Love or Math

Re Drayton’s Endimion and Phoebe:

He would like to have depicted the Platonic ascent from carnal to intelligible love, but has really no idea of what one would find at the top of the ladder.  He has to fill up with astronomy and the theory of numbers.

-- C.S. Lewis, English Literature in the Sixteenth Century (1944), p. 532

Actually, not a bad expedient.

The idea is developed  here:

https://worldofdrjustice.blogspot.com/2014/01/the-ladder-of-abstraction-with-added.html 

Note:  The title of this post alludes to Edward Frenkel’s book, Love and Math, considered here:

http://worldofdrjustice.blogspot.com/2013/11/gradus-ad-parnassum.html

Ontological discontinuity (at the nether end)

As was known to all until perhaps the 18^th century (when certain mechanists would strive to deny it, as do certain homunculi  even in our own day), there is an ontological discontinuity between Man and Beast; a matter touched upon here.  But what is the case, within the order of the Beasts;  or indeed, between the least of these, and the mere madness of atoms?

We have  from time to time  penned odd little poems, or perplexing parables,  which in fact do seek to glimpse that phase-change, from random atoms  to the least of God’s creatures -- the infinum of the scala naturae.  For a representative sample, try this:

https://worldofdrjustice.blogspot.com/2012/10/gabriels-pep-talk-to-bugs.html

For the full roster of such things, here:

https://worldofdrjustice.blogspot.com/search/label/bugs

Beginning at the Bottom

Supreme perfections  in each insect  shine;

each shrub is sacred,  and each weed  divine.

-- Richard Blackmore

Such is the scene  at the foot of the scala naturae.

For glimpses of that lofty natural ladder, try these posts:

   The Great Chain of Being

And for God’s least creature, the lovable bug, these:

   Bugs

The Beasts: a post-Benthamite assessment (Take Two)

The Beasts:  a post-Benthamite  assessment (Take Two  Three)

[The Sage of Houndsditch, updated]

The question about animals is not, “Can they reason?”,

but  “Can they handle algebraic geometry  Hodge theory  motivic cohomology?”

[Note:  We have further updated this post, in an attempt to outpace the rise in cleverness  among the squirrels, which has been growing at an alarming pace. Already they have laid waste to algrebraic topology]

Principle of Sufficient Reason

The Principle of Sufficient Reason

together with its Corollary,

that All is Right with the World

“An object has been posited:
 a cat.”
 -- W.V.O. Quine

The cat is there

to be a cat.

And we are here

to greet that cat.

Non cogito -- sed sum

(QED)

Scala naturæ (downward direction)

Jesus preached to the people;

then Francis, a man, catechized the birds of the air

(as Saint Maël, we read, preached to the penguins).

Shall a penguin, then,

take pulpit to the ocean-folk,

filling them with such dim visions as might fit,

proportional to their subaqueous understanding?

On What There Is (expanded)

Existence is -- what existential quantification expresses.

       -- W.V.O. Quine, “Existence and Quantification”  (epigrammatic punctuation added) 

And, contra a couple of celebrated slogans of Quine:

The locution ‘ontological commitment’ is not one I have any use for, and neither do I care to ask or to answer the curious question “What is there?”  I say “There are chairs in the room,” and if someone wants to say “Therefore there are chairs”, tout court, it sounds odd… If this is ontology, then ontology is a mouthful of air.

-- Paul Ziff, Semantic Analysis (1960)

The subject of this essay is ontology; we gave a foretaste of the subject here.

By its dictionary definition, ontology is the study of Being.   Now, for me, “What is Being?” is the ultimate conversation-stopper;  that question, like Being itself in so bald an encounter, is like a diffuse and vaguely repugnant blancmange, filling all space.   It is questions like that which persuaded me early on that I was not interested in Philosophy.   And at that level, I still am not.

Quine, it turns out, is of like mind, for he remarks, of his epigram above, “This is as unhelpful as it is undebatable, since it is how one explains the symbolic notation of quantification to begin with.  The fact is that it is unreasonable to ask for an explication of existence in simpler terms. … Explication of general existence  is a forlorn cause.”

(Similarly, this, self-stultified by its own generality:

      The meaning of a remark in any language.

