Questioning Math: A Look at Proving Math with Math

[cosmic_chemical_nova]

This might be a stupid question but I'm a stupid person so it kind of fits. I'm a fan of Mr. Kaku's website and his articles. He makes me think about all the possibilities out there in our universe. My question is how do we know our mathematics is correct? The most math I ever took were a few algebra classes so I'm the last person to question brilliant people like Newton or Einstein. How can you prove math with math? Any comments good or bad would be appreciative so I can maybe look at this in a different way. Thank You.

 

[Hurkyl]

The scientific method can be used to evaluate how well a particular mathematical model reflects reality, if that's what you mean.

 

[mathwonk]

well the idea of "proof" in mathematics is based on simple but arguable assumpotions of logic. E.g. if you believe that all men are mortal and that I am a man, does it follow that you must believe I am mortal? if so you are granting one of the basic techniques of logical proof.

On the other hand, the validity of mathematics for interpreting physics is based on its success rate. E.g. if you see a wagon full of corn and multiply lengfth by width by height of the wagon, to calculate the volume of the corn, you can check your accuracy by weighing the corn afterwards, or counting the ears or soemething.

if the mathematics gave essentially the right answer thenh you have more faith in the usefulness of mathematical predictions about the world. Peopl have made matheamtical calculations to rpedict the retuirn of hales comet, and then they see whether it comes back riughly on time. All the moon shots are also based on calculations, as well as continual feedback of corrections to the data.

the theorems of calculus are in a sense completely false for the real world as they are based on assumptions which are in fact not at all true, like the completeness of the real line, a purely theoretical construct not at all valid in real life.

for instance there is no rteason at all to believe that a real life max min problem actually ahs an exact solution. this is noly true in amth calss because we believe the intermediate value theorem, an absurdity when one is talking about physical quantities like fences, bricks, and rope. still experience shows these calculations ahve practical use as approximations.

as pointed out above, the famous result that moving projectiles like bullets and thrown rocks, follow the path of a parabola is actually false, not just because of the influence of the air resistance, but also because gravity does not act downwards in a family of parallel lines.

it is have been observed that it is amazing that math does iluminate pohysics, the phenomenon that has been called something like "the unreasonable effectiveness" of mathematics.

 

[cosmic_chemical_nova]

Thank you for your post. Searching around the forums I have found some great opinions that touch on the same subject. I think my question is more towards not so much the scientific method and looking at mathematics that way but to look at how we use or more importantly name objects. When thinking about other advanced life forms through out the universe it might be possible for them to see a part of the problem that we have not seen. This goes to my question on questioning math and how it could easily go into the wrong direction. I'm not in anyway a religious person and please don't think that's were I wanted this to go but my interest in mathematics is growing everyday but it seems that compared to a much more advanced civilization everything we could be working for is wrong and all it takes to get it right is just another way of looking at it. I think I might be rambling off and making no sense but I just trying to get across my question. Thanks again.