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

[ continued here …]

Realism: What

[We choose a classic Lockean title-style for this chapter, and begin with a middle-of-the-road definition.]

Realism Defined

First, a plain-man's characterization of what we might call minimalist/core/plain-vanilla Realism:

Outside our heads  there is freestanding reality.  Only madmen and a scattering of constructivist philosophers doubt its existence.

-- Edward O. Wilson, Consilience (1998)

This view may also be referred to as Platonism (though cf. a special restriction of this term  to mathematics, outlined below):

One issue that has traditionally divided philosophers  is whether ther are abstract objects.  Nominalists have held that there are not;  realists (in a special sense of the word) or Platonists (as they have been called  to avoid the troubles of ‘realist’), have held that there are.

-- W.V.O. Quine, Word and Object (1960), p. 233

Now for the more careful distinctions of professional philosophers:

A. E. Taylor, Elements of Metaphysics (1903; page references to the University Paperback reprint), p. 67:

By Realism is meant the doctrine that the fundamental character of that which really is, as distinguished from that which is only imagined to be, is to be found in its independence of all relation to the experience of a subject.  What exists at all, the realist holds, exists equally  whether it is experienced or not.

Within this, he distinguishes (p. 68) two subtypes:

Agnostic Realism, while asserting the ultimate dependence of our experience upon a reality which exists independently of experience, denies that we have any knowledge of the nature of this independent reality.

Sic:  not “full knowledge”: any knowledge.  Which is absurd.

Contrast (p. 69):

Dogmatic Realism, of which Leibnitz and … Herbart are the most important representatives … while maintaining that real being is independent of experience, at the same time  holds that it is possible to have positive knowledge  not only of its existence, but of its nature.

Sic:  postive knowledge, but not necessesarily full knowledge.  And in this form I heartily subscribe, despite the invidious label conferred by its opponent Mr. Taylor.   As William James put it, in The Principles of Psychology (1890), vol. II, p. 634:

Reality exists as a plenum.  … But we can neither experience nor think this plenum.  What we experience, what comes before us, is a chaos of fragmentary impressions  interrupting each other;  what we think is an abstract system of hypothetical data and laws.

Full knowledge -- plenary knowledge of Reality in all its fullness -- not only in some rarefied ding-an-sich sense, but in the ordinary sense of the sciences -- is doubtless impossible in the case of anything so wildly complex as an acorn or a rock;  but in the case of something simple, like Hilbert Space (infinite-dimensional, it is true, but tamed by a norm that is based upon an inner product), we can perhaps come close to exhausting most of what can be known of it.

Taylor fancies he has refuted Realism in all its varieties, summarizing his triumph thus:

Produce any instance you please, we said to the realist, … and we will undertake to show that it derives its reality for you  from the very fact that it is not ultimately separable from the experience of a subject.

Note the crucial bait-and-switch!   He whisks Reality -- an ontological category, and a big deal -- under the thimble, and takes out -- “Reality for you”, a miserable psychological gewgaw of no general interest whatever.   This latter concept is, we readily concede, drenched in irreducible subjectivism;  no need to bother to argue the point.  And bearing, it may be, little relation to Reality, whether theoretically or empirically.

For:  Reality, in reality-for-X, is an incomplete symbol.

Thus for instance:   It is not possible to survey the roster of extant men, and finally lay your finger upon one specimen, the average man.  The Average Man is not a real, but an ideal, to which various actual men may approximate to one degree or another.  And Real-for-Joe-Blow is not a real, but a figment, which may reflect more or less of an actual reality, with greater or lesser distortion.

~

~  Posthumous Endorsement ~

"Were I alive today, and in the mood for a mystery,

this is what I would be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(I am John Locke, and I approved this message.)

