What is it like to be a bat??

That classic query by the philosopher Thomas Nagel  has for many decades  stood as an uncrackable conundrum.  Yet, surprisingly, until right now, no-one has thought of simply asking a bat what it is like.

And so, speaking through an interpreter (none other than our good friend and colleague  Dr John Dolittle, of Puddleby-on-the-Marsh), we put the question to an amiable fruitbat of our acquaintance, and he replied as follows:

efInfn2090sdpqoieavnq¥ÉÀ$)Benv8Y$%&JœÃKJGF$##!@!FFGIU^$FDF+(Kk+=ÕefInfn2090sdpqoi‰ÂÊnq¥ÉÀ$)B4rtwgsnY$%&JK?>GKJGF$##!@!FŸU^$FD®+(KÍ+=ÕefInfn2090sdpqoieavnq¥ÉÀ$)BU5¢Y$%&J∆™¿JGF$##!@!F»ñIU^$FDF+(Kk+=Õ …

According to the good Doctor, that all makes perfect sense in bat-language, and presents a fascinating portrait of the cave-dwelling, upside-down-hanging, echo-locating, fruit- and bug-eating, night-flying life (ah what delight, to mate  in mid-flight!) of these nocturnal vespertilians:  but unfortunately the account is not translatable into English.

For more on the excellent Professor Nagel, try here:

http://worldofdrjustice.blogspot.com/search/label/Thomas%20Nagel

[Update]   The bulk of Lofting’s late novel Doctor Dolittle’s Garden (1927)  concerns just such an exercise:  learning the language of insects.   The book is alas not so engaging as its predecessors, simply because such creatures must be forever inscrutable to our psychic understanding.  (Granted, one entomologist could write a fine book titled For Love of Insects.)

The illustrations, however, continue to be outstanding.   He was his own E.H. Shepherd, so to speak, and one increasingly appreciates his qualities as a draughtsman.

[Bibliographic post-note]  In the course of seeking a .jpg to download, I came across a handful of purported “Dolittle” titles  of which I had not previously heard:  for the very good reason that they are latter-day fakes.   I shall not cite any of the titles  even to denounce them -- Nicht gedacht  soll seiner werden.  There is nothing wrong with coming up with new stories about characters broached by deceased authors -- most of European literature has been exactly that, since ancient times, and sometimes as a very explicit continuation as in the two halves Le roman de la rose, respectively by Guillaume de Lorris et Jean de Meung;  indeed, I have essayed such myself, in the matter of the good doctor, here:  Doctor Dolittle voyages to visit the Penguin King.   But these commercial ventures seek lucre by passing some hasty new Machwerk off  as from Lofting’s own hand (for legal reasons, the covers merely say “based on stories by HUGH LOFTING”, hoping you’ll skip over the fine print, and do not name the ghostwriter;  but catalogues, with only an “Author: “ field and no “Fake author: “ field, make no such distinction.)   And the illustrations are of an unspeakably humdrum nature; everything that made the Dolittle pictures special  has been lost.

From Finitude to Infinity

[Good heavens, it’s Sunday, and there isn’t a stitch to post !  Well -- le mieux est l’ennemi du bien, so here’s a stub -- what Malkiel would call a “torso” -- to be worked on further  if I am spared.  Think of this as a construction site, which the curious stroller may peer at through a knothole in the fence, checking back in a week or so, to see how the thing is coming.]

~

Thomas Nagel, The Last Word (1997), p. 71:

We draw this access to infinity  out of our distinctly finite ability to count, in virtue of its evident incompleteness.

Leave off, for the nonce, your incessant wrestling with the Riemann Hypothesis, and return to the simplicity of thought outlined in our parables of addition, one of the woodchuck, and one of Farmer John.

