Elective Acolytes

 

Not many are called, and even fewer chosen.

Tales from the Vienna Circle Woods

 

Vienna, 1927:

After several more appointments with Schlick alone, Wittgenstein had been persuaded to get together with a select group from the Circle, though he had never once attended an official Circle gathering.

Waismann began, subconsciously, to imitate Wittgenstein’s speaking-patterns.  Schlick began to attribute  some original ideas of his own  to Wittgenstein, though they had been expressed before he had even read the Tractatus.  Wittgenstein must have approved of this submissive attitude:  by the fall of 1929  he was choosing to restrict his discussions to Schlick and Waismann alone, usually at Schlick’s home.

-- David Edmonds, The Murder of Professor Schlick (2020), p. 48-52

 

Though recalcitrant about joining, or even really following the lead of, the Wiener Kreis, Wittgenstein did attend their summer 1930 congress in Königsberg (the one-time hometown of Kant, who was the Circle’s Aunt Sally), which honored him with a presentation re “The Nature of Mathematics:  Wittgenstein’s Standpoint”.  Here he again encountered a couple who had known him as a teen:

Present too at Königsberg  were Professor Stanislaus Jolles and his wife, Adele.  They were the couple with whom Wittgenstein had stayed during his spell in Berlin, 1906-08.  Their relationship with their lodger had been affectionate;  Stanislaus felt protective and paternal toward “Little Wittgenstein”, as they called him.  But, as so often with Wittgenstein, there had been a rupture, and, typically again, it seems to have arisen from Wittgenstein’s perception that his hosts had fallen short of his exacting standards.

-- ibid, p. 97

It didn’t take much for the prickly master to cancel you (or, to use the term current among Berkeley lefty groupuscules during the 1970s before cancel acquired its later flavor among the Woke, to “break with” you).

If you did manage to remain in Wittgenstein’s good graces, it was a mixed blessing, for he tended to treat such scholars as acolytes or thuriféraires, rather than full colleagues.  Consider the case of Friedrich Waismann, mathematician and physicist, and a core member of the Kreis, who enjoyed the rare privilege of occasionally being closeted with Wittgenstein alone:

 

Waismann’s principal function was prompt and note-keeper.  One philosopher later described his relationship to Wittgenstein as one of “glove puppet to controlling hand.”  … There was something shocking about the degree to which he subordinated his interest to those of Wittgenstein, and the ingratitude with which his efforts were rewarded.
-- ibid, p. 101, 104

 

Bertrand Russell, a supremely well-established logico-philosophical panjandrum himself, could not be so scanted; yet “Bertrand Russell, according to Ayer, was now downgraded to being merely ‘a forerunner of the Christ (Wittgenstein)’.” (ibid, p. 109)

Russell himself was cordial to Wittgenstein, who had read Russell’s mathematical philosophizing, and who in 1911  showed up at Russell’s rooms in Cambridge, and soon formed a close relationship.   Russell had been a member of the Cambridge Apostles, cosily known to themselves as simply “the Society”;  and Wittgenstein received an invitation to join.  But once again, Wittgenstein held back from any circle whose multiplicity made them unwieldy to dominate as a whole.

In a letter of 1913, Russell wrote to a friend:

My friend Wittgenstein was elected to the Society, but thought it was a waste of time, so he imitated henry john roby and was cursed.

-- The Autobiography of Bertrand Russell (v. 1, 1951), p. 364

 

(The reference is to an earlier selectee, who disdained ever to attend the Apostle conventicles;  the miffed members promptly canceled him by decapitalizing his name for all eternity, and pronouncing a ritual malediction from time to time.)

 

~  ~  ~

 

In 1932, through the good offices of Gilbert Ryle, the Oxford philosopher A.J. Ayer  was introduced for the first time  to Wittgenstein at Cambridge.   Queried by the eminent Austrian  as to what was the most recent book he’d read (cocktail-party filler or ice-breaker, one would have thought),  Ayer replied, La Vida es Sueño, adding modestly that he hadn’t understood it very well.  That was actually owing merely to his shaky Spanish; but Wittgenstein apparently took it as trenchant skepticism, much along the lines of the logical empiricists protesting that they “didn’t understand” (i.e., considered as rubbish) a great many everyday non-scientific statements.  “From then on he treated me as a protégé.” (-- A.J. Ayer, Part of My Life (1977),  p. 120.)