~         ~

~

A more carefully phrased importation of subjectivism into the debate  is provided by Michael Dummett, in “Truth” (1959), repr. in Truth and other enigmas (1978), p. 23f:

The claim … should be rejected by a realist, who might, and I think ought to, agree to the following weaker principle:  that a statement cannot be true unless it is in principle capable of being known to be true. […]  The anti-realist interprets ‘capable of being known’ to mean ‘capable of being known by us’, whereas the realist interprets it to mean ‘capable of being known by some hypothetical being  whose intellectual capacities and powers of observation  may exceed our own.’ … The issue between realism and anti-realism … is one of the most fundamental of all the problems of philosophy.

Thus here, clearly, we are once again confronted -- like it or not -- with the question of theism.

So:  if you are a full-bore, double-barrelled, two-seed-in-the-spirit, dyed-in-the-wool copper-bottomed no-holds-barred Cantorian Realist … (we pause for the roars of approval to subside) … even in merely so much as mathematical Platonism:  must you therefore, of logical necessity, be a theist in the sense of the Abrahamic tradition?

Perhaps not; certainly, we have not shown so.  (Nor do we so much as aim to show more that that.   At this point, Jews Muslims and Christians are all singing kumbaya together in one big tent.)   You could, in principle, construct for yourself an ontological halfway-house, with a non-omnipotent, though omniscient, Knower.   And who might this Wiser Being be?  Why -- none other than Babar, the Elephant King!

*

In these essays, we are concerned mainly with the form of Realism known (in a mathematical context) as Platonism, defined as

the theory that mathematics describes a realm or system of real and independently existing objects, whose nature is known to us through proof, but which are entities  over and above the proofs by whch we discover them.

-- Roger Scruton, Modern Philosophy (1994), p. 384

(This is fine, but don’t let’s squabble about “objects”; “patterns” or “truths” would do just as well.  We’re not interested in reifying anything, but merely in claiming that we’re not just making it all up arbitrarily.)

And:

… the subject matter of mathematics as realistically (i.e. platonistically) construed.

-- Colin McGinn, “Truth and use”; in: Mark Platts, ed.  Reference, Truth and Reality (1980), p. 35

 But for the record, there are other sorts:

Do we wish to say that there is a moral reality, which underpins our moral judgements  and guarantees their truth?  Some philosophers have argued for such a view (‘moral realism’).

-- Roger Scruton, Modern Philosophy (1994), p. 98

Scruton himself is far from being a moral relativist or nihilist, but seems not to feel forced to accept such moral realism, at least in the form of what we might call a moral Correspondence Theory (as opposed to the real existence of God and his laws).    Indeed, he compares (we shall call it) Aesthetic Realism:

It is obvious that St Paul’s Cathedral is beautiful, and the new Lloyd’s building  repulsive;  but is there some ‘aesthetic reality’ that makes these judgements true?

You readily see how problematic this is:  Saint Paul’s to a Wahhabi or an Iconoclast would lack appeal; and presumably the people who designed and paid for the Lloyd’s building  felt it had a certain something.   I still think a kind of case might be made out (which I’ll not pursue), that it’s not just all a matter of personal preference, and that de gustibus certe dispundandum est.   The argument would proceed by analogy with the much tighter case in mathematics.  Many things are obvious to the expert which are not obvious at all to the general public, and which indeed could never be made obvious to them no matter how hard you tried (sheaf theory, for starters).  And more tellingly,  the Man on the Clapham Omnibus will pronounce some things obvious (particularly in the area of probability, e.g. the Monty Hall problem) that are demonstrably mistaken (though some will understand the demonstration, and some will not).    We do not conclude from this that mathematical truth is just a matter of taste (“we” here referring of course to men of good sense, and excluding the Postmodernists).   Some are more qualified than others to grasp certain truths.    Quite possibly the same is true in the field of aesthetics, as regards, say, the Goldberg Variations, on the one hand, and “Who Let the Dogs Out”, on the other.   One would have to make out the case that those who exalt the latter and contemn the former have certain other things wrong with them as well.

