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.

Interlude:   A physicist depicts the arithmetical state-of-play in a papyrus from Egyptian/Babylonian times:

It records the resolution of a great number of fractions  into a sum of aliquot parts,  the original numerator always being 2:  as, for instance,

2/97 = 1/56 + 1/679 + 1/776

But no rules are given for effecting such resolutions, and the whole treatise seems to be a mere compendium of results obtained by repeated trials.

-- James Jeans, The Growth of Physical Science (1947 [posthum.]; 2^nd edn. 1951), p. 11

            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 question 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 Austrian  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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We now return you to your regularly scheduled essay.

*     *     *

            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"

.

The Whole of Modern Mathematics, turned into verse

Well now -- Here’s a project for my retirement.  To versify the entire mathematical corpus,  from Gauss to Groethendieck.

Herewith an earnest upon that enterprise.

Clerihew on the Cantor Set

One day, for a wager, Cantor,

after much ribbing and banter,

took an axe to a set, its middle lopping,

chopped it some more, and kept on chopping.

Und ...   so ....  weiter ...  ...    ..    ..    .      .

Freud vs. Cantor

.

In his Psychopathology of Everyday Life, 1904, Freud gave an early expression to his naturalistic outlook on religion and allied topics.  “I believe in fact that a great part of the mythological view of the world, which reaches far into the most modern religions, is nothing other than psychological processes  projected into the outer world.  The obscure apprehending of the psychical factors and relationships of the unconscious  is mirrored -- it is hard to put it otherwise; one has to use here the analogy with paranoia -- in the construction of a supersensible reality.”

-- Ernest Jones, Freud: The Last Phase (1957), p. 353

As for the actual existence of this or that supersensible reality, I shall have nothing to say here.  As a matter of mere logic, Freud, having assumed (A) their non-existence (on principle), seeks to explain their prevalence in people’s belief, and finds the answer in psychological projection.   (I have myself attempted a similar explanation, in the related case of penguins.)

Very well.  But what of that vast, enduring, ever-evolving, and richly articulated  suprasensible reality  known as mathematics ?   A Naturalist (or, in our terms Nominalist, as opposed to Platonist or Realist) account  must either hopelessly scumble the actual distinctness, variety, and logical interrelation of its explicandum, or reduce to absurdity (which part of the Oedipal complex gives rise to the Urysohn Metrization Theorem?).

[We’ll refrain from writing yet another anti-naturalist satire, and merely point the reader who has an appetite for such things, to the following, directed against the overreachings of ultra-Darwinism/evolutionary-psychology:  The Urysohn MetrizationTheorem:  an Adaptationist Account. ]

The real point of that observation has, of course, nothing to do with Freudian psychology per se, for we count ourselves (on many points) among its defenders, but rather takes aim at Assumption A -- the axiomatic non-existence of suprasensible realities.   If that assumption is infirmed in the case of mathematics, other questions are re-opened as well.

(In actual fact, I believe that even coffee-cups -- and certainly rabbits -- are largely supersensible, ourselves having access to but the intruding tip here below;  but that is a matter for a later seminar.  For the nonce, consult our discoveries concerning the elusive snow bunnies, who are uncontroversially supersensible.)

We have expounded and defended the Platonist account, in a long series of essays beginning here:

            Theologia mathematica

We hold -- to make the point precise -- that psychology has nothing whatever to say, nor ever could, about the transcendental organon -- timeless, independent of species and even of embodied consciousness -- of abstract mathematics itself.   Where psychology may have something to say, is about the vicissitudes of mathematical discovery, as a human (or Venusian) activity:

http://worldofdrjustice.blogspot.com/2012/02/psychology-of-mathematical-invention.html

So far, however, no-one but mathematicians themselves (as opposed to professional psychologists) have had anything of interest to say on the subject; and even they, not much.

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 !

A Minimum Axiomatization for Reality (Part II)

[The following continues an essay begun here.  Read that first or the rest won't make sense.]

