Tag: Georg Cantor

A mathematical paradox of Christmas

Rembrandt nativity

An aspect of Christmas that is closest to my heart is that, where A is any set and where H is the set of all humans, it is also about the following:1, 2

!&#8707 A: A &#8836 Q &#8658 Q &#8834 Q &#8743 Q &#8835 Q
H &#8834 Q
&#8707 h: h &#8712 H &#8743 h = Q &#8658 Q &#8712 H &#8743 H &#8712 Q

In other words,3 God (an aspect of whom is represented here by Q), who includes and exceeds everything (which is beautifully put by the Islamic expression Allāhu Akbar – الله أكبر): “God is greater”), has become an individual human person (h) – i. e., a member of one of the myriad sets whose strict superset He is. The resulting recursive, infinite regress of divine in human in divine … is a fundamental aspect of what Christians believe and celebrate at Christmas. Yes, there is a lot more to it and, no, I am not attempting either a reduction or any sort of proof here, but, I believe the mathematical representation of what happened at Christmas just makes the enormity and irregularity of it stand out more starkly than using any other means.

God – the transcendent beyond, has become a man – a finite creature, while retaining His transcendence and immanence simultaneously. This paradox (and it strictly is a paradox since it amounts to a finiteness and infinity that cannot be resolved) is a mind-boggling event regardless of the specific circumstances under which it took place and would in any case have shown God’s love for humanity. His choice of coming into being as a human from the moment of conception (which Christians celebrate on the feast of the Annunciation and when the above paradox actually took place), being born the way all humans are and doing so under the humblest of circumstances just eases our comprehension of this extraordinary event and of the inexhaustible love that God has for us – a love that can also be read from the above set theoretic notation.

Another level of recursive infinity is introduced when Jesus institutes the Eucharist (making his divine-human self present in a transubstantiated, finite quantity of bread and wine), when it is He Himself who is present among any set of two or more humans who follow his words, and when it is He Himself who is also present in each human being. This makes not only God be transcendent-immanent but every single one if us too, by participation in His life. The ontological configuration that the incarnation introduces is truly a source of constant wonder for me and I hope that I managed to share some of that with you here.

Merry Christmas!


1 Thanks to my bestie PM for fixing my initially muddled notation!
2 Yes, I know I am using Cantor’s naïve set theory here and not Zermelo–Fraenkel (ZF or ZFC), but I do so since the latter has been designed to avoid the occurrence of paradox, which in most cases makes good sense, while Cantor’s system is more unconstrained and is precisely what the expression of the Christmas paradox calls for. Also, note that the term “naïve” here merely means non-axiomatic as opposed to implying any derogatory connotations.
3 In case you are not familiar with (this) set theoretic notation, it reads as: “No set A exists such that A is not a strict subset of Q (where a set X is another set Y’s strict subset if and only if all of the members of X are also members of Y and Y has at least one member that is not a member of X), therefore Q is a strict subset and also a strict superset of itself. H is a strict subset of Q. There exists a h that is a member of H and that is identical with Q, therefore Q is a member of H and H is a member of Q.”

Does the word “infinity” make you uncomfortable?

The Dominoes Are Falling

I learned a lesson today: never read the parish newsletter before the start of mass. This morning I did and it just lead to regret as I spent the vast majority of mass being distracted1 by it and trying to reconstruct in my mind the reasons against an argument put forward in it. What is even worse is that this wasn’t just the regular newsletter that our parish priest prepares (and that has as yet lead neither to disappointment, nor elation), but a newsletter – “Our Faith on Sunday” – prepared by the company who provides the weekly mass sheets and who ought to know better.

The argument in question is that of Aristotle’s unmoved mover (or first cause), which is a form of the cosmological argument. The basic idea is the following: since all change (motion, temperature variation, …) is the result of a previous change, there are two possibilities: either a causal chain stretching back into an infinite past or a first, “unmoved mover” that triggered a finite chain of causal links leading to the present. The possibility of infinite regress is dismissed as ridiculous, ergo there had to be a first mover. So far Aristotle’s argument from over 2300 years ago, which at that time was unarguably brilliant and which has survived without chinks into the 18th century (this by itself being pretty impressive too!). So, Aristotle comes out pretty well from this incident. The same cannot be said about the nameless author, who not only sticks it into a parish newsletter in 2012 without attribution, but who – to add insult to injury – finishes the piece with saying that the “unmoved mover” is God.

