Modus tollens tollendus est ! (iterum re-updated anew)

In philosophy there are very few  and perhaps no valid  logical-impossibility  or reductio ad absurdum  proofs.

-- Alasdair MacIntyre, After Virtue (1981; ²1984), p.  101

Wer A sagt, muss auch B sagen.

-- old folk-saying

Die bürgerliche Stellung des Widerspruchs

-- L. Wittgenstein, Philosophische Untersuchungen, #125

An example of epistemological ‘character armor’:

All scientific research programmes may be characterized by their ‘hard core’.  The negative heuristic of the programme  forbids us to direct the modus tollens at this ‘hard core’.  Instead, we must use our ingenuity to articulate or even invent ‘auxiliary hypotheses’, which form a protective belt around this core, and we must redirect the modus tollens  to these.

-- Imre Lakatos, “Morphology of Scientific Research Programs”, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 133

In political or religious ideology, there are obvious analogues of this ‘hard core’.  (E.g. 'sacred cows'.)

~

[This exercise originally began as a simple application of cool, unruffled logic  to a sociopolitical topic too hot to handle in less rarified terms.  The point was, If you were logically consistent, you would … (taceo).  But a working-out of the general intellectual idea  led, inevitably, towards that domain (mathematics, logic itself) where the very idea of consistency is most at home;  and led, less obviously, to a kind of backhanded defence of inconsistency  (wherein we follow our countryman Emerson, in his depreciation of that ‘hobgoblin of little minds’).

Modus tollens is that figure whereby, when a proposition entails a falsehood, that proposition is infirmed.   The procedure finds general acceptance among the logically inclined.  This essay considers cases where otherwise logical people nonetheless kick against the pricks of this restriction.

We now return to the original plan.]

~      ~      ~

It should be obvious that the fundamental objections to racism and sexism … apply equally to speciesism.

-- Peter Singer,  Animal Liberation (1975), p. 6; quoted in Stephen Schwartz, A Brief History of Analytic Philosophy (2012), p. 282.

Singer’s proposition has the form  “P => Q”;  we shall call it the “Singer Sentence”.  It is, so presented, a proposition, and no mere proposal, since he claims that it is “obvious”, presumably because founded upon a generally-accepted principle:  basically, What’s sauce for the goose  is sauce for the gander.    Since that principle is reasonable (we have often invoked it ourselves), though extra-logical, we have no a-priori quarrel with the Singer Sentence at all.

Suppose, however, that you yourself eat meat, wear wool and leather, and are content that rabid dogs should be shot.  Suppose further that -- selfish cad that you are -- you do not, like the saintly naked Jainist, wear a veil before your lips when drinking tea, lest you accidentally imbibe some supernatant insect.   Nor (for shame!)  do you enjoin your dentist and your physician  to refrain from administering anaesthia or antibiotics, on the grounds that these, having been developed with the aid of experiments on helpless animals, are fruit from a poisoned tree.   And suppose that (blast your impudence!)  you do not intend to amend your ways.  Well then, in effect, objectively, you hold that “not-Q”.  What follows from this?

What follows (by the most elementary logic -- the rule known as modus tollens) is that, if you assert the Singer Sentence, then you must needs conclude:  not-P.   -- That, of course, would be a political catastrophe.

Short of adopting Jainism, there are only a few ways out:

(1) Have recourse to a kind of meta-tollentic principle, to the effect that any set of propositions which entail not-P  must itself be denied, P being unantastbar.

(2)  Proclaim the truth of Q, even while continuing your carnivorous habits;  shrug apologetically; reference Emerson re the "hobgoblin of little minds".

(3) Boldly hold the joint and several validity of :
P;  P=>Q; and not-Q. 
Then utter not another word upon the subject.

Such a mindset is known as “doublethink”.  It is not so bad once you get used to it, judging by its millions of satisfied customers.

Cf:

Whether the ethic of ‘speciesism’, to use Richard Ryder’s term, can be put on a logical footing  any more sound that that of ‘racism’, I do not know.

-- Richard Dawkins, The Selfish Gene (1976; 2^nd edn. 1989), p. 10

(That ‘any more’ sounds rather half-hearted …)

~

There is a different sort of logic-chopping that may be operative in a case like this.   Someone wishes, for non-logical reasons, to assert that Q; then trumps up (fallacious) reasons for asserting that P => Q.   (Thus, currently, in a certain political fringe, the desired outcome is “It’s all Obama’s fault”.  Whatever P may arise in the world, a hasty P => Q is asserted, to reach the desired conclusion.)   But, for psychological and political reasons, I do not believe that Singer himself is here guilty of this:  he in all likelihood does feel ethico-logically compelled to deduce Q, however unwelcome that conclusion may be in many quarters.   For he likewise (by different steps) reaches a quite distinct and socially unmentionable conclusion (this time concerning deformed fetuses or the disabled), and one which is precisely of a nature to enrage that segment of society which would embrace his earlier conclusion about animals.  So, no, Singer was not trying to win any popularity contests.

