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

.

Veracity, Verifiability, Vindication

[That is an ascending series.  Some things that are true, may yet be inaccessible to us -- temporarily, or forever;  a veracious person is someone who asserts only what he sincerely believes to be true (and -- for this quality to have any practical value -- has reasonable warrant for so believing, reasonable relative to the contemporaneous state of the art.)

Verifiable means that the assertion is ‘in the running’ for being experimentally (or proof-theoretically) confirmed, even though these happy results have not yet eventuated.

Vindicated means that the result has been supported, whether by (theory-supported) experiment, formal proof, or revelation.]

[Original post from 7 VI 2015]

A crisp, concise op-ed, in this morning’s NYTimes, “A Crisis at the Edge of Physics” by Adam Frank and Marcelo Gleiser (both professors of physics), states the case against (over-cantilevered or under-buttressed) speculation, not only for string theory (which has notoriously come in from a lot of pushback;  see Lee Smolin, The Trouble with Physics), but for supersymmetry generally.  The authors note that, as of even date, “no supersymmetric particles have been found” (perhaps they are hiding out in a back room, playing poker with the Higgs boson).  So far as that goes, not a problem;  plenty of propositions in math and science took centuries or even millennia to settle.  What disturbs the authors is that some champions of supersymmetry -- the jusqu’au-boutistes, we might call them -- may simply move the verificationist goalposts.

Some may choose to simply retune their models to predict supersymmetric particles at masses beyond the reach of the Large Hadron Collider’s power of detection -- and that of any foreseeable substitute.

Exactly the same concerns were voiced, a good decade earlier, by the mathematician-physicist Roger Penrose, in the section “Can a wrong theory be experimentally refuted?” (p. 1020 ff.), in The Road to Reality (2004), concerning “un-Popperian” practices in physics.

If that were only a problem for one avenue at the forward fringes of physics, that would not be a problem for most of us  as we bustle about our daily chores.  Yet the authors further suggest that certain well-traveled avenues  are actually cul-de-sacs:

The standard model, despite the glory of its vindication, is also a dead end.  It offers no path forward to unite its vision of nature’s tiny building-blocks with … gravity.

What really bothers the authors is something that goes well beyond physics:  “the specter of an evidence-independent science”.   And that specter has been haunting the West for some time, and increasingly reaches into the headlines, as witness the countermovements to the theses of natural selection or of global climate change.

Not being a physicist, I have no right to comment;  but, at the margins, this:

(1) The larger cultural worry, is the dissociation of the notion of Truth überhaupt  from that of Evidence and Argument. In that perspective, we would deplore the demand to dissociate theory from experiment.

(2)  Yet -- Do not forget  Einstein’s classic crack in 1919, anent the possible negative results of an experiment purporting to validate or refute General Relativity:  “Da könnt’ mir halt der liebe Gott leid tun.  Die Theorie stimmt doch.”  (Informal translation:  "I'm right.  Bite me.")

Die Theorie stimmt doch!

The consensus of scientific history (for right reasons or wrong)  has been to applaud  those cheeky remarks .

~

Einstein was speaking of his theory of gravitation.  But similarly for particle physics.

Compare, re Feynman and Gell-mann’s joint article “Theory of the Fermi Interaction” (written in 1957, and subsequently published in Physical Review):

The V - A theory was in disagreement with more than a half dozen experimental results on beta-decay,  but it was so beautiful  that the authors proposed it anyway, suggesting that all those results were wrong.

-- Harald Fritzsch, introduction to Murray Gell-Mann: Selected Papers (2010), p. 5

And:

The Standard Model … has been driven largely by certain powerful consistency requirements, hard to satisfy in such theories.  In order to appreciate something of the force behind these consistency requirements (which continue to drive the more modern speculative theories, such as string theory), we shall need to look at the structure of quantum field theory. … The theoretical requirements appear to be so tight  that it might seem almost incidental that these answers are actually in excellent agreement with experiment!

-- Roger Penrose,  The Road to Reality (2004), p. 655-6

And more generally:

Polanyi delighted in drawing attention to cases where the scientific community ignored or waved aside or explained away  seeming counter-evidence to accepted theories.  He seems to have felt that a scientist would abrogate his personal responsibility for his beliefs  if he allowed them to be at the beck and call of experimental results.

-- John Watkins, Science and Skepticism (1984), p. 29

Nor must we wait until our own extravagant age of post-modernism and M-theory, to find ourselves confronted with an “All is Permitted” ethos in the realm of physics.

