What did you do in the Linguistics Wars, Daddy?

Awhile back I offered an essay in Mathematical Lexicography:

What is Mathematics?

It led off with this:

    First, a sociological/tautological non-definition:

  What is mathematics?  One proposal, made in desperation, is ‘what mathematicians do’.

  -- -- Ian Stewart,  How to Cut a Cake (2006), p. 27

That stab in fact fails to offer even the virtue of a tautology, since it isn’t even true, without the further qualification that it is what mathematicians do… when they are doing mathematics.  -- If you’d tried similarly, without qualification,  to define linguistics based simply on the activities of linguists (ex officio: faculty and students in the Linguistics Department) at Berkeley during the years I was there, you would conclude that the field consisted of:  fixing your Volkswagens[**];  eating Chinese food;  and dabbling in neighboring fields like psychology and philosophy (later all these fields hopped into the hot-tub together and were newly baptised as Cognitive Science) -- all this while studiously ignoring most of the work done in the previous centuries of philology and language-sciences.

[**]  Sprawling supine beneath one's VW Beetle or Bus  was known locally as the "Berkeley lotus position".

[Note:  The title of this post  alludes to an insightful, delightful sociohistory of the era,
Randy Allen Harris, The Linguistics Wars (1993)]



[For further reflections on Berkeleyana, click here.

For the place of linguistics among the sciences, here.

For sociohistorical reflections on other sciences, here.]

Sigmund Freud: R.I.P.?

The writings of Frederick Crews  have delighted me  ever since I was in Junior High, when The Pooh Perplex came out in 1963.  It gave me not just delight, but lasting literary influence (cf. our posts labeled sotie or pastiche).  Then later, essays collected in Skeptical Engagements and The Critics Bear It Away.     I have not especially followed his evolution from Freud-embracer to Freud-basher,  but it is a notable trajectory, and (being self-critical) is at the very least  entitled to a certain respect.

And now, after long incubation, he has just published a book which … the world was not exactly waiting for, holding its breath:  Freud: The Making of an Illusion.  It replows, and resows -- nay, re-salts -- old ground.   In a front-page review in the current New York Times Book Review, George Prochnik poses the inevitable question:

Crews has been debunking Freud’s scientific pretensions for decades now;  and it seems fair to ask what keeps driving him back  to stab the corpse again.

The most creditable answer would be that, if the case to be made is important, it is worth doing in full, taking into account new developments:  No-one questions when, say, a classic biology text or physics text  is given further editions.   Yet that doesn’t seem to be all that is going on in this case (cf. Chomsky, Freud, and the Problem of Acolytes).

~

Prochnik’s review is workmanlike, with several well-put observations; but Louis Menand’s essay in the current New Yorker, taking off from the same publication, is magisterial.  We’ll not go over any of his widely-informed insights, since the essay is well worth reading in full;  but only address a couple of points that pursue the theme, “Freudianizing the (anti-)Freudians”.

Menand too sees this new volume as something other than an updated and sturdier consolidation and regimentation  of arguments made reasonably well before:

His criticism of Freud is relentless to the point of monomania.

Menand then applies his magnifying glass to that vexed and vexatious bone of contention, Freud’s relations with Minna Bernays.

Crews imagines assignations in the family home in Vienna as well.  He notes that Minna’s bedroom was in a far corner of the house, meaning that “the nocturnal Sigmund could have visited it with impunity in predawn hours.”  Could he have?  Apparently.  … Did he, in fact?  No one knows.  So why fantasize about it?  A Freudian would suspect that there is something going on here.

As Menand here implicitly concedes, that sort of psychological second-guessing of possibly unconscious motives, in which he himself has just indulged, is rightly reckoned to the legacy of the Viennese master;  and is a permanent Errungenschaft of our cognitive culture, slate ye the master howsoever ye may.