-- section-heading in: Jonathan Cohen, The Diversity of Meaning (1963), p. 154 )

A pre-philosophical, psychosociological observation:  The opportunity of waffling-on about Being with a capital B, seems to bring out the worst in writers.  As, Emerson (well, not quite fair to ontologists, since it doesn’t take much to bring out the worst in Emerson -- an ordinary pen-nib will do), in his celebrated essay “Compensation” (1841):

There is a deeper fact in the soul than compensation, to wit, its own nature.  The sould is not a compensation, but a life.  The soul  is.   ….  Being is the vast affirmative, excluding negation, self-balanced, and swallowing up all relations, parts and times  within itself.

Now that is ten pounds of horse-doody  in a five-pound bag.

~

At only one level down of abstraction, philosophers have traditionally brooded upon the ontological status of qualities or attributes or essences -- “whether concepts have a supramundane, or only a psychological existence;  whether they are transcendent intuitables or only private instrospectibles.” (Gilbert Ryle, “Ordinary Language” (1953).)   For hardcore Realists like Meinong and even early Russell, “Consistently with the assumed equation of signifying with naming, they maintained the objective existence of all sorts of abstract and fictional entia rationis.” (id., “The Theory of Meaning” (1957).)

The question becomes more compelling in the context of the philosophy of quantification (“To be is to be the value of a variable,” quoth the Quine); and shapelier still in the context of what counts as an ‘item’ for, say, physics.
(Incidentally… I have borrowed the title of this post from a well-known work (1948) of the eminent Harvard philosopher, who taught me logic when I was but a wee lad.  Quine, having passed to a different realm of quantification, will doubtless not object.)
(Sudden update:  Just browsing around, I notice that this is actually the second time I stole  the title -- Shakespearian in its simplicity -- of my former magister;  earlier effort here.)

Thus, the focus shall be now  not on Being, but on beings -- on what counts, for us, as entities, when we pursue science, and why.   As Quine nicely puts it:  “Ontology … is a generalization of somatology.” (Roots of Reference (1973), p. 88).  The top-down approach to ontology -- “What is Being?” -- is baffling;  but starting from what we understand, we might build upwards.

Thus, we confront more tractable matters of hypostasis (reification) and individuation.  

By getting down in the weeds concerning what choices have been hit upon by the various scientific enterprises that have had to deal practically with such matters, we might in time return refreshed to the general question.

Consider this analogy.  The ancient Greeks asked themselves, “What is motion?”, and discovered that, once you dig into the matter, it’s more puzzling than it looks  -- cf. St. Augustine’s celebrated quip about the meaning of Time:  If you don’t pose the question, I know perfectly well;  if you ask me point-blank, I am flummoxed.  Some philosophers even came to feel that the very notion of motion was paradoxical, or impossible : compare more contemporary thinkers with similar doubts about Free Will.  (Eppur’, in both cases, si muove.)  Once one has studied the matter, however, in classical dynamics and in special relativity, and understood how (Achilles and the tortoise) an infinite series may yet sum to a finite value, you return to the matter with new confidence.

~     ~     ~

We must concede at the outset that ontological quandaries seldom arise in daily life.  Only very occasionally, and that not systematically, do you pose What-There-Is questions.  Things like:  Does Bigfoot exist?  Does Dark Energy?  (Everyday life if you’re a physicist, that is.)   True, a questing undergraduate may once in a while trouble himself with questions such as the Existence of Other Minds, and Is the Universe an Illusion;  but such queries cease once he gets himself a proper girlfriend.

 

~     ~     ~

The atomists, in their purest and here somewhat idealized form, imagined a world in which indivisible particles were the basic Things, all else being combinations of these, and thus, in the most parsimonious view, ontologically subaltern.   (Leibniz imagined something rather like this for the noösphere, with his ineffable monads.)  And indeed, we can well imagine a world, in which such entities entered into but fleeting congeries, without definite or lasting outline, and crucially, with no emergent properties for the ensembles (thus, in particular, no reproduction of atomic ‘clouds’).   Such a world would have an essentially unambiguous, monolevel ontology.

Now, however, consider a different world:  a pool table.  And -- for this is necessary too, and we rather finessed the question in the fable immediately above -- consider that we have been given a task:  viz, to characterize the perambulations of matter atop it.   In this scenario, it is the billiard balls themselves we must consider, and in no wise the atoms that constitute these.

 

Consider now Euclidean geometry.  Here, fundamental ontological status was posited for just two entities:  the point, and the line.   (Notoriously,  one can present this geometry in a ‘substrate-neutral’ way that professes agnosticism as to the nature of these posited ‘points’ and ‘lines’ -- beer-mugs and beer-mats, we could call them just as well.  And, more tellingly, styles of geometry in which the point and the line are dual to each other, thus interchangeable.  But to consider this further, were to sail afield.)