[Update]   I have just come across an intriguing analysis relevant to the case of the “Lloyd’s building”, and that buttresses the Aesthetic Realist conjecture above, to the effect that it’s not all just a matter of mutually incomparable ‘tastes’, but that there are objectively different levels of aesthetic competence, owing to the importance of informed appreciation.  The analysis appears in a delightful essay, “The Gherkin”, by John D. Barrow, in his book 100 Essential Things You Didn’t Know You Didn’t Know (2008;  the book is better than its title).  The “gherkin” in question is a new building in the City of London, a.k.a. “the Pine Cone” for its pocked appearance:   “Prince Charles sees it as symptomatic of a rash of carbuncular towers on the face of London”.   Yet it won the Stirling Prize for architecture.  What gives?

It turns out the building is, from an engineering standpoint, quite ingeniously designed, and that principally with a view towards eco-friendliness.  Thus it narrows somewhat at the bottom, to defeat the wind-tunnel effect that annoys pedestrians; and tapers again towards the top, which “opens up more of the sky and reduces the dominating effect of the structure because you can’t see the top from close-by on the ground.”   Even the pocks have their point:  “They bring light and natural ventilation deep into the heart of the building”, saving greatly on energy costs.  And more details in this vein.

I’ve never seen it, but it sounds … beautiful …

~

Realism in psychology:

According to his friend and biographer, Freud “had a high and serious respect for the reality of psychological facts.  They were as real and concrete to him  as metals are to a metallurgist.” (Ernest Jones, Freud: Years of Maturity (1955), p. 432)

[continued here]

Further juicy quotations:

As a Platonist, he saw everything on earth as broken arcs, which merely suggested the perfect rounds above.

-- Louis Auchincloss, The Rector of Justin (1964), p. 92

Realists [with a capital R] are not the same thing as ‘realists’ in daily life, who are men who expect neither themselves nor others to be any better than they ought to be, and generally much worse.

-- Ernest Gellner, “The crisis in the humanities” (1964), collected in The Devil in Modern Philosophy (1974), p. 15

To mathematicians who study them, moduli schemes are just as real as the regular objects in the world.

-- David Mumford, Forward to Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. xi

REALIA & IRREALIA

[This is a continuation of a thread begun here.]

            How, then, are the prime numbers – or Hilbert space – different from a unicorn?

            (That reads like a joke, to which the reply would be:  “They lack a horn.”
Or:  Hilbert and this unicorn  walk into a bar… Yet the question is meaningful enough.)

            Unicorns do not (alas) exist, or so I’m told; but people do talk about them, and some (such as myself) may even believe in them.  Descriptions and depictions do exist, which purport (wrongly; or tongue-in-cheek; or in some extended sense, accessible only to those who believe in snow-bunnies) to depict them – in all their monocerous, silver-shanked magnificence.  (There are worse purposes, to which silk or ink might be put.)
A picture-of-a-unicorn (so goes the rote) is as much an unfissionable entity as a hotdog, which does not split neatly into a hot and a dog:  what at first glance seem  components (<picture> + <unicorn>)  are in reality synsemantic (“incomplete symbols”).  A picture-of-a-unicorn differs from a picture-of-a-phoenix, but you cannot extract the quantifier and give it wide scope:  there exists no thing which eíther is a picture of.  When we say (straining the usage of “exists” just a bit) that “The unicorn exists in heraldry”, we mean, nothing ontological, but merely: If you wish to find a picture-of-a-unicorn, go to the heraldry section of your local library,  and not to the section on pets.

