The Double Truth

It is clear by now that the old Averroist doctrine of the “double truth” (veritas duplex)  must be revived:  this time, not along scientific-vs-theological lines, but political.  Radio talk-shows have long recognized the incommunicability of one set of truths, to partisans of the other, by offering separate Republican vs. Democrat call-in lines.

So, let us consider that simplest possible area of human belief and inquiry:  arithmetic, the discipline that offers up such hoary verities as “2 + 2 = 4”.  

In modern mathematics, the study of such numbers (called integers)  has exfoliated quite a bit, and indeed has bifurcated into named subdisciplines.   Thus we have, on the one hand, analytic number-theory,  which deals with such things as the Riemann hypothesis and the Goldbach conjecture, using techniques from complex analysis, and on the other, algebraic number-theory, involving extensions of the integers such as algebraic number-fields.   Both subfields have developed to such an extent, that specialists in the one will be scarcely able to understand, let alone prove, theorems in the other;  thus reproducing, unintentionally, a simulacrum of the current partisan Dialogue of the Deaf.

And now we have two new entrants to the disciplinary matrix:   Democratic number-theory and Republican number-theory.

Democratic number-theory is characterized (up to isomorphism) by its claim that all numbers are equal.   The denial of that doctrine is termed ‘fascism’.

Republican number-theory, by contrast, generalizes the ancient concept of a perfect number (equal to its aliquot sum) to that of the awesome numbers, which are greater than the sum of everybody else.  The leftovers are called the  pathetic numbers (also termed ‘losers’).  This theory focuses exclusively on the former subset (although the membership in this set  can vary capriciously  from day to day).

So far, neither body of theory has produced interesting theorems.

[Update 8 Jan 2018]   No sooner had I posted that would-be satirical thought-dream, than I happened upon this,  by the philosopher Stewart Shapiro:

Hartry Field takes mathematical language at face value.  Since he holds that mathematical objects do not exist, mathematical propositions have objective but vacuous truth-values.  For example, he maintains that ‘all natural numbers are prime’ is true, since there are no natural numbers.

-- Thinking about Mathematics (2000), p. 226

So once again, reality beggars satire.  (Apparently this Field fellow is some sort of an academic, rather than an inmate or homeless-person.)

Note, a propos, that the Fieldian assertion “all natural numbers are prime”  is actually an axiom of Democratic number-theory.   (“Everybody gets a trophy.  Everyone is special.”)

[Historical footnote]    If satire is the production of humor by clever exaggeration of a real situation, then here again the satire lacks its bite:  for such ideologically infected strains of the hard sciences  are already exemplified in history -- not merely in the musings of philosophical fantasists, but under dictators with the power of life or death.   Thus Hitlerian notions of Aryan science vs. Jewish science, or the various scientific orthodoxies under Stalin.    Under those two regimes, mathematics, fortunately, was relatively spared distortions of actual content (so long as the formulaic obeissances to ideology were packed into the preface),  though  in human terms  there were profound and pernicious effects on who was allowed to be mathematical practitioners.   The neo-Stalinoid zealots of “anti-racist mathematics” go further, prescribing severe limits to content and presentation as well.

Truth and Provability (expanded)

[A footnote to this.]

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

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

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

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

~     ~     ~

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

~ Whatever can be meant,  can be expressed.  ~

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

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

In the words of a Neothomist philosopher:

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

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

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

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

~

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

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

~

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

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

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

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

What is proof? -- he asks.  There is no such thing as proof in general, he answers himself.  There is only proof  persuasive for this, that, or the other kind of man.   Cogency of proof  is relative to what you are.  he notices that this does not seem to apply to mathematics, and brazenly comments that just this has always made him suspicious of mathematics.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 180

Actually Oakeshott  put his case too weakly:  varying standards of proof are relevant in mathematics -- indeed, it is only within mathematics  that such scruples have structure and are in point.   In pre-Cauchy/Weierstrass analysis, proof was a bit of a kludge.   Later on, Constructivist qualms  came into play.  And in our own day, we distinguish between theorems whose proof requires the (disputed) Axiom of Choice, from those that can dispense with it.

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

Ernest Gellner on Truth

“Truth” is, on the one had, a bland and boring concept:  “Paris is the capital of France” is true, “Las Vegas is the capital of France” is false.

Yet in other venues, fraught:  as in, Pravda.  Shading into metaphysical mysticism (“The Search for Truth”).  If I am trying to find out, for which X the sentence “X is the capital of Albania” is true, then in a sense I am Searching for Truth; but really, only for a truth; and indeed, not really under that description:  I merely wish to know what Albania’s capital is called.

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Ernest Gellner on Truth

Relevant quotes, bridging the gap, from works by Ernest Gellner.  (For our full essay on the subject of truth, click here.)