No higher figures were distinguished as fundamental -- neither the triangle nor the ten-million-and-seventeen-gon  enjoy axiomatic status as part of the furniture of the Euclidean universe.  And indeed, in the broader perspective (the Erlangen Program) which sorts out and makes sense of a variety of geometries, it is not the individual figures  on which all things hinge, but their transformations -- their symmetries, and the way these form an algebraic Group.

~     ~     ~

The prototypical example of an indisputably extant entity is you.  You are physically coherent, you have purposes and plans, you are self-aware from moment to moment; ontologically, it doesn’t get better than this.  And if you’re Donald Trump, you’re done:  end of ontology.  You slide through life like a bubble down the duodenum, a blob of solipsism.

For the rest of us, we embrace the existence of Other Minds, and indeed quite on a par with our own.  

And now comes (as Blessed Pope John Paul II put it ) an ontic discontinuity or  “ontological gap” between ourselves and the beasts.   There is a spiritual truth to this, but biologically, it does not cut nature at the joints.

(Note:  That gap itself should not be over-emphasized, since, in the grand mediaeval vision of the scala naturae, it is just one of several such.  Roughly:

archangels -- seraphim -- cherubim -- penguins -- mankind -- critters -- Protista -- sludge.)

[Click that image for more exciting details!]

So we admit the biosphere -- only, just where to draw the lines among individuals gets murky, the more you learn about what-all is out there.  Herd animals, species all of whose members are genetically identical, parasites, incorporated former parasites such as plasmids and mitochondria, slime mold, elm forests (one giant subterraneanly-connected plant), and even such exotica as the cast-off arm of a male cuttlefish:  as Darwin put it, “So completely does the cast-off arm resemble a separate animal, that it was described by Cuvier as a parasitic worm”.

There is no fact-of-the-matter about such cases; their intershadings show that our question, Which are the functional individuals?, must be more sharply posed.

~

~  Posthumous Endorsement ~

"Were I alive today, and in the mood for a mystery,

this is what I would be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(I am Quine, the great and powerful;

and I approved this message.)

~         ~

Let us revisit the examples of the pool table, and the geometries.   Here the basic entities were identified relative to certain transformations of the roster of potential entities:  the billiard balls caroming about, rebounding, never blending, proved to be the units to reckon with here.   And in modern mathematics, the symmetry transformations of the individual geometries proved more important that the various squiggles and shapes (or collections of squiggles and shapes) that undergo them.    So perhaps the way forward is to consider the kinematics of life. Ecology, that is, and Evolution.

When the theory of Natural Selection was introduced to the world in 1859, species rose to prominence in our conception of the way the world really is, right in the title of that great work, The Origin of Species.  Individuation can be problematic when we contemplate such things as animals undergoing complete metamorphosis, sessile vs. vagile stages,  and so forth:  but at each moment the species are (in the somewhat idealized classic view) sharp in outline, non-interbreeding, reliable entities.  (From a NeoPlatonist perspective, the species may even be more real than any of the variously imperfect and misshapen individuals that instantiate that ideal.)  For a time, Nature red in tooth and claw was conceived as a battle among these  supra-individual entities, competing, going extinct -- tyrannosaur versus triceratops. predator and prey, the early mammals peering out discretely from the prehistoric underbrush, waiting their chance.

Yet no sooner had we managed to wrap our heads around the notion of the species   as the fundamental unit of biological accounting (which in particular, delightfully,  cleared up the mystery of sex), than a pot of cold water was flung in our face:

Why should a female  produce offspring carrying only half her genes, when by parthenogenesis … she could produce clones…?  The simple answer, that the variability produced by sexual recombination makes for greater adaptability, and is therefore ‘for the good of the species’, will not serve.  Darwinian natural selection … has to do … with individuals,  and selection for group characteristics  has no simple place.

John Bonner & Robert May, introduction (1981) to a reprint of Darwin’s Descent of Man.