[Footnote] For those of you in the quandary (2), here is the place 4 U to shop (courtesy of Garrison Keillor):

People’s Meats

Most of us accept strict vegetarianism as the best way,  but many find it difficult to change their eating habits.  People’s Meats is an interim solution.  All of our meat comes from animals who were unable to care for themselves any longer.  Hoping to phase out the operation, we do not advertise hours, prices, or location.  We do not deliver.

~            ~            ~

The analysis above was clad in sociopolitical raiment;  but its skeleton is logical, which is subject-matter-neutral.  Consider the following (which is skeletally somewhat distinct, but in ways  unimportant  for our purpose):

Let P be standard mathematical praxis.   (And here -- as seldom -- we actually are referring to the human practice of mathematizing, rather than to the timeless and species-independent truths of mathematics itself, whatever these may be.   For more on the distinction, see here: http://worldofdrjustice.blogspot.com/2012/08/on-nature-of-mathematical-knowledge.html.)

Something similar to the  P => Q step was broached about a century ago; it concludes (while using reasonable background metamathematical assumptions comparable to the “what’s sauce for the goose is sauce for the gander” enthymeme above) that this standard practice leads to paradox.  As John von Neumann put it,

A closer study of the merita of the case, undertaken by Russell and Weyl, and concluded by Brouwer, showed that the way in which  not only set theory  but also most of modern mathematics  used the concepts of ‘general validity’ and of ‘existence’  was philosophically objectionable.

-- quoted in James R. Newman, ed. World of Mathematics (1956), p. 2058

Must we then give up P?   Brouwer (a Dutch mathematician who had previously proved important results) now plays the role of Singer, and went on to his logical conclusion:

A system of mathematics which was free of these undesirable traits, “intuitionism”, was developed by Brouwer.  In this system  the difficulties and contradiction of set theory  did not arise.  However, a good fifty per cent of modern mathematics, in its most vital -- and up to then unquestioned -- parts, especially in analysis, were also affected by this “purge”:  they either became invalid, or had to be justified by very complicated subsidiary considerations.

-- id.

(That final clause is a far more dreadful consequence than might be apparent to those outside mathematics, since mathematicians prize elegance and generality of proof. )

Du muss dein Leben aendern ...

So -- shall we bow to these strictures, and surrender our mathematical meat? 

Von Neumann goes on:

Only very few mathematicians were willing to accept the new, exigent standards for their own daily use.  Very many admitted that Weyl and Brouwer were prima facie right, but they themselves continue to trespass, that is, to do their own mathematics in the old, “easy” fashion -- probably in the hope that somebody else, at some other time, might find the answer to the intuitionistic critique and thereby justify them a posteriori.

In short, the bulk of mathematicians adopted strategy (3) above.

Brouwer, like Singer, went on to make a pest of himself for many years.  

[Footnote]

Intuitionism -- initially a sort of mathematical vegetarianism -- is by no means dead.  Michael Dummet, no crank, espouses intuitionistic logic (I am currently painfully working my way through his essay on the subject, line by line.)   And it has subsequently morphed in ways that are quite beyond me, e.g. in topos theory.

[Footnote 2, a half hour later]  In his essay “The Philosophical Basis of Intuitionistic Logic” (1973), Michael Dummet writes (for our present purposes, the context is unimportant), concerning a proposal that he has just put forward:

What is involved is a thesis in the theory of meaning  of the highest possible level of generality.  Such a thesis is vulnerable in many places:  if it should prove that it cannot be coherently applied  to any one region of discourse,  to any one class of statements, then the thesis cannot be generally true,  and the general argument in favor of it  must be fallacious.  [dbj:  That last phrase has rather a Sherlockian cast to it.]  Construed in this way, therefore, a position in the philosophy of mathematics  will be capable of being undermined by considerations which have nothing directly to do with mathematics at all.

This amounts to offering a hostage to fortune -- specifically, a hostage to modus tollens, in its strong quantified form:    P => ∀x Q(x):  the existence of but a single exception (∃x  ¬Q(x) )  blows the whole game.

~

We need not have recourse to anything so rarified as ethics or metamathematics  to be confronted with an apparently (morally, intellectually) scandalous state of affairs:  It stares us in the face in that favorite tow-headed boy of the philosophy of science, physics.   And here, in that most venerable part of it,  Maxwell’s theory of electromagnetism.  Rock-solid in its own day, it survived (and even helped inspire) the introduction of Special Relativity.  It fit smoothly with the new developments in non-EM forces, graciously uniting with the Weak force to form the Electro-weak, and so forth.  And yet, in the context of the atomic theory, it was (if only in whispers) an intellectual scandal, since it predicted that atoms were unstable:  the whirling electrons would radiate energy away and quickly collapse into the nucleus.  Thus, the material world as we know it, could not exist.