Re Orwell’s Nineteen Eighty-Four:

Freedom is the recognition that 2 plus 2 makes 4 :  not because there is no escaping such necessity, but because only such necessity is a refuge from arbitrary social power. [It is] an extra-social objective truth, which account for why such fuss should be made  of a morally and emotionally rather neutral piece of arithmetic.

-- Ernest Gellner, Contemporary Thought and Politics (1978),  p. 4

On a strategy of self-validating beliefs (which he dubs “auto-functionalism”, a term which seems not to have caught on):

It consists of establishing the soundness of one’s beliefs, not directly, in the ordinary and straightforward way, by showing them to be true, but, on the contrary, of deriving their soundness by showing them to play an essential role, to be ‘functional’, in the internal economy of one’s own personality or society … The first step is to put forward a theory of truth: truth ‘really is’ the fulfilment of a biological, or social, linguistic, etc., function.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 14-15

(Truth, soundness, validity, provability -- actually an interesting Wortfeld;  a topic for another day.)

And, re the egregious Althusser:

He argues, in effect, not that Marxism is true, but that the Marxist epoch is still with us.  What is defended, in the end, is not the truth of a doctrine, but its alleged role.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 17

Now,  that all sounds rather feckless and po-mo; but to add some perspective, it is reminiscent of the “regressive justification” in axiomatics, particularly for set theory.  (More here.)

A somewhat more degenerate version of this auto-functionalist approach, endemic to the America of “pot, pop, and protest” -- degenerate in that, unlike that of Althusser et alia, it makes little reference to the world outside the speaker’s individual ego-bubble (indeed, it works best for pure solipsists, for whom the external world need not exist):

In America, it possesses a theory of knowledge, and above all an associated style of expression, which goes back to populism and beyond it … Its basic idea is that sincerity is the key to truth.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 82

(It’s amusing to hear such a stance referred to as a “theory of knowledge”, but social scientists really do talk that way, speaking  for instance  of a baby’s “theory of the world”.)

“The hysteria of complacency of the postwar period,

and the hysteria of protest of more recent times.” (1971)

And again, back to the math connection, reporting the fantasies of Michael Oakeshott:

What is proof? -- he asks.  There is no such thing as proof in general, he answers himself.  There is only proof  persuasive for this, that, or the other kind of man.   Cogency of proof  is relative to what you are.  he notices that this does not seem to apply to mathematics, and brazenly comments that just this has always made him suspicious of mathematics.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 180

Actually Oakeshott  put his case too weakly:  varying standards of proof are relevant in mathematics -- indeed, it is only within mathematics  that such scruples have structure and are in point.   In pre-Cauchy/Weierstrass analysis, proof was a bit of a kludge.   Later on, Constructivist qualms  came into play.  And in our own day, we distinguish between theorems whose proof requires the (disputed) Axiom of Choice, from those that can dispense with it.

The ultimate selbst-aufhebung of all such alethic egalitarianism is plain:

If almost everything is true in its own fashion, truth cannot matter very much.

-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 16

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Bonus nuggets, from the bottom of Gellner’s crackerjacks-box:

It is a travesty to say that martyrs die for Truth.  Real truths seldom require such dramatic testimony.

-- Ernest Gellner, The Devil in Modern Philosophy (1974), p. 55

the feminine theory of cognition:  that truth is not a matter of exploring or penetrating an external reality, but of gestation and parturition.

-- Ernest Gellner, The Devil in Modern Philosophy (1974), p. 62

Only psychoanalytic interpretations that tally with what is real in the patient, can mediate veridical insight.

-- Adolf Grunbaum, quoted in Ernest Gellner, The Psychoanalytic Movement (1985; 2^nd edn. 1993), p. 185

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On Tarski’s “Convention T” (expanded)

Accolades:

Convention T embodies our best intuition as to how the concept of truth is used.

-- Donald Davidson

My lone dissent:

More boredom thou shalt never see

than Tarski’s Truth Convention T.

“Snow’s white” is true -- let’s get this right --

if, but only if,  snow is white.

A.T.,  logician and skirt-chaser

[Footnote]

When Tarski’s logical machinery is used to provide languages with an interpretation, it should not be seen as giving us a definition of truth. [Emphasis in original.]  When used in this way, it gives us a theory that, for each of the infinitely many sentences of the language, assigns a condition that obtains if and only if that sentence is true -- where truth is something we takes ourselves to understand antecedently, without definition.

-- Scott Soames, Philosophical Analysis in the Twentieth Century (2003), vol. II, p. 293

For a major empirical research program, investigating the validity of Convention T, click here.
 

 ~

A fellow dissenter:

See for example the panegyrics in Wallace:  “It may strike the reader that Convention T is an astonishingly powerful intellectual device…” (etc.)  Wallace’s expressed admiration for the achievements of modern logic  is of course sincere, and I share it (not always for his reasons).  But these achievements should not be used to terrorize the reader …

-- Saul Kripke, “Substitutional Quantification”, in Evans & McDowell, eds., Truth and Meaning (1976), p. 340