Dueling Epigrams

Clashing stances re noögenesis:

   Materialist:   Mind is an emergent property of the  brain.
   Theist:   Mind is an emergent property of the  soul.

A New Proof of the Existence of Coffee-Cups

But the math had not clarified one basic question:  what the hell did it all mean?  How could the world of atoms be nothing but a puff of probabilities, and yet conglomerations of those atoms could create something as strong and unbending as the  chair on which he sat?

-- David Kaiser, How the Hippies Saved Physics (2011), p.  54

[Update to an earlier essay.  For the original, with readers’ comments, click here.]

Re the authors of “The Stability of Many-Particle Systems” (1966):

They couldn’t explain why it was  that ordinary everyday objects wouldn’t just go up in smoke.  It was quite anomalous.  Inert matter just lies there in a kind of steady state, ticking over silently in a metaphysical sort of way, but this ordinary and plain fact of experience  was incomprehensible to physics.

-- Ed Regis, Who Got Einstein’s Office? (1987), p. 185

There comes a time in the life of every young person, when they doubt the existence of coffee-cups.  So here comes jolly old Doctor J, to prove it for you;  and more importantly, to reassure you that, even should you someday forget the details of the proof, you may continue  to place entire faith in the existence of these useful objects, asking no questions for conscience’ sake.

(We earlier sketched the idea of the proof, but there the argumentation was informal.)

So!  It’s a quiet Sunday, the desk and the mind are clear, let us pour out a generous helping of this fresh-ground, new-brewed, steaming darkling French roast, into the convenient receptacle which (to all appearances, at any rate) sits within easy arm’s reach upon the solid oaken desktop, to stimulate the grey cells, and set ourselves to this great task  of confounding the nominalists and solipsists.

[Excursus:  Historique du problème

Dr Johnson famously proved the existence of stones  by kicking one (a martyr to science -- he reckoned without his gout).  The Cambridge philosopher of banality, G. E. Moore, satisfied himself (I choose the verb with care) of the existence of physical reality, by noticing his own hands.  (The choice was not a happy one, being potentially seen as solipsistically self-centered, not to say onanistic;  further, the phenomenon of phantom limbs complicates the picture in irrelevant ways.)
Accordingly, we shall proceed by proving the existence of coffee-cups:  both because they form the prototypical physical object for philosophers (apart from mats, which are ignored unless they have cats on them), and because, for mathematicians, who in general care not a fig for the physical, do care crucially about this one concrete object, since coffee is their essential fuel.]

But before embarking upon the abstract part of this argument (since some folks are uncomfortable with abstractions -- though really, they need not be;  what follows makes a lot more sense than most of what gets uttered around the water-cooler), I shall offer what critics have called the objective correlative:  a description of the actual coffee-cup whose space-time coordinates are My-Here and My-Now [** But now see below!].  (Rather than the ambiguous “here and now”;  for such coordinates must indeed be relativized to some observer, though not necessarily a mortal observer.)  After all, we wouldn’t want you to wonder whether we might not be just making the whole thing up.

[**Horrified update, August 2019:  Testing the link, I clicked, winding up  indeed at the sales-site for the "Catholic Attitude" mug;  but -- It has been censored!  Shorn of its knight, along with his attitude! -- Fortunately I -- Fahrenheit-451-fashion -- saved the original image, and included it in this essay; vid. inf.]

From the standpoint of the chemist, the object in question would appear to consist in some sort of ceramic, whatever that may be, though I really couldn’t say -- anyhow, something sturdily non-porous.  In general outline, it forms a cylinder, sealed off at the bottom and evacuated at the top.  Somewhat spoiling this sleek Platonic profile, a handle protrudes from the side, a concession to the physical infirmities of hominoidal incarnation.

It is thus strictly speaking a ‘mug’ rather than a cup;  but still I say “coffee-cup” because, for philosophers (who don’t get out much), that is the prototypical object in the cosmos.  (At least this is true for hypercaffeinated Americans; more leisurely Oxonians look to "the tree in the Quad").  In similar fashion, the propotypical Contingent Truth for the Oxbridge set is:  “The cat is on the mat.”  (That last one is not true at the moment, b.t.w.;  she seems to have wandered off.)

