Doctor Justice names the penguins

[Prefatory remarks:
In this space, we intend, favente deo, to begin a quest of world-historical import:  the Naming of the Penguins.   The project is to appear in installments.  To prepare yourselves, read (or re-read),  L’Ile des pingouins, by Anatole France.]

(1) We read in Scripture, how that Adam did name the beasts:

And out of the ground the Lord God formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof.
And Adam gave names to all cattle, and to the fowl of the air, and to every beast of the field ..

Further research reveals, however, that our Original Parent did not name each beast individually, but only by kind:   Behold, these are the lions, these the lambs;  these the hawks and kites and crows … 

Thus, much work remains to be done.  And as a linear descendant of Adam (on my mother’s side -- father’s too, in fact), I feel it is incumbent to me to take up this cross.

Where to begin?   Well, more numerous than all the beasts of the field, are the Penguins, of pure repute:

Appellavitque Adam nominibus suis cuncta animantia

Accordingly, all the penguins of Antarctica have lined up single-file, and are passing patiently beneath my hand, as I sain each one, and give each one its name before God.

“Fluffy; Chubby; Tumtums; Blackie; Whitey; Roly-poly; Poly-Roly …”

As of press-time, Dr Justice has individually named eighty thousand penguins; just five million more to go.

(2)  [Update, 28 April 2017]

Five hundred thousand and counting:

“….  Fishsnitcher, Egghuddler, Iceberg Bertie, Lollybop,

 Antarctic Archie, Austrobird,  Gus the Glacier Guy, Snowmelt…”

Te baptizo, Nitide !

[TBC, DV….]

Climbing Mount Ineffable

[A Lenten meditation]

Darwinians use a nicely heuristic image for evolution as blind climbings of a fitness-landscape.  Richard Dawkins sharpens the metaphor with the title of his (excellent) book, Climbing Mount Improbable : thus recognizing that, though he is a staunch proponent of Natural Selection, evolving something as nearly perfect as a Penguin  is not a slam-dunk.

The pinnacle of Natural Selection  so far

(Indeed, penguins represent a classic case of Irreducible Complexity:  remove one single feather, and the creature is not nearly so cute.)

Our own essays have recurred to a mountaineering metaphor, in support of Platonism (the Realism position in math).  Namely:  Team A sets out to conquer Mount A from its forbidding southern face;  Team B sets out to conquer Mount B  from its frigid north one.  They meet at the summit, to their mutual surprise.   A equaled B, all along!  This attests to the reality of the mountain, prior to and independent of all human endeavor.  (For ‘mountain’ read:  the truths of mathematics, arrived at independently  by various researchers  using quite disparate methods.)

Now, Imre Lakatos  is neither (statically) Realist nor Nominalist -- he is a Dialectician, and juggles both views.   In the course of a quite intricate examination of the evolution of the Euler characteristic  (don’t imagine you really understand the following sentence unless you have worked through that monograph), he remarks in a footnote:

As far as naïve classification is concerned, nominalists are close to the truth when claiming that the only thing that polyhedra have in common  is their name.  But after a few centuries of proofs and refutations, as the theory of polyhedra develops, and theoretical classification replaces naïve classification,  the balance changes in favour of the realist.

-- Imre Lakatos, Proofs and Refutations (1976), p. 92  (**)

Presumably what motivated this formulation, was the experience of beginning with a hopeful conjecture that soon is drowned in a welter of disparate counterexamples; yet with time and hard analysis, we do progressively manage to sort things out -- as though our fumblings were being guided by something real, though unseen.

Similar remarks, I would cautiously submit (under correction), might apply to the multimillennial evolution of theology, in the Historical Church.

Apart from the dogmatic mouthpiece ‘Episilon’ in his classic dialogue/sotie, Lakatos suggests that we have not reaching the summit of any mathematical mountain, and perhaps never will.  With that I concur wholeheartedly;  only admonishing, that the summit is there.  Nor are we likely to get much insight into the internal workings of the Godhead, this side the eschaton;  but those workings are there as well. 

And so we strategically retreat  to the more modest metaphor of the base-camp.  We never quite reach the summit, but with luck and elbow-grease, we might climb high enough that we can detect the smoke from the campfires of the North Face team.

(The theological analogue here is one of the favorite themes of C.S. Lewis, in The Abolition of Man and other works:  the anticipation of Christian insights in other traditions.)

The mathematical upshot of all this, goes back to a repeated theme of these essays:  the distinction between our (human, contingent, fallible) mathematicizing, and the (antecedently existent, transcending) mathematical truth.   (For anyone who might dispute that, answer this:  Did the Universe even exist, prior to Newton?)

The theological upshot -- Well, longstoryshort:  Don’t go cutting off heads, merely because you imagine you perceive some straw in your neighbor’s eye.

(**)  More pointedly: 

… the problem of finding out where God drew the boundary dividing Eulerian from non-Eulerian polyhedra.  But there is no reason to believe that the term ‘Eulerian’ occurred in God’s blueprint of the universe at all.

-- -- Imre Lakatos, Proofs and Refutations (1976), p. 68

Amen.  But He has a blueprint, of which our notion of “polyhedra” is a primitive glimpse.

Similar remarks might apply to the Trinity.