~

Now we ourselves, in turn, shall get down on the intricate Persian carpet  with our magnifying glass,  searching-out such fragments of analytic tobacco as might be telling, now  not for Freud  nor for his critic Crews,  but for the meta-critic  Menand. 

That salacious item from the gossip pages of history is one in which neither Menand (avowedly) nor I  have any particular interest.   But, oddly, amidst an exemplary essay, Menand now drops the logical ball.  In the very next paragraph, he reports:

Some Freud scholar floated the suggestion that  since Minna’s bedroom was next to Freud and Martha’s, there would have been few opportunities for hanky-panky.

So:  diametrically opposite assertions about the floorplan.   Yet Menand in his own words presents the contradictory assertions with idioms that, linguistically, are “factives”:  that is, they presuppose the truth of their predicate.   “He notes that (X)” and “Since (not-X)”.   Odd, from such a careful stylist.

And now let us wiggle the scalpel a bit, under the skin.

On page 79 of the magazine, Menand refers to a well-known doctrine of Freud, call it P (und den wir nicht nennen, daß wir selbst nicht angepöbelt werden; siehe aber dies, das, und jenes.)   Now, P may well be false, for all we know;  or, true only to a limited extent.   But Menand goes farther, calling it

patently absurd

Absurd is a very strong epithet.  A hypothesis may be provably, definitively false -- as, say, that of the postulated primality of Fermat numbers -- without ever having been properly describable as "absurd", even in retrospect.  And, patently absurd -- scarcely any proposition once held as true by some community, in context, can justifiably be called that:  Not astrology, not the ether, not the geocentric theory, nor even the flat earth.  Our Freudian-Sherlockian thus here arches a brow.  (Actually I suspect that Menand’s protest-too-much formulation here  does not reveal anything unsuspected about his own unconscious, but merely reflects the pressure of political correctness.   Likewise the nervous parenthetical qualifier “justifiably” on page 78.)

So, Freud, R.I.P.  Requiescat in pace?  No, they will not let him rest in peace.

Resurrexit in potentiâ?   Possibly.

Refutation vs. Disconfirmation

We continue our observations about Verification (essay here) and Modus Tollens (essay here).

It has been maintained that there is an important asymmetry between the verification and the refutation of a theory in empirical science.   Refutation has been said to be conclusive or decisive, while verification was claimed to be irremediably inconclusive. … The falsity of [the theory] is indeed deductively inferable by modus tollens.

-- “The Falsifiability of Theories”, in: Adolf Grünbaum, Collected Works, vol. I (2013), p. 62

That is basically because an empirical law is, in general, a universal: ∀x P(x).   Given any  ¬P(a), that is falsified.    Whereas successive confirming instances, while perhaps increasing one’s confidence in the hypothesized law, do not deductively entail it.   Nor need that fact be, in practical terms, a mere quibble, in cases of an infinite domain:  thus, a conjecture in number theory might be verified for all values of n up to a billion -- a trillion -- a googolplex -- yet still turn out to be false.  (Indeed, toy conjectures of this sort  are trivially easy to dream up.  E.g. “No integer is evenly divisible by a googolplex.”)   Taking confirming instances ‘too seriously’, is the Fallacy of Affirming the Consequent.

Grünbaum then, however, in a Duhemian vein, weighs in against such modus-tollentic tyranny, with the observation that, in empirical science, you are really assuming (ex hypothesi) not merely a theory, but (in unspoken conjunction therewith) various Auxiliary Assumptions:  and one or more of these, rather than your pet theory, might be the ones to suffer the fury of the tollendum.  Thus,  “Isolated component hypotheses of far-flung theoretical systems are not separately refutable but only contextually disconfirmable” (id., p. 63).

The same idea, more sociologically stated:

Typically  a theory is abandoned, not because of recalcitrant observations on their own, but because of these together with some, perhaps quite complex, reasoning from them:  in the extreme case, the observations may all be quite well known already, and all that is new  is the reasoning.

