The Ontology of Physics (updated)

The question at issue here is, What sort of entities are fundamental in physics?  E.g., in the late nineteenth century through early twentieth centuries,

whether atoms were real objects or only mnemonic devices for coding chemical regularities.

-- Abraham Pais, Subtle is the Lord (1982), p. 80

The idea of atoms in some sense  goes back to Ancient Greece; and today, they are taken for granted.  So it is surprising to laymen, how long opposition to a Realist take on atoms lasted among some philosophers and physicists (e.g. Ernst Mach).  An opposing view, from a leading German chemist:

Ostwald’s ‘Energetik’, according to which molecules and atoms are but mathematical fictions, and energy, in its many forms, the prime physical reality.

-- ibid, p. 83

Good bluff Rutherford, by contrast, had sucked atoms with his mother’s milk, and claimed that he “could see the little buggers as plainly as a spoon”.

Oddly, the debate on this seemingly practical laboratory matter,  had elements more characteristic of political or theological controversy, in which neither side has a prayer of of convincing the other by rational argument:

The most remarkable fact about the nineteenth century debates on atoms and molecules   is the large extent to which chemists and physicists spoke at cross purposes, when they did not actually ignore each other.

-- ibid, p. 80

~

Since that time, a host of new physical building-blocks have been proposed, and in some cases observed: from the unexpected and rather unwelcome muon (I.I. Rabi: “Who ordered that?”), to the massless chargeless neutrino (rather like Bishop Berkeley’s “ghosts of departed quantities”), through quarks, gluons, gravitons & gravitinos, selectrons & electrinos, axions, preons, virtual photons, phonons … such as might make even Rutherford gag.  But several of these are now widely accepted -- not as mere bookkeeping mechanisms, but as entities with further properties to be discovered;  so that the smart money has been on Realism, so far.

Ernest Rutherford,  swallowing an atom  but straining at a quark

Footnote:

We shouldn’t be too hard on old Ostwald  for backing the wrong horse in the Atoms Affair.   Dissident voices today would declare  as epiphenomenal not only atoms, but even the elementary particles of which those are admittedly merely bundles:  demoted to excitation-states of superstrings;  or emerging from the combinatorial-automaton structure of the world. 

Additionally, his Energetik has enjoyed a bit of a revival in some quarters:

  https://de.wikipedia.org/wiki/Energetik_(Philosophie)

More recently, information (or, solemnly, “The Information” -- Wheeler's "It from Bit") has emerged among some as a skeleton-key to everything else in the physical world.

Thus, at the bottom of everything, behind and beyond the Maya of the particle zoo, lies:

Thales:  Water.
Oswald:  Energy.
John Wheeler:  Information.

Stewardess:  Coffee tea or mi-ilk?
The Milesian:   Just water, thanks.

Or, a more recherché candidate, from philosopher Hilary Putnam:

Nothing has more physical significance than spectral measure.

(That might strike the layman as rather a … spectral candidate for the role of firmest substrate of all.)

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The word “ontology” is not one you are likely to overhear at the bus-stop;  nor indeed does a physics-major typically run across it.   But the more we see physics as science (Wissenschaft) rather than a special kind of engineering (the “shut up and calculate” ethos of the years around WWII), the more we meet questions traditionally treated under that rubric -- and indeed, contemporaneously, even under that very name. 

As:

My own position is that the issue of ontology is crucial to quantum mechanics.
-- Roger Penrose,  The Road to Reality (2004), p. 785

Let us now return to the ontology of the consistent-histories approach.  The theory operates with entities called coarse-grained histories.  … The ontological status of the insertion of such a projector set  is still not fully clear. …  A history from a maximally refined set seems to me to provide a strong candidate for what might be regarded as ontologially ‘real’.

