Circular Definitions in Newton

 

Re Time:

http://worldofdrjustice.blogspot.com/2025/08/sur-la-pluralite-des-temps.html

 

Re Matter:

http://worldofdrjustice.blogspot.com/2020/05/informative-tautologies-updated.html

 

Informative tautologies (updated)

Technically, for a logician, or a semanticist of the Snow-is-White school, tautologies convey no information;  but to linguists and pragmaticians, in context they often do.  In fact, we may state that they usually do, since otherwise why utter them?  “Business is business” is flint-hearted;  “Boys will be boys”, tenderly exculpatory.

The opposite of an utterance that pretends to contain no information (and thus, in particular, to be inexpugnable) but actually does (and often of a trenchant sort), is a definition that, ex cathedra, is all about informing, but which melts to the touch.  Cf. our essay here:

http://worldofdrjustice.blogspot.com/2015/01/pugnaciously-vacuous-definitions_24.html

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Covertly vacuous uses of technical-sounding terms  have been scouted in histories of science.  As, “Things fall because of gravity, and rise because of levity.” Or Molière’s virtus dormitiva.  But more may be packed into such terms than may be initially apparent.  As, one philosopher pointed out that “The ball rebounded to the height that it did  because of its resiliency.”  But this is informative:  the height is owing to internal characteristics of that ball, rather than from the ball’s having been dropped from a greater height, or having been dropped on a more resilient surface.

Additionally, the initial tautological character of a sentence can  so to speak  “age off”, in accordance with semantic evolution of its terms. Thus, “Atoms are indivisible” was initially as circular as “Bachelors are unmarried”, since they were defined from the outset as indivisibilia (as their name, a-tom, etymologically implies).  But on its current interpretation, the sentence would qualify as false.

A mathematician looks at Newton’s Definition 1, in the Principia:

Quantity of matter is a measure of matter that arises from its density and volume jointly.

Great acumen is hardly needed to realize that this definition is hopelessly circular, since density is normally defined as the ration of mass to volume;  but Newton’s unhelpful phrase  does have some implicit implications.  For example, we expect the mass of an object to remain unchanged if we change its shape.

-- Michael Spivak, Physics for Mathematicians: Mechanics I (2010), p. 9

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A 2015  example of the uses of tautology:

Robert Buissière on Médi1, re Presidential candidates:

Jeb Bush, frère de son frère,

et Hillary Clinton,  épouse de son époux.

As they stand, these statements are “analytic”; but we understand the import:  Jeb and Hillary got where they are today, largely owing to family association.

Cf. & contrast the common expression “He is his father’s son.”  Normally this means that he takes after his Dad, and not that he is getting any special favors from other people owing to that filiation.  To imply the latter, you might say “Daddy’s little boy” or something.  By contrast, the French phrases in the above context  do not imply that Jeb’s politics are a close match to Dubya’s, let alone that Hillary’s are a close match to Bill’s.

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A bare tautology like “Business is business”, as a free-standing statement, invites contentful interpretation via a “Gricean implicature” (specifically, the Maxim of Quantity).   The following is a syntactically more complex case, there the tautology is embedded in a subordinate clause:

Ever since self was self, nature been keepin’ folks off of red-hot stoves.

-- Zora Neale Hurston, Their Eyes Were Watching God (1937)

The meaning is:  Common sense has been extant  from time immemorial.  Operationally, the (tongue-in-cheek) interpretation is:  Go back and back in time, sampling as you go. For each sample-point, verify whether  “self = self” holds at that time; and if so, then evaluate “Common Sense is in effect? Y/N”.  The sentence, for all its folksiness, has a kind of philosophy-class spin to it; the moreso as “self = self” calls up First-Order Logic with Identity.

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Stylistic appreciation

Usually there is a summary “That’s that” finality to tautologies, whether used informatively or not;  stylistically, they are bare-bones.  But consider this:

Herod:  The moon has a strange look tonight. … She reels through the clouds like a drunken woman. … Does she not reel like a drunken woman?  She is like a madwoman, is she not?

Herodias:  No;  the moon is like the moon, that is all.

-- Oscar Wilde, Salomé (1891)

Here the barrenness of the pale white, plain round  far-floating body, is reflected in the unyielding tautological formula.

~

Enten-Eller

Philosophers have scribbled  much ink, and later worn out many a typewriter ribbon, and finally expended great bushels of pixels, discoursing upon the status of “logical truths”; such as, paradigmatically, the following:

            (I) Every man is either married or a bachelor.

