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I added a reference in the section on terminology to Makkai’s ’Towards a categorical foundation of mathematics’, where he defines what he calls the ’Principle of Isomorphism’. This is essentially what ’evil’ captures, I think, and it is handy to have a published version with a sensible name to which to refer people.
Here’s a wild thought: what about renaming the page principle of isomorphism and having evil redirect there. It would necessitate a rewrite of the page, but still contain material about the jokey names (evil, kosher etc). I recall that someone here told how some of these in-jokes are off-putting to outsiders or newcomers (Zoran, maybe?). Just an idea.
I think the wild idea, or something like it, is a good idea. It makes me uncomfortable that one of the most controversial pages (judging from the categories list) is also one of, perhaps the top of the looked-at pages according to Google. (Not that I am persuaded by those who find it off-putting; I just don’t think the name ’evil’ is important enough to justify upsetting people like Joyal, and David’s suggestion seems quite acceptable to me.)
Another source: Aczel calls this idea the Structure Identity Principle: isomorphic structures have the same structural properties. (see Oberwolfach report 52/2011, talking in the context of the Univalence Axiom).
This to me seems a little weaker, in that is doesn’t ask that all ’sensible’ properties, or properties we care to talk about are isomorphism-invariant, just that structural properties are. Although one could augment this by saying that we only allow structures to be structural (!!) - see, e.g. Barr saying that the maps between the structure determined by ZF sets don’t preserve that structure, and so we should not use such a definition of a set.
I’ve added a reference to Aczel’s version to evil#terminology_20
What about the “univalence principle”? (-:O
Mike - why? Whenever I hear univalence I think HoTT, not so much structuralism/kosherness/whatever, even if the name is warranted. Do you have some sources or are you playing Bourbaki? (well, someone has to at some point, hopefully they pick an intuitive name)
If we go for “principle of isomorphism” I would suggest we pay tribute to the in “Lab” and make it “principle of equivalence”.
While I like the idea of relating the entry to the univalence axiom, what I find unfortunate is that this term (“univalence”) is not descriptive of the main point here. What is univalent about univalence refers to a size issue: an object has at most one classification up to equivalence (namely if it is small enough, none if it is too large).
If we renamed “evil” to “univalence principle” we would have to explain this, and it would seem to be quite a distraction form the main point of “evil”.
The route by which Voevodsky chose the word “univalence” refers to a size issue, but the meaning of the word “univalence” doesn’t really have anything to do with that. We should all just forget about where it came from and regard “univalence” as a newly coined word which means what it means: namely, that equivalences satisfy the elimination rule of equality. Since the elimination rule of equality is essentially just “substitution of equals for equals”, this expresses exactly the principle under consideration: all properties (or structure) of any object are invariant under equivalence.
Since the elimination rule of equality is essentially just “substitution of equals for equals”, this expresses exactly the principle under consideration: all properties (or structure) of any object are invariant under equivalence.
While I entirely agree, what I still find unfortunate is that, taken at face vaue, this is more an argument for renaming the univalence principle into “principle of equivalence” than for the converse.
Of course I understand that “univalence principle” is by now too well established to mess with it. But it remains unfortunate, to my mind.
Or maybe I take that back. Just thought about it a little more.
If we put it right, then it could neatly make the point of “non-evil”:
So reading “univalence” explicitly “single-valuedness” it really is “single-valuedness up to eqivalence”, of course. So we’d have:
It’s wrong (“evil”) to regard as multi-valued what is uni-valued (up to equivalence).
Ah, the joy of playing with words. In any case, I strongly agree that it would be good to rename the entry and make the discussion of the term “evil” a side issue in some sub-paragraph while otherwise having the entry concentrate on more sober discussion of the issue.
“Univalence” is an anagram of “evil nuance”. ;-)
It’s wrong (“evil”) to regard as multi-valued what is uni-valued (up to equivalence).