-- Penrose,  The Road to Reality (2004), p. 788

Present-day quantum mechanics has no credible ontology … The importance of having an ontologically coherent quantum mechanics cannot be over-estimated.
-- Penrose,  The Road to Reality (2004), p. 860, 865

Re the notion of macroscopic quantum superposition being unproblematic:

This is taking a ‘pragmatic’ stance  that does not really address the ontological issues.

-- Roger Penrose,  The Road to Reality (2004), p. 812

And, full-bore:

Many contemporary thinkers seem to have supposed that, in discarding its mechanist ontology, physics had discarded its ontology:  matter had been dematerialized … The very progress of physics itself  seemed to them to call for the renunciation of mechanism and materialism  in favour of the de-ontologised view of science presented by Mach.

-- John Watkins, Science and Skepticism (1984), p. 138

 

In philosophy proper, ontology is fundamental, being prior to anything else.   In its application to or rather analogue within  physics, by contrast, it historically comes behindhand, as a setting in order of what-all several centuries of reflection and experiment have come up with:  We may think of it as a kind of cast of characters, not fully drawn-up until the play has been written:  in the course of writing it, you find out you need a ladies-maid, and so eventually she is placed upon the prefatory page of Dramatis personae, that typographically precedes the play itself -- and as a nice afterthought, you name her Lisette.  Similarly, particle physics did not begin by being defined, a priori, as (back among the Greeks) the Science of Atoms, or (later) as the Science of the Proton, the Neutron, and the Electron, or (later still-- the Barock Age) as the Menagerie-management of the Particle-zoo (with a fixed given roster of inmates), nor as the Curating of the Wiggling of Strings.  There is a thematic continuity throughout all these stages, but the staffage keeps changing.

As for the role of this Ontology, or Cast of Characters, it is not (despite the spectral example of traditional metaphysics per se) just something to admire from afar, like Mount Rushmore, but rather, as Goedel said pragmatically re which axioms we should adopt for math and logic, they should themselves possess generative potential -- by their fruits ye shall know them.  Thus, hard-headedly:

Kepler’s theoretical ontology, unlike Gilbert’s, was not organically related to his laws;  even if it could be squared with the latter, which seems doubtful, it failed to make any contribution to the testable content of his system.

-- John Watkins, Science and Skepticism (1984), p. 197

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As remarked earlier, I may well go to my grave without ever grasping the concept of an observable, much less physical ontology in general.   Still, it is helpful towards organizing my thoughts, to have an online scribble-space, so that the matter is, so to speak, officially a topic, a project under way.   For now, this is just a whiteboard on which to stow some juicy quotes.  Your own juicy contributions are more than welcome.
For a more general surview of the ontology of the various sciences, click here.

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Physics may be defined as the art of saying things about stuff (or stuff about things -- predications concerning entities, for the fastidious).  But what are these entities, whereof we predicate?  In the first place -- observables.

P.A.M. Dirac, The Principles of Quantum Mechanics (1930; 4^th edn. 1958), p. 116:

From our assumption that the energy is an observable, there are sufficient stationary states for an arbitrary state to be dependent on them.

For a layman, this is bemusing.  The assumption that it’s an observable?   Can you observe it, or can’t you?  -- Evidently there is much more to qualifying as  “an observable” than merely being … observable.
(Compare Einstein, in one of his Zen moments: "It is the theory that decides what we can observe.")

P.A.M. Dirac, The Principles of Quantum Mechanics (4^th edn. 1958), p. 458 (re certain eigenstates):

Science contains many examples of theoretical concepts which are limits of things met with in practice  and are useful for the precise formulation of laws of nature, although they are not realizable experimentally, and this is just one more of them.

Emphasis added.  “Limits” in the mathematical sense.

Note that “not realizable experimentally” does not constitute much of a disability.  What, after all, is?  “Carthage lost the Punic Wars”; “I love you”; “E8 is a 248-dimensional rotation-space”:  no, almost nothing is.

Robert Lindsay & Henry Margenau, Foundations of Physics (1936), p.402:

Quantities such as position, energy, momentum, and the like, capable of measurement… will be called observables,  although it is not intended to imply that they are observable directly.

