.
Die bürgerliche
Stellung des Widerspruchs
-- L. Wittgenstein,
Philosophische Untersuchungen, #125
In philosophy
there are very few and perhaps
no valid
logical-impossibility or reductio
ad absurdum proofs.
-- Alasdair
MacIntyre, After Virtue (1981; 21984), p. 101
wer A sagt, muss
auch B sagen
-- old folk-saying
An example of epistemological ‘character armor’:
All scientific research programmes
may be characterized by their ‘hard core’. The negative heuristic of the programme forbids us to direct the modus tollens
at this ‘hard core’. Instead, we
must use our ingenuity to articulate or even invent ‘auxiliary hypotheses’, which form a protective belt around this core, and we must redirect the modus tollens to these.
-- Imre Lakatos, “Morphology of
Scientific Research Programs”, in I. Lakatos & A. Musgrave, eds., Criticism
and the Growth of Knowledge (1970), p. 133
In political or religious ideology, there are obvious
analogues of this ‘hard core’. (E.g. 'sacred cows'.)
~
[This exercise originally began as a simple application of
cool, unruffled logic
to a
sociopolitical topic too hot to handle in less rarified terms.
The point was, If you were logically
consistent, you would … (
taceo).
But a working-out of the general intellectual idea led,
inevitably, towards that domain (mathematics, logic itself) where the very
idea of consistency is most at home;
and led, less obviously, to a kind of backhanded defence of
inconsistency (wherein we follow our countryman Emerson, in his
depreciation of that ‘hobgoblin of little minds’).
Modus tollens is
that figure whereby, when a proposition entails a falsehood, that proposition
is infirmed. The procedure
finds general acceptance among the logically inclined. This essay considers cases where
otherwise logical people nonetheless kick against the pricks of this
restriction.
We now return to the original plan.]
~ ~ ~
It should be obvious that the
fundamental objections to racism and sexism … apply equally to speciesism.
-- Peter Singer, Animal Liberation (1975), p. 6;
quoted in Stephen Schwartz, A Brief History of Analytic Philosophy
(2012), p. 282.
Singer’s proposition has the form
“P => Q”;
we
shall call it the “Singer Sentence”.
It is, so presented, a
proposition,
and no mere proposal, since he claims that it is “obvious”, presumably because
founded upon a generally-accepted principle:
basically,
What’s sauce for the goose is sauce for the gander.
Since
that principle is reasonable (we have often invoked it ourselves), though
extra-logical, we have no a-priori quarrel with the Singer Sentence at all.
Suppose, however, that you yourself eat meat, wear wool and
leather, and are content that rabid dogs should be shot. Suppose further that -- selfish cad
that you are -- you do not, like the saintly naked Jainist, wear a veil before
your lips when drinking tea, lest you accidentally imbibe some supernatant
insect. Nor (for
shame!) do you enjoin your dentist
and your physician to refrain from
administering anaesthia or antibiotics, on the grounds that these, having been
developed with the aid of experiments on helpless animals, are fruit from a
poisoned tree. And suppose
that (blast your impudence!) you
do not intend to amend your ways.
Well then, in effect, objectively, you hold that “not-Q”. What follows from this?
What follows (by the most elementary logic -- the rule known
as modus tollens) is that, if you assert the Singer Sentence, then
you must needs conclude:
not-P. -- That, of
course, would be a political catastrophe.
Short of adopting Jainism, there are only a few ways out:
(1) Have recourse to a kind of meta-tollentic principle, to the effect
that any set of propositions which entail not-P must itself be denied, P being unantastbar.
(2) Proclaim the truth of Q, even while continuing your
carnivorous habits; shrug
apologetically; reference Emerson re the "hobgoblin of little minds".
(3) Boldly hold the joint and
several validity of :
P; P=>Q; and not-Q.
Then utter not another word upon the subject.
Such a mindset is known as
“doublethink”. It is not so bad
once you get used to it, judging by its millions of satisfied customers.
Cf:
Whether the ethic of ‘speciesism’,
to use Richard Ryder’s term, can be put on a logical footing any more sound that that of ‘racism’, I
do not know.
-- Richard Dawkins, The Selfish
Gene (1976; 2nd edn. 1989), p. 10
(That ‘any more’ sounds rather half-hearted …)
~
There is a different sort of logic-chopping that may be operative in a
case like this. Someone
wishes, for non-logical reasons, to assert that Q; then trumps up (fallacious) reasons
for asserting that P => Q.
