Metrization, Mensuration, Measurement

We earlier treated the matter of a metric on a topological space, in a series of essays beginning here:

The Urysohn Metrization Theorem:  for real this time

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.

http://www.nytimes.com/2016/01/17/opinion/sunday/how-measurement-fails-doctors-and-teachers.html?ref=international

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 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.  It 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 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 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 figure  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

The Realist Vernacular

            What follows is neither proof nor argument, nor philosophy of any sort, but rather an exercise in sociolinguistics.  [And as such, a sort of sociological preparation for the thread announced here.]  The point is simply to exhibit a common way of talking among contemporary mathematicians, as well as some physicists and philosophers -- an easy style of conversation  that you would never imagine exists, if most of your acquaintance with science and its philosophy is mediated by figures like Daniel Dennett and Richard Dawkins.

            To cite evidence of theistic talk from the learned men of history, from antiquity through the nineteenth century, would be pointless, since the mode was well-nigh universal, at all times and in all realms.  Yet in our present day, most of the habitués of faculty clubs and coffee-houses  have managed to satisfy themselves, that all those who ever lived, in history, from Pythagorus  to the Einstein of “Der Herrgott würfelt nicht”, were, without exception,  imbeciles, and simply didn’t know what they were talking about, when they talked that way.  At last mankind has seen the light (or the darkness, rather);  we speak only of what is sensible and sniffable, like Donald Trump.  And who should be so rash as to cite the testimony of a mere Plato, or Galileo, or Cantor, or Gödel, against the brass certitude of so eminent a scholar as Sir Christopher Hitchens, Ph.D?

            Therefore I shall limit myself to recent citations.  A few examples from among many, snatched at random from  recent reading.

            Again, note:   I am not implying anything about the theology, per se, of the people here quoted, let alone suggesting that they never miss Mass.  Indeed a Dennett -- a roaring atheist -- can still indulge in such turns of phrases as "we can take advantage of the God's-eye perspective we have temporarily adopted" (the Devil can quote Scripture to his purpose).   So far, this is just corpus linguistics;  but more anon.

(1) Theistic language

 (a) mathematics

Arthur Koestler, The Act of Creation (1964):

Karl Friedrich Gauss described how he finally proved a theorem on which he had worked unsuccessfully for four years:  “At last, two days ago, I succeeded, not by dint of painful effort, but so to speak  by the grace of God.”

Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 219:

In one of the most cited discussions in his much-quoted book, Kuhn [1962] talks of scientific decision in terms of “conversion experience” and “faith”.

Richard de Millo et al., “Social Processes and Proofs”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 271:

The classical view does not require that an ordinary proof be accompanied by its formal counterpart; on the contrary, there are mathematically sound reasons for allowing the gods to formalize most of our arguments.

Similar language is often used in formulating the contemporary concept of a “hypertask”.  Cf. likewise

Shaughan Lavine, Understanding the Infinite (1994), p. 55:

            For Cantor … “countable” meant countable by God.

and similarly

Michael Potter, Set Theory and its Philosophy (2004), p. 250, re the plausibility of the Axiom of Choice:

… generalizing to the uncountable case  by appeal to the idea than an ideal being could achieve the choices required of him (or perhaps Him).

If we adopt some particular postulate system for abstract set theory, and agree that the criterion for accepting an intuitive set-theoretic argument  is that its analogue can be justified by the postuates, then we are  in efect  agreeing that the set of all intuitive sets  endowed with the membership relation  is a configuration satisfying the postulates.  We can have at best  intuitive reasons for believing this.  It is really an act of faith.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 153

Gregory Chaitin, “Gödel’s Theorem and Information”, in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 306:

If God tells one how many different programs of size less than N halt, this can be expressed as an N-bit base-two numeral, and from it  one could eventually deduce  which of these programs half  and which do not.  An alternative divine revelation would be knowing that program of size less than N which takes longest to halt.

Robin Wilson, Four Colors Suffice (2002), p. 214, recounting a mathematician’s reaction to the immensely long and repetitive, unsurveyable, computer-aided proof of the Four Color Conjecture:

God wouldn’t let the theorem be proved by a method as terrible as that!

