The Continuum: Mainstay or Menace? (erweitert)

The Continuum:  the original sin, from whose fecund loins
came all that is non-constructive in mathematics.

-- Anon.

Kronecker dismissed mathematical entities beyond the natural numbers as “Menschenwerk”.  An average practicing mathematician (who uses such entities all the time) may  agree with him to this extent:

(1)  Our intuitions about the natural numbers are clear and solid.   So long, indeed, as one deals only with some set of actual numbers (thus, a finite set), nothing especially surprising  or even all that interesting  turns up.  If we extend our horizon to the actual infinite of the set of all natural numbers, we meet some concepts that take getting used to (Hilbert's hotel):  but once we’ve done so, they seem natural enough.

(2) The rationals and negative integers  definitely, the algebraic numbers  probably, pretty much come along for the ride (that is, you can hardly exclude them once you’ve accepted N), and they still bring in no paradox – being, after all, of the same cardinality as the natural numbers themselves.  Though, a case could be made that these are not “entities” of the same standing as the integers, which in a sense we can hold in our hands (embodied in oranges, say), but rather abbreviations for operations on integers.  Thus, we cannot hold minus-two oranges in our hands; minus-two is not a thing, but a bookkeeping device. 

(3)  The real numbers, by contrast, are … a piece of work.  Maybe even Menschen-work, except that one could hardly imagine Menschen coming up with anything so intricate and even bizarre.  Their very cardinality baffles intuition  -- and the independence of the continuum hypothesis  shows that we are right to be baffled.  [Note:  The simple infinity of the integers already baffles *untutored* intuition;  but eventually you get the idea.  Click on the Label "Hilbert's Hotel" for further exemplification.  Whereas, the cardinality of the continuum is more like... Hilbert's Nightmare...] All sorts of queasy consequences arrive for simple quantification (cf. Quine re.  objectual vs. substitutional quantification).  The reals were invented (discovered?) for purposes of analysis, which in turn was developed largely for the sake of physics: but it now appears that physics (whether in its quantum cast, where Uncertainty provides a certain indissoluble granularity; or in the Wolframesque finite-automata approach) might not actually require, or afford, a continuum.

And yet standard mathematics speaks indeed ontologically of the reals, not merely pragmatically.  Thus for instance, Rudin’s standard text (Principles of Mathematical Analysis, 3^rd edn. 1976, p. 8):

We now state the existence theorem [emphasis in original] which is the core of this chapter.

Theorem. There exists an ordered field R which has the least-upper-bound property.

The author then mentions that the proof actually constructs the Reals out of the Rationals.  This is, of course, the most solid sort of proof of all – not one of those Cantorian diagonalization thingies that has you winding up assenting to the Infinite Woodchuck, without ever quite knowing how you got into such a fix.  It gives you an actual recipe for the construction of these extended numbers, as concrete and explicit as for baking a cake.  And yet… all kinds of things can be thus “constructed”, at will, including items which presumably are not part of the furniture of the universe, in the sense that angels actually sit on them.

~

A roaring vote of confidence in the continuum  is voiced by the noted mathematician René Thom:

“God created the integers and the rest is the work of man.”  This maxim spoken by the algebraist Kronecker  reveals more about his past as a banker who grew rich through monetary speculation  than about his philosophical insight.  There is hardly any doubt that, from a psychological and, for the writer, ontological point of view, the geometric continuum is the primordial entity.

-- “’Modern’ Mathematics: An Educational and Philosophic Error?”, in American Scientist (1971), repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 74.

That is in-your-face Platonism, with which, quâ Realism, we have no quarrel.  But the psychological claim seems dubious:  Our intuition of the continuum is probably no more than a vague notion of a smear (and not very infinite at that, neither going out nor going down).   And as for the ontology … When we first meet the Real numbers mathematically (that was the very first thing we did in first-year calculus, with the opening chapter of Spivak’s text), we conceive them as the completion of the rationals.  And such they are indeed:  only, with respect to the metric provided by the absolute value.   With a p-adic valuation, you get a different completion of the rationals, the p-adic numbers.   Lastly, the surreal numbers augment the continuum in yet a different unexpected direction.  (I have less than no intuition about any of this.)

