I did not want to deny the need of mathematical thought, but I found this example of topos theory emerging unexpectedly quite pleasing. Note that the mail is *not* about Isabelle. Admittedly, I do not know anything about the quoted system HOL light, so I have not verified Slind's claim. Till Elwood Wilkins wrote:
Isabelle's internal logic is not constructive. The existential clause of the BHK-interpretation is violated so that a consquence of "all unicorns have horns" is that "some unicorn has a horn". The moral (biased towards theorists): software engineering considerations are not enough, a coherant philosophy of mathematics is also needed.
Elwood
----- Original Message ----- From: Till Mossakowski <till@Informatik.Uni-Bremen.DE> To: Categories <categories@mta.ca> Sent: Tuesday, May 27, 2003 11:44 AM Subject: categories: Topos logic arises naturally
You see: even with just aesthetic and software engineering considerations you are eventually lead to topos logic...
-- Till Mossakowski Phone +49-421-218-4683 Dept. of Computer Science Fax +49-421-218-3054 University of Bremen till@tzi.de P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
Message-Id: <200305270616.h4R6G3w8008134@plxc2089.pdx.intel.com> To: Randy Pollack <rap@inf.ed.ac.uk> cc: John Harrison <johnh@ichips.intel.com>, isabelle-users@cl.cam.ac.uk Subject: Re: HOL without description operators Date: Mon, 26 May 2003 23:16:03 -0700 From: John R Harrison <johnh@ichips.intel.com> Sender: Larry Paulson <Larry.Paulson@cl.cam.ac.uk>
Hi Randy,
| Perhaps more usefully, how do you suggest I do this task of developing
HOL
| without description operators.
You might find it interesting to look at HOL Light. This gives a different axiomatization of the HOL logic, developed precisely because I was dissatisfied with the one in the original HOL system, on which I believe the Isabelle/HOL logic is based.
Although I do eventually introduce the description operator, quite a lot of the basic logic --- including the automation of inductive definitions --- is developed without it (and indeed without excluded middle or extensionality). You may find it a more congenial starting-point.
As Konrad Slind later pointed out, what I settled on is remarkably close to the internal logic of a topos, as presented for example in Lambek and Scott's book. This was a surprise to me since at that time I knew next to nothing about toposes and was motivated by a mixture of aesthetic and software engineering considerations.
Cheers,
John.
-- Till Mossakowski Phone +49-421-218-4683 Dept. of Computer Science Fax +49-421-218-3054 University of Bremen till@tzi.de P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till