A Dive to the Depths (expanded)

A phrase you will often meet in higher mathematics, and almost nowhere else, is:

“a deep result”

O loveliest of monostichs, thou !

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:

            On Depth and Breadth

            On Scope and Difficulty

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 the 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, 2^nd 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; 2^nd 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, 2^nd 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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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, 2^nd 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

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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)

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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:

http://www.youtube.com/watch?v=s5xDEyuXaOY&feature=fvwrel

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

[Kindle]  [Nook]

~     ~     ~

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

 

.

Adventures in Algebraic Geometry

The closest I ever came -- and that  unwittingly -- to a brush with algebraic geometry,   was in freshman calculus.  (This was back in the ‘sixties -- before your time.)  The instructor was Robin Hartshorne, a young and winsome elf of a man.  He was an engaging lecturer, teaching from  or at least in parallel to  a beguiling text (Spivak’s Calculus, so utterly different in spirit from the dry Thomas treatise that had repelled me in high school to the point of dropping the class);  moreover he was -- now that I think back on it -- the first deeply intelligent teacher I’d ever had (though there were to be others -- notably Gleason):  up through high school, there had been no hint that such creatures even existed.

Still, he wore his learning lightly, like his tweeds.  The class was fun.  Particularly endearing -- though also startling, at the time -- was one day in the second semester, when he was presenting the topic of definite integrals -- painful but necessary, rather like a rectal exam.   He was chalking away, when all at once he seized up, staring in bemusement at the blackboard;  then turned to us with a sheepish grin.

“It’s been a long time since I’ve done one of these,” he said.

~

Now -- if the anecdote stopped there, it would be just one more ultimately stupid instance of Genius Porn :   the populace cooing contentedly when told that Einstein flunked grade-school math,  or that Gauss was late to learn to speak, or that Erdös tried to cut a grapefruit with a butter-knife  (which indeed he did, though it was a craftily calculated move).   These falsely flatter our vanity.  They are the opposite of a much better genre of joke, which you need a bit of math to actually understand, such as the one about von Neumann and the summation of infinite series.

For the incident, trivial until viewed in the light of later developments, did  there and then  plant the seed of doubt and wonder, as I sat theretofore clueless in the second row.   His being momentarily at a loss  struck me  at the time  as quite surprising, almost inexplicable:  as though a test-pilot, stepping from his aircraft into his roadster, were to stare at the ignition and say, “Remind me how to start one of these.”

Had I continued with a chemistry major as originally planned (well, originally-originally an English major, until I realized, with chill horror, the error of my ways), and thus retreated or perhaps advanced  depending on how you look at it, into ever-more-technical intricacies  and mechanical practicalities, the significance of that incident would never have become apparent.  But as it was, Hartshorne’s class (and Spivak’s sparkle) were instrumental in turning me towards a concentration in pure mathematics -- which was the only kind of mathematics they really taught at Harvard (golden memories of that climate of abstraction here), and eventually towards having a go at Berkeley towards a Ph.D.   Accordingly I was to be introduced to as-yet-unsuspected levels of intellectual depth, such that each, compared with the one before (I speak loosely;  they are basically incomparable), is as the definite integral to 2 + 2:  rising like the serried ranks of angels.  And though I myself never progressed beyond the level of the cherubim, it was sufficient to glimpse that empyrean wherein, indeed, you might forget the particular monkey-tricks used to solve thorny individual definite integrals (basically you just memorize these, storing them for reference like tools in a toolkit, unless you’re Euler or von Neumann, in which case you re-derive them instantly from scratch, or simply perform a brute-force numerical calculation in your head).

As each glowing level is added, the one below  becomes obsolete ...

Moreover, the tired old cart-horse of the calculus was, it turns out, very far from the centers of Hartshorne’s research interests, which are almost unimaginably abstract.   In that pokey little classroom in Massachusetts, he was really only on loan to us from Sagittarius, having once studied with Grothendieck, a confirmed extraterrestrial.  It was bruited about that he had something to do with something called projective geometry;  but only much later was I to learn that he is one of the pioneers of …

sheaf theory

… a topic so ferociously abstract, that even to define what sheaves are  is utterly beyond me.  (Wikipedia doesn’t even try, observing that “their correct definition is rather technical”.)  Nay, wert thou to gaze upon this theory naked -- not even to speak, not even to breathe of its further generalizations in topos theory -- ‘twould make thine eyes, like stars, to start from their spheres, and thine each particular hair -- nay more, thou wouldst  in sooth  explode in flames,  as did dame Semele,  when  all unheeding she beheld,   unveiled,  great Zeus in all his lightning !
 

~

~  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.

(Je m'appelle Evariste Galois, and I approved this message.)

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~

~

As so often when some movement of math has been seen streaking off westwards  out into the void, presumably never to been seen again by mortal man, it reappears shining in the east, reborn in some applicable form -- thus suggesting, you will notice, that the global topology of the noösphere is toroidal.  In the present case, algebraic topology has come to be crucial in such applications as: string theory (via its prior discovery of Calabi-Yau manifolds), coding theory,  cryptography and steganography -- which means that the juicy bits are probably highly classified.

And this raises the interesting paradox, which Epimenides would have relished, whether someone like Hartshorne is Cleared for the contents of his own head.