The physicist Schrödinger is less sure:

The idea of a continuous range, so familiar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is really accessible to us.

-- Erwin Schrõdinger, “Causality and Wave Mechanics”, repr. in translation in: James R. Newman, ed. World of Mathematics (1956), p. 1059

And from an Intuitionist (close kin to a physicist):

This could be done  by seeing the continuum as something that is infinitely becoming, instead of already being.

-- Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 333

(Compare our old friend the actio/actum distinction.)
Might be fine for physics, doesn’t work for math.  ‘See’ it however you like; that uncompleted-account doesn’t jibe well with Cantor-style constructions.

~

One might say:  The continuum feels unproblematic enough, so long you take it for granted, as just some kind of smooth dense slippery thing, like mud.  Yet so soon as you pause to enquire more nearly, you are back in Saint Augustine’s predicament with regard to Time: “Quid est tempus? Si nemo a me quaerat, scio …”

~

Even in a universe which (like Wolfram’s) abjures the continuum, the continuum might turn out to be mathematically indispensible for its treatment.   Cf. the indispensible role of “imaginary” numbers in electromagneticsm or quantum mechanics, even though all observables must be real-valued.

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

On “The Nature of Mathematical Knowledge” (enlarged)

That question is about as interesting as the nature of our knowledge of elephants.  We are interested in the zoology of elephants, not in the specificities of classroom biology lessons, or the economics of zoos.  We are uninterested in each blind man’s subjective and partial report upon the individual organs of these splendid creatures.

Mathematical knowledge, like pachydermal knowledge, is imperfect knowledge of something real that exists independently of us. By contrast, just which images we manage to form of these objects  are very much dependent upon ourselves – and to that extent, of interest only to unemployed social workers.

As our former math teacher put it:

Mathematics has a real content which transcends the inadequacies of our efforts to formalize it.

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

Or, from a philosopher:

For most mathematicians most of the time, having a feel for what is evident is important, but it is also enough: There is no further need for a theory of what that feel is.

-- Shaughan Lavine, Understanding the Infinite (1994)

*

Actually, it should not be deduced from the above, that I am somehow dissing our big friends the elephants.  A Theory of Elephants is considerably more promising than that idle grail of the physicists, a Theory of Everything.

The outstanding problem in the Theory of Elephants is the ontological status of BABAR – THE KING !!!!!

His Majesty ... The King !!!

An agnostic or non-Realist version, wilfully po-faced:

Our deductivist proposes clean answers to philosophical questions.  What is mathematics about?  Nothing … What is mathematical knowledge?  It is knowledge of what follows from what.  Mathematical knowledge is logical knowledge.

-- Stewart Shapiro, Thinking about Mathematics (2000), p. 150

*

*

There is a systematic ambiguity (roughly that of actio versus actum) to the term mathematics:

(I) The praxis of mathematizing.  This is a human pastime, comparable to needlework or basketball.

(II) The truths of mathematics.  Or, more or less synonymously: The real (though invisible) world.  These, in themselves, bear no dependency upon human practice or to any species whatever;  they existed before we were born.

The above may count as a polemical reformulation of roughly the dichotomy in the title of Hao Wang’s fine essay, “The Theory and Practice of Mathematics”.

*

In 1950, Raymond Wilder gave an address, “The Cultural Basis of Mathematics”, reprinted in  various places, and later wrote a whole book on the subject, Mathematics as a Cultural System (1981).   Bien-pensant commentors treat these with grave respect;  but the notion is practically nonsense.  For, if we take mathematics in sense (II) – the only sense of interest to us here – that is like saying “The cultural basis of elephants”:  there is none.  There is a cultural basis of circus stunts, of mouse- and peanut-myths, of Dumbo, but not of elephants themselves.  Their basis is their own four feet.