Calling our toy system of the physico-noöspheric world  “ZFC” (borrowing the label from bloodless, disembodied set-theory)  is of course  in the first instance  just a fun pun: the “choice” is rather different in either case.  Nevertheless,  the analogy may be more circumstantial than is afforded at first glance.  For, the indubitability of free-will  depends upon ourselves being actual incarnated instances of same; some extraterrestrial scientist, reviewing the labnotes on time-series behavior-sequences of H. sapiens terrensis, might not be driven so forcefully to this conclusion.  As for Choice in the set-theoretical sense, we ourselves live outside of Mathland – most of us, very far outside it – and are thus  at best  in the position of that transgalactic analyst.  But for that special few of us who actually did emigrate to Mathland, and gain status as a Landed Immigrant, that axiom has been more familiar.  Michael Potter again (p. 291):

The centrepiece of Zermelo’s original axiomatization  was the axiom of choice.

And p. 259:

Cantor made frequent use of the axiom of choice in his work on cardinal arithmetic.  Indeed there is no evidence to suggest that Cantor ever doubted the validity of the axiom  for a moment:  it was a principle which, in Zermelo’s words, he ‘unconsciously and instinctively used everywhere, and expressly stated nowhere’.

Such fideism will scarcely be found  even in a Saint Augustine.

Indeed, even from outside, lacking all insight into what goes on inside our human pretty-little-heads, that alien sage (after examining reams of printout)  might be led to the positing of free will in H. sapiens, simply to make any sense of the data at all. Compare (Potter p. 260):

Many branches of abstract mathematics  are very much streamlined by the assumption of the axiom of choice.  A good example  is general topology, which becomes decidedly disconcerting in its absence (Good and Tree 1995).

The article referenced, incidentally, is entitled “Continuing horrors of topology without choice.”  Replace “topology” with “philosophy” and the phrase remains apt.

[Continued here]

The Urysohn Metrization Theorem (concluded)

(The continuation to this.)


Is there any distinction between a metrizable space and a metric space?   Seen naively, it’s the difference between a barn that hasn’t been painted yet, and one that has.

            Mathematically, the difference is insignificant.  Notice how one of the statements of the theorem  quoted above  slurs over the distinction:

     A compact Hausdorff space that is second countable is a metric space.

There is no mathematically interesting category of metrizable spaces prior to actual imposition of some specific metric -- analogous, say, to entangled quantum particles prior to collapse of the wave-packet, which are very interesting indeed, both philosophically (EPR Theorem, Bell’s experiments) and practically (quantum computing, quantum cryptography).  (For a quick course in the Uncertainty Principle, click here.)  If there actually were an analogy, how neat it would be, since in both cases  the final step involves (in some sense) “measurement”.

            There is, though, a lesson here for our larger project of Cantorian Realism, and the ontology and epistemology of mathematical objects.   Thus, consider a space (given initially as a base set and a defined neighborhood-system) which, after fiddling awhile, we find to be regular and second-countable.  Aha, so it’s metrizable, though knowing this does not by itself hand us a workable metric;  we’ll have to see what works.   Here, clearly, the metaphor of the unpainted barn breaks down.   For if barns -- which we build -- were like mathematical objects -- which (it is our contention) we discover -- some of them would prove recalcitrant to painting -- purely and simply unpaintable;  much as the Long Line can never be metric, howsoever it twist and turn.   Further, some paintable barns would admit more than one hue of paint, though not indefinitely many.

For let us emphasize:  Being metrizable is not a property of a bare set, but of a topological space -- that is, a base set together with a roster of which subsets count as open -- this roster itself is referred to as the “topology”.  The question then is whether a metric can be defined on the base set that will induce that roster of open-sets.  We have already been given the open sets we’re ‘aiming for’;  if the metric fails to yield these, then it is not a metric for that topology.  If no metric yields the right open sets, then that space (with that topology) is not metrizable.

Example 1:  Take the real plane, R x R, and let the interior of circles (i.e., open discs) be a basis for the topology.  Now define a metric on this set such that d(x,y) = 1 for all pairs of distinct points in the set.  This metric induces a topology all right -- the discrete topology, in which every pointset is itself open -- but it is not the Euclidean topology;  no cigar.  (Note:  The discrete topology is that of Leibnizian monadology, where every man is an island unto himself.)  The space itself is metrizable, however;   just use the usual Euclidean metric.

Example 2:  Now take a countably-infinite product of the set of reals with itself, R x R x R …  (You can pronounce this “R to the omega”.)   Assign the usual product topology to this (in which all but finitely many of the projections of an open set onto the individual R’s  must be all of that R).  You can induce this topology via a modification of the uniform norm.   But now instead assign the box topology (in which there is no restriction on how many of the slices may be less than all of R).  No metric induces that topology.