Now, one stance of reaction in the face of so bald a challenge, is to say:  “R-r-right!  Science cannot err;  ergo, the world as we know it  does not exist.  Meta-ergo, we all are just brains in a vat.”   Such, in effect, is the path taken by contemporary eliminative materialists -- a shuffling tribe of hunchbacked ne’er-do-wells, who, faced with their irrefutable failure to derive free will, faith, thought, beauty, aspiration, or much of anything of value, from their prolonged and proctoscopic vivisections of sea-slugs and the like (to which the World of Dr Justice, friend to all creatures great and small, responds with the evisceration of eliminative materialists)  -- these gentlemen (stretching that word to its breaking-point), these … bipeds, conclude that Free Will is an illusion, faith and beliefs and reason  all one great gigantic joke, and that we are all just robots, brains marinating in a vat of simulation, or (barely) glorified sea-slugs.   (In this they are partly correct:  the eliminativists -- mark the name -- are but brains in a chamber-pot.)

So, are those who do not take that bold blind path, whenever some contradiction turns up, in a state of intellectual bad faith?

The answer is a subtle and qualified mmmm….n-n-no-o-o …..  For relief, we turn to Quine.

~

We frequently do turn to Quine for relief,  to savor his elegant pellucid prose, when the cacophony of the agora grows too rebarbative.    Yet it is not for his literary qualities that we seek him now, but for his (jointly with Pierre Duhem) holism.  Specifically, his celebrated doctrine that theses face the tribunal of experience, not single-spies, but as a corporate body.  There are (so to speak) concentric shells of propositions relatively dispensible and relatively central:  but in principle, none is immune to revision.  (Intuitionists have even gone dicking about with the Law of the Excluded Middle, and you don’t get more central than that. -- In that case, Quine was unimpressed:  “When you change the logic, you are only changing the subject.”  Rather a Platonist remark, that, Van.)

Thus, consider again the plight of electromagnetic theory, faced with the atomic paradox.  Its bacon was eventually saved, not by any refinement of that theory itself, but by an entirely unanticipated development:  quantum mechanics.

Now, there is no sense in which a pre-quantum physicist could have said “We saw that coming” or “Toleja so”.   Until the quantum theory arrived from nowhere, physicists had been content to simply live with the contradiction, in the classic fashion of Walt Whitman (“Do I contradict myself?  Very well then, I contradict myself.”)   And their quietism was justifiable, even during the years when no resolution was in sight.  For, Maxwell’s electromagnetic theory had done sterling service  both practically and theoretically, in a host of ways.  [This, I am aware, is intellectually comparable to the classic defense of Mussolini, that he made the trains run on time.]  The fact that it predicted an anomaly on the atomic level … well okay, an anomaly involving THE ENTIRE UNIVERSE BLOWING TO BITS -- but still, an atomic anomaly, not a macroscopic one (save secondarily), suggesting that one might, for the duration (until this beast be slain), simply wall-off the subatomic level (“there be dragons”) and get on with our lives.  It certainly did not make sense to throw out the Maxwellian baby with the anomalous bathwater.

In the case of EM, it turned out that there wasn’t even any bad bathwater to discard:  electromagnetism survived intact.   In the case of the aether, the anomaly was the Michelson-Morley experiment, whose results were later explained by yet another where-did-that-come-from new revolutionary theory, Special Relativity;  and in this case, the aether theory had to be discarded.   But both cases illustrate the thesis of Quine-Duhem, that when a body of doctrine is challenged, it is not initially evident which pieces must eventually give, and some may be close to the core of the structure.   In the case of Relativity and quantum theory, the transmogrification went deeply into the core indeed, upending our notions of space, time, causality, and continuity.   In retrospect, those curiously stable atoms seem not so bad;  the explanation is harder to live with than was the thing that was unexplained.

~

A word on our less-than-effusive approval of Quinean holism.

It is all too easy to imagine self-serving uses of such a principle.  As, Dennis the Menace, caught with his hand in the cookie-jar, exclaims:

“Mother, do not prejudge!  Granted, your B-fibres seem to present an image of someone resembling young Master Dennis with his arm hovering above a receptacle of some sort.  The hand itself -- which you suspect of larceny -- is not visible;  perhaps it has been tragically amputated, in which case the young fellow is more to be pitied than blamed.  Yet, how are we to reconcile this dubious alleged image with the far more desireable thesis that his character is pure as the driven snow?  Remember:  Propositions face the tribunal of experience as a corporate body!  Perhaps 'tis but an illusion of swamp-gas;  nay, perchance the fault lies somewhere in that oft-critiqued principle of Induction …”

(Later, as he sits with his pookie-bear in the familiar corner, he steams:  “But I had her epistemologically …”)

~

Of all human endeavors, surely mathematics is the most sensitive to refutations, however slight.    Yet behold this brawny scoffing attitude, specifically as regards the central and indispensible Calculus (famously the target of Bishop Berkeley’s barbs -- which it shrugged off):

If the calculus had not been ‘justified’ Weierstrass-style, it would have been ‘justified’ anyway.  The point is that the real justification of the calculus  is its success.