Why should the culture of mathematics  (necessarily, sensu I) – that is, the foibles of mathematicians – retain our attention?  The purely “human side” of mathematicians  is generally less interesting than that of country music stars.  A lot of mathematicians are pretty Asperger’s, frankly.
(For a poignant illustration of this, read The Genius in My Basement.)

There is, we grant, a certain interest in the sociology of mathematics, or in biographies of the great mathematicians. Intellectually, it is on a level with gossip about the off-court antics of basketball stars.  Fun, but of no mathematical (or basketball) interest.   It’s just a way for the mind to chew gum while it’s too exhausted to do anything substantial.  To get real, do math (or play basketball).

Not to come down too hard on the small geeky community that does follow the doings of math and physics whizzes; I number myself among them.   It would even be neat  if, instead of collecting baseball cards, people collected mathematician cards (“Trajea two Steven Smales for a John Milnor!”) .  -- By “people”, I here mean “eight-year olds”.

*

More interesting is the purported “reduction of mathematics to logic”.  It is not initially clear, however,  to what extent this program, if successful on its own terms, would enlighten us as to mathematics-sensu-(II), as opposed to the sense-(I) territory of our own mathematical formulations and formalizations (these being, after all, largely for mere convenience).  It might be more along the lines of the demonstration of the equivalence of the Heisenberg-style matrix-mechanics formulation with that of the Schroedinger-style wave formulation, of quantum mechanics. That feat didn't tell us all that much about the actual phenomena of physics,  apart from the fact that the world is a many-splendored thing, and can be described -- blind-man-fondling-elephant-fashion -- in a variety of ways.  It’s more like deciding whether today’s symposium shall be conducted in English or in French.

*

That said --

We argued here that the axiomatic method is cognitively post-hoc, and that  only in cases where (as with the Euclidean axioms) their positing is transparently motivated by our experience of the sensible world, is a top-down, axiomatic presentation  pedagogically sound.   Thus similarly in physics:

In lecture after lecture, and essay after essay,  Einstein began, not with an introduction to the subject at hand, but with an overview of how he’d arrived at that subject, or of how scientists in general  arrive at subjects in general. … For Einstein himself, the results of science had become incomprehensible without an understanding of the processes that led to them.

Richard Panek, The Invisible Century (2004), p. 153-4

C'est exact;  and the farther physics wanders from our human experience, and the father math develops beyond anything the world has seen before, the more necessary such a psycho-cognitive ladder does become.

*

Something like the dichotomy outlined above  must have been behind André Weil’s tart remark, in “History of Mathematics” (reprinted in Collected Works v. III as (1978b)):

Some universities have established chairs for “the history and philosophy of mathematics”;  it is hard for me to imagine  what those two have in common.

For:  the one is situated and contingent, the other timeless and beyond place.

*

Footnote:   These remarks about mathematics  apply  mutatis mutandis  to the Deity.  Deliberately confusing the distinction between truth and praxis, Karen Armstrong wrote a book -- a minor best-seller -- with the impudent title A History of God.  (At least she put A, not The; probably saved herself an extra millennium in Purgatory right there.)

*

Lakatos’ classic dialectical-dialogue Proofs and Refutations (you see the Hegel-style paradox already in the title), though focussing on the (as he persuasively argues, in the course of a detailed case-study spanning many decades) micro-level mess of actual mathematical progress, is yet Realist at its core:  the subtitle is “The Logic of Mathematical Discovery”, not “The Sociology of  ‘Mathematical’ Invention”.   We quoted him in another context  thus:

As far as naïve classification is concerned, nominalists are close to the truth when claiming that the only thing that polyhedra have in common  is their name.  But after a few centuries of proofs and refutations, as the theory of polyhedra develops, and theoretical classification replaces naïve classification,  the balance changes in favour of the realist.