~

Another parallel:

Consider a theological scholar  working on an apparent inconsistency between two Biblical passages.  Theological doctrine assures him that the Bible, properly understood, contains no inconsistencies.  His task is to provide a gloss that offers a convincing reconciliation of the two passages.  Such work seems essentially analogous to ‘normal’ scientific research as depicted by Kuhn;  and there are grounds for supposing that he would not repudiate the analogy.

--  John Watkins, “Against ‘Normal Science’”, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 33

Cream for your Coffee

[The following paragraph has just been added to our essay, "A New Proof of the Existence of Coffee-Cups".]

Having at length satisfied ourselves as to the reality, or at least reliability, of coffee-cups, would should not  on that account sink back into an attitude of Moorean complacency (“I’m all right, Jack;  I’ve got hands”).  For our commitment to these  suggests yet further commitments, which we had not realized were there to assess.  Such as :  Realism with regard to quantum state vectors.

The question of ‘reality’ must be addressed in quantum mechanics -- especially if you takes the view that the quantum formalism applies universally to the whole of physics -- for then, if there is no quantum reality, there can be no reality at any level.

-- Roger Penrose,  The Road to Reality (2004), p. 508

And:

The question of the objective existence of the objects of mathematics … is an exact replica of the question of the objective existence of the outer world.

-- Kurt Gödel, “What is Cantor’s continuum problem?”, in American Mathematical Monthly, 1947.

A complex Schrödinger equation,

after the Collapse of the Wave Packet

In for a penny, in for a pound.

On “Realism”

That is to say:  Not directly concerning that deep philosophical subject (which lies, alas, beyond our reach), but dealing more with just the socionoëtic and lexicographic aspects of the thing.    Our principle essay-series on the matter  begins here:   Realism:  What;  and here:  The Realist Vernacular.   Here are the latest stray quotations  appended to that.

As a Platonist, he saw everything on earth as broken arcs, which merely suggested the perfect rounds above.

-- Louis Auchincloss, The Rector of Justin (1964), p. 92

Realists [with a capital R] are not the same thing as ‘realists’ in daily life, who are men who expect neither themselves nor others to be any better than they ought to be, and generally much worse.

-- Ernest Gellner, “The crisis in the humanities” (1964), collected in The Devil in Modern Philosophy (1974), p. 15

To mathematicians who study them, moduli schemes are just as real as the regular objects in the world.

-- David Mumford, Forward to Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. xi

Thoughts 'n' Things

  Rudy Rucker, Infinity and the Mind, p. 38:

(**) Just as a rock is already in the Universe, whether or not someone is handling it,  an idea is already in the Mindscape, whether or not someone is thinking it.

This is itself a pleasant thought, recalling the ditty about God-in-the-quad; but in actual fact – I don’t think so.

(So you see—I am not an uncritical Platonist.  Platonic heaven must be so gerrymandered, as to exclude such things as cheese doodles and Sponge Bob Squarepants.)

The actual universe has (for example) -- whatever geometry it has:  regardless of whether there are rational creatures capable of understanding it, let alone deriving it.  Likewise the landscape of math.  But particular formulations of physics, and perhaps even of math – matrix mechanics v. wave mechanics, Cauchy analysis vs. non-standard analysis – do not exist in complete independence from their proponents.  They are, one might say, propositions, not objects.  The objects (or patterns, or whatever they are)  exist  even in the absence of  a person to spout propositions about them; but the propositions require a proposer.  – Nothing specially abstract here; the same thing is true of rocks.  This rock exists independently of any finite mind, but: “There lies a rock” and “Behold that rock!” and “What a rock that is!” must come out of some actual someone’s mind or mouth.

            The unbridledly idealistic view in (**) conjures up a skyscape of untethered thought-balloons.  It is pleasant to contemplate, in a comic-strip sort of way, but not to be taken too seriously.  For one thing, unlike the situation with mathematical truths, where anyone at any place or time might discover them, there is no way for a rational creature in another galaxy or dimension to reach out and grab one of those thought-balloons by the tail;  he is required to blow his own bubbles.  Whereas the structures of mathematics are like fixed landmarks, which one encounters again and again, from different approaches.  For instance:  Yang-Mills gauge theories, discovered by the physics expedition; and connections on fibre-bundles, discovered by the math team; and lo, they meet in the middle.  Likewise group-theory.  Different body-parts of this have been grabbed onto by matrix theory, algebra (symmetries of solutions to equations), geometry (the Erlangen program), particle physics (glad you could get here; meet Sophus Lie), and in time it becomes clear that it’s all part of the same elephant.  Whether they come from physics, or mathematics, or computer science, two such explorers may not realise that they have come upon the same mountain, till they have circled around it a bit and compared notes.  And this happens repeatedly.  We may summarize in an epigram:  The mindscape of mathematics is a multidimensional torus:  whatever direction you set off in, you eventually wind up back at Hilbert’s Hotel.

It turns out that Shing-Tung Yau likes this montane metaphor as well.  Cf. The Shape of Inner Space (2010), p. 103:

A mathematical proof is a bit like climbing a mountain.

And he nicely outlines the Yang-Mills case (p. 290):

The physicist Chen Ning Yang was similarly astonished to find that the Yang-Mills equations, which describe the forces between particles, are rooted in gauge theories in physics  that bear striking resemblances to ideas in bundle theory, which mathematicians began developing three decades earlier, as Yang put it, “without reference to the physical world”.  When he asked the geometer S. S. Chern how it was possible that “mathematicians dream up these concepts out of nowhere,” Chern protested, “No, no.  These concepts were not dreamed up.  They were natural and real.”

