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”.)
~
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 19th-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
~
.
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