You awake in the night, in a cold sweat:  Might addition eventually break down?  For, aways down the number line -- somewhere we have never traveled, gladly beyond any experience -- there are these great big lumbering numbers -- the cube of googolplex, and whatnot.  Might they not eventually creak and crack beneath their own immensity?  Might not gigamegagoogolplex  simply refuse to have one digit more added to him, lest (like the glutton in Monty Python’s “The Meaning of Life”) he simply explode, splattering integers throughout the cosmos, in an arithmetical Big Bang ?

(Wittgenstein used to worry about that sort of thing, bless his heart.)

We do not, of course, believe anything of the sort;  though we’d be hard-pressed to explain just why we rule this out.  After all, some cosmologists, to account for certain perplexities in the red-shift, once put forward  in all seriousness  the hypothesis of “Tired Light”.  (Hey, if you had been stuck to the same old geodesic for thirteen billion years, wouldn’t you be tired too?  Or more likely, bored.)

Our metaphysical certainty  that finity is, so to speak, the same throughout, with no surprises down the line, is psychologically similar to various metaphysical assumptions in cosmology (uniformity in the large), but much more absolute.  The universe has, after all, so far proved lumpy at every scale, from the quark to galactic clusters and cosmic strings;  we can easily entertain the hypothesis that it might be gnarly all the way down.  Not so with the integers.  Our characterization of these as never ‘breaking down’  is a panoptical statement about their finished totality, thus about actual infinity.  Our faith in the well-behavedness of the finite  rests ultimately in our trust in the infinite.

Oh, and those photons ?  They do not get tired.  They whiz forever,
 tiny and trusting,

   communing with their Maker,

       in ways proportional to their understanding.

~

Other cases of proceeding from the finite to the infinite (if only heuristically):

The main purpose of the study of operator theory is to discover, formulate, and prove  the proper generalizations, valid for all Hilbert spaces, of the powerful results known in the finite-dimensional case.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 74

It turns out that, for instance, every compact operator on a Hilbert space is the norm-limit of some sequence of finite-rank operators.  But properties do change in the process:  thus, a compact operator need not have any eigenvalues.

~

A compact topological space is one in which every open cover has a finite subcover.  This turns out to be a very nice property indeed:  the classification theorem for compact surfaces (i.e., 2-manifolds) is one of the neatest and simplest and most satisfying and most startling in all of mathematics;  it completely settles that problem  once and for all.  All such shapes you can imagine  boil down to just:  a sphere; a sphere with handles; a sphere with cross-caps.  That’s it.

But as we leave compactness, we leave behind that finiteness -- and things get weird.

The classification theorem for compact surfaces  does not extend in any way to noncompact surfaces.  There is an incredible variety of noncompact surfaces.

-- Michael Henle, A Combinatorial Introduction to Topology (1979), p.  129

If, however, we retain compactness (and thus a kind of finiteness), but allow now surfaces with boundary, then everything settles back into place.  The choices are the same as before, only now you’re allowed to cut some holes.   As Henle puts it:

Every compact, connected surface with boundary  is equivalent to either a sphere or a connected sum of tori or a connected sum of projective planes, in any case with some (finite) number of disks removed.

~

A platonist conception of the infinite as a completed actual totality  misrepresents, according to [the intuitionists], the very nature of the infinite, by illicitly assimilating its character to that of finite collections.

Colin McGinn, “Truth and use”; in: Mark Platts, ed.  Reference, Truth and Reality (1980), p.  34

Note that that critique is not McGinn’s own, but is that of the sloth-like finitist/constructivist tribe known to the préfecture de police as the “Intuitionists”;  nor are the preternaturally sophisticated working mathematicians of today  guilty of any such puerile reductionist assimilation.  If anything (for all I know), the smart money now views individual integers as (more or less tragic) restrictions of an original infinitude : much as we creatures here below, though fashioned by His transfinite hands, yet mostly muck about the malls and brothels, not that different, all told, from our lesser cousins  feathered or furry.