-- Hilary Putnam, “What is Mathematical Truth?”

Breezy, that!  Huey Long couldn’t have put it more pithily.

~

And now let us bring it all on home:  confronting the challenge of modus tollens, when a refutation or contradiction or paradox is met, in its home territory of mathematical Logic…

Celebratedly, the great German logician Frege  fell into despair (his masterwork already in galleys), upon being informed by Russell  of the latter’s eponymous Paradox. 

There we see  the logical conscience  at its most delicate.  For  Russell’s paradox, worthy though it be, is rather far-fetched, involving sets-that-are-members-or-not-members-of-themselves  (to which your average mathematician, let alone physicist, will say:  Huh??), all too reminiscent of the well-known but trifling Barber Paradox, involving a purported barber who “shaves everyone who does not shave himself”.  Paradox:  Does he shave himself???   Answer:  Fageddaboudit;  ain’t no such barber.

Rather other was the case of Quine’s Mathematical Logic.   In the version of its first edition (1940), this was shown to entail a contradiction.

Now:  an axiomatic system of logic, such as ML, is so tightly knit, that one bad apple really does spoil the whole barrel -- you can’t just shrug and say, “Nobody’s perfect.”  The situation in question, is generally held to be a catastrophe -- that any system which can derive a contradiction, can derive any proposition at all.

However, Hao Wang stepped in (ever the gentleman), and tidied things up, and all was well:  put right  in the second edition.

Haec fabula docet:  Contradictions are a bummer,  but don’t commit suicide  on their account.

[Footnote]   The Duhem-Quinean corporate-body doctrine  can be stated in terms of modus tollens, thus.  Given

            P1 + P2 + … Pn => Q

and

            ¬Q

we conclude

            ¬P1  ∨¬ P2 …  ∨  ¬Pn

That is, at least one of the co-conspirators whose conjunction led to a falsity, must itself be false.

But this does not imply  that the eventual valid conjunction will involve most or even any of those Pi;  we might even toss the whole lot of them overboard, and usher in a whole different set of conjuncts, to accomplish what we previously attempted with the P’s.  Such, roughly, describes the introduction of quantum mechanics, or the refutation of astrology, or any other paradigm-switch between incommensurables.

In his section on Quine-Duhem, Lakatos writes: 

Some people felt intuitively that the modus tollens from refutation  may ‘hit’ very distant premisses  in our total knowledge,  and therefore were trapped in the idea that the ‘ceteris paribus clause’ is a premiss which is joined conjunctively with the obvious premisses.  But this ‘hit’ is achieved, not by modus tollens, but as a result of our subsequent replacement of our original deductive model.

-- Imre Lakatos, “Morphology of Scientific Research Programs”, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 186

~

On a related note, another look at the notion that, should you ever derive a contradiction -- not as a tollentic hypothetical, but as the result of a deduction -- it’s Game Over for everything you’ve ever done.  A noted logician writes:

In an inconsistent system, every proposition is a theorem.

-- Hao Wang, From Mathematics to Philosophy  (1974), p. 42

Yet a few pages later  he qualifies this:

There is a generally accepted principle  that a contradiction implies everything.  We may yet distinguish proofs of the system which go through contradictions  from those which do not.

-- Hao Wang, From Mathematics to Philosophy  (1974), p. 47

~

Purely linguistic note:  A savorsome turn of phrase for the modal-tollentic move, one which I had not previously encountered:

Assume  towards a contradiction  that there is a set U  ∈ T that is not in T’.

-- Volker Runde, A Taste of Topology (2005),

~    ~     ~

There is a curious parallel, or at least an analogy, between, on the one hand,

(1) the foundational challenges to mathematics  alluded to above, together with, not a refutation of that challenge, but a broadly dismissive (and, arguably, pragmatically well justified) response on the part of professionals in the field;

and on the other hand

(2) recent broad-bore philosophical challenges to adaptationism (neo-Darwinism), along with the overall reaction of evolutionism’s professionals: indifference or -- the socio- and noö-politics of the thing being what they are -- foaming hostility.

Consider in particular  a recent (2010) volume co-authored by Jerry Fodor and Piattelli-Palmarelli, the well-written if pugnaciously titled What Darwin Got Wrong.

