Saturday, March 27, 2021

Maximum generality isn't always the same as mathematical depth

An ATLAS excess: a new LHC paper has an interesting excess on Figure 6, Page 13, in the decay \(H\to aa\to bb\mu\mu\). The invariant mass \(m_{\mu\mu}\approx 52\GeV\). The confidence level is 99.95% (3.3 sigma) locally, 95% (1.7 sigma) globally. Note that in 2018, the CMS saw a dimuon excess at \(28\GeV\), close to a 2016-retro-seen \(30\GeV\) dimuon excess at ALEPH (LEP).
After he read my thoughts about the supremacy of physics-like proofs, Mr H. Turtledove (CZ) has exchanged a dozen of interesting e-mails with me that revolved around the importance, depth, mathematical ideas, physical ideas, and their relationships. I have tried to persuade him that the "fundamental depth in theoretical physics or string theory" is an entirely mathematical characteristic which is nevertheless important both in physics and in important mathematics. A mathematical characteristic may be crucial in physics because the laws of physics are constrained primarily (and perhaps only) by mathematics.

One obvious defining trait of mathematical thinking is the tendency to generalize. If you are asked to compute the total area of three yellow triangles and there is an obvious generalization to \(N\) yellow triangles, it is natural for you as a mathematically inclined person to solve the problem for a general \(N\in\ZZ\).

If you are a baby and you know how to generalize an insight to complex numbers, you should do it because that's what babies are all about. Similarly, if you're a schoolkid, you invent an interesting theorem, and you feel it could be generalized to \(D\)-dimensional shapes or cohomology or K-theory classes or derived categories of coherent sheaves, you should establish the generalization because that's what schoolkids normally do. ;-)

The lack of generalizations in mathematics classes is a sign of "non-mathematicians" in charge. Kids who learn too special things (e.g. fractions where the denominators have to be at most six) are not really learning full-blown mathematics. They are memorizing a few isolated factoids. A general integer denominator \(q\) in a fraction \(p/q\) is one "mandatory" way to generalize the concept of fractions. But adults' mathematics often generalizes things much more intensely – it builds several and sometimes dozens of floors of a skyscraper whose majority is incomprehensibly abstract for the laymen. Category theory is normally considered the "extreme outcome" of this desire to generalize mathematical ideas.

But the e-mails have reminded me of the fact that the non-experts' bias may go in the opposite direction, too. Some folks (I guess that I am mostly talking about the laymen but maybe some category theorists will see themselves in the mirror, too) understand that the generalization is a characteristic attribute of mathematics – but they overstate the importance of this trait and assume that it is the recipe for "everything" in mathematics, especially for the importance of the results. These people (I am not sure that Mr Turtledove is an example but I guess he is) believe that when you want to do really great mathematics, you should always generalize the results, drop some assumptions in your theorems, and the success is guaranteed.

Well, I beg to differ. As I wrote in the title, the maximum generality isn't the general recipe for the mathematical depth. Sometimes the pressure to generalize may be useless, unconstructive, pushy, shallow, annoying, and even stupid. One obvious example of these ideas is the mathematical concept of a group. A group \(G\) is a set of elements that can be denoted \(g,h\in G\) – so far I call them elements which is a tautological name because "the things inside any set" are called its elements but we may adopt other names for the elements (numbers, operations, matrices...) – that are equipped with a unary operation "identity" (which chooses a special element); and a binary operation \(f(g,h) = g\cdot h\) (various names and symbols for the operation may exist: addition, multiplication, composition; dot, plus, little circle, diamond...) that obeys associativity. There must also exist an inverse element \(g^{-1}\) for each \(g\). OK, once again, the axioms are:
  • \(\exists 1\in G:\quad \forall g\in G: \quad 1\cdot g = g\cdot 1 = g\)
  • \(\forall g\in G: \quad\exists g^{-1}\in G:\quad g\cdot g^{-1} = g^{-1}\cdot g = 1\)
  • \(\forall g,h,k\in G: \quad (g\cdot h) \cdot k = g\cdot (h\cdot k) \)
Simple enough and nice. Examples of groups \(G\) obeying these definitions are the integers \(\ZZ\) with addition, real numbers \(\RR\) or complex numbers \(\CC\) with addition, \(\ZZ_K\) i.e. the addition modulo \(K\), vector spaces, nonzero real numbers with multiplication, and non-trivial, non-commutative examples such as the permutation group \(S_N\), the even permutation group \(A_N\), groups of isometries of Platonic polyhedra, orthogonal, symplectic, unitary groups, exceptional groups, finite groups including 26 or 27 sporadic groups (the Monster group is the largest one among those), and many more. Clearly, groups are "deep mathematics" and they are also "the kind of mathematics that tends to be important in theoretical physics". It hasn't always been like that; the importance of group theory in physics only started in modern physics, basically in the 20th century.