That is pretty much all you need to be  to call yourself a coffee-cup.  But these days, most cups are not quite so minimalist:  most of them sport some wacky slogan of office lore, or the logo of a sports team.  This particular specimen displays the image of a medieval knight, standing in contrapposto, his tabard emblazened with a cross pommée.  On his face is an expression difficult to read, some blend of troubled arrière-pensées and manly determination.  He is set largely within a blue circle, which his mailed headgear slightly overtops; at the bottom -- a detail I just noticed only now -- the end of his belt (bulging slightly at the tip) laps down and over, in a way that, hm,  might be misinterpreted.  And written boldly above him, the words:

Catholic  Attitude

Well.  So much for the objective correlative.  Now for the proof, deo volente -- while mindful of the strictures of Kant:

It remains a scandal for philosophy  and for human reason in general, that the existence of things outside us  must be taken only on faith,
and that, if it occurs to someone to doubt it,
we can produce no counter-argument  sufficient to prove it.

-- Kant, Critique of Pure Reason

~ ~ ~

We must confess at the outset (well, it’s a bit late for that) that we cannot actually prove the existence of coffee-cups, by the lofty standards of proof first intuited by Euclid, and refined  in our own day  to a high sheen.   Outside indeed of mathematics, such a capability appears not to exist:  Whenever I try to read a physics article reporting recent research, it always appears at some point to be hand-waving, “You’ll just have to trust us on this.”   And within mathematics itself (trade secret; don’t whisper this to Muggles) we seldom explicitly set out all the steps, even in the final published results.   As for the actual process of mathematical discovery, it normally bears no relation at all to Hilbert’s formalist program;  at its deepest and most mysterious, it is more like…. (and here we’ll have to whisper very softly) … Revelation…

Anyhow, we obviously can’t actually prove the existence of coffee-cups, because logically, strictly speaking, they might not exist:  We might all be just brains in a vat, hallucinating the whole thing.   So to clear the air, we frankly state our Auxiliary Assumption:

(AUX)  We’re not just brains in a vat.

If you personally reject that assumption -- well, happy marinating.  For the rest of us -- to proceed.

[Excursus:  Among the merits of d’Abro’s history The Rise of the New Physics (1939), is that he repeatedly points out the (sometimes unacknowledged) auxiliary assumptions that were required in the development of even the most successful programs in the physical sciences, by the most celebrated researchers, during physics’ Golden Age.

Almost nothing we do in everyday life  is free of unspoken assumptions, sheer force of habit, and random whims.  So, simply by listing such propositions as (AUX), we are making some modest progress.]

The principal premises of the proof, and the only ones that would occur to most people, are evidential.   This is where the hard work of science is done.  Fortunately, in the case of perceptions of ordinary middle-size objects, Nature does it for us, that we need not continually trouble our little heads:  all sorts of cross-connections and epistemological assumptions are hard-wired into our brains.  (These brains, b.t.w., reside in a cranium, vice a vat.)  We shall nevertheless lay out some of the perceptions and observations that serve to buttress the conclusion to objectual existence -- not so much to convince you yet further that your mug is real, but rather, virtually the contrary:  by such explicit exposure of our evidential grounds, to make plain their essential poverty, absent certain grounding metaphysical principles, along the lines of (AUX) though more substantial -- or rather again, more abstract and as it might be insubstantial, since (AUX), though it has the look of a metaphysical assumption, might in some circumstances be actually demonstrable, and thus empirical, as happens in the movie “The Matrix”.

(PE) Perceptual Evidence

(1) I seem to see before me a colour-patch (Note to the lay reader: These colour-patches compose the whole of the world for the positivist empiricists, who never quite manage to convince themselves that coffee-cups are real, and therefore cannot drink the coffee, and die of thirst), roughly cylindrical in outline -- though the exact projection upon the retina depends on the viewing-angle in complex ways (blah blah blah; insert usual empiricist verbiage here).

(2) When I reach out with my hand (for reassurance as to the existence of your own hands, consult the works of G. E. Moore), groping in the general direction of the above-named coloured patch, I abruptly encounter what seems to be a solid object -- a bit too abruptly, it turns out, as some sort of dark hot liquid is now pouring into my lap.

(3) (Insert more such evidence here -- crucially, cross-modally, involving the sound of the mug as you strike it with your pen; the smell and the taste of the coffee within).

(4)  When I pour a small amount of coffee into this apparent container, it does not quantum-tunnel out:  thus recalling, by uniformity and analogy, such similar objectual posits as the Water-Glass.