            Contrast the case with “thoughts”.  Supposititious entities of the mindscape, even some popular thought-balloon, tethered to a billion different heads, need never be rediscoverable by another explorer, nor acknowledged as real should he simply be grabbed by the lapel by one of the thinkers, and treated to an exposition of same.  For example, the notion held dear by countless generations of schoolboys around the globe, of the uniquely funny nature of flatulence, will never appear among the gravely ellipsoidal thought-balloons of the solons of Fdrmrphlandia; even “funny”, for them, is not well-defined, and not particularly worth defining.

Now, probably Rucker meant to restrict the realm of “ideas” to just some of them.  Not, “Wouldn’t it be fun to dip Suzy’s pigtail into the inkwell!”, but things like “The square of the hypotenuse is equal to the sum of the squares on the other two sides.”  Fine; but careful, here.  The Pythagorean theorem has  as its basis  a fact about Euclidean geometry, in every possible world; just as Fermat’s Last Theorem expresses (in a possibly somewhat contingent and imperfect way) a fact about the natural numbers.   But a fact is not the same thing as an idea.  As a matter of fact, there is a coffee stain on this shirt; but “the idea of this coffee-stained shirt” is no strut or girder of God’s architectonics.  An idea concerning a fact of mathematics, in a finite mind,  may bear – must bear -- but an imperfect relation to the fact itself (‘fact’ here used broadly: it may refer to a wildly transfinite complexus of relations, some of them perhaps perceptible only to angels).   Most people’s ideas of mathematical truths bear as much relation to the truths themselves  as does a crayon scribble to the Sistine Chapel  which it might (based merely upon memory of a fleeting ill-lit glimpse) attempt to depict.  To posit that all truths of mathematics exist as Ideas in God’s mind, is logically allowable, but really adds nothing, and is in any case unknowable. To identify these truths with the neuronal states of the pitiful meat-wads sloshing around in our half-cracked crania, is to add nothing at all, but is rather to detract.

[Appendix]  Karl Kraus apparently entertained a notion of independent or pre-existent thoughts.  He speaks of someone being

von der Präformiertheit der Gedanken  überzeugt, und davon daß der schöpferische Mensch  nur ein erwähltes Gefäß ist; und davon, daß die Gedanken und die Gedichte da waren  vor den Dichtern und Denkern.

-- “Heine und die Folgen”, reprinted in J. Franzen, The Kraus Project, p. 88

The whole ‘meme’ idea (itself a meme) is similar -- not that the various Chiclet-thoughtlets were truly Platonically pre-existing, but that, once hatched, they lead a promiscuous existence, wandering into people’s minds  like pollen into our air-passages.

~

Footnotes from the 19^th century:

Dedekind … allowed his philosophy of mind  much reign, with a ‘proof’ that “there are infinite systems”;  for he gave  as evidence “the totality S of all things, which may be objects of my thought”, since  as well as any of its elements s,  it contained also “the thought s’ that can be the object of my thought …This ‘proof’ did not gain a good reception.”

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 105

For Frege,

In contrast to subjective ‘ideas’ (Vorstellungen), ‘thought’ was intended in an objective sense, rather like state of affairs, sharable among thinkers  and indeed independent of anyone thinking then.

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 190

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 Urysohn Metrization Theorem: an Adaptationist Account (updated)

People in labcoats have been puzzling over the preponderance, over a wide range of far-flung and disparate societies, of belief in the Urysohn Metrization Theorem, to the effect that every regular topological space with a countable basis is metrizable.  How to account for this strange coincidence?

A rear-guard of Platonists and theists would persist in maintaining, that every such space is, as a matter of sheer fact, metrizable;  that the fact is “out there”, like a mountain, whether or not you or I are aware of it, and whether or not we can assemble some semblance of a demonstration to “climb” it -- to clarify the assertion, make it plausible, or to ‘prove’ it in some sense.

This, however, is not the method of modern science, which spurns the affordances of mere reason, and denies the evidence of our eyes, relying instead on various  techniques and equipment in well-funded laboratories.  Accordingly, herewith an account of how belief in the Metrization Theorem arose spontaneously, by the proven processes of Natural Selection.

You see, many many years ago, a number of tribes roamed the savannah.  Some went picturesquely naked, others were draped in animal skins.  And one of these tribes, fancying that the stronger separability criterion of normality was required, whereas the only spaces to be found in their ecosystem at the time were merely regular, despaired of ever metrizing anything; sickened, and died.   Another tribe failed to reckon with the necessity of a countable basis (mere first-countability being insufficient), and promptly went extinct.  Still another lowballed the separation condition, imagining that merely being Hausdorff was enough;  their metrizations went awry, and they were eaten by mastodons.

In this way, in the fullness of geological time, the only tribes remaining  possessed an innate belief  in the so-called Theorem (which is itself, of course, completely meaningless.)   Current estimates place the gene for this Theorem on Chromosome 14.

*     *     *

~ Commercial break ~

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

We now return you to your regularly scheduled essay.

*     *     *

[That was a philosophical satire.  For something a bit more substantive concerning the theorem in question, click here.]

