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.

The Ontology of Biology (mise à jour)

We treated of this in some detail in our core essasy, “On What There Is”.  Indeed, the details of this particular discipline threatened to overwhelm that general survey.  Accordingly, we remove to this adjacent seminar-room, for a few additional remarks.

Let it be said, that biology, dealing with actual palpable furry waddling entities, would not appear a priori  a likely arena of serious ontological doubts.   Physics, by contrast, can’t avoid such scruples, since it deals largely with the unseen, and postulates just about everything.  Mathematics, freilich:  it being (as some would see it) about Being itself.   The only reason, then, that the study of biology figures so prominently in our remarks, is the extraordinarily high level of methodological and even philosophical thinking that has been invested in the evolutionary enterprise, beginning with Darwin himself. 

We shall begin, as we love to do, with a quote from Quine:

When we say that some zoölogical species are cross-fertile, we are committing ourselves to recognizing   as entities  the several species themselves, abstract though they are.

-- Quine, “On What There Is”.

This is true -- indeed, true virtually as a matter of sheer quantificational logic; but its proper biological content is nil -- as Quine well knows.   We may avoid this purely formal commitment by “so paraphrasing the statement  as to show that the seeming reference to species on the part of our bound variable  was an avoidable manner of speaking.”  (The substantive, as the saying goes, can be pegasized away.)

~     ~     ~

In our essay above-referenced, we focussed on the ontological tug-of-war between species and gene as the favored stockkeeping unit of evolution.  But whatever the upshot of that, species themselves are obviously central.  Only … what are they?

As so often, Darwin was ahead of the time he himself largely created.  Coyne & Orr quote his Origin of Species (1859):  “We shall have to treat species in the same manner as those naturalists treat genera, who admit that genera are merely artificial combinations  made for convenience.”

And to similar effect:

J.B.S. Haldane observed that “the concept of a species  is a concession to our linguistic habits  and neurological mechanisms”…

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 11

And again:

Ridley notes:  “The fact that independently observing humans  see much the same species in nature  does not show that species are real rather than nominal categories.  The most it shows  is that all human brains are wired up with a similar perceptual cluster statistic.

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 14

Why are organisms apportioned into clusters separated by gaps? … Dobzhansky (1935) found this question intractable:  “The manifest tendency of life toward formation of discrete arrays  is not deducible from any a priori considerations.  It is simply a fact to be reckoned with.”  … While history can create discrete clusters containing groups of species, we do not see how it can produce species themselves.

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 49

Biologists… even questioned whether species exist.

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 5

…whether species are real entities  or arbitrary constructs of the human mind.  [This is the Nominalist or ‘hocus-pocus’ position. -- dbj]  Several lines of evidence  show that species are real.   …  One asks whether assemblages of individuals -- populations -- are partitioned into discrete units that are objective, not subjective.

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 7, 10

(To see how far- or over-reaching such scepticism is, for species substitute coffee-cups.)

We pause to marvel at such self-critical ontological nominalism about the central practical entities of one’s own discipline.  As though:

Dentists:  When we speak of ‘teeth’, this is a largely arbitrary demarcation among the hard structures of the body.

Football coaches:  To counterpose ‘offense’ and ‘defense’ is to indulge a false dichotomy -- which is, however, useful for certain purposes.

Patrolmen’s Benevolent Association:  To talk of ‘crime’ is, of course, merely a façon de parler.

Systematists, whose task is unraveling the history of life, often prefer species concepts  different from those used by evolutionists more interested in evolutionary processes.

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 10

~

~  Posthumous Endorsement ~

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

this is what I'd be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(I am Carl Linnaeus, and I approved this message.)

~         ~

~

Another ontological posit:

Cluster analysis distinguished 32 fairly discrete groups in phenotypic space (“phenons”).

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 23

(One’s heart rather sinks, reading this.  Recall the profusion of dubious -emes in linguistics.)

… the essential dialectical unity of the biological and the social,  not as two distinct spheres … but as ontologically coterminous.

Lewontin, Rose, & Kamin, Not in Our Genes (1984)

~ 

All species concepts require some subjective judgments.