The caveat is troubling enough;  but now this:

In quantum mechanics, the state of a system is no longer defined by means of a number of variables having an immediate intuitive appeal … In fact, it is not defined in terms of observables at all;  it is simply a function in configuration space.

Carl Hempel, “Problems and Changes in the Empiricist Criterion of Meaning” (1950):

Green, soft, liquid, longer than  designate observable characteristics, while bivalent, radioactive, better electric conductor, and introvert do not.

This odd assertion, by a well-known philosopher of science, seems more psychological than scientific.  It is reminiscent of Locke’s distinction between simple and composite ideas.

Eugen Merzbacher, Quantum Mechanics (1961, 2^nd edn. 1970), p. 153:

Following Dirac, we call observable any Hermitian operator which possesses a complete set of eigenfunctions.

This might sound opaque to some, but for a math guy it’s the clearest statement yet, by far.  Of course, what it amounts to physically, intuitively, is something else…

Gerald Holton, The scientific imagination (1978), p. 202:

The idea of making quantitative indicators of anything at all  fascinates some persons, and repels others as dangerous or absurd.  This difference is caused largely by thematically incompatible -- and therefore often unresolvable -- personal views concerning the ability of quantifiables to lead to … the deepest reality.

Note the silly dichotomy -- as though failing to lead to "the deepest reality" (a deeply suspect term) meant that they couldn't be "indicators of anything at all".

~

I had some fun above, playing with a rumpled old word like stuff, shoving it before the microphone of science.  Here a gifted popularizer  makes similar play  with pronouns:

[In its] Einsteinian reframing … is spacetime a something?

-- Brian Greene, The Fabric of the Cosmos (2004), p. 39

In that historical context, the question concerned the ontological status of (the novelty) ‘spacetime’, as opposed to the traditional notions of the independent entities, space, and time.

(More recently, spacetime has been demoted in some theories -- not returning to a Cartesian product of space and time, but being derived as an epiphenomenon of more fundamental items.  Thus, twistor theory, among others.)

If there is no aether to provide the standard of rest, what is the what  with respect to which this speed is to be interpreted?

-- Brian Greene, The Fabric of the Cosmos (2004), p. 45

If an individual electron is also a wave, what is it that is waving?

-- Brian Greene, The Fabric of the Cosmos (2004), p. 88

(Here the wordplay inheres not in the pronoun what, but in the verb.  He could more conventionally have written, “What is the medium for the wave?”, but the startling verbal formulation ‘makes it strange’, confronting us with something more fundamental.)

~

Stephen Hawking, A Brief History of Time (1988; 2^nd edn. 1996) p. 75:

The fact that confinement prevents one from observing an isolated quark or gluon  might seem to make the whole notion of quarks and gluons as particles   somewhat metaphysical.  However, there is another property of the strong nuclear force, called asymptotic freedom.  The concept of these entities  was already well-defined, or not, as the case may be:  certainly well-defined as bookkeeping conventions, if nothing more.   Asymptotic freedom -- “at high energies, the strong force becomes much weaker, and the quarks and gluons behave almost like free particles” -- simply adds a further mode of observing their effects:  and in this case, their effects when they are relatively ineffectual -- quarks on holiday.

Failure to be observable in isolation certainly doesn't make a thing "metaphysical" (in the colloquial bad sense intended here).  You cannot observe a "brother" in isolation:  dissect him down to his last tissues, nothing will reveal his brotherhood but the historical context.  Nor, perhaps, can you observe Coulomb attraction in a single isolated particle -- it takes two to tangle.  (I might be wrong on this -- the photon cloud and all that.  But how does the cloud tell you whether you've got an attraction or a repulsion?)

Steven Weinberg, Dreams of a Final Theory (1992),  p. 181:

The positivist concentration on observables like particle positions and momenta  has stood in the way of a “realist” interpretation of quantum mechanics, in which the wave function is the representation of physical reality.