~ ~ ~

Analytical appendix:

‘Twere a mug’s game, to cite specific instances of sociobiological overreaching -- just-so stories that purport to explain Love, Music, Art, what have you.  Chesterton already skewered these several generations back.   More worth noting are the (rare) cases where such thumb-sucking is found among mathematicians themselves.

Thus Reuben Hersch (apparently during a brief psychotic episode) wrote (“Some Proposals…” (1979); repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 23):

Our mathematical ideas fit the world  for the same reason that our lungs are suited to the atmosphere of this planet.

Yet the author knows better.  Just a bit further up the page (with his customary lucidity) he wrote:

Consider the theorem  2^c < 2^(2^c), or any theorem in homological algebra.  No philosopher has yet explained in what sense such theorems should be regarded as referring to physical ‘possibilities’.

[Update 2 March 2013]  There is yet another, and deeper, level, to this Gedankensotie, which has only just now  become apparent.

At the very most superficial level, the piece is an attack of mathematical Realism.  Hopefully  none of my readers  understand it as such.

At the next, and still obvious, level, it is a defense of the same against Ultra-Darwinism (namely:  No conceivable considerations of mere individual survival can attach to these arcane topological considerations; ergo, the fact that the qualified international community is unanimous in embracing the truth of e.g. the UMT, is evidence for the transcendental truth of the latter.).  This is the spirit in which it was written, quite parallel to our essay “On the Existence of Penguins”.

But now (having re-read Freud, and been reminded of the role of the Wish-Fulfilment in dreams), I notice something quite further in this scenario of tribes perishing for misprizing the higher truths of topology:   namely, the wish that our evolution had been guided, not merely by such paltry contingencies as tricks of climate and the bite of the sabertooth, but by the very beckoning of such transcendant truths.  In pursuit of such ends,  gladly, I, and my tribe, would strive and maybe die!

A Hearty Welcome to Space Aliens

Most of us, sensibly, seldom dwell on these.  And those who do, tend to obsess about them -- and that, in a remarkably trivialized way, culminating in tales of alien abduction and anal probes.  That meme is a sort of narcissism for retards:  An elaborately advanced civilization sends its best and brightest through the vast distances and perils of space  to our own small planet -- and the first thing these voyager-scientists want to do is to look up your sorry-ass bung-hole??

A moment’s contemplation shows that an actual encounter would be quite different.  For indeed, the only obvious reason I can imagine  for galactically advanced beings to come visiting our neck of the woods, is to see whether we have settled the Riemann Hypothesis.  (On their world, this is known as the “Blorf Conjecture”, and is phrased rather differently, but a bit of analysis shows them to be logically interequivalent.)

For:  Any matter of biology, or arts, or psycho/socio/economics,  would be just too different to bear fruitful comparison;  and as for physics, well, not to hurt anybody’s feelings, but if they’re the ones motoring over to us, we probably have little to teach them in that regard.  Whereas the truths of Mathematics fill all time and all space, indwelling each phyllo-leaf of the multiverse, equally for all.

The Realist Vernacular

            What follows is neither proof nor argument, nor philosophy of any sort, but rather an exercise in sociolinguistics.  [And as such, a sort of sociological preparation for the thread announced here.]  The point is simply to exhibit a common way of talking among contemporary mathematicians, as well as some physicists and philosophers -- an easy style of conversation  that you would never imagine exists, if most of your acquaintance with science and its philosophy is mediated by figures like Daniel Dennett and Richard Dawkins.

            To cite evidence of theistic talk from the learned men of history, from antiquity through the nineteenth century, would be pointless, since the mode was well-nigh universal, at all times and in all realms.  Yet in our present day, most of the habitués of faculty clubs and coffee-houses  have managed to satisfy themselves, that all those who ever lived, in history, from Pythagorus  to the Einstein of “Der Herrgott würfelt nicht”, were, without exception,  imbeciles, and simply didn’t know what they were talking about, when they talked that way.  At last mankind has seen the light (or the darkness, rather);  we speak only of what is sensible and sniffable, like Donald Trump.  And who should be so rash as to cite the testimony of a mere Plato, or Galileo, or Cantor, or Gödel, against the brass certitude of so eminent a scholar as Sir Christopher Hitchens, Ph.D?

            Therefore I shall limit myself to recent citations.  A few examples from among many, snatched at random from  recent reading.

            Again, note:   I am not implying anything about the theology, per se, of the people here quoted, let alone suggesting that they never miss Mass.  Indeed a Dennett -- a roaring atheist -- can still indulge in such turns of phrases as "we can take advantage of the God's-eye perspective we have temporarily adopted" (the Devil can quote Scripture to his purpose).   So far, this is just corpus linguistics;  but more anon.

(1) Theistic language

 (a) mathematics

Arthur Koestler, The Act of Creation (1964):

Karl Friedrich Gauss described how he finally proved a theorem on which he had worked unsuccessfully for four years:  “At last, two days ago, I succeeded, not by dint of painful effort, but so to speak  by the grace of God.”

Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 219:

In one of the most cited discussions in his much-quoted book, Kuhn [1962] talks of scientific decision in terms of “conversion experience” and “faith”.

Richard de Millo et al., “Social Processes and Proofs”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 271:

The classical view does not require that an ordinary proof be accompanied by its formal counterpart; on the contrary, there are mathematically sound reasons for allowing the gods to formalize most of our arguments.