Their basic thesis is extensively and subtlely argued; hopefully I don’t overmuch crush it by fitting it into the following nutshell:

=>  The (currently hegemonic) gene-centered adaptationist theory  empirically does not work; indeed, for quite general reasons (having little to do with biology per se) it could not work even in principle;

=> A holistic, phenotype-cum-ecosystem-cum-kitchen-sink-centered approach  might -- might -- prove valid, at least in principle;  but in practice, the problem is intractable from a nomologically explanatory standpoint, apart from occasional lucky breaks.

Or, in the author’s own undistorted prose (p. 127):

To be sure, none of that actually shows that there aren’t laws of selection:  there may be, on the one hand, units of phenotypic change, and, on the other hand, units of ecological change;  and there may be laws that connect the two.  But there’s no reasons to suppose, as adaptationists routinely do, that the units of phenotypic change  are anything like what we generally think of as individual phenotypic traits.

(For connoisseurs of academic-polemical rhetoric, there is actually a sly move here.   While rhetorically conceding the possibility of a pheno-unit/eco-unit correlation, thus retrodictively legitimizing the latter two theoretical posits, these supposititious entities “units of ecological change” are by no means as familiar as other entities in the ontology of biology, and on the face of it  sound sort of bogus…)

~

We must distinguish two categories of challenge to any received body of doctrine:  The Anomaly; vs. Foundational.   The former (speaking just psychologically now) can either represent a mere annoyance  (or even:  Something to be hidden from the fickle general public at all costs, lest they overestimate its importance -- e.g.,  Evolution, Climate Change, anything medical), or a (possibly career-making) challenge.   The latter, to almost everyone, from the man in the street to the faculty lounge, tend to be just d*mned annoying.

Thus, consider  the Lasting Atom Scandal of the pre-quantum years, a poster-child example of Anomaly. (Nobody called it that, but  in all candor  they should have.)  This was phenomenologically egregious (in ways even a layman could understand), and had to be resolved somehow (some day, by someone else) -- indeed, imagine that the problem existed today, rather than a hundred years ago:  You’d have Republicans calling for the de-funding of physics, demanding e-mails archives, etc.  But it did not necessarily challenge -- certainly it did not set out to challenge, chin-foremost -- the nature of (say) Time, or Energy, or  for that matter  the existence of atoms (which had indeed been doubted by scientists, much later in history than most folks realize, but on quite different and less sophisticated grounds).

This anomaly, as we have seen, turned out to be handled in the most gratifying way possible:   We got to retain all the Maxwellian E-M we had laboriously learned, and now the new Quantum Mechanics stepped in (Jeeves-like)  to handle the anomaly.

More recently, there actually have been some challenges to physical theory  at a more basic level:  Time, which hoped to have escaped further challenge  by being subsumed into Space-Time, is once again on the carpet, from string theorists and others, who maintain that such parameters should fall out of a final theory, and not be input to it.  

Had such a challenge been posed, say, in the nineteenth century, it would have been an impudent kick at the foundations.  But now it (allegedly)  grows out of theory:  The theory may be mistaken, but it calls Time into question because it thinks it has something better. 

Much of the history of physics has been like that -- which is partly why it has been (politically) such smooth sailing.   Michaelson-Morley presented an Anomaly to the aether theory -- but no-one had been going around snarkily dissing ideas like distance and simultaneity until Special Relativity came along and re-conceived these.  This was (to use the Hegelian terminology, which here does fit) not a destruction, but a sublation  of the old ideas.

Sometimes there will be a simple anomaly, such as the discovery of continuous-but-nowhere-differentiable-functions, or space-filling curves.  These are eventually gobbled up and incorprated into a more robust and sophisticated mathematics.

Often an original sui-generis gnarly anomaly  will suggest a more general research program:  As, We need to pay closer attention to matterns of continuity and convergence (pointwise, uniform, almost-everywhere, etc., on the analytic side; and eventually the full exfoliation into the neighborhood-systems of topology).

Quite otherwise are challenges that come out of nowhere and that threaten to knock the stilts out from under you.  Russell’s Parodox, reaching Frege at press-time, did not strike him as another delightful puzzle to wrestle with some Sunday morning, but as a poisoned dart in his life-work.   Though they individually made numerous important positive contributions, the lytic work of Gödel and of Brouwer can be read this way, challenging the very notion of validity, truth, and proof.   Denial of the Excluded Middle indeed!  “Sirs, there you go too far!”  (“When you change the logic, you are merely changing the subject.” -- Quine, dyspeptically.)

Schematically (in your worst nightmare):

“Miller has reduced mathematics to set-theory;  and Spiller has shown that set-theory is rotten at its foundations.  Therefore everything that you are doing, or have ever done, or ever could do in your sorry life, is utterly worthless.”

Or:

(Pontius Pilate; Brouwer; post-Modernists):  “What is truth, anyway?  Huh? -- Meh.”

