Back to mathematics.
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.