Wiki, "Quantum field theory" (excellent article, btw):

In quantum field theory, unlike in quantum mechanics, position is not an observable.

From the point of view of quantum field theory, particles are identical if and only if they are excitations of the same underlying quantum field.  Thus, the question ‘Why are all electrons identical?” arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental.

The global phase of the wave function  is arbitrary, and does not represent something physical.

Wiki, "Implicate and explicate order" (of interest only to those who are already devotees of guru-physicist David Bohm):

 Bohm’s paradigm is inherently antithetical to reductionism … and can be regarded as a form of ontological holism.

Wiki, “Introduction to Gauge Theory”:

The electric field and the magnetic field are observable, while the more fundamental electromagnetic potentials V and A  are not.

~

In this ontological context, it is far from clear how the phrase ‘more like’ is to be applied.  Comparison of historical theories gives no sense that their ontologies are approaching a limit:  in some fundamental ways, Einstein’s general relativity resembles Aristotle’s physics more than Newton’s.

-- Thomas Kuhn, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 265

Cf. too Dirac’s remarks (1951) that the aether concept was ripe for resuscitation.

[Update 8 May 2012] And now this:

http://www.nature.com/news/a-boost-for-quantum-reality-1.10602

The philosophical status of the wavefunction — the entity that determines the probability of different outcomes of measurements on quantum-mechanical particles — would seem to be an unlikely subject for emotional debate. Yet online discussion of a paper claiming to show mathematically that the wavefunction is real has ranged from ardently star-struck to downright vitriolic since the article was first released as a preprint in November 2011.

The paper, thought by some to be one of the most important in quantum foundations in decades, was finally published last week in Nature Physics

They say that the mathematics leaves no doubt that the wavefunction is not just a statistical tool, but rather, a real, objective state of a quantum system.

I told you so...

Physicists reify space-time. They elevate it from a four-dimensional diagram used to record their experience into the kind of “real essence” that Bohr warned us not to seek.
-- David Mermin (March 2014), at:

http://www.nature.com/news/physics-qbism-puts-the-scientist-back-into-science-1.14912?WT.ec_id=NATURE-20140327

~

Not the same as the question of the building-blocks (ontological bricks) of physics, but related to it, is that of the Boundaries of Disciplines:  between physics and neighboring fields (chemistry, mathematics, …) and within physics itself (mechanics, astronomy, electromagnetism, condensed-matter, nucleonics, quantum theory, …).   In one sense, the question is idle -- you are working on whatever project you are working on, with methods appropriate thereto, however outsiders might classify them.  But it also has practical consequences, e.g. in the writing of textbooks.  As:

The traditional teaching of thermodynamics and statistical mechanics  as distinct subjects,  has often left students with their knowledge  compartmentalized, and has left them ill-prepared to accept newer ideas such as spin temperature or negative temperature  as legitimate and natural.

-- F. Reif, Fundamentals of statistical and thermal physics (1965), p. viii

That, from the textbook we used in stat mech at Harvard -- in the physics department, though previously I had only met notions of enthalpy, temperature, free energy, and entropy, in a chemistry course.

Similarly, Lindsay & Margenau remark, in their historical overview Foundations of Physics (1936), that they are moving away from treating optics and electrodynamics as distinct disciplines, “the former being, since Mawell’s time, really a branch of the latter.”

~

God’s-truth vs Hocus-pocus:

It is tempting to dismiss these quantum waves  as mathematical contrivances … but in the laboratory these “probability waves” can be manipulated with mirrors …

-- George Johnson,  A Shortcut Through Time (2003), p. 38

~

The prototypical example of an ontological ‘bit’ of chemistry and physics, is the atom (the ‘indivisible’ in its Greek etymology).  But later perspectives can get quite unprototypical:

A neutron star … is basically a giant atomic nucleus, stabilized by gravity.