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 34

This, from a pair of very hard-headed writers.  Cf. similar points by Keynes in his Treatise on Probability (1921).

Because we rejected ecological differentiation as part of the Biological Species Concept in sexually reproducing groups, we obviously endorse the use of different species concepts in different groups.  We do not consider this pluralism to be a weakness of the BSC.  Because the causes of discreteness may well differ among taxa, so may the concepts appropriate to addressing the species problem.

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 52

Since this astute observation strikes me as very sound, I shall give it a nice name:  not Nominalism about species, but Polytypic Realism.

~ 

Just as it may be no simple matter to posit the right entities, it may not be evident how to ask the right questions:

The main reason we have had a hard time answering “How many genes cause postzygotic isolation?” should now be clear:  It is not a single question.  Instead, this query masks a large number of questions that may have very different answers.

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 306

And indeed, the authors conclude:  “There has been nearly endless discussion of species concepts.   This vast and stupefying literature has produced little new or interesting biology.”

[Update 22 March 2012] And indeed … Andrew Hamilton, reviewing Richard A. Richards, The Species Problem: A Philosophical Analysis, Cambridge University Press, 2010, writes:

There are four books on my shelf and countless papers in my electronic and physical files with the title "The Species Problem." Many of these were written by philosophers, but almost as many were written by biologists. …  None of the books and papers on this topic contains anything like a consensus solution.

The reviewer then rejects this latest attempt:

As a solution to 'the' problem or as a theory-based definition, however, Richards' strategy isn't going to work. … While it may be true …  that all contemporary species concepts pick out population-level evolutionary lineages, they do so in different and often incompatible ways. Agreeing that species are lineage segments goes no distance toward helping to find the boundaries or to settle the host of arguments about what boundaries to seek. This approach swaps one hard question (what are species?) for another (what's an evolutionary individual?), as is the case with most or all of the other proposed solutions to date.

Metasolution:

Having rejected the solution, I would now like to reject the problem.

That, actually, has pretty much been the history of Anglo-Saxon analytic philosophy, from G. E. Moore on out:  you don’t solve a problem; you unmask it as metaphysical (or some other handy word), and dissolve it.

[Update 26 May 2013]  Even more radically rejectionist is What Darwin Got Wrong (2011), by Jerry Fodor and Massimo Piattelli-Palmarini.

The latter author (who largely wrote the former half) reviews (p. 57) the entities that have been proposed as the axis upon which all selection turns:

Pleas have been made … for revising the traditional neo-Darwinian thinking that either the individual, or the population as a whole, are the sole units of selection.   There are selective processes … also at the level of genes, chromosomes, whole genomes, whole epigenomes, cells, developing tissues, kin groups, societies and communities;  and, of course, organisms and populations.

(Notice that he doesn’t even mention “species”, as in The Origin of Species, yo?)

The former author, laying about him with a broadsword, writes (p. 126)

Phenotypes aren’t bundles of traits; they’re more like fusions of traits.  Prima facie, the units of phenotypic charge are whole phenotypes.

Compare the famous hyperholistic thrown-gauntlet of Quine: “The unit of empirical significance is the whole of science”.

Remarks on Nominalism

Donald Trump and a Nominalist  walk into a bar.  The Nominalist (who, to his relative credit, is also a Minimalist) orders a glass of water.  Trump downs a fifth of Baccardi and stumbles off to the loo to vomit; trips, falls head-first into the turd-filled toilet and drowns.  The Nominalist, too dazzled by the phenomenalistic surface of things to perceive the underlying Reality, fails to notice that, instead of genuine water, he has been served a cheap substitute.

Q:  How many Nominalists does it take to screw in a light-bulb ?

A:  Nominalists doubt the existence of light.  Consequently, they live in the dark.

Q:  Why did the Nominalist cross the road ?

A:  He didn’t.

On Aboutness

We sometimes meet dismissive statements like this:

“Mathematics appears to be not about anything at all.  On such a view, it has no subject-matter.”