(Thus, currently, in a certain political fringe, the desired outcome is
“It’s all Obama’s fault”. Whatever
P may arise in the world, a hasty P => Q is asserted, to reach the desired
conclusion.) But, for
psychological and political reasons, I do not believe that Singer himself is
here guilty of this: he in all
likelihood does feel ethico-logically compelled to deduce Q, however unwelcome
that conclusion may be in many quarters. For
he likewise (by different steps) reaches a quite distinct and socially
unmentionable conclusion (this time concerning deformed fetuses or the
disabled), and one which is precisely of a nature to enrage that segment
of
society which would embrace his earlier conclusion about animals. So, no, Singer was not trying to win
any popularity contests.
[Footnote] For those of you in the quandary (2), here
is the place 4 U to shop (courtesy of Garrison Keillor):
People’s Meats
Most of us accept strict
vegetarianism as the best way, but
many find it difficult to change their eating habits. People’s Meats is an interim solution. All of our meat comes from animals who
were unable to care for themselves any longer. Hoping to phase out the operation, we do not advertise
hours, prices, or location. We do
not deliver.
~ ~ ~
The analysis above was clad in sociopolitical raiment; but its skeleton is logical, which is
subject-matter-neutral. Consider
the following (which is skeletally somewhat distinct, but in ways
unimportant for our purpose):
Something similar to the P => Q step
was broached about a century ago; it concludes (while using reasonable
background metamathematical assumptions comparable to the “what’s sauce for the
goose is sauce for the gander” enthymeme above) that this standard practice
leads to paradox. As John von
Neumann put it,
A closer study of the merita of the case, undertaken by
Russell and Weyl, and concluded by Brouwer, showed that the way in which not
only set theory but also most of modern mathematics used the concepts of ‘general validity’ and of ‘existence’ was philosophically objectionable.
-- quoted in James R. Newman, ed. World
of Mathematics (1956), p. 2058
Must we then give up P? Brouwer (a Dutch mathematician
who had previously proved important results) now plays the role of Singer, and went
on to his logical conclusion:
A system of mathematics which was
free of these undesirable traits, “intuitionism”, was developed by
Brouwer. In this system the difficulties and contradiction of set
theory did not arise. However, a good fifty per cent of modern
mathematics, in its most vital -- and up to then unquestioned -- parts,
especially in analysis, were also affected by this “purge”: they either became invalid, or had to
be justified by very complicated subsidiary considerations.
-- id.
(That final clause is a far more dreadful consequence than
might be apparent to those outside mathematics, since mathematicians prize
elegance and generality of proof. )
 |
| Du muss dein Leben aendern ... |
So -- shall we bow to these strictures, and surrender our
mathematical meat?
Von Neumann goes on:
Only very few mathematicians were
willing to accept the new, exigent standards for their own daily use. Very many admitted that Weyl and
Brouwer were prima facie right, but they themselves continue to trespass, that
is, to do their own mathematics in the old, “easy” fashion -- probably in the
hope that somebody else, at some other time, might find the answer to the
intuitionistic critique and thereby justify them a posteriori.
In short, the bulk of mathematicians adopted strategy (3)
above.
Brouwer, like Singer, went on to make a pest of himself for
many years.
[Footnote]
Intuitionism --
initially a sort of mathematical vegetarianism -- is by no means dead. Michael Dummet, no crank, espouses
intuitionistic logic (I am currently painfully working my way through his essay
on the subject, line by line.)
And it has subsequently morphed in ways that are quite beyond me, e.g.
in topos theory.
[Footnote 2, a half hour later] In his essay “The Philosophical Basis of Intuitionistic
Logic” (1973), Michael Dummet writes (for our present purposes, the context is
unimportant), concerning a proposal that he has just put forward:
What is involved is a thesis in the
theory of meaning
of
the highest possible level of generality.
Such
a thesis is vulnerable in many places:
if it should prove that it cannot be coherently
applied
to any one region of
discourse,
to any one class of
statements, then the thesis cannot be generally true,
and the general argument in favor of it
must be fallacious.
[dbj:
That last phrase has rather a
Sherlockian cast to it.]
