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; 2^{nd}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; 2^{nd}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:

One 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 pronouns,*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. Hence, in order to retain Euclideanism,
it would then be necessary to

**the space. …. Einstein’s geometric articulation of that thesis does not leave room for saving it by resorting to a***remetrize***remetrization**in the sense of making the length of the rod*vary*with position or orientation 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 cite the passage as
though on one of those “citation slips” we used to rely on at Merriam-Webster.

~

I just now 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.
~

A philosophically alert historian of physics calls attention to a
linguistico-philosophical subtlety in the word

*measurement*as it is used in quantum theory:
In all cases, an observation is
accompanied by a

**measurement**. The converse is not true, however, for we may quite well perform a measurement and yet fail to observe the result. [ndlr: That much is true but trifling, but then he goes on to make his point.] Now in considering the disturbance generated by an observation, we must make clear that the disturbance is caused by the physical measurement, and not by the cognitive act whereby the result of the measurement is comprehended by the percipient. … The observation is rendered possible by the collision of the photon with the particle, and hence it is this collision which constitutes the measurement.
-- A. D’Abro,

__The Rise of the New Physics__(1939), vol. II, p. 667
Here he is not making an

*actio/actum*distinction in the term*measurement*(though one exists; for the*actum*, “Her measurements are 36, 28, 36”), for we are still dealing with*actio*here: but he is at pains to remove the connotation of a (human)**action**-- a human**act**, which one foggy school of thought has deemed central and essential to all of quantum physics.**Observation**, D’Abro is saying, is a human action;**measurement**is whatever triggers the collapse of the wave-function.
~

This whole question of

*measurement*is, for the man meditating over brandy, frankly pretty annoying. We want to know the scheme of things -- if equations be at the base of it, well and good, the more general the better. (As: Hamiltonian dynamics; Set Theory; Topology.) Beholding Saint Peter’s or the Taj Mahal, we wish to savor the whole, and perhaps to penetrate to the aesthetic and formal ideas behind them; but we do not wish to know the length or this or that member in centimenters, nor how much the materials cost, etc. Such matters are distinctly*hypo-ouranian*.
In latterday musings upon quantum theory,

*measurement*has been lifted to a role rather like that of (in earlier days)*action*, or*conservation of energy*-- or rather, like that of the Demiurge, bringing entities (or the values of their parameters) into existence. Yet in practice, they don’t always even tell you much about what you are trying to measure.**Measurements**on the force of attraction between two electric charges will not in general verify Coulomb’s law. We observe that the force depends in some peculiar way upon the position of external charges, which suggests to us that the measured effects are not entirely due to the system in question, namely, the two test charges.

-- Robert Lindsay & Henry
Margenau,

__Foundations of Physics__(1936), p. 524