Karl Pearson, as a pioneer of statistical thinking in a wide range of fields, is in that respect  a representative of a hard-headed, just-the-facts-ma’am, shut-up-and-calculate approach to messy realities.  But when doing (what he thought of as) physics, his Romanticism, which early on was a major strain of his make-up, got the better of him.   In the years around 1890, he theorized about atoms in terms of the then-regnant ether theory:

He spoke  not of causation  but of analogy, indeed “analogies … of the vaguest description”, and in the context of this paper, his doubts about the human capacity to get at real objects or real causes  functioned as a license to invent … Since his ether model  so far  had strange, almost inconceivable properties, he discarded physical plausibility as a criterion of a good theory.

-- Theodore Porter, Karl Pearson (2004), p. 187

"You see it's all simply a matter of ether-squirts ..."

~

Working the equations of physics to their long-reaching logical conclusions, continually leads to apparent absurdities:  negative energies or frequencies, unobserved particles, particles moving backwards in time, a Hobson’s choice between acausality or indefinitely-proliferating alternate universes, and miscellaneous infinities.  Some physicists shudder at such;  others grin and say “Bring ‘em on.”  (Unfortunately, the latter are the ones favored in the popular media -- the phenomenon of Physics Porn.)  The problem is deciding when that is just the way Nature (inscrutably) actually works (in which case you have made a major discovery), and when it is merely absurd.  Will the Higgs boson turn out to have been more like the positron (born from the forehead of Dirac’s mathematics) and the pion (brain-born from Yukawa), eventually found in everyday space, or  instead  like the cute-sounding but still-missing photinos, squarks, and pentaquarks?

[Update July 2015]  Bzzt!  No sooner had I posted that, than experiments claim to have spotted one of the elusive critters:

http://www.nature.com/news/forsaken-pentaquark-particle-spotted-at-cern-1.17968

So here is the larger temptation -- the intellectual Occasion of Sin:

Beginning several decades ago, comparing the results of experimentally well-verified physics  with the predictions of the equations, scientists marveled at what was memorably dubbed “the unreasonable effectiveness of mathematics”, summed up by the epigram “the equations seem to give us more than we put into them;  they seem to be wiser than ourselves”.   But The Edge beckons when we start to conclude, that if our favorite equations predict something, then that something must be so (if only in the Multiverse) -- even if, to the guys in the lab, it doesn’t seem physically reasonable.

As Penrose puts it:

What is the physical justification in allowing oneself to be carried along by the elegance of some mathematical description  and then trying to regard that description as describing a ‘reality’?

-- Roger Penrose,  The Road to Reality (2004), p. 670

That question takes us back  to a very old debate -- as old as poetry :  What is the relation between Beauty and Truth?

In that essay, we concluded that, in a sense, ‘beauty’ (in a rather austere sense of crisp symmetric elegance, having more to do with the Parthenon than with a Miss America pageant or a Turnerian sunset) does characterize any deep theory in the mathematicized sciences -- but only in retrospect, after years and decades of its practitioners coming to appreciate its depth; the theory does not wear its beauty on its sleeve.

Thus, even in the case of the (relatively) well-behaved, now-long-familiar poster child of particle physics, QED:  Paul Dirac, the pioneer of QFT, wasn’t buying it.   In response to a 1936 experiment (by Shankland) which suggested (incorrectly, as it turns out) that energy need not be microscopically preserved,

Dirac immediately jumped at this opportunity to disown QED, claiming “because of its extreme complexity, most physicists will be glad to see the end of it."

-- Matthew Schwartz, Quantum Field Theory and the Standard Model (2014), p. 247

~

Taking physics on faith

Penrose again, concerning a couple of signature contributions by Richard Feynman -- probably the educated public’s favorite hip physicist since Einstein:

The path-integral approach is, it seems, almost wholly dependent upon a faith that the wildly divergent expressions that we are presented with (like the divergent series above) actually have a deeper ‘Platonic’ meaning  that we may not yet properly perceive.

-- Roger Penrose,  The Road to Reality (2004), p. 670

Theophysical note:  Here we see a reference to “faith”;  its truth-functional content may be roughly equivalent to “working assumption”, but since we are indeed dealing with such deep and ultimate matters of Platonism, the theological overtone is not actually out of place. 