Construed in this way, therefore, a
position in the philosophy of mathematics
will be capable of being undermined by considerations which have
nothing directly to do with mathematics at all.
This amounts to offering a hostage to fortune --
specifically, a hostage to modus tollens, in its strong quantified form: P => ∀x
Q(x): the existence of but a
single exception (∃x ¬Q(x) ) blows the whole game.
~
We need not have recourse to anything so rarified as ethics
or metamathematics to be
confronted with an apparently (morally, intellectually) scandalous state of
affairs: It stares us in the face
in that favorite tow-headed boy of the philosophy of science, physics. And here, in that most venerable
part of it, Maxwell’s theory of
electromagnetism. Rock-solid in
its own day, it survived (and even helped inspire) the introduction of Special
Relativity. It fit smoothly with
the new developments in non-EM forces, graciously uniting with the Weak force
to form the Electro-weak, and so forth.
And yet, in the context of the atomic theory, it was (if only in
whispers) an intellectual scandal, since it predicted that the whirling
electrons would radiate energy away and quickly collapse into the nucleus. Thus, the material world as we know it,
could not exist.
Now, one stance of reaction in the face of so bald a
challenge, is to say:
“R-r-right!
Science cannot
err;
ergo, the world as we know
it
does not exist.
Meta-ergo,
we all are just brains in a vat.”
Such, in effect, is the path taken by contemporary
eliminative materialists -- a shuffling tribe of hunchbacked ne’er-do-wells, who, faced
with their irrefutable failure to derive free will, faith, thought, beauty,
aspiration, or much of anything of value, from their prolonged and proctoscopic
vivisections of
sea-slugs and the like (to which the World of Dr
Justice, friend to all creatures great and small, responds with the
evisceration of eliminative materialists)
-- these gentlemen (stretching that word to its breaking-point), these …
bipeds, conclude that Free Will is an
illusion, faith and beliefs and reason
all one great gigantic joke, and that we are all just brains in a
vat.
(In this they are
partly correct:
the
eliminativists -- mark the name -- are
but
brains in a chamber-pot.)
So, are those who do not take that bold blind path, whenever
some contradiction turns up, in a state of intellectual bad faith?
The answer is a subtle and qualified mmmm….n-n-no-o-o …..
For relief, we turn to Quine.
~
We frequently do turn to Quine for relief, to savor his elegant pellucid prose,
when the cacophony of the agora grows too rebarbative. Yet it is not for his
literary qualities that we seek him now, but for his (jointly with Pierre
Duhem) holism. Specifically, his celebrated doctrine
that theses face the tribunal of experience, not single-spies, but as a
corporate body. There are (so to
speak) concentric shells of propositions relatively dispensible and relatively
central: but in principle, none is
immune to revision. (Intuitionists
have even gone dicking about with the Law of the Excluded Middle, and you don’t
get more central than that. -- In that case, Quine was unimpressed: “When you change the logic, you are
only changing the subject.” Rather
a Platonist remark, that, Van.)
Thus, consider again the plight of electromagnetic theory,
faced with the atomic paradox. Its
bacon was eventually saved, not by any refinement of that theory itself, but by
an entirely unanticipated development:
quantum mechanics.
Now, there is no sense in which a pre-quantum physicist
could have said “We saw that coming” or “Toleja so”. Until the quantum theory arrived from nowhere,
physicists had been content to simply live with the contradiction, in the
classic fashion of Walt Whitman (“Do I contradict myself? Very well then, I contradict
myself.”) And their quietism
was justifiable, even during the years when no resolution was in sight. For, Maxwell’s electromagnetic theory
had done sterling service both
practically and theoretically, in a host of ways. [This, I am aware, is intellectually comparable to the classic defense of Mussolini, that he made the trains run on time.] The fact that it predicted an anomaly on the atomic
level … well okay, an anomaly involving THE ENTIRE UNIVERSE BLOWING TO BITS --
but still, an atomic anomaly, not a macroscopic on, suggesting that one might, for
the duration (until this beast be slain), simply wall-off the subatomic level
(“there be dragons”) and get on with our lives. It certainly
did not make sense to throw out the Maxwellian baby with the anomalous bathwater.