(We simplify, since the matter is not really of interest; leaving out of account, for instance, the curious case of Schrödinger’s groom.)

It is agreed that such a sentence tells us nothing about the world, unlike that time-honored exemplar of informativeness,

            (II)  The cat is on the mat.

which has been so oft repeated. down the years, that said cat has achieved the immobility and timelessness of an Egyptian idol.  (Presumably the mat lies in a patch of sunlight, so why ever move?)

            And yet its affordances are quite different from those of another statement of the same logical form; say:

            (II) Every number is either even or odd.

For, although the sentence (I) does not perhaps baldly state anything substantive about the world, its presuppositions speak volumes.  For one thing, it gives us to understand that there is a sharply defined institution, Marriage, into which a man may enter or not; and that his resultant state is either-or. Even so much will give our Martian anthropologists  sufficient grist  for many turns of the mill.

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The above are mostly individual linguistic parlor-tricks.  Much more generally, there is the matter of the status of the equations of mathematics.  A view put forward by the dessicated and ennervating tendency called formalist (nominalist)  maintained that, being mathematical, they are tautologies, and being tautologies, they are uninformative -- semantically vacuous.  Practical experience shows that doctrine to be false.  As a way to see how such equations manage to be informative, consider the number pi.

Pi can be defined, qualitatively, in a number of different ways, most familiarly as the ratio of a circle’s circumference to is diameter.  It  is, moreover, a Given of the invisibilia, woven into the fabric of the noöspheric pattern;  and -- crucially -- it can be reached in a startling variety of precise quantitative ways, along this strand or that of the warp and the woof -- and the woorp and the wahf, along any of the dimensions of the multidimensional mathematical textile.   Pi is itself ever one, the peak of the Golden Mountain;  but the paths that climb to it are numberless.  It can be expressed as a definite integral; as an infinite continued fraction; as an infinite product; as an infinite sum; with many variations of each.  (Behold some of them at https://en.wikipedia.org/wiki/Pi.)  That fact that any one of these equals π is informative and indeed amazing, for in effect each one constitutes hiking instructions -- a trail-map -- to the summit, each from a wildly different base-camp.  And each one of these infinitely intricate expressions  is equal to any other, in an equation which is as distant from tautological or uninformative as can be imagined.

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[Update 13 May 2020]  An aborbing article by Evan Osnos, in the 11 May 2020 issue of The New Yorker, maps out the paths by which the Greenwich Connecticut  tennis-and-boating-club crowd  came to support Trump.  The notables of that town have a patrician heritage, in principle at variance with the flashy style of the vulgarian from the Bronx, but as one blue-blood testifies, his conversion came while witnessing an early speech by the candidate:  “He had that line that he would use: ‘Folks, we either have a country or we don’t.’ And I felt the chill .. I’m, like, ‘Oh, my God, this is a really good line.’

Apart from the formally tautological character of that line, it puzzles by its vagueness:  out of context, it is unclear what at all is being hinted at.  Presumably the line is a dog-whistle, a bit of Rorschach rhetoric from which the listener will extract whatever meaning he likes.

The line does nothing for me; but it does recall such successful political antecedents as “The business of America is business.”

~

[Update 14 May 2020] 

Some of you may be familiar with the British sport of trainspotting.  That may or may not be in accordance with current U.K. guidance on coronavirus lockdown.  But here’s a hobby you can practice in the safety and comfort of your own home:

Tautology-Spotting !

As:

Headline in this morning’s New York Times:

The People Behind the Counter Are People

Remember this the next time you order takeout.

That one recalls those sleep-inducing example-sentences from introductory logic class (All brave Athenians are Athenians).  But in this case, it has a punch, and the source of that punch is not logical but lexical.  For, people has a variety of senses;  from the neutrally classificatory

(a)  an instantiation of the species Homo sapiens, near-cousin of Pan troglodytes, sometimes known as “a forked radish”;

to the “pregnant sense”  (I phrase it freely)

(b)  an ensouled being created in the image of the Lord of Hosts, whom Christ died to redeem.

In the cited sentence, the first occurrence of the word people has the (a) meaning; the second occurrence, (b).