 (b) physics

Hermann Weyl, Symmetry (1952):

Contingency is an essential feature of the world.   Clarke  in his controversy with Leibniz  admitted the latter’s principle of sufficient reason, but added that the sufficient reason often lies in the mere will of God.  I think, here Leibniz the rationalist is definitely wrong, and Clarke [the theist] on the right track.  But it would have been more sincere to deny the principle of sufficient reason altogether, instead of making God responsible for all that is unreason in the world.

Abdus Salam (1988):

God created just two dimensions -- one of space and one of time. … At a later epoch, there was a phase transition to four dimensions, plus six internal ones.

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 91, alludes to Roger Penrose’s paraphrase of cosmic censorship:  “God abhors a naked singularity.”   The (jocularly) theistic language is especially odd and noteworthy, since the epigram here being echoed -- “Nature abhors a vacuum” (Latin:  horror vacui) does not use it.

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 126f.:

These laws [of physics] may have originally been decreed by God, but it appears that he has since left the universe to evolve according to them  and does not now intervene…

That, then (for that author) leaves only the explanation of the settings of the parameters for initial conditions:

One possible answer is to say that God chose the initial configuration of the universe for reasons that we cannot hope to understand.  This would certainly have been within the power of an omnipotent being, but if he had started it off in such an incomprehensible way, why did he choose to let it evolve according to laws that we could understand?

Simon Blackburn, Think (1999), p. 69, in a section called “A Scientific Model”:

Classical physics identifies the temperature of a gas with the mean kinetic energies of the molecules that compose it.  So in making hot gases, God has only one thing to fix…

Blackburn is a philosopher, not a physicist; as commonly in that community, he is writing in an “as if” mode;  but still, the choice of words is noteworthy.

John Gribbin & Martin Rees, quoted in Edward Harrison, Cosmology (2^nd edn. 2000), p. 489:  Absent the strong anthropic principle,

If there is a unique ‘theory of everything’, then … we would have to accept it as genuinely coincidental, or even providential, that the constants determined by high-energy physics happen to lie in the narrowly restricted range that allows complexity and consciousness to evolve…

(“Providential”… a lovely word…)

Paul Davies, The Goldilocks Enigma (2006), p. 3:

It appeared to Hoyle as if a superintellect had been ‘monkeying’ with the laws of physics. … Like the porridge in the tale of Goldilocks … the universe seems to be ‘just right’ for life.

Paul Davies, The Goldilocks Enigma (2006), p. 236:

Most theoretical physicists are Platonists in the way they conceptualize the laws of physics as precise mathematical relationships  possessing a real, independent existence…

Shing-Tung Yau, The Shape of Inner Space (2010), p. 101:

As Robert Greene puts it, “you’re trying to find the one metric given you by God.”

Michael Atiyah re string theory, quoted in Shing-Tung Yau, The Shape of Inner Space (2010), p. 292:

They’re onto something, obviously.  Whether that something is what God’s created for the universe  remains to be seen.  But if He didn’t do it for the universe, it must have been for something.

This statement is reminiscent of the Principle of Plenitude; and recalls Einstein’s celebrated quip, anent the possible failure of an experiment to confirm his prediction: "Da täte mir halt der liebe Gott leid; die Theorie stimmt doch."

Less seriously, but using a religious metaphor: J. L. Synge, Relativity:  The General Theory (1960), p.  ix:

It is to support Minkowski’s way of looking at relativity that I find myself pursuing the hard path of a missionary.

More serious is the following.  The author is discussing the vexed question of the Collapse of the Wave-Packet upon observation (related to:  If a tree falls in a forest, and no-one’s around, does it make a sound?  -- here rewritten in Oxford terms, replacing the forest by a quad), and quotes the old limerick:

Dear Sir, Your astonishment’s odd;

I am always about in the Quad.

And that’s why the tree

Will continue to be,

Since observed by Yours faithfully, God.

He comments (J. C. Polkinghorne, The Quantum World (1984), p. 67):

Divine reduction of wavepackets would be an overkill, since it would operate everywhere and always, forcing the electron each time to go through a definite slit.  The point about measurement is that it only occurs spasmodically.

Note the predicted empirical consequences of God in the Quad!   Then, between square brackets, the author adds:

This observation is in accord with the classic theological understanding of creation, which sees God as the ground and support of all that is (in our terms, the guarantor of the Schrödinger equation), but not as an object among objects (no collapser of wavepackets).