In the case of EM, it turned out that there wasn’t even any
bad bathwater to discard:
electromagnetism survived intact. In the case of the aether, the anomaly was the
Michelson-Morley experiment, whose results were later explained by yet another
where-did-that-come-from new
revolutionary theory, Special Relativity;
and in this case, the aether theory had to be discarded. But both cases illustrate the
thesis of Quine-Duhem, that when a body of doctrine is challenged, it is not
initially evident which pieces must eventually give, and some may be close to
the core of the structure.
In the case of Relativity and quantum theory, the transmogrification
went deeply into the core indeed, upending our notions of space, time, causality,
and continuity. In
retrospect, those curiously stable atoms seem not so bad; the explanation is harder to live with
than was the thing that was unexplained.
~
A word on our less-than-effusive approval of Quinean holism.
It is all too easy to imagine self-serving uses of such a
principle. As, Dennis the Menace,
caught with his hand in the cookie-jar, exclaims:
“Mother, do not prejudge! Granted, your B-fibres seem to present
an image of someone resembling young Master Dennis with his arm hovering above
a receptacle of some sort. The
hand itself -- which you suspect of larceny -- is not visible; perhaps it
has been tragically amputated, in which case the young fellow is more to be pitied than
blamed. Yet, how are we to
reconcile this dubious alleged image with the far more desireable thesis that
his character is pure as the driven snow?
Remember: Propositions face
the tribunal of experience as a corporate body! Perhaps 'tis but an illusion of swamp-gas; nay, perchance the fault lies somewhere in that oft-critiqued
principle of Induction …”
(Later, as he sits with his pookie-bear in the familiar
corner, he steams: “But I had her epistemologically …”)
~
Of all human endeavors, surely mathematics is the most sensitive to refutations, however
slight. Yet behold
this brawny scoffing attitude, specifically as regards the central and
indispensible Calculus (famously the target of Bishop Berkeley’s barbs -- which
it shrugged off):
If the calculus had not been ‘justified’
Weierstrass-style, it would have been ‘justified’ anyway. The point is that the real
justification of the calculus is
its success.
-- Hilary Putnam, “What is
Mathematical Truth?”
Breezy, that!
Huey Long couldn’t have put it more pithily.
~
And now let us bring it all on home: confronting the challenge of modus
tollens, when a refutation or contradiction or paradox is met, in its home
territory of mathematical Logic…
Celebratedly, the great German logician Frege fell into despair (his masterwork
already in galleys), upon being informed by Russell of the latter’s eponymous Paradox.
There we see the logical conscience
at its most delicate. For Russell’s paradox, worthy though it be,
is rather far-fetched, involving
sets-that-are-members-or-not-members-of-themselves (to which your average mathematician, let alone physicist,
will say: Huh??), all too reminiscent of the well-known but trifling Barber
Paradox, involving a purported barber who “shaves everyone who does not shave
himself”. Paradox: Does he shave himself???
Answer: Fageddaboudit; ain’t no such barber.
Rather other was the case of Quine’s Mathematical Logic. In the version of its first edition (1940), this was shown to
entail a contradiction.
Now: an
axiomatic system of logic, such as ML, is so tightly knit, that one bad apple
really does spoil the whole barrel -- you can’t just shrug and say, “Nobody’s
perfect.” The situation in
question, is generally held to be a catastrophe -- that any system which can
derive a contradiction, can derive any proposition at all.
However, Hao Wang stepped in (ever the gentleman), and
tidied things up, and all was well:
put right in the second
edition.
Haec fabula docet: Contradictions are a bummer, but don’t commit suicide on their account.
[Footnote]
The Duhem-Quinean corporate-body doctrine can be stated in terms of modus tollens, thus. Given
P1
+ P2 + … Pn => Q
and
¬Q
we conclude
¬P1
∨¬ P2 …
∨
¬Pn
That is, at least one
of the co-conspirators whose conjunction led to a falsity, must itself be
false.
But this does
not
imply that
the eventual valid
conjunction will involve most or even
any of those Pi;
we might even toss the whole lot of
them overboard, and usher in a whole different set of conjuncts, to accomplish
what we previously attempted with the P’s.
Such, roughly, describes the introduction of quantum
mechanics, or the refutation of astrology, or any other paradigm-switch between
incommensurables.
