From: Pat Hayes <phayes@ihmc.us>

Date: Fri, 20 Sep 2013 00:39:26 -0700

Cc: Jeremy J Carroll <jjc@syapse.com>, Sandro Hawke <sandro@w3.org>, www-archive <www-archive@w3.org>

Message-Id: <03FFD8F2-BFCF-4A5E-82D7-3EE434D07E06@ihmc.us>

To: Gregg Reynolds <dev@mobileink.com>

Date: Fri, 20 Sep 2013 00:39:26 -0700

Cc: Jeremy J Carroll <jjc@syapse.com>, Sandro Hawke <sandro@w3.org>, www-archive <www-archive@w3.org>

Message-Id: <03FFD8F2-BFCF-4A5E-82D7-3EE434D07E06@ihmc.us>

To: Gregg Reynolds <dev@mobileink.com>

On Sep 19, 2013, at 11:25 AM, Gregg Reynolds wrote: > On Wed, Sep 18, 2013 at 1:33 PM, Jeremy J Carroll <jjc@syapse.com> wrote: > > Something of an aside … > > On Sep 18, 2013, at 1:29 AM, Gregg Reynolds <dev@mobileink.com> wrote: > >> The suggestion that a pair of mathematical entities with exactly the same extension are not equal doesn't help - it reads like an attempt to redefine mathematics. > > > Gregg > > I think you misunderstand mathematics ... > > It's called the Extensionality Axiom. If x and y have the same elements, then x is equal to y. According to Ciesielski, it's one of the two most basic axioms of set theory. Right. It is true for *sets*, but there are many mathematical entities which are not sets. Two groups, for example, may have the same elements but be distinct (because they have different group product operations.) The conclusion for the present discussion is that these things we are trying to talk about are not sets, which is hardly surprising, since the thread began when we set out to distinguish them from RDF graphs, which *are* sets. > You may object that set theory does not exhaust mathematics, but I remind you that we're talking about a technical specification whose audience consists of working programmers who are virtually guaranteed to take this axiom for granted. I would be very surprised if working *programmers* will take this for granted. The idea of having isomorphic but distinct data structures is built into the very fabric of almost all programming languages. > So if you're going to claim it does not hold, you have some explaining to do. In my experience, the problem is to explain the idea of mathematical extensionality to programmers, for whom this idea is quite alien. > I attach two pictures. > > The first is my copy, of Jones' copy of a diagram in a book in the vatican library which is a tenth century, maybe fifth generation, copy of a diagram drawn by Pappus of Alexandria in the 4th century, which may in turn have been a (n-th generational) copy of a diagram drawn by Euclid a few hundred years earlier. > The copy in the vatican library, has, according to Jones, got a mistake in it: which he corrected, assuming it to be a copyist's error and not an error of Pappus or Euclid. > > All these copies will have minor variations .. such as angles and distances and sizes being slightly different > > In some sense there is one diagram, even the one with a mistake in it, which refer to the same mathematical concept. In another sense there are multiple diagrams - the one in this e-mail is even in some sense different from the one at > http://oriented.sourceforge.net/images/jones139.jpg which is bitwise identical. The intent of the two pictures is quite different - within this e-mail I am discussing the meaning of pictures, on that website I am interested mainly to compare and contrast with my own picture, the second picture here, which, I boldly assert is a picture of the same mathematical concept as Pappus' original picture. (My picture and Pappus' original form the same arrangement of pseudolines in the projective plane - concepts which had not been invented when Pappus drew his picture) > > And you would be wrong. That's a matter of (bad) historical judgment, not mathematics. If you see your contemporary mathematical concept in a tenth century diagram, its because you put it there. It's a variation on the Fallacy of Anachronism. You might find Quentin Skinner's "Meaning and understanding in the history of ideas" enlightening. You can find a copy on the web. If not email me privately and I'll send you a copy. Another fascinating account of how this happens is "Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin" by Jens Høyrup. He shows how the scholars who first reconstructed Babylonian mathematics radically misread the texts in spite of their brilliance, because they treated their own historically situated concepts of mathematics for timeless universals and therefore read them into texts written by people who had no such ideas. > > ... > > The point of all this is that even in mathematics we need to be able to talk about the very human tasks relating to copying and changing and making my own copy … where notions such as identity move from being obvious to being really very subtle. > > No, the point is that mathematicians make bad historians. No, the point is exactly what Jeremy said it was. Your response here to his final side remark about mathematical concepts is irrelevant to his main point about copying and identity of diagrams. Pat ------------------------------------------------------------ IHMC (850)434 8903 home 40 South Alcaniz St. (850)202 4416 office Pensacola (850)202 4440 fax FL 32502 (850)291 0667 mobile (preferred) phayes@ihmc.us http://www.ihmc.us/users/phayesReceived on Friday, 20 September 2013 07:39:56 UTC

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