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 , 2^nd 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; 3^rd 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?

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.

Saint Anselm’s Proof of the Perfection of Penguins

Lemma.  Penguins have every excellence.

Proof.   Suppose there were some excellence which some one penguin  lacked. -- But that is absurd.   Whence the lemma. ∎

Thus, using this result:  Create an ascending K-chain of Excellent Penguins, and use transfinite induction on P.   

Q.E.D.

“Zorn’s Penguin”:

The most perfect penguin of them all

[Biographical footnote:  Over brandy and cigars, Dr J was led to confess, that the proof was of his own devising, but that, out of modesty, he had attributed it to the saint.]

Saint Sherlock’s Ontological Argument

A reader going by the handle of “solspot”  has a very fine comment  here:

http://blogs.scientificamerican.com/cross-check/2014/03/03/did-sherlock-holmes-believe-in-god/#comment-9185

My dear Horgan, it’s elementary:
Neither atheist, theist nor agnostic could prove or disprove their beliefs to Holmes’ satisfaction (otherwise, it would lack the primary quality of faith). Holmes would require deduction; that is, eliminate the impossible.
1. It would be impossible for a human to be accountable for their actions without free will. Holmes obviously believes that humans are accountable for crimes; ergo, Holmes believes that humans have free will.
2. Likewise, Holmes believes in an absolute morality because it is entailed by free will.
3. It is impossible to have an absolute morality without a transcendent purpose of that morality; otherwise the morality is simply relative to each person (this was Moriarty’s logical error!).
4. The transcendent purpose is Holmes’ God, his raison d’etre, without which logic itself does not exist!

As the shamus Jerome once said:  "Ratio donum divinum est,  et sic ipsa est divinae consors naturae."

Feathering-out the details of the Ente-logical proof

a couple of ducks,

each individually sufficent to witness God’s goodness;

but -- --  two  of them,   together ! !

[For a concise statement of the theorem, click here:

http://worldofdrjustice.blogspot.com/2011/08/ocularoracular-proof-of-existence-of.html ]

[PS:  Ente means 'duck' in German. Wordplay on ontological.]

A New (Non-)Proof of the Existence of God

Actually an old one, of eighteenth-century vintage.  Let D. E. Smith tell it (History of Mathematics, 1923, vol. I, p. 523):

Diderot had somewhat displeased the Czarina  by his antireligious views, and so she persuated Eurler to assiste her in suppressing him.

Diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of God… Euler …said gravely, and in a tone of perfect conviction:

     Monsieur, (a + b^n)/n = x, donc Dieu existe; répondez !

Diderot, to whom algebra was Hebrew, was embarrassed and disconcerted, while peals of laughter arose on all sides …

For my own brilliant demonstration that modern syntactic theory implies the truth of the Nicene Creed, click here.

These are, of course, nonsense;  and are offered simply as counter-weights on the other side of the balance-pan, to equally ludicrous (though seriously meant) would-be scientific demonstrations of the opposite (chemicals go sploosh when you think of a tree, therefore we’re all just robots).

~

Denis Diderot, penning one of his zingers

Kripke offers a useful correction to the hoary anecdote:

In fairness to Diderot, it should be mentioned that the incident surely never took place as described.  In fact  Diderot was the author of several learned mathematical essays. 

If we do not take care, however, some of our philosophical discussions are in danger of coming to resemble the legendary confrontation.  I have seen cases where a very simple, almost mathematically trivial technical trick  has captured a philosopher’s imagination, and been used as if it were the key   that easily and mechanically unlocked  doors that were forever closed to ordinary philosophical investigation.

-- Saul Kripke, “Is There a Problem about Substitutional Quantification?”, in: Evans & McDowell, eds., Truth and Meaning (1976), p. 415.

Keynes makes the same point in some detail in his philosophically-attuned Treatise on Probability.

A Minimum Axiomatization for Reality (Part V -- finis)

[A completion of the essay begun here.)

