At the beginning of his preface to the introductory
exposition Philosophy of Logic (1970), W.V.O. Quine quotes Tweedledee:
“Contrariwise, if it was so, it
might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.”
That was a bit of nineteenth century donnish waggery (though
it does give rather the flavor of modal logics and possible-world
semantics in our own day).
Quine then drolly goes on:
If pressed to supplement Tweedledee’s
ostensive definition of logic with a discursive definition of the
same subject, I would say that logic is the systematic study of the logical
truths.
We pause for Homeric laughter.
Anyone schooled in the lore of Russell’s paradox and impredicative definitions, will
recognize that the professor is having his fun.
He then has some more:
Pressed further, I would say that a
sentence is logically true if all
sentences with its grammatical structure are true. Pressed further still, I would say to read this book.
(Ah! ‘Tis a
plaint we ourselves have often
made: Buy my books!)
[Note: Contrast
the equally terse but quite
unselfreferential definition of logic
in
R. Goldblatt, Topoi
, 2nd edn. 1984: “the
study of the canons of deductive reasoning”. Word for word, that’s very good.]
The jest is not without a subtle content, assuming as it
tacitly does that there are such things as logical truths (a subject we see in a
different light after Quine’s own extended attack on the analytic/synthetic
distinction). And it is
quite in line with Quine’s celebrated existential bon mot:
“To
be is to be the value of a variable.”
To non-initiates, that will be mystifying; to semi-initiates, cheeky; to familiar
navigators of Quinespace, an ultra-compact allusion to his perennial concern
with ontology in relation to quantification.
Put more pugnaciously:
Existence is -- what existential
quantification expresses.
-- W.V.O. Quine, “Existence
and Quantification”
[Further Quinefun available here:
http://worldofdrjustice.blogspot.com/2015/01/quine-on-bunnies.html ]
~
The celebrated Cambridge Philosopher of Bland, G. E. Moore,
wrote in his Principia Ethica:
If I am asked, “What is good?”, my
answer is: Good is good; and that
is the end of the matter.
That is: If you
have to ask, I can’t tell you.
(Even so, that pseudo-definition is preferable to the
nihilist/reductionist dismissal by the Eliminative Materialists.)
Indeed, I shall now venture a definition quite in Moore’s spirit:
Truth is … what is true
(and known to God as such), quite apart from human-knowability, let alone
provability (whether by finitistic, intuitionistic, or classical means).
~
From a college textbook:
A measure space consists of a set X equipped with two fundamental
objects:
(1) a σ-algebra M of “measurable”
sets, which is a non-empty collection of subsets of X closed under complements
and countable unions and intersections.
(2) A measure μ: M -> [0, ∞] with the following defining property …
-- Elias Stein & Rami Shakarchi, Real
Analysis (2005), p. 263
The bolded terms are there defined; but the term measurable sets is not -- the clause
that follows reminds us of the definition of a σ-algebra instead. The authors acknowledge their
sleight-of-hand (effectively remedied by material elsewhere in the book) by
placing the offending word in quotation marks. -- Note that this typographic
care exemplifies the semantic Akribie
of math writers, which we have elsewhere praised.
Not all math authors are as onomastically aware as
those. As:
Old-fashioned text-books tend to start off with mystifying
definition of these terms: Euclid’s
own definition, “A point is that
which has no part”, is a good example.
After a perfunctory discussion of these, the author clears his throat,
begins a new chapter, and gets going with some concrete examples: the definitions are mercifully
forgotten.
-- Stephen Toulmin, The Philosophy
of Science (1953), p. 72
~
To define --
literally, ‘delimit’ -- is by way of penning-in the definiendum with
antecedently familiar landmarks.
To say that none such exist, is to refuse definition; as in this classic hymn:
There is nothin' like a dame,
Nothin' -- in - the - world,
There is nothin' you can name
That is anything like a dame!
Nothin' -- in - the - world,
There is nothin' you can name
That is anything like a dame!
-- “South Pacific”
By this point
we have passed definitely from the realm of Oxbridge drollery to that of
Joe on the Boat. And here, indeed,
we discover a whole subculture of definitional legerdemain, noticed in the
delightful Dictionary of American Slang, by Wentworth & Flexner
(1967). In the Supplement we
find this entry:
blivit (n.)
Anything unnecessary, confused, or annoying. Lit. defined as “10 pounds of shit in a
5-pound bag.” Orig. W.W. II Army
use. The word is seldom heard
except when the speaker uses it in order to define it; hence the word is actually a joke.
As a former harmless-drudge chez Merriam-Webster, I salute
that as a gem of the lexicographic art.
A classic development of the pugnaciously uncooperative
definition -- Lexicography with an Attitude -- is The Devil’s Dictionary,
by Ambrose Bierce. (Sample: “fork:
an instrument for putting pieces of dead animals into the mouth.”) Similarly impish was Hobbes’ definition
of paradox: “an opinion not yet generally received”.
A Pugnaciously Vacuous definition of the meaning of life
can be viewed here:
~
Quite other than such conscious humor, are cases like this:
Hamilton follows the Kantian notion
of time closely in his “Essay on Algebra as the Science of Pure Time”. Since the inner sense of time is more
general than the outer sense of space, Hamilton concludes that algebra is a
more general and fundamental branch of mathematics than geometry.
-- Thomas Hankins, Sir William
Rowan Hamilton (1980), p. 268
As the discoverer of quaternions, Hamilton has as much right
as anyone to deliver himself of after-dinner remarks (a genre of public
speaking to which he was particularly devoted) about the nature of
algebra; but this one is
horse-hockey.
