A math foundation problem, of sorts

[jeremy22511]

So I was wondering why a single problem can have two entirely different approaches (based on contradictory axioms) to solving it? For example, consider sunlight. Sunlight always comes from a slightly different direction than we see because of refraction. They can be studied with Euclidean geometry (e.g. Snell's law), but they can also be studied with a specific non-Euclidean geometry made for them. They are based on completely different sets of axioms so why can they arrive at the same answer and conclusions?

I can't think of other examples right now. Would appreciate it if someone would answer :)

 

[zoki85]

Science, and math is a science, allows and encourages creativity. That's why.

 

[jeremy22511]

Thank you for your answer, but that doesn't answer my question. Creativity is a crudely defined word that does not fit in any model of math or science. It's like saying 'it's luck' when all it is is probability.

 

[symbolipoint]

Understanding your question is difficult. Maybe you are confusing physical sciences with Mathematics. The behavior of light is very mathematical, but it is not itself mathematics.

 

[HallsofIvy]

It is impossible to answer your question as written because it is bases on incorrect assumptions. You might well be able to answer a question using "Eucidean" geometry and "non-Euclidean" geometry but they will NOT, in general, give the same answer and at least one must be wrong.

Of course, in dealing with "real", physical, problems, our data is based on imperfect measurement and we will always have a region of possible error. It might happen that both answers are with that "region of error" but that does not mean that the both (or either) are "correct" in the sense in which you are using the word.

 

[jeremy22511]

@HallsofIvy
I still remember you answering some of my questions a few years back. Thank you again :)

That's not what I mean though, and frankly I'm still a bit confused. Riemann's non-Euclidean geometry can be used to study a sphere surface, and get properties of that sphere surface, but Euclidean geometry can also be used to study a sphere surface, albeit a lot harder (that's what we have been trying to do, isn't it?). But two of their axioms are completely different. Feels like I'm missing something here.

 

[jbriggs444]

The axioms are different, but the mapping of the mathematically defined objects to corresponding physical objects is also different. The physical world can then conform to both sets of axioms simultaneously.

Of course, one must be careful not to transliterate a result obtained from the one set of axioms, map that result onto the physical world using the other mapping convention and then pretend that the resulting physical claim ought to be true.

 

[HallsofIvy]

@HallsofIvy
I still remember you answering some of my questions a few years back. Thank you again :)

That's not what I mean though, and frankly I'm still a bit confused. Riemann's non-Euclidean geometry can be used to study a sphere surface, and get properties of that sphere surface, but Euclidean geometry can also be used to study a sphere surface, albeit a lot harder (that's what we have been trying to do, isn't it?). But two of their axioms are completely different. Feels like I'm missing something here.

No. "Riemann geometry" is a lot more general than Euclidean geometry, but the axioms of Riemann geometry, restricted to the surface of a sphere are exactly the same as those of three dimensional Euclidean geometry, restricted to the surface of a sphere.

 

[Stephen Tashi]

So I was wondering why a single problem can have two entirely different approaches (based on contradictory axioms) to solving it?

What's your definition of a problem? A mathematical problem must be specific enough to specify what axioms are assumed when we state it - or else it isn't a mathematical problem. A single real life problem, or puzzle or riddle can be stated without indicating what assumptions can be made in solving it. To turn a real life problem into a mathematical problem, we must make assumptions. So different people can choose to make different assumptions.