In fact, neither the thesis of axioms as foundational, nor the antithesis I have presented with the label “regressive strategy”, is the whole truth.  We stand before a dialectic, well described by Bertrand Russell, in Introduction to Mathematical Philosophy (1919; 2^nd edn. 1920), p. 1, after distinguishing the (so to speak) synthetic from the analytic approach:

Early Greek geometers, passing from the empirical rules of Egyptian land-surveying  to the general propositions by which those rules were found to be justifiable, and thence to Euclid’s axioms … were engaged in mathematical philosophy …; but when once the axioms … had been reached, their deductive employment … belonged to mathematics ….

~ ~ ~

A couple of further oddities about axiomatics, not conforming to their traditional status as epistemological bedrock:

Shaughan Lavine, Understanding the Infinite (1994), p. 47:

[Cantor] did not work axiomatically.  He believed in the reality of his ordinal numbers and sets, and he saw himself as discovering their properties.  Therefore, no axioms were necessary.

Joseph Ullian, in Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 585:

Truth accrues to an axiom, if at all, from the success of the system in which it participates.

What an extraordinary phrase -- “truth accrues”.  And what a surprising thing for it to “accrue” to -- axioms, which one had rather imagined to be beyond that whole dimension of assessment:  they are (the common thought had run) foundational -- stipulated, not assessed.

~ ~ ~

A curious coda to all this.

We have argued that, to be adequate to our experience of the world, we must posit the existence of Free Will as an axiom:  this, since  without it  no aspect of our experience makes sense, and since (so materialists assure us) it cannot be itself derived from the rest of science.

(At this point, the materialists are content to deny Free Will altogether, and to lapse into a robot coma;  where we shall leave them.)

We have further argued that no such apodictic necessity adheres to the thesis of the existence of God;  there are logically possible alternatives, though they are all horrible.

All this, in the course of a sort of Gedankenexperiment or finger-exercise, whereby we set up a (distant, but beguiling) metaphorical connection between, on the one hand, the standard set-theoretical foundations known as ZFC, where the “C” refers to the Axiom of Choice (in what is intended to be a strictly mathematical sense, but which, in its explication, often gives rise to images of voluntarism), and, on the other hand, our own worldview as rational beings incarnated in this cosmos, where the (relatively uncontroversial) counterparts of the “ZF” basics are now the metaphysical underpinnings of the scientific enterprise itself (we discussed these here), and the opposite-number to the Axiom of Choice is now…. Choice itself -- Free Will.

Now.

It is a remarkable milestone in mathematical logic  that three fundamental postulates (none of them theorems in themselves, and indeed later shown to be unprovable in any ordinary sense) -- the Axiom of Choice, the Well-Ordering Principle, and Zorn’s Lemma -- each arrived at independently in the course of mathematical history (by which I mean, of course, the history of our own mathematizing, the truths of mathematics themselves being timesless)  have been shown to be logically inter-equivalent.  That is, any one of them can be logically derived (via sophisticated arguments) from either of the others.

The fact is surprising enough in itself;  more surprising still, in light of their psychological inequivalence.   A classic joke runs: 

The Axiom of Choice is obviously true;  the Well-Ordering Principle, obviously false;  and Zorn’s Lemma -- who can understand it?

It would be worth your while to obtain a Ph.D. in mathematics, simply to be able to get that joke (which contains deep truths).   Nothing else in the universe is nearly so funny.

With that excursus -- back to our original program.

Zorn’s Lemma (as we know now, as a “lemma” it is misnamed, not being provable without assuming one of the other two principles) may be stated thus:

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, the structure of this proposition is (mutatis naturally mutandis) isomorphic to the Ontological Argument for the existence of God.   Notoriously, that argument as Anselm stated it  is unconvincing in isolation (as philosophers pipe-puffingly put it, it “does not go through”).   As is Zorn’s “Lemma”.  But !  Given the assumption of the Axiom of Choice (which mathematicians have, in the course of their practice, come to feel largely indispensable), you do get Zorn’s Lemma.   The parallel being (in our metaphorical or possibly not-so-metaphorical thought-experiment) that, given the axiom of Free Will, you get … Anselm’s Lemma, as we shall call it in this form.

Not a proof;  just a thought.  But what a thought !