Ryle appears later to have some regrets about those good-offices:

 

Ryle had met  and got along with  Schlick, but his encounters with Wittgenstein left him with serious reservations.  This was evidently a man who needed acolytes, not colleagues;  someone always on the brink of an explosion,  too quick to divide the world into the saved and the damned.

-- Nikhil Krishnan, A Terribly Serious Adventure:  Philosophy at Oxford  1900-1960, p. 54)

 

In the same memoir, the mild-mannered Ayer recounts the brusque reception that met Waismann, who had so long sedulously served Wittgenstein, when they later became colleagues at Cambridge University:

Waismann was Jewish, and when Vienna fell to the Germans  he fled with his family to England.   He went to Cambridge, which was willing to accept him, but Wittgenstein did not desire that what he regarded as a deceptive echo of his own thought  should be audible in the same university, and therefore announced that anyone who attended Waismann’s lectures  would not be allowed to come to his.

-- A.J. Ayer, Part of My Life (1977),  p. 132

 

Horresco referens, but such petty and revanchist behavior reminds me of Tr*mp.

~

 

The incomparable logician Kurt Gödel  played a notably honorable role in this Kreis of often quarrelsome prima donnas.   While faithfully attending their gatherings, he was content so remain modestly in the background, and be taken for a subaltern, while all the time excogitating, leading to work more important more lasting than anything any of the others in the Kreis would accomplish.   Again, let the portraitist tell it:

 

Although almost all Circle members became convinced that, drawing on Wittgenstein and Ramsey, they had solved the problems of mathematics -- that mathematical truths were a type of tautology -- Gödel had sat quietly at Circle meetings  without believing a word of this.  He was a mathematical Platonist.
-- David Edmonds, The Murder of Professor Schlick (2020), p. 148

 

Those affordances of the Circle, and of Wittgenstein in particular, were worse than useless: positively stultifying for the practice of mathematics.  [For our essays on the subject, consult

=> https://worldofdrjustice.blogspot.com/search/label/Platonism  

]

 

As for the excesses of acolytism, Gödel kept a level head:

 

While Schlick and Weismann revered Wittgenstein, Gödel was among several  bemused by the cult-like deference he inspired in his acolytes. … [And later, when Wittgenstein reigned at Cambridge:]  Many students became disciles -- who, like Waismann in Vienna, subconsciously came to mimic his mannerisms.

-- ibid, p. 149, 247

 

The socio-historian and polemicist Ernest Gellner  provides a glimpse of the Cambridge period of Wittgenstein’s ascendency.  There grew up

 

… the first set of ‘companions of the prophet’.  Initially, there was a small, carefully vetted, conventicle of devotees in Cambridge, in the years preceding the Second World War. … But the movement grew …

Maor premise:  all cultural cocoons, all forms of life, are valid and self-sufficient, and Wittgenstein has shown this to be the case.  Minor premise, never spelt out or discussed, but operationallly taken for granted:  only our cocoon is of any interest. … Entry to Wittgenstein’s seminar was restricted at the master’s whim, and the ideas circulated in privately-copied typescripts which Wittgenstein himself refused to have published.  This esotericism greatly enhanced the appeal of the ideas, which were treated as a major revelation by the adepts.

 

-- Ernest Gellner, Language and Solitude (posthum. 1998), pp. 160-165

 

(Parallels from the history of linguistics  during the Chomsky years, could be adduced…)

~  ~  ~

 

Let it not be supposed that the field of philosophy is at all atypical  among academic specialties, as regards such interpersonal rugosities.  Parallels from the history of linguistics  during the Chomsky years, could be adduced.  As, the mathematician Mark Kac, father of the linguist Michael Kac, remarks in his memoir:

 

Linguistics is a strange field, full of cliques and fiefdoms, each fiercely attached to its staked-out territory, and consumed with enmity toward the others.