Something possibly a bit along those lines, in the supra-empyrean context of sheaf theory, is tantalizingly broached by Edward Frenkel  here:

The [finite] number 3 is a mere shadow of the [infinite] 3-dimensional space …

~

For a merely morphological look at this word finitude, try this:

     http://worldofdrjustice.blogspot.com/2012/07/on-finitude-and-truthitude.html

The Honorable Atheist

In a recent post, I praised the book of Thomas Nagel, The Last Word (1997); and do here reiterate that praise.  Yet towards the end of that volume (pp. 129-131) he delivers himself of some most interesting musings.  In an observation recalling one we quoted previously from Michael Dummett, he embraces Rationalism, but sees that as leading inexorably on to Theism, and shrinks back.  Then, becoming suddenly quite strikingly autobiographical in what is otherwise a mostly patrician book:

I want atheism to be true. … It isn’t just that I don’t believe in God and, naturally, hope that I’m right in my belief.  It’s that I hope there is no God!  I don’t want there to be a God;  I don’t want the universe to be like that.

Ecce homo.  Ich kann nicht anders.  Eppur’ si muove.

These are deep words, of honorable men; and poles apart from the snarky sneers of a Hitchens.   The silence of those infinite spaces terrifies him, as it did Pascal.

What God himself may think of such a plea, we cannot know.  But we know what Chesterton would think, because he stated it himself, in his novel The Ball and the Cross And we know what C. S. Lewis would think, because he told us, in his autobiography Surprised by Joy.

Anyhow, despite his understandable qualms, I do hope that Professor Nagel finds Paradise to his liking, because I’m pretty sure that that is where he is headed.

What Is At Stake

Realism about mathematics is just a part -- and not the chief part -- of Realism about Reason.   One can actually get along pretty well, from week to week, without having recourse to Algebraic Geometry.  But Reason is at the core of our being and our freedom:  and it is Reason that, in these dark days, finds itself under attack.

The case for Reason has recently been ably and gracefully put by Thomas Nagel, in The Last Word (1997);  I shall not repeat his arguments, but simply urge you to buy his book.

Sheer Reason, however, is difficult to reason about.  Mathematics is thus a useful test case for the larger thesis, since  when the truths of mathematics come to be known, it happens  only by Reason (occasionally supplemented, it may be, by Revelation, which then however feeds smoothly into the usual operations of Reason itself:  exactly like a theorem that has been proved, to your satisfaction, by someone else).   Math has the added advantage of being species-neutral (unlike Ethics, and much else).  It has also proved useful in opposing what Nagel calls “Darwinian imperialism” (p. 133), namely “the ludicrous overuse of evolutionary biology to explain everything about life, including everything about the human mind”.   (I satirized this in an earlier post, “The Urysohn Metrization Theorem:  an Adaptationist Account”.) 

            A psychologist or philosopher, soddened by overlong splashing in the swamps of Raw Feels, will tend to be satisfied, in his Gedankenexperimenten, with the most trivial sort of elementary arithmetical facts, for the organism’s tacit recognition of which  a Darwinian account may seem not too far-fetched, especially to someone who doesn’t really care about mathematical reality anyhow, and thus is easily satisfied.  But hyperDarwinists (not a slur -- that is Dawkins’ self-chosen label) can less readily account for our familiarity with E8. -- Though, to be sure, a deep acquaintance with this object will prove crucial for our species’ survival  when, in the year 30,906, we shall be forced to flee our imploding galaxy for a cosmos in which -- but that is of no account, for Darwinian orthodoxy emphasizes that Natural Selection cannot peek into the future.