This aside is in fact central:  by the time he wrote this, Polkinghorne had quit his endowed Chair in physics to become a village vicar.

(c ) analytical philosophy

Michael Dummett, Truth and other enigmas (1978), p. 15, discussing character as (it may be, untested) hidden propensities:

If B still wishes to maintain the necessity of ‘Either Jones was brave or he was not’, he will have to old  either that there must be some fact of the sort to which we usually appeal … or else that there is some fact of an extraordinary kind, perhaps known only to God.

Dummett’s use of the term, to characterise the viewpoint of a hypothetical philosopher inclined to Realism, is so to speak opaque, not representing his own view, which is rather (id, p. 150) that “only a philosophically quite naïve person would adopt  realist view of statements about character”.  Only such, or Saint Peter.

(2) Realist language

Arthur Koestler, The Act of Creation (1964):

Gauss is reported to have said: “I have had my solutions for a long time, but I do not yet know how I am to arrive at them.”

Klaus Jänich,  Topology (1980; Eng. trans. 1984), p. 35:

When topological groups are found in nature [emphasis added], they  are generally not given abstractly as a set G with a composition law and a topology, but concretely, as a group of transformations…

This is not really anymore outrageously realist than saying “When a set of six objects is found in nature…”   (say, the familiar six-pack; although the one at my side is already down to five, and it’s not even noon).

And again, p. 157:

Covering spaces very often “occur in nature”:  that is, one comes across them spontaneously, while studying entirely different problems.

Shaughan Lavine, Understanding the Infinite (1994), p. 160: 

We seem to have nontrivial intuitions concerning the infinite, going far beyond simple things like Extensionality, Pairing, or even Power Set.

Shing-Tung Yau, The Shape of Inner Space (2010), p. ix:

The strength of this discipline [i.e., mathematics] lies not simply in its ability to explain physical reality…, because to a mathematician, mathematics is reality.

If all you know of math is elementary arithmetic, this statement may lack punch.   But in the upper reaches of set theory, topology and so on, it embraces a world of miracles and of monsters.

~ ~ ~

It is important to note, that this is the way Realists talk en famille.  The talk is casual, often not literal, yet is meant in some serious sense.  It is not to be compared with the tawdry pseudo-theological, pseudo-mystical sort of gibberish that popular authors (and even some serious ones, yielding no doubt to the Satanic promptings of the marketing department) use to gin up their pap for the masses -- like “God particle” for the freaking Higgs boson.

It may be objected (it will be objected;  it has been objected) that such expressions, in a modern mouth, are a mere façon de parler.  To which we reply (with Whorf), that a façon de parler tends to cohere with a façon de penser. 

Nor is it a refutation of the point here made, to adduce agnostic pseudepigrapha from any of the gentlemen here quoted.   We are each a walking contradiction;  we contain multitudes.   C.S. Lewis himself  confessed that he tended to be a cranky agnostic at dawn, but a theist later, when he’d had tea and was more himself.

Once again:   The point here is not to assert that so-and-so among our near contemporaries  is or is not a believer.  Indeed it will strengthen my eventual case, if many of them are not in fact believers, yet find themselves attracted, or guided, or driven, to Realist or Theistic language, whether for convenience, or (in the case of Cantor, Gödel, and Einstein) something deeper.

So, a very modest initial move, a sort of pawn to king’s four.  (Only later -- much later -- shall we see if we can capture Satan’s Queen.)   So far we hold simply, that occasional use of Realist or Theist language, in serious discourse, is not  in and of itself  diagnostic for Trisomy 21.

The Urysohn Metrization Theorem: for real this time

Background:   Topology is familiarly, informally characterized as “rubber-sheet geometry”.  That is, unlike the Euclidean geometry that we learned in school, which applies to flat rigid surfaces,  you’re allowed to stretch and bend the space, so long as you don’t tear it or let it intersect itself.

