

The Mighty Mathematician You’ve Never Heard Of - emmanuelory
http://www.nytimes.com/2012/03/27/science/emmy-noether-the-most-significant-mathematician-youve-never-heard-of.html

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rbehrends
If anything, the article probably understates her importance (and doesn't
really explain why she is so important). You do get an idea when you start at
her bibliography; not many scientists have their papers subdivided into epochs
[1].

To begin with, Einstein didn't describe Emmy Noether (in his obituary of her,
sent to the New York Times) as "the most 'significant and 'creative' female
mathematician of all time". The exact quote is instead: "In the judgment of
the most competent living mathematicians, Fräulein Noether was the most
significant creative mathematical genius thus far produced since the higher
education of women began." [2]

In essence, he is describing her as the greatest mathematician of the past
decades, male or female. That may have been a slight exaggeration (she was a
friend of his and visited frequently at Princeton during her time at Bryn
Mawr), but not much: she was definitely one of the most influential
mathematicians of the 1920s and early 1930s. Many mathematicians and
physicists visited Göttingen specifically to interact with her.

What made her so important were not just her contributions to abstract algebra
(groundbreaking as they were, essentially redefining the field), but probably
most of all her axiomatic "method of abstraction", pioneered in several papers
starting with 1921's "Idealtheorie in Ringbereichen". For non-mathematicians,
it is roughly the same concept as abstract data types; instead of proving
theorems about very specific mathematical constructs (such as integers and
polynomials), you prove them for more general constructs, abstractly defined
through operations over them and axioms connecting those operations. It is one
of those things that seem blindingly obvious in retrospect, but the approach
was actually hotly debated in the 1920s; these days, it's foundational for
much of mathematics.

A good example of this is the so-called Lasker-Noether theorem, which was
initially proven by Emanuel Lasker for a couple of specific rings, a result
that Emmy Noether later generalized for a wide class of rings (rings that we
now call, for other reasons, Noetherian rings).

Another reason why she is little known these days even among mathematicians is
probably that she didn't care much about getting credit for her ideas (even
though she was painstakingly careful about giving others credit in her own
papers). Her own publications, numerous as they were [1], are probably only
the tip of the iceberg; it is known, for example, that volume 2 of van der
Waerden's "Modern Algebra" (the standard textbook for abstract algebra for
decades) was essentially her work (van der Waerden never made a secret of it).
A lot of her other work was likewise published under the name of her students
and colleagues. (Unlike what the article states, she never published under a
man's name.)

This is the final way in which she influenced mathematics -- as a teacher. She
never was a very good teacher when it came to actual lecturing (being too much
of an absent-minded professor), but she was generous with sharing her ideas
and an inspiring influence for her students on colleagues (her first Ph.D.
student, Grete Hermann, went on to lay the foundations for Computer Algebra,
for example).

As van der Waerden wrote, "Each of her lecture series was a paper. And nobody
was happier than herself when such a paper was completed by her students.
Completely unegotistical and free of vanity, she never claimed anything for
herself, but promoted the works of her students above all."

[1] <http://en.wikipedia.org/wiki/Emmy_Noether_bibliography>

[2] [http://www-history.mcs.st-
andrews.ac.uk/Obits2/Noether_Emmy_...](http://www-history.mcs.st-
andrews.ac.uk/Obits2/Noether_Emmy_Einstein.html)

~~~
jhales
No doubt she is a very signifigant figure, but (Although you do qualify your
statement later) I have to disagree with:

"If anything, the article probably understates her importance ..."

In the second paragraph of the first page the author subtly suggests that she
is the greatest mathematician as well as the greatest physcicist (at least a
bunch of her contemporaries, who are smarter than you, the reader, thought
so).

Also, in my personal experience, she is quite well known.

I would add to your list of her accomplishments: I've been told that she was
one of the first to push for Homology to be treated as groups and not simply
just betti numbers.

~~~
rbehrends
> In the second paragraph of the first page the author subtly suggests that
> she is the greatest mathematician as well as the greatest physcicist (at
> least a bunch of her contemporaries, who are smarter than you, the reader,
> thought so).

This isn't quite what it says. That paragraphs talks about her being the most
significant and creative _female_ mathematician of all time, and attributes it
to Einstein. That's a significant weaker statement, given how few women
mathematicians there were before modern times.

Conversely, if the article had described her as the greatest mathematician and
greatest physicist of all time, that would be a massive overstatement, of
course.

