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

~      ~      ~

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:

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

 

.

What is Truth?

[Once again, we shall begin, not with any essay  matured to fruition in the womb of time, but with a bare space to write on, as ideas arrive.  And again, we shall begin at the linguistic -- even the lexicographic end of things:  noting terminological and semantic oddities, like passing around a plate of hors d’oeuvres.  But if the past is any guide, at some point insights might congeal.]

~  ~  ~

What is Truth ?  -- At so lofty a level, speech fails, just as for What is Being.  No more than Pilate  do I stay for an answer.

Quid est veritas?

Nor did the Ancients, really:  as George Pitcher puts it in his introduction to the collection Truth (1964):

The great philosophers of history said surprisingly little [about Truth]:  they were far more interested in truths than in ‘truth’.

Similarly:

If, instead of asking the question, what makes this or that proposition true, I ask the question, what makes any proposition true, then I can find no answer:  the question is over-generalized. (Compare ‘How much does this book weigh?’ with ‘How much does anything weigh?’)

-- Roger Scruton, Modern Philosophy (1994), p. 108

(Actually, that straw-man example could well be given a sense:  “Any thing weighs:  its rest-mass times a constant proportional to the strength of the gravitational field in which you are weighing it, times a velocity-dependent relativistic correction.”   And that statement, far from a tautology, does contain a lot of hard-won physics.)

About modern theories, the linguistic philosopher John L. Austin wrote, in his article “Truth” (collected in the volume just mentioned):  “the theory of truth is a series of truisms”.  And, even more epigrammatically (Anglo-American philosophers tend to be good at coining these):

~ In vino, possibly, veritas; but in a sober symposium, verum. ~

To this I would only add that, in a symposium, there should also be vinum, since the Greek word means literally ‘drinking together’.

And so, hoisting a chalice of the blushful  in a salute to Truth -- may she ever remain spotless ! -- We proceed to the matter at hand.

~    ~

There is a use  of the predicate true  for grudging acceptance-- “True enough, but--"  “That’s all very true, but--" -- which demotes it.   Mathematics sharpens our sense of what it might mean to be true without such reservations.

Thus  the philosopher and logician Bertrand Russell (“My Mental Development”), upon discovering the “timeless world of Platonic ideas”:

This world, which had been thin and logical, suddenly became rich and varied and solid.  Mathematics could be quite true, and not merely a stage in dialectic.

Yet few things are ever so simple.  For one frequently meets statements along these lines  (in the present instance, reporting the work of Freedman and Donaldson on h-cobordism):

     It’s true topologically, but not smoothly, for dimension four.

(Well... "frequently", depending on which pool-halls you hang out in.)

Now:  We are taught at our nanny’s knee:  Let your answer be:  Yea, yea; and nay, nay:  Whatsoever is more than this,   cometh of evil.    Or, equivalently, from Grandpa Quine, arguing against logics with nonstandard notions of truth:  When you change the logic, your are actually changing the subject.  -- So, what:  are the modernists here positing some abstruse new varieties of truth -- topological and smooth?

Not a bit of it.  That adverbial shorthand, unpacked, means that, in four dimensions, under certain conditions, it is

* unreservedly true that there exists a homeomorphism between the structures in question;

* unreservedly false that there exists a diffeomorphism between these structures.

But in that case (cannily you ask), why demote the two domains of truth-assessment to mere adverbs upon a single predicate?   And the answer is again mathematical, for homeomorphism and diffeomorphism are variant instantiations of a unitary notion of isomorphism.

~

That example was clear because math is, and because the unfamiliar example did not evoke siren-calls of preconception.   But syntactically similar instances are less clear:  One reads that something P is, say, “true economically but false politically”, while Q is “true literally but false psychologically”.  Here the grammatical test does not furnish unambiguous results, for “political truth” and (especially) “psychological truth” are idiomatic coin of the realm.  Nonetheless, I suggest that the correct analysis is identical to the one above:  P -- a statement about economics -- is true (without qualification), but politically unpalatable;  and, Q is true simpliciter,  but … and here there are many possible pragmatic though not alethic failings:  counterintuitive;  true-as-far-as-it-goes but it’s kind of an idiot-savant thing to say in the circumstances, the formally-correct tin-eared observation of a visiting Martian.