But at some point, we might like -- without returning quite to the simplicities and rigidities of the Euclidean picture -- to make our space… a bit less rubbery.   As the godfather of Calabi-Yau manifolds puts it:

We start with some raw topological space, which is like a bare patch of land that’s been razed for construction.  On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 77

[Note from Jan 2011]  We were nonplussed to learn that this site comes up on the first page of Google search on “Urysohn Metrization Theorem”. 
[Note:  This has since changed, owing to a hack attack by the Nominalist Internationale.]
[Metanote:  It's back again, thanks to a counteroffensive by the Realist Underground.]
And ill at ease, since our post of that title is a satire on sociobiological/ultraDarwinistic  overreach, a satire of a sort practiced almost a century ago by G.K. Chesterton in his book The Everlasting Man.  Pity the unsuspecting physicist or math major who winds up there in hopes of learning the first thing about metrization, Urysohn or otherwise.   So we feel we owe it to these blameless Internauts to offer them at least a little something for their trouble.   Here, then, for the non-mathematician, or (God willing) the mathematician-to-be, is a thumbnail sketch of what led to this theorem in the first place. 

The intuitive content of the theorem is as follows.   If you have a space with enough structure to keep things apart which ought to be, and if the space itself is not too huge, then you can define a distance between any pair of elements.  The function that specifies this distance is called the “metric”, from the Greek word for 'measure'.

Thus, you mightn’t be able to do this if you lived in an oozy sort of world, where the minimal entities were like blobs with sometimes inextricably intertwined tentacles; nor if your world were scattered among separate universes.

(Footnote:  the idea is that you can come up with a nontrivial metric.  After all, any set whatever can be regarded as a (trivial) metric space, given the discrete topology.)

As for formal statements, these vary somewhat.  Here is a sampling.  (The following assemblage is an atavism from my days as a lexicographer at Merriam-Webster;  we worked from piles of attestation-slips, called "cites".)

John Kelley, General Topology (1955), p. 125:

Metrization Theorem (Urysohn)

A regular T1-space whose topology has a countable basis  is homeomorphic to a subspace of the [Hilbert] cube and hence metrizable.

James Dugundji, Topology (1965), p. 195, formulates it as

In 2-countable spaces, regularity is equivalent to metrizability.

and he labels this merely a “corollary” of

Theorem (Nagata and Smirnov) A topological space is metrizable if and only if it is regular and has a basis that can be decomposed into an at most countable collection of neighborhood-finite families.

The same can be said for the Bing metrization theorem, which likewise sharpens the sufficient condition into one both sufficient and necessary:  “a topological space X is metrizable if and only if it is regular and T0 and has a σ-discrete basis.” (Wiki)

Other formulations:

Stephen Willard, General Topology (1970), p. 166:

Urysohn’s metrization theorem.  The following are equivalent for a T1-space X:

(a)  X is regular and second countable

(b) X is separable and metrizable

(c )  X can be embedded as a subspace of the Hilbert cube.

James Munkres, Topology: a First Course (1975), p. 217:

Urysohn’s metrization theorem.  Every regular space with a countable basis is metrizable.

Michael Henle, A Combinatorial Introduction to Topology (1979), p.  283:

Metrization Theorem (Urysohn)

A compact Hausdorff space that is second countable is a metric space.

That one uses a stronger condition to reach the same conclusion, and is thus a weaker theorem; the same version appears here:

Boto von Querenburg, Mengentheoretische Topologie (3^rd edn.  2001):

Ein kompakter Hausdorff-Raum is genau dann metrisierbar, wenn er eine abzählbare Basis besitzt.

Something of an odd-man-out, possibly importing the stronger “normality” condition from the Urysohn Lemma, is this:

Seymour Lipschutz, General Topology (1965), p. 142:

Urysohn’s metrization theorem. Every second-countable normal T1-space is metrizable.

But cf. this:

George Simmons, Introduction to Topology and Modern Analysis (1963), p. 138, which offers a slightly stronger version, and names it differently:

Urysohn Imbedding Theorem.  If X is a second-countable normal space, then there exists a homeomorphism of X  onto a subspace of R-to-the-infinity, and X is therefore metrizable.

And indeed, the Lipschutz formulation is echoed much more recently in the October 2010 American Mathematical Monthly (“A Tale of Topology”, by Gerald Folland):

    Every second-countable normal space is metrizable.

If all that  already makes sense to you and seems obvious, you’re done.  If not, read on.

*

            The first order of business is to motivate the theorem.   What does it mean for a space to be metrizable, and why should we care?