-- Mark Kac, Enigmas of Chance (1985), p. 107

 

For an extended look at academic acolyte relations, pour yourself a brandy  and relax with these:

 

=> http://worldofdrjustice.blogspot.com/2013/02/chomsky-freud-and-problem-of-acolytes.html

and its appendix

=> https://worldofdrjustice.blogspot.com/2012/12/the-agony-and-acolyte_28.html

 

These include anecdotes about my fondly former Berkeley Doktorvater  in Rom. Phil.,  Professor Yakov Malkiel, including portraits of the (pro tem) Malkielitas, and one (canceled) Malkielito.  One will detect, in the telling, a Nabokovian tone;  which is only just, as the Malkiel family and the Nabokov clan  were BFFs in Berlin, back in the ‘30’s (while it lasted).

 

Truth and Provability (expanded)

[A footnote to this.]

Trying to suss out the nature of Truth by staring straight at it  is like attempting heliology by staring at the Sun.   In both cases, more assimilable enlightenment  comes from the corona.

Thus the related but distinct notion of Provability. Gödel was the first to neatly delineate the two notions:  the Propositional Calculus is deductively complete (i.e., all truths may be derived via the defining rules of the system), whereas anything as robust as the integers is deductively incomplete :  one can, within that more expressive system, formulate statements that are true but (intra-systemically) unprovable .  Previously, the Formalists (Hilbert et al.) had seen provability as an analytic explanation of what is meant by ‘truth’ itself.

Kurt Gödel
("I'm looking throuuuugh you...!")

Pre-scientifically, and indeed theologically, we are not surprised that the two notions should not be equivalent (though of course we had no notion of the precision afforded by Gödel’s results).  Some things, existing from before we were born, and lasting ever after,  just are true;  why should they be logically derivable, or even humanly comprehensible?

~     ~     ~

There is a perhaps related notion within the philosophy of language:

~ Whatever can be meant,  can be expressed.  ~

(Intended-meaning : truth  ::  expression : provability)

Yet the very existence of our word ineffable, suggests that we at least entertain the possibility that this may not be so.

In the words of a Neothomist philosopher:

La communicabilité de la pensée  est un fait immense, incontestable  et elle n'est possible que par le langage;  mais tout suggère que, dans le langage, la pensée reste  par nature  essentiellement autre que son moyen de communication.

-- Etienne Gilson, Linguistique et philosophie (1969), p.  39

Taking this in a maximalist sense  would imply, not merely that certain thoughts are ineffable, but that no thought is quite equivalent to its verbal expression.

A further discussion of this topic may be consulted here:
The "idea" idea.

~

It is clear that we need a notion of truth independent (or partially independent) of provability (however vexed and vague that notion must necessarily be, without such buttress), else how to assess the validity of what purportedly is proved.   In the Awful Warning against dividing by zero, delivered to pupils in their tender years, the young scholars meet a Falsidical Paradox:  an impressive display of mathematical handwaving, purporting to show -- there it is on the blackboard, plain as day --  that zero is equal to one.  Yet even a lad in short-pants surely resists such demonstration, if only with an incoherent “Uh, no, it’s not.”

Compare the smoke and mirrors with which philosophers and neuroscientists demonstrate that consciousness is an illusion, and free will  a will-o’-the-wisp.  Equally we reply: “Uh, no, it’s not.”

~

Relativist conceptions of truth  are familiar.  As the Harvard philosopher wittily put it, mimicking the cant of the post-‘60’s undergraduates:

“You may not be coming from where I’m coming from, but I know relativism isn’t true for me.”

-- Hilary Putnam,  Reason, Truth, and History (1981), p. 119

Less familiar is a relativist conception of provability.   Here, Ernest Gellner comments on the position of fellow-philosopher 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.

Constructivist Angelology

But yet when considered, may help us to enlarge our thoughts  towards greater perfections of it  in superior ranks of spirits. … The several degrees of angels  may probably have larger views.

-- John Locke, An Essay Concerning Human Understanding (1690)

Man’s understanding, though allied to the angelical, operates differently.  The angels understand intuitively, man by the painful use of the discursive reason.