-- Michael Dummett, Truth and other enigmas (1978), p. 409

Compare:

… that falsity, not truth, is the primary notion.

-- Michael Dummett, Truth and other enigmas (1978), p. xl

~

It is seldom that a theory is decisively k.o.’d by a Popperian uppercut to its predictive jaw.   Often the process happens slowly, like a river silting up;  there is no single Aha! moment (or rather, Oy veh! moment).   In science, as in life,  sometimes love grows old and love grows cold.  As, assessing a certain philosophical program:

The further one goes down the reformist path, the more implausible the consequences of one’s theory are likely to become, until  at some point  the implausibility of the consequences  comes to outweigh the initial attractiveness of the theory.

-- Scott Soames, Philosophical Analysis in the Twentieth Century (2003), vol. I, p. 299

That gradual, einsickernd, process of disenchantment, lacks the drama of Popperian refutation, or of Kuhnian paradigm-shift;  it is more reminiscent of the dissatisfying phenomenon (if it even is one) of truth decay.

~

The history of science -- though not the historiography of science -- is littered with once-flourishing theories that somehow just … faded, without necessarily ever having been decisively confronted or refuted.  It is no fun to research and write about these, since
(a) they are losers in the lottery (rightly or not), and scarcely known today (so that one’s potential readership is infinitessimal -- and indistinguishable from zero among working scientists); and
(b) there is seldom a clear-cut moral in the tale.

The occasions on which one does get an insight into some defunct theory  tend to be biographies of otherwise-great scientists -- e.g.  Theodore Porter’s fine account of the (as we now reckon him) statistician Karl Pearson, which must needs notice that hero’s period of infatuation with, mm, ahh, … ether squirts (pas devant les enfants!);  or Thomas Hankins‘s  of William Rowan Hamilton.  Or, sometimes, a historian of science rolls up his sleeve and deconstructs/resurrects  some particular thread of experiment.  As, Gerald Holton’s commendable study of “Subelectrons and Presuppositions”, involving the “classic” Millikan experiment, of which we all get a telescoped and sanitized version in high school physics.  The actual saga is more anfractuous, and less rich with easily digestible lessons.

[Millikan] found himself working with the wrong presupposition, but he knew how to rid himself of it eventually.  Millikan launched into that work with the same energy and obstinacy  as into his earlier work on the quantization of the charge of the electron, yet with the opposite assumption.

-- Gerald Holton, The scientific imagination (1978), p. 72

And re the related experiments of Felix Ehrenhaft:

There was never a direct laboratory disproof of Ehrenhaft’s claims.  … Lorentz … remark[ed]:  “The question cannot be said to be wholly elucidated.” In his review of the case, R. Bär noted …: “the experiments left, at the very least, an uncomforable feeling.”  Like most such controversies, this one also faded into obscurity, without anything as dramatic as a specific, generally agreed-upon falsification taking place at all.  Indeed, Ehrenhaft continue to publish on subelectrons into the 1940s, long after everyone else had lost interest in the matter.

-- Gerald Holton, The scientific imagination (1978), p. 79

And that, within (central, classical) physics.  The less rigorous (natural and social) sciences  are much pocked with such as well.

~

Science, not as logically implying some new phenomenon (such as the gravitational bending of light), but as suggesting likely places to look.   Thus:

The search, which has proved so successful, for chemical atoms  having specific nuclear and electronic constitutions, and for chemical molecules having specific atomic constitutions, has been stimulated by the previous construction of theories to explain the consitution of atoms or of molecules already known, theories which had, as it were, ‘gaps’ in them.

-- Richard Braithwaite, Scientific Explanation (1953), p. 70

And indeed, not only the existence of, say, atoms or isotopes with a given atomic weight and atomic number  are thus -- not exactly ‘predicted’ in the deductivist sense, but pointed to -- but additionally, we shall expect certain behaviors, or ranges of behaviors, based upon the place in the periodic table of that species (should it exist).