That’s nice. I was trying to think of some good fictional etymology, but you beat me to it. (-:
By the way, I presume that Voevodsky took “univalence” from the notion of univalent function in complex analysis, where it just means “injective” (and holomorphic). But that is actually a bizarre usage if you look at the meaning of the root: if “uni-valued” refers to the output of a function, then any function is “uni-valued”, whereas if it refers to the input, then it should refer to bijections/isomorphisms/equivalences, not injections (the latter’s input being more “zero-or-one valued”).
I also support the proposal to rename evil, by the way. “Univalence principle” is a serious proposal, but if the majority leans another way that’s fine. (“Principle of equivalence” makes me think of general relativity, however.)
(“Principle of equivalence” makes me think of general relativity, however.)
I was wondering about this earlier today, too.
As a further piece of trivia, I’ll share the following thought.
While these two principles are about very different things, they share one common property:
while they were/are both regarded as being a major guiding principle while the theory was/is developed, after a while the principle became/will become so tautological as to be essentially empty.
Because, if you think about it, what does the “equivalence principle” in general relativity really say, from the modern perspective of differential geometry? It really just says “there are tangent spaces to a manifold”. That used to be an important insight while humanity was still figuring out that GR is about differential geometry. But once you accept this, the “equivalence principle” is not a big deal.
Same here: while higher category theory is still in its adolescence, we go on about the meaning of equivalence. But in 100 years people will find it hard to imagine how anyone could have ever made an “evil” statement in the first place.
I thought the “equivalence principle” in GR referred to the equivalence of inertial and gravitational mass. Which is an empirical observation, not a logical tautology.
There are many “equivalent” ways to state the equivalence principle, all of them informal of course. To make full sense of the statement as you quote it, one needs to say what “inertial mass” is, and this is done with respect to momentum / acceleration in flat space.
A formulation of the statement that more directly goes to this point is (I am quoting from the list given in the Wikipedia entry):
The local effects of motion in a curved space (gravitation) are indistinguishable from those of an accelerated observer in flat space, without exception.
That flat space appearing in that sentence is the tangent space to the spacetime manifold at the given locus. The statement says: a manifold looks locally like its tangent space at that point.
But, of course, initially this is an empirical observation, true. It is not logically necessary (well, as far as we know, we are working on it… ;-) that gravity is described by pseudo-Riemannian manifolds. But once you have established this, the original fact that brought you to this insight sits so deep at the root of the very formalism that describes it all, that in retrospect it is hard to see it as a deep statement. There are tangent spaces.
(Well, of course you may disagree. I am drawing an analogy, and by nature of analogy, you can, if you want, push it to the point where it breaks. So if you feel that point is too close, that’s okay with me, but it’s maybe not worth discussing further. This just to prevent us from spiraling into another philosophical argument not related to formal mathematics. ;-)
Okay, I think now I get it. The fact that mathematics is basically all isomorphism-invariant is (or at least, could be said to be) also an empirical observation. So in both cases we start with an empirical observation, but then that observation gets built into our formal framework at an extremely basic level, so that when expressed in terms of the formal framework the original empirical observation looks like an obvious tautology. Thanks for explaining!
Links to evil#terminology_20 may break in the future as the page is redesigned. I have now made a permanent anchor (which indeed broke the old links) to evil#terminology, which should work forever. Also principle of isomorphism now redirects. (I would vote for that title over the other suggestions, other than evil itself.)
Urs and Mike, you seem to telling a similar story to Michael Friedman in his ’Dynamics of Reason’. He’s intrigued by the physical theory of a time being composed of principles, coordinating principles,and empirical laws and regularities, and how their status changes through a revolution. E.g., what is an empirical regularity for Newton that gravitational and empirical mass are identical becomes a consequence of a a more fundamental principle - the equivalence principle - for Einstein. What was constitutive for Newton, that space is flat, becomes an approximate empirical consequence under Einstein.
Today “reddit” linked to our entry evil, see here.
While the term does trigger attention, it is not clear that it is the kind we want.
