The Mighty Mathematician You’ve Never Heard Of(nytimes.com)
nytimes.com
The Mighty Mathematician You’ve Never Heard Of
http://www.nytimes.com/2012/03/27/science/emmy-noether-the-most-significant-mathematician-youve-never-heard-of.html
11 comments
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.
"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.
> Also, in my personal experience, she is quite well known.
True in my experience, also. My very first college math professor took 15 minutes out of lecture, in Calculus I, to tell us about her. He culminated by naming his dog after her, which I guess was his highest form of praise.
Don't get me wrong. Deriving conservation of energy from time invariance, conservation of momentum from linear translation invariance, etc. is totally cool. But it's not really accurate to say that she's not well-known.
True in my experience, also. My very first college math professor took 15 minutes out of lecture, in Calculus I, to tell us about her. He culminated by naming his dog after her, which I guess was his highest form of praise.
Don't get me wrong. Deriving conservation of energy from time invariance, conservation of momentum from linear translation invariance, etc. is totally cool. But it's not really accurate to say that she's not well-known.
> 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.
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.
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.
"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.
Thank you for taking the time to share this. I didn't appreciate that her work was a far-reaching as it was.
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.)
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.)
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).
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).
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.
Leonard Suskind gives a nice and accessible Stanford lecture based on this line of attack at http://www.youtube.com/watch?v=FZDy_Dccv4s#t=27m12
I think you mean action(A : A') = - action(B' : B)?
Sorry, yes. I went back and forth between -action(B' : B) and +action(B : B') and I apparently settled on an inconsistent phrasing! Silly me...
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.
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.
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".
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".
I learned Sierpinski's name in high school (he had a pretty carpet, don't you know); I'm pretty sure I learned Hilbert's name from a Nova episode at around the same time.
I first heard Noether's name last year, I think, from some article like this one. Of course, the notion that "where there is a symmetry, there is a conservation principle" came up half a dozen times during my physics/EE Ph.D. studies, but nobody ever suggested that this might be a theorem with a name, let alone the name of Noether.
Your point is well taken, though. A major reason why the female scientists and mathematicians of history are largely invisible, unless you make an effort to see them, is that all scientists and mathematicians are invisible to first order. To get your name out there, even to fans of such things, you have to do a bunch of self-promotion. And self-promotion is a nonlinear endeavor: A little additional friction – from, say, having to lurk quietly at the back of lecture halls and slink in and out the back door of the office, lest you be forced to defend your very presence from a handful of people who are made uncomfortable by it – and you're behind in the fame game.
I first heard Noether's name last year, I think, from some article like this one. Of course, the notion that "where there is a symmetry, there is a conservation principle" came up half a dozen times during my physics/EE Ph.D. studies, but nobody ever suggested that this might be a theorem with a name, let alone the name of Noether.
Your point is well taken, though. A major reason why the female scientists and mathematicians of history are largely invisible, unless you make an effort to see them, is that all scientists and mathematicians are invisible to first order. To get your name out there, even to fans of such things, you have to do a bunch of self-promotion. And self-promotion is a nonlinear endeavor: A little additional friction – from, say, having to lurk quietly at the back of lecture halls and slink in and out the back door of the office, lest you be forced to defend your very presence from a handful of people who are made uncomfortable by it – and you're behind in the fame game.
On the contrary, I think being a woman probably helped Noether's fame. Off the top of my head, I can think of precisely two algebraists: the woman (Noether) and the guy who was killed in a duel (Galois).
So you think that being a woman helped her fame, when the whole point of the report was how little she is known, even in many mathematics circles?
As other commenters have mentioned, she probably isn't as unknown as the article suggests. I'd have thought that her symmetry work is pretty well-known in the field of Physics, and credited under her name.
I'm not a mathematician by any stretch, but am a reasonably well-qualified geek. I've run across Sirpinski, Voronoy, Hilbert, Nash, and quite a few other mathematicians in my day (mostly on account of being a bit of a Mandelbrot groupie as a teenager -- I met him once after a lecture, and he seemed baffled by the existence of teenaged groupies...) -- but I've never run across Noether before. She sounds pretty incredible, and I'm grateful to have read this article.
I'm sure you're correct that mathematicians will never be truly "famous" amongst the general public, except to the extent that they're played by Russel Crowe. But my own experience would suggest that Noether could use a bit more exposure within the non-mathematician general-purpose geek populace.
I'm sure you're correct that mathematicians will never be truly "famous" amongst the general public, except to the extent that they're played by Russel Crowe. But my own experience would suggest that Noether could use a bit more exposure within the non-mathematician general-purpose geek populace.
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?
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.
I guess most who do know him as a mathematician also know Leibniz.
As a mathematician, Pythagoras probably is way better known than Newton.
Newton is a famous mathematician in the same way that Jesus is a famous Jew.
Surely most everybody who has been to high school has at the very least heard of calculus?
Also, it strongly depends on where in the world you ask, many Londoners will have heard of Christopher Wren for instance. And loads of people know of classical mathematicians such as Archimedes and Da Vinci. Also you get the crossovers such as people like Ada Lovelace, who will be known to many fans of Byron.
Also, it strongly depends on where in the world you ask, many Londoners will have heard of Christopher Wren for instance. And loads of people know of classical mathematicians such as Archimedes and Da Vinci. Also you get the crossovers such as people like Ada Lovelace, who will be known to many fans of Byron.
Gauss? Pascal? Descartes? Russell? Fermat? Knuth?
You probably need to take a survey.
You probably need to take a survey.
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.
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.
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)
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
People generally seem to assume that mathematical and CS theorems are named after men rather than women. I'd be surprised if most programmers could tell you that Liskov (of Liskov's substitution principle) was female.
Was it relevant? Maybe not.
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/...
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/...
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.
I made it a point- following something like Eugene Hecht's lead in his (brilliant) Physics Algebra/Trig text.
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 2 times 2 is 3
2 times 3 is 1
3 times 3 is 2
3 times 2 is 2 times 3
"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.
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.
Also, if anyone has concerns because they don't agree with Derbyshire on his political views, rest assured that both books are entirely apolitical.
The shadow-sister of Paul Erdos I guess
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_...