# On Ada Lovelace Day

Recently there’s been a bit about Ada Lovelace and setting aside a day for noting “important women” in science. Why Ada and not another woman? Some ask, if not Ada, who? I say, not Ada. The only rational choice is Emmy Noether. There was nobody like her. Ever. This started as a comment (at my personal blog) on today’s link thread were this was noted. But it grew into post size, so I’ve promoted it.

The point I’m trying to make if had the name the top 5 most influential people in 20th century physics (ignoring their sex), Emmy Noether would be a top candidate for that list … or possibly even the top 3. The Ada Lovelace thing is for “famous women scientists”. Other names are suggested but … none of which have that stature. The big question is why don’t people recognize her? Have you heard of Emmy Noether? If not, one might ask is the reason why not due to sexism or anti-semitism (she was a Jew)? Is that a factor? Einstein was a Jew … and it didn’t diminish him .. but it’s a possibility I raise, especially noting in the 30s and 40s anti-Semitism was far more common than it is now.

One other possibility was that it was territorial, i.e., Noether wasn’t a physicist. One might think that it’s embarrassing (for physicists) that one of the biggest theoretical discoveries in your field to be made by some one who just stopped in looked at the maths in your playground for a bit and said, you know “I had this little idea, so I wrote it up.” And subsequently this little paper becomes the cornerstone of your whole science for the next century and counting. In part this is why I find the “Ada Lovelace” kind of thing questionable, there isn’t any question of who the most important women thinker/scientist of the last N years has been, where N is a number larger than 100 (1000? or 10000?). There’s only one candidate, and the other question might be was there anyone male *or* female who was more influential … perhaps there’s a short short list. There is not a single other woman who has dominated *two* separate fields of study and wrenched them both around in such a fundamental way. What men might you make the same claim for, what male scientist revolutionized two separate scientific fields? If you think there is a better candidate, put that name out there .. link or comment .. your choice.

So, was it scientific jealousy? Anti-Semitism? Or sexism? Or something else?

My commenter (this started as a comment response), noted he watches Discover/Cosmos type shows. So, in the nature of a quick “Cosmos” style precis, where does Ms Noether’s work fit in the grand scheme of things? (that explanation goes below the cut)

Einstein’s theory of General Relativity was one in which space-time became part of the thing which is malleable by matter in a back/forth interaction between space and matter. Kaluza-Klein in the early 20s (without checking the date) suggested putting General Relativistic equations in 5 dimensions instead of 4, but taking that 5th dimension and “curling” it up small. Think of putting a little “circle” at each point in space. This was the first modern theory that we now call a “gauge theory”. So after setting the problem up, they began working out field equations for interactions in that funny higher dimensional space. The results that fell out were surprising, to say the least, because Maxwell’s equations (electricity and magnetism) dropped out of those in a natural way. Remember from those shows on GR that the “curvature” of space time is related to the mass at each point in GR … well to “curve” in the one direction that sharply so it is really really little, you have to figure that the coupling and “masses” forcing interaction between fields is stronger, and not related to what we normally think of as mass. Electric charge fits the bill. You’ve probably heard of the “Standard model” and the other two forces (strong and weak which add to electricity and gravity) from those Cosmos-type shows. The theories that bring those all together are all gauge theories. How do we get from the Kaluza/Klein model to the Standard Model. Well, instead of just circles at each point, you put more complicated spaces. Remember those circles at each point.

If you suggest that the physics, the universe should not depend on exactly how those circles are set up, what that means in maths/physics speak is that this is a symmetry of theory/universe. Here is where Emmy Noether’s theorem comes in. She showed that whenever there is a symmetry… then the theorem tells you that along with that there is a conjugate “current” is which is conserved. What is this “current” thing? Well, for the Kaluza/Klein model (circles) this meant electric charge was conserved, which is something we see in nature. This connection and this way of linking geometry, charges, and fields together drove all of 20th century Physics and Noether’s theorem was at the root of it. This doesn’t even touch on the really important fundamental stuff she did *in her* real field of study (rings), which is truly astounding. Can you name another person who made one of the most important discoveries in a field that they didn’t work in?

Thus all gauge theories depend on Emmy Noether pointing out the very important very deep link between symmetries and conserved quantities. Experimentally this works the other way. When you start to figure out that you have something conserved, like charge, energy, or color (strong force) that means you have a symmetry.

Yang and Mills (mentioned in the aforementioned comment thread) were the ones who worked out how to put a more complicated non-abelian groups into those “little” spaces at each point. A group is a generalization of “an operation” like addition. Non-abelian means operations aren’t commutative. Recall from early math that commutative operations are those in which the order doesn’t matter. 2+5 is the same as 5+2. But turning left and taking 3 steps and then turning right and taking three steps isn’t the same as turning right and taking 3 and then left. Rotation+translation is not commutative. Putting on pants and then underwear is another example, the order of operations matters. The current “Standard Model” is a Yang-Mills theory in which groups called U(1)xSU(2)xSU(3) are the description of the “space” you put at each point. U(1) as noted is the electromagnetic part, SU(2) corresponds to the weak force and SU(3) the strong force, roughly speaking. The Standard Model, just vindicated by finding the Higgs boson, is our best, although kinda fugly, understanding of how the universe is put together.

One more final thing, recall in the previous paragraph we noted that groups are “like operations” such as addition. Well, groups and group theory are simple studies of just one operator. A “ring” is just a simple another step up from groups in that a ring is two groups working. Two operations instead of one. We all know about from arithmetic, where the two operations are addition and multiplication. Mathematicians were trying to generalize arithmetic from simple things like integers, rational number, to other things like polynomials and weirder things. So a ring is a set (like numbers or whatever) with two operations.

A “field” in the mathematical sense is a ring in which both operations have inverses (you can undo each of the operations). A ring is one in which some of the operations like * don’t have an inverse. What are called, Modular or Finite fields and rings are the simplest examples of these sorts of things. Do you know the “mod” operation in computer languages or spreadsheets? Modding by a number means taking the remainder after dividing by that number (8 mod 5 = 3 for example). If the number you “mod” by is a prime number then the result is a field (multiplication always has an inverse). If the number you mod by is not an prime number then what you are studying is a “ring”. For example in if you take the integers “mod 6”, you have the numbers {0,1,2,3,4,5}. One of those multiplicative operations is going to cause problems. Look at all the non-zero numbers in that set, if you multiply them (modding by 6 if they get bigger than 6) … all is OK except …. 2 * 3 = 0, which is going to cause problems when you try to invert that operation. But if you took the integers mod 5 (0,1,2,3,4) doing the same you find your multiplication table has no zeros. Everything can be safely “undone.”

The abstract theory describing rings is the field of study which was Emmy Noether’s primary field of study.

**Filed under: **Mark O. • Science

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