A phrase you will often meet in higher mathematics, and almost nowhere else, is:
“a deep result”
The
very notion of what ‘deep’ means, in such a context, is itself deep;
indeed, too deep for me, at present. This, owing to a crippling
condition of mathematical
oligophrenia.
-- which, however, I pray that time and diligence might partly
palliate. (For a glimpse into the terrible sufferings of mathematical
oligophreniacs, click
here,
if you dare.) Yet I am putting up this skeletal promissory-note of a
post, so that there will be a space to scribble insights as they wake
me in the night.
First off -- the term "deep" does
not mean simply ‘
difficult’;
indeed, though such results lie in the depths, and are not to be had
for the asking, once you have somehow managed to fish one up, it may
seem clarity itself. Nor does merely being difficult make anything
deep. Any humongous brute-force calculation falls into that category;
for a more-substantive example, consider the Four-Color Hypothesis,
which people suspected should be deep, but the proof that changed
"Hypothesis" to "Theorem" is a combination of clever tricks and
elbow-grease. The response of the mathematical community was disappointment: "So, it turns out it wasn't an interesting conjecture after all." (Of course, it may yet prove to be "interesting" in our cognitive human sense; that awaits a proof of an entirely different kind.)
We
may go further, and put forward that an overarching purpose of
mathematical research is to reveal something previously difficult as
now simple, when seen in the right way. Again and again this has
happened in history, beginning with the replacement of finger-counting
by symbols, and of clunky symbols like Roman numerals by decimals. For a
more recent example:
Although
Beurling’s own proof [characterizing invariant subspaces of an
operator on Hilbert space] was quite involved, it is by now simple to
prove; it depends on hardly anything more than the geometry of Hilbert
space. The profitable point of view is not sequential but
functional.
-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed. Lectures on Modern Mathematics, vol. I (1963), p. 17
Nor is there any simple similarity or opposition between being “deep” and using what are now called “elementary”
methods in number theory (as opposed to analytic methods), e.g. Alte
Selberg’s proof of the Prime Number Theorem by elementary methods,
compared with earlier analytic proofs by Hadamard and others.
Typically, proofs that restrict themselves to “elementary” methods are
harder than those that permit themselves a more capacious toolkit; but
whether they ever, or generally, gain depth via this austere
discipline, I have no idea.
In the meantime, some related posts outside of a mathematical context are these:
As
appetizers, try the following hors-d’œuvres platter -- to follow
which, however, we have as yet prepared no meal (as with our early
essays on t
he Realist vernacular, this is more by way of linguistic warm-up):
The connection between linear transformations and bilinear functionals goes quite a bit deeper …
-- Paul Halmos, Introduction to Hilbert Space (1951, 2nd edn. 1957), p. 38
The Riesz representation theorem depends on some of the deeper parts of the theory of measure and integration.
--George Simmons, Introduction to Topology and Modern Analysis (1963), p.
The analogy between cyclotomic fields and fields formed from the points of finite order on elliptic curves is very deep.
-- Neil Koblitz, 1993
The notion of Kan extensions is the deeper
form of the basic constructions of adjoints. We end with the
observation that all concepts of category theory are Kan extensions.
-- Saunders MacLane, Categories for the Working Mathematician (1971; 2nd ed. 1998), p. vii
The continued-fraction representation of real numbers is deeper than the decimal expansion.
-- Roger Penrose, 2004
There are deep ties between enumerative geometry and Ramanujan’s tau function.
Contrast:
The useful fact about products of projections lies near the surface.
-- Paul Halmos, Introduction to Hilbert Space (1951, 2nd edn. 1957), p. 47
In
the same work, while acknowledging the possibility of a
lattice-theoretic formulation of the subspaces of Hilbert space, he
dismisses this possibility as “trite” (p. 22).
* * *
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Relief for
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We now return you to
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* * *
Much commoner than “trite” is trivial, which is virtually a terminus technicus of mathematical practice. Let one quote stand for all:
One
of the useful conclusions we can draw from Theorem 2 [to the effect
that the norm of a Hermitian operator equals the supremum of its
eigenvalues] is that the spectrum of a Hermitian operator is not
empty. This is not a trivial conclusion. We shall obtain the
corresponding fact for normal operators only after the application of a
lot more relatively deep analysis.
-- Paul Halmos, Introduction to Hilbert Space (1951, 2nd edn. 1957), p. 55
Psychosociological
note: Attempt to imagine the effect on our pumped-up math-major
freshman or sophomore brains, hearing dismissed as “trivial” (with a
wave of the hand) propositions which, but a year before, we ourselves
would not have begun to understand, and which indeed most of our
countrymen will go peacefully to their graves without understanding.
(For more on the hubris involved, click
here.)
Also
note: What counts as “trivial” is relative to where you stand. Thus,
in the very next sentence, Halmos adds: “We hereby report that the
spectrum of an arbitrary
operator is also not empty; since we shall have no occasion to make
use of this fact, we shall not enter into its proof.” The proof, one
gathers, is more difficult still. But when once you have mounted, and
stand upon that summit, the fact that Hermitian operators in particular have eigenvalues, is trivial indeed.
