Thursday, May 29, 2014

Categories and Sameness (with linguistic appendix)

The free product is a sum in the category of groups;
and the direct sum is a sum in the category of R-modules.
-- old folk saying

We earlier had some fun with (in effect) the notion of hierarchical and cross-cutting patterns of Natural Kinds, in the essay Categories for the Working Mom.  And now that the laundry is fresh and folded and tucked away, we can all relax with a cup of your favorite beverage, and take another look.

[Update -- a note to our readers.
It has come to our attention  that a number of you have reached this oft-viewed post   under perhaps a misapprehension:  not out of ontological or taxonomic curiosity, but  by searching for a jpeg of "Homer strangling Bart".   This delicious image  we do indeed offer below;  but it is by no means the meat of the essay.  For those of you whose interest runs to cartoons rather than ontology, you may consult our meagre offerings here: ]

~ Sigmund Freud  und  Sherlock Holmes: ~

“Feynman,” said Wheeler, “I know why all electrons have the same charge and the same mass.”
“Why?” asked Feynman.
“Because," said Wheeler, “they are all the same electron!”
-- Martin Gardner, The New Ambidextrous Universe (1964; third revised edn. 1990), p. 305

Less Zen:

Particles transforming in the same representation of the Poincaré group  and having the same additional quantum numbers  are said to be identical particles.
--  Matthew Schwartz, Quantum Field Theory and the Standard Model (2014), p. 207

Linguistic philosophy is familiar with the caveat that to say two things are “the same” must be interpreted under some description.  (The technical name for this:  sortal identity.)

Thus, it is commonly said that the Chevy Whatsis and the Toyota Whomever are the “same” car, because they are built on the same chassis and are mechanically identical, just differently branded.  Now imagine an automotive expert given a pile of paired photographs, each depicting one vehicle; for each pair he must give thumbs up or thumbs down as to whether the vehicles are the “same” in this sense.  He will give thumbs-up to a Whatsis/Whomever pair, though now his assessment is in two steps:  First identifying the particular vehicles depicted as a representative  each of a specific brand;  and next, assessing those two brands as being “the same” under the applicable metric.

More narrowly -- this, say, for a car-dealer rather than a mechanic -- two vehicles are “the same” only if they are in the same SKU -- a Ford Taurus now being categorized separately from the Mercury Sable, though they are the same for most practical purposes.

More narrowly still, two examples of a Ford Taurus might be accounted equivalent  iff -- we are back in the auto-repair shop -- they are both in the same set of models, these being drawn up according to which parts they require.  The sets will be relative to the part in question:  this group uses drive-train X1, this other requires X2; cross-cutting these, another group uses such&such style of alternator, another another…

Even more narrowly:  the police want to know whether this vehicle sitting here in Arthur J. Mungo’s garage  is the same as the one used as the getaway car in last night’s robbery. 

More narrowly still:  I buy a certain car from you, new, for $20,000.  Five years and many fender-benders later, I demand you buy it back for the same price -- after all, it’s the “same car”.  And indeed, in the sense immediately above, it is, and will still serve to convict you if the robbery happened five years ago rather than last night.  But as seen by the eyeshades of an insurance company, it is not the same at all.

We could keep going, until  ultimately  the only pair of photos that will pass the green-eyeshade test  is ... two identical photos.

I have my mother's eyes ... He has his father's nose

Epigrammatic reflections of the paradoxes of ‘sameness’:

The successive unstable cabinets of the 3me République, were mostly a reshuffling of the usual suspects.  Thus,

There was some truth in the quip of Clemenceau, when he was criticized by a deputy for having overthrown so many governments.  “I have overthrown only one,” he replied.  “They are all the same.”
-- Wm Shirer, The Collapse of the Third Republic (1969), p. 101

Mocking Anglo-American journalists who take the Russia tour and become pundits:

After a year  they went home … sat down at their typewriters, and hastily wrote the same book.  I have been reading that book all spring, under several different titles.
-- Malcolm Cowley, The Flower and the Leaf (misc., collected 1985)

~     ~     ~

In our discussion of “analogy” and “sameness” in the essay Consilience in Mathematics, we saw that this informal term gets formalized in various ways -- isomorphism, homeomorphism, diffeomorphism, bijection … -- that, despite the variegated terminology, are really just one central idea, namely:  detailed pairings (“maps”) between objects, which preserve structure of some sort.  Which sort, depends on the mathematical category you are working in.

