We earlier treated the matter of a metric on a topological space, in a series of essays beginning
here:
Now, as lagniappe, we offer a pair of Metrization Epigrams, which the budding mathematician, stuck for an
opener with that leotard-clad vision at the espresso bar, can use for a pick-up
line:
A discrete space
is like the autistic atomism of the Tractatus, or Leibnizian monads.
An indiscrete space is (as one writer charmingly put it), “really
quite crowded: each point is an accumulation point of
every other set.” (One pictures
the Jellyby children in Bleak House, ever tripping over one another’s
legs.)
(Believe me, chicks go wild over such things. Or at least, if you are like most
gangly Adam's-apple-challenged graduate-students in math, it’s your last best shot.)
~
It is by no means only topological
spaces that one might wish to subject to a metric: all kinds of things,
really: Which species
lie how close to which others (and different metrics -- phenotypic, cladistic,
etc. -- yield different results); which
languages are neighbors in linguistic space (again there is a
phenotypic/cladistic distinction:
descent vs. Sprachbund); which people have a natural affinity with which
other (seating-plans at dinner-parties; blind dates; etc.) And more generally, what is the
curvature tensor of the noösphere?
As (from a philosopher of science):
One sometimes wants to say that a
theory has much more testable content
than some other statement: for
instance, the Newtonian theory has much more testable content than ‘The moon
orbits the earth’. But our
apparatus, as it stands, does not entitle us to say this. We have no metric for testable content.
--John Watkins, Science and
Skepticism (1984), p. 185
Compare Keynes’ critique of relative subjective
probabilities.
Here a noted philologian on the notion as applied to
languages:
Was nun die Sache selbst anlangt,
so meine ich daß immer Sprache und
Sprache, mögen sie auch noch so weit auseinander liegen, in wissenschaftlichem
Sinn enger zusammengehören als Sprache und Literature, seien es
auch die desselben Volkes.
-- Hugo Schuchardt, “Über die
Lautgesetze” (1885), in Leo
Spitzer, ed., Hugo Schuchardt-Brevier (1921; 2nd edn. 1928),
p. 85
And here, indeed, he polemicizes against the nominalist treatment
or ‘indiscrete toplogy” of diachronic linguistics:
Ist es denn nun nicht an sich ganz
gleichgültig, ob rom. andare von adnare oder addare oder ambulare oder
einem keltischen Verbalstamm herkommt;
ob in diesem Dialecte l zu r und in jenem r zu l
wird usw.? Welchen
Sinn haben alle die Tausende etymologischer und morphologischer
Korrespondenzen, die Tausende von Lautgesetzen, solange sie isoliert bleiben,
solange sie nicht in höhere Ordnungen aufgelöst werden?
-- Hugo Schuchardt, “Über die
Lautgesetze” (1885), in Leo
Spitzer, ed., Hugo Schuchardt-Brevier (1921; 2nd edn. 1928),
p. 84
~
This metaphor of ‘metrization’, outside the exact sciences,
is very loose, as it is not strictly needed -- for taxonomic purposes, a more
approximate neighborhood-system will suffice (a “Uniformity”) so to speak --
and still less is to be obtained.
As, a pair of British linguists comments:
Once recent attempt by French
researchers has given us the term dialectometry, which describes a
formula for indexing the dialect ‘distance’ of any two speakers in a
survey. So far, the utility of the
index has not been demonstrated.
--J.K. Chambers & Peter
Trudgill, Dialectology (1980), p. 112
This, in the synchronic arena, is reminiscent of the glottochronology
of Morris Swadesh, who
attempted a sort of carbon-dating of linguistic evolution, based on an
assumed universal rate of lexical decay, in the absence of
direct evidence.
[Update 17 January 2016] Another cautionary tale about the fetishization of metrics:
TWO of our most vital industries,
health care and education, have become increasingly subjected to metrics and
measurements. Of course, we need to hold professionals accountable. But the
focus on numbers has gone too far. We’re hitting the targets, but missing the point.
Philologisches:
Whether, in that opening paragraph, the author wrote “metrics
and measurements” intending to refer to two distinct though related concepts,
or whether it was just an idle bit of synonymic accumulation like “bequeath and
bestow” for those who might be unfamiliar with the somewhat technical word metric, is there unclear. But it does raise a linguistic point.
The word metric,
and its close kin meter, metrical, metrization, derive from Greek.
