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