            The space we’re best familiar with is the one we live in;  but the one we have studied most analytically, traditionally in high school geometry class, is the nice flat one, called the Euclidean plane.  This we studied  first by Euclid’s own methods, which date back over two thousand years, with axioms and proofs that justify each step -- the best possible mental exercise -- and lots of diagrams.  Later (if we stay the course) we take up a new approach, using analytic methods, which largely began with Descartes, in the seventeenth century.   Here we add a grid of axes, which measures exactly where each point is and how far apart they are, and prove things about figures: now not just triangles and circles and rectangles, but hyperbolas and cycloids and any shape we want, by means of equations.  You don't have much in the way of equations with Euclid;  for that, you need numbers -- given by the metric.

            And lo -- already, in these simple memories of high school, we have, in miniature, a picture of what has happened at the forefront of mathematical research over the past century or so.    For geometry,  in the sense with which you are all familiar, came to be generalized to a new subject, topology (originally called analysis situs -- both mean ‘the study of place’, as geometry means ‘the measuring of the earth’).   Whereas the Euclidean plane is rigid, we let these spaces get all stretchy and bendy.  In that case  we can no longer say what the circumference of a circle is, because by the time we wake up in the morning it may have stretched and drooped like one of Salvador Dali’s watches (in his painting, “The Persistence of Memory”):  but some things do remain true, such as the fact that that curve has an inside and an outside, meaning you can’t get there from here without crossing that curve.  (Note:  Such entirely general, almost naively simple-sounding statements are typical of topology.  The content of that one is called the Jordan Curve Theorem, and it's a real bear to prove.)

            Now topology was originally point-set topology, which mentally is rather like Euclid’s geometry:  you set up the ground rules for a space, then you ponder and visualize and reason things through, using pictures if you possibly can, and your own intuition.    Meanwhile, behind the scenes, a new view of topology was taking shape, somewhat analogous to what Descartes did for (or to) geometry:  instead of reasoning, half-intuitively, with spaces and shapes, you come up with an algrebra whose structures manage to reflect what is going on in those spaces in more detail, yielding numbers and equations and things you can calculate with.   It could have been called “analytic topology” by analogy with “analytic geometry”, but instead it is called algebraic topology.    Though very powerful, it is somewhat bloodless (at least for the beginner), and requires different habits of mind.   (Habits I alas lack.  Readers of my tales of woe will recall my bruising encounter with that subject;  the spot still smarts  in frosty weather yet.)

            So:  Cartesian geometry takes us, from shapes,  to the antecedently familiar realm of equations involving numbers.   Homology (a part of algebraic topology) takes us from more general shapes to the relatively tractable algebraic structures called Abelian groups.  

            The distance function in Cartesian geometry is what you get from the Pythagorean theorem.  The criteria for the general topological notion of a metric are a straightforward abstraction from this:  mainly, if you make a beeline from here to there, and another beeline from there to yonder, the distance traveled must be at least as much as had you simply gone straight to yonder from here.  As to what-all can meet the criteria -- ah, there lie surprises.

            Euclidean geometry is described as what you can do with a straight-edge and compass.   Sometimes people say “ruler” and compass, but that is a mistake:  we have no measurement-markings on our straight-edge; there are no pre-established units of measurement.   We can still determine that two different line-segments are the same length -- just take our trusty compass, measure the first segment with it, and now see if that compass-setting matches the endpoints of the second segment.  You might say that, in this world, length itself is not absolutely defined, whereas being-as-long-as is.   (This observation could be pursued in a syntactic direction -- that of incomplete symbols -- with interesting results.)

            Now, in the Cartesian approach -- analytic geometry -- we want to work with actual numbers, because that speeds things up.   So we turn the plane into a metric space -- “metric” just means ‘measurement’.   And the reason it is possible to do so is that the (pre-Cartesian) Euclidean plane was already rigid:  you do not change the length of something simply by moving it about; you can slide one triangle over to another one and see if they’re congruent.   