I am inclined to pick up the suggestion that Todd, I think, used to make forcefully: let’s rename the page and change the lead-in discussion. This evil-business is just a vast distraction. The title should be “equivalence invariance” or similar, and the lead-in paragraph should be something like the following:
Sensible reasoning in mathematics in general, and in higher structures such as category theory, homotopy type theory, etc., is enforced to be or should be invariant under the respective notion of equivalence. This is an issue whenever some structure is presented by other structures, since the notion of equivalence of the presentation will be finer than that of the notion being presented.
For instance a category can be presented by a simplicial set, but isomorphism of simplicial sets is much finer than equivalences of their corresponding categories. It is a mistake to mix up these two levels and for instance assign to a category properties that are shared only by some of the simplicial sets representing it. Such “breaking of equivalence invariance” has sometimes, half-jokingly, be called “evil in category theory”. As in “It is evil to single out an object from its isomorphism class.”
Unless somebody similarly forcefully disagrees, I will change the page title in a few days.
In ahead with the renaming. I’m sorry I didn’t do it when I brought this up in March.
The change looks good. How about the Principle of Equivalence Invariance? This strongly hints at the analogy with physics’ Equivalence Principle, discussed in #13-15 above, which I’d like to see included in the entry, without being identically named.
What about the principle of covariance? Aside from that only properties are invariant while more general structures (stuff etc) are covariant (as explained in the article if you search for that term), it also suggests something from GR that is precisely an example.
Although I’d still probably prefer principle of isomorphism (or principle of equivalence if you want to immediately include higher-dimensional language), in part for having been used in published papers on this topic.
I would also prefer one of ’principle of isomorphism/equivalence’. The notion Urs is referring to seems to me to be secondary to the original idea.
We can even do a bit of negative thinking with this concept. If we work with presets, or some notion of collection with an equality predicate, then we can say we shouldn’t ask whether two elements are one and the same, only that they are equal via our predicate. There is probably a h-type interpretation of this with , sameness being ’a same as b’ is of h-level 0. Up a level we then shouldn’t use equality, but isomorphism. And so on.
As I said before, I support renaming the page. I would be happy with any of the proposals.
Okay, I have
renamed the entry,
fought the cache bug,
pasted #19 into the Idea-section
removed paragraphs on how the term “evil” is a bad idea and suggestions what to replace it with;
removed every single occurence of “evil” (except one that briefly mentions the idea) and replaced them by suitable variants of “breaks equivalence-invariance” etc.
Phew. That used to be a pretty embarrassing entry. Hope that it is a bit better now. But we could do with a dedicated author who looks into polishing the whole thing.
Thank you!!!!!!
Back to the digression, the thing that bugs me about equivalence principle in GR is that there may actually be three masses to consider for any given object: these are its inertia (as a perturbable particle), its weight (as a test particle), and its induction of curvature (as a source particle); the GR picture says inertia and weight are the same; that inertia and curvature induction should be the same I suppose replaces what would have been conservation of momentum? I seem to recall hearing that an assymptotically-flat two-black-hole universe does act like the proper pair of masses, at least if they aren't too close together; but, then, not everything looks like a black hole up-close. Doubtless the local specialists have better wisdom.
You are referring to the fact that much of the old discusison on the equivalence principle focused on the situation where a fixed gravitational background field is assumed, and the motion of a small test particle inside it is studied ignoring the back-reaction of that particle on the spacetime.
This is true, this is a typical approximation. In the full theory there is just the spacetime with a metric and some matter fields on it.
But this full setup does reduce to the respective approximations in the suitable limits, so there is nothing to be bugged about.
For instance you write
but, then, not everything looks like a black hole up-close.
This is true, but what matters is the opposite: everything isoloated does look like a black hole from far enough away, or rather: everything isolated looks like a Schwarzschild spacetime from far enough away, and this is one way in which ordinary particle mechanics is recovered in a suitable limit.
everything isoloated does look like a black hole from far enough away, or rather: everything isolated looks like a Schwarzschild spacetime from far enough away
What does that mean ? You likely assume some flatness and free space asymptotically “far away”. There are likely solutions which are globally not such and I am not sure if the statement makes sense then. What is the formal assumption on the global solution which makes your statement true ?
Yes, “isolated” means that.