Mensuration and commensurable go back to Latin mensura.
Measure comes
ultimately from that Latin word as well, but via the phonetics of medieval
French.
The same Indo-European root is said to lie at the base of all of them.
English, an etymological patchwork, has some tendency to layer
its vocabulary by origin, Greek roots being reserved for the most technical,
followed by Latin, with the Saxon
vocabulary as jack of all work. In
this, it contrasts with German: to
Graeco-English oxygen, hydrogen, nitrogen correspond homely-sounding
Germanic compounds Sauerstoff (‘sour-stuff’), Wasserstoff, Stickstoff. And Freud’s German originals for
the English ego and id, were nothing but nominalizations of
the ordinary pronouncs, das Ich &
das Es.
Roughly such tiering is at work in our Wortfeld of ‘measure’.
Measure sounds
reasonably English (though partly just because it chimes with pleasure and leisure, which are likewise French words in disguise), and is in
everyday use for all purposes (though it also has technical specializations, as
in mathematical measure theory).
Latinate mensuration
(little used) is scarcely more than a stuffy synonym for ‘measuring’; commensurate
is literate and has everyday though businesslike uses (“a salary commensurate
with the job responsibilities”); while commensurable
is mostly technical, whether in its mathematical sense, or its more recent
philosophic sense (“commensurable discourses”).
Metric is kind of
a green-eyeshade/clipboard sort of word at best. If becomes fully technical in mathematical uses like metric space and semi-metric, finally
soaring off into the intellectual empyrean with metrizable.
~
Still wearing our lexicographer’s hat, here is an
attestation for a word with which I had previously been unfamiliar, used by a
philosopher of science. After a
rather confusing Gedankenexperiment judging the verifiability of a physical
geometric hypothesis, involving all sorts of skulduggery with measuring rods,
and “tampering with the semantic
anchorage of the word congruent”
(that’ll get you two weeks in the clinky without the option), and in which Albert Einstein (from
beyond the grave) plays a role like that of Fantomas, battling the equally post-mortem
shade of Pierre Duhem, our
professor writes:
The required resort to the introduction
of a spatial dependence of the thermal coefficients might well not be open to Einstein. Hende, in order to retain Euclideanism,
it would then be necessary to remetrize the space. …. Einstein’s geometric articulation of that thesis does not leave room for saving it by
resorting to a remetrization in the sense
of making the length of the rod vary
with position or irientation even after it has been corrected for
idiosyncratic distortions. But why
saddle the Duhemian thesis as such
with a restriction peculiar to Einstein’s particular version of it? And thus why not allow Duhem to save
his thesis by countenancing those alterations in the congruence definition
which are remetrizations?
-- “The Falsifiability of Theories”,
in: Adolf Grünbaum, Collected Works, vol. I (2013), p. 72-3
As indicated, I couldn’t really follow the dialectical
taffy-pull in that conterfactual-strewn discussion, but simply the passage as
though on one of those “citation slips” we used to rely on at Merriam-Webster.
~
Just happened upon a passage which we quoted earlier in
another context (here): a use, by
a mathematician (or if you prefer, a logician) of the
in-itself-not-expressly-mathematical term measurement, not in a technical mathematical sense
such as measure zero or measure theory, but sliding mathwards towards concepts
very far from any plain man’s conception of “measurement” (as: wholly
non-numerical, non-quantitative fundamental
group):
Mathematics is, as it has always
been, largely the science of measurement. But “measurement” must here be
understood as referring to more than the meter stick. The genus of a topological figures measures one of its
aspects; objects of genus
zero are in a sense simpler than
those of higher genus.
There are many dimensions of
measurement ….: characteristic,
transcendence degree, cardinality, fundamental group … Occasionally we are so
successful in the science of measurement
that we can completely characterize an object … by giving, as it were,
its latitude and longitude: its
measurements in the relevant dimensions.
-- Herbert Enderton, “Elements of
Recursion Theory”, in: Jon
Barwise, ed. Handbook of Mathematical Logic (1977), p. 554
I originally cited that as a not-especially-successful
attempt (in its first sentence) at a one-line characterization of What
Mathematics Is. Measurement, in the usual sense, is common to a great many studious
activities, from chemistry to engineering to dressmaking. He only manages, in what follows,
to make that characterization more
or less work, by moving the goalposts
and re-defining measurement in
his own Pickwickian sense.
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