And for whoso should say, this is but a shadow-play,  we offer this envoi, from a Thomist, anent the ontological agnosticism of certain linguists:

L’oreille la moins exercée  perçoit aussitôt   sous de tels énoncés  la présence des problèmes qui, sous les noms de réalisme et de nominalisme, ont agité les écoles du moyen âge  pendant au moins trois siècles.  Aujourd’hui, on se contente de tenir ces problèmes philosophiques pour résolus  en vertu d’un simple décret préalable  de ne pas philosopher.  Il ne suffit pourtant pas  qu’une solution ne soit pas philosophique  pour qu’elle devienne scientifique.  Qu’est-ce que cet « object concret »  qui ne serait qu’un « exemplaire »  du concept ?  Platon, Aristote, Abélard et Ockham  demandent aussitôt la parole,  et on ne peut aujourd’hui que se taire  ou reprendre le problème  au point où ils l’ont laissé.

-- Etienne Gilson, Linguistique et philosophie (1969), p. 49

*

Travaillant au noir,

le détective  se trouve aux prises

avec le Saint-Esprit

C’est quoi, le péché irrémissible???

*

Updates re the language of Realism

[An update to this ]

Gilbert Ryle, “The Theory of Meaning” (1957):

The difficulty is to steer between the Scylla of a Platonistic  and the Charybdis of a lexicographical account of the business of philosophy and logic.

Here he uses lexicographical in an idiosyncratic sense, influenced by the etymology of Nominalist, from Latin nomen ‘name, noun’.

Colin McGinn, “Truth and Use”, in Mark Platts, ed.  Reference, Truth and Reality (1980), p. 19:

Realism is the thesis that truth (falsity) is an epistemically unconstrained property of a sentence; there is nothing in the concept of truth (falsity) to exclude the possibility that a sentence be unknowably true (false).

Incidentally -- for those not yet indoctrinated into the philosophical patois, those parentheticals are not intended as synomyms or definitions of the word to which they are appended --“truth = falsity”, à la 1984;  our philosophical faculty are not quite so nihilistic as all that.  Rather, this is the beziehungsweise way of speaking, which you can easily acquire at the cost of tens of thousands of dollars in tuition and several years of your life.

Simon Blackburn, Think (1999), p. 260:

A true realist or opponent of idealism  wants to contend for facts and states of affairs that are entirely independent of the mind.

Simon Blackburn, Think (1999), p. 260:

REALISM (sometimes PLATONISM).  These rules have a real, objective existence.

CONCEPTUALISM.  These rules are creatures of the mind.

NOMINALISM.  There aren’t really any rules at all.

Simon Blackburn, Think (1999), p. 268:

What is often called ‘postmodernism’ is really just nominalism, colourfullly presented as the doctrine that there is nothing except texts.

Thomas Nagel, The Last Word (1997), p. 87:

“internal realism”, according to which our apparently objective world picture should be understood as essentially a creative product of our language and point of view.

Thus, “internal realism” is to realism what “National Socialism” is to socialism (the parallels are many).

Rom Harré, The Philosophies of Science (1971, 2^nd edn. 1984), preface to the first edition, uses the term in a sense very far from what we mean by it.  He contrasts “the positivist position” with “the realist point of view, which emphasizes the work of the human imagination in leading to conceptions of the realities behind sense-experience.”  I’d call that Idealism, not Realism in our sense at all.

On p. 87, he offers a different contrast, namely with skepticism;  “Copernicus  was a realist.”

Page 92 presents yet a third dichotomy: realism vs. phenomenalism:

If mechanics  with its eliminable concept ‘force’  provides a model science for phenomenalists, the virus theory of disease provides a counter-model for realists.

Note:  only the term “constructivism” was (so far as I know) specifically confected to contrast with “Platonism”, in the way that nominalism is the antonym of realism.  But intuitionism is a whole program, not just an anti-stance.  Likewise Hilbert’s formalist program.  According to Reuben Hersh (“Some Proposals”, repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 16), 

Hilbert’s writings and conversation display full conviction that mathematical problems are questions about real objects

-- i.e., full-bore Platonism.  But for foundational reasons he championed a program called formalism.  As such, they need not logically stand in contradiction;  but in practical, psychological terms, they do tend to.  Again Hersh:

We can see the reason for the ‘working mathematicisans’s’ uneasy oscillation between formalism and Platonism.