Der Fall Rorty (Explication de texte)

 Richard Rorty, Philosophy and the Mirror of Nature, p. 19:

If we try to clarify the orthodox notion of ‘the divine’, we seem to have either a merely negative conception, or else one explicated in terms of the notions of ‘infinity’ and ‘immateriality’. Since reference to infinity explains the obscure by the more obscure, we are left with immateriality.

            Rorty seems to have solved the Packing Problem:  This is ten pounds of bullsh*t in a five-pound bag.

            The first clause first, though it is the least exceptionable.  It neither says nor presupposes anything false; the bias is merely rhetorical.  We “try to clarify” (evidently doomed in our efforts, so obscure is the clarificandum) – as though clarification were not likewise a necessity for everything from rocks to snowflakes to the Continuum Hypothesis.  Bishop Berkeley tried, and failed, to clarify Newton’s fluxions; Cauchy succeeded.  Next!  -- And then those little tweezers, the squotes:  I use them myself, so I know what they’re used for.  Preferably not to denigrate divinity before we’ve even begun (“…of ‘the divine’ ….”).   

            The “merely negative conception” does point to a real current in theology (both Christian and Islamic), but a current so rich, that the “merely”, while essentially an emotion-word, almost rises to the status of a lie.  For the “negative conception”, the  via negativa, (discussed elsewhere in this series;  just click on the label) is nothing like nihilism, nothing like negativism, nothing even like skepticism.  It is a stance arrived at after a long – centuries-long – attempt to characterize, in positive terms, what is… well, it turns out, very difficult to characterize, at least in terms of the predicates we inherited from our hunter-gatherer and pastoral past.  We might compare it to the history of our contemplation of the Continuum Hypothesis.  In the early days, red-blooded mathematicians naturally set out on their stallions to prove it (positively) true; or, failing that, to prove it (positively) false.  The current position (after much intermediate agony)  is:  It is true in some models; false in others; and independent of the axioms as usually deployed in set theory.  This whole “independent of the axioms – neither true nor false, exactly, but not nonsense either” is a rich notion, almost too rich to digest,  which required millennia to arrive at.  It is indeed a kind of (unanticipated) negative result; but not “merely negative”.  And since the problem of God is a superset of the problem of the Continuum Hypothesis, we cannot well expect a simple etch-a-sketch portrait of the guy (as it were, posing along the boardwalk at Ocean City).  (Indeed, if you follow me closely, my adducing the C.H. for comparison, so far from being some typical theistical opportunistical move, if anything undermines our naïve conception of the Creator; at least, its initial effect on myself was sickening, as bad as the Beagle on Captain FitzGerald.  And there will be no post-initial effect for quite some time, until – God grant it – I gain greater insight into the thing.)

He goes on:  ”or else one explicated in terms of the notions of ‘infinity’ and ‘immateriality’.”

(Note again the sneer quotes, offered in lieu of argument by this ‘philosopher’.)

Being passably ignorant of theological history, I cannot say whether these two epithets, among the many that might be hazarded as regards the Godhead, are the two statistically most prominent ones, but let us suppose they are.  No theologian, and no widow lighting a candle, ever imagined that these were the crux of the thing.  The integers are infinite and immaterial; so is the Infinite Penguin; we worship neither.  How about “Creator”, “Redeemer” and “the ground of our morality”, yo?

  “Since reference to infinity explains the obscure by the more obscure…”    Good… Lord.  I wish it were so.   If the nature of the divine were actually clearer than the nature of infinity – I wouldn’t need to write this essay, it could be left to the kindergarten teacher, while the rest of us dance table-top, champagne in hand.   Infinity – by which I shall here always mean, the very teensiest flavor of same, aleph-nought – has been very well studied by now.  It is virtually suitable for the nursery.

            If I have hesitated to push the infinity-of-the-integers business too far (their infinity, as opposed to their Necessity), it is not because it is too obscure:  it is rather too simple, too almost shallow.  I can tell you a lot more about little-omega (that least, most modest infinity, with the ordinal type of the natural numbers) than I can about, say, something really difficult, like a duck.