Similarly, my casual reference to “revelation” above, as denoting one of various routes to knowledge, was not flip.  Compare, from our hard-headed flinty-eyed philosopher of science:

If  we had a hot line to the Author of Nature, and if we had a clearly formulated IP [for which see below], an excellent question to put to him would be:  Is our IP true?  If he answered ‘Yes’, we could happily set a computer to work to print out all those h[ypothese]s that are singled out by our evidence  in conjunction with this authoritatively endorsed IP.
-- John Watkins, Science and Skepticism (1984), p. 93

And if that strikes anyone as credulous, note that most of us largely treat computers as oracles as well (e.g. in the proof of the Four-Color Theorem, or any of innumerable unsurveyable and possibly preposterous simulations).

Back to the sadder-but-wiser Penrose:

Even that archetypal renormalizable theory, QED, is not actually a finite theory, even after renormalization.  How can this be?  Renormalization refers to the removal of infinities from finite collections of Feynman graphs.  It does not tell us that the summation of all these resulting finite quantities is actually convergent. … In fact it is not finite, but has a ‘logarithmic divergence’.

-- Roger Penrose,  The Road to Reality (2004), p. 680

(Logarithmic divergence is comparatively mild, but it still gets where it's going -- an unphysical infinity -- in the end.)

As for the next step beyond QED (which is part of the Standard Model), QFT:

Strictly speaking, quantum field theory … is mathematically inconsistent.

-- Roger Penrose,  The Road to Reality (2004), p. 610

~

But let us set aside quantum mechanics, that known maze of paradox, along with its ever-more-speculative successors.  Surely matters stand better in the case of classical mechanics and electromagnetism, along with their tool-of-all-work, the venerable Lagrangian, which dates back to the eighteenth century. 

Yet even here, Penrose demurs:

In modern attempts at fundamental physics, when some suggested new theory is put forward, it is almost invariably given in the form of some Lagrangian functional. … However, I must confess my unease … The choice of Lagrangian is often not unique, and sometimes rather contrived … Even the Lagrangian for free Maxwell theory … has no obvious physical significance. … Moreover, the ‘Maxwell Lagrangian’ does not work as a Lagrangian unless it is expressed in terms of a potential, although the actual value of the potential, A, is not a directly observable quantity. … In most situations, the Lagrangian density does not itself seem to have clear physical meaning.

-- Roger Penrose,  The Road to Reality (2004), p. 491

Nor is Penrose a professional maverick or skeptic.   After all, the book we’ve been quoting from clocks in at over a thousand pages, and is subtitled “A Complete Guide to the Laws of the Universe”;  you wouldn’t do that if you thought physics was a crock.

~

Simply as an assertion, the Weyl curvature hypothesis  is perhaps more like a claim for ‘an act of God’  than a physical theory.

-- Roger Penrose,  The Road to Reality (2004), p. 769

Rule of Thumb:

Physics advances by dint of Physicists’ Encyclicals.

These are almost never arrived at purely deductively;  nor as the result of conclusive, slam-dunk experiment, leaving no leeway for doubt of validity nor variation in interpretation(**); yet neither do they come out of nowhere.

[**:   That classic 1919 experiment, about which Einstein was so dismissively cocksure, was not actually so probative as the newspapers made out.  Worse yet, that alltime- irreproachable über-icon of experimental virtuosity, the Michaelson-Morley experiment, traditionally cited as crucial for Special Relativity, did not really quite have the widely-advertised null result.  Lindsay and Margenau, in their history of physics, note this fact with some embarrassment, since so much of what they have to recount -- and which they do recount -- rests upon that pedestal;  they keep mentioning it with a proviso.  And at an even more elementary level in cosmology, the experimental results that led to the Red Shift principle: when Steven Weinberg painstakingly went over the actual original data, he pronounced himself baffled as to how Hubble ever extracted his famous monotonic relation from them.]

Schrödinger’s equation -- Feynman’s path-integrals -- the laws of thermodynamics:  inspired guesses, which awaited the mathematicians to tidy things up.

-- Not trying to debunk, here;  simply being historical.

(Feyerabend, too, was long  historical  in this sense, before he went over to the dark side.)

~

Donning our old lexicographer’s hat, let us look a bit more into the matter of vocabulary, the Wortfeld of terms for justification. 

Our negative result so far, entirely in line with Hume’s, is that, without an inductive principle, there can be no legitimate ascent from level-0  [i.e., things like “yellow patch, for me, here, now”] to level-1, or from level-1 to level-2, and that any inductive principle strong enough to “legitimise” the ascent  could not itself be legitimised.   If that is so, then it is obvious that there can be no legitimate ascent to still higher levels.

-- John Watkins, Science and Skepticism (1984), p. 105

In the following, we see a fine distinction drawn between justification and “vindication”.