For more on perspectival semantics and pregnant senses, check out this essay:

https://worldofdrjustice.blogspot.com/2019/10/words-of-day-sublunary-postlapsarian.html

~

In all the instances above, tautology is used in its traditional logico-philosophical sense.  But technical terms sometimes get picked up by a wider audience, where their use may be lax.  Thus, the literary critic V.S. Pritchett, in his article on the novelist Anthony Powell, wrote:

Mr. Powell is excellent with the raffish…. I think the sententious irony succeeds. … It adds a very English flavor, either of comic tautology or deflation.

The sense of tautology is unclear here.  Perhaps it refers to mechanical repetition, a standard component of broad humor.

Adventures in Juvenile Lexicography

Katherine Nelson, in Keith Nelson, ed., Children’s Language (1978), p. 66,  offered the following glimpse into the orismological instincts of budding lexicographers.  Asked a “What is it?” question about the word tiger (“a large, fierce, flesh-eating animal (Panthera tigris)…” to you and me), the tots responded:

(1) “at the zoo”
(2) “animal”

(3) “it’s like a lion:

(4) “lives in the jungle and runs a lot”

(5) “animal with stripes and it eats a lot of things”

(6) “to run”

(7) “someone growls”

(8) “hair on its head”

I then polled our son (aet. su. 4 years,  3 months), who offered this:

   “It’s something that is big, and it eats people, and it runs around in the jungle.”

His assessment of her other examples:

apple: “it’s juicy; it’s big; it’s round”

car:  “Something that’s big, and not so tall -- it’s this tall [shows with his arms];  it can kill someone that stays in front of it, if it’s moving.”

coat: “It’s something that keeps you warm, is big and sort of smooth, and has little furry stuff”  [Note:  Our family was at that time facing an Edmonton winter]

bed:  “It has legs, or doesn’t, and it has a pillow, and it stands up on its posts”.
[Note: That first idea in the definiens, at once oddly precise and maddeningly vague, probably meant:  “Prototypically a bed has legs (the ones you see in books), but ours doesn’t” -- the family, indeed, then in exile and furniture-poor, slept on a mattress on the floor.]

Striking  is this repeated note of ‘big’, present in every definition except the last:  extending even to the humble apple -- even a baby is bigger than an apple, let alone a robust four-year-old.   But in light of the lad’s subsequent specialization in differential geometry, an explanatory hypothesis presents itself.  What may well have struck him was the apple’s unabashed convexity -- round, not like a thin dime, but round all around:  having everywhere positive and (roughly) constant Riemannian curvature, as he would no doubt rephrase the definition upon more mature reflection.  Such an apperception of an apple was indeed the Eureka moment of the founder of differential geometry, Carl Friedrich Gauss, as depicted in the movie “Die Vermessung der Welt”.

A (non)Definition of Depth

We’ve posted a number of reflections about the idea of “depth” in (especially) mathematics and related science (for the complete list of these, click here: http://worldofdrjustice.blogspot.com/search/label/depth ), without ever really defining the term.  And this, for a reason:

Two interrelated ideas that have been widely assumed to be unanalysable  are those of one scientific theory being deeper and more unified than another.

-- John Watkins, Science and Skepticism (1984), p. xiii

The idea of theoretical depth has considerable importance in Popper’s philosophy of science.  “If at all possible, we are after deep theories.”  But he was pessimistic about the possibility of any sharp characterisation of the idea.

-- John Watkins, Science and Skepticism (1984), p. 188

And that, not necessarily for any ‘deep’ reason -- not correlating intimately with the intricacies of physics, say -- but much as it is hard to characterize sharply such multifaceted (or blobby) concepts as beauty or game.    And here, I must sympathize with the archetypal philistine of a hundred New Yorker cartoons,  genially conceding, “I don’t know much about art, but I know what I like.”   A mathematician or physicist may not be able to define depth in a way that would satisfy the notoriously finicky tribe of philosophers:  but he knows it when he sees it.  And smiles.

So, sorry, no necessary-and-sufficient conditions, nor even a rough-and-ready dictionary definition;  but anyhow, an epigram:

Whewell’s requirment that a deep hypothesis, one that gets hold of nature’s ‘alphabet’ as he put it, must enable us ‘to explain … cases of a kind  different from those which we contemplated in the formation of our hypothesis.

-- John Watkins, Science and Skepticism (1984), p. 190

(A similar metaphor, more popular since Whewell’s day:  a good theory must “cut Nature at the joints”.)