~

[Afternote]   It is a highly useful feature of the Blogspot interface, that it allows hot-linked Labels.    So for instance, were you disposed to read more by or about Mr. Hao Wang, for example, you would simply click on his name in the Labels field, and you will see every post so Labeled, in reverse chronological of posting.   But, annoyingly, there is a stringent limit on how many Labels any given essay is allowed.  We filled this one up with mathy stuff and now have no slack left over for the Darwiny bits.  So here you go:

http://worldofdrjustice.blogspot.com/search/label/adaptationism

http://worldofdrjustice.blogspot.com/search/label/philosophy%20%20of%20science

http://worldofdrjustice.blogspot.com/search/label/ultra-Darwinism

http://worldofdrjustice.blogspot.com/search/label/Charles%20Darwin

Note:  The last two are not redundant upon each other, and indeed have (I believe) zero overlap.  Darwin was not an ultra-Darwinist;  “Je ne suis pas marxiste” -- Marx.

Additional relevant Labels, which (boo, blogspot) would not fit into the Label field  for this post:
http://worldofdrjustice.blogspot.com/search/label/Ernest%20Gellner

[Post-Afternote]  Other examples of modus drasticus tollens.

A mittel-europäischer rationalist recalls “those golden, and, all in all very peaceful final decades of the colonial system”, and adverts ad the hermeneuts:

The argument seems to be -- Descartes led to Kipling.  We repudiate Kipling, so we must repudiate Descartes as well.   The expiation of colonialism must include the repudiation of clarity, for that had been but the tool of domination.

-- Ernest Gellner, Language and Solitude (posthum. 1998), p. 176

We, by contrast, raise a glass to Kipling;  so if this Descartes fellow had anything to do with it, chap can’t be all bad.

“During five literary generations, every enlightened person has despised him,  and at the end of that time  nine-tenths of those enlightened persons are forgotten, and Kipling is  in some sense  still there.” -- George Orwell, 1942

Modus tollens has surprisingly many enemies. Intuitionists, indeed, reject it:

The intuitionist position is that one can only state “P or Q” when one can give either a constructive proof of P  or a constructive proof of Q.  This standpoint has the consequence that proofs by contradiction (reductio ad absurdum) are not valid.

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

Holism à la Duhem-Quine  skirts it with a flanking maneuver:

… arguing that a given observational consequence is deduced  not from an empirical hypothesis alone, but rather from the conjunction of this hypothesis with the relevant set of auxiliary assumptions.   Hence, the failure of the observational consequence does not deductively refute the hypothesis by modus tollens, when taken by itself, but discredits only its conjunction with the pertinent auxiliaries.  

--  Th. Kupka, “Introduction”, in: Adolf Grünbaum, Collected Works, vol. I (2013), p. 2

~

Another example from a political context, where propositions are surrounded by swarms of sensitivities --  Re series of premises and conclusions relating ethnicity, nationalism, and industrialism:

The argument is impeccable.  Its premises are valid.  How can a valid inference from true premises  yield a conclusion which appears to be wholly refuted by historical reality?

Ernest Gellner, Nationalism (1997), p. 32

~

Quite different in purpose and detail from the classical modus tollens, is the assumption of a premise known to be contrary to fact, but where you have antecedently proven (at great expense of elbow-grease) that the assumption of this simple premise does not effect the results of calculations which otherwise must be carried out laboriously.  As, when (having slogged through a bit of integral calculus) you prove that, in calculating the gravitational effects of a ball, you can pretend that the entire mass of the ball is concentrated at the center:  an enormous simplification.  Or again:

We can get the correct answer for the probability of partial reflection  by imagining (falsely) that all reflection comes from only the front and back surfaces.

-- Richard Feynman, QED (1985), p. 107

The elegant idiom for introducing such a foredoomed hypothesis is “Suppose, per impossibile, ...”

~

At the antipodes from the “tollendus” camp, are the celebrants of falsification or falsifiability, associated  in particular  with the name of Karl Popper. ...

~

Miscellaneous additions: 

(1) Psychological observations

People seldom intuit what is unpalatable to them.  [Moreover, one can] eliminate any undesirable indirect implication of their special insight  by means of an additional hilfs-intuition, liquidating the embassassing logical relation.

-- Ernest Gellner, The Devil in Modern Philosophy (1974), p. 95

In short, one ‘answers’ the sceptic  by striding across logical gaps  to conclusions inconsistent with premises that one does not contest.

-- John Watkins, Science and Scepticism (1984), p. 34

(2) The reductio ad absurdum /”self-mate” gambit:

The traditional argument for the primacy of acceleration-retardation  rests on the absurdity of denying it.

-- Stephen Jay Gould, Ontogeny and Phylogeny (1977), p. 216

Chomsky’s book Syntactic Structures, which is regarded by some as a foundation-stone for this kind of activity, has been described by no less an authority than Roman Jakobson  as an argumentum a contrario (Jakobson, 1959), showing the impossibility of the whole enterprise.