            Furthermore, the way we shall conveniently measure things  was already suggested to us by the celebrated truth of Euclidean geometry, expressed in the Pythagorean Theorem.   The earlier formulation of this was:  “the square on the hypotenuse is equal to the sum of the squares on the other two sides”:   meaning, the area of a square figure,  one of whose sides is the long side (the ‘diagonal’) of a right-angled triangle,  is equal to the sum of the areas of two other triangles likewise sticking off the shorter sides that are perpendicular to each other.    This was still a ‘point-set’ geometric view.   But now we start writing it in symbols, saying that, if x is the length of the one leg, and y is the length of the other, then the length of the diagonal is the square root of the sum of x-squared plus y-squared.  (This is known as the “Euclidean metric”.)  That’s algebra.  And the new viewpoint is reflected in the way you’ll here the theorem quoted nowadays: “the square of the hypotenuse is equal to the sum of the squares of the other two sides”:

            The rest  you are familiar with.  We briskly mark off a bunch of equal lengths along one direction, which we call the x-axis, and likewise along the y-axis that is perpendicular to it, and from this we get graph-paper, with its familiar grid.   And now our old friend the plane, which we first came to know as a tabula rasa -- the plane itself, and our own childish minds -- wears its Metric Space status on its sleeve, so to speak.

            This is all so familiar, that we are in danger of letting our memories do the thinking for us.   For in fact there are many different ways of deciding to set of a scheme of measurement on a flat surface.  We might stipulate that the ‘distance’ between two points shall be simply whichever is larger, the difference in the x-value or the difference in the y-value.  Or we might say instead that the distance shall be the square root of the difference of x-squared and y-squared, rather than their sum.   This is what Minkowski did, and it turns out to be the key to uniting space and time into a single Metric Space -- spacetime.   These and others are alternative possible metrizations of a plane.  (In one of them, it is possible to draw a round square -- the paradigm example of what philosophers tell us is impossible.   You can read about one here.)

            Of course, we don’t ourselves live inside of piece of paper (as Flatlanders do), we live in nice big rooms -- three dimensions rather than two.   We set up a third axis, the z-axis, like a tent-pole, to give us some breathing-space:  and now the grid shapes are little cubes instead of little squares.  Using this Euclidean metric, it turns out that an analog of the Pythagorean Theorem holds here as well:  we can consistently define the distance as the square root of the sum of x-squared plus y-squared plus z-squared.   Analytic geometry proceeds as before, with barely any change in methods (I originally wrote, "since the ones we used in the two-dimensional case were already so powerful"; but actually that puts the cart before the horse:  it is precisely such (unexpected) generalizability of a method that leads us to call that method 'powerful'.).    So now, instead of just circles and parabolas and so forth, we have a richer world of shapes, like cones and spheres and ellipsoids and helices, and on and on.  (Actually these were already known to the Greeks, though how they managed it with the pre-Cartesian methods they had, is something of a miracle.)  And it continues to be easy to prove things about these, since we still have basically the same metric, which is well adapted to equations and their numerical solutions.    Likewise in four dimensions, and on up as high as you like.
(Note:  People get all spooky when they hear things like 'fourth dimension', but these metric methods absolutely tame them. -- Children:  Study math.)

            Bottom line:  A metric space is a very convenient thing to work with.  You can pretty much know where you are and do what you want, even when the space gets hairy in other ways, like being infinite-dimensional, or very curvy.

            But.

            In the inexhaustible splendor of the Creation, there are many many different spaces, more numerous than the stars.    They sprang full-blown from the Creator’s brow, and it is up to us to discover their structure.  Unfortunately, when we first meet up with one of these, it may not be wearing a nice convenient metric on its sleeve.  It may be a very confusing, huge, menacing, squishy blob.  -- Recall that when we first met the plane, it too came without a pre-drawn grid.  But a particular grid was already implicit, because Euclid’s axioms imply the Pythagorean theorem.   (There are other metrics you could adopt where that theorem wouldn’t be true, but these would not conform to our everyday local experience, which is why Euclid chose the one he did, and why it took two millennia to generalize the naive notion of "distance" to the mathematical notion of "metric".)