(5)  Jones here -- a stout fellow of sound mind -- affirms that, egads, he too perceives a coloured patch in what, calculating the parallax by triangulations, dum-de-dum, doing the math, appears to correspond to the same space-time locus that I myself have identified.

(6)  (etc. etc. etc.)

(C ) Conclusion

Coffee-cups really do exist.

(They do, but this one's just an image of such a cup)

Now:   The thoughtful reader will already have noticed how grotesquely speckled with gaps  such reasoning is -- why, it shouldn’t convince a child.  For, though it pretends to consist of but simple reports of perceptions -- “observation sentences” in the lingo -- rather than any abstract reasoning that might be open to critique, it actually smuggles in a great deal of unbuttressed assumptions.  Thus -- what ties (1) and (2) into any sort of connection with each other?  Why,  the unexamined assumption that the visual coordinates of (1) map smoothly and without controversy to the kinesthetic coordinates of (2):  a fact by no means obvious  -- intellectually, that is;  of course, the identity is more or less hard-wired, though it may take baby a certain amount of groping and spoon-dropping to get the respective ordinates and abscissae to finally match up.  Nor is the correlation in any sense necessary -- indeed, it can be easily overturned in simple experiments involving funny spectacles.

And as for that “Jones” there -- you are assuming the existence of Other Minds, about which great vats of ink have been spilled!  (There is, in fact, one sense in which philosophers at any rate  are brains-in-a-vat.)  And indeed you need vastly more than that -- for the Other Minds projected by mere analogy, might be only Other Monads, and incommunicable among themselves.  All right, wire them one to another -- still not enough:  you are assuming that you each mean ‘the same thing’ when you utter the same (or: “similar”) syllables:  an assumption demonstrably false in every political summit or marital argument.  All right, plow ahead and assume that:  you’re still not there.  With all the intelligence and good-will in the world, your private knowledge might not be transmissable intact: witness the celebrated case of determining whether or not your extragalactic pen-pal resides in a world made of matter or of anti-matter, or whether he is right- or left-handed.  -- And here my grey-cells throw in the towel; but were you (younger, and keener) to pursue this line of speculation further, things would probably only get worse.

No, for the various clauses of (PE) to have any cohesion and probative force at all, we require a vast apparatus of non-evidential, metaphysical assumptions, almost never made plain.  The above syllogism,  (PE) => (C ), is thus in reality an enthymeme .  Suppressed is what we might dub the enthymematic assumption, which we now state here:

(EA -- full version)  Background metaphysical premises

(Insert the entire body of philosophy here, along with the whole of science.  Do not omit to mention the Assumption of Cosmic Uniformity (spatial; temporal; spatio-temporal), the Problem of Induction, the Reliability of Deduction, along with some still-unnamed rules of thumb concerning our right to ignore pesky quantum-mechanical paradoxes when speaking of mid-level objects, etc. etc.)

Now, that is rather a tall order, and even were it somehow to be accomplished for this particular case (which might or might not "go over" towards founding the existence of pickles, say), no individual mind could survey the results and verify their correctness and inter-consistency.   And yet, this is the heavy machinery that we need to conclude to so modest a proposition as the existence of coffee-cups.  So let us superadd, as a concession to our feeble powers of ratiocination and memory, the following Concise Version:

(EA -- concise version)  God is Good.

This one works for me;  but if you’re an epistemological stickler, you can probably get by with this weaker assumption, as stated by Einstein, and applied, not only in his life, but in the practice of his physics:

(EA -- weakened version)  Boshaft ist er Nicht.

Which is to say, anglicê:  at least He is not plain mean.   By which he meant:  The cosmos is not just some vast joke, jerry-rigged to baffle our senses.  Overall, it ultimately makes sense -- even if what we perceive here below, is but the thread-gnarly underside of the grand patterned carpet.

(Note:  The basic point is more logical than theological;  still, the realms do tend to intertwine.)

Historical footnote:

Descartes derived the principle of inertia from two premises:

* the homogeneity of the straight line, and

* the immutability of God, of which the constant quantity of motion in the world is an expression.

-- Charles Gillispie, The Edge of Objectivity (1960), p. 90

Note incidentally that the first of these Cartesian premises  is at least as fraught as the second (the problems and paradoxes of The Continuum).