When confronted with such a pronouncement, we resort to the Weierstrass Penguin Test:  namely, does the same objection apply to penguins?  What are they “about”?  Well, in the colloquial sense, they are all about :  playing, swimming, sliding on their tummies, and all the rest;  but that is not a meta-level aboutness.  They are not the less funny and fat, for all that.

When someone speaks of mathematics, dismissively, as not being “about” anything, we suspect he never made it past simple arithmetic, and finds the integers pale and spectral  next to nice red apples.   And it is true, you cannot eat the number 2 [**]; but then, neither can you really take the square root of a couple of apples.

There is plenty for mathematics to be about. Geometry is about spatial structures.  Number Theory is about Our Friends the Integers.  Topology is about sets together with a neighborhood system. Algebra is about relations that generalize those familiar from simple arithmetic.  And so on.  The reason it is hard to say what mathematics overall is “about”, other than the tautological-sounding “mathematical objects”, is that it is so rich and deep and broad.  What, for that matter is literature ‘about’, or politics, or scholarship?

[**]  Actually, according to the frantically nominalist view of Bertrand Russell,  you can -- or a part thereof at least.  Namely, those two apples, which, along with the Everley Brothers and much else, go towards making up the unimaginable vastness of duple things, which his Lordship once fancied were simpler than das, was der liebe Gott gemacht. To such extremities is the atheist obliged to travel.

A Taxonomy of Nominalism

Among my favorite books in the field of logic is Susan Haack, Philosophy of Logics (1978).  The plural in the title might be surprising -- isn’t Logic simply the formalization of the way things are?  -- but it is by no means a typo.  She distinguishes no fewer than five families of logics:  traditional logic (i.e., the Aristotelian syllogistic), classical logic (what you were  taught in college, unless you spent all your time on panty raids), extended logics, deviant logics (no no, it’s not what you imagine), and inductive logics.  These each come in various flavors:  thus, under “extended logics”, we understand:

            modal logics

            tense logics

            deontic logics

            epistemic logics

            preference logics

            imperative logics

            erotetic logics  (no, that’s not what you imagine either)

Note the plurals:  These each have subflavors too.

Are we having fun yet?  We should be.  William James remarks (The Principles of Psychology (1890), vol. II)  that humanity seems to have a kind of taxonomic instinct, reveling in setting up systems of pigeon-holes:  rather like the sex instinct but even better because you don’t get AIDS.  Surely you remember the delight you felt as a ten-year-old, putting your return address on the envelope of a letter to a friend (well, that was before e-mail;  you’ll remember if you are over about eighty years old):

Timmy Thompson

322 Mulberry Street

Blissville

Indiana

The Midwest

America

The Western Hemisphere

The Planet Earth

The Solar System

The Milky Way …

Nowadays it’s even better, since you can keep going all the way up to the Multiverse.  (Only, no-one writes letters anymore, and there are no more ten-year-olds.)

In her later work, Evidence and Inquiry (1993), in the course of arriving at her new style of epistemology --- “double-aspect foundherentism”, no less -- Susan Haack coins or refines nomenclature for a great many epistemological schools and subschools, and even for individual varieties of ideas, labeling these with such tags as 

            FD1 superscript-E subscript-EXT

and

            FD2 superscript-P

rather reminiscent of the old joke about the comedy club, for whom the jokes were so familiar they all had numbers.  (Punchline:  “933.”  Nobody laughs.  Why not? “He didn’t tell it right.”)

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Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

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In this spirit, we shall here embark upon an ambitious project of anatomizing Nominalism.  (Funded by USAF grant #586-77-E8-99999.)

(1)  On Nominalism

There are a great many individual theories beneath the big tent of Nominalism.  Since, however, they all suck, these need not detain us.

Much more rewarding is the following.

(2)  On Anti-Nominalism

There are several venerable schools in this vein.  Most familiar is Classical Anti-Nominalism, also known as Anti-Nominalism simpliciter, whose central thesis reads as follows:

(C.A-N.)  Nominalists just suck.  End of story.

