Friday, March 6, 2015

Internal, External, Universal

[Today’s theologico-mathematical analogy may be stretched, far-fetched;  but ‘tis the Lord’s day, a time meet for meditation  more at large.

For more extensive reflections, focusing on Realism in both domains, consult the essay series that begins here.]


Instead of defining the properties of a collection by reference to its members -- its internal  structure -- one can proceed by reference to its external relationships with other collections.
-- R. Goldblatt, Topoi , 2nd edn. 1984

I am reminded of the Christian critique of narcissistic individualism, so telling for our own day, when it has become a very plague, both sapping the individual character, and corrupting the polity as it forms an algal bloom as identity politics.  This view was made more acute, and very contemporary, by C.S.Lewis in The Four Loves and elsewhere, with its metaphor that health lies neither in religious solipsism (the “inner light”, which he decries) nor in that solipsism-à-deux of “looking into each other’s eyes”, but rather in mutually apprehending some external thing, of which we each see aspects, though along different sight-lines.

There are traditional notions of something large and out-there, above us and beyond us;  but these are vague and unstructured, and have perhaps grown stale through overfamiliarity (though we have never understood them well enough to have leave to dismiss them out of hand).   So let us turn to consider a mathematical notion of something containing -- something larger than what you started with, yet perfectly contained within itself:  not spreading over us like a fog, but rounding us out.  The technical name for this is comforting, downright cozy:  compactification.  (The Water Rat of Wind in the Willows  pictures his snug and tidy den.)

Compactness has turned out to be one of the most central notions of topology, a field which itself is about as central as you can get.  For details, see Wikipedia (that paradisal repository of all that is known, or could ever be known);  but the takeaway is, that it is a quite vaunting generalization of the idea of finiteness.  Such spaces are nice to work with.

Thus for instance:  consider the open interval (0,1).  It is not too intimidating (apart from its harbored continuum), but it is irksomely incomplete, in that a well-regulated sequence of points -- ½,¼, 1/8 … -- can march off towards nullity,  yet nullity they find not, nor unity neither  should they march the other way.  We can complete this space, and simultaneously compactify it, in an obvious way:  just add the points zero and one at either end, to get the closed interval [0,1].  Now all is well.
But there exists a less obvious kind of compactification, involving the addition of but one point (we pause, that you might wonder:  Yet how can this thing be?).  In turns out to be deeper, in that such a one-point compactification (via Alexandroff extension) is available for any locally compact Hausdorff space.  In the simple case of our open interval, conceptually you add a point at one end and bend the segment around to meet it.  The result is a little ring:  like all round things, it is ever so perfect and pleasing.

And our pleasure at this maneuver  is more than aesthetic, for the move applies as well to the entire real line R.  This space is complete in the standard Cauchy-sequence sense, yet it too is “incomplete” in a way, namely, in the sense that an infinite sequence might have no convergent subsequence (R is not 'sequentially compact', as they say in the trade):  the series (such as 1,2,3, …) may march off forever towards infinity, but “infinity isn’t there”.  We can both ‘complete’ and compactify it  by adding a “point at infinity”, replacing the standard metric with a bounded one (the resulting space being homeomorphic to what we started with), and then “round it around” to a ring-shape as before.
You see where we’re going with this.
Ah, but do you.  For mathematics has latterly progressed in ways considerably more intricate than simply sharpening our intuitions of infinity, so that, when we say that “God is infinite”, we can have something much more incisive in mind than simply “way bigger than an elephant”, with which our grandsires had to make do.  For geometry has been algebrized: beginning with Descartes, but zooming off in unexpected new directions with algebraic topology.

We have seen that there are varying ways of compactifying a given space.  In the context of Universal Algebra, a question arises:  For any given space, is there one way that is, in some sense, universal or canonical -- the “Mother of all compactifications” (to speak with Saddam Hussein)?  Indeed there is:  it is known as the Stone–Čech compactification. The result is universal in that any continuous map whatever, from our original space to a compact Hausdorff space, can be factored through the Stone–Čech compactification.  (Thus, the closure of (0,1) into [0,1] does not rate as Stone–Čech, since e.g. sin (1/x), defined on the open interval, does not extend to the closed.) -- Whoever can grasp this, will never consort with Nominalists again.
We have considered this matter in a particular area of point-set topology, but the notion of universality, as made precise by this notion of lifting a given map to procede through the universal, is quite general -- hair-raisingly general, in fact.  In general, “a morphism [is said to be] universal  [iff]  any other morphism into a system with this property  factors uniquely through the universal morphism.” (Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3rd edn. 1999), p. 129.)

~   ~   ~

So much for the math.  And now for our dominical metaphor, offered in all humility.
We are, according to Scripture, but now also in a sense which might possibly someday be made relatively precise, made in (or better:  from) the image of our Maker.  Only, not visually (that were absurd, and gives rise to all the idolatries), nor yet (abstractly, or spiritually) isomorphically,  but rather: homomorphic images, of various types and sizes.  (Bonus:  homomorphic now becomes a graeco-latin pun.)  Whatever can apply to us, can apply to and through Him, in a manner made familiar by Category Theory.
And by what seems a kind of anticipation of the functorial view, the Historical Church chose precisely universality as its defining epithet:  catholicus.