The next step (and the consensus of current thinking) settles neither on individuals nor on groups, but on a unit which, in Darwin’s day, was not even known specifically to exist:  the gene.  The argument has been superbly laid out for the general public in Richard Dawkins The Selfish Gene, so we needn’t walk through the reasoning here.  The upshot is as follows:

Richard Dawkins, The Selfish Gene (1976; 2^nd edn. 1989), p. 34:

In sexually reproducing species, the individual is too large and too temporary a genetic unit  to qualify as a significant unit of natural selection.  The group of individuals is an even larger unit.  Genetically speaking, individuals and groups are like clouds in the sky or dust-storms in the desert.  They are temporary aggregations or federations.

(This reminded me curiously of a suggestive passage from a historical-espionage novel by Tim Powers, Declare:

You know what the djinn tend to be made of, from moment to moment -- wind, dust, snow, sand, agitated water, swarms of bugs, hysterical mobs. )

Anyhow, Dawkins goes on to make clear that his definition is functional not anatomical:

The largest practical unit of natural selection -- the gene -- will usually be found to lie somewhere on the scale between cistron and chromosome.

This functional/structural rather than physical definition  is reminiscent of the notion of phoneme, as opposed to a phone or sound.

Edward Wilson concurs:

The average differences between people in different localities … are narrowing.  Genetic homogenization has similarities to the stirring together of liquid ingredients.  … But the most elemental units, the genes, remain unperturbed.  They stay about the same  in both kind and relative abundance.

-- E.O. Wilson, Consilience (1998), p. 273

Now we feel we are back on familiar ground.  These genes are rather like biological analogs of atoms, in the old Greek well-behaved, billiard-ball-like conception of these.  They just take some getting used to.

Yet even after the first ontological question has been answered (What is there?) in favor of the gene, there is still the second (What is it?).  As.

Should we think of a gene … as a structure that is replicated, or as information that is copied and translated?

J. Maynard Smith & E. Szathmáry, The Origins of Life (1999), p. 10

Actually, Dawkins makes a much simpler and apparently unanswerable argument for thus privileging the gene as a unit of accounting:

The true unit of natural selection has to be a unit of which you can say it has a frequency.

(Individuals and groupings obviously don’t fit the bill.)   This argument is completely general, and is independent of the details of biology.   Thus in particular, it should apply (if it is valid) mutatis mutandis  outside of biology.

For the style of thought, though not the detailed content, cf. Quine (“On What There Is”), maintaining that quantification is “the only way we can involve ourselves in ontological commitments”.

Further, compare this:

Gerd Gigerenzer et al, The Empire of Chance (1989), p. 246, quoting Read Tuddenham:

To the statistician's dictum that whatever exists can be measured, the factorist had added that whatever can be 'measured' must exist 

[For a brief and untendentious survey of the various candidates for status as a Unit of Selection, click here.]

~     ~     ~

Having persuaded us that the gene, rather than the individual or the herd or the species, is the fundamental reckoning-unit of life, Dawkins then complicates matters in a way reminiscent of those extended and ill-individuated entities like elm forests and slime molds, or even Bertrand Russel’s definition of the number ‘four’ as the set of all foursomes:

What is the selfish gene?  It is not just one single physical bit of DNA, it is all replicas of a particular bit of DNA.  … ‘It’ is a distributed agency, existing in many different individuals at once.

By this time it is clear that, the more you look into it, the ontology of biology looks more like biology and less like Ontology -- in that original maximally abstract metaphysical program to whose allurements we confessed ourselves deaf.

Still, this business of the gene, defined as a functional rather than a spatiotemporal unit, does get us back to old-fashioned ontology as practiced by philosophers.   Quine sparkles at this.   He wastes no time on “What is the Nature of Being?”, but rather rolls up his sleeves, and, in the chapter “The Ontogenesis of Reference”, constructs a plausible, insightful, and wittily-told fable of how we acquire our notions of objects, and what it is that we acquire.   (A wry tribute to the style of mind involved in such exercises  can be appreciated here.)   By page 98 of Word & Object (1960), he has made a case for the ontological respectability of “a single sprawling object”, and admonishes:

There is no reason to boggle at water as a single though scattered object, the aqueous part of the world.  Even the tightest object, short of an elementary particle, has a scattered substructure  when the physical facts are in.

To which, Amen;  adding only that, when even more surprising physical facts are in, concerning indistinguishable elementary particles (bosons, at any rate), there is a sense in which these too could be considered a single scattered object.

The genes (or bosons), as thus conceived, are only, so to speak, accidentally scattered;  things empirically might have been otherwise.  Consider now rather entities that are scattered by construction, by definition:  higher-level entities, sets or collections of lower ones.