Then there is Aetiological Anti-Nominalism, characterized by the following insight:

(Ae. A-N.) Nominalists were all dropped on their heads as infants;  a fact that explains, while it does not excuse, their behavior.

And lastly Pediculotic Anti-Nominalism, which maintains:

(P. A-N.)  Nominalists can best be understood as a form of body lice.

This popular view breaks into three schools, according as the insects in question be specifically head lice (Pediculosis capitis), lice infesting the armpit hair, or some even naughtier lice of which decorum forbids mention.

(This will all be on your mid-term.)

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Dames, dames, dames !

We now return you to your regularly scheduled essay.

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[update October 2011]

I just stumbled upon this spicily-titled item:

Scientific Realism and Mathematical Nominalism:  A Marriage Made in Hell

Here we learn that “Nominalists hold that there are no numbers”;  this involves them in certain epistemological difficulties.   But en revanche (ô doulce revanche), numbers hold that there are no nominalists.  This  involves them in no difficulties at all.

~     ~     ~

Nevertheless, in that broad spirit of tolerance so characteristic of Realists, we here list those instances in which a Nominalist did (against all odds) manage to come up with something witty or interesting.  We believe this list to be complete.

Unum, verum, bonum -- the old favorites deserve their celebrity.  There is something odd about each of them.    Theoretical theology is a form of onomatolatry.

-- J. L. Austin, “Truth” (1950)

 

~     ~     ~

 

[Update 2016]   Well okay -- one synonym from a linguistically-inclined Nominalist in his treatise about ambiguity and vagueness:

Applied to semantical topics, the nominalistic attitude here outlined  may be labeled  inscriptionalism.  The label implies no theoretical contrast with nominalism;  it simply calls attention to the favored status of tokens, and the exclusion of types, classes, meanings, forms, and attributes  from our semantic apparatus.

-- Israel Scheffler, Beyond the Letter (1979), p. 8

Cf. Realism, in a math context, being called Platonism.

Realism: Contrasting Terms

 [This continues a lexicographic/definitional thread begun here.]

Rather the way information about an analytic function in the interior of a region  may be gained from an examination of its values on the boundary,  the nature of Realism -- a multifaceted body of thought -- may be illuminated by various contrasts with what it is not.

Michael Dummett writes, in “Realism” (1963), repr. in  Truth and other enigmas (1978), p. 145:

I was told at school that the scholastic doctrine known as realism, and opposed to nominalism, had nothing whatever to do with that opinion known as realism in later philosophy, as opposed to idealism.  It was only much later that it struck me that the two disputes bore to one another an analogy which made the use of the same designation ‘realism’ for one side in each of them  more than a pure equivocation:  although the subject-matter of the two controversies differed, there was a rememblance in the form of the disputes.

(We heartily concur in this formulation.)

Dummett goes on (p. 147) -- to the admiration of this former lexicographer -- to present a list of dichotomies:

realism about material objects, opposition to which has traditionally taken the form of phenomenalism;
realism about the theoretical entities of science, which is opposed by scientific positivism:
realism about mathematical statements, for which I shall use the standard name ‘platonism’, employed (not altogether happily) by Bernays and Quine, opposition to which is known as ‘constructivism’;
realism about mental states … to which is opposed behaviourism.

He finishes this useful roster with “realism about the past and about the future”, but provides no antonym.  We might name it for him:  “relativistic Wal-Martia".

Further dichotomies from Dummett:

Michael Dummett, “The Structure of Appearance” (1955), repr. in Truth and other enigmas (1978), p. 29:

… nominalistic [vs.] platonistic:  … those which do not, and those which do, use class-theory as part of the logical framework …

Michael Dummett, “Nominalism” (1956), repr. in Truth and other enigmas (1978), p. 42:

The platonist is he who employs in his formalism  the machinery  either of set theory  or of higher-level quantification;  the nominalist is he who dispenses with these  and uses at most  the calculus of individuals.

Michael Dummett, “Truth” (1959), repr. in Truth and other enigmas (1978), p. 18:

Intuitionists speak of mathematics in a highly anti-realist (anti-platonist) way:  for them it is we who construct mathematics; it is not already there ….

