The Art of Abstraction

enToday is the birthday of the magnificent Emmy Noether, known as the “mother of modern algebra”, and one of my mathematical heroes. She is one of the few mathematicians in history who fundamentally changed what mathematics is about.

It was Emmy (I use her first name in order to distinguish her from her mathematician father Max) who first fully recognized the power of abstraction, which became the driving force of 20th century mathematics. She demonstrated time and again that it can be easier to solve a general problem than a specific one, and therefore the best way to attack a specific problem is often to generalize. Do you want to prove a fact about polynomial functions? First notice that polynomial functions can be added together, and they can be multiplied, and they obey certain laws along the way (like associativity and commutativity). Now prove a theorem that applies to anything that can be added and multiplied subject to those laws. Do it right, and you’ll replace intricate calculations with simple logical deductions. What was hard becomes easy. You get your result for free, and a whole lot of other results as a bonus.

Or, if you that doesn’t quite work, figure out what additional properties you’re using about polynomials, beyond associativity and commutativity, and prove a theorem about everything that has those properties.

To get a sense of how revolutionary this was, consider the Hilbert Basis Theorem, one of the foundational results of modern algebra. Have a look at Hilbert’s original proof — though you might not want to work through every detail in the 62 pages of equations and formulas. By contrast, Noether’s proof of a more general, more powerful and more useful version occupies all of one paragraph on Wikipedia.

I daresay most of the great mathematical triumphs of the past 100 years came about because people took Noether’s dictums and ran with them. The greatest of all those triumphs, the oeuvre of Alexandre Grothendieck, takes all its inspiration and all its power from the philosophy and the work of Emmy Noether.

As a minor avocation, Noether dabbled in mathematical physics, where her main contribution was the insight that conservation laws (like conservation of mass) and symmetries (like “the laws of physics don’t change from one location to another”) are two different ways of looking at the same thing. Uncover a new symmetry and you can automatically deduce a new conservation law. Much of what we know about particle physics was discovered using Noether’s principle, which, according to the Nobel-prize-winning physicist Leon Lederman, is “certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem”. And physics was her sideline!

Emmy Noether’s portrait hangs in a place of honor on my home office wall. Every day I look on it with awe.

Share/Save

4 Responses to “The Art of Abstraction”


  1. 1 1 Babinich

    Steve,

    Great post!

    I believe that you’ll find few have achieved so much because in that time at that time Emmy Noether was looked down upon by her male dominated profession.

    In this regard she reminds me of Rosalind Franklin.

    I believe it was in the John Derbyshire book ‘Unknown Quantity’ where Einstein himself approaches Noether to get him out of a mathematical “jam”.

  2. 2 2 Tom

    Hear hear.

  3. 3 3 gaddeswarup

    I think that it was also Emmy Noether who pointed out to Alexanderoff and Hof that the numerical invariants of spaces that topologists were lookig at were actually the invariants of the homology groups of the corresponding chain complexes. having said that I think that Galois influence was greater.

  4. 4 4 Steve Landsburg

    gaddeswarup:

    I think that it was also Emmy Noether who pointed out to Alexanderoff and Hof that the numerical invariants of spaces that topologists were lookig at were actually the invariants of the homology groups of the corresponding chain complexes

    I believe this also, and had a paragraph about it in the original blogpost, which I deleted because I couldn’t figure out how to make it even marginally comprehensible to the non-mathematical reader (at least not without putting in more time than I had available). But this one observation, all by itself, was more productive than most entire careers, though it was, like the physics work, tangential to the main body of Noether’s work.

Comments are currently closed.