Leave it to mathematics to recruit even the notion of triviality into some highly non-trivial constructions. E.g.
A topological space over X is called a locally trivial fibration if every x in X has a neighborhood over which Y is trivial.
-- Klaus Jänich, Topology (1980; Eng. trans. 1984), p. 129
And:
Sard’s Theorem … is … a highly non-trivial theorem which is elementary
in the sense that it uses only the notion of a differentiable map.
-- Shlomo Sternberg, Lectures on
Differential Geometry (1964)
The terms deep and elementary
(here in the everday sense, and not the special number-theoretic
meaning mentioned above) are not antonyms, but they do contrast:
The
spectral theorem implies that every normal operator has a large supply
of invariant subspaces; this is classical and can be considered
elementary by now.
-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed. Lectures on Modern Mathematics, vol. I (1963), p. 17
This
dependence of “depth-perception” upon experience accumulated with the
passage of time, does not refute the notion of relative mathematical
depth as ill-defined. What is intuitive though difficult to put into
words is a notion of “deeper than” rather than of absolute depth. To
the giant, neither the pond nor the puddle appears deep; but the pond
is deeper for all that.
~
Re the classification of simple algebras (“simple”, to be sure, in a certain technical sense, meaning roughly: incredibly complex and difficult):
The tools employed are not deep; they are just, so to speak, linear algebra raised to the nth power.
-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed. Lectures on Modern Mathematics, vol. I (1963), p. 122
~
Further attestations.
A
pioneer of Hilbert space theory, expounding the then-contemporary
state-of-the-art for a nonspecialist mathematical audience, particularly
as regards dilations and extensions of operators:
There
do not seem to be any conspicuous and challenging yes-or-no questions
that serve to indicate the direction in which the search for new
results might begin, but I have faith. There is depth in the
subject; the trouble is that the surface has not been explored enough
to show where the deepest parts lie.
-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed. Lectures on Modern Mathematics, vol. I (1963), p. 17
Writes a premier algebraic-topologist:
There
are geometric problems which require the use of the multiplicative
structure of the topological invariants. Such problems are deeper than
those which can be solved by considering the additive structure alone.
Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed. Lectures on Modern Mathematics, vol. I (1963), p. 105
Re Lie algebras and a closed subgroup H of the full linear group:
It
is furthermore true (and this is deeper) that these one-parameter
subgroups fill a neighborhood of the identity in H, and consequently
generate H if H is connected. …The converse part of the correspondence
involves a subtlety of the type that makes the study of Lie groups a quite sophisticated topic.
-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed. Lectures on Modern Mathematics, vol. I (1963), p. 118
Re the Hodge conjecture:
It arises deep within the subject, at a high level of abstraction; and the only way to reach it is by way of those layers of increasing abstraction.
-- Keith Devlin, The Millennium Problems (2002), p. 9
This is a truly adult depiction of depth: as lying deep within the layers of the enigmatic onion. It is not a case where you can just swallow some peyote and see it all in a flash.
(For more, compare: The Ladder of Abstraction.)
Writes a philosopher:
The axioms are not logical truths ... Their truth is established by intuitions which lie too deep for proof, since all proof depends on them.
-- Roger Scruton, Modern Philosophy (1994), p. 392
An example from outside the field of mathematics -- though
it is a mathematician who is writing this:
Just how a protein manages to
organize itself in space, using only the sequence of its own amino acids,
remains a mystery, perhaps the deepest in computational biology.
-- David Berlinski, “What Brings a
World into Being” (2001), collected in:
The Deniable Darwin (2009), p. 243
~
Related vocabulary:
Related to the concept of depth (which focusses on the root of things) is that of richness (regarding the blossoms that bloom from this root). Hadamard adopts this metaphor explicitly:
Application’s
constant relation to theory is the same as that of the leaf to the
tree: one supports the other, but the former feeds the latter.
-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 125
Further examples:
The algebra of composition of maps resembles the algebra of multiplication of numbers, but its interpretation is much richer.
-- Lawvere & Schanuel, Conceptual Mathematics (1997), p. 11
… a recent, and surprising,
theoretical advance by Winkler. He shows that, for many …
algebraically closed fields, the “free Skolemization” has a model
companion.
-- Angus Macintyre, “Model Completeness”, in: Jon Barwise, ed. Handbook of Mathematical Logic (1977), p. 164
The concept of thickness [in graph theory] is the deep mathematical idea that underlies the recreational puzzle of Earth/Moon maps.
-- Ian Stewart, How to Cut a Cake (2006), p. 126
~ ~ ~
Having convinced ourselves, by the examples of mathematics, that there is more to this assessment-word deep
than an emotional or impressionistic grunt, we look to some cases
outside of mathematics where an idea has been similarly assessed.
Some discoveries provide answers to questions. Others are so deep that they cast questions in a new light, showing that previous mysteries were misperceived.
-- Brian Greene, Fabric of the Cosmos
We are not at home in the world, and this homelessness is a deep truth about our condition.