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Hmm, now I’ve piqued my own curiosity.  How does a practicing lexicographer go about characterizing the term deep in the sense(s) of interest here?

Okay, for starter’s, a British one -- Collins English Dictionary (1979):

deep:  … (6) difficult to understand or penetrate; abstruse

(7)  learned or intellectually demanding: a deep discussion

Sense (6) is of no interest.  We went to some effort in this post (“A Dive to theDepths”) to distinguish depth from difficulty.  (For the list of posts re difficulty: http://worldofdrjustice.blogspot.com/search/label/difficulty .)   Sense (6) would be used by a lazy man, giving up -- “Too deep for me.”   Here he is not even using the word with its native literal resonance -- he could quite as well speak with poor Mr Tulliver, who often confessed that the world was “too many” for him.   Indeed, sense (6) might deserve one of those non-semantic/non-grammatical, sociolinguistic labels like “ -- Not in polite use”:  here,  “-- Not used in this sense by serious thinkers.”

Actually, by a twist of pragmatics, the phrase does get used by serious thinkers -- but typically as an ironic put-down.  Thus:

The towering 19^th-century mathematician Hamilton  labored hard on his “Law of Hodographic Isochronism”,

but when he sent it to John Herschel (whom Hankins calls “the best-known and best-regarded British scientist of his time” [p. 134]), he got the reply:  “You are fairly got out of my depth”.  (This was Herschel’s regular response when he did not have the time or inclination to follow Hamilton’s long analytical excursions.)

-- Thomas Hankins, Sir William Rowan Hamilton (1980), p.

And again, roughly a century later:  After presenting a rather absurd and convoluted, goalpost-moving series of proposals from Lakatos and Morrall:

I am out of my depth with a claim of this kind.

-- John Watkins, Science and Skepticism (1984), p. 334

Here he is being disengenuously self-deprecating;  and his reply is all the more biting.

~

A more general thought, though, on depth versus difficulty.   I almost wrote that the latter was “much less interesting” -- though really, that depends on your day-job.  If you teach primary school, you need not (ex cathedra) worry your head one bit about depth in our sense;  whereas you must ever be alert to the perils of difficulty.  To epigrammatize into a dichotomy:  Depth (again, in our privileged sense) inheres in the subject itself;  Difficulty is relative to the practical limitations of some species (be it human, chimp, or the poor fly stuck in the fly-bottle) when grappling with the problems in that subject.   Since the philosophy of this blog is Platonist, we have little interest in the latter (no intellectual interest;  though some emotional interest, maudlin or morbid, as here).

From that perspective, sense (7) is also disappointing.   A subject itself (such as algebraic geometry or M-theory) cannot be called “learned” (i.e., learnèd):  that epithet might only be applied to whoever is gassing on about it.  “Intellectually demanding” could be applied either to a subject or a particular discussion or presentation thereof.  And that quality might be due to anything from the intellectual limitations of the audience (“The concept of evidence is too intellectually demanding for Trump voters”)  -- thus, back to sense (6) -- to an (overly) condensed presentation on the part of the lecturer, to actual depth inherent to the subject (and which would still be apparent to an angel, who understood the subject perfectly well).   Thus, neither sense goes far towards elucidating what mathematicians mean when they refer to a “deep result”.

~

Curious now whether my old alma-mater Merriam-Webster  did any better, I looked it up in their Collegiate Dictionary (Eleventh Edition),  I found something quite different.

First, their treatment of the geospatial, ‘literal’ sense of the term, from which all others ultimately derive, is unexpectedly rich and reticulated (I almost wrote:  “deep”), containing sub-subsenses like

deep  1 b (1) : extending well inward from an outer surface <a ~ gash>

(That is the sort of distinction you come up with when you are working from a generous deskful of carefully chosen citation-slips, rather than copying other dictionaries  or pulling the definition out of your butt.)

But then things sort of fall apart.  The sense “difficult” is not treated as a top-level numbered sense, as in the Collins, but as a subsense of a sense not defined save as the sum of its (rather disparate) subsenses:

deep  3 a : difficult to penetrate or comprehend : recondite < ~ mathematical problems>

3 b : mysterious, obscure <a ~ dark secret >

3 c : grave in nature or effect <in  ~est disgrace>

3 d :  of penetrating intellect : wise <a ~ deep thinker >

along with several more lying well off our axis of interest.   And oddly, despite all the careful hair-splitting, nothing really corresponding to Collins’ (7).