~

~ For a theo-philosophical fantasia, try this:

Murphy and the Magic Pawnshop

[Kindle] [Nook]

~

~ ~ ~

To add some punch to our insistence on the Importance of the Enthymeme, with all its tacit metaphysical assumptions, let us instance another, less familiar syllogism.

(P) Evidential Premises:

1. Something funny’s going on here.

2. What’s that smell?

3.  I can’t find my car keys -- probably somebody swiped them.

4.  What’s that noise?!

5.  So where does Jones get the money for a fancy car like that?

6. No way I’m letting that bastard merge into my lane.

7. Why are all those people staring at me?

8. Ouch!

9. It’s a conspiracy.

(C ) Conclusion:  The world is ruled by giant lizards.

That conclusion is an actual doctrine, firmly held by some people:  non-institutionalized people, actual citizens who have the right to vote (and who seemingly exercise that right with disproportionate frequency).  I’d provide a link to the relevant Web sites, but fear lest these poison your computer.

What, though, must be the missing enthymeme, which licenses this deduction?  Apparently something like this:

(E)  The cosmos is ruled by Satan.

Compared with (E), (C) begins to look relatively inviting.

~  ~  ~

But enough of such stuffy studies!   The sun, renewed, has broken through the clouds, the bird-sounds rise in chorus, and a breeze bestirs itself among the leaves.   Let us then fare forth, into the wide world, whose splendor bears the imprint of its Maker, as plainly as had He signed it, John-Hancock-style, with His celestial pen.

~  ~  ~

[Afterthought]  But what then, of the java, that flows from the mug whose reality we have just proved???  Why, it flows to fuel the brains of mathematicians! 

[Appendix, Jan 2015]  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.

In for a penny, in for a pound.

~

This notion of ‘enthymeme’ deserves further comment, and here is not really the place;  yet I can find no other home for it, among these essays.  Thus:

This is characteristic of ancient informal logic -- that is, of the logic of proof or of thought-experiment …. ; we regard it as enthymematic  only through hindsight:  it was only later that an increase in content  became a sign, not of the power, but of the weakness, of an inference.

-- Imre Lakatos, Proofs and Refutations (1976), p. 81

For more from this pen, try this:
http://www.linguasacrapublishing.com/justice.html

IQ/GQ

[Note:  These presents are not particularly intended to impugn the validity -- well okay, yes, the ‘validity’, but not the (potential) usefulness (for this purpose or that) -- of IQ tests -- : it’s just a thought that somehow occurred to me.]

(1)  Consider the following extraordinarily cost-efficient proposal for measuring IQ.  It applies around the globe, and requires no expensive testing.

Criterion:  IQ, as assessed by this test,  is based exclusively on surname.  Those with surname beginning with “A” are classified as less intelligent (by a factor of 8.9) than those with a surname beginning with “B”;  and so on, through the alphabet.

Sociological result:   Aaron Aardvark hates this;  Zoe Zzyzzyva  thinks it’s fine.  -- But then, they would, wouldn’t they?

[NB:  This is no idle fable, but a mnemonic representation of what has seriously been objected to present practices.  For order in the alphabet, read class standing.  Yet already, even at this caricatural level, there is a serious epistemological problem involved.]

(2)  OK so -- we seriously try to do better  in devising a test that is not self-fulfilling.  Only … before we can come up with clever schemes to measure something, and evaluate each test on how well it accomplishes its goal, we need to know what we are measuring -- antecedently, independently of the tests (which, at this stage, are themselves being tested.)

The scientific datum for which this insufficiently appreciated epistemological problem  has most starkly been brought home to me, is the matter of the Age of the Earth.   Estimates have fluctated by many orders of magnitude over time;  nevertheless, current scientists are confident of the value they give (to several purportedly significant figures).   But the problem is … how can you even talk about such a thing, unless you have (a) a suite of several planets  (b) a known accurate value (necessarily, given by God) for the age of each.  Given that, you could test your methods (though that would still be hard);  absent that, what are we even talking about.

(Note that the concept of God is here brought in, not for theological reasons, but for epistemological reasons.)

So, posited, that we mysteriously receive, from a cloud or Mt Sinai, a list containing the names of everyone on earth, labeled with a number:  a number which purports (though the ancient pre-Mosaic proto-Hebrew is somewhat difficult to interpret) to represent the absolute level of Intelligence, as viewed by God.