~

It may be, that in the miasmic swamps of Postmodernism, the very truth-predicate itself is under assault, along with all standards of tradition and decency.  Quite possibly, in their orgiastic symposia on Bald Mountain, the various adepts of this doctrine or passle of doctrines -- hunchbacks, dwarves, and other infrarational minispawn -- shuffle forth (blinking at the daylight) to proclaim that there are as many meanings of True as there are pressure groups to squabble tooth-and-pinkynail for them -- True for Feminists; True for Autists; True for the Transwhatevered -- motleys over which it is difficult to quantify.  Perhaps even they  have not yet sunk this low:   but they will, they will.

*

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Relief for beleaguered Nook lovers!

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*     *     *

[Update 16 February 2012]

Now this:

Facts Are Stupid: “story-truth” vs. “happening-truth”.

We comment on that article here:

http://worldofdrjustice.blogspot.com/2012/02/truth-vs-truthiness-updated.html

~

There are many propositions  for which truth is problematic;  most everything we say  may be thus attaindered.  “It’s love that makes the world go round.” “Business is business.”  “Quadruplicity drinks procrastination.”  “Those Mets are amazing.”   But -- not problematic in a fruitful way.

It is in mathematics that the whole question of Truth becomes actually interesting again.   Take the matter of the derivability of the Parallel Postulate in Euclidean geometry; and relatedly, the status of Euclidean space as true a priori.  As posed, these problems did not call Truth itself into question; but their brilliant and surprising resolution  did:  We are now intimately and concretely familiar with the notion of a proposition being true in a model.   Which is but one step away from that of Truth, simpliciter, in a model.

Here, though, once the smoke had cleared (and the landscape was smoky enough, that Gauss refrained from publishing his results concerning non-Euclidean geometry, for fear of the howls of the Boeotians), the question settles into serenely clear form, accessible to any undergraduate.  Yet -- within mathematics -- there lie areas problematic  even for professional philosophers and mathematicians.

As:

            (1)  Problems of the various infinities (you might stomach some of them -- but are you cool with measurable cardinals?)  and non-constructive “proofs”, attacked by the Intuitionists.  (Their challenge is not dead;  cf. Michael Dummett, and topos theory.)

            (2)  The unsettling results of Gödel’s Incompleteness Theorem:  things known to be true but unprovable.  As Dummet puts it (“Wittgenstein’s Philosophy of Mathematics”, 1959),

Gödel’s Theorem shows that provability in a single formal system  cannot do duty as a complete substitute for the intuitive idea of arithmetical truth.

(Such an “intuitive” idea  of truth beyond proof, is Realist, it would seem, despite Dummet’s championship of anti-Realist Intuitionism.   And the Theist, at this point, has surely perked up. -- but I’ll grind that axe another time.)

            (3)  The equally unsettling class of Independence results, such as the independence of the Continuum Hypothesis.   So-o-oo … is  it nevertheless true?  Or -- if false, then we could exhibit -- or an angel could -- a subset of the reals with cardinality less than that of the reals  and greater than that of the integers.  Only … if you could exhibit such a thing -- you’d have a proof ?  right ??  Which means it would not be independent after all.  Only, Cohen/Gödel proved that it was.   Which means … ???

            (4a)  The allegorical but not unrealistic case of supersheaves.   [At time of writing, I made that word up.  But so rapid is the advance of math, that by the time you read this, something by that name may be the subject of seminars at MSRI.  Just pretend otherwise.] Only one mathematician in the whole world professes to intuit the truths of these;  his intuitions are unfortunately incommunicable, the rank-and-file of everyday unionized Algebraic Geometers  avowing themselves baffled.  So, Supersheaf Theory:  True; not true?  -- And before you too quickly dismiss this allegory, consider that it applies every day, everywhere, in a million ways.  There will often be only one person in the room who undertands some given thing.