Thus, we still have come no further towards our goal.

~

The subject of depth, unlike that of mathematics, or Christianity (or oahspe),  tends not to attract disquisitions of the “What  is ….?”  sort.   We tried our hand at one for math (here), basically coming up with little more than a florilegium of blind-men-and-the-elephant stabs at it, for the overly general definiendum mathematics itself;  more fruitful was the task of characterizing topics within mathematics, like affine connection or topology, since here (at least for the former example) the definer’s intention is more in the nature of targeted enlightenment  than an after-dinner speech :  the result was a nice bouquet of epigrams.

~

Back to Depth vs Difficulty.    

(1) A deep remark or insight  is associated, not with presenting difficulties (as in Collins sense (6) ), but -- quite the contrary -- with resolving them.

A humble but poignant case  has been recounted here (Induction/Recursion), where a problem that had seemed difficult (to New Jersey third-graders, back in the complacent days before the impact of Sputnik  had filtered down to elementary school) -- that of multiplying multi-digit numbers -- suddenly became transparent, under the impact of an insight which (relative to what we had learned so far, most of it from the Mickey Mouse Club) might qualify as (qualifiedly) deep. 

(2) Above, we made something akin to an actio/actum distinction between difficulty and depth (human activity vs. the subject itself);   yet now we may make an additional distinction, on the same -- human -- side of the Platonic/psychological divide.  You might call it horizontal/vertical,  syntagmatic/paradigmatic :  judging words (concepts) by the company they keep.

~

Let us recur to that subsense in Webster’s Colleagiate,  “3 b : mysterious, obscure”, and consider the idea of  deep as it appears in company with that of being hidden:

The mind is in a sad state, when Sleep, the all-involving, cannot confine her spectres within the dim region of her sway, but suffers them to break forth, affighting this actual life, with secrets that perchance belong to a deeper one.

-- Nathaniel Hawthorne, “The Birthmark”

(I.e., a deeper, hidden something, that somehow itself  amounts to a “life”.)

And from a mathematical physicist:

It is indeed true that we can prove, from this kind of Euclidean argument,  that squares, made up of right angles, actually do exist.  But there is a deep issue hiding here.

-- Roger Penrose,  The Road to Reality (2004), p. 28

Namely (tying in with cosmology):

His fourth postulate asserts the equality of all right angles.  … In effect, the fourth postulate is asserting the isotropy and homogeneity of space.

-- Roger Penrose,  The Road to Reality (2004), p. 29

~

.

The Ontology of Linguistics (expanded)

[I have noticed that some people fight shy of long essays -- doubtless an effect of the syndrome lamented here.   Accordingly, as an experiment, we shall intitially put up just a bit -- a “stub”, in Wikipedia’s terminology;  or, as Professor Malkiel loved to say, a “torso” -- adding to it as the days go by,

as the moon rises and the sun sets,

as the leaves  fly off the calendar  in the winds of Time ...]

The ontology, then, of “linguistics”:  and not, note, of “language”.   Similarly, we may speak of the Ontology of Psychology (and not of the psyche), the Ontology of Geology (and not of the Earth).   All That Is, is what it is;  “I am that I am”.   But for purposes of this or that variety of study, we structure things.  Such structures are constrained by what is Out There, but are not straightforwardly or uniquely determined.)

So, first of all:  ‘What is linguistics?’

(We cannot pose this tiresome question, save in squotes, just as we did for ‘What is Mathematics?’)

Two candidates present themselves:

(I)  Linguistics is the study of language.

(II)  Linguistics is the study of languages.

In many modern perspectives,  these are distinct.   And each, in its own way, is problematic.

(I)  Already with Saussure if not before, the nature of the pre-theoretical notion ‘language’ (langage) was split, for precision, into langue and parole.  Subsequently, this basic bifurcation acquired theoretical heft with, in one corner (in the red trunks) those championing “I-language” and the innate “Language Acquisition Device”, versus (in the blue trunks) corpus-mavens, connectionists, frequentists, “usage-based” grammarians, and other sundry nominalists.  For a riposte to the latter, confer Frederick Newmeyer, whose tautologically-titled essay “Grammar is Grammar and Usage is Usage” (in Language, 2003), makes many useful points.

(II)  This formulation is less problematic, as being less ambitious, and more traditional.  Indeed we may say:  Philology is to languages, as linguistics (in the contemporary sense) is to language-tout-court.    But then we are faced with the question, what is “a language”  -- as opposed to a different one, or a dialect, or some other semantic signaling system.  And there we meet disagreements once again.

