Dear all, Thank you for your answers and for the stimulating discussion so far. I agree that identity, to the extent that it is distinguished from equality, is more of a philosopher's concept. One could perhaps say that in mathematical logic there is an overlap of two terminological traditions: a philosophical one, in which "identity" and "equality" are distinguished, and a mathematical one, where they are not. Regarding judgemental equality/identity: if by this one understands definitional identity, then a case can be made that it is found also outside type theory. Namely, instead of saying that a and b are definitionally identical, we can also say that they are identical by definition. And this phrase, "identical by definition", could well be used by a "generic mathematician", could it not? There is also the common practice of using a special identity sign when writing definitions. One might say that the notion of definitional identity captures the semantics of this special sign. With kind regards, Ansten Klev On Thu, May 7, 2020 at 8:58 AM Thorsten Altenkirch < Thorsten....@nottingham.ac.uk> wrote: > Dear Andre, > > It seems that Egbert already gave the perfect answer. When we about Type > Theory we don't only talk about syntax but about the relation of syntax and > semantics. And then judgemental equality is modelled by propositional > equality in the metatheory. Not a reversal but a level shift. > > I wouldn't say that propositional equality is a purely formal game, indeed > we explain the nature of equality by modelling it. In a set theoretic > setting it is enough to say that equality is a congruence, i.e. an > equivalence relation that is preserved by all functions. But this is not > sufficient to explain structural equality which is not propositional. As > you know best (because you have invented many of the important notions) a > better explanation of equality is that is an omega groupoid and that all > functions are modelled as functors on these groupoids. > > Cheers, > Thorsten > > On 07/05/2020, 00:29, "Joyal, André" wrote: > > Dear Thorsten, > > I have naive question for experts. > > I believe that judgemental equality is on the syntactic side of type > theory, > while propositional equality is on the semantical side. > The homotopical interpretation of type theory is mainly > concerned with propositional equality. > What is the semantic of judgemental equality? > (independantly of the semantic of propositional equality). > Could we reverse the role of the two equalities? > Could we regard judgemental equality as the true meaning > of type theory, while regarding propositional equality > as a purely formal game? > > André > > > > > > ________________________________________ > From: Thorsten Altenkirch [Thorsten....@nottingham.ac.uk] > Sent: Wednesday, May 06, 2020 6:54 PM > To: Michael Shulman; Steve Awodey > Cc: Joyal, André; homotopyt...@googlegroups.com > Subject: Re: [HoTT] Identity versus equality > > I agree but let me try to make this more precise. We cannot talk about > judgemental equality within Mathematics it is not a proposition. > Judgemental equality is important when we talk about Mathematics, it is a > property of a mathematical text. The same applies to typing: we cannot talk > about typing because it is not a proposition it is a part of the structure > of our argument. > > This becomes clear in the example: we talk about numbers but x+y is an > expression, hence talking about judgemental equality only makes sense when > we talk about Mathematics. To say that x+y is not judgemental equal to y+x > doesn't make any sense within our argument it is a sentence about it. > > When I say that 0+x is definitionally equal to x I don't prove > anything but I just point out a fact that follows from the definition of +. > That is I draw attention to it. > > Hence clearly there is no reason to use any other word that equality > to mean that two objects are equal which means that the equality type is > inhabited which is signified by using =. There is the issue that we may > have a more that one way in which two objects can be equal which creates > the need to talk about elements of an equality type. I don't like the > word "identifications" because it seems to suggest that the two objects are > not properly equal but just "identified" artificially. > > Thorsten > > On 06/05/2020, 20:19, "homotopyt...@googlegroups.com on behalf > of Michael Shulman" shu...@sandiego.edu> wrote: > > As I've said before, I strongly disagree that the standard > interpretation of "a=b" as meaning "a equals b" clashes in any way > with its use to denote the identity type. Almost without > exception, > whenever a mathematican working informally says "equals", that > *must* > be formalized as referring to the identity type. Ask any > mathematician on the street whether x+y=y+x for all natural > numbers x > and y, and they will say yes. But this is false if = means > judgmental > equality. Judgmental equality is a technical object of type theory > that the "generic mathematician" is not aware of at all, so it > cannot > co-opt the standard notation "=" or word "equals" if we want > "informal > type theory" to be at all comprehensible to such readers. > > > On Wed, May 6, 2020 at 12:01 PM Steve Awodey > wrote: > > > > Dear Andre’ (and all), > > > > The sign a = b is pretty well established in mathematics as > meaning “a equals b”, > > which does indeed clash with our choice in the book to use it > for the identity type, > > and to call the elements of this type “identifications”. > > Thorsten has rightly pointed out this clash. > > > > Although I am personally not eager to make any changes in our > current terminology and/or notation, > > I’m certainly glad to consider the possibiiy > > (we did agree to return to this question at some point, so maybe > this is it : - ). > > > > We need both symbols and words for two notions: > > > > - judgemental equality, currently a\equiv b, > > - propositional equality, currently a = b, short for Id(a,b). > > > > There seems to be a proposal to revise this