-- Imre Lakatos, Proofs and Refutations (1976), p. 92

In an appendix to the main work, he offers a Hegelian formulation, one which (by the time the reader has progressed this far) has a certain paradoxical piquancy:

Mathematics, this product of human activity, ‘alienates itself’ [in the sense of Hegel and Marx] from the human activity, which has been producing it.  It becomes a living, growing organism, that acquires a certain autonomy [emphasis in original] from the activity that produced  it.  … The activity of human mathematicians, as it appears in history, is only a fumbling realisation of the wonderful dialectic of mathematical ideas.
-- Imre Lakatos, Proofs and Refutations (1976), p. 146

Plato, in his Paradise, smiles.

An update re Mathematical (and Natural) Definition (re-updated)

[Continuing this essay:

http://worldofdrjustice.blogspot.com/2013/12/theorems-propositions-dumb-questions.html ]

While we’re on the subject, let us consider further the question of definition in mathematics.

Re Hilbert’s approach to the axiomatization of geometry:

Rather than defining points or lines at the outset  and then postulating axioms that are assumed to be valid for them, a point and a line were not directly defined, except as entities that satisfy the axioms postulated by the system.

-- Leo Corry , “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 139

This is not quite so radical or ‘post-modernist’ as it might sound, since traditional grammar recognizes many analogous cases in natural language, under the rubrics of synsemantic,  syncategorematic and implicit definition.  [See posts with the Label "incomplete symbol".]  It is a relative notion, with a sliding scale;  but analysis will suggest that a very large set of words and multiword expressions (as, the use of a word in an idiom, especially in an opaque idiom) partake of some degree of syncategorematicity.   However, in the particular perspective of mathematics, this idea harmonizes especially well with a logicist or formalist approach to the subject:

The use of undefined concepts  and the concomitant conception of axioms as implicit definitions  gave enormous impetus to the view of geometry as a purely logical system.

-- Leo Corry , “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 139

Again, this is much less disorienting and self-bootstrapping than it may seem, since -- outside, indeed, of formal contexts -- virtually all of natural language works exactly like that; and not only expressions like whereas, the moreso as, French ne, German doch, which wear their syncategorematic character on their (empty) sleeves, either:  but plain words like bunny.   You do not learn to use such words on the basis of a definition, formal or informal -- however much it might please linguistic philosophers to invent a terminus technicus such as “ostensive definition”, which labels a phenomenon (or passle of phenomena) without explaining it.   For, as we have seen in our discussions and parables related to matters Quinean, these don’t really work, not logically;  they work pragmatically, to the extent that they work at all, because (since we are all molded from the same clay; or  if you prefer, since our bloodlines have all been subjected to the rigors of Natural Selection) we are all cut to the same cloth.  (To the extent that some individuals fall outside the innate cognitive norms, they fail to acquire the same semantics that the rest of us do:  or else, like some gifted and industrious autists, they acquire this only by dint of an artificial study, like someone learning Sumerian logographics.)   Thus, the following Onomastic Primal Scene does not actually obtain in any real nursery:

“That, Timmy” (pointing -- but at or towards what?) “is a rabbit (noun count, singular).   And by this -- attend now, and please do not misunderstand me -- I do not intend to indicate the entire scene embracing carrots and furballs and playpen and binky (who left that there?) etc., let alone the cosmos as a whole (after all, one has to point somewhere), whether by itself or considered as but one flaky layer in the whole baclava-like complexus known as the multiverse;  but only the, er, furball-related entity.   And by this, I do not mean, so much, (although I do not literally not mean it, either), a pointlike or infinitessimal space-time slice of a leporiform trajectory along the world-sheet, nor a “thickened” (perceptually available) neighborhood of the same;  nor a sort of puddle of rabbit-stuff, undifferentiated from the rest of the puddle; nor a concrete instantiation of the Platonic Form, ‘Rabbit’;  nor a subobject in the Category Leporidae;  nor an agnostically structured pointset consisting of Undetached Rabbit Parts (although I sort of mean that, since, at some point, once you have detached the poor critter to bits  and scattered its disjecta membra over the face of the earth, to be eaten by vermin and recycled as independent atoms, -- at some point, we can no longer confidently say, “That is a rabbit”, in the sense of noun count, singular),  nor -- well, dash it all, I mean just Fluffy, okay?  Fluffy and other creatures that look and hop and act like her.  And by the way it looks like Fluffy wants a cuddle or something, because she is spritzing the wood-shavings in a semantophobic panic.”