            So:  Faced with such a blob, can we come up with a consistent measuring-scheme that will tame it, by turning it into a metric space?  That is, is it metrizable?  -- And the answer is:  Sometimes you can, and sometimes you can’t.   So, we want to come up with ways we can tell, whether the project is doable or hopeless.  It’s a bit like figuring out whether you can tame a given kind of animal.   Long ago, people figured out that you could do that with dogs, and later with horses, to a huge extent, so that instead of being wild beasts they are actually useful.  Cats, it turns out can be tamed to a lesser extent -- tamed to tolerate us, so long as we are not late with the cheeseburgers. They’re not useful, but they’re decorative.  (Meanwhile in Catland, the lecture reads:  “Peeps are redonkully e-z 2 tame;  goggies, not so much.”) So, you see a puppy and, no matter what breed it is, the mere fact that it is Canis familiaris tells you that you have an excellent chance of taming it:  though just how to do so may vary with the breed.  In a similar fashion, when we first meet an untamed space at the Space Store, we can know, by certain signs (which Urysohn specifies in his theorem) that the thing is in principle metrizable -- though it doesn’t give us a useful metric just for free;  for that we still have to do some work.

*

            If you and I are points in a metric space, the metric tells us how far apart we are:  that’s like analytic geometry.   But topology lets out the sails a bit.  In the most general topological space, you can’t say how far apart two points are -- but you can always say what bunks with what.  The bunks are known as “neighborhoods” or (roughly synonymous) “open sets”, and are given as part of the very definition of the space.   Thus, the most featureless space of all, justly (in this metaphor) called the “indiscrete” space, everybody bunks with everyone else, no privacy at all.  In the opposite, the “discrete” space (discrete, not discreet), each man is an island.  But most spaces, and every space of interest, is in between.

            To describe just where they fall on this in-between spectrum, we name various “separation” properties.   The simplest common one is that any two points can be separated (any two people can sleep in separate bunks).  In a metric space, this is easy.  If you and I are a certain distance apart, then if I draw (if we’re in a plane) a circle or (if we’re in a fatter space) a sphere  around me, with a radius less than that distance, then I’m in my own special bubble and you’re outside it.   Such elementary capacity of separation is also available in most non-metric spaces. This basic degree of separation is called T1.  If you and I can each draw such a neighborhood simultaneously, even better -- the space is “Hausdorf”.

            A more demanding requirement is that I can fix around myself a bubble (a neighborhood) which keeps me clear, not of just a single point, but of any collection of points called a “closed” set.  A set is closed if, for any point you can creep up on, in an infinite sequence, that point is in the set.  So for instance, on the number-line, take all the reciprocals of the natural numbers, ½, 1/3, ¼, etc:  these creep up on zero -- they get as close as ever you please, so the set of these reciprocals is not closed:  to close this set, you have to add zero.  Or, take everything inside a circle.  You can creep up to any point on the boundary, from within the circle, so the interior is not closed.  Add the boundary, now it’s closed.   -- So:  given a point, can that point stay clear (hide inside a bubble), not just from any other point (that’s easy), but from any other closed set that doesn’t contain that point -- no matter how pushy and encroaching?   Well, if that set is closed, it can’t keep creeping up on me indefinitely:  at some stage, it can’t come any closer, otherwise I’d be a limit point of that set, and since I’m not in that set, it wouldn’t be closed.  So at some point it keeps its distance -- I’m safe in my neighborhood (say, Beacon Hill), where the menacing set cannot encroach.  (In a metric space, this is easier to visualize:  it keeps its distance -- say, d.  So I draw a little bubble round me, of radius smaller than d, and I’m safe.(  -- Spaces that are like this are called regular.  Obviously, every regular space is Hausdorf, but not vice versa.

            (The next step up is:  Can any two closed sets -- not just one closed set and a point -- be kept apart by disjoint neighborhoods (neighborhoods that don’t intersect)?  If so, that space is called normal.  Normality is used in the Urysohn lemma, basically unrelated to the UMT.)

  

            So, being regular suffices to keep things separate enough to define distances between things.  But regularity by itself is not sufficient -- the space might be too ‘big’ to fit in a metric.  How big is too big?  Bigger than second-countable, the other premise of the U.M.T.  In that case, points can be just too far apart to have a finite distance.

*

None of these definitional and formal considerations  gets across the real power of the metric-space idea.   This arises when we begin to consider a more abstract sort of space, in which the “points” are not characterless, dimensionless ideal dots, but … functions.   (This is an example of the “Ladder ofAbstraction”.)   Andrew Gleason puts the matter well:

The assignment of a metric to a set of functions  gives this set an intuitively geometric character.  The success of the theory of metric spaces in analysis  can be attributed to the remarkable insight into the nature of functions  which has come from exploiting the geometric point of view.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p.  226

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