Thread: Are there any theorems that deserve the title "fundamental" in these branches of math?

I asked in a different forum why there is no fundamental theorem of geometry. In the answer, I learned that no "fundamental theorem" is really fundamental to its branch of mathematics. So now I wonder, is there a theorem, or a collection of theorems, that could arguably be called "fundamental" for, say, arithmetic, algebra, geometry, and calculus?

Originally Posted by GreatBigBore

I asked in a different forum why there is no fundamental theorem of geometry. In the answer, I learned that no "fundamental theorem" is really fundamental to its branch of mathematics. So now I wonder, is there a theorem, or a collection of theorems, that could arguably be called "fundamental" for, say, arithmetic, algebra, geometry, and calculus?

It all depends on what you take "fundamental" to mean. See https://en.wikipedia.org/wiki/Fundamental_theorem for a reasonable discussion (especially the comment that application of the word is a reflection of tradition, rather than satisfaction of some formal set of criteria).

I have a fundamental theorem of fundamentality, but no one seems to use it.

Originally Posted by tk421

It all depends on what you take "fundamental" to mean.

Rats. I was hoping to get a bunch of opinions about what all of you guys consider "fundamental". Any ideas on what question I should have asked? Are mathematicians just not that opinionated?

Fundamental theorem of algebra - every nth degree polynomial has n roots in the complex plane. Fundamental theorem of arithmetic - every integer is the unique product of powers of primes (not using 1). Fundamental theorem of calculus - (rough description) taking derivative and integrating are inverse operations of each other.

Originally Posted by GreatBigBore

Originally Posted by tk421

It all depends on what you take "fundamental" to mean.

Rats. I was hoping to get a bunch of opinions about what all of you guys consider "fundamental". Any ideas on what question I should have asked? Are mathematicians just not that opinionated?

No, really, my answer was completely serious. Unless and until you define precisely what YOU mean by the word, there's no point in a discussion. You'll just get lists, like mathman just gave you (and which are among those listed in the wikipedia article I linked to).

So, again, what do you mean by fundamental? What are your criteria? As I've already explained, mathematicians have applied the word to certain theorems out of tradition.

Originally Posted by tk421

So, again, what do you mean by fundamental? What are your criteria?

Ok, now I get what you're saying. I have one criterion, and as I try to type it here, I realize that it's probably incredibly naive. I'll surely need your help formulating it to any precision. So, my naive vision is that the boundaries between arithmetic, algebra, geometry, and calculus are sharp enough to say that certain activities belong to certain fields, simply by definition. If that's totally wrong, then maybe I don't have a question at all. If that's even close to right, then I would think of a fundamental theorem as a theorem from which the rest of the field can be derived. Or maybe, a theorem that the whole field needs, as a theoretical underpinning. Like if you pulled it out, the whole edifice would crash down. Maybe there are multiple such theorems holding up each edifice?

A few years back, I looked up the fundamental theorem of algebra in Wikipedia because I had forgotten the proof that I was taught at school. I was shocked to discover that the proof is actually very difficult, far more difficult than anything I would have been taught at school. One thing in the article struck me: that there is no known purely algebraic proof, that all known proofs rely on analysis in some way. However, it did occur to me that there are actually two distinct ways to interpret the theorem. One way, the way that is best known, considers the roots of the polynomial equation. The other way, the way I think may be more appropriate for a theorem called "the fundamental theorem of algebra" is to consider the factorisation of a polynomial expression into linear and quadratic factors (which in turn can be factorised into linear factors over the complex field).

Is there a fundamental ** theorem that deals with the applicability of mathematics to physical reality?

Would singularities be an example where the relationship breaks down?

Is "diverge" a better description ?(perhaps the mathematics of singularities will turn out to be useful in the long term)

** if "fundamental" is too much to ask then "any".....

Originally Posted by GreatBigBore

Originally Posted by tk421

So, again, what do you mean by fundamental? What are your criteria?

Ok, now I get what you're saying. I have one criterion, and as I try to type it here, I realize that it's probably incredibly naive. I'll surely need your help formulating it to any precision. So, my naive vision is that the boundaries between arithmetic, algebra, geometry, and calculus are sharp enough to say that certain activities belong to certain fields, simply by definition. If that's totally wrong, then maybe I don't have a question at all. If that's even close to right, then I would think of a fundamental theorem as a theorem from which the rest of the field can be derived. Or maybe, a theorem that the whole field needs, as a theoretical underpinning. Like if you pulled it out, the whole edifice would crash down. Maybe there are multiple such theorems holding up each edifice?

From the wording of your answer, I suspect that what you are really after are the fundamental axioms that constitute the foundations (fundaments). Theorems derive therefrom, so aren't fundamental in that sense. So do you really mean axioms?

Originally Posted by geordief

Is there a fundamental ** theorem that deals with the applicability of mathematics to physical reality?

Would singularities be an example where the relationship breaks down?

Is "diverge" a better description ?(perhaps the mathematics of singularities will turn out to be useful in the long term)

** if "fundamental" is too much to ask then "any".....

I know you're curious about a great many things, but it would be less impolite if you were to start your own separate threads.

Originally Posted by geordief

Is there a fundamental ** theorem that deals with the applicability of mathematics to physical reality?

Would singularities be an example where the relationship breaks down?

Is "diverge" a better description ?(perhaps the mathematics of singularities will turn out to be useful in the long term)

** if "fundamental" is too much to ask then "any".....

http://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf