Tuesday, March 4, 2014

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

[Continuing this essay:

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 synsemanticsyncategorematic 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.

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?
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.

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