Intermediate between the anti-realist just-so-story view, and the fullbore realist account:

Fregean notion of a mathematical reality waiting to be discovered
-- id.

To all this, Dummett offers a compromise (one  to my mind, incoherent):

If we think that mathematical results are in some sense imposed on us from without, we could have instead the picture of a mathematial reality not already in existence, but as it were  coming into being as we probe.  Our investigations bring into existence  what was not there before, but what they bring into existence  is not of our own making.

Michael Dummett, “Frege’s Philosophy” (1967), repr. in Truth and other enigmas (1978), p. 88:

Frege would have to be classified as a member of the realist revolt against Hegelian idealism … but apart from his assault upon psychologism, Frege barely troubled to attack idealism at all; he simply passed it by.

*

Miscellaneous further dichotomies:

William James, The Principles of Psychology (1890), vol. II, p. 617:

This structure is supposed by the apriorists to be of transcendental origin, or at any rate  not to be explicable by experience;  whilst by evolutionary empiricists  it is supposed to be also due to experience, only not to the experience of the individual, but to that of his ancestors as far back as one may please to go.

Note that one can consistently hold both positions, in the following sharpened sense:  and in this sense it is in fact the position of Chomsky and his followers vis-à-vis our linguistic apparatus.   John’s linguistic (tacit) knowledge is not (fully) explicable by his own, personal experience;  one is however at liberty to suppose that his connate Language Acquistion Device may have been moulded by Natural Selection, acting upon untold generations of his ancestors.

Quine, in Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 619:

Duhem’s fictionalistic attitude toward physics  and my realistic attitude

Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. xvi:

the dichotomy between realism and constructivism:  is mathematics discovered or invented?

Susan Haack, Evidence and Inquiry (1993), p. 188:

At the strongly irrealist end, there is Rorty's proposed identification of 'true' with 'what you can defend against all comers'.

 

Reuben Hersh, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 11:

The typical working mathematician is a Platonist on weekdays  and a formalist on Sundays.

Our Friends the Integers

What, after all, is a natural number?  There are Frege’s version, Zermelo’s, and von Neumann’s, and countless further alternatives, all mutually incompatible, and equally correct. … There is no saying absolutely what numbers are;  there is only arithmetic.

W.V.O. Quine, “Ontological Relativity” (Journal of Philosophy, 1968)

Randbemerkung:  The repeated reference to the integers in these notes  might mislead the reader into imagining that I accord them the least importance, mathematically or ontologically (let alone theologically).  Not so.  I only recur to them on the rhetorical grounds that the arch-nominalist Kronecker gave them up for free.  Had he instead said:

            Die Mengen hat der liebe Gott gemacht; alles andere ist Menschenwerk

then we should have instead busied ourselves with the necessary Menschenwerk of building up the integers out of set theory in any of the usual ways, pointing out that this new epigram eventually commits you to the integers in any event.  And had he said (oh would that he had):

            Die offenen Mengen hat der Liebe Gott gemacht…. ,

then the touchstone would have been topology.  Yea, had he instead, like certain of our very ethereal contemporaries, posited rather Category Theory at the base, sets and numbers and all the rest to be developed out of that – well, I might personally have balked, because I don’t understand Category Theory.  Still, a donkey does not understand the Goldbach Conjecture, so intellectually that is no objection. 

            The infinities of the Creation  -- and a fortiori, of the Creator – are --- whatever they might be, we may never know, but in any event, nothing particularly to do with whatever a handful of our own species happens, at any particular instant, to latch onto.  The integers are one toenail in one foreleg (or is it hindleg) of the Infinite Elephant.  It matters not what an infinitessimal portion this may be, of that great beast (to Whom even Babar tips his hat), nor how ineptly we manhandle it:  what matters is that the Elephant is Real.  Praise Him!

