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

“Red Mercury”

Re the age of alchemy:

Many authors chide the reader, telling him that, if he uses common mercury, he will be wasting his time;  instead he should use ‘the mercury of the philosophers’, our ‘our mercury’.

-- M. P. Crosland, Historical Studies in the Language of Chemistry (1962; ²1978), p. 26

There is a curious contemporary parallel to this, in the terrorists’ will-o’-the-wisp “red mercury”, a mythical WMD.

More on Renaissance cover-terms -- alchemical COMSEC:

There were many cases where the name of a substance … was definitely misleading, by falsely suggesting its composition.  For example, the term cinnabar of antimony suggests a compound of antimony, whereas the name was applied to red mercuric sulphide, prepared from antimony sulphide and mercuric chloride.

-- M. P. Crosland, Historical Studies in the Language of Chemistry (1962; ²1978), p. 88

How Physics Saved the Hippies

I am currently reading, with great pleasure and instruction, a recent book of David Kaiser, How the Hippies Saved Physics (2011).   The title rather put me off, as did the dust-jacket, featuring a naked man standing on his head;  but an excerpt on the Net proved excellent, and the work as a whole is proving even better, both from a scientific and even from a simply literary standpoint.

[Trigger Warning:  I tossed out the offending dust-jacket, but the sensitive reader must be steeled to encounter another affront, this one on page 55, where a physicist poses topless before a whiteboard, suitable for sexting.]

The title of the book, and that of this note, are both (as it might be) infinitely silly;  but their infinitudes are opposite in sign, or complementary.   Hence in posting this, so subject-lined, I may hope to usher in a welcome renormalization, in which the twin infinities  mutually cancel.

At all events, the canny author is well above the tawdriness of  his title, which turns out to be tongue-in-cheek (p. xii):

Hence the brashness of my title, How the Hippies Saved Physics.  Readers may well note a tinge of the same bravado, equal parts ironic and defiant, that animated Thomas Cahill’s well-known study, How the Irish Saved Civilization.  The similarity is by design.

I sympathize with such titular ludic thrusts, as the titles of several of my own efforts  attest:  An introduction to cinematic literacy;  Rootabaga exegetics;  A New Proof of the Existence of Coffee-Cups.

Despite the cheekiness of those he profiles, Kaiser’s book is by no means an example of “physics porn”, but is a work of solid journalism, and sociologically alert.  I was at Berkeley during the years of which he writes -- how well he captures the feel of it all!

~

Kaiser’s book, portraying American physics on the philosophical periphery, at a time of economic and academic retrenchment,  is a perfect pendant to Ann Finkbeiner’s engrossing history of American physics while it was feeling its financial and political oats:  The Jasons. 

[To be continued …]

[Update 16 Dec 2014]  This week's New Yorker has an informative article about the discovery and development of graphene, which might as well be titled "How the Hippies Saved Chemistry".

What is Algebraic K-Theory *really*, Grandpa?

Mathematicians, physicists, chemists, and hamster-fanciers  do not wonder how to define their subject (not until they have tenure, at any event), but simply roll up their sleeves and dig into it.

Later, in their rocker,  on the front porch,  bourbon on the sideboard and surrounded by the admiring upturned faces of their numerous grandchildren (always providing that such creatures shall still exist in years to come, rather than being all solipsistically oblivious  as, fondling their J-Pads or whatnot  with downturned personae and sightless eyes behind their encasing of Google-glass, they text their various virtual-avatar BFFs in unbreakable code, never stirring from the vat of amniotic fluid in which they float forever wihout sensation, all their internal organs having long since been harvested by their Reptilian Overlords for use in necromancy --

-- but I wander from my topic -- forgive it -- an effect of age),

later, as I began to say, when, fallen into the sere, the yellow leaf, and sicklied o’er with the pale cast of philosophic doubt, exposed to the bitter winds of (and so forth),

some of them may indeed eventually turn their minds, not to mah-jong, but toward certain Questions which, while not in any easily specifiable sense more “fundamental” than the deepest truths of their own chosen subject, are yet, in a precise sense, “meta” to them:  exempli gratiâ :  What are we really about, when we mathematize / botanize / admire our hamster?

Other than to those who have themselves delved deeply into such matters -- in particular, to those actively engaged in clearing new forests in their own fields, and with no time for such marginalia-- the answer may strike one as purely post-hoc, decorative, suitable for framing, worthy of being enshrined in a museum, but in no wise analogous to a fingerpost, or to fuel in your tank.   

Yet consider the following passage from a research mathematician and (for my generation) prominent textbook writer:

The main problem of topology is to decide when two spaces are homeomorphic.  From this point of view, the main problem of Operator Theory is to decide when two operators are unitarily equivalent.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963).

Here, the answer to “What is Topology?”  is not simply knitted into a sampler and tacked to the drawing-room wall, but used as an actual heuristic for finding your way in a different field.

Soberingly, he adds (aye, and has the wounds to prove it):

“But usually this is too hard.”

Still, that did not mean that the field of Operator Theory was stymied; similar questions in the topology of low-dimensional manifolds were “too hard” until the work of Thurston,  Perelman, etc.
 

~

All of which is simply by way of invitation to our  essay on the subject,

“What is Mathematics?”

where these matters are discussed in greater depth and detail.

[Note:  The quotation marks are largely ironic.]

~

Appendix: 

“What is Chemistry?”

Walter White, in the pilot episode of Breaking Bad, introducing the subject to his high school class:

Teacher: “Chemistry… is the study of … what.”

Pupil: “Chemicals.”

Teacher: “ ‘Chemicals’ -- N-n-n-n-NO!

Chemistry is: technically, the study of matter. 

But I prefer to see it as the study of:  change.”

 

(That  presages the very great change that he himself is about to under go -- a veritable transition to a new phase-state.)

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