-- E. Tillyard, The Elizabethan World Picture (1942)

It is presumably not obvious to the chimpanzee (or, if this be setting his smarts too low, to the humble woodchuck) that for all m, n in Z, m + n = n + m.  Nevertheless, in his daily scurryings and burrowings, he will repeatedly meet up with particular instantiations of this modest truth.

            For the woodchuck (at any event the southern northeastern lesser striped variety) builds a number of nests and other temporary dwellings, each of which has the framework of a variously triangulated  polyhedron, built tinkertoy-fashion from a fixed number of sticks.  Now, gathering them one by one would take too long, nor can the tidy woodchuck stand to have any sticks left over.  So when constructing his summer dwelling -- an icosahedron, which needs thirty sticks (did I get that right? My calculating powers are not much beyond those of a woodchuck) -- he normally harvests a jubjub bush, which has twenty-two sticks of exactly the right specs and which blooms in the spring, then rounds it out with the eight-sticked glubglub bush, which sprouts slightly later. 

But then one year, the blooming of the jubjub was delayed, and the woodchucks despaired.  All but one, the enterprising Willie, who went doggedly (or groundhoggishly) ahead  and harvested the available glubglub, supplementing this  when the jubjub arrived slightly later.  This remarkable exploit was recorded in the annals: for 22 then 8, one may substitute 8 then 22.

            It was subsequently found that a mubmub bush (18 sticks) followed by a nubnub bush (12) would do just as well – und zwar, in either order!  This fact too was recorded.

            The years went by, then the centuries, and the millennia, and the annals grew to seven times seventy stout volumes, densely filled with such arcana as: a cube-for-cubs may be constructed of a lublub (7) plus a rubrub (5), and this in either order; and so on for billions of examples.  All this was considered a branch of botany, a purely empirical science.

            By this means, the woodchucks arrived at an analogue of Babylonian mathematics.

Interlude:   A physicist depicts the arithmetical state-of-play in a papyrus from Egyptian/Babylonian times:

It records the resolution of a great number of fractions  into a sum of aliquot parts,  the original numerator always being 2:  as, for instance,

2/97 = 1/56 + 1/679 + 1/776

But no rules are given for effecting such resolutions, and the whole treatise seems to be a mere compendium of results obtained by repeated trials.

-- James Jeans, The Growth of Physical Science (1947 [posthum.]; 2^nd edn. 1951), p. 11

            Until one day one Wisedome Woodchuck, a distant descendant of Willie, figured the whole thing out, and in a remarkable demonstration of only eighty pages (rather hard to follow, but sound), showed that m + n = n + m  was a perfectly general fact, replacing the seven-times-seventy volumes at a stroke, and freeing up his brethren for yet further architectural innovations, which previously had been shunned, as their particulars were not yet in the book.  The annals were placed in a museum, which the elder woodchucks might still visit, marveling at favorite exhibits (as who could forget that remarkable winter, when 5,878 + 519 turned out to be equal to 519 + 5,878?  A tour de force!). Meanwhile generations of young woodchucks (the pride and despair of their parents, who could not follow them into Canaan, with their aging brains) studied Wisedome’s proof, breaking their little heads against it.

           

Meanwhile in Metropolis… The humans, learning of this, politely saluted Wisedome’s modest accomplishment, and experienced a pang of sympathy for woodchuck-kind; yet felt no inclination to visit their Museum of Particular Results: for which they felt, indeed, a kind of horror.  And even the general result, while true, is somehow to us not truly interesting. In any case we are all too busy wrestling with the Riemann Hypothesis, to have time to look back.

Meanwhile in Elysium, where throne the angels sensu strictior, the lowest order of angelic beings sensu lato, a mock compliment is paid to Andrew Wiles, who finally figured out that little Fermat puzzle, with which the angel-kind  are wont to amuse the nursery.  Not that the angels arrived earlier at his proof, nor any refinement thereof.  They simply scoop up a few infinities of integers with their fractal fingers, twist them this way and that—and see, it doesn’t fit!  Simple.

            Moreover, all facts about all structures of ordinal type omega, whether or not deducible by any finite axiomatization, are equally transparent to the angels. They just look.