            Anyhow, the integers are fine, but nothing special.  Frankly, in fact, the Riemann Hypothesis bores me to tears (mainly because I still cannot truly intuit its significance). The Poincaré Conjecture is much  more inciting – though, having been finally, after a century, proved in its entirety, it serves less well than the R.H. as an image of the blaue Blume.  Compare, indeed, the final chapter of Russell’s Introduction to Mathematical Philosophy, for a nice dissing of the integers.  Likewise Wittgenstein (Zettel 706): “Die Zahlen sind der Mathematik nicht fundamental.”

Scorecard

In these essays, I am not concerned to tussle in detail with any of the panoply of agnostic philosophies currently on offer in the cultural Mall of America, nor even particularly to distinguish among them.  Their various designating terms may even be used loosely, as the phrase requires -- “nattering nominalists”, “sodden solipsists”, "deviant deconstructionists", "prurient proctoscoptists", and so on.  The point is rather to explicate, and make more plausible, Cantorian Realism, with a nod in the direction of its nice fit with theism.   I capitalize the word Realism, not to exalt it, but to distinguish it from the everyday use of the word, which can shade over into almost an opposite meaning, as in Realpolitik (the capitalization there is an artefact of German).   I don’t capitalize nominalists because, well, they suck.

Nor, despite the keystone role of “visibilium omnium et invisibilium”, am I the least interested in any philosophical debates about Abstract vs. Concrete.   The integers could be made of green cheese, for all that matters.

So, here’s all you need to keep track of the action:

The Good Guys (in the golden jerseys):

Realists

A typical Realist, in native dress

The Bad Guys (in the purple jerseys):

nominalists, solipsists, subjectivists, perspectivalists, epiphenomenalists, eliminativists,  phenomenalists, behaviorists, occasionalists;
finitists, intuitionists, constructivists, formalists [these, all in the mathematical sense -- in particular, game-formalists];
fatalists, predestinarians; instrumentalists [by which I mean, not strummers of the ukelele, but the theory-dissing attitude to science],
idealists [in the non-idealistic sense],
sociologismists [an awkward coinage, but we need to distinguish these from actual sociologists, who are  in principle  honest souls  like plumbers];
gnostics;  cultural constructionists, moral relativists, ‘internal’ ‘realists’, situation ethicists, irrationalists, tribalists, identity-politicians, deconstructionists, post-modernists, nihilists, diabolists, atavists, primitivists, Donald Trump, bestialists, methodological onanists, proctoscopic introspectionists, ……

A clowder of Nominalists

Guys we bar, though there is something to be said for them :

empiricists, verificationists, logical positivists; existentialists

Guys that maybe get a bad rap:

mysterians, mystics, quietists; coherentists

The score so far:

Good Guys: 700000.    Bad Guys:  0.

Credo

[Theologia Mathematica, ch. 2.   Continues this.]

        

There is a phrase within a clause within a prayer, of which I am particularly fond, and upon which I ponder ceaselessly:  “…the Father almighty, Maker of Heaven and Earth,

and of all things visible and invisible…”

… visibilium omnium et invisibilium.  Doubtless each line of the credo might profitably exfoliate into a tome,

http://en.wikipedia.org/wiki/Mass_%28music%29#III._Credo

http://www.youtube.com/watch?v=fjtxKpHSXzg&feature=related

but let us now, here, pause for a moment at this one.

           

            The phrase is no mere afterthought.  There are in fact a whale of a lot of invisibles out there, and it’s not just ghosts, or disembodied ectoplasm, or ethereal unstructured mush.  Nor do I mean “dark matter” or “dark energy”, though apparently there are gobs & gobs of that as well: more than of ordinary matter, which now rattles about in the cosmos like spare change.  (Or, to coin a phrase, like angry candy.) No, dark-whatever, doubtless all jolly stuff, and quaint in its way, but no more inherently fascinating than, say, rabbits.  It’s merely the library-paste and plasticine that happened to be lying about when God got around to making this particular universe on some particular day, possibly with leftovers from some earlier practice project; and now it hangs about, drifting moodily hither and yon, like so much unemployed blancmange.  Being invisible doesn’t make it significant or interesting.  Let us have no fetish about the invisible.  If for some reason the credo had said, “… and of all things probable and improbable,” and if the improbable had somehow been mostly ignored, yet contained most of what was of interest in the universe, then we’d be talking about the improbable; or the fantastical; or the ironical.  The entities  I shall be getting at here  are not significant because they’re invisible; I’d be even willing to concede that they’re significant despite being invisible, that visibility would be one further and delightful perfection, one which we may someday hope to glimpse.  In any event, what is meant here is the mathematical scaffolding, on which the sun and the moon and the quarks hang  like so much laundry.  That is, the plan of the thing, so much more permanent and pervasive than the things themselves.  I mean the symphonic score  from which our ephemeral melodies derive.