In his section on Quine-Duhem, Lakatos writes:
Some people felt intuitively that
the modus tollens from
refutation may ‘hit’ very distant
premisses in our total
knowledge, and therefore were
trapped in the idea that the ‘ceteris
paribus clause’ is a premiss which is joined conjunctively with the obvious premisses. But this ‘hit’ is achieved, not by modus tollens, but as a result of our subsequent replacement of our
original deductive model.
-- Imre Lakatos, “Morphology of
Scientific Research Programs”, in I. Lakatos & A. Musgrave, eds., Criticism
and the Growth of Knowledge (1970), p. 186
On a related note, another look at the notion that, should you ever
derive a contradiction -- not as a tollentic hypothetical, but as the result of
a deduction -- it’s Game Over for everything you’ve ever done. A noted logician writes:
In an inconsistent system, every
proposition is a theorem.
-- Hao Wang, From Mathematics to
Philosophy (1974), p. 42
Yet a few pages later he qualifies this:
There is a generally accepted
principle that a contradiction
implies everything. We may yet
distinguish proofs of the system which go through contradictions from those which do not.
-- Hao Wang, From Mathematics to
Philosophy (1974), p. 47
~
Purely linguistic note: A savorsome turn of phrase for the modal-tollentic move, one which I had not previously
encountered:
Assume towards a contradiction
that there is a set U ∈
T that is not in T’.
-- Volker Runde, A Taste of
Topology (2005),
There is a curious parallel, or at least an analogy,
between, on the one hand,
(1) the foundational challenges to
mathematics alluded to above, together with, not a refutation of that
challenge, but a broadly dismissive (and,
arguably, pragmatically well justified) response on the part of professionals
in the field;
and on the other hand
(2) recent broad-bore philosophical
challenges to adaptationism (neo-Darwinism), along with the overall reaction of
evolutionism’s professionals: indifference or -- the socio- and noö-politics
of the thing being what they are -- foaming hostility.
Consider in particular
a recent (2010) volume co-authored by Jerry Fodor and
Piattelli-Palmarelli, the well-written if pugnaciously titled What Darwin
Got Wrong.
Their basic thesis is extensively and subtlely argued;
hopefully I don’t overmuch crush it by fitting it into the following nutshell:
=> The (currently hegemonic) gene-centered adaptationist
theory empirically does not work;
indeed, for quite general reasons (having little to do with biology per se) it
could not work even in principle;
=> A holistic,
phenotype-cum-ecosystem-cum-kitchen-sink-centered approach might -- might -- prove valid, at least in principle; but in practice, the problem is intractable
from a nomologically explanatory standpoint, apart from occasional lucky breaks.
Or, in the author’s own undistorted prose (p. 127):
To be sure, none of that actually shows that there aren’t laws of
selection: there may be, on the
one hand, units of phenotypic change, and, on the other hand, units of
ecological change; and there may
be laws that connect the two. But
there’s no reasons to suppose, as adaptationists routinely do, that the units of
phenotypic change are anything
like what we generally think of as individual phenotypic traits.
(For connoisseurs of academic-polemical rhetoric, there is
actually a sly move here.
While rhetorically conceding the possibility of a pheno-unit/eco-unit
correlation, thus retrodictively legitimizing the latter two theoretical posits,
these supposititious entities “units of ecological change” are by no means as
familiar as other entities in the
ontology of biology, and on the face
of it
sound sort of bogus…)
~
We must distinguish two categories of challenge to any received
body of doctrine: The Anomaly; vs. Foundational.
The former (speaking just psychologically now) can either represent a
mere annoyance (or even: Something to be hidden from the fickle general public at all
costs, less they overestimate its importance -- e.g., Evolution, Climate Change, anything medical), or a (possibly
career-making) challenge. The latter, to almost everyone,
from the man in the street to the faculty lounge, tend to be just d*mned
annoying.
Thus, consider the Lasting Atom Scandal of the
pre-quantum years, a poster-child example of Anomaly. (Nobody called it that, but
in all candor they should have.) This was phenomenologically egregious
(in ways even a layman could understand), and had to be resolved somehow (some
day, by someone else) -- indeed, imagine that the problem existed today, rather
than a hundred years ago: You’d
have Republicans calling for the de-funding of physics, demanding e-mails archives, etc. But it did not necessarily challenge --
certainly it did not set out to
challenge, chin-foremost -- the nature of (say) Time, or Energy, or for that
matter the existence of atoms (which had indeed been doubted by scientists,
much later in history than most folks realize, but on quite different and less
sophisticated grounds).