to something like: > > > > - judgemental equality: written a = b and pronounced “a equals > b”, > > - propositional equality, written maybe a \cong b, and pronunced > ”a iso b”, > > (the elements of this type are called “isos"). > > > > Another (partial) option would be: > > > > - judgemental equality: written a = b and pronounced “a equals > b”, > > - propositional equality, written Id(a,b) and shortened somehow > a ? b, > > and pronunced ”a idenitfied with b” > > (the elements of this type are called “identifications"). > > > > Do either of these seem preferable? > > Are there other proposals? > > And how should one decide? > > > > Regards, > > > > Steve > > > > > > > > > > > > On May 6, 2020, at 12:02 PM, Joyal, André > wrote: > > > > Dear all, > > > > A few more thoughts. > > We all agree that terminology and notation are important. > > > > I love the story of the equality sign = introduced by Robert > Recorde (1512-1558). > > "because no two things can be more equal than a pair of parallel > lines". > > It took more than a century before been universally adopted. > > René Descartes (1596-1650) used a different symbol in his work > (something like \alpha). > > We may ask why Recorde's notation won over Descartes's notation? > > Of course, we may never know. > > I dare to say that Recorde's notation was *better*. > > Among other things, the equality sign = is symmetric: > > the expression a=b and b=a are mirror image of each other. > > Recorde's motive for introducing the notation was more about > > convenience and aesthetic than about philosophy and history. > > The notation was gradually adopted because it is simple and > useful. > > It was not because Recorde was a powerful academic, > > since he eventually died in prison. > > > > There is something to learn from the history of the equality > sign. > > I guess that it can also applied to terminology. > > A new notation or terminology has a good chance to be adopted > universally > > if it is simple and useful, but it may take time. > > > > André > > ________________________________ > > From: homotopyt...@googlegroups.com [ > homotopyt...@googlegroups.com] on behalf of Ansten Mørch Klev [ > anste...@gmail.com] > > Sent: Tuesday, May 05, 2020 4:47 AM > > To: HomotopyT...@googlegroups.com > > Subject: [HoTT] Identity versus equality > > > > The discussion yesterday provides a good occasion for me to pose > a question I have long wanted to put to this list: is there a convention > generally agreed upon in the HoTT-community for when (if ever) to use > 'identity' instead of 'equality'? > > > > Here are some relevant data. > > > > A Germanic equivalent for 'identity' is 'sameness'. > > A Germanic equivalent for 'equality' is 'likeness'. > > > > For Aristotle equality means sameness of quantity. This is how > one must understand 'equal' in Euclid's Elements, where a triangle may have > all sides equal (clearly, they cannot be identical). The axiom in the > Elements that has given rise to the term 'Euclidean relation' and which is > appealed to in Elements I.1 is phrased in terms of 'equal' rather than > 'identical'. > > > > In Diophantus's Arithmetica, on the other hand, the two terms of > an equation are said to be equal, not identical, and this would become the > standard terminology in algebra. For instance, the sign '=' was introduced > by Robert Recorde as a sign of equality, not as a sign of identity. The > explanation for this apparent discrepancy with the Aristotelian/Euclidean > terminology might be that when dealing with numbers, equality just is > identity, since for two numbers to be identical as to magnitude just is for > them to be the same number. Aristotle says as much in Metaphysics M.7. > > > > Hilbert and Bernays might be one of the few logic books in the > modern era to distinguish equality from identity (volume I, chapter 5). > 'Equality' is there used for any equivalence relation and glossed as the > obtaining of "irgendeine Art von Übereinstimmung". Identity, by contrast, > is "Übereinstimmung in jeder Hinsicht", as expressed by indiscernibility > within the given language. > > > > Frege, by contrast, explicitly identifies (sic!) equality with > identity, and glosses the latter as sameness or coincidence, in the first > footnote to his paper on sense and reference. Kleene and Church do the > same in their famous textbooks: if one looks under 'identity' in the index > to any of those books one is referred to 'equality'. > > > > Clearly the two cannot be assumed to mean the same by analysts > who speak of two functions being identically equal! > > > > -- > > You received this message because you are subscribed to the > Google Groups "Homotopy Type Theory" group. > > To unsubscribe from this group and stop receiving emails from > it, send an email to HomotopyT...@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAJHZuqYLY-_DB9uh-0FW0jr2KSoQ%2BpwRGD0PPjq%2BxyQvaJFN2A%40mail.gmail.com > . > > > > -- > > You received this message because you are subscribed to the > Google Groups "Homotopy Type Theory" group. > > To unsubscribe from this group and stop receiving emails from > it, send an email to HomotopyT...@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/8C57894C7413F04A98DDF5629FEC90B1652F515E%40Pli.gst.uqam.ca > . > > > > > > -- > > You received this message because you are subscribed to the > Google Groups "Homotopy Type Theory" group. > > To unsubscribe from this group and stop receiving emails from > it, send an email to HomotopyT...@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/05375057-883F-4487-8919-2579F5771AFC%40cmu.edu > . > > -- > You received this message because you are subscribed to the Google > Groups "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, > send an email to HomotopyT...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQwTR4qK094%2Bkmgj%3D2547UxGgmbjW9k5MgLe%2Bne9xPef3w%40mail.gmail.com > . > > > > > This message and any attachment are intended solely for the addressee > and may contain confidential information. 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