            Properly apprehended, it is a structure of – crystalline palaces, transparent and thus largely invisible to the untutored gaze, save as the light of insight  glances off them at an angle, and so catches the inner eye.  These ideal edifices are as hard and as chiseled and as real, as our own makeshift dungeons of stone: nay, more real, for these intricate perfections are the prototypes, whereof our own poor earthbound shantytown is but the fallen, partial, semi-crumbled, quasi-scrambled, half-forgotten misremembered afterimage.  They are, it is true, invisible: but in part (in increasing part) -- not unimaginable.  Through intense and lifelong study, we may – by luck, or grace, or mental sweat – eventually acquire a glimpse of their upper ramparts, from which turrets rise, from whence pennants flutter – flutter in a plenitude, an infinitude of dimensions, one upon the other like palace halls; so that our own most swirling ballet or crashing waves  are but as the slogging of an ant  trapped between the narrow glass walls of the ant-farm.

            These diaphanous entities, being (as we shall argue) a part of the Creation, display a different side of God, from what we customarily encounter.  Or rather, as it may be, many different sides: the mystery of the Trinity becomes the mystery of the Infinity.  For we must not think of “Math” as just some subject in school, or as a section at the bookstore, beyond “Gardening” and next to “Pets”.  For one thing, there is just so much of it,  acres of math like fields of wheat, with more unfolding with each passing day, and much which, when first met with, seems qualitatively, drastically diverse:  not like different species, say a wolf and a fox, but like different phyla – a microbe and a mastodon.  And even as science has discovered some of the commonalities between mastodons and microbes, in the process deepening our appreciation of each, so too does the steady, then accelerated, and finally springing advances of our collective understanding – as it might be, the Mathematical Overmind – deepen and widen and heighten and… beyonden  our sense of the unitary structure of All There Is.  The whole enterprise is so fantastic, with such unity-in-diversity (again, compare the Trinity) that whole new fields have evolved at a metalevel, just to keep tabs on it all:  Set Theory and Proof Theory, to police our reasonings, and Category Theory, to provide display cases for all the genera of the menagerie, in the museum of the mind. 

            At the bookstore-cum-giftshop in Hilbert’s celebrated Hotel, you will find aisles for:  History; Fiction (including Astrology and Economics); Physics ‘n’ Chemistry; Biology; Number Theory; Point-set Topology; Algebraic Topology; Algebraic K-Theory; Topological K-Theory; Real Analysis; Complex Analysis; The Riemann Hypothesis; Sheaf Theory; Topos Theory; The Poincaré Conjecture.;  and Miscellaneous (i.e., gardening, computers, self-help, sports, celebrities, stamp-collecting, and all the rest).  There is no separate section for Theology, since that overlaps all of them.

            Now, none of this is exactly new: it is paleo/retro-NeoPlatonism. A retread, you may say, and twice-refried.  Yet there is now much more concrete substance to the view, than was available to Plato or Plotinus. They might imagine the cube and the icosahedron, and Kepler might attempt to stuff the planetary orbits into such homely and visible (risible) boxes, but they never encountered a Riemann manifold – or rather, they did, because we live in one, but they couldn’t see it  so they couldn’t imagine it, any more than they imagined fibre bundles or E8. 