            As for “immateriality”, that’s… immaterial.  Is He immaterial, or nonmaterial, or intangible, or  ethereal, or abstract, or funereal  -- wholly or partly so? Some say He once walked through Jerusalem, leaving footprints in the sand.  That is intriguing, if true (trust Rorty to latch on rather to the boring possibility); but as for immateriality, that is no part of our interest in, or devotion to, Him.  Smurfs are immaterial for the matter of that, and I don’t go to Smurfs on Sunday.  In fact, given a choice (say, by a dating service) between a Material entity and an Immaterial, I’d go with the Material every time.  It might tickle our idle curiosity to find out, whether He is – or rather, for that is nonsense, to what extent and in which ways He is ` ` ` immaterial ‘ ‘ ‘ (and here it is the philospher’s use, not the possible pis-aller use by the theologian, that I am punctuationally excoriating), -- how like a ghost, like the air, like the integers, like the non-algebraic numbers, like the angels, like aleph-one, like our late great-grandfather, like a poem once spoken but never written down, how like a prayer we would have uttered, but that the Reaper came too soon …. You’d need a lot of terminological tidying-up for the question to even make sense, and it isn’t worth doing.  Those who believe they have had some actual experience of God, have a number of tart things – be they true be they false things, but – a number of sharp and hard things, to say, about these experiences, which in no way resemble the mumblings of a sociophobic agnostic concerning the silences of a fog-enveloped all-encompassing blancmange. To ignore all this and focus on “immaterial”, is like never bothering to learn Relativity, yet loudly wondering (peevishly) about just what was Einstein’s favorite flavor of ice cream. (My understanding, incidentally, is that he preferred minkowskian, with sprinkles.)

            Nothing hangs on this red-herring of the “immaterial”.  Most folks who have had (as they imagine; again, righly or wrongly) any immediate experience of God, tend to emphasize the personal.   My own conception of the Creator is actually more immaterial than most, because I’m emphasizing the actual nature of the Creation, than which the Creator is logically-necessarily more complex: a Creation which has – I mean just plain in terms of Measure Theory -- far more of the mathematical (call that abstract or immaterial or whatever you like) than it does of rocks, or turds, or mxlnthnkxs. (These last are certain purely material items of Universe #138; they greatly outnumber our atoms, but there are far fewer of them than there are  integers.)  Were I someday actually to run into the chap, and were He to appear in the quite tangible aspect of Mr. Natural seated beneath a tree, I should, to be sure, feel mildly surprised, but philosophically neither cheated nor refuted.

(Key to that last paradox:

 While we dub the Lord the “Necessary Being”, that description, like that of “Father” or “Creator” or the “Lord of Hosts”,  does not exhaust Him.  It is impossible to conceive of Him as being nothing but the Necessary.  For in that case He  really would be just a giant math book. Hence the boring quality of the ontological argument (whether valid or not). We can conceivably deduce certain things about God along those lines, but nothing of His concreteness.)

*

Rorty adds, later down the page:

If it makes any sense to speak of the existence of universals, it would seem that they must exist immaterially.

First:  It does indeed make sense to talk of universals, if it makes sense at all  to talk about integers, or modus ponens, or “all men are mortal”, or “the most wonderful mom there ever could be”   -- this “existence of” pre-modifier is something of a rogue.  It makes sense to talk of Hamlet, unicorns, democracy, love, and prime numbers; what is added by this “existence of”?  “Please pass the existence-of  salt.”  No, nonsense.

Next: These universals may “exist”, if you like, `immaterially’, or in any other fashion – I wouldn’t insist on the point, nothing hangs on this. For all I care, they exist in a pickle-jar; none of their properties (in universe after universe) are affected thereby.  As it happens, I personally tend to see Hilbert space as rather more concrete and well-defined, and mountains as rather more abstract and ill-defined, than has hitherto been customary. Likewise, Schubert’s piano sonata in B flat is – abstract /ideal/ immaterial/ call-it-what-you-may, it is still acoustically-ontologically tangible, and each recording or performance thereof is quite concrete.