~

The scenario above  comports more naturally with a coherence theory of truth, rather than a correspondence theory.

~

The quirky, philosophically-minded Intuitionist mathematician Brouwer, harbored similar “mysterian” views on ultimate indefinabilty:

In Brouwer’s opinion, mathematical definitions should not be looked upon in a mathematical way, but should only be used as a support for our memory.  Basic concepts, such as ‘continuous’, ‘once again’, ‘etcetera’, have to be irreducible.

-- Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p.  45

Nooit nog heeft door de taal iemand zijn ziel aan een ander meegedeeld;  alleen ein verstandhouding, di toch reeds is, kan door de taal worden begleid.

(Caption quotation:  op. cit., p. 32.)

~

Since, outside of the classroom, new words are almost never introduced explicitly, let alone lexicographically or metalinguistically, we must conclude that the language-learner somehow gets the right idea “from context”.   This notion is more problematic than might appear.

For, the history of contexts met-with over the course of a learningful life,  varies considerably from person to person (my own nursery school was wonderfully bunny-rich -- unless the creatures were actually guinea-pigs, come to think of it:  I no longer recall, and after all had nothing to compare them with at the time, they were simply our class mascots and Furry Friends -- but sadly penguin-deficient (of that I am quite sure);  nay, my lifelong platypus-deprivation has been nothing short of absolute), and the fact that we can happily chatter away  among our fellows  about all creatures great and small, without needing to resort to pointing at picture-books (although I do always carry a bunny-book about with me, just in case I should run into Wittgenstein) or red-faced arm-flapping exasperation as we attempt just one more time to make ourselves understood to our perversely thick-witted interlocutors (“Not a ‘triplex of mutually orthogonal rabbit-slices’, dammit!  I mean three  separate  rabbits !!”) strongly suggests that we come from Nature’s Nursery with a lot of shared ontology inborn.  (Chomsky’s school reached similar conclusions many years ago, by a somewhat different path.)

Two-dimensional representation of an imaginary rabbit.  Question:  What is the dimensionality of the *actual* imaginary rabbit?

~

Back to mathematics.

Here definition, in contemporary use, is the intuitive idea, to which axiomatization is the formal counterpart.    You define the term group by simply listing the axioms which any set endowed with an operation must satisfy  if it is to aspire to that dignity.

Yet, having made this move, we see that a vagueness was lurking in our original intuition:  since being ‘axiomatizable’ comes in various flavors:  finitely axiomatizable, axiomatizible in first-order bzw. second-order logic, etc.   And we find surprises, such as when so familiar an item as a torsion group  turns out not to be finitely axiomatizable within first-order logic.   Yet we know what we mean by it, for all that.

Compare: 

mammal:  definable in (cladistic) terms of shared descent

reptile: not so definable

water: definable in terms of molecular composition

blood, wine : not so definable

quartz : definable in terms of mineral composition

granite :  only approximately so definable, or definable at one remove.

~

How you define a mathematical item -- we may even say, how you go about defining it, the tack you take in trying to define it -- depends upon what you are intuitively aiming at.

For example:  How to extend the definition of the multiplication of a finite set of factors, to the infinite case?  (We did so for the case of convergent infinite sums without difficulty.)

Because of the special properties of zero with respect to multiplication, the most obvious definition of a convergent infinite product is not the valuable one.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966)

Or again, from the great Gleason,  ever alert to the lexicographic aspects of mathematics:

The Bolzano-Weierstrass property is often taken as the defining property for compactness, since it is frequently the handiest property  for dealing with compact metric spaces.  However, it is not equivalent to the Heine-Borel property in general topological spaces, and it turns out that the latter is the more valuable in the general case.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 269

Coming up with a useful definition (and here the “coming up with” does seem closer to invention than discovery) becomes an interesting question in its own right, and not a matter of mere fiat. 

