The equal sign is innocent, it doesn't prevent us from studying some very abstract geometric structures
A few days ago, Kevin Hartnett wrote a provoking article about mathematics for the Quanta Magazine,
The human hero of the story is Jacob Lurie, a mathematician who recently moved from Harvard to IAS Princeton. I do believe that he is an extremely good mathematician – also because of his classifications of topological field theories, research into exceptional groups, and more. But the Quanta Magazine article is about Lurie's somewhat idiosyncratic hobby, namely his holy war against the equal sign.
There shouldn't be any equalities like \(2+2=4\), the article argues. Instead, all equalities should be loosened to equivalencies, like in category theory, which are better because equivalences come in many flavors and degrees. And the original category theory of Eilenberg and MacLane (1945) wasn't loose enough which is why we need to switch to "infinity categories" and Lurie's articles that are some 8,000 pages long in total.
The text makes it very clear that the number of mathematicians who actually understand the argumentation with all the relevant details is very small. The rest of us must unavoidably guess. We're told by another mathematician that "No one goes back once they’ve learned infinity categories". Sounds like a far-reaching statement. But isn't the reason why no one goes back that he would be crucified for apostasy by the "infinity categories believers"? Is there an actual conceptual reason why it's better to switch to the "infinities categories"?
I think that if there existed a rational reason why we "shouldn't ever go back from infinity categories to a simpler framework", we would have already heard it – or at least its sketch – just like I can give you a sketch of the rational reason why "we never want to return from string theory to quantum field theory again".
Another mathematician reported that people often tell her "it's in Lurie somewhere". She would ask "Oh really? You're referencing 8,000 pages of text. That's not a reference, that's an appeal to authority." Even though I have also provided others with a similar vague "reference" in the past, I agree with her. It's sensible to expect at least that most of such answers are just appeals to authority, somewhat demagogical tricks to manipulate the person who asks and stop her from asking.
Equality or equivalence
Lurie is a typical guy who tells us "equivalences are good, equalities are bad". Equivalences are supposed to be better because they have many flavors and subtleties etc. Such comments may be presented as virtues if you deform your lips in a happy way. But this flexibility may also be framed as a potential vagueness or a related weakness. Hartnett's article reports that some mathematicians admit that proofs within the "infinity categories" paradigm may be heuristic and potentially incomplete.
What is the difference between an equality and an equivalence? An equality is something like\[
\] (there's a much faster MathJax 3 but my upgrade from 2.7.6 just didn't work last night, probably because of some other scripts or widgets, it is so frustrating) and it says that both sides are exactly the same. Note that "equation" isn't quite the same thing as an "equality". An equation is a "problem" to find some values of variables fo which an equality is obeyed. OK, in the equality above, we assume that a priori, both sides are elements of some set, in this case a set of numbers (integers, real numbers, complex numbers, the equality above works in all of them) and generically, the left hand side and right hand side could be two different elements of the set. But it's being said that they are the same element which is a nontrivial statement.
An equivalence is a relation between objects – elements of a set – that is reflexive, symmetric, and transitive. For each pair of elements in the set, the relation says "yes" or "no" – whether the relation between these two elements exists – and it must be true that \(x\) is in relation with \(x\) for any \(x\); \(x\) is in relation with \(y\) if and only if \(y\) is in relation with \(x\); and if \(x\) is with \(y\) and \(y\) is related to \(z\), then \(x\) must be in relation with \(z\). OK?
By this definition, an equality is clearly a special case of an equivalence. It's the "equivalence with the finest resolution". It distinguishes things maximally. But there are also "looser" equivalences. For a general equivalence, you may divide the set into "equivalence classes" such that the relationship holds within each equivalence class, but never holds in between two different classes. For this reason, the relation \(x\heartsuit y\) is logically equivalent to (I really mean that the truth value is equal to the logical value of)\[
C_\heartsuit(x) = C_\heartsuit(y)
\] i.e. to the statement that the equivalence class \(C_\heartsuit\) of \(x\) and the equivalence class of \(y\) are the same. For example, the equivalence "two real numbers \(x,y\) are equal up to the sign that may differ" may be concisely written as \(x^2=y^2\). So just like the "anti equal sign" warriors may say that "every equality should be replaced with an equivalence", I may at least equally convincingly argue that equivalences should be banned and they should be replaced with equalities. If you allow a special "function", the equivalence class for any element (the function depends on the relation, that's why there is the heart in the subscript), then the general equivalence class is nothing else than another equality – an equality between equivalence classes.
