Is mathematics a separate entity from nature?

[TheDestroyer]

Hello guys:

A very known fact we live is that mathematics is not falsifiable. There hasn't been a day in history when someone came up and made an experiment to prove that the equation "x^2-5x+6=0" has solutions different than 2 and 3.

In science, if we find something that doesn't comply with mathematics, we never doubt the math, but we may change the model of our physics, and then test whether the new model would answer more questions physically.

From this, I find that math is completely and totally a separate entity from nature. Nature can be represented by mathematics, however, because of it's consistency. No day will ever come when we have a single apple that becomes two apples for no reason; and bam, there you get your simplest conservation law in nature... "The conservation of apples :)"; In other words (in my view, I'm not sure whether there's a theory that supports this, please let me know), every conserved system can be described by mathematics.

Gödel then comes and says that mathematics isn't complete; why is that the case, Mr. Gödel? Because mathematics isn't completely provable...

ah, really? Big deal! We know this already! And I actually find this very natural! No one ever stated that mathematics should be proven by experiments or by any kind of complete test before axioms! I work with physics for more than 10 years and haven't seen this ever. Mathematics is naturally not completely provable because:

Mathematics, by its nature, is not experimental, and we are experimental entities. In order to realize the existence of something independent of nature, we have to interact with nature; therefore: I can't expect that I'm going to be able to prove its completeness, since my connetion to it is through something which is achieved by experimental interaction. (Did I miss any piece of the puzzle?).

Now my question is: is mathematics independent from nature (physics), or is it dependent on it? In other words: do you find my conclusion above logical, and is mathematics falsifiable in any way?

In other words: if we exit the universe, can we ever find out that 2+2=5?

Notice that whole mathematics is built and based upon simple ideas like addition and multiplication, and you generalize that further with more dimensions, which provide us with matrices, which then we use as operators, and then when you move to continuous systems you realize that sums become integrals and differences become derivatives... So all mathematics came up just from the simple 1+1=2 and other simple statements!

Thank you for any efforts. I really need to hear specialists opinions about this.

 

[micromass]

Mathematics is absolutely falsiable and experiments can absolutely be done.

Mathematics is used in physics, engineering, biology, etc. All the time. The mere factt that it works there and gives correct conclusions should be enough proof that it works.

Mathematics is not separate from nature. We have "invented" mathematics exactly by observing nature and abstacting it into laws. It's not some arbitrary game that we invented to have fun. It is actually grounded in experimental facts. For example, our notions of area and volume came from dividing up lands for farmers. Our notions of counting come from counting sheep in the field.

I agree that mathematics is so advanced now that it's difficult to see the experimental roots. In part this is also the fault of the mathematicians who try to teach things at the most abstract level possible.

Read this: http://pauli.uni-muenster.de/~munsteg/arnold.html
Especially: " Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap."

Also, about being falsiable. You say that "x^2-5x+6=0" has solutions different than 2 and 3." is not falsiable. I disagree. I can actually draw the graph $f(x) = x^2 - 5x + 6$" I can choose points on the graph, and if I have enough points, I got an actual representation of the graph. And then I can see for myself that it doesn't appear to have any roots. This is experimental evidence! It's not a rigorous proof, but experiments and rigorous proofs are two different things.

Finally, there are a lot of things in physics that I have no experimental evidence for. For example, I have no direct evidence for the Shrodinger equation. That is alright. We use the Schrodinger equation in our physics, and we derive things that we actually can test. This also experimentally verifies the Schrodinger equation.

In short, I fundamentally disagree that mathematics is not experimental and independent of nature.

 

[TheDestroyer]

Dear micromass, thank you for your answer and your time.

First, I would like to express my extreme disagreement with the article in the link you provided. The guy in the article is completely mixing two different things:

1- The mathematics as an entity,
2- and the way we understand mathematics.

Of course, with no doubt, if we separate mathematics and physics in our lives, we'll not understand mathematics. We're, by our nature and evolution, are experimental entities. Our brains are nothing but learning machines, that mostly learn things using grouping and comparison. That's why when you look at the signal, the first thing you'll catch is any kind of periodicity in the signal. Our brains have evolved to learn compare chunks of information together very efficiently.

However, this has nothing to do with the point I'm trying to make. We have to distinguish between having mathematics being fundamentally falsifiable in some other nature (some parallel universe?); with the case when we, as experimental entities, fail to understand mathematics without an experimental reference.

I'm sorry. But I see that the guy talking in the article in the link is mixing stuff; and I wish I could hear an opinion about this from someone else, too. Maybe I'm wrong.

About the quadratic equation, you have also have to keep in mind, that the solution of a quadratic equation:

1- isn't derived from a graph, but rather using an analytical formula;
2- and you should not falsify a method by introducing systematic errors because of your physical evaluation method, like having low resolution plots. This is informal and I think it's even scientifically invalid. And if you state error propagation calculations you'll still be on the verge of the correct result within the error you introduced. The solution provided with the graph is simply incomplete!

The answer I was expecting, with all due respect, is something like a reason why 1+1 shouldn't be 2 in some other world/universe/nature. In very simple terms, I see that mathematics is built brick after brick based on very simple and very fundamental ideas like addition and multiplication (right?).

Please don't misunderstand my impatience, I still would like to here counter-opinions, but I've been thinking about this for a very long time.

Thank you.

 

[WannabeNewton]

Lol I would think thrice before saying Vladimir Arnold is mixing up anything. He's Vladimir freaking Arnold.

 

[jackmell]

Now my question is: is mathematics independent from nature (physics), or is it dependent on it? In other words: do you find my conclusion above logical, and is mathematics falsifiable in any way?

In my opinion there is no mathematics without biological evolutionary history. That history, through hundreds of millions of years and razor-sharp selection pressures, evolved a complex neural machine as a crowning achievement of survival. Mathematics is in my opinion, an emergent property of that machine.

 

[Evo]

I am afraid that this is too speculative.