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Hello,
I wonder how laws of physics, mathematics and logic influence each other. What I mean is the following: In Quantum Mechanics (probably even in general physics), only very few non-linear problems can be explicitly solved. The most important ones are 1.the harmonic oscillator (potential r^2) and 2. the Hydrogen atom (potential 1/r). This is the reason why almost any other non-linear problem is first reduced to a r^2 or 1/r-potential problem.
This seems like lucky coincidence or a divine act or whatever you might call it: 2 very basic physical problems can be expressed and solved in a very basic mathematical way.
Now I keep on wondering: If our mathematics was based on some different algebra than the one it actually is, say, elliptic functions (=objects that are reasonably hard to express in "our" mathematics), would our understanding of physics be different?
(For example: would we better understand physical facts that are now "too complicated" (because of their mathematical complexity), and -maybe- fail to really understand the harmonic oscillator, just because there is no means to express this problem mathematically?)

Hello,
I wonder how laws of physics, mathematics and logic influence each other. What I mean is the following: In Quantum Mechanics (probably even in general physics), only very few non-linear problems can be explicitly solved. The most important ones are 1.the harmonic oscillator (potential r^2) and 2. the Hydrogen atom (potential 1/r). This is the reason why almost any other non-linear problem is first reduced to a r^2 or 1/r-potential problem.
This seems like lucky coincidence or a divine act or whatever you might call it: 2 very basic physical problems can be expressed and solved in a very basic mathematical way.
Now I keep on wondering: If our mathematics was based on some different algebra than the one it actually is, say, elliptic functions (=objects that are reasonably hard to express in "our" mathematics), would our understanding of physics be different?
(For example: would we better understand physical facts that are now "too complicated" (because of their mathematical complexity), and -maybe- fail to really understand the harmonic oscillator, just because there is no means to express this problem mathematically?)

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Very interesting query. But is it necessarily a 'lucky coincidenceor divine act' that basic physical problems can be solved using basicmathematics? Maybe the reason is that is that our mathematics has beendeveloped in order to deal with the physical world.

True, thereare many familiar cases where techniques which were were firstdeveloped by pure mathematicians later turned out to be useful fornovel physical applications. For example, complex numbers are great fordealing with alternating currents in electrical circuits. This may make it seem that the physical applicability of these bits of mathematics is a coincidence after all.

But it may still be true that, at a more general level, mathematics is designed to analyse the kinds of patterns that are displayed by physical phenomena. The same kinds of patterns often repeat themselves in nature, as your question implies, so it is scarcely surprising that we have developed mathematics to deal with these patterns.

Of course, none of the above rules out the possibility that some 'alternative' mathematics would enable us to deal with problems that are present mathematically intractable.