Commercial Break
A private detective  confronts the uncanny;
an ecclesiastical mystery:
Murphy Calls In a Specialist
~     ~     ~

Mathematical structures being immutable and eternal, there is no problem about cross-identifying them across time.  But in the peopled world, things change all the time -- or rather, as we now are careful to notice:  change under certain descriptions, and not under others.
The following is an excerpt from an essay, “Continuity of Identity”, which I hope to post someday.

Here’s a really practical case, of current political relevance.
In the year 19xx, a group of local politicians, and a state or municipal employees union, agreed to keep current wages at a specified modest level, against the promise of lavish pensions in the future:  the latter to be paid by “the taxpayers”.
Now, many of these deals were hatched behind closed doors.  Still, there was little pressure to expose them, for “the taxpayers” ca. 19xx  did indeed benefit from this arrangement:   Their taxes were lower than they would otherwise have been, had the services of police and fire and teachers and what have you had to be bid for on the basis of straight salary and current benefits;  these services were acquired relatively cheaply, the full bill being deferred.
And now, circa half a century later, “the taxpayers” are being asked to pay that bill.  Only… they are not the same people who benefited from the arrangement half a century or so ago-- in many cases, before they were born.  Not the same… as individuals;  but as a corporate body, yes, these are indeed “the taxpayers”.  So: legally, morally:  Can a politician and a union seal an agreement that binds, not themselves, but some third parties in the future, who at the time of the agreement were not even born?
Before you answer too hastily with a resounding outraged No, consider that our entire society is webbed with agreements exactly like that, and would likely fall apart without them.  Every time our nation signs a treaty, or issues a bond, or even launches a project like building highways and bridges (which future citizens will need to keep in repair, or much of the money will have been wasted), we are doing this.

~     ~     ~

An incident earlier this evening reminded me of an old conundrum:  the seemingly uncontroversial matter of what constitutes the same tune.

I happened to overhear on the radio, a snatch of a tune of which I’m very fond, of which I did not know the name.  It is melodic, and simple  apart from some rhythmic oddities.  Emotionally, its effect on me is like that of the brief beautiful interlude in waltz time, in Corelli’s Christmas Concerto.
As the piece segued into the next movement, I realized it had to be Copland, probably quoting someone else -- as he quoted “Simple Gifts” in Appalachian Spring.  Fortunately the announcer eventually came on and said the overall suite of which that piece was a part  was Rodéo
Looking at the program for this, I was puzzled.  “Corral Nocturne”?  Surely, for the dreaminess.  But no, it turned out to be -- “Saturday Night Waltz”.
Now, I would never have thought of searching for the piece under that name, since, firstly, it didn’t at all seem to be what some farmer-cowboy couples would be up to on a Saturday night, and futhermore -- okay, this is embarrassing -- I did not realize it was a waltz.  There’s an insinuating syncopation;  certainly if I ever tried to actually waltz to it, I would fall on my punkin haid…

Seeking to learn the mystery of this exotic composition, I got together a safari team -- seasoned explorers and native bearers -- and made my way through malarial jungle and crocodile-infested swamps  over the space of many months, finally scaling an icy peak and putting the question to the loincloth-clad hermit who sits at its summit and --  Well, that’s how it would have been in the old days;  to save time, I simply looked it up in Wikipedia.
And learned, to my astonishment, that the quotation in question is from … “I Ride an Old Paint”.   Astonishing because I absolutely grew up on that song, as sung by Burl Ives.  Our family only owned a half-dozen vocal recordings, so I listened to them over and over; and of these,  “I Ri-ide… an old Paint;  I le-ead…. an old Dan” was one of an even smaller handful at the top.  Never did learn what the lyrics meant, but I heard it and sang it   over and over.  And, being told -- like a puppy by the scruff of the neck -- that the tune right in front of me, in Copland’s ballet, is that; well, yes, I can hear the resemblance.

How could I not have heard the sameness in the first place -- the sameness beneath all the strangeness?  The simplest hypothesis is:  Musical retardation.  To which I partly plead guilty, only -- there are other, even grosser cases of mental compartimentation, which require a different explanation (nobody’s that retarded).

Thus, from childhood:  Consider “Twinkle Twinkle Little Star” and the Alphabet Song.  The tunes are identical, note for note;  failure to perceive similarity might be chalked up to incapacity, but failure to note identity (which was my failure until recent times) -- refusal to assent to "A = A" … something else is going on.