-- J. Richard Gott, The Cosmic Web (2016), p. 29

.

On Crunchy Numbers

In the Ike era, we grew up on Wonderbread® :  a sort of Brot ohne Eigenschaften whose edulcorated transmogrification is known as Twinkies.

Since that time, we have learned to abjure, not only such treif, but anything not calling itself wholegrain.   Or, better still, multigrain: some brands boast seven grains, a few claim twelve; disparate mixtures  full of gritty, grainy, crunchy goodness.

 

Now  our local upscale supermarket offers a variety of own-branded bread, that boasts (in large letters) no fewer than

27 GRAINS

That really surprised me.   It’s one of the main points of Jared Diamond’s Guns, Germs, and Steel  that digestible, domesticable, feasibly growable grains  are not to be had for the asking;  there just aren’t that many of them.   The explanation is that, a little lower down and in smaller font, the label reads

AND SEEDS

In other words, this brand of bread is equally at home in the bakery and in the bird-feeder.

Our son, scoffing at this terminological legerdemain, inquired why the market chose “27” of all things.   Unhesitating I replied, “Because it is three to the third power -- the trinitarian pinnacle of the Perfect Cubes”.  -- Said heir and offspring, himself a nascent mathematician, appreciated the point, but doubted that your average shopper was aware of such things.

And indeed, there is a larger point.   For number-theorists like Ramanujan, each integer has its own flavor, its own biography and backstory -- cf. the famous incident of the taxicab numbers (in which G.H. Hardy plays the straight-man or fall-guy).  But for ordinary folks, like rocket scientists (who deal with contingent analogue quantities, rather than integral transcendent entities) or English professors (surrounded by a midge-cloud of  pre- or sub-arithmetical post-modernists), all but a very few -- small -- integers  will be featureless.   Some will be visually familiar:  3 and 4 (the triangle, the square), 5 (the quincunx on dice), 6 (boxcars, ditto),  7 (a “lucky” number for the superstitious) or 23 (ditto, for the more cerebral), 10 (count your fingers), 20 (with its portrait of Jackson).   Some will be familiar for incidental, non-mathematical reasons, like 100 and 1000 (which owe their prominence to the accident of base-10 notation -- a case of decimal fetishism).   --  Those associations are widely shared;  but there may be others  more individual;  as, (mostly for girls), “sweet sixteen” (or in Latin America, la quinceañera).   Indeed,  any integer small enough that you have lived that number of years (particularly those for which you still kept track of your birthdays).    As, the poem(-collection) “Now We Are Six” (by A.A. Milne), an anniversary in memory still green (and which I teach to the neighborhood children when they reach that delightful threshold).  Or…. 27;  which, even before I had attained that age, always seemed numinous (probably, indeed, for its prime-power nature), and which was ratified as such by my marrying at exactly that age, quite close to my birthday.    Whereas, for most folks, a number like 81 (pourtant a perfect square, as well as  3^2^2, to boot) tells no tale, sings no melody.

~

The preceding remarks are of psychological or anthropological interest, but of no moment for mathematics itself:  They present an external, human-centered view of the integers.  But further considerations suggest a subtle distinction between two ways a given integer may be “interesting” (a more bloodless equivalent of our anthropomorphic term crunchy).   We might dub these internal and external:  both times internal to mathematics as a whole, but in one case only, internal to number theory in its most elementary sense.

Thus, consider 5.  Number-theoretically, it’s a prime and that is pretty much that; but in the geometry of three-space, it is the number of Platonic solids.  Or 17:  Number-theoretically it is both a prime and a Fermat number; but in the geometry of two-space, it is the number of crystallographic planar symmetries.  

https://en.wikipedia.org/wiki/Wallpaper_group

Or: 230, the number of space groups.

Here, the integer in question is not a creature or crystal considered distinct and in itself, but a stopping-point one arrives at by calculating and counting.  There might have turned out to be, say, 18 plane symmetry groups, without upending the world; but the internal structures of 17 and 18 are unrelated.