            As Dauben reports, Cantor himself eventually discovered the strange gap between our meeting a mathematical object for the first time -- presumably full-blown, yet still partially inscrutable -- and any eventual fullness of understanding. “Cantor no longer assumed that every set is born well-ordered.”

*

            Though the superficial similarity of the quantum case and the U.M.T.  doesn’t hold up, there does appear to be a rather arresting analogy with post-Chomskyan linguistics.

            The traditional view of language learning was that it involved general learning-strategies:  learning to make relative clauses was not radically different from learning your colors or the names of the kings of England (I caricature somewhat):  and just as different peoples conceive the color-palette in apparently incompatible ways, and the order of the kings might have been different (or no kings at all), so languages could differ indefinitely.

            Chomsky then challenged all this in ways much deeper and more philosophical than appeared to most people at first.   Many were surprised when, after laboring for a while at the forefront of fashionable linguistics, he out of the blue published a study of the time of Descartes, far outside the intellectual horizons of most of his followers.   But indeed, his project coheres, and always has.  Following his thought over the years, and finally getting the point, is a bracing intellectual experience.

            What initially attracted people was the positive expressive power in the slogan “Generative Grammar”;  yet very soon, those at the heart of the enterprise began to emphasize the theme of constraint. 

            In Chomsky’s view, as language-learners we must contend with certain hard (as in: hard-wired), quasi-algebraic parameters, each with a small finite range of possible values (often just two).  By our exposure to the particular ambient language in which we find ourselves, we (unconsciously) flip the various switches to their contingent, discovered position.   Certain combinations of settings will have further structural consequences.

            If we were as happily wired for topology as we are for language, we would meet a space, play with it in our cribs, learn in time what is the setting for its Separation parameter (T1, Hausdorf, regular, normal…), its Countability parameter (first-countable, second-countable, or neither) -- and having found that it is regular and second-countable, we would know it to be metrizable.

            Chomsky’s approach has been said, including by his fans, to involve an “innateness hypothesis”, a term at which he sometimes bridled.   And indeed, I called this roster of pre-existent parameters simply “hard”, where the imagery could be that of crystals (a full complement of Platonic solids, say) rather than that of a wiring diagram.  The default assumption in our scientific culture is, of course, that they reside on some gene or other;   but their actual nature renders problematic (not impossible) their visibility to the usual processes of Natural Selection.   (And again, to the puzzlement of his friends, Professor Chomsky never leapt with one bound onto the Darwin bandwagon.)

Also, if these parameters were coded for separately, one might expect a richer panoply of language-related mutations than is in fact observed.  What is the linguistic equivalent of lactose intolerance?

(Click here for our satire on the subject, which led to this whole U.M.T. thread in the first place.)  We might leave it open, just where these structures do reside:  perhaps upon that same hillside where the qualities of being Abelian, distributive, semi-simple, etc., may be found.

THEOLOGIA MATHEMATICA

 Synopsis/Zusammenfassung/ précis :

            Beginning with a parsimonious outset of only two Postulates,

            (1)   Die ganzen Zahlen hat der liebe Gott gemacht … [Kronecker]

            (2) …visibilium omnium et invisibilium [the Credo]

we conclude to the Realist position in mathematics, associated with Cantor and Gödel. We note the nice fit with theism.

Der hlg. Cantor

Der hlg. Gödel

More broadly, we distinguish the Truth of mathematics, from human mathematical practice.  Our Realist position applies only to the former;  our view of mathematical praxis is historical.

In parallel, our view of the Creator is Realist, but we take a historical position as regards the practice of the various religions.

It is not our goal, as José Ferrerós remarked regarding Hilbert’s program, “to employ Kroneckerian means for a justification of modern, anti-Kroneckerian methodology” (in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 151)  First, we could scarcely hope to succeed where the Hilbertians failed.  More to the point, our argument is essentially epistemological, rather than deductive, constructive, or even properly mathematical.  We contend, and hope to make plausible, along a meandering path of insights and fables, that our knowledge of the (eternal) truths behind the (contingent) practices of mathematicians, is not discontinuous in kind from the way in which we come to know anything which we know not directly or experientially but only inferentially:  the reality of atoms -- of quarks -- of Mars -- of Novosibirsk -- of the Pelopponesian War -- of Other Minds -- or … coffee cups.

[To begin the thread, click here.]

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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            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.

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

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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"

.