            Okay, so no unicorn; darn.  What about its Meinongian doppelgänger, the “Idea of a Unicorn”?  The modern consensus, to which I overtly subscribe (with certain arrière-pensées concerning the unicorns themselves, whose hooves, like flint, strike fire from the stones  whereon they gallop, never heeding the wind) declines to admit that either unicorns or the “idea” of a unicorn does exist, in any concrete or meaningful sense – that is, as a thing with some independence, an idea that is there, somewhere, even if no-one is presently thinking it; a thing, so to speak, with a front and a back.  (And here I do truly, and passionately, agree:  if the forests lie sad and silent, unroamed by unicorns, then away with such rubbish as the “unicorn-idea”!)   I may be thinking of unicorns (as indeed I usually am), or entertaining (second-order) beliefs about them (their mane is like silk, like sand), but this is more like a verb than a thing.  (Again: There exists no unicorn, whereof I am thinking; I am merely daydreamicorning, unicontemplating…)  People may well continue to entertain ideas about unicorns (and by now you can supply the punctilious punctuation: entertertain-[ideasaboutunicorns]) so long as the (foolish, folk) tradition is passed down:  but once let the Earth disappear in a puff of smoke (which, bear in mind, it may do at any instant), and all ideas of unicorns die with it. (And here, though at my most sentimental, I make no objection:  They lie, with Thor, in a common grave.) Travelers from another dimension, roaming the ruins of our world, will never encounter a trace.

            The case with integers is quite otherwise.

            Let all sentient beings perish in the Big Crunch (this is meant, not as an optative, but as a [contrafactual] hypothesis); then let the universe evolve anew.  There may or may not, in that cycle, evolve any rational beings at all; or they may evolve, but be such sobersides as to have (like your boss) no interest in fables and fairy tales; or they may develop some folklore of their own – in all likelihood, nothing resembling a unicorn (these new rational beings  being themselves, for one thing, perfect spheres;  mothers frighten their children with ellipsoids):  yet let any one of them turn his hand (or flipper) to enumerating the stars of the firmament, or the ways in which he loves his sweetheart:  and the precise same integers as before  will stream to his aid.

(Yet let us pause for a bit, for I am seized with sudden sadness.  In principle very glad, that our sturdy friends the integers  survived that  cosmo-catastrophic transition;
and yet I do lament,
and sorely miss,
the brightness and that brashness,
of those   proud
                      white
                                 steeds ….)

*

Let us look  at the same thing  from another angle.  (A nice diagnostic, for things that are really real,  in the round.)
People have been tempted, and were scolded by Russell for so doing,  to say that Hamlet is real  in Shakespeare’s imagination, the way (or at least rather like) Napoleon in ours.  Russell’s retort is classic:

       When you have taken account of all the feelings roused by Napoleon in writers and readers of history, you have not touched the actual man; but in the case of Hamlet  you have come to the end of him. If no-one thought about Hamlet, there would be nothing left of him; if no one had thought about Napoleon, he would have soon seen to it that some one did.

Spoken like an Englishman!

Stop ...  being ... silly ..... ..... ...

I still have a sneaking sympathy with the reality of the gloomy Dane, in the sense that there are propositions about him (/it/whatever) that are true or false in our world, never mind in Platonic paradise, or in that of the late Shakespeare.  If, on a test, you identify Hamlet as the prince of Macedonia, you will be marked down; nor is his ladylove named “Buffy”.  (In southern California, that answer might get you half-credit.)  So, Hamlet is not real (unlike “Hamlet”, a play by the bard of Avon), but…real-ish.  (Compare the concept, discussed elsewhere, of truthitude.)  And contrariwise, the constructs of our world are – mostly  exactly that:  constructs, and thus ideal or even -- fake-ish.  -- I won’t emphasize this latter point, as it has been  if anything  overemphasized of late  by post-modernists and deconstructionsists, not to mention earlier and more honorable  skeptical attacks.  The upshot is simply that the two levels of Things  tend to edge towards  meeting in the middle.  We can still, faced with the following examination-question,

            Choose the odd man out:   

            (A) unicorn   (B) dog    (C) robin

correctly pick (A).  But we must concede that “dog” is a concept held together with duct tape, consisting as it does of an ever-evolving medley of breeds, beginning with one insensibly different from the ancestral wolf, and fanning out into some that may be no longer interfertile, and differing the one from another, in both appearance and behavior, more than do some allied species among themselves.  And as for “robin”, the term in lay use designates polysemically (though not really homophonically, the way “pen” designates both a female goose and a writing implement) a variety of superficially similar species, depending on where you hail from.  In terms of being a “natural kind”, the unicorn may have more uniformity than the canine.