~

This dichotomy of internal versus external interest  chez the integers, represents ideal poles, which are not exhaustive, but bookend a spectrum.  As, probably intermediate: the “crunchiness” of natural numbers as viewed by students of finite groups.    (Here the masticatory metaphor  returns unbidden, for I always imagine finite groups along the lines of a wrinkly-surfaced walnut.  Finite simple groups -- those for which no homomorphism can split out any subchunk as its “kernel” -- the hardest nuts, the ones you can’t crack.)  Simon Norton seems to have had such an intimate gustatory appreciation of individual finite groups, before he Threw It All Away and went off to ride the bus.

~

In light of our training, we know at least one thing to do, should we ever be given an unfamiliar integer and locked in a room, with no other toys to play with.  Namely, we can probe for primality.  (An activity that becomes, indeed, crucial, for cryptographers.)

But now imagine that instead we had been handed some fraction like

      (2 +  √137) /4081

Untrained, we react with dismay.

Yet later, learning of the Golden Section, (1 + √5)/2, and its many remarkable properties -- not the least of these being that it can be represented as the infinite continued fraction consisting of nothing but ones -- we conclude that the critter is crunchy indeed;  and that there are more things in heaven and earth, than are dreamt of here below, and that we must anticipate the afterlife, before we could begin to embrace them.

~

Our treatment focused on what the innumerate are missing, much like what the Daltonist, unbeknownst, lacks of the hues.   But there is such a thing as unearned crunchiness --  a bogus significance assigned to certain numinous numbers, like 19 among the Baha’i’s; the “23 enigma”; 666; 1000 AD as the Millennium; the mumbo-jumbo numbers in “Lost” and "Touch"; for all which, cf. Wikipedia on apophenia.   In the face of such things (which have snared some otherwise rational people -- a close friend of mine, critically brilliant but unmathematical, fell for the “23” business), we are inclined to say, along with Nulla  extra ecclesiam  salus,  that  Nulla  extra mathematicam  ratio.

~

For a rather recondite example of crunchiness, consider the ‘amicable numbers’  (  الآعداد المتحابة) reported by the medieval historian Ibn-Khaldun in his Muqaddima .   Apparently only two of these were known to the Arabic medievals (or: they are the only two numbers characterized by a theorem of Thabit ibn Qurra), and they are not much to look at:  220 and 284;  but they meant something to contemporary practitionars of the talismanic art.

The modern view is summarized here:

https://en.wikipedia.org/wiki/Amicable_numbers

A (non)Definition of Depth

We’ve posted a number of reflections about the idea of “depth” in (especially) mathematics and related science (for the complete list of these, click here: http://worldofdrjustice.blogspot.com/search/label/depth ), without ever really defining the term.  And this, for a reason:

Two interrelated ideas that have been widely assumed to be unanalysable  are those of one scientific theory being deeper and more unified than another.

-- John Watkins, Science and Skepticism (1984), p. xiii

The idea of theoretical depth has considerable importance in Popper’s philosophy of science.  “If at all possible, we are after deep theories.”  But he was pessimistic about the possibility of any sharp characterisation of the idea.

-- John Watkins, Science and Skepticism (1984), p. 188

And that, not necessarily for any ‘deep’ reason -- not correlating intimately with the intricacies of physics, say -- but much as it is hard to characterize sharply such multifaceted (or blobby) concepts as beauty or game.    And here, I must sympathize with the archetypal philistine of a hundred New Yorker cartoons,  genially conceding, “I don’t know much about art, but I know what I like.”   A mathematician or physicist may not be able to define depth in a way that would satisfy the notoriously finicky tribe of philosophers:  but he knows it when he sees it.  And smiles.