            Stone-Čech compactification is a bit like this.  Its truth is clear, in a general way, to all who understand topologies and categories.  Yet the Stone-Čech compactification of something as basic as the natural numbers is at present beyond clear-eyed human comprehension.  (Wikipedia has an entry on this  that will turn your hair white.)

            (4b)  The case of…. meta-mega-hyper-supersheaves.   Avowedly, every single mathematician on the planet pronounces himself utterly baffled by these, without so much as a shadow of an intuition about what things even might be (let alone are) true.  And yet and yet -- Again without exception, they profess to glimpse a glimmer of a hint, of, that, which is to say … it cannot be put into words but … Adoremus !!!

~

Apart from and beyond such detailed considerations, the very truth-predicate itself has been questioned within mathematics (albeit, by a rabble of Nominalists).   Thus, for a comparatively straightforward proposition “Catalan’s constant is transcendental”,

A constructivist will not accept that this is either true or false.  This may seen odd, or even obviously wrong, until one realizes that constructivists have a different view about what truth is.   For a constructivist, to say that a proposition is true  simply means that we can prove it in accordance with the stringent methods that we are discussing.

-- José Ferreros, “The Crisis in the Foundations of Mathematics”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 150

Here we are reminded of Quine's gibe about changing the logic vs changing the subject.

~

One topic that sharpens our perceptions of Truth is its relation to Provability.

In “Wittgenstein’s Philosophy of Mathematics” (1959), Dummett misstates the Platonist position with regards to mathematical statements:

A platonist will say that there exists either a proof or a disproof of the statement;  the fact that the statement is true, if it is true, consists in the existence of such a proof  even though we have not discovered it.

In “Realism” (1963), he states the matter correctly.  Taking as a concrete example Fermat’s “Last Theorem” (which at the time was still an unproved conjecture):

There seems no reason to assume, from a platonist standpoint, that the statement could not be true  even though there did not exist any such uniform proof:  it might be that, as it were, the inequality should just happen to hold for each quadruple [of integers].  For each particular quadruple, the inequality could not be accidental:  but there might be no finitely stateable reason why it was the case that it held for every quadruple.

Exactly right.   Some open problems that just might fall into this category:  Goldbach’s Conjecture; the existence of an odd perfect number.

Dummett’s anti-realism is primarily directed at the objectivity of truths  and not at the existence of objects;  but one can readily appreciate how a platonist picture of mathematical objects  should be presupposed to a proof-transcendent view of mathematical truth.

-- Colin McGinn, “Truth and use”; in: Mark Platts, ed.  Reference, Truth and Reality (1980), p.  35.

Dummett himself notes that the mathematical notion of provability  has a broader epistemological counterpart of knowability (at a minimum, justified true belief):

One who adopts a [R]ealistic view of any problematic class of statements  will have to interpret “in principle possible”  in a fairly generous way.  He will not hold that, whenever a statement is true, it must be possible, even in principle, for us to know that it is true, that is, for beings with our particular restricted observational and intellectual faculties …; it may be possible only for beings with greater powers …

But even the most thoroughgoing [R]ealist  must grant that we could hardly be said to grasp what it is for a statement to be true  if we had no conception whatever of how it might be known to be true;  there would, in such a case, be no substance to our conception of its truth conditions.

-- Michael Dummett, “What is a Theory of Meaning? (II)”, in: Evans & McDowell, eds., Truth and Meaning (1976), p. 100

Dummett has counterfactuals principally in mind;  but his observations are valid as well  for our Parable of the Supersheaves.   For even though, in that thought-experiment, one actual human being does profess to understand the truths of this new theory (of his own discovery or -- invention), and fills many folio pages with elaborate scribbles that may or may not be some analog of “formulas”, the ruck of ordinary pencil-wielding Algebraic Geometers are as clueless as to what it all might mean, as is the ordinary iPhone-wielding businessman confronted with the truths of algebraic geometry.  Leaving the rest of us  none the wiser.

