Comments on: The arity of equality http://bit-player.org/2008/the-arity-of-equality An amateur's outlook on computation and mathematics. Thu, 17 May 2012 09:36:23 +0000 http://wordpress.org/?v=2.6.3 By: Rob Mayoff http://bit-player.org/2008/the-arity-of-equality#comment-1826 Rob Mayoff Fri, 24 Oct 2008 20:01:48 +0000 http://bit-player.org/?p=170#comment-1826 In Mathematica, Min[] returns Infinity and Max[] returns -Infinity. In Mathematica, Min[] returns Infinity and Max[] returns -Infinity.

]]> By: brian http://bit-player.org/2008/the-arity-of-equality#comment-1825 brian Thu, 23 Oct 2008 19:01:57 +0000 http://bit-player.org/?p=170#comment-1825 Further to Aaron's and Carl Witty's comments on (max): The last time this came up for me in a real program, I was applying max to a list of numbers representing the pairwise distances between points in a plane. In this context there is a readily available least element: 0, or perhaps 0.0. But even if I had a version of max set up so that (max) ==> 0.0, I wouldn't be happy with it. Because then I'd have to check every 0.0 result to see if it came from (max) or from (max 0.0 0.0 ...). Of course this is an instance of a more widespread problem of distinguishing exception or error signals from normal results, but it raises the question of whether the at-least-one-argument approach might not have some practical justification, even if there's no good, principled rationale for it. Further to Aaron’s and Carl Witty’s comments on (max):

The last time this came up for me in a real program, I was applying max to a list of numbers representing the pairwise distances between points in a plane. In this context there is a readily available least element: 0, or perhaps 0.0. But even if I had a version of max set up so that (max) ==> 0.0, I wouldn’t be happy with it. Because then I’d have to check every 0.0 result to see if it came from (max) or from (max 0.0 0.0 …).

Of course this is an instance of a more widespread problem of distinguishing exception or error signals from normal results, but it raises the question of whether the at-least-one-argument approach might not have some practical justification, even if there’s no good, principled rationale for it.

]]> By: brian http://bit-player.org/2008/the-arity-of-equality#comment-1824 brian Thu, 23 Oct 2008 18:48:19 +0000 http://bit-player.org/?p=170#comment-1824 My apologies for the primitive comment interface. I'm embarrassed by it, enough so that I have marked my calendar to attempt a Wordpress update this coming weekend. (I'm not sure the update is all that I need, but it's a necessary first step.) My apologies for the primitive comment interface. I’m embarrassed by it, enough so that I have marked my calendar to attempt a Wordpress update this coming weekend. (I’m not sure the update is all that I need, but it’s a necessary first step.)

]]> By: Mike Stay http://bit-player.org/2008/the-arity-of-equality#comment-1823 Mike Stay Thu, 23 Oct 2008 17:46:53 +0000 http://bit-player.org/?p=170#comment-1823 And I meant distributive poset, not distributive category: clearly the disjoint union of all sets is not the one-element set. And I meant distributive poset, not distributive category: clearly the disjoint union of all sets is not the one-element set.

]]> By: Mike Stay http://bit-player.org/2008/the-arity-of-equality#comment-1822 Mike Stay Thu, 23 Oct 2008 17:44:40 +0000 http://bit-player.org/?p=170#comment-1822 That should have been - initial object is less-or-eq <= every object - terminal object is gt-or-eq >= every object (Darn parser thought it was a tag...) That should have been
- initial object is less-or-eq <= every object
- terminal object is gt-or-eq >= every object

(Darn parser thought it was a tag…)

]]> By: Mike Stay http://bit-player.org/2008/the-arity-of-equality#comment-1821 Mike Stay Thu, 23 Oct 2008 17:42:04 +0000 http://bit-player.org/?p=170#comment-1821 - poset=partially ordered set - toset=totally ordered set - morphism A->B in a poset is A less-or-eq B or A divides B or A subset B etc. - initial object is = every object in a poset - product x is intersection (gcd) in a poset, min in a toset - coproduct + is union (lcm) in a poset, max in a toset - in a distributive category, x distributes over + - also in a d.c. the product of all objects is initial and the coproduct over all objects is terminal. - so gcd of all integers is 1, lcm of all integers is 0 - poset=partially ordered set
- toset=totally ordered set
- morphism A->B in a poset is A less-or-eq B or A divides B or A subset B etc.
- initial object is = every object in a poset
- product x is intersection (gcd) in a poset, min in a toset
- coproduct + is union (lcm) in a poset, max in a toset
- in a distributive category, x distributes over +
- also in a d.c. the product of all objects is initial and the coproduct over all objects is terminal.
- so gcd of all integers is 1, lcm of all integers is 0

]]> By: Mike Stay http://bit-player.org/2008/the-arity-of-equality#comment-1820 Mike Stay Thu, 23 Oct 2008 17:41:42 +0000 http://bit-player.org/?p=170#comment-1820 Well, crap. There were seven more paragraphs full of examples and illustrations that got cut, with no recovery! A warning would have been nice. Here's the terse version: Well, crap. There were seven more paragraphs full of examples and illustrations that got cut, with no recovery! A warning would have been nice. Here’s the terse version:

]]> By: Mike Stay http://bit-player.org/2008/the-arity-of-equality#comment-1819 Mike Stay Thu, 23 Oct 2008 17:32:01 +0000 http://bit-player.org/?p=170#comment-1819 Congratulations! You've rediscovered the idea of initial and terminal objects in a category. If your category has at most one morphism between any two objects, then you've got a poset (partially ordered set), and we say that A is less than or equal to B if there's an arrow from A to B. If for any pair of objects A, B it's true that either A Congratulations! You’ve rediscovered the idea of initial and terminal objects in a category.