            So, is Elysium the mathematical Paradise?  Not quite…

            In a remarkably lucid and accessible article*, which should be packed into every pupil’s lunchbox by a considerate mom, Gödel observes that our continuing failure to resolve Cantor’s continuum problem, left over from the previous century, is quite an embarrassment.  It means that we are unable to wrap our minds around the very simplest multiplication problem possible, beyond the finite ones that these days can scarcely stump a woodchuck. Namely, two times two (times two, times two – keep going).  He writes:

            “It is easily proved that the power of the continuum is equal to 2^(aleph-nought). So the continuum problem turns out to be a question from the ‘multiplication table’ of cardinal numbers: namely, the problem of evaluating a certain infinite product (in fact the simplest non-trivial one that can be formed).  There is, however, not one infinite product (of factors > 1) for which so much as an upper bound for its value can be assigned. […] It is not even known whether or not m < n implies 2^m < 2^n.” 

            We are  so to speak  staring helplessly  at a pile of sticks.

            Nor does the subsequent Cantor+Cohen demonstration of the independence of the continuum hypothesis  from a particular system of axioms for set theory   set the matter aside. Gödel had already anticipated Cohen’s result, and wrote:

A proof of the undecidability of Cantor’s conjecture from the accepted axioms of set theory (in contradistinction, e.g., to the proof of the transcendency of pi) would by no means solve the problem.  For if the meanings of the primitive terms of set theory … are accepted as sound, it follows that the set-theoretical concepts and theorems describe some well-determined reality, in which Cantor’s conjecture must either be true or false.

            Indeed Gödel suspects that the Cantor conjecture is actually, factually false: which means that somewhere, among the actual literal real numbers, there is hiding a set of cardinality intermediate between aleph-nought and its power set, with definite members which the angels could name.  Not, however, the lowest order thereof; this lies beyond them.  But at the next step up, the archangels hang these sets from mobiles over their infants’ cribs.  In fact a woodchuck may somewhere inadvertantly have used one of these sets for nesting materials, and even now lies sleeping on it – a night of troubled dreams.

            So much for a simple pancake-stack of omega-many deuces – the limit of the lower-angels’ ken.  What about the square root of omega-to-the-omega; or cross sections of fibre bundles on toroidal cap-omega-cross-theta space? For each level of angels, there will be something beyond them that they just don’t get.

*

There are two poles of the range of approaches to the problem of infinities.  One is that of the badger-like Brouwer, who simply sweeps the chessmen to the floor, folds up the board and goes home.  (An only somewhat more amenable figure, says Gödel, is Weyl, who allows as how there might be something to board games, but suggests we play checkers – or Chutes ‘n Ladders – rather than chess.)  The other pole says:  Infinities are tricky, but they all exist, and are present to the Infinite Mind. Gödel himself uses that term, e.g. noting that Ramsey’s admission of formulae of (countably) infinite length  might be constructivistic for an infinite mind  but not for our own.  Gödel does not, however, seem to feel much need for any desperate appeal to such a mind, in the course of an ordinary day, since he -- like Badger’s amiable friend the Water-Rat-- is a thoroughgoing Realist, and comfortable as such in his own skin.  For him the assumption of infinite classes “is quite as legitimate as the assumption of physical bodies, and there is quite as much reason to believe in their existence.”  The outwardly gloomy Austrian  is really the jolly Dr. Johnson of set theory.

            Only now there’s a problem, of a sort which did not confront the schoolmen, who never counted on the uncountable:  the Infinite Mind is all very well, but -- Which infinity did you have in mind?

            Who comprehends *everything*? God does, by definition. Yet He cannot be simply the crown on a tower of constructively ascending intelligences.  He is like an “inaccessible cardinal” – and not the first.  Nor perhaps ‘the last’, if there is no last.  Whatever He might be, there is Cantor in the wings, grinning, waiting to perform a Power Set on God, yielding – what?  -- Nothing one can begin to commence to pretend that we can approach with our sadly finite understanding.