~

Most attacks upon classical accounts of Truth, such as you stumble upon in today’s cultural gutter, stem from somewhere on the continuum from Nominalism to Nihilism, often with a particularist or paraphiliac flavor.   But there exist as well  deeply thought-out alternative accounts, such as offered by Dummett in the essay above-quoted.  Here he returns to his core interest in mathematics and logic:

A theory of meaning in terms of truth conditions  cannot give an intelligible account of a speaker’s mastery of his language;  and I have sketched one possible alternative, a generalization of the intuitionistic theory of meaning for the language of mathematics, which takes verification and falsification as its central notions, in place of those of truth and falsity.

-- Michael Dummett, “What is a Theory of Meaning? (II)”, in: Evans & McDowell, eds., Truth and Meaning (1976), p. 115

This is on quite another plane from those who languidly maintain that “pi equals two” is true-for-the-Amazonians.

~

Related but extra-logical uses of the term true:

There are two kinds of practical “truths”, the truth of craft or art, and the truth of prudence.

-- James Schall, S.J., The Order of Things (2007), p. 103

The first sense is reflected in our idiom out of true (‘out of alignment’); the second in things like “a brave man and true”.

~

A related ambiguity in the verb believe:

In English  we have a peculiar difficulty here  because, in popular speech, “believe in” has two meanings:

(a) To accept as true;

(b) To approve of -- e.g. “I believe in free trade.”

Hence when an Englishman says he “believes in” or “does not believe in “ Christianity, he may not be thinking about truth at all.

-- C.S. Lewis, “Modern Man and his Categories of Thought” [unpublished MS, 1946], printed in Present Concerns (ed. Hooper, 1986)

~

A perhaps innocuous-sounding  but actually radical proposal (and radically misconceived):

We must replace the notion of truth, as the central notion of the theory of meaning for mathematical statements, by the notion of proof:  a grasp of the meaning of a statement consists in a capacity to recognize a proof of it when one is presented to us.”

-- Michael Dummett,  “The Philosophical Basis of Intuitionistic Logic”, in: Truth and other enigmas (1978), p. 225

On one reading, that statement is (idle but) unexceptionable, though devoid of interest to mathematicians:  namely, that upon which the clause following “truth”, despite being set off by commas as though parenthetical, is restrictive, and with the term “meaning” meaning: meaning-for-us:  in which case, we are back in the dank damp realm  of hominoid-sapiential psychology, quite superfluous to any philosopher, or even to any psychologist  outside of the forked-radish clan.   (Hamsters react differently to mathematical truth:  their whiskers twitch.)

That business about “capacity to recognize a proof”  is even more weaselly:  do you mean, correctly recognize?  In which case we are back to the notion of Transcendental Truth.   If all you mean is a capacity for some featherless biped to (for whatever reason) often hit upon a good thing (much like Jimmy the Greek), then this purported “capacity” to “recognize” a “proof” would be less useful and probatory  than a tendency to get an erection whenever (transcendentally) a mathematical statement is (in fact) True.

[Footnote] Further material here:
http://worldofdrjustice.blogspot.com/2015/06/on-tarskis-convention-t-expanded.html

[Appendix]

Ernest Gellner on Truth

 “Truth” is, on the one had, a bland and boring concept:  “Paris is the capital of France” is true, “Las Vegas is the capital of France” is false.

Yet in other venues, fraught:  as in, Pravda.  Shading into metaphysical mysticism (“The Search for Truth”).  If I am trying to find out, for which X the sentence “X is the capital of Albania” is true, then in a sense I am Searching for Truth; but really, only for a truth; and indeed, not really under that description:  I merely wish to know what Albania’s capital is called.

~

Relevant quotes, bridging the gap, from works by Ernest Gellner. 