A hint at what this is, is provided by cases in which we children were not mere consumers, but producers of isomelodic products.
Thus:  “Nyaah, nyahh, nya-nyaah nyaah.”
And, on the same ‘tune’, any number of smug worthless despicable taunts:  “TIM-my’s Got a GIRRRL-friend…!!!” or what have you.
[This just in -- probably from the Vatican:  
Homer Simpson, delivering a stern lecture to his wayward boy
The taunts here cited, are sufficient proof that the Fall of Man spares no-one, not even little children, fancied by sentimentalists  to be innocent.   From a secular perspective, such productions suffice to justify strangling the little bastards in the cradle.  That such action is not in fact justified, is a miracle, unexplained by any reasoning within the purely secular realm.  The overriding fact is, that God loves us -- Lord only knows why…]

Clearly, the melodic medium and its message form a gestalt.  To the child, and to some extent later, it would be as artificial to peel off the tune, as to say that two quite distinct people are actually identical (as regards having two eyes, a nose, etc. etc.), their secondary differentia being whatever is left over.

The nec plus ultra of such a perspective  is spoken sentences:  for these each have an intonation which is partly conventional, like a tune:  differing between French and English, or between British and American English.  Hearing a sentence, we do not  on any level  equate it  with all the (thousands of) other sentences we have heard  similarly intoned.

~     ~     ~

To avoid the distraction of same in the sense ‘self-identical’, let us continue the discussion using the term equivalent.  The latter is actually more mathematical, in that there it has a special sense, that of equivalence classes.  (Check Wiki and read all about ‘em.)
So:  Items will be equivalent or not, depending on whether they are viewed sub specie this category or that.  As, in biology:  When working with multiple species, two individuals are equivalent if conspecific.  When working within a species, two individuals might be identified  if, beyond being conspecific, they are of the same sex, and -- for species which show distinctions such as larva vs. adult, sessile vs. vagile, pupa/chrysalis/imago -- in the same stage of life.

*     *     *
~ Commercial break ~
We now return you to your regularly scheduled essay.

*     *     *

In mathematics, two objects can be topologically equivalent -- homeomorphic; smoothly equivalent -- diffeomorphic; algebraically equivalent -- isomorphic;  metrically equivalent -- isometric; numerically equivalent -- equinumerous; and so forth.  An example of such usage, in a Hilbert space context:

…the latter direct sum itself is isomorphic (unitarily equivalent) to H
…It is unitarily equivalent (I might as well say that it is the same as) the bilateral shift.
-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), pp. 7, 12

[Remarkably, Wikipedia seems not to devote an article to this subject, the closest being this rather abstract one:
So I’ll say a little more.]

Further, we speak of two norms on a space as being equivalent if they generate the same topology on that space.  Two group-representations are equivalent if they are interrelatable via conjugation by a nonsingular matrix.  Two signed measures are equivalent if each is absolutely-continuous with respect to the other.  Two vector bundles are equivalent if they are related by a homeomorphism that preserves fibres. Two normal series of a group are equivalent if there’s a bijection between the factor groups with these being isomorphic.

There are even finer distinctions:  Two set-theoretic structures are “equivalent if just the same sentences are true in each, and elementarily equivalent if just the same first-order sentences are true in each.  Isomorphic structures are evidently equivalent, and hence  in particular  elementarily equivalent.” -- Michael Potter, Set Theory and its Philosophy (2004)

As it begins to appear, these different labels are not simply definitional and contentless;  sometimes you arrive at them only after doing some work.  E.g. one reads:

A fundamental fact of differential topology is that the notion of isomorphism in the categories Top[ology], P[iecewise]L[inear] and Diff[erential], is the same in dimensions three and below.  In dimention four, PL and Diff agree, but Top differs.  In dimensions above six, they all differ.

In particular, things go hog-wild in dimension seven.  We thought we knew what roundness was, it turns out we did not.

~ ~ ~

[Update 3 Dec 2011]
This phenomenon of tunes known  but hiding their identities beneath disparate lyrics and thus subjectively unconnected,   deserves a name:  we shall dub it cryptomelodia
(©2011 Dr J Worldwide Enterprises Inc.)