            All of which suggests, if nothing else does,  that God is something more and other than an alternately wrathful and affectionate granddad  with a perfectly enormous white beard – however much longer that beard might be, than the stubble which disfigures your chin or mine.  Who one day, apparently from sheer idleness, as one might choose chocolate, chose the Jews.  Who later, some say, cast a Jove-like eye  on a certain Palestinian virgin.  And who at present is very angry indeed with the Democrats (or the Ravens, or whomever).  Yet what He in fact might be, we cannot even begin to imagine anyone’s beginning to conceive.  (Cf. the suggestion of 1 Kings 8:27  that the heavens themselves have heavens (and so on up); and that the whole omega-tower of them  cannot encompass God.)

*     *     *

~ Commercial break ~

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

We now return you to your regularly scheduled essay.

*     *     *

            We actually wind up with a sort of hamstringing of the Ontological Argument. Notoriously its conclusion does not really follow from its premise;  but now even its premise limps: “Since we can imagine a Perfect Being…”  But that’s just it, we can’t!  Not even little infinite bits of one! Yet paradoxically (and God reportedly loves paradox – at least Chesterton does, His publicity agent on Earth), this seeming stomping on the prostrate corpse of the offspring of Anselm, this despairing cry that somehow even Infinity does not suffice, so far from opening the agora  to legions of snickering atheists chanting “Toleja so!”, points somehow upward, -- outward,   -- onward ….  Praise Him!

Postscript:

John Locke himself, normally regarded as the Poster Boy for Empiricism, of I'm-from-Missouri common-sensicality, yet delivers himself of this (Essay, III.vi.12):

That there should be more species of intelligent creatures above us, than there are of sensible and material below us, is probable to me from hence:  that in all the visible corporeal world, we see no chasms, or gaps.

That is to say:  The gap between ourselves, and God, must somehow be filled, according to the Principle of Plenitude.

And again (IV.iii.23):

He that will consider the infinite power … of the Creator of all things, will find reason to think, it was not all laid out upon so inconsiderable, mean, and impotent a creature, as he will find man to be;  who  in all probability, is one of the lowest of all intellectual beings …

Angels of all sorts are naturally beyond our discovery, and all those intelligences, whereof ‘tis likely there are more orders than of corporeal substances, are things, whereof our natural faculties give us no certain account at all.

Since theism is far from central to Locke’s Essay, it is curious to see the emphasis on this scala naturae idea.

--------------

*”What is Cantor’s Continuum Problem?”, repr. Benacerraf & Putnam, eds., Philosophy of Mathematics.

~

Postscript:  For the possibility that the structure of certain mathematical truths relating to an infinite domain  might resist any but a case-by-case “Babylonian” approach, cf. the quotation from Michael Dummett towards the end of this post:

http://worldofdrjustice.blogspot.com/2012/01/what-is-truth.html

Compare further (re ascending ranks of abstraction and generality):

"The Ladder of Abstraction"

.

Cream for your Coffee

[The following paragraph has just been added to our essay, "A New Proof of the Existence of Coffee-Cups".]

Having at length satisfied ourselves as to the reality, or at least reliability, of coffee-cups, would should not  on that account sink back into an attitude of Moorean complacency (“I’m all right, Jack;  I’ve got hands”).  For our commitment to these  suggests yet further commitments, which we had not realized were there to assess.  Such as :  Realism with regard to quantum state vectors.

The question of ‘reality’ must be addressed in quantum mechanics -- especially if you takes the view that the quantum formalism applies universally to the whole of physics -- for then, if there is no quantum reality, there can be no reality at any level.

-- Roger Penrose,  The Road to Reality (2004), p. 508

And:

The question of the objective existence of the objects of mathematics … is an exact replica of the question of the objective existence of the outer world.

-- Kurt Gödel, “What is Cantor’s continuum problem?”, in American Mathematical Monthly, 1947.

A complex Schrödinger equation,

after the Collapse of the Wave Packet

In for a penny, in for a pound.

The Ladder of Abstraction (with added rungs)

The following  logically belongs in the “Abstraction” section of our essay Consilience in Mathematics.  But as that effort is growing overlong,  we begin to cultivate here a particular idea  building upon that of abstraction simpliciter :  namely, the tendency, in modern mathematics -- and indeed this may serve virtually as the defining characteristic of modern (even: modernist) mathematics -- to abstract from any given abstraction, layer upon layer, rise upon rise, to a virtual (topless/cloud-topped) Babel, reaching to the Beyond.