What I was trying to get at here (and what Einstein probably was trying to
convey to a lay audience, too) was that she was very (probably critically)
important for the development of early 20th century mathematics, when many of
the foundations of how we do mathematics today were laid, and that this was
the case regardless of her gender.

~~~
jhales
I get what you are saying, one has to read between the lines a little, but if
you read carefully I think that is what is implied:

"Albert Einstein called her the most “significant” and “creative” female
mathematician of all time, and others of her contemporaries were inclined to
drop the modification by sex."

If you drop the female from that sentence then it reads "most significant and
creative mathematician of all time."

Another point that, which I think you pointed out as well, was that this is a
misquote of Einstein. The original statement included the clause 'since the
education of women...' which would make the dropping of gender much more
reasonable.

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drostie
The simplest way I've seen to explain Noether's theorem comes from Feynman's
Messenger Lectures.

Suppose -- as we have in modern physics -- that the laws of physics can be
phrased in a certain way. Namely:

(1) There is some number we can calculate for all of the possible paths that
an object could take from point A to point B. We call this number the "action"
of the path.

(2) Of all of the paths that an object _can_ take, the one which it _does_ has
a stationary action -- meaning that if you wiggle the path a little bit, the
action remains the same. This can happen for example if the action is at a
minimum for that path -- if you're at a minimum then you can't go any lower,
and so there cannot be a proportionate response to any slight change in path.

This work was already done before Noether -- the observation, I mean, that
there is a definition of "action" for all known laws in physics such that you
can phrase those laws as a stationary-action principle. An example is light.
All of these complicated rules of lenses and light bending when it enters
water can be phrased as the much simpler rule that "light always takes the
least-time path." The action of the path is then the time it takes for light
to travel along that path from A to B.

Now, suppose the action has a continuous symmetry. For example, it might be
the case that if A and B were moved a centimeter to the right, to two new
points A' and B', then the actions on the corresponding paths between them
would remain the same.

Now construct a new path: We go from A to B like so: we go from A to A',
follow the physical path to B', then go to B, rather than following the
physical path from A to B.

Call these two paths:

(1) A → B

(2) A : A' → B' : B

You have two competing facts. First off:

(1) the symmetry: action(A' → B') = action(A → B)

(2) stationary action: action(A : A' → B' : B) = action(A → B).

In other words, as long as the little wiggles described by ":" are very small,
the stationary action principle holds and we don't change the action when we
move from a physical path.

It must then be the case that action(A : A') = - action(B : B') in some sense.
In other words, there is some quantity local to both A and B which is exactly
the same -- it is the susceptibility of the action to changes if you shifted
in one direction or another. The big symmetry also forces some little number
to be constant while this thing is taking whatever path it takes.

Whenever you have a stationary-action principle -- which we have for gravity
and electromagnetism -- then Noether's principle holds and symmetries in the
action become conserved quantities of the physics.

We can also calculate some of these susceptibilities for the most common
action principles: we find that symmetry with respect to space-translation
(moving 1cm to the side) conserves momentum in that direction. We find that
symmetry with respect to rotations about a point conserves angular momentum
about that point. We find that if the action remains the same from from one
second to another, then the energy is conserved -- literally if someone tells
you they've got a "free energy device," they're saying "the laws of physics
that this thing uses won't be the same tomorrow as they are today."

But it can get even more interesting. For example, wavefunctions have a phase,
but shifting the phase does not change any observables. This shift turns out
to have relativistic consequences, and can be associated with conservation of
charge. (That's as I understand it, but I only saw a half-sketch of the proof
and I haven't worked it out myself, as I have for these other examples.)

~~~
kmm
Indeed, phase symmetry provides conservation of charge. In fact and this is
one of the most amazing things in physics, if you demand the system to be
invariant under a change of phase everywhere, a symmetry called U(1), you will
automatically find electromagnetism! A fundamental force of nature is a
consequence of symmetry.

This is the unifying principle of physics as we know it: the Standard Model.
Except of course for gravity on one hand and the fact that everything has mass
on the other hand (which these symmetries forbid). That is why the search for
Higgs boson is so important. It provides a way of breaking symmetries so
particles can have mass but still interact with various forces. (The reasons
why symmetry forbids massive particles are complicated).

~~~
creamyhorror
You've piqued my interest here with the linking of symmetry to the fundamental
forces and symmetry breaking to the existence of the Higgs boson. Thanks to
you and drostie for your intriguing posts.

------
xyzzyz
Her contributions are core foundations of a theory of commutative rings -- one
of the most important properties we usually want from a ring we study is for
it to be Noetherian, which means that it satisfies certain finiteness
condition, which lets one apply finite and computational methods to study it.
Many important theorems in abstract algebra state properties of Noetherian
rings, for instance famous Hilbert' basis theorem states that the ring of
polynomials over Noetherian rings is Noetherian itself. Other important
theorem, Hilbert's Nullstellensatz is frequently proved using a tool called
Noether normalisation lemma. Because of that, Noetherian rings, algebras and
modules are crucial to many subfields of algebra and algebraic geometry.

One of her other insights, which is especially important in the field I'm
interested in, was to notice that homology forms a group -- this led to
explosion of new results and marked the beginning of one of the most important
and influential field in XX century math, algebraic topology, as we see it
today.

------
yummyfajitas
The title "mighty mathematician you've never heard of" is odd. Aren't most
"mighty mathematicians" people we've never heard of?

It isn't surprising people in the mainstream have never heard of her.
Virtually everything she did was abstract algebra and the only connection her
work had to real life was studying some invariants of relativity (read: very
high level, very specialized). Similarly, how many people have heard of
Sierpinski, Voronoy or even Hilbert?

While important, the sort of work she did doesn't lead you to fame. As an
applied mathematician I never directly used any of her theorems. The word
"ring" (let alone "noetherian") doesn't even appear in my thesis.

There is only one famous mathematician in the world. His name is "that crazy
guy played by Russel Crowe".