Great. The three axioms above is a rather small number of axioms which makes the concept robust. But some people know even smaller numbers than three – such as one or two. And I am not even talking about zero; the number of axioms is probably not going to be negative. (The set of such smart people included Joe Biden when he came to the U.S. Senate 120 years ago.) OK, can't you and shouldn't you drop some of the three axioms of the group and create a more general, grander mathematical concept? And of course, you may drop an axiom – but it will have various consequences and the happy adjectives that you started with won't necessarily be accurate attributes of your "product".

If you drop the axiom about the existence of the inverse element, you will obtain a generalization of a group that is called a monoid. If you also drop the existence of the identity element, you get a semigroup. Yes, help yourself, you may also drop the associativity and then you get a grupoid, binar, or a magma. I wanted to be specific enough in the case of the group because it's a good idea to understand an example in detail. (There are no other at least semi-interesting combinations because the identity element's axiom requires the inverse elements to exist; and the inverse element axiom is only useful if there is an associative binary operation.)

Because a grupoid assumes less than a group, it must be more general and therefore more important than a group, right? I think that while you might call it more general for obvious reasons, it is not more important. (And monoids and semigroups aren't more important than a group, either.) On the contrary, a group is much more important than the three generalizations. While the generalizations as "mere structures" are more general than the concept of a group, you can't say the same thing about the "products" that actually justify the work with the definitions of a group – the theorems. Why not? Because most of the interesting lemmas and theorems that can be proven for a group don't exist at all for monoids, semigroups, let alone grupoids.

How is it possible? Well, it's because most of the interesting, nontrivial theorems require all the operations in the group to exist and to obey the axioms. Without these operations and/or axioms, most of the interesting, nontrivial theorems don't hold – and many of them can't even be formulated. You may imagine that a group is like an organism; the identity element is like its heart that is needed to pump blood into the body and bring oxygen, energy, and nutrients to various organs; the binary operation is like sex that is needed for reproduction (which is needed for survival of the structures after a single lifetime); and the associativity is a condition for the sex to work. I sincerely hope that the TRF readers whose average age is 81 aren't like children and won't think about the associativity as a sexual position (of three people?). The metaphor isn't supposed to work at this precision although it is possible that some of you will actually prove that it does surprisingly work at this precision, too.

So a group is a much deeper mathematical concept than a monoid, semigroup, or a grupoid. Everyone should think about this assertion, whether he agrees with it, and why it is true.

You know, the recommendation "you should generalize all your results from groups to monoids" may sound like a good policy for mathematicians but the problem is that it is an extremely cheap advise. It is cheap because it is prepared for any situation. It only says "always generalize". At this level, "always generalize" is just a stupid slogan. It simply doesn't universally work. If it worked, all concepts would already be known in their generalized form and we wouldn't even discuss the need to generalize things further.

It's simply a fact that the generalizations such as monoids don't represent the same rich fountains of interesting structures and mathematical wisdom like groups, the full-blown objects. My analogy makes the reasons clear. A monoid is a group without the identity element which I said to be the "heart". Without a heart, an organism dies. You want the things to be living, thriving, and reproducing, so the heart – the identity element – is a useful if not essential feature. Similar comments apply to the existence of the binary operation and its associativity; and the existence of the inverse elements.

So in many cases, there exists a Goldilocks location or a golden ratio or the best compromise which maximizes the mathematical usefulness, fertility, or depth – and these three attributes, while similarly desirable, aren't necessarily equivalent to each other. If you added some extra axioms to the group axioms, you could get some special groups that may even be non-existent. If you omitted some axioms, you would get "dead or impotent" mathematical structures, as we have already explained.

But similar "optimal compromises" in the middle exist in many situations. For example, higher-dimensional geometry looks increasingly hard for humans to visualize and the laymen assume that things become really insane if you consider 6-dimensional or 11-dimensional manifolds. However, the complexity computed in a more professional way says something else. The topology of the 3-dimensional or 4-dimensional manifolds is actually the "peak" and things simplify if you consider fewer-dimensional but also higher-dimensional manifolds than that! To say the least, this surprising "peak near our home" exists in the sense that it was harder to prove the Poincaré conjecture for this seemingly mundane dimension \(D=3\) than for other dimensions. Some other attributes of the spacetime dimension (perhaps those that include black holes) are maximized for \(D=10\) or \(D=11\), the critical dimensions of string theory and M-theory, respectively. There are lots of such examples.