Similar language is often used in formulating the contemporary concept of a “hypertask”.  Cf. likewise

Shaughan Lavine, Understanding the Infinite (1994), p. 55:

            For Cantor … “countable” meant countable by God.

and similarly

Michael Potter, Set Theory and its Philosophy (2004), p. 250, re the plausibility of the Axiom of Choice:

… generalizing to the uncountable case  by appeal to the idea than an ideal being could achieve the choices required of him (or perhaps Him).

If we adopt some particular postulate system for abstract set theory, and agree that the criterion for accepting an intuitive set-theoretic argument  is that its analogue can be justified by the postuates, then we are  in efect  agreeing that the set of all intuitive sets  endowed with the membership relation  is a configuration satisfying the postulates.  We can have at best  intuitive reasons for believing this.  It is really an act of faith.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 153

Gregory Chaitin, “Gödel’s Theorem and Information”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 306:

If God tells one how many different programs of size less than N halt, this can be expressed as an N-bit base-two numeral, and from it  one could eventually deduce  which of these programs half  and which do not.  An alternative divine revelation would be knowing that program of size less than N which takes longest to halt.

Robin Wilson, Four Colors Suffice (2002), p. 214, recounting a mathematician’s reaction to the immensely long and repetitive, unsurveyable, computer-aided proof of the Four Color Conjecture:

God wouldn’t let the theorem be proved by a method as terrible as that!

 (b) physics

Hermann Weyl, Symmetry (1952):

Contingency is an essential feature of the world.   Clarke  in his controversy with Leibniz  admitted the latter’s principle of sufficient reason, but added that the sufficient reason often lies in the mere will of God.  I think, here Leibniz the rationalist is definitely wrong, and Clarke [the theist] on the right track.  But it would have been more sincere to deny the principle of sufficient reason altogether, instead of making God responsible for all that is unreason in the world.

Abdus Salam (1988):

God created just two dimensions -- one of space and one of time. … At a later epoch, there was a phase transition to four dimensions, plus six internal ones.

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 91, alludes to Roger Penrose’s paraphrase of cosmic censorship:  “God abhors a naked singularity.”   The (jocularly) theistic language is especially odd and noteworthy, since the epigram here being echoed -- “Nature abhors a vacuum” (Latin:  horror vacui) does not use it.

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 126f.:

These laws [of physics] may have originally been decreed by God, but it appears that he has since left the universe to evolve according to them  and does not now intervene…

That, then (for that author) leaves only the explanation of the settings of the parameters for initial conditions:

One possible answer is to say that God chose the initial configuration of the universe for reasons that we cannot hope to understand.  This would certainly have been within the power of an omnipotent being, but if he had started it off in such an incomprehensible way, why did he choose to let it evolve according to laws that we could understand?

Simon Blackburn, Think (1999), p. 69, in a section called “A Scientific Model”:

Classical physics identifies the temperature of a gas with the mean kinetic energies of the molecules that compose it.  So in making hot gases, God has only one thing to fix…

Blackburn is a philosopher, not a physicist; as commonly in that community, he is writing in an “as if” mode;  but still, the choice of words is noteworthy.

John Gribbin & Martin Rees, quoted in Edward Harrison, Cosmology (2^nd edn. 2000), p. 489:  Absent the strong anthropic principle,

If there is a unique ‘theory of everything’, then … we would have to accept it as genuinely coincidental, or even providential, that the constants determined by high-energy physics happen to lie in the narrowly restricted range that allows complexity and consciousness to evolve…

(“Providential”… a lovely word…)

Paul Davies, The Goldilocks Enigma (2006), p. 3:

It appeared to Hoyle as if a superintellect had been ‘monkeying’ with the laws of physics. … Like the porridge in the tale of Goldilocks … the universe seems to be ‘just right’ for life.

Paul Davies, The Goldilocks Enigma (2006), p. 236:

Most theoretical physicists are Platonists in the way they conceptualize the laws of physics as precise mathematical relationships  possessing a real, independent existence…

Shing-Tung Yau, The Shape of Inner Space (2010), p. 101:

As Robert Greene puts it, “you’re trying to find the one metric given you by God.”

Michael Atiyah re string theory, quoted in Shing-Tung Yau, The Shape of Inner Space (2010), p. 292:

They’re onto something, obviously.  Whether that something is what God’s created for the universe  remains to be seen.  But if He didn’t do it for the universe, it must have been for something.

This statement is reminiscent of the Principle of Plenitude; and recalls Einstein’s celebrated quip, anent the possible failure of an experiment to confirm his prediction: "Da täte mir halt der liebe Gott leid; die Theorie stimmt doch."

Less seriously, but using a religious metaphor: J. L. Synge, Relativity:  The General Theory (1960), p.  ix:

It is to support Minkowski’s way of looking at relativity that I find myself pursuing the hard path of a missionary.

More serious is the following.  The author is discussing the vexed question of the Collapse of the Wave-Packet upon observation (related to:  If a tree falls in a forest, and no-one’s around, does it make a sound?  -- here rewritten in Oxford terms, replacing the forest by a quad), and quotes the old limerick:

Dear Sir, Your astonishment’s odd;

I am always about in the Quad.

And that’s why the tree

Will continue to be,

Since observed by Yours faithfully, God.

He comments (J. C. Polkinghorne, The Quantum World (1984), p. 67):

Divine reduction of wavepackets would be an overkill, since it would operate everywhere and always, forcing the electron each time to go through a definite slit.  The point about measurement is that it only occurs spasmodically.