So, sorry, no necessary-and-sufficient conditions, nor even a rough-and-ready dictionary definition;  but anyhow, an epigram:

Whewell’s requirment that a deep hypothesis, one that gets hold of nature’s ‘alphabet’ as he put it, must enable us ‘to explain … cases of a kind  different from those which we contemplated in the formation of our hypothesis.

-- John Watkins, Science and Skepticism (1984), p. 190

(A similar metaphor, more popular since Whewell’s day:  a good theory must “cut Nature at the joints”.)

~

Hmm, now I’ve piqued my own curiosity.  How does a practicing lexicographer go about characterizing the term deep in the sense(s) of interest here?

Okay, for starter’s, a British one -- Collins English Dictionary (1979):

deep:  … (6) difficult to understand or penetrate; abstruse

(7)  learned or intellectually demanding: a deep discussion

Sense (6) is of no interest.  We went to some effort in this post (“A Dive to theDepths”) to distinguish depth from difficulty.  (For the list of posts re difficulty: http://worldofdrjustice.blogspot.com/search/label/difficulty .)   Sense (6) would be used by a lazy man, giving up -- “Too deep for me.”   Here he is not even using the word with its native literal resonance -- he could quite as well speak with poor Mr Tulliver, who often confessed that the world was “too many” for him.   Indeed, sense (6) might deserve one of those non-semantic/non-grammatical, sociolinguistic labels like “ -- Not in polite use”:  here,  “-- Not used in this sense by serious thinkers.”

Actually, by a twist of pragmatics, the phrase does get used by serious thinkers -- but typically as an ironic put-down.  Thus:

The towering 19^th-century mathematician Hamilton  labored hard on his “Law of Hodographic Isochronism”,

but when he sent it to John Herschel (whom Hankins calls “the best-known and best-regarded British scientist of his time” [p. 134]), he got the reply:  “You are fairly got out of my depth”.  (This was Herschel’s regular response when he did not have the time or inclination to follow Hamilton’s long analytical excursions.)

-- Thomas Hankins, Sir William Rowan Hamilton (1980), p.

And again, roughly a century later:  After presenting a rather absurd and convoluted, goalpost-moving series of proposals from Lakatos and Morrall:

I am out of my depth with a claim of this kind.

-- John Watkins, Science and Skepticism (1984), p. 334

Here he is being disengenuously self-deprecating;  and his reply is all the more biting.

~

A more general thought, though, on depth versus difficulty.   I almost wrote that the latter was “much less interesting” -- though really, that depends on your day-job.  If you teach primary school, you need not (ex cathedra) worry your head one bit about depth in our sense;  whereas you must ever be alert to the perils of difficulty.  To epigrammatize into a dichotomy:  Depth (again, in our privileged sense) inheres in the subject itself;  Difficulty is relative to the practical limitations of some species (be it human, chimp, or the poor fly stuck in the fly-bottle) when grappling with the problems in that subject.   Since the philosophy of this blog is Platonist, we have little interest in the latter (no intellectual interest;  though some emotional interest, maudlin or morbid, as here).

From that perspective, sense (7) is also disappointing.   A subject itself (such as algebraic geometry or M-theory) cannot be called “learned” (i.e., learnèd):  that epithet might only be applied to whoever is gassing on about it.  “Intellectually demanding” could be applied either to a subject or a particular discussion or presentation thereof.  And that quality might be due to anything from the intellectual limitations of the audience (“The concept of evidence is too intellectually demanding for Trump voters”)  -- thus, back to sense (6) -- to an (overly) condensed presentation on the part of the lecturer, to actual depth inherent to the subject (and which would still be apparent to an angel, who understood the subject perfectly well).   Thus, neither sense goes far towards elucidating what mathematicians mean when they refer to a “deep result”.

~

Curious now whether my old alma-mater Merriam-Webster  did any better, I looked it up in their Collegiate Dictionary (Eleventh Edition),  I found something quite different.

