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What makes a theorem "fundamental"?
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I've studied three so-called "fundamental" theorems so far (FT of Algebra, Arithmetic and Calculus) and I'm still unsure about what precisely makes them fundamental (or moreso than other theorems).
Wikipedia claims:
The fundamental theorem of a field of mathematics is the theorem considered central to that field. The naming of such a theorem is not necessarily based on how often it is used or the difficulty of its proofs.
So I am left wondering: what is the main criterion for a theorem to be considered fundamental?
Why, for instance, is there no "Fundamental Theorem of Complex Analysis" (say, de Moivre's formula, for example), or "Fundamental Theorem of Euclidean Geometry" (e.g. Pythagoras' theorem)?
Does anyone have a source of a more-rigourous definition of "fundamental", in this context?
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edited Sep 28, 2014 at 11:04
asked Sep 26, 2014 at 19:39
beep-boop
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$\begingroup$<br>I suppose you could call Pythagoras' theorem the Fundamental Theorem of Euclidean Geometry<br>$\endgroup$
vadim123
vadim123
2014-09-26 19:42:07 +00:00
Commented<br>Sep 26, 2014 at 19:42
$\begingroup$<br>The fact that all others are build upon them.<br>$\endgroup$
Lucian
Lucian
2014-09-26 19:43:25 +00:00
Commented<br>Sep 26, 2014 at 19:43
$\begingroup$<br>For logic there are de Morgan's laws. For number theory, see Arithmetic. In probability and statistics, there is the law of large numbers.<br>$\endgroup$
Lucian
Lucian
2014-09-26 19:51:07 +00:00
Commented<br>Sep 26, 2014 at 19:51
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$\begingroup$<br>I think the idea of a rigorous definition of "fundamental" is misguided. Category Theory gives us some rigorous ideas about what we mean by "natural", but most of us encounter the word in relation to natural numbers long before we hit category theory. "Fundamental" has all the foibles and ambiguities of a word in the english language, most of the time. The meaning derives from use rather than definition. It is, though, indicative of theorems which capture basic mathematical relationships and ideas.<br>$\endgroup$
Mark Bennet
Mark Bennet
2014-09-26 20:01:30 +00:00
Commented<br>Sep 26, 2014 at 20:01
$\begingroup$<br>If Complex Analysis had a fundamental theorem, I would nominate the Cauchy Integral Formula. If the field had a second fundamental theorem, I would nominate the Riemann Mapping Theorem. Those two really are the sine qua non of the field.<br>$\endgroup$
Emily
Emily
2014-09-26 20:37:05 +00:00
Commented<br>Sep 26, 2014 at 20:37
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Let's examine some of these so-called "fundamental" theorems.
($1$) The Fundamental Theorem of Arithmetic.
"Every natural number greater than $1$ has a unique representation as a product of primes."
This theorem about factorization establishes the primes as fundamental building blocks for studying numbers. This idea and the obsession with these numbers (who have an entire theorem named after them) which are the core building blocks of all integers has sparked multiple entire fields of research. Almost all problems about integers in Diophantine equations makes use of this theorem, because it makes proving an enormous body of results vastly simpler and in many cases makes them possible where direct proof is impossible. Most fields in mathematics would probably be crippled without this theorem. And because of the immense advantage of unique factorization (which not all settings have) number theorists were led to develop the field of algebraic number theory where this theorem finds its ultimate generalization in the context of Dedekind domains. In fact, to underscore just how good unique factorization is: if it were true that every number ring had unique factorization, there are old proofs of Fermat's last theorem which would have given us the result ages ago, instead of making us wait 350 years!
($2$) The Fundamental Theorem of Algebra.
"Every polynomial of degree $n$ with coefficients in the complex numbers has exactly $n$ complex roots (counting multiplicity)."
Algebra has its roots in the study of polynomials (no pun intended). You learn about other topics in a high school course named "algebra," but good old polynomials were some of the first objects considered by mathematicians in this field. Ring theory uses polynomials extensively, group theory was created to deal with the problem of understanding solutions to polynomials. Field theory would not be a thing without...