Note the predicted empirical consequences of God in the Quad!   Then, between square brackets, the author adds:

This observation is in accord with the classic theological understanding of creation, which sees God as the ground and support of all that is (in our terms, the guarantor of the Schrödinger equation), but not as an object among objects (no collapser of wavepackets).

This aside is in fact central:  by the time he wrote this, Polkinghorne had quit his endowed Chair in physics to become a village vicar.

(c ) analytical philosophy

Michael Dummett, Truth and other enigmas (1978), p. 15, discussing character as (it may be, untested) hidden propensities:

If B still wishes to maintain the necessity of ‘Either Jones was brave or he was not’, he will have to old  either that there must be some fact of the sort to which we usually appeal … or else that there is some fact of an extraordinary kind, perhaps known only to God.

Dummett’s use of the term, to characterise the viewpoint of a hypothetical philosopher inclined to Realism, is so to speak opaque, not representing his own view, which is rather (id, p. 150) that “only a philosophically quite naïve person would adopt  realist view of statements about character”.  Only such, or Saint Peter.

(2) Realist language

Arthur Koestler, The Act of Creation (1964):

Gauss is reported to have said: “I have had my solutions for a long time, but I do not yet know how I am to arrive at them.”

Klaus Jänich,  Topology (1980; Eng. trans. 1984), p. 35:

When topological groups are found in nature [emphasis added], they  are generally not given abstractly as a set G with a composition law and a topology, but concretely, as a group of transformations…

This is not really anymore outrageously realist than saying “When a set of six objects is found in nature…”   (say, the familiar six-pack; although the one at my side is already down to five, and it’s not even noon).

And again, p. 157:

Covering spaces very often “occur in nature”:  that is, one comes across them spontaneously, while studying entirely different problems.

Shaughan Lavine, Understanding the Infinite (1994), p. 160: 

We seem to have nontrivial intuitions concerning the infinite, going far beyond simple things like Extensionality, Pairing, or even Power Set.

Shing-Tung Yau, The Shape of Inner Space (2010), p. ix:

The strength of this discipline [i.e., mathematics] lies not simply in its ability to explain physical reality…, because to a mathematician, mathematics is reality.

If all you know of math is elementary arithmetic, this statement may lack punch.   But in the upper reaches of set theory, topology and so on, it embraces a world of miracles and of monsters.

~ ~ ~

It is important to note, that this is the way Realists talk en famille.  The talk is casual, often not literal, yet is meant in some serious sense.  It is not to be compared with the tawdry pseudo-theological, pseudo-mystical sort of gibberish that popular authors (and even some serious ones, yielding no doubt to the Satanic promptings of the marketing department) use to gin up their pap for the masses -- like “God particle” for the freaking Higgs boson.

It may be objected (it will be objected;  it has been objected) that such expressions, in a modern mouth, are a mere façon de parler.  To which we reply (with Whorf), that a façon de parler tends to cohere with a façon de penser. 

Nor is it a refutation of the point here made, to adduce agnostic pseudepigrapha from any of the gentlemen here quoted.   We are each a walking contradiction;  we contain multitudes.   C.S. Lewis himself  confessed that he tended to be a cranky agnostic at dawn, but a theist later, when he’d had tea and was more himself.

Once again:   The point here is not to assert that so-and-so among our near contemporaries  is or is not a believer.  Indeed it will strengthen my eventual case, if many of them are not in fact believers, yet find themselves attracted, or guided, or driven, to Realist or Theistic language, whether for convenience, or (in the case of Cantor, Gödel, and Einstein) something deeper.

So, a very modest initial move, a sort of pawn to king’s four.  (Only later -- much later -- shall we see if we can capture Satan’s Queen.)   So far we hold simply, that occasional use of Realist or Theist language, in serious discourse, is not  in and of itself  diagnostic for Trisomy 21.

Mathematics and Morality

In our series of essays “Theologia Mathematica”,  we have focussed on philosophical and epistemic issues, very far from the sort of thing discussed in the pulpit of a Sunday.   We have striven both to enrich theism and to defend it against its intellectual detractors -- yet in this special vein of argument, do not aspire so high as Christianity, or indeed morality in general.  Nor have I, really, any insights to share upon this head:  in cases where I do make so bold as to venture onto the public square,  such political or ethical statements as I might hazard  make no claims at all upon any results from physics or from mathematics -- although, as with philosophers and theologians in all times, they are of course informed by logic.

Still, as the subject lies inevitably near at hand, it seems meet to at least put up a bulletin-board, on which we can post any quotes we may come across, which illustrate cross-fertility between the moral and the mathematical spheres.

~

Koestler, Sleepwalkers , I.2.ii, characterizing the philosophy of Pythagorus:

Numbers are eternal, while everything else is perishable … They permit mental operations of the most surprising and delightful knd, without reference to the coarse external world … -- which is how the divine mind must be supposed to operate.  The ecstatic contemplation of geometrical forms and mathematical laws  is therefore the most effective means of purging the soul of earthly passion, and the principal link between man and divinity.

Koestler, Sleepwalkers , I.2.iv [39]: “It is said that Pythagoras, like St Francis, preached to animals.”

~

John Henry Newman, Apologia pro Vita Sua (1864):

He who made us, has so willed, that in mathematics indeed we arrive at certitude by rigid demonstration, but  in religious inquiry  we arrive at certitude by accumulated probabilities.