Again compare:

Finding a definition (or, really, “characterization”;  yet ultimately the ink-stained lexicographer must needs still define) of:  Romanticism, Minimalism, Idealism;  joke, game; mollusc, microbe, plant;  silver, beige;  etc.

~

So for instance, let’s take logicism.

There is evidence that, in 1899, Hilbert endorsed the viewpoint that came to be known as logicism.  Logicism was the thesis that the basic concepts of mathematics are definable by means of logical notions, and that the key principles of mathematics are deducible from logical principles alone.

-- José Ferreirós, “The Crisis in the Foundations of Mathematics”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 143

So there you are:  A nice clean definition.  Think what you will of the thesis, and come what may by way of later evidence pro or con, the definition is what it is, right?

Wrong.

Our author goes on:

Over time, this thesis has become unclear, based as it seems to be on a fuzzy and immature conception of the scope of logical theory.  … Historically speaking, logicism was a neat intellectual reaction to the rise of … the set-theoretic approach.

So!  In addition to being confirmed or refuted, apparently a thesis can decay, lose its sharp edges, like an unrefrigerated vegetable.   For:  Any definition of X  itself takes for granted the well-definedness of certain understood entities Y, Z …  Should the latter fall foul of better understanding, X itself can be left high and dry.

The consider the following definitions:

phlogiston:  a material which is the source of light and heat attendant upon combustion

phlogisticated air:  air mixed with phlogiston

monokeratic phlogisticene :  phlogiston mixed with powdered unicorn hoof  (cures scrofula and gout)

These delightful definienda, whose delineation was once so clear, have each met with a sad fate.

Definitions, like dephlogisticated unicorn-hoof, are liable to crumble into dust with the passage of time.

Thus, in mathematics:  Newton’s fluxions, etc.

~

Example of a definition  introduced in full awareness that it is merely provisional:

This definition of an affine algebraic variety should be considered only a working preliminary definition.  The problem is that it depends on considerations extrinsic to the objects themselves, namely the embedding of the affine variety in the particular affine space Cⁿ.

-- Karen Smith et al., An Invitation to Algebraic Geometry (1998/2010), p.

This definitio (taking this in the actio rather than the actum sense) is in the spirit of Lakotos’  Proofs and Refutations.

~

Mathematics often sharpens our understanding of any pre-existing conception (“continuity”, “dual”) that comes to swim within its ken.  And so it is for the very notion of definition :  long assumed a matter of free choice, until Russell’s Paradox brought matters up short.   Whereupon he and Poincaré worked out their understanding of impredicative definition or impredicativity. 

Thus, in one formulation of Poincaré’s predicativist  approach:  “All mathematical objects (beyond the natural numbers)” (these being, as even Kronecker concedes, God-given) “must be introduced by explicit definitions.”  And, not just any definition you take a fancy to will do: 

If a definition refers to a presumed totality  of which the object being defined is itself a member, we are involved in a circle:  the object itself is then a constituent of its own definition.

-- José Ferreros, “The Crisis in the Foundations of Mathematics”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 146

And this, you understand, is Very, Very Bad.  (We might cheekily dub it Definitional Incest.)

~

Mathematicians, like philosophers, and unlike anyone else (including even lexicographers), are given to a certain semantic Akribie --  an extraordinary self-critical attention to their own use of language.   As, consider this:

The conservation “laws” of momentum and angular momentum  are also readily introduced …

-- Robert Hermann, Differential Geometry and the Calculus of Variations (1968), p. 100

I have no idea what subtle mental reserve caused to author to quarantine the word laws in sneer-quotes, nor why he felt it necessary so to caveat -- so to signpost the approach to a possible Occasion of Semantical Sin -- in a work aimed (according to the preface), not at philosophers, nor Jesuit spiritual directors, nor even mathematicians, but to engineers and physicists (those are the grease-stained guys tinkering under the accelerator).   But the fact is, if you move in mathematical circles, your semiotic conscience becomes exquisitely sensitive and attuned.