Questions about the ontological status of such things can arise even in the everyday pre-philosophical world.  In what way can we say that the following are genuine entities, with lasting contours and cross-temporal identification, despite the changing roster of the individuals that make them up? -- Your (nuclear/extended/….) family; the Boy Scouts; the nation-state to which you belong.   This is a moral and practical matter, not simply ontological:  having pledged allegiance to any one of these at t=0, are we likewise bound at a later time, despite their ever-shifting membership (and foreign policy)?  (I address such questions in a projected essay, “Continuity of Identity”.)

So, we have noticed an actual ontological question within the cares of daily life.  Still, it is not to a metaphysician that you would turn for clarification, should your eighth cousin thrice removed suddenly show up on your doorstep, claiming ties of kin that give him the right to move in with you and to borrow your car,  nor to an ontologist, were the Boy Scouts ever to get the Bomb.

In Biology -- the fons et origo of structured higher-level objects in scientific practice -- such entities include:  species (made up of conspecific individuals);  genus (made up of species); family (made up of genera); order; class; and so on up.  Here the ‘atom’ is the individual animal or plant;  there is no place in the traditional taxonomy of considering an individual as a congeries of genes -- and indeed the set-theoretical structure is completely different, the gene-sets in question being radically non-disjoint, whereas an animal is either in one species or another, not both.

 

In the nature of the case, it is clear that the higher taxa of biology are not ontologically given as such, but are confections of convenience, based  to be sure  upon what’s out there, and proceeding via sound and defensible principles.  Thus in particular, whereas a species as a whole does pretty much hang together or hang separately (say, in a sexually reproducing species, if the survivors are too sparsely scattered to hook up), there is no such selective linkage among the various n-level groupings in a taxon at level n+1.  Should the echidna ever bite the dust, ‘twill be a sad day for all lovers of monotremes; but the valiant platypus  still will soldier on.

There have also been major revisions in higher-order taxa;  even some quite familiar ones (reptiles, insectivores, puffballs) have  upon closer inspection  been dismissed as polyphyletic.

~     ~     ~

So much for the entities of biology.  What of Chemistry -- which is “the next level down” in terms of the agenda of Consilience?

Here we are in for a pleasant surprise.  No such agonizing and backtracking will be necessary as it was before.  The answer is:  atoms.   And not just atoms, in a row as it were, but, stacked, structurally stacked, in a most revealing way.  This is the Periodic Table of the Elements, first unveiled to a grateful world by Mendeleev, of blessed memory.  It is possibly the single most satisfactory scientific object on the planet.  Moreover its elements and its structure reach directly, consiliently, straight down to basic physics.  It is a wonder to behold.

There is even a loose analogy between atoms-and-molecules, on the one hand, and genes-and-individuals, on the other.  Loose, but better than most of those cited by Wilson in his ambitious book.

~     ~     ~

Physics, by contrast, is in no such happy case.  Such subjects as cosmology or thermodynamics or hydrodynamics don’t seem to have ‘basic-level objects’ in any obvious way.   There are, to be sure, the “elementary” particles, but these have been as troublesome as they are helpful, referred to distastefully as the “particle zoo”.   What with quarks and various subtle symmetries, these have now been regimented into something more satisfactory, though still nothing like as self-explanatory as the Periodic Chart.  Further, they do not span the whole of physics, but only of Particle Physics, a subfield.

There are, nonetheless, deep ontological questions within physics, with still-tentative but sophisticated answers.   I am not currently competent to comment on these, but a selection of intriguing quotations may be consulted here.

~     ~     ~

Astronomy affords relatively little by way of ontological interest; but consider this wise observation by astronomer  Mike Brown (quoted in The New Yorker for 24 July 2006):

Planets are like continents. ‘Continent’ is a good geological word, but, like ‘planet’, it has no scientific meaning whatsoever.

That is an epigram, and thus permits itself a breezy way with words; meaning here really means ‘ontological status’.

The point is of course lost on the layman;  witness the heavy coverage of Pluto’s “dethronement” by the latest Kuiper-belt detritus, as though this were of the least importance for the understanding of our cosmos.  But far more important, the opposite assumption seriously misled some of the finest minds of the Middle Ages.  For Galileo’s misadventure with circular planetary orbits, click here.  For Kepler’s fine failed vision of the planetary distances reflecting nested Platonic solids, here.  Their basic insights were sound, even brilliant; but planets (i.e., floating lumps of dirt) simply don’t have the ontological status to deserve such angelical constructions.

