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.
“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 specific representative each
of a specific brand; and next, assessing those two models 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
A similar use of same
in the sense ‘same-old’, 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)
(Here, the determiner "that" actually becomes witty, in context.)
These recall Wheeler’s quip about the electron, although
there the meaning was more subterranean, having to do with fermion
statistics; cf. clonal colonies of
trees: One aspen, or many?
[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:
http://worldofdrjustice.blogspot.com/search/label/cartoons ]
~ Sigmund Freud und Sherlock Holmes: ~
~ ~ ~
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:
http://en.wikipedia.org/wiki/Equivalence_of_categories
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. μελῳδία : 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 !
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.
Chinese and English, for example,
may have the same case system as Latin, but the phonetic realization is
different.
(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?!?”)
-- D. Hymes & J. Fought, American
Structuralism (1975), p. 214