~

In focusing on definition, I am inadvertently revealing the déformation professionelle of one who used to earn his bread (or rather his hardtack; the profession is ill-paid) as a lexicographer.   For, rather than trying to say what a thing “is” (and here the Korzybskian strictures against the copula  have their full force), we may say, pragmatically rather than ontologically, what a thing is for.   Thus, a hammer “is” a manufactured object of a certain range of shapes and weight, classically with a metal head and wooden handle, (etc. etc. -- “Etc.”, as the Korzybskians have it), if that is helpful to you;  but it is for driving in nails.

Thus -- to take a couple of concepts that always somehow puzzled me definitionally :

Chains and partitions of unity  free our proofs  from the necessity of chopping manifolds into small pieces.

-- Michael Spivak, Calculus on Manifolds

Now that is something a kitchen-maid could understand.

~

[Weiteres zum Thema]

On provisional/dialectical definition:

Menger wrote, in a series of papers on foundational questions  published in 1928:

Dabei möchte ich betonen, daß ich das Wort ‘Konstruktivität’ für ein  wenn überhaupt, so  vermutlich  auf verschiedene Arten und in verschiedenen Abstufungen  präzisierbares (bisher noch nicht präzisiertes)  Wort halte.

-- quoted in Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 199

(For logophiles only:  Let us here salute and savor  that phrase,  “ein  wenn überhaupt, so …”   Impossible to translate this into English  in so compact a compass.)

~

Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 14, quotes Lebesgue:

Bien que je doute fort  qu’on nomme jamais un ensemble qui ne soit  ni fini, ni infini,  l’impossibilité d’un tel ensemble  ne me paraît pas démontré.

Quite aside from the mathematical content to this, as sheer semantic content  that will baffle anyone who

(a) has learned the terms finite and infinite as simple contradictories (infinite iff not-finite);  and who

(b) accepts the tertium non datur

it seems a mere tautology, like the analytical-philosophical lore  of bachelors and married-men.

But this is from Lebesgue, note, as familiar with the intricacies of the various infinities  as anyone on earth.  Clearly something subtler here is meant.  Something I’d never heard of before -- the first worry of the Continuum Hypothesis, so I had understood, concerned the possible existence of wiggle-room between countable infinite and the cardinality of the continuum.

Quite possibly, however, since Lebesgue and Brouwer sometimes shared an intellectual orbit, the explanation may be sought in the following hint (op. cit., p. 66):  “Brouwer distinguishes between species which are abzählbar, zählbar, auszählbar, durchzählbar, and aufzählbar, where some of the distinctions  are related to the question of decidability.”

~

Another parallel between mathematics and (e.g.) biology, as regards a certain type of ‘definition’.

Sometimes you are not trying to focus on a new concept in splendid independence, giving necessary and sufficient conditions to ‘be an X’, de-fining (demarcating) its boundaries (Jordan-curve-fashion) between what-all is inside  and what-else is out;  but, rather, starting from some homely, antecedently-familiar item Y, to define this new X as being similar to that Y.    Sometimes you say they’re similar, and leave it at that:

            A hare is like a rabbit.

            A coot is kind of like a duck.

Sometimes you add differentia:

            A zebra is like a horse with stripes.

Or, you may say that the new concept X generalizes Y, without giving necessary or sufficient conditions for membership in the generalization, with or without further examples of members of X:

            Amphibians form a taxon of animals that includes frogs.  (They ‘generalize’ the frog.)

            Amphibians form a taxon of animals that includes frogs and salamanders.

All these strategies are (so to speak) topologically distinct, the one from the other.

Compare, in math (an actual textbook example):

Locally convex spaces are topological vector spaces that generalize normed spaces.

Here the relatively exotic new concept “locally convex spaces” plays the role of amphibians in the example above, with the normed spaces (familiar from the nursery) filling that of our friends the frogs;  with an additional delimiter, topological vector spaces, basically saying:  “generalize, but not too far”.   Thus, if we said

Vertebrates form a taxon of animals that includes frogs.

that would still be a true statement, but the belt would have been let out too many notches to hold up the conceptual trousers.