~     ~     ~

In Mathematics, the conundrum concerns, not so much the existence of thís (class of) object versus that (class of) object, let alone which are the ‘basic-level’ objects (I know of none), but the existence of any objects überhaupt.  That is, we have retreated from the question of beings, and are back at the bad old topic of Being.   At best:  for in fact, the question is probably not best posed in terms of “the existence of mathematical objects”, which threatens to involve us in fruitless discussions of what they are exactly (e.g. the integers as really sets of one sort or another, including Russell’s extravagant suggestion), whereas in fact,  mathematical objects or entities or thingums or whatever they are, are the very plume and prototype of substrate-neutrality;  a less contentious formulation would be “the transcendence (epistemological independence) of mathematical truths”.  (One is less likely to wonder whether a “truth” is, say, pink, than whether an object is.)

Let us consider a specific question, with an at least superficially ontological aspect, that is more localized than that vast barely-answerable question about the ontological status of mathematics as a whole.  (My attempt at a Realist answer to that one begins here.)

For example:  Does there exist a topological object of the following description:  It is regular, second-countable, yet could never be assigned a metric?   Urysohn looked into the matter, and concluded that none exist.  But it wasn’t by looking around, or by exhaustive search, that he reached this conclusion, the way you might drag every inch of Loch Ness and finally conclude that it contains no monster.  Never quitting his armchair, he deduced the result, in a way in which things were never really serially considered.

Indeed we had to strain a bit to cast the problem in the form of a question about ‘existence’ at all:  it’s not like finding Bigfoot, or failing to find him.   If biology were like mathematics, then we could infer the existence or non-existence of Bigfoot, without ever actually spotting him, nor searching the wooded hills, based upon abstract patterns elsewhere in the system.  This is one of the very many ways in which biology and mathematics are not the least bit alike (I mention this only because of the counter-program of consilience - a nice idea, but a will-o’-the-wisp.)

Mathematics does nonetheless afford good grist for the ontology-mill, indeed more clearly ontological than anything we have yet seen.  Namely, the entities posited by what are known as “existence proofs”.    There is no properly (intra)mathematical doubt about these purported objects -- they uncontroversially have such&such properties, if indeed they are there to bear properties at all.  The problem is with the special sort of purported demonstration that says, although we may never see such a thing, yea verily, it doth exist.  Such proofs can be purely deductive, non-constructive; so that, although we are assured of the existence of something fitting a given description, we are given no hint as to how to find the item in question.  Understandable ontological qualms about such spectral beings led to the founding of a school of mathematics that rejects such non-constructive proofs:  Intuitionism.  This dog-in-the-manger school gets vastly less play in actual day-to-day mathematical practice, than it does in philosophy books.

The one place within mathematics where ontology is definitely at home is Set theory.  A typical credo:

I have written this book from an uncompromisingly realist or platonist position; that is, I have taken the viewpoint that  in some sense  sets do exist,  as objects to be studied, and that set theory is just as much about fixed objects as is number theory.
Frank Drake, Set Theory (1974), p. 18

Indeed, this subject is often practiced by ontologically-inclined philosophers (such as Quine) and taught in the philosophy department.  (That other mathematical outlier -- logic -- is likewise often so housed.  I took Intro Logic -- “Phil 140” -- from Quine.)  It is from this milieu that we got the slogan “To be is to be the value of a variable.”

Quine’s quip, suitable for recital to the babe in the cradle, is actually trickier than it sounds, owing to his notion of substitutional quantification, which does not express existence, vs. objectual quantification, which does.  There is a grey area of entities which, like most nonalgebraic real numbers, are assumed to lead just as robust an existence as the algebraic irrationals, but which are not finitely specificable.
(Further remarks on the ontology of logic and set theory  here.)

Note, incidentally, a certain resonance between this last distinction, and the notion in physics of observables -- an attempt to get a firm handle on What There (Really) Is, amid the welter of mathematical abstractions.

