Comments for A Dialogue on Infinity http://dialinf.wordpress.com between a mathematician and a philosopher Mon, 23 Nov 2009 19:49:13 +0000 http://wordpress.com/ hourly 1 Comment on Ordering the infinite time by rflurbaniak http://dialinf.wordpress.com/2009/11/15/ordering-the-infinite-time/#comment-390 rflurbaniak Mon, 23 Nov 2009 19:49:13 +0000 http://dialinf.wordpress.com/?p=292#comment-390 Could you elaborate on how this is supposed to be different from, for instance, Aquinas? Could you elaborate on how this is supposed to be different from, for instance, Aquinas?

]]> Comment on A circle with the center everywhere by timBoucher » Parking is free (but difficult to find) http://dialinf.wordpress.com/2008/04/03/a-circle-with-the-center-everywhere/#comment-388 timBoucher » Parking is free (but difficult to find) Sat, 21 Nov 2009 23:29:10 +0000 http://dialinf.wordpress.com/?p=29#comment-388 [...] “God is an intelligible sphere, whose center is everywhere, and whose circumference is nowhere.” [...] [...] “God is an intelligible sphere, whose center is everywhere, and whose circumference is nowhere.” [...]

]]> Comment on Fraisse Amalgams as Limits by Aidan A. Kelly, Ph.D (Theology, GTU, 1980) http://dialinf.wordpress.com/2009/11/05/fraisse-amalgams-as-limits/#comment-384 Aidan A. Kelly, Ph.D (Theology, GTU, 1980) Sat, 07 Nov 2009 18:10:11 +0000 http://dialinf.wordpress.com/?p=286#comment-384 Following up on my comment further down the list: You guys are obviously serious mathematicians. I can't claim that. I got shot down by differential equations at UC Berkeley in the fall of 1961; went back to San Francisco State; got my BA and MA in poetry. After working for five years as an editor for Scientific American Books, I went to the GTU and got my Ph.D. in theology (actually, advanced humanities) exactly on my 40th birthday. What's the relation between math, poetry, and theology? I see all three as systems for manipulating abstract symbols in order to generate maps of possible realities. It's nice to find a website where people might understand what the hell I mean by that. Following up on my comment further down the list:

You guys are obviously serious mathematicians. I can’t claim that. I got shot down by differential equations at UC Berkeley in the fall of 1961; went back to San Francisco State; got my BA and MA in poetry.
After working for five years as an editor for Scientific American Books, I went to the GTU and got my Ph.D. in theology (actually, advanced humanities) exactly on my 40th birthday.

What’s the relation between math, poetry, and theology? I see all three as systems for manipulating abstract symbols in order to generate maps of possible realities. It’s nice to find a website where people might understand what the hell I mean by that.

]]> Comment on The Invisible Dialog Between Mathematics and Theology by Aidan A. Kelly, Ph.D (Theology, GTU, 1980) http://dialinf.wordpress.com/2009/04/29/the-invisible-dialog-between-mathematics-and-theology/#comment-383 Aidan A. Kelly, Ph.D (Theology, GTU, 1980) Sat, 07 Nov 2009 17:48:24 +0000 http://dialinf.wordpress.com/?p=234#comment-383 I've just stumbled across thia, and have not read your archives, but I've been contemplating the relations between the mathematics and theology of infinity for several decades now. Let me pose a few suggestions, hoping you haven't already exhausted these concepts. I suggest that: there is not just one infinity, but an infinity of infinities, since Kantor demonstrated that there are at least two different infinities even in a simplified mathematical model; human concepts, such as same and different, one and many, spirit and matter, simply do not apply to the infinite; each infinity is itself the ultimate reality; the ultimate reality is an infinite compassionate consciousness that is both male and female (yes, I am suggesting that gender, unlike matter energy, or spacetime, is an ultimate reality): I look forward to seeing what responses I might get to this. I'll be happy to explain the reasoning that led me to these hypotheses. I'll try following this as a blog, if I can figure out how. I’ve just stumbled across thia, and have not read your archives, but I’ve been contemplating the relations between the mathematics and theology of infinity for several decades now. Let me pose a few suggestions, hoping you haven’t already exhausted these concepts.

I suggest that:
there is not just one infinity, but an infinity of infinities, since Kantor demonstrated that there are at least two different infinities even in a simplified mathematical model;

human concepts, such as same and different, one and many, spirit and matter, simply do not apply to the infinite;

each infinity is itself the ultimate reality;

the ultimate reality is an infinite compassionate consciousness that is both male and female (yes, I am suggesting that gender, unlike matter energy, or spacetime, is an ultimate reality):

I look forward to seeing what responses I might get to this. I’ll be happy to explain the reasoning that led me to these hypotheses.

I’ll try following this as a blog, if I can figure out how.

]]> Comment on Case Study I: Zilber’s Field by Fraisse Amalgams as Limits « A Dialogue on Infinity http://dialinf.wordpress.com/2007/12/15/case-study-i-zilbers-field/#comment-381 Fraisse Amalgams as Limits « A Dialogue on Infinity Thu, 05 Nov 2009 09:34:52 +0000 http://dialinf.wordpress.com/2007/12/15/case-study-i-zilbers-field/#comment-381 [...] as Limits November 5, 2009 Posted by dcorfield in Uncategorized. trackback Our very first post here spoke about the Fraïssé amalgam, a way of constructing a universal object out of a countable [...] [...] as Limits November 5, 2009 Posted by dcorfield in Uncategorized. trackback Our very first post here spoke about the Fraïssé amalgam, a way of constructing a universal object out of a countable [...]