This anomaly, as we have seen, turned out to be handled in
the most gratifying way possible:
We got to retain all the Maxwellian E-M we had laboriously learned, and
now the new Quantum Mechanics stepped in (Jeeves-like) to handle the anomaly.
More recently, there actually have been some challenges to
physical theory at a more basic
level: Time, which hoped to have
escaped further challenge by being
subsumed into Space-Time, is once again on the carpet, from string theorists
and others, who maintain that such parameters should fall out of a final theory, and not be input to it.
Had such a challenge been posed, say, in the nineteenth
century, it would have been an impudent kick at the foundations. But now it (allegedly) grows
out of theory: The theory may
be mistaken, but it calls Time into question because it thinks it has something better.
Much of the history of physics has been like that -- which
is partly why it has been (politically) such smooth sailing. Michaelson-Morley presented an
Anomaly to the aether theory -- but no-one had been going around snarkily
dissing ideas like distance and simultaneity until Special Relativity came
along and re-conceived these. This
was (to use the Hegelian terminology, which here does fit) not a destruction, but a sublation of the old
ideas.
Sometimes there will be a simply anomaly, such as the
discovery of continuous-but-nowhere-differentiable-functions, or space-filling
curves. These are eventually
gobbled up and incorprated into a more robust and sophisticated mathematics.
Often an original sui-generis gnarly anomaly will suggest a more general research
program: As, We need to pay closer
attention to matterns of continuity and convergence (pointwise, uniform, almost-everywhere,
etc., on the analytic side; and eventually the full exfoliation into the
neighborhood-systems of topology).
Quite otherwise are challenges that come out of nowhere and
that threaten to knock the stilts out from under you. Russell’s Parodox, reaching Frege at press-time, did not
strike him as another delightful puzzle to wrestle with some Sunday morning,
but as a poisoned dart in his life-work. Though they individually made numerous important
positive contributions, the lytic work of Gödel and of Brouwer can be read this
way, challenging the very notion of validity, truth, and proof. Denial of the Excluded Middle
indeed! “Sirs, there you go too far!” (“When you change the logic, you are
merely changing the subject.” -- Quine, dyspeptically.)
Schematically (in your worst nightmare):
“Miller has reduced mathematics to set-theory; and Spiller has shown that set-theory
is rotten at its foundations.
Therefore everything that you are doing, or have ever done, or ever
could do in your sorry life, is utterly worthless.”
Or:
(Pontius Pilate; Brouwer; post-Modernists): “What is truth, anyway? Huh? --
Meh.”
~
[Afternote]
It is a highly useful feature of the Blogspot interface, that it allows hot-linked
Labels. So for
instance, were you disposed to read more by or about Mr. Hao Wang, for example,
you would simply click on his name in the Labels field, and you will see every
post so Labeled, in reverse chronological of posting. But, annoyingly, there is a stringent limit on how
many Labels any given essay is allowed.
We filled this one up with mathy stuff and now have no slack left over
for the Darwiny bits. So here you
go:
Note: The last
two are not redundant upon each other, and indeed have (I believe) zero
overlap. Darwin was not an
ultra-Darwinist; “Je ne suis pas
marxiste” -- Marx.
Additional relevant Labels, which (boo, blogspot) would not fit into the Label field for this post:
http://worldofdrjustice.blogspot.com/search/label/Ernest%20Gellner
[Post-Afternote]
Other examples of modus drasticus tollens.
A mittel-europäischer
rationalist recalls “those golden, and, all in all very peaceful final decades
of the colonial system”, and adverts ad
the hermeneuts:
The argument seems to be --
Descartes led to Kipling. We
repudiate Kipling, so we must repudiate Descartes as well. The expiation of colonialism must
include the repudiation of clarity, for that had been but the tool of
domination.
-- Ernest Gellner, Language and
Solitude (posthum. 1998), p. 176
We, by contrast, raise a glass to Kipling; so if this Descartes fellow had
anything to do with it, chap can’t be all bad.
 |
| “During five literary generations, every enlightened person has
despised him, and at the end of
that time nine-tenths of those
enlightened persons are forgotten, and Kipling is in some sense
still there.” -- George Orwell, 1942
|
.