 

~     ~     ~

Linguistics and Anthropology come each in two flavors: on the one hand, traditional mostly-European philology and Völkerkunde; and on the other, a present-day, typically American scientistical approach.  The former fall under the Humanties.  They were not much concerned with positing abstract analytical entities;  the “parts of speech” go back to ancient times, and were defined intuitively, largely morphologically, which is something you can get away with in the highly inflected classical languages like Latin or Sanskrit or Greek.   The latter, by contrast, strives (or, in the case of Anthropology, strove; now it strives only to be politically correct) to be honest-to-goodness sciences like physics or anything else.   Many intricate and closely-argued entities were posited and fought over, for phonology and syntax (some in morphology and semantics too, of course, but those weren’t worth fighting over);  and anthropology became algebraically structural in its analysis of kinship systems.  Both fields were self-aware of what they were up to, and there was a running discussion of the ontology of the theories, under the genial rubric “God’s Truth vs. Hocus-Pocus”.  The God’s Truth faction took a Realist stance towards the posited analytic entities; the Hocus Pocus faction, a Nominalist.

~     ~     ~

Somewhat surprisingly, the study of Folklore is also distinguished by the positing of abstract analytic entities, known as motifs; and this, already in the early years of the twentieth century.  These were carefully and exhaustively catalogued in the Stith-Thompson Motif Index.  Their combinatorics determine the tale-types around the world.   They are reminiscent, not really of atoms (since the characteristics of molecules are so wildly ‘emergent’ above anything visible in the atoms that make them up;  cf. H2O, I rest my case), but rather of genes.   Okay, the analogy is loose, but no worse than that of genes & memes.  Indeed, motifs were the forerunners of the meme idea, and already much better thought out.  There is even a sort of folkloristic analogue of the allele:  the oikotype.

 

~

There is no reason to boggle at water as a single though scattered object,  the aqueous part of the world.  Even the tightest object, short of an elementary particle, has scattered substructure  when the physical facts are in.

-- W.V.O. Quine, Word and Object (1960), p. 98

In support of this:

(1) “the aqueous part of the world”:  cf. “empty space”, an anything but simply-connected entity (object).

(2) “scattered substructure”:   Unsure quite what he meant by this -- quarks are substructure of hadrons, but were unknown -- nay, unhypothesized -- in 1960, the publication-date of Quine’s classic.   However, a “smeared-out” (not really ‘substructural’)  nature of something so tiny-tight as the electron (still regarded as truly elementary) was suggested already

~     ~     ~     ~     ~

Postscript:   These are the posts so far that have touched on ontology. These largely concern mathematical Platonism, which we won’t focus on here,  other than to say that Quine’s quip ("To be is to be the value of a variable"), suitable for recital to the babe in the cradle, is trickier than it sounds, owing to his notion of substitutional quantification, which does not express existence, vs. objectual quantification, which does.

[Footnote] Contra-Quine:

It is not true that ideas face the bar of reality as corporate bodies:  rather, in the past, they evaded reality  as corporate bodies.  …  This word has, so to speak, a turnover ontology.  The “objects” (ie. the terms in which we classify the continuum of experience  into “things”), are not there for keeps.  In trying to handle … the continuum of experience, it is … proper to experiment with … diverse ways of clustering the flux into “objects”.

-- Ernest Gellner, Plough, Sword, and Book (1998), p. 64f.

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.

(ode to bug)

We here salute   the least of God’s creatures,

the  (humble bug),

beside whom   e’en the Humble Woodchuck

doth seem proud.

Nay,  but for the bug,

the Ladder of Nature  had collapsed,

lacking a footing.

Oft kroch ein Käfer kribbelkrab

Am hübschen Blümlein  auf und ab.

-- Wilhelm Busch

And here we have a little poem written by an actual bug.
(Here you can see him actually writing it.)

O Noes !!

O ...  Noes!  Da it-tle bug!

Bug  not  know!   Bug all confused!

Meb-beh   kit-teh    eat     da bug !!!

Allgone bug   go allgone !!!!!!   (runrunrun)

(Fragments of consciousness, like sparks --

nay   like  fireflies

in the dark …)

Thumbnail summary for the busy business-man:

All U really need 2 know about bugs

Flash!

Recent figures released by the Department of Agriculture

reveal that there remain only 934 Americans

who have yet to savor the pleasures

of the pistol-packing, wise-cracking

Murphy Brothers, P.I.s.

If you are among these unfortunate few,

make up for lost time here:

I Don't Do Divorce Cases

Murphy on the Mount.

[Update -- or rather, retrodate]
Cf. James Thomson's poem "Summer" (1727):
   the mighty chain of beings, lessening down
   from infinite perfection  to the brink
   of dreary nothing, desolate abyss...

(Comments Bug:  "Well, I like that !!")

Moral:  Do ye not look down  on Peter Paramecium !

.