The philosopher John Watkins imagines a principle, call it the Inductive Principle, which would answer Hume’s objections, to the satisfaction of inductivists.  What would then be the status of the IP?   He distinguishes several possible theses :  that it is “synthetic and true a-priori”, “synthetic and provable by a transcendental argument” (which latter turns out to be little more than “Well, it seems to work”), and:

*  IP is synthetic and empirically justified.

*  IP is synthetic, and, although it cannot be justified either a priori or a posteriori, it can be vindicated.

-- John Watkins, Science and Skepticism (1984), p. 93

The verb vindicate is slightly odd here;  usually it has moral overtones, of someone having been right against opposition or against the odds.   You verify someone’s age on his driver’s-license;  you validate a parking-stub; you vindicate a statesman’s course of conduct.

The term justification also has a richly complex ethico-theological usage in Christianity, quite opaque to an outsider.

~

More fine distinctions, this one semi-defined on the fly:

One’s degree of rational assent to a hypothesis should be controlled by its degree of confirmation (‘confirmation’ being understood in some quasi-verificationist or probabilist sense).

-- John Watkins, Science and Skepticism (1984), p. 118

Watkins then spins off into the world of proofs-and-refutations, abduction, and the like:

The sought-for relation between e[vidence] and h[ypothesis] is now inverted:  instead of an upward, quasi-verifying inference from e to h, we have a downward, explanatory derivation of e from h.

-- John Watkins, Science and Skepticism (1984), p. 119

Note those squirrely “quasi”s, by the way.  For all the wealth of the verificatory Wortfeld, no term seems quite to fit.

~

We have been focusing on physics;  but analytic philosophers have long discussed these matters in great depth.  A few representative teaser-quotes, giving some extra vocabulary, and the flavor of the debates:

The great contribution of [Quine’s “Two Dogmas of Empiricism”] was that it offered an essentially verificationist account of language  without committing the logical-positivist error of supposing that the verification of every sentence could be represented as the mere occurrence of sense-experiences. … Proof, which is verification by inference alone, thus becomes a limiting case, or a distinct species.

-- Michael Dummett, “What is a Theory of Meaning? (II)”, in: Evans & McDowell, eds., Truth and Meaning (1976), p. 111

… notorious problems about the connection between corroboration and verisimilitude …

-- Susan Haack, Evidence and Inquiry (1993), p. 105

Note here that, within the genus of positive instances, the term “confirming instance” is disastrously ambiguous  as between a supportive  and a nonsupportive species of positive instances.  By the same token, logical mischief has been wrought by the weasel word “verification”  and the equivocal verb “verify”.

-- “Is Falsifiability the Touchstone of Scientific Rationality”, in: Adolf Grünbaum, Collected Works, vol. I (2013), p. 15

[Update Dec 2015]  Physicists are beginning to get seriously perturbed by all this.

A Fight for the Soul of Science

https://www.quantamagazine.org/20151216-physicists-and-philosophers-debate-the-boundaries-of-science/

String theory is at the heart of a debate over the integrity of the scientific method itself.

http://www.nature.com/news/feuding-physicists-turn-to-philosophy-for-help-1.19076

[Update 19 Jan 2017] Philosophically on a more modest level than any of these "V's", is simple reproducibility of experiments --a minimum requirement for verifiability.  But there are problems even with that.  A new study of reproducibility released its results for the first five classic cancer-related experiments whose re-performance was attempted:  five experiments, five failures.   Background:

Cancer reproducibility project releases first results

The Reproducibility Project: Cancer Biology launched in 2013 as an ambitious effort to scrutinize key findings in 50 cancer papers published in Nature, Science, Cell and other high-impact journals. It aims to determine what fraction of influential cancer biology studies are probably sound — a pressing question for the field. In 2012, researchers at the biotechnology firm Amgen in Thousand Oaks, California, announced that they had failed to replicate 47 of 53 landmark cancer papers2. That was widely reported, but Amgen has not identified the studies involved.

Perhaps the clearest finding from the project is that many papers include too few details about their methods, says Errington. Replication teams spent many hours working with the original authors to chase down protocols and reagents, in many cases because they had been developed by students and postdocs who were no longer with the lab. Even so, the final reports include long lists of reasons why the replication studies might have turned out differently — from laboratory temperatures to tiny variations in how a drug was delivered.

http://www.nature.com/news/cancer-reproducibility-project-releases-first-results-1.21304?WT.ec_id=NATURE-20170119&spMailingID=53225513&spUserID=MjA1NjgwMjUyOAS2&spJobID=1083504884&spReportId=MTA4MzUwNDg4NAS2