[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-Cech
Stone-Čeck
Stone-Ček
Stone-Czech
Stone-check
Stone- tchèque
Stone-Tscheck
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?
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.