(Oh, and here again we have a term from the arts, Modernism, which, as it includes “abstract art”, metaphorically applies to mathematics.  Compare our earlier essay on Minimalism in Mathematics.)

In normal practice, mathematicians mostly talk to one another -- and indeed, mostly just to those within their own hyperspecialized neck of the woods.  But occasionally, one writes an undergraduate textbook, and thus must descend to earth, if only for the nonce, and address the laity.  Thus:

This “intrinsic” formulation of Calculus, due to its greater “abstraction”, and in particular  to the fact that, again and again, one has to leave the initial spaces, and to climb  higher and higher  to new “function spaces” (especially when dealing with the theory of higher derivatives), certainly requires some mental effort, contrasting with the comfortable routine of the classical formulas.  But we believe that the result is well worth the labor, as it will prepare the student to the still more general idea of Calculus on a differentiable manifold.

-- Jean Dieudonné, Foundations of Modern Analysis (1960), p. 141

We dub this the “ladder of abstraction”, taking the phrase from our teacher of yore. Referring likewise to ascent into functions-of-functions, and function spaces, and functions from one function space to another, and to the duals of all that:

Detached from any context, this construction is a pointless formality.  But as we move up the ladder of abstraction, we find that constructions such as this  become commonplace …

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 43

The metaphor of ascent is well attested.  Gödel speaks of

the infinite series of ever stronger axioms of infinity, each of which expresses a new idea or insight.

-- quoted in Hao Wang, From Mathematics to Philosophy  (1974), p. 325

A mathematician writes of

the inferential staircase  leading from the laws of physics  to the world that lies about us …

-- David Berlinski, “The End of Materialist Science”, collected in:  The Deniable Darwin (2009), p. 160

~

Saunders MacLane,  in his book Mathematics:  Form and Function (1986), p. 36ff, has a section called “Mathematical Activities”, structured somewhat like our own in the Consilience essay.  Some of the topics are the same (analogy, abstraction, generalization), while others, not relating to consilience especially, differ (conundrums, axiomatization, proof).  One, intrinsic structure, seems to relate to consilience, but is only briefly developed; and the last, completion, we have treated under the more Quinean label of rounding out.

Now, abstraction and generalization are related notions, but neither entails the other.  MacLane acutely adduces the example of group theory.  Originally, this grew out of the concrete examples known as groups of transformations.  Later, algebraists abstracted into abstract groups.  Whether a generalization has thereby been achieved, is (as Chomsky likes to put it) “an empirical question”;  and in this case, it turns out, it has not. “No new groups turn up in this process, in view of the famous theorem of Cayley, which asserts that every (abstract) group is isomorphic to a group of transformations.”  Thus, in this case, the ladder of abstraction has only one rung.  (By contrast, abstract rings do turn out to generalize upon their original model, rings of integers.)

~

Seeking analogues of the Ladder of Abstraction  outside of mathematics proper, I happened upon this:

Quine suggests that levels of abstractness, modeled on Russell’s Theory of Types, might be established.  “In the beginning  there are only concrete objects.”  These constitute type zero and are the values of bound individual variables.  “To be is to be a value of a variable.”  Next comes first-order classes and relations:  they constitute entities of type 1 and are the values of bound predicate variables.  Classes of classes, and relations, constitute entities of type 2;  and so on.

-- Harold Lee, “Discourse and Event”, in: Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 297

The resemblance to Russell’s Theory of Types had not escaped me, but I rejected mention of it, since, rather than leading -- as the Ladder does -- to ever greater depth (the metaphor is here in distress -- maybe think of it as a ladder down a mineshaft), it seems to lead mostly to More of the Same.  In other words, forming those strata, as described above, is less like the dizzying and ethereal Abstract Ascent of mathematics, than simply forming new sets via the Power Set operation (a new and larger set consisting of all the subsets of the original set).  Now this, if we start with a finite set, leads absolutely nowhere.  It’s just like counting.  If you start with the whole of the Natural Numbers, now the Power Set operation does become more powerful, leading to new and incomparable levels of infinity.    Whether this leads to true new depth, or is rather a mere formal exercise, I do not know, since I lack all intuition of any infinities beyond the countable, let alone the Power of the Continuum or Measurable Cardinals.  Perhaps it does;  espresso-sodden Berkeley conversations about Quality emerging out of Quantity, return to mind.