-- Hilary Putnam, “Some Issues in the Theory of Grammar”, in  Mind, Language, and Reality (1975), p. 85

.

Minimalism in Mathematics (further updated)

A disclaimer:   What follows is not a substantive proposal, but a suggestive meditation, turning over this minute but multifaceted notion of “minimalism” and seeing how the light glints off.  It is neither better nor worse than a metaphor.

A couple of years ago,  a book-length treatment was published  that similarly plays with the notion of (in this case) “modernism”  -- which, like “minimalism”, is originally a term of the arts -- in relation to math:  Plato’s Ghost:  The Modernist Transformation of Mathematics, by Jeremy Gray.   To the extent that such an enterprise is worthwhile, it is in casting a bit of light from innovative angles, rather than deepening one’s understanding of math itself (though it did manage to get published by Princeton University Press):  it is more like a bull-session than a milestone.    Reviewing the book for American Scientist (Sept 2009), the mathematician Solomon Feferman sums up by quoting a remark by the historian Leo Corry, to the effect that

Extending the appellation modernism to mathematics … is like “shooting an arrow and then tracing a bull’s eye around it.”

Our own effort, in seeking resonances with the prior notion of minimalism, in mathematics, physics, and linguistics, is open to the same remark;  but it is what it is.

In the stylistic spirit of minimalism (and of that pointilliste Wittgenstein), we shall begin with a Delphic  epigram:

Logicism:  a kind of reductionist minimalism.

*

Considering that he took on the whole universe, in his methods  Newton was surprisingly Spartan.  Not only as regards “hypotheses non fingo”, but methodologically:

Newton consistently preferred Euclidean-style proofs.  He used his own calculus only where strictly necessary, and barred algebra from his treatise  entirely.

-- Leo Corry, “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008).

(Cf. a laborious non-analytic “elementary” proof in number theory.)

As fastidious as an Intuitionist!

*

Not a matter of method, let alone of taste, but sheer fact (albeit initially so counter-intuitive as to have been dubbed a "paradox"):

The Löwenheim-Skolem theorem: if a first-order theory has a model, then it has a countable model.

*

The attempts, lasting centuries, to do away with the Parallel Postulate by deriving it from the other Euclidean axioms, represent a remarkable early manifestation of the minimalist instinct.  Success would not have added to our fund of theorems about geometry, nor led to more perspicuous proofs.  The impulse was in part aesthetic.

A topic to explore:  the relation between abstraction in mathematics (an intellectual quality) and mathematical minimalism (which is not antecedently defined, but I have in mind the aesthetic, even spiritual side).

Contrast Finitism, Intuitionism, etc.:  Not Minimalism, but self-castration.

Zijn lange, magere  maar gespierde gestalte,  zijn scherp ascetische gelaatstrekken…

There is also a sterile sort of minimalism:  as, the replacement of the standard set of logical symbols AND, OR, NOT, by a single one --   NOR or  NAND (Sheffer’s stroke).  It led nowhere.

*

A variety of the Minimalist instinct  characteristic of abstract mathematics  is the notion of elegance.   Its role in mathematical practice (it has no purchase on mathematical fact) is reminiscent of, though practically distinct from, that of beauty in the practices of physics.

This, from a man with one foot firmly in either camp, math and physics:

The development of mathematics may seem to diverge from what it had been set up to achieve, namely  simply to reflect physical behavior.  Yet, in many instances, this drive for mathematical … elegance takes us to mathematical structures and concepts  which turn out to mirror the physical world in a much deeper and more broad-ranging way…

-- Roger Penrose,  The Road to Reality (2004), p. 60

(This is the "unreasonable effectiveness" motif.)

*

We earlier noticed what we called “the Dialectic of the Topological Enterprise” -- abstracting-away from rich familiar entities, extracting what seem the essentials, and seeing what happens.   The first step might seem Minimalist, but the consequence is an effusion and exfoliation of new spaces which meet the newly relaxed criteria, and which turn out to have an even richer riot of properties than we began with.   Per se, there is little in all this that might justify bringing in the aesthetically-tinged label of “Minimalist” (not a traditional term in mathematics; the closest you get is “abstract”):  but the aesthetic ethos is there, for all that.  Thus Shing-Tung Yau, The Shape of Inner Space (2010), p. 77:

 

We start with some raw topological space, which is like a bare patch of land that’s been razed for construction.  On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways.

[Footnote 2026:  

> like a bare patch of land that’s been razed for construction

 

Terrain vague, quand tu nous tiens!