And yet further a bit (Rorty goes on):

“…the immaterial – the mystery beyond the bounds of sense…”

Now this is a nice phrase, suggesting in particular  the existence of something beyond the confines of the lavatory, where nominalists spend so much time; so we receive it with respect.  Still, conscience oblige, we are compelled to observe, that the “bounds of sense” are a purely relative, species-bound, even individual- and moment-bound notion. A pickle-jar exists beyond the bounds of the blind bat, that doesn’t make it a mystery.  Hilbert space exists beyond the bounds of certain uninstructed individuals; whose bad?  The Blorks of the planet Fnoid can neither see nor touch a stationary object, but they can move it, and then sense it, using their solutions to the equations of motion, and this more accurately than with the eyes.   Cantor and Gödel had a sense for the infinite that compares favorably with many an ear for music or nose for wine. 

            And as for “mystery”, a term often used as a sort of hand-waving dismissal: many things, both physical and immaterial, are mysterious until you study them; others, unmysterious until you study them.  Rocks are much more mysterious after the discovery of atoms (my my, mostly empty space. And yet so solid).  Fractions, which every kindergartner now takes for granted, spooked the Egyptians, who for some reason expressed them, not in the simple form of today, but as an elaborately calculated sum of reciprocals.  Algebraic numbers, imaginary numbers, lose their mystery in a single intellectual wedding-night.  Transcendental numbers like e and pi, retain – or rather have gained, in mystery, but in a specific sort of mystery in each case: indeed, we have a solid handle at the hither end of it.  It is a mystery we never would have discovered, let alone elucidated, had we adhered to a Rortyan agnostic-proctological underview.

*

Edward O. Wilson, Consilience (1998), p. 190, re Rorty’s replacement of epistemology with hermeneutics:

Discourse among scholars, in short, can proceed without worrying about consilience.  About rigor too, it would seem.  Although this concession is welcomed by postmodernist scholars, it is a premature surrender that would drain much of the power and joy from scholarly inquiry.

Wilson counsels eschewing such a replacement, “except at cocktail time” -- a wise proviso, since indeed, prattling on about hermeneutics is much more likely than epistemological discourse  to get you laid.

 

A Proof of the Non-Existence of God

The Catholic Church was for centuries distinguished by the devotion of its Schoolmen to logic, developed along Aristotelian lines.  Among their efforts, not especially successful, were purported Ontological Arguments for the existence of God.  But to be fair, atheists too have been given the divine gift of Reason.  Now,  I have just discovered an ingenious proof of God’s non-existence, which we may dub the Proctological Argument.  And since, to my knowledge, no-one has yet published it, at least not in so perspicuous a form, we do so here.

Premise.   I am the Center of the Universe, the fons existentiae, Lord and Master of all I survey.

Lemma.  Ergo,  any Deity worth his salt  would cater to my every need instantly.

Fact.  Yet I feel blue/ my hamster died/ etc.

Conclusion.  There is no God.

The syllogism above is perfectly valid -- given its premise.   Such, one suspects, is the logical understructure beneath many of the cogitations of our fellows, though seldom expressed with such dazzling clarity.

THEOLOGIA MATHEMATICA: The M&Ms Edition

[This is a continuation of a thread begun here.]

The response to these ongoing ponderings has been not as vocal as I might have expected.  True, Bernie Riemann, writing in from the beyond, dropped a nice note; but overall there seems to be a hush, if not actual fidgeting, among the audience.  And yet I know that present company dwells uninterruptedly upon eternal things.  Dwells on them, possibly, tankard in hand (indeed these present notes might not have been so copious, but for the promptings of the cold and blushful); but dwells on them.  Perhaps it is all just too much at once.

So instead I’ll emit, from time to time, a bite-sized Thought for the Day, and maybe keep emitting them if I detect the sound of munching..  They all circle around the same basic idea, and can be consumed individually.

And so… on to our sermonette!

*  *  *

Today’s M&M:  The Ontological Argument

B. Russell, Introduction to Mathematical Philosophy, chapter 18:

If we reject the ontological argument, we seem driven to conclude that the existence of a world is an accident.