Naturally concerned whether “intelligence” be the best translation of F’n(b)rkk(hhh)blx,  we run a sanity check:  and sure enough, Shakespeare and Einstein come out scoring much better than Donald Trump.  So far so good. (Oddly, P.G. Wodehouse turns out to outrank them both;  but that is an enigma for another time.)  

Now, however, our task is more problematic.  Instead of cooking up some clever-seeming tests (think: tests to measure gravity-waves in physics, while somehow discounting the rumble of the passing milk-truck), we have to come up with … something … which will somehow manage  more or less to match  desiderata given ex cathedra.   -- And no, the problem here has nothing to do specifically with psychology.  Physics sets itself the same task when it tries, on principled theoretical grounds, to predict (retrodict) the measured values for e.g. electron, proton, and neutron mass or radius-- these values being taken, for practical purposes, as God-given (though they are subject to change, and have indeed recently been challenged).  -- The problem in psychology is, to be sure, more difficult again on another dimension, in that the God-given number here is, as given, Delphic and ineffable;  we have no prior certainty about, say, the IQs of familiar objects such as coffee-cups, pigs, and Donald Trump, whereas we do feel quite at home weighing such items, and serenely reckon that the same notion of ‘mass’ applies to electrons as well, without changes. 

That serenity may be ill-founded.  In a flavor of the philosophy of science that had its run of celebrity, the meaning of a measured quantity was inextricably tied in with the method of its measurement.  (This is a subclass of verificationism.)  By that criterion, the “mass” of the proton  and the “mass of a galaxy”   bear no simple relation to each other, nor to the mass of a cat.
 

*     *     *

~ Commercial break ~

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.

*     *     *

[Note:  Early in 1967, word came down from the scientific mount, that an objective intellegence touchstone  had indeed been discovered, upon which Wechsler and Stanford-Binet and all the rest  were but parasitic:  this being, how fast the neurons could react.

At first reading, that sounded convincing;  although a back-brain reservation said:  Doesn’t the pattern of reaction matter?  We know plenty of fools who respond on a hair-trigger.

-- And how do I recall the date, after all these years?  Ah, thereby hangs a tale!!  (Shout-out to W.S. of Leonia.)]

(3)  And now supposed that, along with the values of what we agree to call an “Intelligence Quotient”, God gives us -- as a bonus -- a GQ or Goodness Quotient  for each listed individual.  -- A hasty check again is somewhat reassuring, with Shakespeare and Einstein coming out slightly above average, and Donald Trump at rock bottom.   But we never imagined there could be a number for such a thing.   And how shall we ever devise tests to measure it?

    

(4)  That last was a Gedankenexperiment, of uncertain application.  Yet here, from a former trader at Salomon Brothers, is testimony to the power of a single number to symbolize a vast value:

Being paid was a sheer misery for many. [Note:  And this, despite the fact that they were paid very well.]  On January 1, 1987,  1986 would be erased from memory  except for a single number:  the amoung of money you were paid.  That number was the final summing-up.  Imagine being told you will meet with the divine Creator in a year’s time  to be told your worth as a human being.  You’d be a little edgy about the whole thing, wouldn’t you?  That’s roughly what we endured.

-- Michael Lewis, Liar’s Poker (1989)

(Notice again how the Almighty is brought in -- not by any means for religious reasons, but simply as a touchstone, to help make sense of it 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.

Anencephalic Epigrams

On religion, I lean towards deism, but consider its proof  largely a problem in astrophysics.

-- Edward O. Wilson, Consilience (1998).

Professor Wilson would probably feel mortified to behold this bone-headed bon mot  displayed naked and defenseless on the printed page, without the protection of dust-jacket accolades from fellow professors.   Not wishing to be in any way hurtful, we therefore provide it a camouflage or protective coloration of similar sallies.

On the existence of Beauty, I remain an Aesthetic Minimalist, but consider the resolution of this trifling problem  to be a mere matter for Optics.

-- Casey Stengel

The postulated existence of the material world is a challenging problem for further research.  We propose to settle the matter  one way or the other, by scientifically examining a hypothetical chunk of the stuff (a Twinkie, as chance would have it) in our up-to-date laboratories.  But for that we shall require a very large grant indeed.