Re Orwell’s Nineteen Eighty-Four:

Freedom is the recognition that 2 plus 2 makes 4 :  not because there is no escaping such necessity, but because only such necessity is a refuge from arbitrary social power. [It is] an extra-social objective truth, which accounts for why such fuss should be made  of a morally and emotionally rather neutral piece of arithmetic.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 4

On a strategy of self-validating beliefs (which he dubs “auto-functionalism”, a term which seems not to have caught on):

It consists of establishing the soundness of one’s beliefs, not directly, in the ordinary and straightforward way, by showing them to be true, but, on the contrary, of deriving their soundness by showing them to play an essential role, to be ‘functional’, in the internal economy of one’s own personality or society … The first step is to put forward a theory of truth: truth ‘really is’ the fulfilment of a biological, or social, linguistic, etc., function.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 14-15

And, re the egregious Althusser:

He argues, in effect, not that Marxism is true, but that the Marxist epoch is still with us.  What is defended, in the end, is not the truth of a doctrine, but its alleged role.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 17

Now,  that all sounds rather feckless and po-mo; but to add some perspective, it is reminiscent of the “regressive justification” of axioms in mathematics, particularly in set theory.

A somewhat more degenerate version of this auto-functionalist approach, endemic to the America of “pot, pop, and protest” -- degenerate in that, unlike that of Althusser et alia, it makes little reference to the world outside the speaker’s individual ego-bubble (indeed, it works best for pure solipsists, for whom the external world need not exist):

In America, it possesses a theory of knowledge, and above all an associated style of expression, which goes back to populism and beyond it … Its basic idea is that sincerity is the key to truth.

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

(It’s amusing to hear such a stance referred to as a “theory of knowledge”, but social scientists really do talk that way, speaking  for instance  of a baby’s “theory of the world”.)

And again, back to the math connection, reporting the fantasies of Michael Oakeshott:

What is proof? -- he asks.  There is no such thing as proof in general, he answers himself.  There is only proof  persuasive for this, that, or the other kind of man.   Cogency of proof  is relative to what you are.  he notices that this does not seem to apply to mathematics, and brazenly comments that just this has always made him suspicious of mathematics.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 180

Actually Oakeshott  put his case too weakly:  varying standards of proof are relevant in mathematics -- indeed, it is only within mathematics  that such scruples have structure and are in point.   In pre-Cauchy/Weierstrass analysis, proof was a bit of a kludge.   Later on, Constructivist qualms  came into play.  And in our own day, we distinguish between theorems whose proof requires the (disputed) Axiom of Choice, from those that can dispense with it.

The ultimate selbst-aufhebung of all such alethic egalitarianism is plain:

If almost everything is true in its own fashion, truth cannot matter very much.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 16

~

Bonus nuggets, from the bottom of Gellner’s crackerjacks-box:

It is a travesty to say that martyrs die for Truth.  Real truths seldom require such dramatic testimony.

-- Ernest Gellner, The Devil in Modern Philosophy (1974), p. 55

the feminine theory of cognition:  that truth is not a matter of exploring or penetrating an external reality, but of gestation and parturition.

-- Ernest Gellner, The Devil in Modern Philosophy (1974), p. 62

.

New Year’s Resolutions

Here’s what’s on-deck for 2012 !!!

(1)  To contemplate the blessed doctrine of the Trinity.

(2)  To, um, ….

(3) To ….    ?????

(x)  ???????????????????????????????????????????????????????????

~ ~ ~

My mind’s as yet too weak to compass this.  But the pen may yet mechanically record, such glimpses as I come across in the writings of others.

~

Father Schall, S.J., in The Order of Things (2007), calls the Trinity “the inner life of the Godhead.”

~

Roger Scruton, in whom the light of logic burns  brightly alive, draws an intriguing conclusion:

God, says the Christian, has three natures, and we come to him by three separate paths:  when we worship him as transcendental law-giver;  when we encounter him incarnate;  and when the Holy Spirit moves through us in its work of concord.  It follows that there are three modes of rebellion against God:  the repudiation of law;  the assault on the sanctity of the human person;  and the desecration of the work of the spirit.

Modern Philosophy (1994), p. 474