Saturday, January 21, 2012

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.]

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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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~ Commercial break ~
Relief for beleaguered Nook lovers!
We now return you to your regularly scheduled essay.

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[Update 16 February 2012]
Now this:
Facts Are Stupid: “story-truth” vs. “happening-truth”.
We comment on that article here:

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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

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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 intuitionstic 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

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