-- Professor von Milchmoustache

Whether it be true that Truth itself exists, as such,  can only be resolved by an expedition to the North Pole, to determine whether, in point of empirical fact, snow is white.  Should it turn out to be the color of polar-bear pee, then Tarski’s entire enterprise  falls to the ground.

-- Yogi Berra

Support, of a sort, from an Unexpected Quarter

Michael Dummett, who has penned several essays of an anti-Realist cast, yet confides in the Preface to a major collection of his works (Truth and other enigmas (1978), p. xxxix):

I once read a paper, which I have not included in this collection, arguing for the existence of God, on the ground, among others, that anti-realism is ultimately incoherent, but that realism is tenable  only on a theistic basis.

This is a brave statement.  We do not disagree.  We might indeed go further, and suggest that the Pathos of Penguins, and the Wonder of Ducks, is comprehensible on that basis only; but that would be wide of our brief.   The stated aim of the essays on this site, is to support Cantorian Realism; the yoking of this to theism  could quite justifiably strike many readers as gratuitous, a long shot.   So it is nice to have such distinguished company as that of Professor Dummett.

*

William James is an ally of a different sort.  The unexpectedness stems simply from his role as a laboratory psychologist;  these tend to be more agnostic than mathematicians (though not perhaps more agnostic than Unitarians, or contemporary Anglican divines).

He argues, not for Realism per se, but for theism:  yet not in the positive way of the Ontological Argument, the Cosmological Argument, or any preacher on TV ;  his approach is not Realist, but pragmatist.  His celebrated predecessor in this approach, is Pascal’s Wager; and James -- ever a man of honor -- has the honesty to cite this explicitly, and to decry it. 

When religious faith  expresses itself thus, in the language of the gaming-table, it is put to its last trump. … If we were ourselves in the place of the Deity, we should probably take particular pleasure in cutting off believers of this pattern  from their infinite reward.

(Sidenote:  Are readers generally aware how witty is William James?   Particularly in comparison with his overrated dull brother -- what's his name -- Henry.  Hank.  Something like that.)

Basically:  If it works at all, it works too well, applying  as it does, without alteration, to belief in the Great Pumpkin.  Whose claims, on that tack, are if anything even stronger:  for the Great Pumpkin offers whatever Yahweh offers plus French fries and unlimited cable;  whereas his wrath towards unbelievers is terrifying to imagine -- you do not want to be in his bad books.

Anyhow.   In The Will to Believe, James offers, not proofs or even probabilities for the existence of God,  but a (well-reasoned) defence of whoso should believe

in spite of the fact that our merely logical intellect  may not have been coerced.

His strategy is, not to prove His existence, but to undermine fallacious arguments against it.   (This is the strategy I took in the chapter “Tumbnail Sketches of Arabic”, in Semantics of Form.)

In the much shorter lecture, Human Immortality:  two supposed objections to the doctrine, he likewise attempts to deflect scientific scorn:  yet, I regret to report, with little success.  His exposition rings, not of the self-delusion or excess of a religious true-believer, but of the plain wackiness characteristic of many a worldview put forth by English-speaking scientists in that period:  e.g. the statistician Karl Pearson, with his “ether squirts”  (see Theodore Porter’s fine biography).

Parsing “Trinitarian Minimalism”

            “Make everything as simple as possible,   but no simpler.”

                   --  attributed to Einstein

As we exulted earlier, your favorite hangout for Cantorian Realism  is also wearing the yellow jersey in the global sweepstakes for googling that soon-to-be-famous phrase,

~ “Trinitarian Minimalism” ~

Now, one other site does come up for this collocation.  It contains the following delightful passage:

If the hidden identity of divine being is revealed within the revelatory content

of the Gospel story, then perhaps there is yet scope for a via media between the Scylla of trinitarian minimalism on the one hand  and the Charybdis of trinitarian maximalism on the other. One possible solution is to unpack further the content of the trinitarian theologoumenon by means of exploring the person and work of the Spirit.

(Sidenote on theologoumenon.  It is not every day that Dr. J. learns a new word;  this one’s a doozer.  Merriam-Webster defines it thus: “a theological statement or concept in the area of individual opinion rather than of authoritative doctrine”. -- Actually, my friend Jennifer taught me another one earlier this morning, equally a keeper:  hooptie. -- It’s all good.)

Of linguistic interest here is the fact that this syntagm of adjective-plus-noun  is, semantically, actually parsed in opposite ways  in either case.

