The 100 Greatest Theorems of Mathematics

As a mathematics teacher, I am often asked what I believe is the single greatest theorem in all of mathematics.  And, depending on my mood, I could claim any one of a dozen theorems to be the greatest.  Talk to other math people and you will probably get a completely different dozen.  In fact, there are probably as many different opinions as there are theorems.  To me that is one of the beautiful things of my subject.

With that being said, I guess there is no point in anyone ever trying to construct a list, right?  Not really. Enter mathematicians Jack and Paul Abad.   In 1999, they set forth on the arduous journey of generating the list of the 100 Greatest Theorems.  In making the list, they used 3 criteria.

  • the place the theorem holds in literature
  • the quality of the proof
  • the unexpectedness of the result

What did they come up with?  Below is their top 12.  For the complete list, click here.

  1. The Irrationality of the Square Root of 2 by Pythagoras (500 B.C.)
  2. Fundamental Theorem of Algebra by Karl Frederich Gauss (1799)
  3. The Denumerability of the Rational Numbers by Georg Cantor (1867)
  4. Pythagorean Theorem by Pythagoras (500 B.C.)
  5. Prime Number Theorem by Jacques Hadamard and Charles-Jean de la Vallee Poussin – separately (1896)
  6. Godel’s Incompleteness Theorem by Kurt Godel (1931)
  7. Law of Quadratic Reciprocity by Karl Frederich Gauss (1801)
  8. The Impossibility of Trisecting the Angle and Doubling the Cube by Pierre Wantzel (1837)
  9. The Area of a Circle by Archimedes (225 B.C.)
  10. Euler’s Generalization of Fermat’s Little Theorem by Leonhard Euler (1760)
  11. The Infinitude of Primes by Euclid (300 B.C.)
  12. The Independence of the Parallel Postulate by Karl Frederich GaussJanos BolyaiNikolai LobachevskyG.F. Bernhard Riemann – collectively (1870-1880)

What do you think?  Did they get it right?  Enjoy the debate!

Remembering Paul Halmos

Sunday October 2 is the 5th anniversary of the death of Paul Halmos.  Remembered for many mathematical contributions in fields such as operator theory and functional analysis, he is probably best known for his amazing ability to write about mathematics – especially mathematical textbooks.  However, what I love best about Halmos was his ability to write about mathematics for the non-mathematician.  He always seemed to have a way to express the beauty and wonder of mathematics in such a way that everyone would stop and think, “maybe mathematics isn’t as bad as I thought.”

Some of my favorite works by Paul Halmos:

  • I Want to Be a Mathematician … an automathography – This is a phenomenal autobiography, taking you deep into the life of an actual mathematician.  Not only do you learn about the research aspects of mathematics, but also about the human side of the subject.  It is a great read!
  • Paul Halmos: Mathematics as a Creative Art – This is a synopsis of a lecture Halmos presented in Edinburgh, Scotland in 1973.  It’s hard to summarize what he speaks about as it includes just about every aspect of mathematics but, if you ever want to know what a mathematician does or what people see in mathematics, then read this.  It is just a beautifully written piece.

Some of my favorite quotes by Paul Halmos:

  • “Don’t just read it; fight it!  Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary?  Is the converse true?  What happens in the classical special case?  What about the degenerate cases?  Where does the proof use the hypothesis?”
  • “It saddens me that educated people don’t even know that my subject exists.”
  • “Mathematics is not a deductive science, that’s a cliché … What you do is trial and error, experimentation, guesswork.”
  • “The heart of mathematics is its problems.”
Paul Halmos is a wonderful ambassador for mathematics, even 5 years after his passing.  In fact, it’s people like him who helped me discover the beauty and wonder of mathematics.  Thank you, Paul.