(Psychohistorical note: Hamilton was for a time utterly immersed in Kant; this characterization of algebra as the
“Science of Pure Time” stems from psycho-philosophical exuberance, rather than
algebraical expertise.
A curious tentative parallel might be made with Hamilton’s
countryman G.K. Chesteron, who likewise was given to flights of literary
exuberance; both were in marital
situations requiring a great deal of self-sacrifice, which they met with infinite patience; and both were given to a kind of
idealism which some might diagnose
as compensatory.
Okay, beyond our pay-grade. Yet as Silvan Schweber affirms in his perceptive review [Isis,
1982] of this exemplary socioscientific biography: “Hankins has eschewed giving psychoanalytical
interpretations, [but] to anyone interested in the psychodynamics of
creativity, William Rowan Hamilton presents a fascinating case study.”)
~
Another subcategory -- already bordered on by Hamilton’s
epigram for the definition of algebra -- is formed by definitions which, while
not vacuous, we might label Pugnaciously Perverse. The poet Coleridge was (unfortunately) Hamilton’s
philosophical mentor, even as regards what Science ought to be; and he defined
that subject thus:
“any chain of truths which are
either absolutely certain, or necessarily true for the human mind, from the
laws and constitution of the mind itself.
In neither case is our conviction derived, or capable of receiving any
addition, from outward experience, or empirical data.”
-- Thomas Hankins, Sir William
Rowan Hamilton (1980), p. 268
(That first sentence does oddly prefigure the sort of prose churned out by the truckload by the
epigones of Donald Davidson. --
Note too the anticipation of Post-Modernism.)
~
Another non-cooperative move in the orismological game, is
pooh-poohing the very definiendum -- denying that there is anything coherent to
define. One Christian writer (C.S.
Lewis or Hilaire Belloc, I forget which) once said testily, that the notion of “the
Renaissance” was a will-o’-the-wisp, used
by secular writers to mean “whatever I like that happened in the
fifteenth and sixteenth centuries.”
~
Composing a truly vacuous definition is harder than you’d think.
Thus, consider the Euclidean definition of a point (“that which has no part”) which
our philosopher friend scoffed at;
and let us phrase it even more egregiously:
point: a point-like
figure
That actually has cognitive content. It means: To visualize what is meant when geometers refer to a punctum (as opposed to a linea, etc.), think of something like a
pencil-point. Do not think of
something like a dance-floor, or the cosmos, or an elephant.
And:
set: a set of elements
Here the definition is so far from vacuous that someone
could reasonably object that it is actually false,
since it excludes the null-set.
(In this ‘definition’, the stress, so to speak, is on “elements”; “set”
could be replaced by “bunch” or “passle” or “bucketload”.)
For indeed:
There is no direct circularity if we presuppose sets in our study of
sets (or induction in our study of induction), since the first occurrence of
the word is in the metalanguage, the second in the object language.
-- Michael Potter, Set Theory
and its Philosophy (2004) , p. 9
In further defense of impredicativity:
Impredicative definitions are
necessary for ordinary mathematics, as they are unproblematic if one adopts a realist attitude about
the objects defined -- realist in just the sense that the objects exist in
advance of the definitions, that they are picked out by the definitions, not
created by them. That imposes a
substantial constraint on any acceptable philosophy of mathematics.
-- Shaughan Lavine, Understanding
the Infinite (1994), p. 107
~
The grandfather of all tautological definitions is the one
given by Yahweh in Exodus:
~ ~ ~ I Am That
I Am ~ ~ ~
Yet at the same time, that is the best definition that could
be given, since, in a common view of the Abrahamic religions, any limitative
predication would be false. (That view led in particular to the via negativa,
which insisted on the denial of all suggested predicates of
the One.)
~
Another style of coyness with definition is illustrated in the following,
immediately after the authors have introduced Maxwell’s Equations for E and H in free space:
At first we do not attempt to give
physical meaning to these symbols.
We merely say: let us
assume that there exist physical quantities represented by symbols having the
indicated properties, and see what these equations say about the
quantities. From the last two
equations [to the effect that E and H -- whatever they might be -- have
divergence zero], it is clear that E
and H are solenoidal …
-- Robert Lindsay & Henry
Margenau, Foundations of Physics (1936), p. 303
After a few pages of discussion, the authors summarize:
In a very real sense, therefore,
these equations may be said to constitute a definition
of E and H.
-- Robert Lindsay & Henry
Margenau, Foundations of Physics (1936), p. 306
Thus, back where we began. The dangled definition in intuitive terms, is ultimately
withheld: No dessert until
you finish your broccoli; then --
Your broccoli was your dessert.
They subsequently reinforce the apophatic stance:
Physical
meaning of E and H: … The only real importance of the
quantities is involved in the fact that they satisfy the field equations. … It
seems most logical to go the whole way and treat E and H as defined by the
field equations in all cases. The
commoner definitions can then be
looked upon as mere picturizations. --
Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p. 311
E & H: They Are That They Are.
~
Lakatos offers his translation of an epigram from Poincaré:
Mathematics
is the art of giving the same name to different things.
Now, that is not vacuous
(i.e., vacuously true, or anyhow only infinitessimally informative), since it
is egregiously false; but it is a witticism, not a blooper,
since we all know that Poincaré was perhaps the leading mathematician of his
time -- he has some cards up his sleeve, which he will slip out when it suits him.
try these:
http://worldofdrjustice.blogspot.com/search/label/definition
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