Thus, when the eka-aluminum that Mendeleev hypothesized  was eventually identified (by a Frenchman, who thus had naming-rights and called it gallium) and turned out to have behavioral values very close to those predicted,  while not proving the theory of atomic chemistry as it then stood, was still much more than just one more danged black raven.

[A footnote on another such predicted discovery, that nicely illustrates consilience in the sciences, combining both forefront atomic physics  and practical geology:

At the time Bohr was developing his theory, the element following lutecium was undiscovered.  But Bohr … asserted that, in this unknown element, the added electron would have to be placed in the level n = 5. This would imply that the unknown element would have an electronic configuration analogous to that of zirconium.  Inasmuch as elements that are chemically analogous  are usually found in the same minerals, Bohr claimed that the missing element should be sought in minerals containing zirconium.  It was indeed in such ores that the missing element, called hafnium [after the Latin name of Bohr’s hometown] was detected.

-- A. D’Abro, The Rise of the New Physics (1939),  p. 549.  ]

This “guiding-light” role of a theory, not being a formal entailment, is likewise not subject to modus tollens (though of course it might be abandoned in the face of an accumulation of recalcitrant instances):

Whereas discovery of observable properties which fill such gaps in a theory  is rightly thought to provide a weighty confirmation of the theory …  failure to discover such properties would not be regarded as weighing against the theory, except perhaps in very special cases.

-- Richard Braithwaite, Scientific Explanation (1953), p. 70

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

Alexander Grothendieck

From the graceful pen of Edward Frenkel  comes this appreciation of the prodigious mathematician Alexander Grothendieck:

http://www.nytimes.com/2014/11/25/science/the-lives-of-alexander-grothendieck-a-mathematical-visionary.html

Frenkel’s summary of Grothendieck’s approach  is phrased in distinctly Platonistic terms:

Grothendieck’s genius was to recognize that there is a “being” hiding behind a given algebraic equation (or a system of equations) called a scheme. The spaces of solutions are mere projections, or shadows of this scheme.

That formulation goes beyond the basic assumptions of mathematical Realism, adding a quasi-personalistic, quasi-spiritual metaphorical dimension.   We might epigrammatize the matter thus:  bare-bones Realism (embraced by most mathematicians as a lower bound) is to the view attributed to Grothendieck, somewhat as deism is to theism.

Frenkel concretely illustrates this (neo-)Platonistic / Plotinean stance of  seeing the relatively concrete and familiar, as but a pale and lesser projection of some higher, transcendent, but paradoxically more solid Reality (which dwells in Platonic heaven, sipping some higher-dimensional nectar), in his justly celebrated book for the layman:

Imagine a world in which natural numbers are replaced by vector spaces;  that is, instead of number 1  we have a line, instead of number 2  we have a plane… Addition of numbers is replaced by the direct sum of vector spaces,  multiplication by their tensor product.  The number 3 is a mere shadow of the 3-dimensional space, reflecting only one attribute of this space, its dimensionality.

-- Edward Frenkel, Love & Math (2013), p. 155

That insight is superficially reminiscent of Bertrand Russell’s definition of, say, the integer three as being not basic, but the rather the (infinite) set of all triads { {Moe, Larry, and Curley}; {Huey, Dewey, and Louie}; {the Andrews Sisters}; ..}  Russell’s move always struck me as, at best, an obscurum per obscurius (or even a clarum per obscurius);  at worst, a parlour-trick.    Frenkel’s suggestion likely goes much deeper -- indeed, to a depth I am in no position to assess.

Finally, without the structural richness of Frenkel’s proposal, but perhaps more accessibly, this from a pair of noted philosophers-of-science:

The abstract objects of thought  such as “numbers” [or] “perfectly straight lines” .. are real parts of nature, even though they do not exist as particular things, but as the relations or transformations of such particulars.