Robin Hartshorne as seen in a recent spectral image.  Since the old days, he seems to have sprouted quite a bundle of fibres over his base-space.

 

~

I recently happened upon an essay by the usually abstruse Samuel Eilenberg (of Eilenberg-and-Steenrod notoriety), written for a collection sponsored by the Office of Naval Research.  In deference, perhaps, to the needs of our seamen, Professor Eilenberg permits himself some observations and analogies   that lie within the reach of the common folk.  Thus:

The analogy between sheaves and covering spaces  is very close.

-- Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 110

Bingo !  Covering-spaces I get.  Of course, I don’t actually see the analogy, but at least it’s a comfort, knowing that there is one.

[Update, Thanksgiving 2014]   Edward Frenkel, noticing my perplexity, kindly sent in this explanation:

Coverings and sheaves are related. And it's not just an analogy. A covering space is an example of a sheaf (the simplest example): It is a sheaf of finite sets (provided that the covering is finite).

Namely, given a covering p: C --> X of a manifold X, and given an open subset U of X, the set of sections of of the corresponding sheaf over U is just p^{-1}(U) [the preimage of U in C under p]. Note that the stalk of this sheaf over a point x of X is just the fiber over x [the set of points in C, which project down onto x under p; p^{-1}(x)].

The covering gives us a way to "glue" these fibers together (indeed, set-theoretically, C is the union of these fibers -- but it's more than that, because C is a manifold, just like X; so C is not a "disjoint" union of these fibers, they are really "glued" together in a particular way).

The simplest covering space is the trivial one: a union of N copies of X, each mapping identically to X under p.

Here is a non-trivial example: let X be a circle. Now take the Moebius strip in which this circle is the circle "in the middle." Take the "edge" of the strip -- this will be your C. Notice that for each point in your original circle X, there are two points in C. But C is NOT the union of two circles (which would be the trivial double covering). In fact, C is just ONE circle, covering another circle (our X) in a non-trivial fashion.

A general sheaf is very similar. The difference is that the fibers could be infinite, or they could be vector spaces, etc. But the idea is the same.

~

We earlier discussed  the memoir Souvenirs d’Apprentissage, by a pioneer of algebraic geometry, André Weil.   Avid for more, we got hold of a copy of the memoir Random Curves (2008), by a contemporary algebraic geometer,  Neal Koblitz, well known to anyone with an interest in Elliptic Curve Cryptography. (It came in via InterLibrary Loan -- interestingly, the lending institution turned out to be the U.S. Naval Academy in Annapolis.    Compare the publishing venue of the Eilenberg article referenced immediately above.  We salute the broad interests of our midshipmen!)

Both authors have a wide range of interests and experience outside of mathematics;  both engaged in extensive foreign travel;  and it is of this that they principally write:  the reader will enjoy these accounts for their own sake, but we set down the volumes with a twinge of disappointment, that we are no closer to insight about algebraic geometry than we were before.  However, one biographical detail did strikingly stand out.  Both authors took a principled stand against unjust wars:  and this, not simply by penning valiant Letters to the Editor from the safety of their studies, but by a brave and almost reckless defiance while actually serving in the armies of their respective countries:  actions that could easily have led to their injury or even death, and which did actually lead to their imprisonment (and, in Koblitz’s case, to a severe beating).  But what is truly remarkable is that, in both cases, they used their time in the slammer far more profitably, mathematically, than most of us  use ours  even in the best of circumstances.   It was there that Weil did his seminal work, and there that Koblitz returned in concentrated form to the practice of algebra which he had largely abandoned during two years of political turmoil.

Intrigued, I wrote to the latter author, inquiring whether, from the standpoint of Kolmogorov-style measure-theoretical probability theory, we may validly generalize from this sample of two (2);  and he was kind enough to reply:

I love generalizations based on small samples.  I'm sure a lot of algebraic geometry was nursed at the Indiantown Gap army stockade!

Thus encouraged, I here make bold to speculate about the martial philosophy of Robin Hartshorne.  I know nothing of his politics, but truly cannot imagine the man wielding an M-16.   Or a flyswatter, for that matter.   Were a mayfly to venture into his office, Hartshorne would no doubt observe the pattern of its flight (musing all the while on the brevity of this earthly life) and calculate whether that trajectory describes an elliptic curve. -- Which, come to think of it, it just well might.   Many mathematical treasures remain to be unearthed in the field of biology!  (For a few of these, see the fine book by Ian Stewart, Life’s Other Secret.)

~

A rather huffy response to modern algebraic geometry, which it is a pleasure to reproduce  mainly because I do not understand the subject, and which suggests (like the characterization of Category Theory back when I was in college, as “the higher macramé”) that (as with these new-fangled things called “computers”) I am perhaps not missing much:

Attempts to extend the geometry of second-order surfaces  and the algebra of quadratic forms  to objects of higher degrees  quickly leads to  the detritus of algebraic geometry, with its discouraging hierarchy of complicated degeneracies, and answers that can be computed only theoretically.

-- Vladimir I. Arnold, Lectures on Partial Differential Equations (Russian edition 1997; English translation 2004), Preface.

Harumph!  Hear hear!