~ Recommendation posthume ~

“Si j’étais encore en vie, et que je  désirais un bon whodunnit,

que lirais-je?"

Micmacs divorce --Pouah!

Murphy sur la montagne

(Je suis Ferdinand de Saussure, et j’ai approuvé ce message)

~

This antinomy of language-per-se versus languages, has become acute since Chomsky, who has little interest in the endless gabble of actual tongues.  If this seems an extreme position, it is nonetheless exactly parallel to that of physicists, who seek general physical laws, rather than endless descriptions of individual objects falling or rolling or spinning or colliding or what have you.

Chomsky bitingly writes:

The grammar is a function-in-intension … the language is epiphenomenal.  Its ontological status is the same as that of a set of pairs of expressions that rhyme.

-- Noam Chomsky, Rules and Representations, p. 83

Here he is using language in the sense of ‘parole’ and not ‘langue’.   But, epigrammatically, it is startling to see language as of but peripheral interest to linguistics.

~

Marveling at the philosopher’s opacity, two linguists write:

In a recent article, Ryle even claims that sentences are not part of language, but only of speech.

--Jerry Fodor & Jerrold Katz, eds., introduction to The Structure of Language (1964), p. 11

Yet Chomsky would later also demote much of what was traditionally thought of as language, to a second-class status as “E-language”.

~

The units of social life are far less clearly defined than those of language … Linguists are fortunate in possessing a domain whose units are at least relatively self-defining and isolable.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 82

The minimal unit of spoken language is not the sentence, but the utterance or intonation unit.

-- William Bright, ed.  International Encyclopedia of Linguistics (1992), vol. IV p. 11

Schuchardt enquires, “What is dialect?” -- the neogrammarians having used the term in formulating their theories -- and shows that it is an abstract notion, with no real existence.

-- I. Iordan & John Orr, An Introduction to Romance Linguistics (1937),  p. 32

With similar vigour, Schuchardt opposes the notion of ‘linguistic periods’ .. Just as there are no fixed boundaries between different vernaculars, so the chronological boundaries between successive periods of a language  are purely fictions of our minds.

-- I. Iordan & John Orr, An Introduction to Romance Linguistics (1937),  p. 33

And indeed:

Die Gesamtsprache ist etwas Abstraktes, ebenso wie die Gesamtseele  gegenüber der Individualseele.

-- Leo Spitzer, ed., Hugo Schuchardt-Brevier (1921; 2^nd edn. 1928), p. 386

The term language is … a relatively nontechnical term.  If we wish to be more rigorous .. we have to employ other terminology.  We shall use variety as a neutral term  to apply to any particular kind of language which we wish … to consider as a single entity.

--J.K. Chambers & Peter Trudgill, Dialectology (1980), p. 5

(Ontologically/methodologically, this picture is not particularly reassuring.)

Structural linguistics has a favored suffix to denote status as a unit in the ontology or theoretical apparatus:  -eme.  Familiar (though embattled) are the phoneme and morpheme; lexeme (roughly: dictionary headword, so that eat and ate are part of one ‘lexeme’ even if you might call them different ‘words’ in one of the senses of that pre-theoretical term).  Hjelmslev proposed seme (or sememe), as the atomic unit of sense (the ‘semantic atom’), though these do virtually no work;  and, getting into the borderline-silly spirit of the thing, from outside of linguistics  Richard Dawkins coined meme. 

~

The ontological status of ‘meanings’:

Quine is quite correct in protesting that meanings are not entities.

-- Harold Lee, “Discourse and Event”, in: Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p.

One might have thought that being an ‘entity’ would be  rather a low bar.

(I recall one smitten young man, back in Berkeley in the 1970s, pleading with her whom he adored, “What do I mean to you?”  -- And she replied, wrinkling her young brow and thinking hard,  “Well, …  you’re a person ….” )

It goes even lower:

Though we sometimes need to conceive meanings timelessly,  we do not therefore need to conceive them as subsistent entitities.

-- Jonathan Cohen, The Diversity of Meaning (1963), p. 161

(“Subsistent”:  cf. Meinong.  Actually, “subsisting” is a very low bar, lower than “existing” entities.)

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 ]

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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).

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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

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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

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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.”)

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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.)

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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.”

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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

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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.)

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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.

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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