(Yet who are these, streaming across the blasted landscape in despair, the wretched remnants of their mockeries  strapped to their backs?  Why, ‘tis the very tribe of atheists, quite put to flight!)

Within Set Theory, there is a notion reminiscent of all this:  the Reflection Principle.  It is very counterintuitive -- but then, so is life.


Appended Epigram
That God is simply the sum of All that Is, is mere pantheism.  We shall posit rather, that He is its Stone–Čech compactification. 

(Here we tread, not on dangerous, but on spongy ground, the sort that led into the swamp of the ‘God particle’.
Various defenses spring to mind, but I have a feeling that they are self-serving.  Taceamus igitur.)

Similar to our image of the lower thing being the homomorphic image of the higher:

The highest things often have “footprints”, as the medievals put it, among the lower things.
-- James Schall, S.J., The Order of Things (2007), p. 22


(All right, now we do something very wrong.  But my character, sapped by whoring after epigrams -- e’en as the bard  was slain by a pun --  cannot resist.
An early post against ultra-Darwinism  mentioned -- purely in passing -- the Urysohn Metrization Theorem;  after which, to my embarrassment, this site received a number of serious enquiries after that worthy result.   Actually  it was kind of cool.  And so, to accommodate surfers who are mathematically advanced but lousy spellers, we add these:
Stone- tchèque
pStone-pČech  [the p is silent ...])

~ ~ ~

All that is rather by way of somewhat remedying the obvious insufficiences of St Anselm’s Ontological Argument, while yet retaining sympathy with his project.

The images/metaphors  of the Scala Naturae, and the Ladder of Abstraction, both point ever-upwards, as if to some final lodestar or ultimate Utmost, without  of course  proving the existence of any such thing.  There is also something empirically amiss, in that both visions are linear -- and reality is generally not like that.    More to the point would be Partially Ordered Sets -- and that gets us straight to the door of Zorn’s lemma:

Suppose a partially ordered set P has the property that every totally ordered subset has an upper bound in P. Then the set P contains at least one maximal element.

Now, that Maximal Element -- remind you of Anyone?

Stairway to Paradise

This is a more robust analogy than that of the long extension-ladder, but it probably won’t buy us anything of theological import.   Note in particular that the various upper bounds referred to must lie in P:   P is already complete.   Whereas a simile for the Godhead would more likely be along the lines of Inaccessible Cardinals, or Proper Classes,  ever beyond iterative reach.

C.S. Lewis drops a remarkable aside, in the final paragraph of his essay “The Language of Religion”:

I sometimes wonder whether the Ontological Argument did not itself arise as a partially unsuccessful translation of an experience without concepts or words.
-- Christian Reflections (1967), p. 141

(Nota bene:  There are intellectual as well as emotional such experiences, as in mathematical insight -- at least, without words.  Brouwer once characterized mathematics as “an essentially languageless activity of the mind”.
More here.)

Lewis’s essay, incidentally, is  gem, developing at length  an idea he has often sketched, concerning the evolving adequacy of language to non-everyday puzzles like theology and math.  In that spirit, we have offered a couple of vizualizable new analogies to play around with:  Universal Compactification, and Partially Ordered Sets.

Lewis’s linguistic point is continuous with his opposition to intellectual “Whig history”.   Thus, if our ancestors spoke of God as though He had a white beard, and depicted him this way in art, it is not because they were morons;  indeed, such a depiction did not, at the time, constitute an asserted denial of the thesis that God is incorporeal:  for that later thesis simply lies (intellectually and chronologically) beyond the original level of discussion.
(In similar fashion, if I state that “the red vehicle was stationary at the time of the collision", that is not meant to deny the thesis that the earth rotates on its axis, and moreover revolves around the sun.)

Exactly the same point can be made with respect to the praxis of mathematics.  (I mean its ever-evolving practice by actual mathematicians, rather than the arguably  timeless, transcendental truths of Mathematics itself, as it resides in the mind of the Creator.)

Thus, Wikipedia (re Imre Lakatos):

Lakatos re-examines the history of the calculus, with special regard to Augustin-Louis Cauchy and the concept of uniform convergence, in the light of non-standard analysis. Lakatos is concerned that historians of mathematics should not judge the evolution of mathematics in terms of currently fashionable theories. As an illustration, he examines Cauchy's proof that the sum of a series of continuous functions is itself continuous. Lakatos is critical of those who would see Cauchy's proof, with its failure to make explicit a suitable convergence hypothesis, merely as an inadequate approach to Weierstrassian analysis. Lakatos sees in such an approach a failure to realize that Cauchy's concept of the continuum differed from currently dominant views.

Lakatos’ dialectical insights are worked out at length in the multisided dialogue (a ‘polygonal’ conversation, as it were), Proofs and Refutations.

[Update April 2017]  I had rather hoped to have added a “Footnote to CSL” with that shtick about creatures as homomorphic images (of various cuts and complexity) of their Creator, a more flexible metaphor than Lewis’ example of the faces of a cube.  But upon re-reading his essay “Transposition”, I learn that Transposition is his term for much the same thing -- he even uses the term algebraic in that connection.  The whole idea is worked-out exquisitely in that place.

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