]]> Comment on Induction and recursion by David Corfield http://dialinf.wordpress.com/2008/03/06/induction-and-recursion/#comment-374 David Corfield Thu, 22 Oct 2009 10:29:52 +0000 http://dialinf.wordpress.com/?p=28#comment-374 What's special about $latex \mathbb{N}$ is that it's an initial object in the category of algebras of the functor $latex X \to 1 + X$, which adjoins a disjoint singleton to a set. That is, there is a map $latex \langle 0, s \rangle: 1 + \mathbb{N} \to \mathbb{N}$, and for any algebra $latex f: 1 + Y \to Y$ there is a unique algebra morphism from $latex \mathbb{N}$. Induction corresponds to the non-existence of a proper subalgebra of $latex \mathbb{N}$. This is a general feature of initial objects: any monomorphism to them is an isomorphism. Recursion corresponds to the existence of a morphism out of $latex \mathbb{N}$. E.g., there's an algebra structure on $latex \mathbb{N}$ which acting on $latex 1 + \mathbb{N}$ sends 1 to 1 and $latex m$ to $latex n.m$. The unique morphism from the initial algebra now defines a function $latex f(m) = n^m$. To take $latex \mathbb{Z}/(p)$ as an initial algebra we'd have to place it in a different category of algebras where the unary function applied $latex p$ times is equal to the identity function. Then induction works. However, now the attempted recursive definition will not work since $latex \langle 1, \times n \rangle: 1 + \mathbb{Z}/(p) \to \mathbb{Z}/(p)$ is not an algebra in this category. What’s special about \mathbb{N} is that it’s an initial object in the category of algebras of the functor X \to 1 + X, which adjoins a disjoint singleton to a set. That is, there is a map \langle 0, s \rangle: 1 + \mathbb{N} \to \mathbb{N}, and for any algebra f: 1 + Y \to Y there is a unique algebra morphism from \mathbb{N}.

Induction corresponds to the non-existence of a proper subalgebra of \mathbb{N}. This is a general feature of initial objects: any monomorphism to them is an isomorphism.

Recursion corresponds to the existence of a morphism out of \mathbb{N}. E.g., there’s an algebra structure on \mathbb{N} which acting on 1 + \mathbb{N} sends 1 to 1 and m to n.m. The unique morphism from the initial algebra now defines a function f(m) = n^m.

To take \mathbb{Z}/(p) as an initial algebra we’d have to place it in a different category of algebras where the unary function applied p times is equal to the identity function. Then induction works. However, now the attempted recursive definition will not work since \langle 1, \times n \rangle: 1 + \mathbb{Z}/(p) \to \mathbb{Z}/(p) is not an algebra in this category.

]]> Comment on Induction and recursion II by David Pierce http://dialinf.wordpress.com/2008/05/28/induction-and-recursion-ii/#comment-372 David Pierce Tue, 20 Oct 2009 13:04:45 +0000 http://dialinf.wordpress.com/?p=35#comment-372 Why? Why?

]]> Comment on Induction and recursion II by wmstem http://dialinf.wordpress.com/2008/05/28/induction-and-recursion-ii/#comment-370 wmstem Wed, 14 Oct 2009 17:15:51 +0000 http://dialinf.wordpress.com/?p=35#comment-370 I am looking for a way which does not involve a truth table. I am looking for a way which does not involve a truth table.

]]> Comment on Induction and recursion II by wmstem http://dialinf.wordpress.com/2008/05/28/induction-and-recursion-ii/#comment-369 wmstem Wed, 14 Oct 2009 17:13:27 +0000 http://dialinf.wordpress.com/?p=35#comment-369 Good explanation. Is it possible to prove modus ponens must be a valid rule of inference? Good explanation.

Is it possible to prove modus ponens must be a valid rule of inference?

]]> Comment on Completions and the Archimedean property by David Pierce http://dialinf.wordpress.com/2009/08/28/completions-and-the-archimedean-property/#comment-361 David Pierce Mon, 31 Aug 2009 01:55:32 +0000 http://dialinf.wordpress.com/?p=254#comment-361 In Dedekind's original conception as I understand it (but I haven't got his book in front of me), if one breaks a geometrical line in two pieces, then on expects to find a point at the end of one of the pieces. There may not be such a <i>rational</i> point; therefore there are gaps among the rationals. But the point that determines the break is itself determined by knowing which rationals fall on which side. The original reason for posting on this topic was the observation that, if an ordered abelian group is completed with respect to the ordering, then the ordered-group structure extends to the completion if and only if the original group is Archimedean. This has lead the the observation that, for ordered fields, completion-by-Dedekind-cut and completion-by-Cauchy-sequence are not equivalent constructions. They give the same result only for Archimedean ordered fields. So it may not be meaningful to claim that one construction is better than the other. Perhaps the Cauchy-sequence construction is of more general interest. In Dedekind’s original conception as I understand it (but I haven’t got his book in front of me), if one breaks a geometrical line in two pieces, then on expects to find a point at the end of one of the pieces. There may not be such a rational point; therefore there are gaps among the rationals. But the point that determines the break is itself determined by knowing which rationals fall on which side.

The original reason for posting on this topic was the observation that, if an ordered abelian group is completed with respect to the ordering, then the ordered-group structure extends to the completion if and only if the original group is Archimedean.

This has lead the the observation that, for ordered fields, completion-by-Dedekind-cut and completion-by-Cauchy-sequence are not equivalent constructions. They give the same result only for Archimedean ordered fields. So it may not be meaningful to claim that one construction is better than the other. Perhaps the Cauchy-sequence construction is of more general interest.

]]>