There is another trait, though, which distinguishes more sharply between unicorns and zoologically more respectable species, than the existence of detailed descriptions (some fictional, some factual) of each.  And that is, the ability to reason with them, inductively.  If, on your trip to Shropshire, you meet a particular bird, let us call him Hoppy, which on visual inspection prompts you to call it a “robin”, an ejaculation which in turn prompts vigorous assent from the circumambient peasants, then, even though Hoppy is not actually quite the same sort of bird as what you used to call a “robin” back in Illinois, nonetheless, you’ll be correct in predicting that Hoppy can fly, and probably that he has a palatal penchant for worms.   Whereas, should you chance upon a monocerous equine in your ramble through Sherwood forest, you can conclude almost nothing.  Whether it would, like the unicorns in books, meekly lie down at the sight or scent of a virgin, is anybody’s guess.  (I’m betting:  Yes.   I mean -- I would….)

So now, let’s try our hand at this one:

            Choose the odd man out:

            (A) unicorn  (B) dog   (C) compact Hausdorf space

The answer is still (A); you can reason with the latter two.  And indeed, (C) is in a sense more tangible, and more reliable, than is (B).  Should you ever wake up one morning in a compact Hausdorf space, you can be quite sure that, should you chance upon an infinite sequence of points running along the path of your morning stroll, then that sequence will infallibly converge to a point in the space. And this would remain true, were the space somehow embedded (this time preferably without your presence) as a subspace of Hell.  For though the Dark Prince (whom God defeat) may have power over reprobates, and even the power  from time to time  to tempt the righteous, yet he has no power over topology.

            This reminds us of the old conundrum:  How can there be an omnipotent Supreme Being, who yet has no power to refute the truths of arithmetic, nor indeed any necessary proposition, such as those of topology?  The simplest answer may be, that topology is part of God.  You don’t refute your own arms and legs.

Straightaway (for conscience pinches) let me hasten to explain what I mean and do not mean by “Topology is part of God”.  It’s a nifty epigram (assonance and all), but it can mislead.  What I mean is – well, all that was said above, which is hard to summarize, but in a nutshell: If you consider the Creation to be part of what characterizes the Creator, then (actually, a fortiori) you should consider the principles by which that creating is regulated (topology among them) to characterize Him.  A simple point.  What I do not mean is – anything a breathless journalist might make of all this.  In particular, it does not directly say anything about what matters most to most of us, day to day, when it comes to God:  Does He bid us do this, or that? Might I be damned? Can I be saved? What must I believe?  Need I believe anything? --  If anything, contemplation of the multitude of invisibilium distracts our attention from such questions:  The more our mind must focus on God as Creator (since, as we come to learn, there is so much more to the Creation than Levittown, so much more still than we could ever behold with our eyes), the fewer neurons are left over for contemplating God as Judge, God as Comforter, God as Redeemer.  Fact is, He is infinite, we’re finite. Only so much bandwidth down here.

            Nonetheless, I must insist on this point -- trivial though, in a moral or eschatological perspective, it may be; if only to help correct the imbalance that has gone before. The Bible (meaning: the sum of the Old and New Testaments) dispatches the Creating in a couple of paragraphs (curiously prescient paragraphs though they be).  All the rest of the text just takes the visible world as given, asking after nothing more; and busies itself with guides to conduct, awful warnings, things admirable but not to be tried at home, before finally culminating in Christ, who if anything (despite a philosophical-sounding, if vague, “In the beginning was the Word” in the odd-man-out of the Gospels, John; an apophthegm  in any case  never really developed) is more centered than ever on the relationship of God to Man, not God and the Plan.