Debating Geometry Reality: Unreasonable Effectiveness & Beyond

[humanino]

I was just having this very old, neverending debate. I would like to have your opinion about this. It all started with geometry, but I think the argument extends to mathematics altogether.

According to my friend, mathematics first come from experiment and thus belong to the category of physical models of the world. In my opinion, mathematics belongs to the ideal platonic world.

I will first re-phrase my opinion since it is eaiser for me
Considering that

• one can study mathematics without knowing anything about the outside world : I know of a blind geometry professor
• the physical world we use as a source of inspiration to choose the geometry we want to study, but we can invent (but see later) as many as we want, as wild as we can think of

I think mathematics exist independent of the physical world. At the very least, mathematics are universal, a circle is a circle no matter if I am a french man, or an alien in another galaxy...

He, on the contrary, argues that everything comes from experiment, and that just like other physical models of reality, geometry somehow belongs to physics. The mathematics we choose to study come from experiment, and are thus a reflection of the physical world

The all debate actually seems to stem from the The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

I also want to push the two previous opinions to their extreme :

One could consider that mathematics are just a toolbox. If a piece of mathematics serves no practical purpose it is useless, should be disregarded and not taught.

I am even among those who think that mathematics is just like art, because I often experience deep esthetics feelings while reading a mathematical proof, intense as one can feel while contemplating a painting of listening to a beautiful musical composition. I do not want to develop to much at this stage and/or go into philosophical consideration, but earlier I would rather have said discover instead of invent since I think mathematics exist per se, before we know them, in the platonic world.

 

[George Jones]

As I said in post #16 of https://www.physicsforums.com/showthread.php?t=102552", I'm with Godel, Hardy, and Penrose, i.e. I am a Platonist.

"... and there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is 'mental' and that in some sense we construct it, others that it is outside and independent of us ... I believe that mathematical reality lies outside of us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations."

G. H. Hardy

From https://www.physicsforums.com/showpost.php?p=1274300&postcount=2", it appears that Hurkyl disagrees.

 

[Aether]

It all started with geometry, but I think the argument extends to mathematics altogether.

According to my friend, mathematics first come from experiment and thus belong to the category of physical models of the world. In my opinion, mathematics belongs to the ideal platonic world...I think mathematics exist independent of the physical world. At the very least, mathematics are universal, a circle is a circle no matter if I am a french man, or an alien in another galaxy...

He, on the contrary, argues that everything comes from experiment, and that just like other physical models of reality, geometry somehow belongs to physics. The mathematics we choose to study come from experiment, and are thus a reflection of the physical world...One could consider that mathematics are just a toolbox. If a piece of mathematics serves no practical purpose it is useless, should be disregarded and not taught.

A nice book on this subject is The Philosophy of Space and Time by Hans Reichenbach.

Geometry is concerned solely with the simplicity of a definition, and therefore the problem of emprical significance does not arise...Properties of reality are discovered only by a combination of the results of measurement with the underlying coordinative definition...It is Einstein's achievement to have applied the theory of the relativity of geometry to physics.

 

[mattmns]

See signature

 

[Thrice]

"It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention."

-Bridgman, P. W.

 

[Chi Meson]

You left out the option of:"merely one of the parts of the whole."

 

[denverdoc]

It remains the most singularly beautiful and powerful invention of mankind. Now the beauty is not always accessibility to the common man, but the utility is without debate. Even our best "art", fails this test.

 

[Hurkyl]

From https://www.physicsforums.com/showpost.php?p=1274300&postcount=2", it appears that Hurkyl disagrees.

"Disagrees" is too strong a word -- I simply think the answer to this question is entirely irrelevant to actually doing mathematics.

 

[Cyrus]

Im an engineer, they are a tool for me. I care about the physics, not the math. All that matters is that I get the right answer. I don't care about things like the limit of 0.9999 =1, it won't explain why an airplane flies, or what the stresses are in a beam.

 

[denverdoc]

Im an engineer, they are a tool for me. I care about the physics, not the math. All that matters is that I get the right answer. I don't care about things like the limit of 0.9999 =1, it won't explain why an airplane flies, or what the stresses are in a beam.

Perhaps, you're being a little narrow minded re this. Engineering w/o math is like your best prayer and a rain dance. You may have no use for exactitude. But the principles upon which you design anything should hopefully come from the heart of mathematic principle. Try it w/o and wait to be sued. Won't be long.

 

[Cyrus]

I think you misread what I wrote. I said I use math as a tool, I don't care about the formalisms behind it. The formalisms have trivial importance to me.

 

[denverdoc]

Most of us do, but I also believe that math sometimes get short shrift when it comes to theoretical mathematics. Often in the history of science and engr, the math has been sitting around for some time waiting for an application. Its any engineer's mother, and not to me dismissed casually as in I have only use for the teats that feed me, and nothing else.

 

[interested_learner]

I also am an engineer. I also use math to solve my problems, but I also see in math a very elegant and beautiful art form. It is the most abstract and austere form of beauty.

 

[Cyrus]

Most of us do, but I also believe that math sometimes get short shrift when it comes to theoretical mathematics. Often in the history of science and engr, the math has been sitting around for some time waiting for an application. Its any engineer's mother, and not to me dismissed casually as in I have only use for the teats that feed me, and nothing else.

Well, if you have a math that is usefull to me, then I'll use it. Otherwise, I won't touch it. Its useless to me, as an engineer. As an engineer, we don't use theoretical mathematics.

 

[denverdoc]

Then you shall be a follower, like the hundreth and one sheep in a flock, which is not all bad as the first hundred may be destined for slaughter and the remainder only shorn for their coats...at least until the next year.

 

[Cyrus]

Then you shall be a follower, like the hundreth and one sheep in a flock, which is not all bad as the first hundred may be destined for slaughter and the remainder only shorn for their coats...at least until the next year.

.....o-kayyyy...this is a bunch of nonsense, but whatever.

 

[denverdoc]

Talking about outsourcing at least in the USA. If one ccan employ some Pakistani or Indian engineer with an IQ of 160 for less than the costs of a USA countertype for twice the $$ and 2/3 rds the IQ, I'd be trying to think outside the box. Advanced math can help.

 

[Cyrus]

Who cares if his IQ is 160, your paying him pennies on the dollar to what you would have to pay me. Its not because he's so much smarter than me, its because he's so much cheaper.

Don't confuse advanced math, with formal math. We have math classes 'for scientists and engineers'. Those are NOT the same as math classes for MATH majors.

 

[Hurkyl]

Then you shall be a follower, like the hundreth and one sheep in a flock, which is not all bad as the first hundred may be destined for slaughter and the remainder only shorn for their coats...at least until the next year.

Since you don't say what you mean, I can only guess. So it's your fault if this has nothing to do with what you meant.

In any pursuit, you need people who look for new useful things, and people who use the useful things we already have. You can't make progress without both kinds of people.