~~~
harshreality
There is at least one more famous mathematician: Isaac Newton. But that proves
your point. How many average people could identify anyone else, even Euclid?

~~~
Someone
Newton a famous mathematician? I know he was, but the general public will only
know him as a physicist.

I guess most who do know him as a mathematician also know Leibniz.

As a mathematician, Pythagoras probably is way better known than Newton.

~~~
scott_s
Newton is a famous mathematician in the same way that Jesus is a famous Jew.

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lusr
Interesting: "being female in Germany at a time when most German universities
didn’t accept female students or hire female professors"

I'm currently reading "The Poincare Conjecture", which suggested that Erlangen
and Gottingen were very open to women, but reading the Wikipedia article it's
clear that the quote is correct and the book (or my memory of it) is
incorrect; she was one of only 2 women in a class of 986 at Erlangen, and she
was actively blocked entrance to Gottingen. It sounds like if it wasn't for
Gordan, Klein and Hilbert she might not have been allowed to attend at all.

~~~
pm90
Yes, even Lise Meitner, who discovered Nuclear Fission, faced similarly
difficult circumstances, and never got the recognition that she deserved (her
colleague Otto Hahn was awarded the Nobel Prize but she wasn't)

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zem
the interesting thing is that i knew who noether was because of her theorem,
but i got all the way through grad school without knowing she was a woman

~~~
philwelch
Was it relevant? Maybe not.

~~~
zem
it was relevant in that it'd help break people of the unconscious assumption
that all the great mathematicians they learn about are male.

~~~
philwelch
Fine, but worthy of an aside at best; mathematics lies before, above, and
beyond society and politics and I rather like it that way.

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sasvari
There's an _Emmy Noether Programme_ from the _German Research Foundation_ [0]
promoting young researchers:

 _The Emmy Noether Programme supports young researchers in achieving
independence at an early stage of their scientific careers. Young postdocs
gain the qualifications required for a university teaching career during a
DFG-funded period, usually lasting five years, in which they lead their own
independent junior research group._

[0]
[http://www.dfg.de/en/research_funding/programmes/individual/...](http://www.dfg.de/en/research_funding/programmes/individual/emmy_noether/index.html)

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dhimes
I will blow my own horn, here, and say that every one of my students has heard
of Emmy Noether, and a bit about her story.

I made it a point- following something like Eugene Hecht's lead in his
(brilliant) Physics Algebra/Trig text.

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dxbydt
Noether is quite famous. My university in New Mexico has a giant picture of
her in the "math TA room" ( this is the room where I spent my years as a grad
student solving the homeworks of undergraduates ..I mean, helping them solve
it :) For the non-math guys, the simplest example of a Noetherian is a Galois
Field GFpn - take p prime, n integer, then GFpn is a Galois Field with
elements from 0 to pow(p,n)-1. Example let p = 2, n = 2, you get the Galois
Field with elements {0,1,2,3}, with the Cayley addition table given by the
usual

    
    
           def add(a,b) = (a+b)%4

and the Cayley multiplication table given by the regular multiplication except
for

    
    
             2 times 2 is 3
             2 times 3 is 1
             3 times 3 is 2
             3 times 2 is 2 times 3
    

That's a _trivial_ Noetherian.

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cgh
"Prime Obsession" and "Unknown Quantity", both by John Derbyshire, contain
significant sections on her (more so in "Unknown Quantity", as it is a popular
history of algebra). Both are excellent books, though I'd have to give "Prime
Obsession" the edge.

~~~
rbehrends
I can second this recommendation. These books are excellent popular primers on
the history of mathematics; "Prime Obsession" is the story of the Riemann
hypothesis, "Unknown Quantity" is the story of modern algebra.

Also, if anyone has concerns because they don't agree with Derbyshire on his
political views, rest assured that both books are entirely apolitical.

------
agumonkey
The shadow-sister of Paul Erdos I guess