(1)  The way I’ve been using it, I start from the broad notion of minimalism (a traditional term of the arts), extend it suggestively to mathematics and the hard sciences (largely in the sense of parsimony, but with retention of the echoes of other uses of the term);  then, alarmed at the possibilities of ‘minimalist excess’ (to coin a phrase) -- Nominalism, solipsism, atheism, "The more the universe seems comprehensible, the more it seems pointless" --   I attempt, in the spirit of Einstein, to buttress it from below, and --  fancifully, metaphorically -- apply the modifier “Trinitarian”.

(2)  Our theologian, by contrast, starts from Trinitarianism, in its usual non-metaphorical sense;  then, seeking to distinguish among stances within this basic viewpoint, avails himself of some nice-sounding nouns lying ready to hand, minimalism and maximalism.   These do not have prior theological standing, and the resultant phrase (which, if I do say so myself, has quite a ring to it) is quite non-standard -- indeed, the page linked to above is the only such use in all of googlespace.

The meanings are thus quite distinct, though not actually at odds.

Now -- what adds a piquant semantic sauce to all this, is that the theological idea of the Trinity was not given to us explicitly in Scripture, but evolved over the centuries, in something like the scenario described in (1) !   It is presumably this fact to which our author alludes, when he applies to the doctrine the modest and cautious term theologoumenon.

The early Church began with monotheism, which it counterposed to the paganisms of the Greco-Roman world:  the which, rather than already-monotheistic Jewry, supplied the bulk of new converts.  But with time and contemplation, it became apparent that there was a danger (if you wish to call it that) that the whole enterprise might bleach out, in the direction pointed by Arianism, and exemplified today by Unitarianism:  a vague feel-good theism without bite. Revelation was sought, and the result is the Credo as we say it today.

Whitehead’s God

 In his inexhaustibly suggestive masterpiece, A Preface to Morals (1929), Walter Lippmann writes:

There is a God in Mr. Whitehead’s philosophy, and a very necessary God at that.  Unhappily, I am not enough of a logician to say that I understand what it means to say that ‘God is not concrete, but He is the ground for concrete actuality.’ …  A conception of God, which is incomprehensible to all who are not highly trained logicians, is a possible God  for logicians alone.

No more indeed than this, can I presume to have advanced, by the arguments in the series of essays towards a Theologia Mathematica.

For the mystic experience, or the calm and certainty that lies at the heart of real faith, I can hardly argue.  You see, or you do not see, the deep red rose.  You feel, or you do not feel, the breath of a breeze on your cheek. 

Further addenda

William James offers an abbreviated version of our Constructivist Fable (the Woodchuck Mathematician / Babylonian mathematics):  William James, The Principles of Psychology (1890), vol. II, p. 340:

By many measurements of triangles, one might find their area always equal to their height multiplied by half their base, and one might formulate an empirical law to that effect.  But a reasoner saves himself all this trouble by seeing that it is the essence (pro hac vice) of a triangle to be the half of a parallelogram  whose area is the height into the entire base.  To see this  he must invent additional lines.

That final observation is key.  To progress, we must advance beyond what is given visibly, and develop our intuitions of Invisibilia.
     These invented additional lines may stand as metonym for many such inspired devices.  Thus, to plumb the propterties of polynomials with exclusively real coefficients, we must consider a much wider world, and treat these polynomials as functions of a complex variable.

~ ~ ~

Although we renounced the project of quoting theistic language from pre-twentieth-century figures, as being but shooting fish in a barrel, the matter does retain some interest when the scientist used theistic ideas in a substantive way, to help guide his science, and not as mere window-dressing.  Thus, Ivar Ekeland, in The Best of All Possible Worlds (2000), p. 72, quotes Euler (1744) arguing for the truth and central role of the Principle of Least Action:

Since the constitution of the universe is perfect, and completed by an all-wise creator, absolutely nothing happens in this world  which canot be explained by some argument of maximum or minimum.  This is why there is no doubt at all that all effects observed in the world  can be explained from final causes, with the method of maxima and minima, with the same success as from efficient causes.

THEOLOGIA MATHEMATICA

 Synopsis/Zusammenfassung/ précis :

            Beginning with a parsimonious outset of only two Postulates,

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

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

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

Der hlg. Cantor

Der hlg. Gödel

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

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

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

[To begin the thread, click here.]