Pop ! goes the Positivist

 [This is a continuation of a thread begun here.]

Another example of fetishizing the visible (cf. now also this)  occurs in Karl Popper’s Conjectures and Refutations: the Growth of Scientific Knowledge (1962; page refs. to the Harper paperback reprint), p. 284:

In the last pages of Testability, Carnap discussed the sentence ‘If all minds should disappear from the universe, the stars would still go on in their courses.’  Lewis and Schlick asserted, correctly, that this sentence was not verifiable.

They so asserted incorrectly.  After all, we have a pretty good handle on those stars in their courses, beginning from not long after the Big Bang  through the present, during most of which there were no human minds.

And here’s a simpler experiment:  Let’s all just refrain from observing the moon for the next two weeks.  Now we open our eyes.  Surpri-ise!  Right where we expected to find it, and at a phase two weeks more mature than when we last checked (rather than the phase as last seen).

One occasionally observes a very young child shut her eyes and announce in a sing-song: “Youuu ca-han’t seee mee….!”  I doubt the child actually believes that, it’s just fun to say.  But suppose she did. She is then entertaining an erroneous optical theory, but not an erroneous ontological one.  She imagines that the lights went out for you as well; but she never entertains the thought that you yourself have been extinguished, or been transfixed into some mute deaf motionless limbo until such time as she shall be pleased to gaze again – after all, she has just shown this, by addressing you.

Incidentally, the whole Schlick-shop with its merry/morose band of logical positivists, appear to have created enormous problems for themselves, and come up with completely counter-intuitive results, from the basic motivation of wishing to exclude all metaphysical propositions from the outset as strictly meaningless.   (The story here is in line with our overarching defense of theism.)  In other words, it wasn’t enough to denounce as wholly unconvincing,  all purported metaphysical statements hitherto; nor to brand it a particularly uninteresting language-game; nor to marginalize the enterprise with the derision usually reserved for circle-squarers.  Metaphysics as such, and for all time, had to have a stake driven through its heart, the stake itself consisting in pure reasoning from selbstverständlich  (metaphysically a-priori? God-given? -- oops, NOT) first principles.  For a demonstration of the absurdities that apparently resulted, cf. Popper, passim.  As (p. 281, op.cit.)

My criticism of the verifiability criterion has always been this: against the intention of its defenders, it did not exclude obvious metaphysical statements, but it did exclude the most important … of all scientific statements … the universal laws of nature.

Why even go near such a disaster, for the sake of excluding-in-principle all metaphysical statements – which they were ignoring anyway?  Popper goes so far as to hazard a guess in a footnote to p. 175:  “One not need believe in the ‘scientific’ character of psycho-analysis (which, I think, is in a metaphysical phase) in order to diagnose the anti-metaphysical fervour of positivism as a form of Father-killing.”  (Note, b.t.w., the capital initial of “Father”.)

*

Since we were obliged, a moment ago, to deprecate Popper, let us hasten to another example, in which we find him on the side of the angels.  First let us note:

             * Our conduct is guided by moral laws.

             * Our scientific activity is guided by metaphysical principles.

             * Both categories are philosophically vexed, and any concrete application of these laws and principles may prove problematic; but we cannot do without them.

            Here is Popper (p. 287) characterizing the logical positivists: “They implicitly accept the rule: ‘Always chose the most probable hypothesis!’  He goes on:

Now it can be easily shown that this rule is equivalent to…’Always choose the hypothesis which goes as little beyond the evidence as possible!’  And this, in turn [amounts to] ‘Always choose the hypothesis which has the highest degree of ad hoc character!’

an undesirable result for science.   (Cf. Chomsky, passim, re the extraordinarily abstract and roundabout way you may have to proceed  to get valid results, by no means simply tip-toeing a bit beyond “the evidence”.)

            The principles which we require instead  may be best illustrated by an old joke (presented here in a slightly revised form):

     A zoologist, a mathematician, and a positivist  are touring the hinterlands of Bohemia by train.  Through the window they spot, standing motionless at some distance, a purple cow.

     “Fancy that!” exclaims the zoologist. “There are purple cows in Bohemia.”

     The mathematician, in a low voice, amends: “There is at least one purple cow in Bohemia.”

     The positivist, raising his finger, crisply corrects to:  “In Bohemia there is at least one cow, purple on one side.”

            The joke is that the positivist has the most precise view of the evidence; and of the three statements, only his own is certainly true (interpreting an unmodified “purple cow” in the natural way, as “purple more or less all over”); yet his scruples are absurd.  Science would not progress, were it bound by such fragmented literalism.  (Indeed, we should have to replace “cow” with “an apparently bovine organism”, and always assuming that we can rule out a mirage, or a collective hallucination, or some effect of swamp gas; or for that matter an analogue to one of Hilary Putnam’s pet entities, the Mechanical Cat from Mars.)  And we would have to gloss Quine’s celebrated interjection, gavagai!, as possibly:  “Lo, a rabbit-stage, seen from the east.”

