Mathematics - Invented or Discovered

[TeV]

But what about all the abstract math that never gets applied to physics? You can assert that it all will be ultimately relevant somehow (I have heard mathematicians do this), but that is faith in things unseen. ..

IMHO,there are SOME differences between abstract math that never gets applied to physics and abstract maths that gets.
My impression:simple reason might be that nature uses the most simple wayouts,paths and mathrules in the base of its' modus operandi,while mathematician sometimes gets astray creating his own rules.Some of these rules might not be too efficient in the basis for the big machine of the universe and Nature rejects them.Example:transfite numbers are rejected by Nature in advance.
That's the freedom of math.Physicist seeks and explores laws of Nature while mathematician creates his *own* rules of game.Note the difference between used words :laws and rules.
Interestingly and quite obvious :mathematicians are also part of nature,beings made of flesh ,water and bones,bulks of matter that are organized in functional system ,and evolutionary quite efficient from the standpoint of mother Nature.Therefore...

 

[matt grime]

"transfinite numbers are rejected by nature in advance"
what does that mean? what does it mean to be accepted, and why are transfinite numbers not accepted. admittedly there is not an infinite number of anyone object in the universe, but even so, what's that got to to with it?

 

[p-brane]

As you admit in this post plane geometry is only an attempt to model that which we see locally. That does not support the idea that it was discovered, and in fact surely it demonstrates that we invented it? The fact that we can take the parallel postulate as true or false without contradicting the other axioms also backs up this assertion.

Here's where definitions of "discovery" and "invention" may come in handy.

When X-rays were discovered they were discovered, not invented. The various uses of X-rays was invented but not the X-ray itself.

When it was discovered that 1+1 objects equaled a group of 2 objects this discovery became one of the basises for several inventions in the realm of mathematical equations.

The difference between discovering X-rays and discovering a pattern in grouped objects is that the objects and their "mathmatical" relationships are judged to be mathmatical by our interpretation, from our perspective. The X-rays are "X-rays" regardless of what we think of that particular electromagnetic spectrum.

In a way it is beginning to look as though math is purely an invention, manifest of the imagination of over-cerebral humans, much like any other language. Thank you.

 

[TeV]

"transfinite numbers are rejected by nature in advance"
what does that mean? what does it mean to be accepted, and why are transfinite numbers not accepted. admittedly there is not an infinite number of anyone object in the universe

There isn't infinite number of any object in the universe I agree,but there's potential infinity in EM and gravitational force reach for instance (according to currently accepted models of these force-field interactions).Also,this is just a possibility and interpretations depend on model of the universe and destiny of the same.Potential infinity isn't the same thing as actual.Hence,no actual infinity-no transfite numbers , ordinals etc.
In sense I hold the universe is sort of "constructivistic machine".
Of course,this is my opinion.You may agree or not.

 

[matt grime]

but TeV, why must numbers only be things which count physical objects? why can they not be used to enumerate the state(s) of a system? and as such there are an infinite number of possible states of a system.

 

[p-brane]

My guess is that there are as many numbers, states, systems and purposes as we can construe out of the material we have available to our five or more senses. This variety and magnitude of states includes our actual senses as well. Its a never ending fractal... well, its never-ending until you stop looking into it. Then it ends.

Imagine that math goes away for the summer. There is sand but no one's counting the grains. There's an infinite party. Try to carry that thought into next fall. Cheers!

 

[TeV]

why can they not be used to enumerate the state(s) of a system? and as such there are an infinite number of possible states of a system.

With finite number of physical objects,recognizing finite number of fundamental force interactions in system,there "is" only possibility* for one infinity-trough the time unlimiting operation procedure:the potential infinity.This is also supported by spacetime quantization requirement in modern era of the science.One can make yourself easier by introducing continuum description of space in aproximation ,but this is just a cheating (a very good one since growth and number of states of the evolving system is astronomical).For * see 3 posts up:we are not sure about the destiny of the universe.Therefore,since the ideas of continiuum and actual infinities are completely human that leaves the debate of the meaning of actual realization in nature fruitless.
But even,from the standpoint of potential infinity sign of singularities in any physical theory are signs of the *sickness* in theory.That was what I wanted to emphasize.