No it ain’t! And that is just the start of a litany of complaints that flooded my mind this morning, with the following being the 800 pound gorillas:

  1. In this context, the gravest mistake is clearly to present a piece of philosophy (however good it may be) and to equate it with God. Not just to say: “Well, this concept gives us hints about some aspects of what God may be like,” but to say “Unmoved mover = God.” Not only is this entirely divorced from Christian theology (giving a false sense of being able to grasp God in His fullness, etc.) but it is positively counterproductive. In essence the argument postulates a God who is relegated to a distant past, who is far removed from us and who just plays the role of a snooker player, hitting the first ball that leads to a vast sequence of knock-ons – a true God of Gaps. This is not the God of Christianity. It is not the loving Father who sent his Son to become one of us and the Holy Spirit to guide us. It is not the God who’s three persons love one another to the point of being one and who invites us to partake in His innermost life. The “unmoved mover” is a cheap imitation and one that is rightly and thankfully the butt of atheist jokes.

  2. Next, taking a philosophical argument made over two thousand years ago and (presumably, hopefully!) not checking whether there have been any significant challenges made against it is pretty sloppy too. And an excuse of obscurity cannot be used here either as the cosmological argument (whose one variant this is) has been debated to death! Furthermore, its critics have included such giants of philosophy as David Hume, who challenged the notion of causation itself (arguing that our senses simply don’t have access to the necessary connection between supposed cause A and supposed effect B – instead, all we have are repeated experiences of event B following event A). With causation undermined, there is clearly no necessity for a “first cause.” Does that mean a disproof of God? No – just of the grotesque God of Gaps of the cosmological argument, and not a disproof as such (those live exclusively in the realm of mathematics or other formal systems – and even there are limited by incompleteness) but a counterargument instead.

  3. Finally, and this is a criticism that I cannot fairly level at the authors of the newsletter, there is also that recurring misunderstanding of infinity that hampers many a philosophical argument from centuries past. Before Georg Cantor’s groundbreaking work on set theory and the concept of cardinality and the subsequent advances in our understanding of infinite sets and their properties (with contributions by pioneers like David Hilbert), an arm-waving approach to infinity and blanket statements about its unintelligibility or impossibility (e.g., by Thomas Aquinas2) were all we could manage. Today these are just not good enough anymore. E.g., a good example of how the impossibility of an infinite sequence of causes can be refuted can be found in Peter Clark’s paper: “Consider the set of events with no first member but a last member: {… an … a4, a3, a2, a1, a0} [where] for every j (aj-1 causes aj). There is no logical contradiction in this supposition whatsoever. […] Every event in the above sequence is finitely accessible from each and every event preceding it.” What this means is that an infinite sequence stretching back in time does not imply the necessity for a member that is infinitely far in the past. No matter how far you go back in the sequence (i.e., an) – and remember that you can’t go back to the beginning, which does not exist – there is a finite number of steps that bring you to the present (i.e., a0). All the infinity of the sequence means is that there is no first member, without necessarily entailing members that are infinitely removed in the past. This may sounds counterintuitive, but presents no logical contradiction.3

So: lesson learned. Next time, I’ll defer reading the newsletter until after mass and especially its “Faith and Reason” section, where, ironically, Aristotle’s argument was plagiarized.

1 I almost missed this gem of a line from St. Paul’s letter to the Philippians: “And this is my prayer: that your love may increase ever more and more in knowledge and every kind of perception, to discern what is of value” (1:9-10), again pointing to an induction of orthodoxy from orthopraxy.
2 “The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited.” (Thomas Aquinas, Summa Theologica, Part 1, Question 7, Article 4).
3 I realize this paragraph barely scratches the topic of infinity, to which I hope to return in the future … Also, please, note that I am not advocating an argument for the universe having existed infinitely – I am merely pointing to the objection to an infinite causal chain being outdated.