(For more on bovine mathematics, see "How Now, Round Cow".)

A gathering of Positivists

Note:  Occasionally one meets with such unilateral positivist agnosticism  in a non-jocular context:

Franz Ferdinand rode erect, his visible foot deep in the stirrup, a saber at his side.

-- Robert D. Kaplan, Balkan Ghosts (1993), p. xxvi

The above scenario has been much discussed among epistemologists, under the rubric “Henry and the barn façades”.   However, the cow version is much funnier.

Ceci n’est pas une grange 

~

To move from this to an outlook useful for science, we need these metaphysical principles:

   (1) a certain uniformity or self-cohesion to nature

   (2) the reliability (not in a logical, but in a practical sense) of induction.

(There are others, of course, such as the usefulness of deduction;  but that applies in every world, not just in this contingent one.)

            Now, both these principles are of vexed and delicate application; as are, indeed, the Commandments.   Just as it is difficult to avoid situation ethics, I see no way around a kind of “situation metaphysics”.   The zoologist, using these principles, together with his discipline-specific knowledge (to the effect that animals generally come in species, not as singletons), concludes the presence of purple cows.  But here I must side with the formulation of the mathematician: there is at least one such cow, but perhaps no more.  For, nature’s uniformity depends on the domain.  Were an experiment ever to identify, say, a Higgs boson, we should conclude the existence of an emphatic plurality of Higgs bosons; that there should only be one in the universe, is inconceivable.  But biology is more mottled than that.  Given our collective prior acquaintance with hundreds of thousands of cows, none of which was ever purple, we would hypothesize here an extremely rare mutation for (bilateral) purple (bilateral because bilaterally symmetric animals generally continue to evolve with bilateral symmetry, pace the occasional narwhal).  Since a cow delivers but a single calf at a time, this one has no identical siblings, so it is probably unique.  Moreover, the bulls will likely shun her for her color, and she shall have no offspring.  The mutation dies with her.

Carnap attempts to avoid the assumption of a metaphysical principle, by declaring (1) to be “analytic” (i.e., more like “2+2 = 4” than like “animals tend to come in species” or “each elementary particle is invariable within its class”).  Popper comments (p.289) that “no such principle of uniformity can be analytic (except in a Pickwickian sense of ‘analytic’)."  In fact, we noticed, the principle is nothing remotely like analytic, having always constraints on its application, which in some domains may be severe.  And as for (2), I concur with Popper (p.292) that it is “a principle of a priori metaphysics.”  And none the worse for that.

[For further notes on ineluctable metaphysical underpinnings of the scientific enterprise, see here.]

Scheinprobleme in der Pinguinologie

(I) Q:  Hamster-fanciers are all in a worrit -- which is the cutest li’l furball.  Do penguinologists face a similar problem?

A:  No.  We understand that, quâ  transitory incarnations of the Ur-Penguin in Platonic heaven, all penguins are cute-o-metrically equivalent.

(II) Q:  Which came first -- the penguin, or the egg?

A:  Why, the egg, obviously, as a local instantiation of the World-Egg, which in turn is subtended by the Perfect Platonic Sphere.

Then  from this  birthed-forth  an endless succession of Perfect Penguins.

Das Weltenei (artist's conception)

*

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*

(III)  Q:  Do penguins have souls?

A:  Of course.   (Very round and nice ones.)

ONTOLOGIA NUMERORUM

[This is a continuation of a thread begun here.]

“Die ganzen Zahlen  hat der liebe Gott gemacht…”

    -- L. Kronecker, in a rare moment of candor

So, God made the integers, good job; but what *are* they, exactly?  -- The question may be ill-posed, as unanswerable as what Matter really 'is', or Time.  A mathematician will be less troubled by this, than a physicist, since mathematicians are accustomed to things being characterized only up to isomorphism.  Even so, a thoughtful mathematician is aware that there is an ontological problem:

“The” natural numbers  have a conceptual existence which is quite independent of set theory [dbj:  Certainly it is cognitively prior], and it is an act of faith on our part  to agree that the set of “the” natural numbers  endowed with the “count-one-more” function  is a configuration satisfying the postulates for a simple chain.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p.  153

The Intuitionist philosopher Michael Dummett points out that, even in this most basic and most intuitive object, unexpected depths and terrors lurk:

The notion of ‘natural number’, even as characterised by the formal system, is impredicative.

-- Michael Dummett, “The Philosophical Significance of Gödel’s Theorem” (1963), repr. in  Truth and other enigmas (1978), p. 199.

