As I understand it, in 1968 a young Italian physicist was attempting to find an equation which would describe the strong force (the force that binds protons and neutrons together). He happened upon a book that contained an equation by a Swiss Mathematician named Leonhard Euler (1707-1783), which seemed to magically describe this force.
The discovery would eventually lead the physics community towards string theory, which has become an explosive channel of theoretical physics (or maybe not, depending on how valid it is).
The point is, isn’t this a really wierd way for it to come about? IANA mathematician, but to me it seems almost like witchcraft. How the heck did one guy’s mathematical musings translate into so-much-of-everything?
I’m sure Euler could not have known about what he was stumbling onto, but then how did he (by chance) begin to describe this fundemental facet of nature? Maybe because IANA Math wiz I don’t know, so please explain it to me. Isn’t it kind of wierd?
I don’t think it’s any wierder than Fibonacci sequences describing plant growth, or fractals describing any number of chaotic natural processes. It’s inevitable for math and nature to coincide from time to time.
I can’t answer your “question” but I can add to your amazement. Euler’s number (2.718281…) also factors into the answer of a seemingly unrelated problem (known as The Secretary Problem). I’ll give you my favorite presentation of the problem:
I’ve got 100 bags of money, each with an unknown quantity, and am willing to let you keep one. You can open one bag at a time and at that time you must decide whether you will keep the amount in that bag or choose another. Once you’ve given up one bag for another you can’t go back and choose a previous bag. How can you maximize the probability of getting the bag with the most money?
Now remember, you don’t even know the range of the distribution. The only info you have is the number of bags I’m willing to let you try and the amount you’ve seen in any bag you’ve opened.
Well the answer is, you should open up
1/e*n bags (where n is the total number of bags to select from) and then select the next bag that has an amount higher than the highest of any opened previously. So in this case, you would open the first 36 bags, noting the highest amount in any of those bags, and then keep the next bag that has any amount higher than that. Interesting and practical when people are handing out bags of cash and only letting you pick…you get the idea.
I’m not familiar with Euler’s equation as it relates to the strong force but is the relationship merely coincidental? I’m asking for clarification because I wouldn’t describe the relationship between the Fibonacci sequence and natural processes like the growth of a nautilus shell or the rings of seeds on a sunflower plant as “just coincidence.”
Well euler’s equation apparently models harmonic subsystems (say " looped strings") and since superstrings act like looped strings the equation is a perfect fit.
Phi is the value of the golden ratio, which doesn’t actually come up in mathematics as much as it does in other fields. e and pi are really the two big ones in my experience.
People in X-ray Crystallography (developed about 90 years ago) use something called Spacegroups to describe how molecules/proteins order themselves in space (e.g. a molecule at the corners of a cube, or in a screw formation). It is my understanding that the 200+ spacegroups were all mathematically solved hundreds of years ago.
Broadly, what Veneziano discovered - the Veneziano amplitude - was surprising at the time, but it wasn’t weirdly surprising.
A bit of context. Physicists describe how particles interact via quantities called scattering amplitudes. Here one can just think of this as a particular function. Ideally, one would like an underlying theory from which one can calculate what this function is. This was a problem in 1968 because nobody then knew what that theory might be in the case of strong interactions. However, regardless of what that theory is, you can guess that the function has to have certain properties. In 1968 some of these came from deep physical principles - things like causes always precede effects - and some were judicious guesses. It was even hoped that these properties by themselves might be enough to uniquely pick out the function involved.
Veneziano was being somewhat less ambitious. By this sort of reasoning, he had a set of properties he thought the scattering amplitude had to satisfy. All he wanted to find was any function that satisfied these properties. He rummaged through the maths library and discovered that a Beta function fitted them.
Now, the thing is, that’s not that uncommon a tactic for theoretical physicists to resort to in certain circumstances - I’ve done it myself and had it work. You figure out that you’re looking for a function with certain properties and you just start hunting through the books to find one. You’re not quite looking at random: experience and an intuition for what sort of thing the answer might be certainly help. Sometimes it’s a matter of spotting a function a bit like the one you want and then seeing if you can fiddle with it to get an answer that fits. There’s luck involved as well and you can easily waste hours and turn up nothing.
I suppose, however, the question really is why we expect this tactic to stand a chance at all? Well, certain functions crop up again and again. Indeed, any theorist will instinctively have a set of tables to hand listing dozens and dozens of such special functions, with all their properties set out. In this case, the Beta function is related to the gamma function (also invented by Euler), which crops up everywhere. Even a Beta function is a pretty common beast to run across.