First, their treatment of the geospatial, ‘literal’ sense of the term, from which all others ultimately derive, is unexpectedly rich and reticulated (I almost wrote:  “deep”), containing sub-subsenses like

deep  1 b (1) : extending well inward from an outer surface <a ~ gash>

(That is the sort of distinction you come up with when you are working from a generous deskful of carefully chosen citation-slips, rather than copying other dictionaries  or pulling the definition out of your butt.)

But then things sort of fall apart.  The sense “difficult” is not treated as a top-level numbered sense, as in the Collins, but as a subsense of a sense not defined save as the sum of its (rather disparate) subsenses:

deep  3 a : difficult to penetrate or comprehend : recondite < ~ mathematical problems>

3 b : mysterious, obscure <a ~ dark secret >

3 c : grave in nature or effect <in  ~est disgrace>

3 d :  of penetrating intellect : wise <a ~ deep thinker >

along with several more lying well off our axis of interest.   And oddly, despite all the careful hair-splitting, nothing really corresponding to Collins’ (7).

Thus, we still have come no further towards our goal.

~

The subject of depth, unlike that of mathematics, or Christianity (or oahspe),  tends not to attract disquisitions of the “What  is ….?”  sort.   We tried our hand at one for math (here), basically coming up with little more than a florilegium of blind-men-and-the-elephant stabs at it, for the overly general definiendum mathematics itself;  more fruitful was the task of characterizing topics within mathematics, like affine connection or topology, since here (at least for the former example) the definer’s intention is more in the nature of targeted enlightenment  than an after-dinner speech :  the result was a nice bouquet of epigrams.

~

Back to Depth vs Difficulty.    

(1) A deep remark or insight  is associated, not with presenting difficulties (as in Collins sense (6) ), but -- quite the contrary -- with resolving them.

A humble but poignant case  has been recounted here (Induction/Recursion), where a problem that had seemed difficult (to New Jersey third-graders, back in the complacent days before the impact of Sputnik  had filtered down to elementary school) -- that of multiplying multi-digit numbers -- suddenly became transparent, under the impact of an insight which (relative to what we had learned so far, most of it from the Mickey Mouse Club) might qualify as (qualifiedly) deep. 

(2) Above, we made something akin to an actio/actum distinction between difficulty and depth (human activity vs. the subject itself);   yet now we may make an additional distinction, on the same -- human -- side of the Platonic/psychological divide.  You might call it horizontal/vertical,  syntagmatic/paradigmatic :  judging words (concepts) by the company they keep.

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Let us recur to that subsense in Webster’s Colleagiate,  “3 b : mysterious, obscure”, and consider the idea of  deep as it appears in company with that of being hidden:

The mind is in a sad state, when Sleep, the all-involving, cannot confine her spectres within the dim region of her sway, but suffers them to break forth, affighting this actual life, with secrets that perchance belong to a deeper one.

-- Nathaniel Hawthorne, “The Birthmark”

(I.e., a deeper, hidden something, that somehow itself  amounts to a “life”.)

And from a mathematical physicist:

It is indeed true that we can prove, from this kind of Euclidean argument,  that squares, made up of right angles, actually do exist.  But there is a deep issue hiding here.

-- Roger Penrose,  The Road to Reality (2004), p. 28

Namely (tying in with cosmology):

His fourth postulate asserts the equality of all right angles.  … In effect, the fourth postulate is asserting the isotropy and homogeneity of space.

-- Roger Penrose,  The Road to Reality (2004), p. 29

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Alexander Grothendieck

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

How Now, Round Cow

~

Sotie : 

le mathématicien  et la conspiration Riemann

Борис Березовский

~

[Update, 30 November 2014]  Further thoughts.

(1) Integers as shadows

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

From Finitude to Infinity

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

(2) “Goodbye to All That”

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

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

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

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

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

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

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

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

De Stultitiâ

 

Other examples of burnout:  

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

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

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

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

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

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

Schrödinger, to Niels Bohr:

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

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

More such anecdotes here:

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

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