. By that time they had already mastered the basic concepts of solid geometry so the earth as a solid sphere and our portion of it as a spherical surface were easily accommodated by Euclid's theories.

We're scientists here and so far the assertion that the universe is a Euclidean space has stood up so well to testing and peer review that it's pretty much a canonical theory by now rather than a hypothesis.

As for the universe making sense, it continues to make more sense as science marches on. For every canonical theory that is occasionally falsified, ten thousand newer and stronger ones arise that explain far more of the universe's mysteries.

Perhaps the universe is not a Euclidean space, but I don't see the evidence for that hypothesis being strong enough to take seriously right now.

 

The universe, by definition, includes everything that there is. If there's something more, something we missed, then that doesn't mean there's another universe; it just means that we were wrong about the extent of this one. It's like the Highlander slogan: "There can be only one."

You can talk about an "alternative universe" because that's only fiction. You're still talking about there being just one, except that it's a different one than the one that exists.