Morris Cohen & Ernest Nagel,  An Introduction to Logic and Scientific Method (1934), p. 372

~

Frenkel reminds us that Grothendieck, dramatically and notoriously, Threw It All Away, resigning his institute and seceding from the mathematical world, upon learning of its defense funding.   Frenkel points indeed to the possibility that even a subject so “gloriously useless” (as it was seen back in G.H. Hardy’s day) as number theory, can have crucial cryptographic applications which, in turn, invite subversion by the intelligence community.   He states, indeed, that such subversion did take place, at the hands of a three-letter Agency too numinous to name, in the case of Elliptic Curve Cryptography.  Yet ironically, that very field was pioneered by a theoretician politically of the far left, Neal Koblitz.  (I knew him slightly when we were both math majors and antiwar activists at Harvard, and recall him fondly here.)

[ A parallel:   For a highly entertaining, though painful, account of Simon Norton,  once Conway’s collaborator, but who likewise Threw It All Away, see the book by Alex Masters, The Genius in my Basement  (2011) ]

I never met Grothendieck, nor so much as approached the foothills of his cloud-surmounting summits;  but my introduction to mathematics did come via one of Grothendieck’s students, Robin Hartshorne, here portrayed:

http://worldofdrjustice.blogspot.com/2011/12/adventures-in-algebraic-geometry.html

We touch upon Grothendieck tangentially in several essays;  here they are:

worldofdrjustice.blogspot.com/search/label/Alexander%20Grothendieck

[Another parallel:  Grothendieck’s abrupt withdrawal from mathematics, and his sometimes surly subsequent comments on its practicioners, recalls that more recent self-reclusion of the Russian topologist Grigori Perelman, who quit his institute in 2005, with the remark that “I have been disappointed in mathematics and I want to try something else.”   Now, to anyone who grasps the subject, disappointment with mathematics is almost inconceivable -- it’s like being disappointed with the Universe, or with penguins, or with life itself.   Other statements he has made suggest that his quarrel is not with math per se  but with certain mathematicians.

For a poignant glimpse of the daily routine of this semi-recluse, try this:

http://www.lepoint.fr/monde/le-genie-qui-s-est-retire-du-monde-30-09-2010-1246189_24.php  ]

~

In other cryptographic news … This week’s New Yorker has a review by Anthony Lane of “The Imitation Game”, which takes us back to the WWII code-breakers at Bletchley Park, with mathematician Alan Turing at the center.   With his welcome skeptical eye, Lane punctures the movie’s superstar approach to the material:

No word is breathed, for instance, of the Polish cryptographers who did much of the heavy lifting on the project  before Turing came on the scene.  As for the cracking of codes, it is shrunk to single, Oh-my-God epiphany, triggered by a comment in a pub.

That last motif is strikingly reminiscent of the trumped-up Eureka moment ascribed to Gauss in the movie “Die Vermessung der Welt” (which we reviewed here), whose insights into differential geometry were supposedly engendered -- in a flash, on the spot -- by a buxom mädchen handing him an apple from the knowledge-tree.

Another warning-sign:   The role of Turing’s  Comely Female Sidekick (de rigueur in Hollywood these days) is played by Keira Knightley, an overactress whose pornographic approach to proximity to great men of science, in the movie “A Dangerous Method”, we earlier had occasion to denounce.

~

[Late-evening update]  And now, fellow devotees of that supreme queen of the noösphere, math -- now that we have all stuffed our tummies with turkey, and are sprawled upon the couch:  as we are probably not just now engaged in settling the Hodge Conjecture (and making but scant progress if we are), let us rather refresh the neurons with a bit of mathematical merriment:

How Now, Round Cow

~

Sotie : 

le mathématicien  et la conspiration Riemann

Борис Березовский

~

[Update, 30 November 2014]  Further thoughts.

(1) Integers as shadows

That epigram, that petite phrase,  “The number 3 is a mere shadow of the 3-dimensional space”, continues to intrigue.  Further thoughts on the subject here:

From Finitude to Infinity

The point being that Frenkel/Grothendieck are here proceeding in the less-familiar opposite direction.