Still, I am inclined to doubt it.  The very fact that the fellow can say “and so on”  virtually proves as much.  For there is no “and so on” to true mathematical abstraction.   There is nothing mechanical about such ascent -- it is more like a miracle.  You can proceed only one step -- nay rather, one leap at a time;  and the interval between leaps may take decades or even centuries.   Above the calculus lies Function Theory;  above that, Topology.  Above that, Algebraic Geometry. Far, far above us, hovers Category Theory, unreachably aloft.  And far, far above and beyond that, soars Topos Theory.  What comes next  is known only to angels.

To vary Nestroy’s celebrated epigram -- “Bis die Topologie gehts noch, aber von da bis sheaf theory  zieht sich der Weg.”

Additionally, Quine introduced the term semantic ascent.  There is some similarity to Gleason’s ladder of abstraction, but the ascent doesn’t go very high, and Quine himself -- perhaps surprisingly for a logician -- is wary of the upper reaches, preferring basic-level entities  behaviourally grounded.

Here the Russian author A. D. Aleksandrov, instead of envisaging a ladder,  uses the metaphor of layers  or (appropriately enough) of nesting, like Russian dolls, in the procession to affine or projective geometry and on to topology:

The properties of space are stratified … with respect to their depth and stability.  The ordinary Euclidean geometry was created by disregarding all properties of real bodies other than the geometrical;  here we perform yet another abstraction within geometry.

-- Aleksandrov et al, eds, Mathematics: Its Content, Methods, and Meaning (publication in the original Russian: 1956;  Eng. tr. publ. 1963), vol. III, p. 133

~

The more I think about it, the more this Ladder of Abstraction idea seems possibly fruitful.  Not so much as in the Theory of Types, but as in the scala naturae, which encompasses angelology.  (Compare also graded algebras.)

By contrast, mere ungraded “abstractness” in itself is of little interest. Thus, to take MacLane’s Group Theory example:  the so-called “abstract” groups (MacLane himself uses the sneer-quotes here) mean to lift aloft from Groups of Transformations, in that they retain the laws (associativity, inverses, and all that) while becoming agnostic as to the nature of the elements of the group.  But, first of all, groups of transformations are, compared with, say, pickles, already quite Abstract;  so the word adds, really, nothing.  Indeed, as soon as you say that two apples plus two apples are four apples, and that in the same sense  two penguins plus two penguins make four penguins (well, and a few more, after a while, if the sex mix is right), you are already indulging in such abstraction.

~

I tried looking up “abstraction” in the index of the various math textbooks and philosophy treatises on my shelves, and basically came up with  bupkes.  Thus, in Dummett’s omnibus volume, Truth and Other Enigmas (1978), we find no reference to abstraction per se, let alone to the Ladder of Abstraction, but only to “abstract objects” -- i.e., pickles versus the Meaning of ‘Pickle”,  the Idea of a Pickle, the set-containing-a-pickle, the… sandwich containing a pickle, the -- but enough.  Mathematics is so far beyond this, no comment is required.

~

The ethic -- even, the aesthetic -- of abstraction for its own sake, sociologically chronicaled here (“On Vulgar Numbers”), eventually evoked a backlash.

The Bourbaki group sought to present the entire abstract structure of all mathematical concepts in one set of volumes, the Eléments de Mathématique. In that treatise, the real numbers, which most of us regard as a starting point, only appeared midway into the series, as a special “locally compact topological group”.

An opposing idea, promoted especially in the Russian school, is that a few well-chosen examples can illuminate an entire field.

-- David Mumford, Forward to Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. xv

~

For the latest in fine reading, check this out:

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

For more about abstraction, here:
         http://worldofdrjustice.blogspot.com/search/label/abstraction

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

.