Pugnaciously vacuous definitions

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 , 2^nd 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!

-- “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:

http://worldofdrjustice.blogspot.com/2012/03/ask-dr-science.html

~

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.

For a series of essays on the art of definition, with especial reference to math,
try these:
http://worldofdrjustice.blogspot.com/search/label/definition

Cream for your Coffee

[The following paragraph has just been added to our essay, "A New Proof of the Existence of Coffee-Cups".]

Having at length satisfied ourselves as to the reality, or at least reliability, of coffee-cups, would should not  on that account sink back into an attitude of Moorean complacency (“I’m all right, Jack;  I’ve got hands”).  For our commitment to these  suggests yet further commitments, which we had not realized were there to assess.  Such as :  Realism with regard to quantum state vectors.

The question of ‘reality’ must be addressed in quantum mechanics -- especially if you takes the view that the quantum formalism applies universally to the whole of physics -- for then, if there is no quantum reality, there can be no reality at any level.

-- Roger Penrose,  The Road to Reality (2004), p. 508

And:

The question of the objective existence of the objects of mathematics … is an exact replica of the question of the objective existence of the outer world.

-- Kurt Gödel, “What is Cantor’s continuum problem?”, in American Mathematical Monthly, 1947.

A complex Schrödinger equation,

after the Collapse of the Wave Packet

In for a penny, in for a pound.

Esse est percipi (redivivus ter)

“To be is to be perceived” -- Bishop Berkeley
[corresponding to the Latin   esse est percipi, pronounced ESS-ay est pare-KIP-ee.  Percipi is the passive infinitive, a delightful category, which does not exist in any of the other languages with which I am conversant.]

“We call a real dynamical variable whose eigenstates form a complete set   an observable.”  -- Paul Dirac

“To be  is to be the value of a variable.”  --  W.V.O. Quine

*

~

~  Posthumous Endorsement ~

"If I were alive today, and in the mood for a mystery,

this is what I'd be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(I am Bishop Berkeley, and I approved this message.)

~         ~

~

Three radically different takes on  being and seeing.

(Cf. "Seeing is believing" -- vs. Berkeley:  Being seen  is being.)

The first and most famous of the epigrams above  applies to the ontology of basic objects, such as coffee-cups -- about which no plain man has any doubts whatsoever.   (Quine's quip  will, however, frequently detain us further;  e.g. here.)

A related but different question  concerns things which, unlike a coffee-cup or its relativistic mass, you cannot directly perceive -- though someone else might.

Scott Soames, Philosophical Analysis in the Twentieth Century (2003), vol. I, p. 17, re the views of G.E. Moore:

For things presented in space, but not things to be met with in space, to exist is to be perceived.  That is, afterimages .. and pains  can only exist when they are perceived or experienced.

Or, classically,  “Beauty is in the eye of the beholder”.

A psychological vice ontological version of Bekeley’s epigram, is the vernacular “Out of sight, out of mind.”  (Cf. French, "Si tu te tais, ni vu ni connu.")  Or, as Fielding wittily put it in Tom Jones,

De non apparentibus, et non existentibus,

eadem est ratio. -- In English, ‘When a woman is not seen to blush, she doth not blush at all.’ 

(Lawfolk interpret this Latin tag more prosaically:  “What is not juridically presented cannot be judicially decided.”   Spoilsports.)

~

A noted intuitionist philosopher, anent “the celebrated [loathsome, leprotic -- ed.] thesis that mathematical statements do not relate to an objective mathematical reality  existing independently of us”, writes that, on such a view, to be is to be conceived:

Unlike material objects,  mathematical objects are, on this thesis, creations of the human mind.  They are objects of thought, not merely in the sense that they can be thought about, but in the sense that their being is to be thought of [or rather, thought into existence -- ed.];  for them, esse est concipi.