More here: 

http://worldofdrjustice.blogspot.com/2026/08/une-promenade-aux-terrains-vagues.html  ]

*

In the arts, Minimalism is a preference:  which, once adopted, is striven for.  In mathematics, you might like to keep things as simple as can possibly be:  but the mathematical facts seem to have a will of their own, at times.   Roger Penrose gives several instances of this, in The Road to Reality (2004).  For instance, with real functions, you can do pretty well as you like; but complex functions have a built-in naturalness.  You can try to define one on a given domain, but they have a mind of their own, and expand to their natural maximal domain by analytic continuation.   Thus, the larger set of numbers, the complex, spanned by the reals and the imaginaries, turn out to be in some sense more ‘real’ -- more round, more natural -- than the “reals” themselves.

Or again:   Suppose, once-bitten by the set-theoretic antinomies, you become twice-shy, and (p. 373)

adopt a rigidly conservative ‘constructivist’ approach, according to which a set is permitted only if there is a direct construction for enabling us to tell when an element belongs to the set.

(I picture this hypothetical constructivist as being played by Graham Chapman doing his officer’s shtick.)   But alas!  Penrose runs through the Turing/Cantor diagonal arguments and concludes (p. 376):

What this ultimately tells us is that, despite the hopes that one might have had for a position of ‘extreme conservatism’, in which the only acceptable sets would be the ones -- the recursive ones -- whose membership is determined by clear-cut computational rules, this viewpoint immediately drives us into having to consider sets that are non-recursive. … We are always driven to consider classes that do not belong to our previously allowed family of sets.

This is either a baffling, even a provoking mystery, or a simple consequence of what the Cantorian Realist indeed believes:  that these things are Out There, independent of ourselves (this might remind you of a certain Deity), and you can’t just methodologically sweep them away.   U B the judge.

(For a similar example applied to physics, click here.)

*

Pedagogical observation from a wise observer, who has been around the block:

Instead of the principle of maximal generality that is usual in mathematical books, the author has attempted to adhere to the principle of minimal generality,  according to which  every idea should first be clearly understood in the simplest situation;  only then can the method developed  be extended to more complicated cases.

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

*

The nec plus ultra  of mathematical minimalism  is probably Category Theory -- which, however, I cannot elucidate, since I do not understand it.  It contains such things as the Forgetful Functor (this pops up in several introductory treatments, so it’s not as though I’m grasping at straws), which, given an algebraic group, “forgets” the group structure, leaving you with just a set  (excuse me: an element of the Category of Sets.)   Great -- die Gruppe ohne Eigenschaften.   The only way this even begins to seem to have a point  is if you then consider the adjoint functor, from sets to… free groups (these being a desolate Last Year at Marienbad landscape, again groups with the flavor removed).   Category theory looks at the bare bones common to many a different area of mathematics -- rather as though one were to study portraiture by looking at stick-figures.
(Actually, there is an analogy with the motif-index in folklore.  So, not knocking it here...)

~

On Ramanujan’s notebooks:

There were thousands of theorems, corollaries, and examples.  For page after page, they stretched on, rarely watered down by proof or explanation, almost aphoristic in their compression, all their mathematical truths  boiled down to a line or two.

-- Robert Kanigel, The Man who Knew Infinity, p. 204

The reasons for this were twofold.  Ramanujan himself was not particularly aphoristic.   But he had never absorbed the modern notion of proof, which would take up so much more space;  and as a poor man in India, he suffered from a shortage of paper.

~

From a logician:

The power-set operation has been interpreted  in the constructible hierarchy  as thinly as possible … We might be tempted to think of [the minimal model] as realizing a sort of contrary of the principle of plenitude -- a principle of paucity, if you will.   …  The principle of ontological parsimony … encourages some authors to eliminate individuals and un-well-founded classes.

-- Michael Potter, Set Theory and its Philosophy (2004) , p. 254

(All so difficult.  Why not relax with a mystery story instead?  Cool ones here: )

http://www.linguasacrapublishing.com/justice.html

Oh, snap!

Par aquit de conscience,  I quote the following against myself -- against the whole thrust of the “Theologia mathematica” series:

Regarding a public lecture by the mathematician L. Brower:

Bei einem Vortrag eines der Führer der Intuitionisten  wurden  diesem  entgegengehalten:  “Ja wenn es auch die Mathematiker heute nocht nicht wissen,  so wird es doch z.B.  der liebe Gott wissen,  wir können also doch annehmen, daß es  entweder das eine  oder das andere ist.”   Darauf erwiderte der Intuitionist:  Dann müßen Sie  den lieben Gott  sehr genau kennen, wenn sie wissen, daß er es weiß.

-- Walter Lietzmann, 1925

(Quoted in: Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 217.)

This, though, from a solipsist;  here he is:  like the face of the damned.

Het eenige ware voor mij  is mijn eigen ikheid van het oogenblik …

(Caption quotation:  id., p. 27)

W-Allahu a`lamu.

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.

*

*     *     *

~ Commercial break ~

Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

*     *     *

[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

.