[Pronounce this krip-to-meh-LO-dee-a.  The word is modeled upon that other stupendous vocable, cryptomnesia ‘hidden memory’, from crypto- ‘hidden’ and -mnesia ‘memory’, as in amnesia (i.e. a-mnesia ‘no memory’).  Those roots are all Greek;  and I wanted to neologize on a proper Greco-Greek model, rather than conjure up some vulgar Greco-Latin hybrid-hippogriff such as Donald Trump would no doubt coin.  But the Latin-looking aspect of “melody” gave me pause. 
So we consulted with Dr Massey, the official philologer for this site, and he reassured us:

Melodia is of Greek origin.  μελδία : from melos, musical phrase, and aoide, song. Cryptomelodia is a great rendering.

This is excellent news.  Dr Massey and I shall split the royalties from this outstanding new word, and retire in opulence.]

A startling example of cryptomelodia occurred at work yesterday.   I work at a pretty patriotic place;  the political spectrum among the employees is much broader than you might imagine, ranging even remarkably far to the left, especially on antiwar issues.   Still, I was startled that morning to suddenly hear the song -- or I should say, the tune -- of “Solidarity Forever” blaring from a computer in the neighboring pod.  I leapt from my seat to see what was going on.  Had the Occupy movement spread to us -- Occupy the Fort ?   -- No, my companions scoffed, that’s the Georgia fight song.  (The woman whose computer harbors this tune on boot-up is from Georgia.)
That only raised another issue:  Why would Georgia -- a pretty conservative neck of the woods -- choose “Solidarity Forever” for their fight song ?  -- Actually, they replied, it’s the fight song of lots of schools.  And then it dawned on me:  The words being different, and the setting (a football stadium) utterly so, probably very few of the fans make the connection -- if indeed they have even heard of the old Wobbly anthem.  (And if they had, they might deem it ironic that the hymn of One Big Union should be hijacked  by gridiron jingoism.)  The tunes inhabit different compartments of the mind.

There was some argument over whether which song actually came first -- some voted for a football origin, which I thought absurd.   So upon reaching home, I burnt some incense before the altar of the All-Knowing -- and Wikipedia promptly informed me that the tune goes back  not only way before football, but way before the IWW:   that I have myself been the victim of a multiple cryptomelodia in this regard.  For, “Solidarity Forever” is simply a re-lyriced “Battle Hymn of the Republic”, which in turn lyrically recycles “John Brown’s Body”.  I am of course quite familiar with both these songs, having sung the first one  in particular  many times in elementary school.  But I never made the connection among these three.
(And upon further reflection … the transposition  to the football field  of the tune that’s held in common by all these anthems, is appropriate after all:  since all of them are  in some sense  fight songs.)

As for the tune itself -- its origins are lost in time;  Wiki traces it to “the folk hymn tradition of the American camp meeting movement of the 19th century”;  before that, all is mist.
Yet hark -- what lyre upon th' aeolian there wafts ?  Nay, ‘tis Aeneas’ bark, plunging westwards from the flames of Troy !  And what air sets the oarsmen bend their strength as one?  Why -- ‘tis … ‘tis that very tune !
~     ~     ~

[Further examples of the notion ‘same’, this time in linguistics.]

If we project backwards to the Pre-Latin form of a third-conjugation infinitive such as dûcere, we note, for Pre-Latin *doukesi,  formal identity with the locative case of a genuine Indo-European s-stem noun.
-- Robert Jeffers & Ilse Lehiste, Principles and Methods for Historical Linguistics (1979), p. 67

Here meaning:  phonetically identical, though with an ultimately different grammatical role.

Another extraordinarily cavalier positing of equivalence:

Chinese and English, for example, may have the same case system as Latin, but the phonetic realization is different.
-- Noam Chomsky, New Horizons in the Study of Language of Mind (2000), p.

(This recalls an anecdote from Charles Fillmore, author of the classic article “The Case for Case”, from back before the idea of abstract/semantic case was as familiar as the traditional notion that hewed to overt morphology.  He began presenting a paper on “Case in Chinese”, using this expanded or deepened sense of the term;  and his elder examiners, leaning forward in alarmed concern, said: “Aren’t you aware, that Chinese doesn’t have case?!?”)

… ‘free variation’:  If two forms are equivalent (that is, freely substitutable for each other), their alternating segments may nevertheless be phonemically distinct.
-- D. Hymes & J. Fought, American Structuralism (1975), p. 214

(Compare that elegant phrase, which slips so smoothly from the pen of Quine:  “substitutable salvâ veritate”.   I have here further spiffed up the expression, by giving the â its neat little ablative hat.)

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