Special functions were typically originally defined in some fairly simple situation. For instance, the gamma function is a way of extending the notion of a factorial
n! = 1 x 2 x 3 x … x n
to values of n other than 1, 2, 3 … Because they’re rooted in these simple situations, the functions usually have at least some nice properties. And it’s those properties of them that one likes in using them. In the Veneziano case, the nice properties of the Beta function relate to physical symmetries. And it’s exactly those symmetries that Veneziano was starting from.
In a sense, the surprise about the result was not that it was this obscure “magical” formula, it was that his answer was so straightforward, simple and familiar. However, it quickly turned out from experiments on the strong force that the Venziano amplitude could only be a rough approximation to the actual scattering amplitude. In the years that followed, QCD came along as the theory of strong interactions and it’s now known that the actual function here must be much messier than Veneziano’s.
As for the maths involved in string theory being invented long ago, the more usual complaint from string theorists is that mathematicians have yet to invent the stuff they need.
In fact, one of the common arguments in favor of string theory is the fact that it’s guiding researchers into finding new mathematics. The idea is that the physicists wouldn’t be smart enough to invent all of this beautiful math on their own, just smart enough to find that math in the real world.
As for Euler and the strong force, it’s hardly surprising that something Euler found is relevant to the strong force. Euler found things that are relevant to a whole heck of a lot of stuff. But it’s worth emphasizing that it’s a far cry from knowing all of the necessry mathematics to putting it together into physics.
Mathematics can be divided into two categories: pure and applied. While these two categories are not totally disjoint, pure mathematics is considered to be theoretical while applied mathematics (as the name suggests) applies to some sort of physical phenomenon. After learning about how Newton’s stumbled upon the foundations of calculus I assumed, like most people, that mathematics is created because of the need to model some physical phenomenon. However, I was surprised to learn that in the vast majority of cases the math usually exists long before it is used to model some physical phenomenon! That’s one of the go to arguments that a pure mathematician uses when applying for grant money! That is, they will claim that the mathematics that they are investigating while merely an intellectual curiosity at the time may eventually be used to model the human genome or a black hole.
How about giving some actual cites from real-life grant applications that theoretical mathematicians do this? If they’re for government grants, they should be publicly available. So go to it. Dig them up and prove this claim.
I know this is a zombie thread, but the topic is a big unresolved philosophical issue, as I understand it. It’s not just a single equation, but mathematics as a whole which is just effective beyond all expectation at modeling, describing and predicting the behavior of the universe.
It is easy enough to imagine a universe that wasn’t so mathematical, or only slightly so. But as it is, the book of the universe is written in mathematics and if you want to understand, or expand our understanding of, the universe, you must know math. Why? I don’t know. That’s just the way it is.
Prove what claim!?!? Do you really doubt that a mathematician would claim that his or her toy ODE or PDE (or whatever) might have some significant real-world implications? Is that something that really warrants corroboration or are you just trying to play gotchya!? If you’re really that interested I can send you one of my own proposals, which btw is not anywhere online as far as I know. I would just need your email address!
The topic of the thread was that it is rather amazing that the Euler equation which was derived around 300 yrs ago applies to the strong nuclear force! My point was that the mathematics oftentimes, if not usually, sits on the shelf well before it is used to model some physical phenomenon! It seems to me that citing grant applications is not germane to this particular thread!
Real-life theoretical mathematicians do not try to claim that their work will yield results that can be used in physics. 99.9% of their work is so abstruse that it will never be put to practical use. The remaining 0.1% gets attention when it does, but that’s a totally unexpected bonus.
I won’t deny that some mathematicians will try to work on known problems in physics. Almost by definition, though, that’s “applied” math not “pure” math. They are not discovering basic principles but using known pathways to give results. Theoretical math is done for the interest of mathematicians and no grant committee would accept a proposal that claimed otherwise.
Um…You’re the one that wanted me to cite specific grant proposals! That is, you brought it up.
You’re really trying hard to confuse my simple point that the math commonly exists in theory before its used to model some real world phenomenon! That’s why you chose to believe that my questions in my previous post are rhetorical. Because you did not want to answer them! You choose to focus your attention on the notion that no mathematician ever claimed, in a grant proposal, that his or her work MAY have more significant repercussions other than that of a theorretical nature. I’ll grant you that someone who works on something like set theory, algebraic topology, or category theory probably does not claim that their work might apply to worm holes or quantum tunneling. I was thinking about some areas of math that could fall into either the pure or applied category. As I mentioned in my original post, pure and applied math are not mutually exclusive. But I was not so overly concerned about being that precise! I just didn’t think anyone would be that sensitive. What I do know for sure is that people in my areas of math are not ashamed to claim that their work MAY apply to something in the real world in spite of the fact that, at the point of writing the proposal, the work is strictly theorretical!
I think most of the people on this thread made some interesting and intelligent points. I thought I had one myself. I’m sorry if you didn’t like it. But I don’t want to waste any more space on this thread with this bickering. You have the last word.