-- Roger Scruton, Modern Philosophy (1994), p. 464
T.S. Eliot affirms that what is past and what is present, even what might have been, indicate a present purpose. This is a metaphysical point of great depth.
-- James Schall, S.J., The Order of Things (2007), p. 69
And, more prosaically, but no less tellingly for all that:
Although
running Bain Capital required a lot more brains and savvy than playing
roulette does -- a lot more brains and savvy than most of us could even
pretend to possess -- the job was not conceptually deep. Romney did not develop a model of the world from the business of private equity. … “He’s not a very notional leader,” [said] Romney’s campaign spokesman …
-- Louis Menand, “Money Pol”, The New Yorker (19 III 2012)
~ ~ ~
This is quite
aside from the path of mathematics, but -- it may be, that such depth
is displayed in quite distantly allied regions: all tracing back to
Him, perhaps by some functorial construction. In that spirit, this:
The final anguish of the Asian bride suggests the depth of the Riemann Hypothesis.
The enigma of a
woman’s heart,
finally espied by a Private Eye,
for less than the
price of a Valentine …
This Rose
~ ~ ~
Somewhat less far off the path …
Deep is indeed the term of
art in mathematics, antonymic to trivial. Now compare,
from another discipline, the word profound,
in reference to Newton’s perplexing, little-known philosophical-speculative opus:
That it is exclusively mystical I do not believe -- that there is a
mystical element seems
certain. I hope that one day some profound
student -- no one less will suffice -- will study this mass of papers.
-- E. Andrade, quoted in James Newman,
ed. World of Mathematics (1956), p. 273
~ ~ ~
Above, we saw the distinction deep
vs. difficult. Here now even the latter concept is
bilayered:
Although Beurling’s own proof
[characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to
prove; it depends on hardly
anything more than the geometry of Hilbert space. The profitable point of view is not sequential but functional.
-- Paul Halmos, “A Glimpse into
Hilbert Space”, in: T. L. Saaty, ed.
Lectures on Modern Mathematics, vol. I (1963), p. 17
In infinite dimensions, it is tedious but not difficult to construct
spaces strictly but not uniformly convex.
-- Prof. Lewis, University of Alberta,
course in Functional Analysis, 1982
The proof that Reidemesiter
moves and planar isotopy suffice to get us from any one
projection of a knot to any other
projection of that knot is not
particularly difficult; however,
it is technically involved.
-- Colin Adams, The Knot Book
(1994), p. 15
And here the original distinction is made even sharper: though they are not interchangeable, depth tracks with generality, which in turn tracks (on the plane of praxis -- of
proof) with simplicity:
Re the Denjoy-Young-Saks Theorem on the derived numbers of functions:
As we would expect in view of the great generality of the
final statement of the theorem,
the proof due to Saks is of extreme simplicity.
-- F. Riez & B. Sz.-Nagy, Leçons
d’analyse fonctionelle [references to the English translation, Functional
Analysis, 1955], p. 17
~
Here a leading mathematician laments the shallowness of his
understanding of something he himself proved (regarding representations of a
Kac-Moody algebra, as it happens):
My proof of this result was technically
quite involved. I was able to
explain how the Langlands dual group appeared, but even now, more than twenty
years later, I still find mysterious why it appears. I solved the problem, but it was ultimately unsatisfying to
feel that something just appeared out of thin air.
-- Edward Frenkel, Love &
Math (2013), p. 181
This is setting oneself high standards indeed. Shakespeare probably did not lie awake o’ nights fretting how
the devil he ever came to write Hamlet; Mozart did not find the bread of pleasure at having written
the Sonata in A turning to ashes
at the thought that it might have been dictated to his unconscious by an angel.
~
A near-synonym of the math-word deep,
but shorn of all irrelevant aesthetic echo, is: “highly nontrivial”. The term is decidedly
commendatory, though to a layman it might sound like faint praise, as were one
to dub one’s lady-love “seriously unugly”. The expression may be extensionally impeccable, but ‘twould
never pass in a sonnet.
Further:
In set theory, a forcing
extension in Cohen’s sense is
reminiscent of algebraic extensions of a field, but
… the forcing method is far more complex, both conceptually and technically,
involving set-theoretic, combinatorial, topological, logical, and metamathematical
aspects.
-- Joan Bagaria “Set Theory”, in Timothy Gowers, ed., The
Princeton Companion to Mathematics (2008), p. 625
Although he does not use the term "deep" here, that expanded characterization
"complex, both conceptually and technically", especially the "conceptually" part, points in that direction.
~
One motive for Frege’s choice was again generality:
Does not the ground of arithmetic
lie deeper than that of all
empirical knowledge, deeper even than that of geometry?
-- I. Grattan-Guinness, The
Search for Mathematical Roots 1870 - 1940 (2000), p. 184
… [Cantor’s] remarks on functions
of several variables (where the provability of theorems was deepening the level of rigour in analysis)
-- I. Grattan-Guinness, The
Search for Mathematical Roots 1870 - 1940 (2000), p. 223
.