(By the way, there should be another extended discussion of a third related term, "isomorphism". It's an equivalence between two objects which must be sets – and this equivalence also comes with some identification of the individual elements of the sets which isn't always uniquely given. For the "isomorphism" to be a full-blown isomorphism and not just a bijective map, a "naked version of an isomorphism", this map between the elements of the sets must preserve some extra structure – operations – that exist on these two sets, e.g. the composition rule on two groups in the group theory sense.)
It seems obvious to me that the difference between these two ways to describe \(x\heartsuit y\) i.e. \(C_\heartsuit (x) = C_\heartsuit(y)\) is purely bureaucratic, it's all about the appearances. These two ways of talking or thinking only differ as much as the Czech language differs from the English language. In some contexts, we may save some space by the first way of talking, in other contexts, it's reversed. It's not a terribly profound difference, it's not rocket science. We don't lose anything and we don't gain anything tangible if we switch from one way of talking to another. The equalities look clearer to me because "they have no strings attached". We know that there are no "looser" ways to obey the condition. The things must be the same elements of a set. They must convey the same information where the word "information" means nothing else than "how to pick an element from a set".
On top of that, we just need equalities or equations in physics. Newton's equations, Maxwell's equations, Schrödinger's and Heisenberg's equations, and other differential and similar equations. This is partly due to the omnipresence of real and complex numbers in physics. However, even when the objects are more complex or more abstract than "real and complex numbers", we still need equations in physics.
In quantum mechanics, the complete physical information about the state of the object is encoded in the density matrix \(\rho\) – which is equal to \(\ket\psi\bra\psi\) whenever the state is pure. And it's important that the method how this \(\rho\) was prepared (or your particular decomposition of \(\rho\) as a sum) is physically indistinguishable. If you prepare a particle in the mixed state \(\rho\) with some values of the matrix elements of this density matrix \(\rho\), you know everything that may affect the probabilities of all future measurements and nothing is missing.
If someone says that \(2+2=4\) is "too strong" because \(2+2\) is "slightly different" from \(4\) because it was prepared differently, he is making the mathematics sloppy. It's true that the preparation protocol may matter for some viewpoints. But it doesn't affect the future predictions in physics if the object is \(\rho\), for example. And even more importantly, well-defined mathematics should also know whether some aspects matter or not.
In other portions of the article, the homotopy is being promoted as an example of the "loosened equivalences". Well, homotopy is surely a nice piece of mathematics and it's important. No doubts about that. A homotopy is some relation between manifolds that may be continuously deformed to each other although I would need to be more specific if I really needed to distinguish homotopy from some similar equivalences in topology.
But when we deal with lots of the interesting facts and general rules involving homotopy, is it really useful to be a "holy warrior against the equal sign"? Is it useful or true in any deep sense to pretend that equivalences (e.g. the homotopic equivalence) are the morally sounder replacement for equalities? I just don't see how it could be true. The homotopical equivalence is something else than the equality. They can co-exist and they should co-exist. They don't need to kill each other all the time. The Ethiopian prime minister would probably agree.
Hartnett's article discusses some mathematics that is even more profound. When you show that two manifolds are homotopically equivalent, you connect them by a path. The very existence of the path is enough for the proof of the original problem and in this sense, it doesn't matter which path you found. On the other hand, it may also be interesting to look at the path – and clump the paths that may serve the role according to their own homotopy, whether they're connected with other paths. And the way how these paths are connected to other paths – by a path on a higher space, a meta-path – is also interesting. And this complexity may be repeated infinitely many times – you may discuss the connectedness on the space of paths of paths of paths... with as many copies of "of paths" as you want. (This may be repeated "infinitely many times" and it's the real reason why there is the word "infinity" in "infinity categories".)
All of this seems very likely to be profound mathematics to me – and probably very important for the ultimate, most universal, understanding of quantum gravity or string theory (this is just a feeling, I don't have any hard or semi-hard proof of the assertion). Some of it sounds close enough to the associahedrons and other cousins-of-stringy-ideas. Well, I also know that some direct efforts to use "higher categories" in physics are just physically misguided. On the other hand, I don't see why the importance of the connectedness on the "space of paths of paths of paths..." should prevent me from authoritatively stating that \(2+2=4\) is a valid equality with no strings attached! ;-)
Maybe Danny from the Alternative Math film insisted that \(2+2=22\) instead because he was working within some infinite categories, on a PhD thesis under Jacob Lurie! Maybe all the bigots who have harassed the old-fashioned teachers weren't just bigots – they could have been infinite category theorists, too. ;-)
You may see that there are certain "ideological tendencies" in the direction where some people want to push mathematics. Category theory and especially Lurie-style "higher categories" and "topos theory" want to unify mathematical statements so that they are as general as possible, and some patterns of proofs may be directly copied from one context to another. And they're actually often doing it.