(Impredicativity is a petri-dish for paradox.)

Wishing to dish up something quotable  and a little less evasive than what St. Augustine said when asked the nature of Time, we coin this epigram:

Z  is the substrate  on which we grow the truths of number-theory. (*)

~

The best-known approach to 'reducing' numbers to something supposedly more basic, is that of set theory, in any of several varieties.

The world waited for Russell, for them to be declared “classes of classes”.   Thus ‘four’ is -- not merely may be analogized to, but is -- a great bulging bag of examples, one of these being the number of the Evangelists (or, in more modern terms, the Beatles).  -- So:  are we commited to an ontology of classes of classes?

            Maybe not.  Even Russell says that “classes…” (let alone classes of classes) “cannot be regarded as part of the ultimate furniture of the world.”  We can in fact, for the present, afford to be completely agnostic as to the best mathematical or philosophical characterization of integers, just as we can afford to be agnostic as to the best characterization of rocks.  For, nothing so far hangs on this characterization.  You can build castles out of rocks, and that didn’t change when rocks were discovered to be atoms surrounded by mostly empty space; they didn’t suddenly become porous, or bounce.  And seven will always be prime, whether it turns out to be best understood as a class of classes, a mess of masses, or the representative on earth of His Holiness Septimus the Seventh. In a sense, I’m not saying we need believe anything abstract or ethereal or (to begin with) even ‘mathematical’ about numbers:  I’m saying they’re already real as rocks.  But if you believe in rocks – if you don’t think you’re just a brain in a vat and are hallucinating them – then you must believe what you discover about the nature of rocks: some are hard, some are friable, etc.  And so it is with numbers: some are composite, some are prime, some are zeros of the Riemann zeta function.  Mathematically, these things just irresistably follow – follow without our pushing them along, like those now-unemployed angels pushing the planets-- follow irrespective of philosophy.  Philosophically, we are at liberty to contemplate the integers with the breezy, boozy indolence of the chimp: one banana good, two banana more good, three banana more gooder still.  What we are not at liberty to do, is to exclude them from our ontology.

            Nothing really hangs on what numbers themselves “are”, or whether the question even has an answer, or has meaning.  Everything hangs on their objective transcendent fixity as positions in a pre-existent pattern.

            Another parable, on the inflexible reality of the integers.  Even if no-one in your entire universe ever ever counts anything, the numbers are present, like the stone guest at the feast.  (Rather as electromagnetic radiation is present, even if everyone is blind.)

            Take for instance planet Xymol.  There, all reference to anything quantitative, or logical, has always been taboo.  The activity of ‘counting’, which (by report) exists in other galaxies, they hold to be a merely contingent and local (and rather disgusting) custom, much like cannibalism.  The inhabitants of Xymol are purely “artistic”, purely emotional and qualitative; numbers, they maintain, are non-existent, and they have no need of them.  And one of the emotional and qualitative things that little Timmi von Xymol loves to do with his cubical blocks, is to arrange them into perfect rectangles (non-trivial ones: each side greater than a unit length).  As Timmi’s guardian, you want him to be happy.  But despite your best intentions, you’d better be careful what you give him.  If you give him 24, or 93784668225, or RSA-576 blocks, he’ll be fine, since these are composite.  But if you give him 17, or Mersenne-41, then after a great deal of fretting and fussing, the poor lad will break down in tears. For these will ever be prime, on the peaks as in the valleys,  though you deny the existence of numbers.

            The question, “What are numbers?”, is  strictly in itself  not very interesting: neither mathematically, theologically, nor philosophically.  Though Carnap’s outlook is in general foreign to the views developed here, his deflationary remarks anent that question, in “Empiricism, semantics, and ontology” [1950; repr. in Benacerraf & Putnam 1983] may stand.

            It’s like – What are penguins, really?  Godlike birds?  – to be sure.  Fusiform instantiations of ornithological perfection? – freilich, freilich.  But what are they quantum-mechanically, cosmologically, mathematically?  -- Oh, away with you, let us simply admire them, as they slide on their tummies along the ice!

(*) Footnote, lest that splendid epigram be taken in a Nominalist sense.

The metaphor is actually fairly exact.  We don’t invent tomatoes:  we discovered them in nature, and now help them propagate their pre-existent kind.  The ‘truth’ of the tomato -- its genetic type -- existed prior to our discovery.   When we grow any individual tomato, it may be stunted, or waterlogged -- we simply do our best to approach the magnificent pre-existent Platonic Tomato.   And likewise with math:  our (imperfect) constructions and theorems are peasant-style approximations to the Type.

God grant that, in Paradise, my resurrected body may feast on a Platonic BLT.

[Further reflections here.]