(2) “Goodbye to All That”

The catchphrase “I Threw it All Away” is from Bob Dylan.   An earlier more familiar expression is “Goodbye to All That”, the title of a memoir by the poet Robert Graves.

To fascinate a public beyond the circle of connoisseurs, a mathematician (or physicist, or chess-player) needs to sport some quirk, some handle onto which the layman can hang his attention.  That is massively true of Turing, first because of his role in the thrillingly clandestine -- and militarily crucial -- cryptographic factory at Bletchley park;  and more recently -- and less relevantly, from any mathematical standpoint -- by reason of his personal predilections (held in common with such extra-scientific figures as Ἀντίνοος and the Baron de Charlus) which at present have reached an apogee of public celebrity.   On a more modest scale (no blockbuster biographies, no movies) the careers of Emmy Noether or Ada Lovelace come to mind.

For anyone not a member of the identity-groups in question, such chance affiliations are mathematically and philosophically boring.   But not so a figure like Groethendieck, who reached intellectual levels that most of the rest of us can only pant and sigh for -- then threw it all away, in a contemptuous gesture.   For that is a challenge to all of us, whatever our private identities;  it calls into question the very value of the ‘it’ for which we so painfully and vainly strive.

A well-known example from chess:  Bobby Fischer, who withdrew from the sport at the top of his game.

Or, in the sixteenth century, the pioneering anatomist Vesalius, who, having published the book that settled his fame forever, and still a young man, left the field and became a simple physician to a valetudinarian Emperor.

A grey-area variant of this motif is found in the case of Simon Norton.   True, he voluntarily withdrew from the field;  but in view of his later pointless eccentricities, we cannot avoid the suspicion by then he had largely Lost It.

That variant motif forms the spine of Rebecca Goldstein’s fascinating mathematico-philosophical novel,  The Mind-Body Problem.

And while Goldstein manages to craft a good read out of the tragedy, in the typical case of gradually fading powers, it is just sad.  Thus Lagrange, and other victims of Oligophrenia mathematica tardiva,  chronicled in the appendix to this essay:

De Stultitiâ

 

Other examples of burnout:  

Newton had a nervous breakdown a few years after publishing his Principia, and never did real science again.

Russell seems to have fried many of his math neurons in the course of writing his Principia (memo to prodigies:  Don’t try to write a Tenth Symphony, and don’t write a book called Principia).   He did much interesting philosophical work after that, but mostly with other areas of his capacious brain.

Somewhat in the spirit of the Stith-Thompson index of folkloric motifs, to I Threw It All Away (TaleType #1729a), we add:

* Taken all away (#1729b):    Persecution took them out of the game, at the height of their powers: Galois (permanently) and, for those who considered that he was hounded to death, Turing.  Temporarily: André Weil and Neil Koblitz (during their imprisonment -- though both managed to use their ‘time inside’ more profitably mathematically  than most of us do with all the free time in the world).

* Would throw it all away, if had to do it over again (#1729c):

Wolfgang Pauli in 1925:
"At the moment, physics is again terribly confused.  In any case, it is difficult for me, and I wish I had been a movie comedian or something of the sort, and had never heard of physics."

Schrödinger, to Niels Bohr:

“If all this damned quantum jumping were really here to stay, then I should be sorry I ever got involved with quantum physics.”

[quoted in: -- Roger Penrose,  The Road to Reality (2004), p. 516]

More such anecdotes here:

http://worldofdrjustice.blogspot.com/2012/06/on-truth-and-beauty-golden-oldies.html

* Should throw it all away, if that’s how they feel (#1729d):  String theorists who have wound up at the dead-end of the Landscape picture.  (“The reductionist voyage that has taken physics so far  has come to an end.  Since that is what they believe, I can’t understand why they don’t take up something else -- macramé, for example.”  More here.)