The cheap recommendation "always generalize" or "always drop some axioms for the mathematical structure" resembles some left-wing policies such as "always increase the budget deficit and pour trillions in the helicopter money everywhere". Well, these aren't great policies. To say the least, they are not universally great policies. The idea that such simple recipes are always great is an ideological prejudice – an unavoidably left-wing one, I would say – which revolves around the ideologue's inability to see (or his dishonesty which manifests itself in his denial or obfuscation of) the other side of the coin, the disadvantages or costs of his recipes (of the generalizations, dropped axioms, helicopter money, and so on).

Aside from its being a cheap recipe, the universal recommendation "always drop some axioms" is just a left-wing ideology for another reason: the idea that this is a "good thing to do" is supported by some extreme egalitarian beliefs. The leftists increasingly often want all the other people to parrot self-evident, ludicrous lies that all the individuals and members of all groups are equal in all respects. The tendency to switch from groups to monoids... or grupoids may be interpreted as examples of a similar egalitarianism that has run amok. The left-wing interpretation of the axioms is that they are things that break the equality between all people or mathematical structures and one has the duty to drop these axioms because one has the duty to fight against the discrimination. So drop the racist requirement that the poor guys (groups) need to have an associative binary operation, the leftists say. (I am only talking about high brow leftists who know and fight against groups in mathematics; the common ones are mainly fighting against the "racist" identity 2+2=4 nowadays.)

I am sorry but the discrimination is overwhelmingly a good thing. Our lives – and the interesting part of the life of the Universe – is all about the discrimination. Everyone who fails to discriminate is a braindead pile of homogeneous manure. In our particular case, we are saying that monoids that are also groups are superior relatively to monoids that are not groups. In my metaphor, monoids that are not groups are people without hearts. It sucks to be a person without a heart – such entities are really emotionless beasts who are also dead, to make it even worse. You can't live without a heart. There is nothing wrong about the discrimination directed against the people without a heart – or against the people who are already dead. For example, it's totally right not to allow the dead people to lie on the sidewalk for many days. They can always whine like other SJW losers, help yourself, corpses! ;-)

OK, so structures with the right collection of operations and axioms that they obey are superior. They are more interesting, more usable in mathematics, you will be able to find a greater number of theorems. This inequality is completely omnipresent in mathematics (and in the real world) and the left-wing ideology prevents many people from appreciating this point – which means that they are totally incapable of any kind of real mathematical thinking. Let me start the final portion of the text with the inequality\[ 196,883 \neq 196,885. \] OK, I hope that you agree even though some hardcore leftists could have a trouble with this inequality right now. The numbers aren't random at all, the first one is the dimension of the smallest nontrivial linear representation of the largest sporadic (and the most interesting finite) group in group theory, the Monster group (click to get reminded of the monstrous moonshine). But my point surely wasn't written just to point out that two large integers may differ by 2 parts per million. Instead, we want to discuss the mathematical depth \(d(N)\) of some integers which is a difficult thing to define. But let's be slightly operational and define it as the number of appearances of the integer in mathematical papers in the following 100 years weighted by the number of citations and other things. OK, again, the depths of the numbers will be different:\[ d(196,883) \neq d(196,885). \] Different numbers are not created equal, not even when it comes to their importance in mathematics. I guess that most leftists' psychological problem may dramatically increase at this point. To make things more troubling for them,\[ d(196,883) \gg d(196,885). \] The number 196,883 is vastly more mathematically important than 196,885. You can scream, you can whine, you can jump from the window, you may elect Joe Biden or Kamala Harris, but that's everything that you can do against this important mathematical fact. The number 196,883 is a dimension of an extremely important representation of an extremely important group; while 196,885 is not. The numbers are simply not equally important at all. They differ by many categories.

To summarize, many people believe in misconceptions such as "it is always great to drop the axioms" or "send the dimensions to infinity" or "all monoids are created equal, whether they are groups or not" or "all numbers are created equally important in mathematics", among other things. Virtually all similar extreme assumptions are wrong and they may be identified as consequences of the people's extreme left-wing ideological prejudices, their tendency to overlook all consequences of some axioms and decisions that have a certain sign, their desire to adopt cheap solutions, and their habit to parrot simple slogans. Mathematics and science don't work like that at all. Inequality of things and objects is omnipresent and paramount and the most interesting things and mathematical structures – the results of an optimization procedure – are often found "in the middle", not in some extreme corners.

And that's the memo.

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