I cite this simply as an amiable proof-text, while disagreeing with basically all of it.

Cardinal Newman

In mathematics, you approach certainty, over the centuries, by experiment, insight, and eventually a proof (or “rigid demonstration”) -- or perhaps a succession of improved proofs, as subtle flaws are found in the first one.

Arriving at -- again not really “certitude”, but what used to be known as moral certitude (“beyond a reasonable doubt”), by accumulated probabilities, strikes me as a nice thumbnail description of the process of science;  theology, not so much.  Theology additionally relies on Revelation -- or “intuition” -- that inward vision -- as it is mystically known in math.

~

For mathematics as the science of reckoning, Chesterton has little feeling;  but the more modern conception of math is the science of pattern, and Chesterton has strong intuitions about the spiritual content of shape.

G. K. Chesterton, Robert Louis Stevenson (1927):

Whether or no we see faces in the carpet, we ought to see a mind in the carpet; and in fact  there is a mind  in every scheme of ornament.  There is as emphatically a morality expressed in Babylonian architecture or Baroque architecture  as if it were plastered all over with Biblical texts.

~

The above concern a fancied moral tinge to mathematics itself.  Quite different is the relatively familiar thesis that moral reasoning can or does proceed more mathematico.  Thus Spinoza.  Or cf. David Hume, Enquiry Concerning the Principles of Morals (1751):  Whether something is to be considered a “crime … consists not in a particular fact … but it consists in certain moral relations, discovered by reason, in the same manner as we discover by reason  the truths of geometry or algebra.”

John Henry Newman, Apologia pro Vita Sua (1864) acknowledges that “every article of the Creed is beset with intellectual dificulties”, yet “ten thousand difficulties do not make one doubt; difficulty and doubt are incommensurate.”   And if you think he is here sidestepping the issue, he adds a telling analogy:

A man may be anoyed that he cannot work out a mathematical problem, of which the answer is or is not given to him, without doubting that it admits of an answer.

Steven Pinker, in The Blank Slate (2002), pp. 192-3, speaks respectfully if non-commitally about the Platonist viewpoint concerning numbers, and goes on to hypothesize that moral truths may have the same status -- “out there” to be discovered, and analogous from planet to planet.  “Our moral sense may have evolved to mesh with an intrinsic logic of ethics  rather than concocting it in our heads out of nothing.”

By the by:  Pinker's richly suggestive book, ranging afar across fields and meadows, adds  in passing  a suggestion of what we might term Aesthetic Realism:  "Even if an art form matured in the West, it may be  not an arbitrary practice  spread by a powerful navy  but a successful product that engages a universal human aesthetic."
Here he has left any area in which I have expertise, so no comment;  read his argument pp. 407-9.

Axiomatics in its Element

[Further thoughts along the lines of the thought-themes treated in the essay that begins here.]

John Locke, An Essay Concerning Human Understanding (1690), §IV.ii.8:

The necessity of this intuitive knowledge, in each step of scientifical or demonstrative reasoning, gave occasion, I imagine, to that mistaken axiom, that all reasoning was ex praecognitis et praeconcessis;  which  how far it is mistaken, I shall have occasion to show  more at large, where I come to consider proposititions, and particularly those propositions, which are called maxims;  and to show that ‘tis by a mistake, that they are supposed to be the foundations of all our knowledge and reasonings.

Quite otherwise is the role of axiomatics in Set Theory.  Here, the axioms are once again not simply arbitrary -- they always have some prior empirical motivation.  But the consequences of assuming any new axiomatization  are initially quite obscure, and have turned out in many cases  to be enormous (or even fatal).  Hence  taking on a new axiom, or retwiddling an old one, is more akin to creating a new universe for exploration, than tidying up our formal description of the old universe.

In physics, if your axiomatization results in predictions refuted by experience -- so much the worse for that axiomatization.  Its role is not truly foundational, it’s mostly just there for show.

In logic and set theory, the case is much more complex, requiring qualitatively deeper criteria for analysis.  You wind up with logically distinct and often incompatible theories of these utterly fundamental subjects.   (Or, such we had taken them to be.  If there exists no canonical description, is their foundational character impugned?)

The proliferation of spaces (finite-dimensional and otherwise) and of surfaces or, more generally, topological objects within them, is all good clean fun -- like discovering new fauna in newly-explored islands.   But to have incompatible characterizations of logic is more like being suddenly troubled by what actually counts as a life-form.  In our own time, the biologist has had actual glimpses of such worries, in the shape of empirical discoveries not known to earlier centuries, such as viruses (alive? not alive?), and of a continuum of creaturely objects of uncertain individuation:  ant colonies; slime molds; clonal species of certain trees and fungi; and, on another dimension, organisms that divide by mitosis, or that clone themselves;  not to mention the newer view that individual genes are the fundamental vital entity, organisms being just their carrying-cases.   Now add further discoveries, such as Quantum Cats -- separate and independent twins, yet joined in an Einstein-Podolsky-Rosen way, so that they are actually not independent at all.   Or the planet in Solaris, a single organism;  and  distant cousin, the self-aware Oort Cloud.  Or:  certain subsets of integers, like the fateful one in the TV series “Lost”, with malevolent propensities of their own, and measurable biological effects.  -- The consequences for our conception of mathematical reality are quite as drastic as that.