-- Michael Dummett,  “ The Philosophical Basis of Intuitionistic Logic”, in: Truth and other enigmas (1978), p.  228

*     *     *

~ Commercial break ~

Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

*     *     *

~

The above is really just a placeholder post, reminding me to someday address the deeper problem of observables.   This may never happen.  To begin with, even in the simplest quantum-mechanical case, an observable is an operator on Hilbert space, so you have to wrap your mind around that.  With some luck and some study, that might be doable.  But just now I encountered the following dismaying sentence:

In quantum field theory, unlike in quantum mechanics, position is not an observable.

I may well have retired to my hamster farm, before ever figuring such things out.

In the meantime, some further thoughts about ontology  here.

~

Bonus quote:

  How do you know you’re having fun  if there’s no one watching you have it?

  --Douglas Adams, The Restaurant at the end of the Universe (1980).

~

Visitors to the site have enquired about the pronunciation of the classic Latin phrase “esse est percipi”.   About this (although I used to be editor of pronunciation at Merriam-Webster, back in the day) I have so far had nothing to say.  Embarrassed, I messaged my colleague and comrade Dr. Keith Massey (officially out of comms, but reachable in the bush), as follows:

More than once, someone has searched the blog via
   pronounce esse est percipi
-- alas in vain.  In my head, I have always pronounced this pear-KIP-ee, but, given that the vowel is short, the Latin would be more like PEAR-kip-ee, no?
Or rather, the Vulgar Latin, since Latin itself, being quantitative, lacked phonemic lexical stress.  Correct so far?
A second question is how philosophers in various countries pronounce it -- anglicized, frenchified, etc.

Swifter than lightning, our colleague replied:

In classical Latin the C is always pronounced as a K. But we don't know when the thing started to change before E and I. I suspect it was already being realized regionally as something else in the Late Imperial period. And speakers of Latin dialects took to pronouncing their classical Latin as if it were their living register. And so, in France,  percipi was pronounced “persipee”. In Italy perchipee, and in Spanish pershipee (attested still in Ladino) and later, in Iberia as perthipee, and, yet later in the New World, as persipee.

Accent in Latin is a hotly debated topic. I personally follow the view that the classical accent is unknowable and therefore all we have to work from is the living dialects and the rules of Ecclesiastical Latin. And so, penultimate except in a few cases. Accent falls in the syllable before -ibus. But I'd say perCIPee.

As for Anglo-Latin, this is an exciting topic. There's some evidence that a Romance language survived in the Isles for a several centuries after the withdrawal of forces in 380. It apparently, from inscriptional evidence, turned long E into a long I, fecit --> feecit. The way we pronounce phrases like Habeas Corpus may actually be representative of the pronounciation of Anglo-Romance, rather than a later scholastic recasting.

Those further interested in this topic can consult the essay by Thomas Pyles, “The Pronunciation of Latin in English:  A Lexicographical Dilemma”, reprinted in his Selected essays on English usage (1979).  Executive summary:  The history is so tangled, it is now an unresolvable mess.

In his “Tempest in Teapot: Reform in Latin Pronunciation”, reprinted in the same volume, Pyles quotes an amusing epigram that alludes to what became of Latin in Spain, where original v and b merged into a bilabial fricative:

~ Felix natio, ubi vivere est bibere ~

[Variant: "o felix iberia, ubi vivere est bibere"]

For further adventures in pronunciation, click here. 

*

Für psychologisch tiefgreifende Krimis,

in pikanter amerikanischer Mundart,

und christlich gesinnt,

klicken Sie bitte hier:

http://www.linguasacrapublishing.com/justice.html

*

Distinguishing discontinuous state-reduction (which he dubs R) from linear evolution as per the Schrödinger equation -- i.e., the “collapse of the wave-function” upon “observation”, a mathematician remarks:

I do not mean to imply that the experimenter deliberately sets up a ‘measurement’ to achieve this.  … Nature herself is continually enacting R-process effects,  without any deliberate intentions on the part of an experimenter or any intervention by a ‘conscious observer’.

-- Roger Penrose,  The Road to Reality (2004), p. 593

Thus achieving the esse of percipi  ‘naturally’ (vacuously).

.