If your category has at most one morphism between any two objects, then you’ve got a poset (partially ordered set), and we say that A is less than or equal to B if there’s an arrow from A to B. If for any pair of objects A, B it’s true that either A

]]> By: Carl Witty http://bit-player.org/2008/the-arity-of-equality#comment-1818 Carl Witty Thu, 23 Oct 2008 17:17:41 +0000 http://bit-player.org/?p=170#comment-1818 @ Brian: It is certainly possible and sometimes useful to reason about the exact behavior of floating point numbers. Granted, it's much more difficult and annoying than reasoning about exact numbers, but floating point is important; we shouldn't just give up on understanding/reasoning about it. @ Aaron: For the answer of "What should (max) return", you have to decide without picking a domain (because the function doesn't know what context it's being called in). Also, we usually pretend that the bignums don't actually have a smallest or largest element (but not always; see http://jwz.livejournal.com/854482.html). @ Brian (again): I still wish these comments had a "preview" button :) @ Brian: It is certainly possible and sometimes useful to reason about the exact behavior of floating point numbers. Granted, it’s much more difficult and annoying than reasoning about exact numbers, but floating point is important; we shouldn’t just give up on understanding/reasoning about it.

@ Aaron: For the answer of “What should (max) return”, you have to decide without picking a domain (because the function doesn’t know what context it’s being called in). Also, we usually pretend that the bignums don’t actually have a smallest or largest element (but not always; see http://jwz.livejournal.com/854482.html).

@ Brian (again): I still wish these comments had a “preview” button :)

]]> By: Aaron http://bit-player.org/2008/the-arity-of-equality#comment-1817 Aaron Thu, 23 Oct 2008 14:30:47 +0000 http://bit-player.org/?p=170#comment-1817 Max and min do have identity elements: The smallest and largest numbers, respectively. These do vary by domain, of course, but so? Max and min do have identity elements: The smallest and largest numbers, respectively. These do vary by domain, of course, but so?

]]> By: brian http://bit-player.org/2008/the-arity-of-equality#comment-1816 brian Thu, 23 Oct 2008 01:49:11 +0000 http://bit-player.org/?p=170#comment-1816 @ Derek Jones: Everything depends on how numbers are represented. With inexact numbers (e.g., floating point), all bets are off. Reasoning about questions like these is pretty much hopeless except in languages that offer some kind of exact number representation. (Most Lisps have exact rationals.) @ Derek Jones: Everything depends on how numbers are represented. With inexact numbers (e.g., floating point), all bets are off. Reasoning about questions like these is pretty much hopeless except in languages that offer some kind of exact number representation. (Most Lisps have exact rationals.)

]]> By: Derek Jones http://bit-player.org/2008/the-arity-of-equality#comment-1815 Derek Jones Thu, 23 Oct 2008 00:20:22 +0000 http://bit-player.org/?p=170#comment-1815 I don't think it is possible to rely on the identity: (≠ x y) = (not (= x y)) where programming languages are concerned. See sentences (number in larger point size in the margin) 577, 602-604, 1197, 1212, 1219 and probably more in www.coding-guidelines.com/cbook/cbook1_1.pdf I don’t think it is possible to rely on the identity:

(≠ x y) = (not (= x y))

where programming languages are concerned.

See sentences (number in larger point size in the margin) 577, 602-604, 1197, 1212, 1219 and probably more in
http://www.coding-guidelines.com/cbook/cbook1_1.pdf

]]> By: John Cowan http://bit-player.org/2008/the-arity-of-equality#comment-1814 John Cowan Wed, 22 Oct 2008 23:46:32 +0000 http://bit-player.org/?p=170#comment-1814 Common Lisp's /= operator is particularly annoying because it's linearithmic (it has to sort its operands to make sure they are all different, although a hash table might be helpful), whereas all the other operators are linear. Common Lisp’s /= operator is particularly annoying because it’s linearithmic (it has to sort its operands to make sure they are all different, although a hash table might be helpful), whereas all the other operators are linear.

]]> By: Jim Ward http://bit-player.org/2008/the-arity-of-equality#comment-1813 Jim Ward Wed, 22 Oct 2008 19:35:09 +0000 http://bit-player.org/?p=170#comment-1813 When you say (+ 3 4 5) this is an implicit (+ 0 3 4 5) because + has an identity operator, so (+3) is really 0 + 3 and (+) is 0. Some operators, like max, min, and = don't have an identity operator. When you say (+ 3 4 5) this is an implicit (+ 0 3 4 5) because + has an identity operator, so (+3) is really 0 + 3 and (+) is 0. Some operators, like max, min, and = don’t have an identity operator.

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