This is indeed a problem:  for the ontological argument, whether as originally given or as subsequently tweaked, seems indeed uncompelling (for one thing it could equally lead you to the existence of the Infinite Penguin – who exists indeed, but independently of the argument). And yet I do believe (discovering this belief lying silently within me, like a present with no indication of donor, beneath the tree), that it is not an accident.  Mostly this is on the basis of what I conceive to be common-sense grounds, virtually on the trivial level of what used to be called “ordinary-language philosophy”:  By “accident” we prototypically mean things like:  The milk-jug tipped over; or, Timmy had an unfortunate occurrence in his pants. To light-heartedly apply this same noun to the intricately evolving Riemannian manifold in which we dwell, with its Hilbertian tangent spaces wherein the Maxwell equations and the Ricci tensor are but chorus-like walk-ons in the whole resounding transfinite Gesamtkunstwerk --  no, sorry, c’est un abus de langage.  If you wish to spit upon the magnificence of the Creation (bearing in mind: all things visible and invisible), you are at liberty to do so, but you’ll have to find some better gob to hawk than “accident”.

“The” ontological argument is in any case a general term; there have been several updatings since Anselm.  Most notably, one by the great logician Gödel.  You can find it summarized in Wang’s Reflections on Kurt Gödel, p. 195. 

I can’t make much of it.  But then, many proofs in formal logic have a kind of baffling quality to a non-logician.  If valid, it only goes to show the bare outlines of a God – nothing about… anything that matters to us, really.  If invalid, that’s interesting; for Gödel was perhaps the world’s most logical man, and not notably devout, so one would not imagine he would perpetrate a fallacy out of mere wishful thinking.

Constructivist Angelology

But yet when considered, may help us to enlarge our thoughts  towards greater perfections of it  in superior ranks of spirits. … The several degrees of angels  may probably have larger views.

-- John Locke, An Essay Concerning Human Understanding (1690)

Man’s understanding, though allied to the angelical, operates differently.  The angels understand intuitively, man by the painful use of the discursive reason.

-- E. Tillyard, The Elizabethan World Picture (1942)

It is presumably not obvious to the chimpanzee (or, if this be setting his smarts too low, to the humble woodchuck) that for all m, n in Z, m + n = n + m.  Nevertheless, in his daily scurryings and burrowings, he will repeatedly meet up with particular instantiations of this modest truth.

            For the woodchuck (at any event the southern northeastern lesser striped variety) builds a number of nests and other temporary dwellings, each of which has the framework of a variously triangulated  polyhedron, built tinkertoy-fashion from a fixed number of sticks.  Now, gathering them one by one would take too long, nor can the tidy woodchuck stand to have any sticks left over.  So when constructing his summer dwelling -- an icosahedron, which needs thirty sticks (did I get that right? My calculating powers are not much beyond those of a woodchuck) -- he normally harvests a jubjub bush, which has twenty-two sticks of exactly the right specs and which blooms in the spring, then rounds it out with the eight-sticked glubglub bush, which sprouts slightly later. 

But then one year, the blooming of the jubjub was delayed, and the woodchucks despaired.  All but one, the enterprising Willie, who went doggedly (or groundhoggishly) ahead  and harvested the available glubglub, supplementing this  when the jubjub arrived slightly later.  This remarkable exploit was recorded in the annals: for 22 then 8, one may substitute 8 then 22.

            It was subsequently found that a mubmub bush (18 sticks) followed by a nubnub bush (12) would do just as well – und zwar, in either order!  This fact too was recorded.

            The years went by, then the centuries, and the millennia, and the annals grew to seven times seventy stout volumes, densely filled with such arcana as: a cube-for-cubs may be constructed of a lublub (7) plus a rubrub (5), and this in either order; and so on for billions of examples.  All this was considered a branch of botany, a purely empirical science.

            By this means, the woodchucks arrived at an analogue of Babylonian mathematics.

            Until one day one Wisedome Woodchuck, a distant descendant of Willie, figured the whole thing out, and in a remarkable demonstration of only eighty pages (rather hard to follow, but sound), showed that m + n = n + m  was a perfectly general fact, replacing the seven-times-seventy volumes at a stroke, and freeing up his brethren for yet further architectural innovations, which previously had been shunned, as their particulars were not yet in the book.  The annals were placed in a museum, which the elder woodchucks might still visit, marveling at favorite exhibits (as who could forget that remarkable winter, when 5,878 + 519 turned out to be equal to 519 + 5,878?  A tour de force!). Meanwhile generations of young woodchucks (the pride and despair of their parents, who could not follow them into Canaan, with their aging brains) studied Wisedome’s proof, breaking their little heads against it.