The result is highly uncomfortable for a Platonist.  It seems as though we are coming after all  to the sterile game of creating arbitrary universes at will -- lifeless and pointless entities, with no connection to the Real (or the divine) -- a nightmare of sterile atheism.  Yet here we see one who did not flinch.  John Dawson (Logical Dilemmas, p. 175) quotes Gödel 1946 on the subject:

He asserted that, even if some new axiom ‘had no intrinsic necessity at all”, its truth might come to be accepted inductively  [dbj: Note, actual “truth”, not mere “usefulness”], on the basis of its “verifiable’ consequences” (those demonstrable without the new axiom, whose proofs by means of the new axiom are considerably simpler and easier to discover”).  Indeed, he declared, “There might exist axioms  so abundant in their verifiable consequences, shedding so much light upon the whole discipline, and furnishing such powerful methods for solving given problems… that they would have to be assumed… in the same sense as any well-established physical theory.”

The style of reasoning here strikes me as theological, quite in the tradition of those views that saw the universe, in all its parts, as having been created just so, for the well-being of God’s favorite creature, Man.  To find such an axiom as that, must indeed strike one as finding a kind of key (like that proverbial watch encountered on the empty strand, left there by the Watchmaker…)

            For:  The invention of the vacuum-cleaner certainly added spring to the housewife’s step;  but we do not therefrom conclude as to the nature of The Vacuum.    By contrast, the mathematical lawfulness of physical phenomena, in a myriad of ways, has indeed impressed observers as saying something about the Universe itself.  In similar fashion, should some axiom (with its associated methods) prove to open up, at a stroke, whole vistas of the already otherwise-intuited invisible world, we would feel it had been left there for us:  that it is as real as rocks.

~

In Modern Philosophy (1994; p.99) the philosopher and theist Roger Scruton  lists some familiar foundational principles of Truth Theory -- things like “If the sentence ‘p’ is true, then so is the sentence ‘ “p” is true’; and vice versa -- but calls these platitudes.  Very nice.  Putting axiomatics in its place.

Another way of putting them in their place  is to observe that no determinate character of ‘axiomaticity’ inheres in any one of them -- a salient illustration of Quine’s point in “Two Dogmas of Empiricism”.  Thus, from a very straightforward and standard textbook, with no philosophical (let alone relativist)  axe to grind:

What is axiom and what is theorem  is often an arbitrary choice, since one is often able to derive each logically from the other.

Sometimes one can go a step further, and make all the axioms of a mathematical system into theorems  by basing the system entirely upon some other system.  Thus, analytical geometry makes it possible to base all of plane geometry upon the properties of the real numbers.  This is also possible with R itself, throwing it back onto modern set theory.

-- Creighton Buck, Advanced Calculus (1956, 3^rd edn. 1978), p. 57

You sketch my hand,  I'll sketch yours

~

Andrew Gleason, by contrast, acutely proposes a distinction between postulates and axioms,  restoring lustre to the latter.  The distinction only makes sense  on a Realist account of mathematical truth :

When we leave the domain of abstract configurations, what we call postulates  take on a different significance.  In the abstract domain  we are in charge; we can frame postulates as we please  and simply exclude from consideration  configurations which fail to satisfy them.  But we cannot take this attitude toward concepts which have any sort of independent existence.  What are appropriately called postulates in the context of configurations  become axioms when we deal with independent conceptions.  They are axioms because they are accepted as true, or at least granted for purposes of argument, for intutitive reasons.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 153

Such a conceptual armature  in effect brings mathematics into comparison with both theology and physics, while increasing the contrast with both postmodernism and finger-painting.

Gleason is not here reporting settled usage, but putting forth a semantic proposal, one of répartition or desynonymization.  As he states earlier, anent the defining properties for an ordered set:

The conditions appearing in the definition are called the axioms or postulates for an ordered set.  The word axiom has long carried the connotation of being self-evident, but it is hard to find a sense in which these conditions are self-evident.  The word postulate (from Latin postulare, to demand) seems more appropriate.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 59

It is, incidentally, refreshing, to hear a mathematician who helped settle one of the Hilbert Problems  freely concede that a (not specially complex) set of condititions  is not, in fact, self-evident.
(In the classroom, he was very much like that:  No trace, either of arrogance or of false humility.)

~

The establishing of first principles  is not a matter for math and science only;  the widest field for its application is the law.  And as we saw in the case of mathematics and of physics, so too in law, the axioms do not historically arise first:

Primitive law is made up of simple, precise, detailed rules  for definite narrowly-defined situations.  It has no general principles.

-- Roscoe Pound, An Introduction to the Philosophy of Law (1922, 1954)

In time, such atomistic empricism is systematized.  (For a fable along these lines, consult our parable of the mathematizing woodchuck.)

In place of detailed rules, precisely determining what shall take place  upon a precisely detailed state of facts,  reliance is had upon general premises, for judicial and juristic reasoning.  These legal principles, as we call them, are made use of to supply new rules [and] to interpret old ones.

-- id.

Spinoza

Once a structured body of legal principles has arisen, judgments may be arrived at more geometrico --  though there will always be a residue of hard cases:

Judicial treatment of a controversy  is a measuring of it by a rule  in order to reach a universal solution for a class of causes, of which the cause in hand is but an example.

Administrative treatment of a situation  is a disposition of it  as a unique occurrence.

-- id.