            The Atheist, viewing a world charged with the grandeur of God -- which is difficult to ignore, since it will flame out, like shining from (to coin a figure) shook foil -- sniffs and dismisses it as tinsel: seeing the shook foil but not the Shaker.  So too the Nominalist, beholding or rather failing to behold  the serried ranks of theorems rising like seraphim beyond sight, regards these as a mere medley of contingent things, simply frothed out of someone’s brain, and which might, like a limerick or a pop-tune, have frothed out into something quite different.  (This is if anything the more charitable of contemporary dismissals, vice the dismissal of math and science as being merely the Eurocentric patriarchal dogma of the racist sexist agist ablist ruling class…)

*

            The nominalist viewpoint was given epigrammatic utterance (1886) by Kronecker, thus:

“Die ganzen Zahlen hat der liebe Gott gemacht; alles andere ist Menschenwerk.”

(“Ganzen Zahlen” means integers, but he may have meant only the non-negative integers, or “natural numbers”.)

Kronecker, rueing the day that ever he denied the transcendence of higher mathematics

(Randbemerkung:  A delightful swipe at Kronecker, well below the belt, and delivered with punch by a pugnacious Platonist, can be savored here:  The Continuum. )

Stephen Kleene translates, “God made the integers, all the rest is the work of man”, and glosses:

We cannot expect that the cognizance of the natural number sequence  can be reduced to that of anything essentially more primitive than itself.

(Introduction to Metamathematics, p. 19.)

The natural numbers – non-negative, non-zero whole numbers -- were, indeed, the only numbers recognized by the Pythagoreans.  One reads somewhere of a Pythagorean  casting himself into the sea in despair, upon encountering the proof of the non-rationality of the square root of 2; in fact, such self-drowning might just as well have been prompted by the sight of half an apple, since once you accept fractions, you are heading straight for E8.

            For in admitting the integers as being in no wise contingent – the work, indeed, of the Necessary Being – while desiring to dismiss the rest (“Menschenwerk” sounds even more like a kindergartner’s art project than the more Biblical “the work of man”), Kronecker has left the castle of mathematical agnosticism unguarded, by leaving open its postern gate.  Suppose we were – setting aside centuries of other riches – to begin by restricting ourselves to the laws of the natural numbers.  We would notice (as Euclid noticed) the primes, and require, for their adequate handling, great heaps of Number Theory.  Now, this already is no small thing.  You could fill a succession of lifetimes with nothing but Number Theory.  New discoveries emerge daily – some of them with such grave implications that they are actually classified, and at compartmented levels well beyond Top Secret, which does well enough for the design of an airplane or the movement of troops.  But the wealth is not just quantitative.  For to adequately handle nothing more than the primes, you need all of Number Theory, including even Analytic Number Theory, which brings in the continuum and the charmingly designated “imaginary” numbers (now much easier to imagine, though they remain as invisible as the number “23”), including indeed the Riemann Hypothesis, which already brings us to the frontier of knowledge with its outstanding unsolved problems.  Soon the whole of mathematics would come tumbling in through the unguarded door. 

[Note:  It’s never that simple.  I am aware that Kronecker himself was even more nominalistic that the famous quotation might suggest, as he did not accept the integers as a finished totality.  He went to some effort to derive results in a way that makes no use of such a totality.  Sort of neat if you can pull it off, like building a castle entirely out of toothpicks.  But if the idea of an actual infinity is problematic, that of the integers somehow running out of breath is even moreso.  I shall accept the natural numbers as given, and shall, for polemical purposes,  portray, as our foil, a sort of idealized Kronecker (who may even now be repenting of his nominalism, in some warm place) – the Kronecker of the quote – as accepting them as well.]

[As to the actual forked-radish of that name,  Joseph Dauben remarks (Georg Cantor, p. 66):

No-one could have been more opposed to Cantor’s ideas, nor have done more to damage his early career, than Leopold Kronecker.

Really, were it not for his key concession that the natural numbers are God-given, Kronecker might well be the villain of the piece.  As George Szpiro wrote of him (in Poincare’s Prize):  “Kronecker would not accept anything if it had not been invented by himself.”   Actually that can’t be quite accurate, since on that account he would not accept the integers… But anyhow, a perfect summary of the solipsist/nominalist epistemology.]




[The essay continues here.]