           

Meanwhile in Metropolis… The humans, learning of this, politely saluted Wisedome’s modest accomplishment, and experienced a pang of sympathy for woodchuck-kind; yet felt no inclination to visit their Museum of Particular Results: for which they felt, indeed, a kind of horror.  And even the general result, while true, is somehow to us not truly interesting. In any case we are all too busy wrestling with the Riemann Hypothesis, to have time to look back.

Meanwhile in Elysium, where throne the angels sensu strictior, the lowest order of angelic beings sensu lato, a mock compliment is paid to Andrew Wiles, who finally figured out that little Fermat puzzle, with which the angel-kind  are wont to amuse the nursery.  Not that the angels arrived earlier at his proof, nor any refinement thereof.  They simply scoop up a few infinities of integers with their fractal fingers, twist them this way and that—and see, it doesn’t fit!  Simple.

            Moreover, all facts about all structures of ordinal type omega, whether or not deducible by any finite axiomatization, are equally transparent to the angels. They just look.

            So, is Elysium the mathematical Paradise?  Not quite…

            In a remarkably lucid and accessible article*, which should be packed into every pupil’s lunchbox by a considerate mom, Gödel observes that our continuing failure to resolve Cantor’s continuum problem, left over from the previous century, is quite an embarrassment.  It means that we are unable to wrap our minds around the very simplest multiplication problem possible, beyond the finite ones that these days can scarcely stump a woodchuck. Namely, two times two (times two, times two – keep going).  He writes:

            “It is easily proved that the power of the continuum is equal to 2^(aleph-nought). So the continuum problem turns out to be a questions from the ‘multiplication table’ of cardinal numbers: namely, the problem of evaluating a certain infinite product (in fact the simplest non-trivial one that can be formed).  There is, however, not one infinite product (of factors > 1) for which so much as an upper bound for its value can be assigned. […] It is not even known whether or not m < n implies 2^m < 2^n.” 

            We are  so to speak  staring helplessly  at a pile of sticks.

            Nor does the subsequent Cantor+Cohen demonstration of the independence of the continuum hypothesis from a particular system of axioms for set theory   set the matter aside. Gödel had already anticipated Cohen’s result, and wrote:

A proof of the undecidability of Cantor’s conjecture from the accepted axioms of set theory (in contradistinction, e.g., to the proof of the transcendency of pi) would by no means solve the problem.  For if the meanings of the primitive terms of set theory … are accepted as sound, it follows that the set-theoretical concepts and theorems describe some well-determined reality, in which Cantor’s conjecture must either be true or false.

            Indeed Gödel suspects that the Cantor conjecture is actually, factually false: which means that somewhere, among the actual literal real numbers, there is hiding a set of cardinality intermediate between aleph-nought and its power set, with definite members which the angels could name.  Not, however, the lowest order thereof; this lies beyond them.  But at the next step up, the archangels hang these sets from mobiles over their infants’ cribs.  In fact a woodchuck may somewhere inadvertantly have used one of these sets for nesting materials, and even now lies sleeping on it – a night of troubled dreams.

            So much for a simple pancake-stack of omega-many deuces – the limit of the lower-angels’ ken.  What about the square root of omega-to-the-omega; or cross sections of fibre bundles on toroidal cap-omega-cross-theta space? For each level of angels, there will be something beyond them that they just don’t get.

