[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. 45Nooit 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

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

*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 C*.*^{n}
-- 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. 146And 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**-- an extraordinary self-critical attention to their own use of language. As, consider this:*Akribie*
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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