But I think that if mathematics becomes at least somewhat ill-defined and ambiguous, it's just too high a price to pay for this generality. When a proof is being imported from one context to another, I think it's totally appropriate that one needs to check in some detail whether the original proof still works in the context. There is nothing wrong about spending a nonzero amount of time by "creating an analogous proof" out of the original template.
After all, I think that if the amount of caution and work needed to transfer the proof from one situation to another were exactly zero and if the transfer were still legitimate, then we should literally say that it's the proof of the same thing, and it would be unjustified to say that the category theory gives us the power to easily prove a "greater number" of statements! It would still be just one statement.
For two situations (like classical physics and quantum mechanics) to be considered different at all, it is important to acknowledge that not every statement or structure may be transferred from one to the other! This is why I consider the ideology in which "proofs should be mindlessly copied through functors into other categories" etc. to be fundamentally misguided.
Many proofs or other structures and constructions are complex enough and their "brute force parts" may be encoded by some diagrams or graphs. And these diagrams and graphs may be equally useful for an analogous proof or structure or construction in another context – where the interpretation of the nodes and links is different. If that's the case, good for you. You may save some time because the part of the problem consisting of the "manipulation with the graphs or diagrams" may be sped up tremendously – the same work has been done before. But it's just a part of the procedure and it's always important to realize that in any situation that deserves to be called different at all, some previous procedures may cease to hold.
Heuristic arguments and "sketches of proofs" are extremely useful and valuable – and this statement is particularly natural for a theoretical physicist. On the other hand, mathematics – by its definition – always tries to turn vague statements and heuristic statements into well-defined, rock solid, and rigorous ones. In mathematics, the heuristic justifications should always be considered as a "temporary state of affairs". Any ideology that has the effect of weakening this clarity of mathematics is deeply counterproductive, I think.
Also, I think that the category theory philosophy was partly motivated by the efforts to "precisely attribute" why some proof works to some minimum assumptions that are needed for that. This is a nice goal but it's less "existentially important" than the clarity and well-definedness of mathematics. On top of that, I think that the behavior of the "category theory believers" becomes highly hypocritical because in their effort to promote categories in the discussion of homotopy and related concepts, they persuade the people that there's something politically incorrect about \(2+2=4\).
Sorry, Danny and Lurie, but \(2+2=4\) is just right and doesn't prevent anybody from deepening the knowledge of homotopy or paths on the space of paths on the space of paths 2019 times repeated. And quite generally, I think that mathematicians should clearly distinguish the "beef of mathematics" from their personal idiosyncratic obsessions about the "preferred formalism or notation that should be used to convey an idea"! Your mental instability notwithstanding, the former is far more important than the latter.
Concerning the previous sentence, one must acknowledge that people's understanding of the depth of proofs may differ. As an undergrad – already firmly decided that theoretical physics was the deepest layer of ideas – I was often provoked by the statements by mathematicians from "set theory" etc. that the axiomatic mathematics, set theory, and Gödel's stuff is the philosophically deepest part of the human knowledge. So I actually did learn this stuff at the level to have the best grade from set theory etc. Did I end up agreeing with them? Well, I surely did learn some interesting enough things about set theory and proofs of completeness and consistency but I still believe that most of these things are about appearances and bureaucracy, not about the nontrivial mathematical beef.
A Gödel-style proof may be viewed as a combinatoric bureaucratic procedure. The mathematicians "imagine" some real infinite sets and types of infinities underlying all these proofs but those feelings – however religiously wonderful they may feel – are just irrational delusions, I think. The axiom of choice can't really be decided physically, not even in principle, which is why it's no wonder that axiomatic systems may be consistent with it as well as with its negation.
For me, the realm with all these extremely arbitrary (hypothetical or postulated) infinite sets is just a "swampland" and interesting thinking only begins once one isolates some interesting "landscape" within this "swampland", a "landscape" of objects obeying rules that are basically independent of the character of the mud that surrounds them! It's the rules that can lead to interesting implications that give "beef" to their ideas – which is also why I am usually unimpressed by "weaker" siblings of algebraic structures such as "groupoids" and "semigroups" that generalize "groups". The "beef" or "potency" goes up dramatically once you demand all the rules for a full-blown group. Transformations become reversible and therefore also reusable. Semigroups don't need to have the identity or the inverses – the latter means that when a semigroup element breaks something, you can't fix it and it's over. ;-)
The excessively generalized mathematics – and Lurie's category theory work seems to be a part of it – is therefore the obsession with zombies and broken gadgets drowning in a huge and muddy swampland!
And that's the memo.