In the General Definition, I see that you’re introducing ‘compatible with equivalence’. While that makes sense, why not ‘equivalence-invariant’ (for properties) and ‘equivalence-covariant’ (for operations), instead of a new term?
Toby, that’s quite fine with me. Feel free to edit the entry. The state I left it in is not the one I consider optimal, but just the one that locally optimized the ratio of improvement over edit-time .
OK; I do intend to go over it, but first I wanted to make sure that you didn’t care about that term.
Google still sometimes thinks that “evil” is among the six most important terms on the Lab. The reason is that even though we renamed the entry “evil” to “principle of equivalence”, there are (or were) still plenty of entries that go on about “evil”.
I just went and opened over a dozen of these to change all occurences of “is evil” to “violates the principle of equivalence” etc.
But it’s tedious, and there are more entries left. Please consider chasing down some. To find them, ask Google to look for
evil site:ncatlab.org
On a related note, notice that even apart from the “evil” it makes no sense to refer, for example, to the “non-evil version of a bo functor”, instead it needs to be the “… version of the concept of a bo functor “.
Cleared 4 more pages.
There’s a discussion at n-fibration which contains the word 7 times. Can we drop the discussion?
I'm not objecting to your edits (which I have not looked at), but it's worth keeping in mind that sometimes (especially in a discussion) when somebody says the word ‘evil’, they really do mean it.
Sure, where you would expect them in pages such as: Baruch Spinoza, Gian-Carlo Rota, Heraclitus, Lectures on the History of Philosophy. Spinoza’s system, Zeno
After reading this comment by Corbin at the nCafé, and the discussion that followed, I visited the nLab and entered the word “evil” in nLab’s search box. That yielded plenty of pages (71!) that contain the word, but none that jump out as defining it. (The knowledgeable searcher might know that principle of equivalence is a good page to read about it.) Is not this word used often enough in a technical sense that it deserves a page of its own in the nLab? From the above thread this has been discussed, but I think the word definitely deserves a page. Complaints about the word can be put in that page. Some of the above Forum discussion could usefully be put in that page. As it stands, the word is used by the “in-crowd”, leaving outsiders scratching their heads.
The usage is mentioned on that page (there). Any remaining occurrences of “is evil” on the Lab should be replaced by “violates the principle of equivalence”. I have just replaced a handful more.
In particular, I saw now that there was an occurrence of the term hidden in mathematicscontents, which is !include-ed as a floating context menu into a bunch of other entries. Now that I have removed this occurrence (and a few others elsewhere), the number of remaining occurrences should be cut down considerably.
Corbin wrote: “I’ve recently been investigating dagger-categories in the context of understanding Evil.” A considerable discussion ensued, in which the word “evil” appears eight times. Now, think of an expert on Hilbert spaces, but not on category theory, who reads that discussion. How can he find out what is meant by “evil” in this context? Can nLab help him learn what is meant? Now, entering “evil” in nLab’s search box yields 60 entries. The principle of equivalence is way, way down on that list, with nothing to single it out as the most relevant page. How can we help that person to know what Corbin and his respondants are talking about? I think that is a worthy question.
It’s the first hit for me when Googling “nLab evil”.
Do people use the internal search engine? I only do it to find all instances of a phrase.
Nobody should say that they are using category theory to understand Evil.
Because it is a gigantic Red Herring with the only effect of distracting attention and energy away from contentful maths to pointless debate; and it is precisely to avoid the “considerable discussions that ensue”, as they always do, that it shouldn’t be said among people interested in getting some maths sorted out.
We don’t want to further this pointless malpractice here and certainly not have the Lab give the impression of embracing it.
This was agreed on back when the entry was renamed and reworded. That there were so many hits for the ill-conceived anology left – thanks for highlighting this to me! – was because somebody had inserted the term even into the context menu for “mathematics” which gets automatically included into all top-level entries on mathematics here.
I have removed that now, as I said above. Once the Google crawler picks this up (we need to wait a couple of days for this to happen now) the remaining number of hits should be considerably reduced. If nobody else does, I’ll eventually remove whatever remains.
Thanks for narrowing down the nLab search results for “evil”!
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