*

There are two poles of the range of approaches to the problem of infinities.  One is that of the badger-like Brouwer, who simply sweeps the chessmen to the floor, folds up the board and goes home.  (An only somewhat more amenable figure, says Gödel, is Weyl, who allows as how there might be something to board games, but suggests we play checkers – or Chutes ‘n Ladders – rather than chess.)  The other pole says:  Infinities are tricky, but they all exist, and are present to the Infinite Mind. Gödel himself uses that term, e.g. noting that Ramsey’s admission of formulae of (countably) infinite length  might be constructivistic for an infinite mind  but not for our own.  Gödel does not, however, seem to feel much need for any desperate appeal to such a mind, in the course of an ordinary day, since he -- like Badger’s amiable friend the Water-Rat-- is a thoroughgoing Realist, and comfortable as such in his own skin.  For him the assumption of infinite classes “is quite as legitimate as the assumption of physical bodies, and there is quite as much reason to believe in their existence.”  The outwardly gloomy Hungarian  is really the jolly Dr. Johnson of set theory.

            Only now there’s a problem, of a sort which did not confront the schoolmen, who never counted on the uncountable:  the Infinite Mind is all very well, but -- Which infinity did you have in mind?

            Who comprehends *everything*? God does, by definition. Yet He cannot be simply the crown on a tower of constructively ascending intelligences.  He is like an “inaccessible cardinal” – and not the first.  Nor perhaps ‘the last’, if there is no last.  Whatever He might be, there is Cantor in the wings, grinning, waiting to perform a Power Set on God, yielding – what?  -- Nothing one can begin to commence to pretend that we can approach with our sadly finite understanding.

            All of which suggests, if nothing else does,  that God is something more and other than an alternately wrathful and affectionate granddad  with a perfectly enormous white beard – however much longer that beard might be, than the stubble which disfigures your chin or mine.  Who one day, apparently from sheer idleness, as one might choose chocolate, chose the Jews.  Who later, some say, cast a Jove-like eye  on a certain Palestinian virgin.  And who at present is very angry indeed with the Democrats (or the Ravens, or whomever).  Yet what He in fact might be, we cannot even begin to imagine anyone’s beginning to conceive.  (Cf. the suggestion of 1 Kings 8:27  that the heavens themselves have heavens (and so on up); and that the whole omega-tower of them  cannot encompass God.)

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http://www.linguasacrapublishing.com/justice.html

We now return you to your regularly scheduled essay.

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            We actually wind up with a sort of hamstringing of the Ontological Argument. Notoriously its conclusion does not really follow from its premise;  but now even its premise limps: “Since we can imagine a Perfect Being…”  But that’s just it, we can’t!  Not even little infinite bits of one! Yet paradoxically (and God reportedly loves paradox – at least Chesterton does, His publicity agent on Earth), this seeming stomping on the prostrate corpse of the offspring of Anselm, this despairing cry that somehow even Infinity does not suffice, so far from opening the agora  to legions of snickering atheists chanting “Toleja so!”, points somehow upward, -- outward,   -- onward ….  Praise Him!

Postscript:

John Locke himself, normally regarded as the Poster Boy for Empiricism, of I'm-from-Missouri common-sensicality, yet delivers himself of this (Essay, III.vi.12):

That there should be more species of intelligent creatures above us, than there are of sensible and material below us, is probable to me from hence:  that in all the visible corporeal world, we see no chasms, or gaps.

That is to say:  The gap between ourselves, and God, must somehow be filled, according to the Principle of Plenitude.

And again (IV.iii.23):

He that will consider the infinite power … of the Creator of all things, will find reason to think, it was not all laid out upon so inconsiderable, mean, and impotent a creature, as he will find man to be;  who  in all probability, is one of the lowest of all intellectual beings …

Angels of all sorts are naturally beyond our discovery, and all those intelligences, whereof ‘tis likely there are more orders than of corporeal substances, are things, whereof our natural faculties give us no certain account at all.

Since theism is far from central to Locke’s Essay, it is curious to see the emphasis on this scala naturae idea.

--------------

*”What is Cantor’s Continuum Problem?”, repr. Benacerraf & Putnam, eds., Philosophy of Mathematics.

~

Postscript:  For the possibility that the structure of certain mathematical truths relating to an infinite domain  might resist any but a case-by-case “Babylonian” approach, cf. the quotation from Michael Dummett towards the end of this post:

http://worldofdrjustice.blogspot.com/2012/01/what-is-truth.html

Compare further (re ascending ranks of abstraction and generality):

"The Ladder of Abstraction"

.