Consilience in Psychology (updated)

We have broached the topic of Whewellian-Wilsonian consilience in general,

    http://worldofdrjustice.blogspot.com/2011/10/consilience.html

and of mathematical consilience in particular:

    http://worldofdrjustice.blogspot.com/2011/10/consilence-in-mathematics.html

And now we shall rather wander from the main path, to notice the role such notions have played in the history of analytic psychology.   They are scarcely central to its development;  we allude to the matter  merely with a view to rounding out -- and consilience is about nothing  if not   rounding out...

Once again, time presses -- We actually do have a day-job, you know -- so instead of a proper essay, we shall simply type out some quotes from books that need to go back to the library soon, so as not to lose them.   Perhaps later these may serve  as pegs whereon to hang an essay, if reader interest warrants.

~   ~   ~

The time in which Freud worked  was already alive to the possibilities of disciplinary cross-connections:

In 1912, Freud was invited by Scientia, an important international periodical  published in Italy and devoted to the study of the relationships between the different branches of science.

-- Ernest Jones, Freud: Years of Maturity (1955), p. 214

Freud’s pupil Fenichel suggests that, so far from psychology being ignominiously reduced to physics, historically there is a touch of the other-way-round:

The idea of looking at mental phenomena as a result of interacting forces  certainly was not derived merely by transferring the concept of energy from the other natural sciences to psychology.  Originally  it happened the other way around:  the everyday assumption that one understands metnal reactions when one understands their motives  has been transferred to physics.

-- Otto Fenichel,  The Psychoanalytic Theory of Neurosis (1945), p. 11

Cf. the historical precedence of philology over biology, in the matter of branching taxonomy.

~            ~            ~

Re the ancient theory that dreams came to us through either the Gates of Ivory, or the Gates of Horn:

Diese vorwissenschaftliche Traumauffassung der Alten  stand sicherlich im vollsten Einklange mit ihrer gesamten Weltanschauung, welche  als Realität in die Außenwelt zu projitzieren pflegte,  was nur innerhalb des Seelenlebens  Realität hatte.

-- Sigmund Freud, Die Traumdeutung (1900), ch. I

(Einklang:  roughly, Consilience.)

Physical physiology … overthrew this philosophy [viz. Naturphilosophie].  … The conquerer introjected the emotionalism of the victim.  “Unity of science” … “physical forces”  were not merely directing ideas  or hypotheses of scientific endeavor:  they became almost objects of worship.  They were more than methods of research -- they became a Weltanschauung.

-- Ernest Jones, Freud: the Formative Years (1953), p. 43

The study of animal behavior with its Innate Releasing Mechanisms  can be seen as a sort of dual to Jung’s endeavors:

Ethology and Jungian psychology can be viewed as two sides of the same coin:  it is as if ethologists have been engaged in an extraverted exploration of the archetype, and Jungians in an introverted examination of the IRM.

-- Anthony Stevens, Jung: A Very Short Introduction (1994), p.  5

The following is as lame as Dawkins’ analogizing of ‘memes’ to genes;  but we cite it by way of examining the hypothesis of Consilience, not by way of defending it:

What Jung was proposing  was no less than a fundamental concept  on which the whole science of psychology could be built.  Potentially, it is of comparable importance to quantum theory …Just as the physicist investigates particles and waves,  and the biologist  genes,  so  Jung held it  to be the business of the psychologist  to investigate the collective unconscious  and the functional units  of which it is composed -- the archetypes.

-- Anthony Stevens, Jung: A Very Short Introduction (1994), p.  47

According to his associate Jones, Freud made

a half-serious prediction that, in time to come, it should be possible to sure hysteria by administering a chemical drug, without any psychological.  On the other hand, he used to insist that one should first explore psychology to its limits, while waiting patiently for the suitable advance in biochemistry, and would warn his pupils against “flirting with endocrinology”.

-- Ernest Jones, Freud: the Formative Years (1953), p. 43

Here Freud was prescient about Prozac and its tribe, while warning against premature horizontal consilience.

~

To the objection, “der Mensch habe noch andere Interessen als die sexuellen”, Freud replies:

Das haben wir keinen Augenblick lang vergesssen oder verleugnet.  Unsere Einseitigkeit ist wie die des Chemikers, der alle Konstitutionen auf die Kraft der chemischen Attraktion zurückführt.  Er leugnet darum die Schwerkraft nicht, er überläßt ihre Würdigung  dem Physiker.

-- Sigmund Freud, “Eine Schwierigkeit der Psychoanalyse” (1917)

Theodor Reik,  a colleague of Sigmund Freud for thirty years, reports:  “He insisted that psychoanalysis, as a science, should adhere to its own methos, and he tried to keep it free of the methods of other sciences.”  (The Search Within (1956), p. 13).

After the Fliess period, one notices a diminishing tendency on Freud’s part  to physiologize -- or, at any rate, to promise eventually to physiologize -- the new discoveries.

-- Philip Rieff, Freud: The Mind of the Moralist (1959), p. 7

Jung never disagreed with Freud’s view that personal experience is of crucial significance for the development of each individual, but he denied that this development occurred in an unstructured personality.  For Jung, the role of personal experience  was to develop what is already there.

-- Anthony Stevens, Jung: A Very Short Introduction (1994), p. 48

This exactly parallels the central tenet of Chomsky.

Chomsky-Halle binarism; ctr. Jungian tetradism:

The categories into which these [characterological] typologies are divided  are commonly four in number.  It is as if the mind has a natural propensity to orientate itself through a tetrad of pair oppositions.  The magnetic compass  is a case in point.

-- Anthony Stevens, Jung: A Very Short Introduction (1994).

For a kindred topic -- here:
http://worldofdrjustice.blogspot.com/2012/09/chomsky-freud-and-problem-of-acolytes_26.html

Consilience

The unity of science consists  alone  in its method, not in its material.

-- Karl Pearson

The unity of science is a pipe-dream.

-- Ian Hacking

The learnèd world  for long has yearned,  that all knowledge might be uniformly regimented beneath the broad skirts of science -- typically led by whatever was in fashion at the time.  For a while, it was Newton.  Thus,

Hume prided himself on doing for psychology  what Newton had done for physics.  He offers a (vacuous) theory of the association of ideas  as the counterpart to the theory of gravitation.

-- Anthony Kenny, The Oxford Illustrated History of Western Philosophy (1994).

In the 1920s, Otto Neurath headed up the influential Unified Science movement, which set the intellectual tone for a whole generation.

Subsequently, various technical developments have proved alluring to architects of airborne palaces -- computers in particular. As, “Hey, maybe the mind is just a Turing machine !  Tape goes to the left, goes to the right, scribble, erase,  and, bingo!,  you get Hamlet and the Credo.”  This eructation was dignified with the title of “machine-state functionalism”.

In our day, in the English-speaking world, the lodestone is Darwin.

In a lengthy work  unfortunately titled Darwin’s Dangerous Idea (1995), Daniel Dennett effectively claims skeleton-key status for Darwinism, comparing it to “universal acid:  it eats through just about every traditional concept”, including in cosmology and psychology.  A clear-headed corrective to his claims, from the ever-trenchant H. Allen Orr, may be read here: http://bostonreview.net/BR21.3/Orr.html.

~     ~     ~

And then Edward O. Wilson weighs in, with his much-talked-about book Consilience: The Unity of Knowledge (1998).   Wilson is the man to beat, when it comes to ants;  has he likewise managed to outline a Theory of Everything?   The dust jacket is certain that he has:  “nothing less than a daring challenge to the prevailing world-view ... a grand, coherent conception,  encompassing the sciences, the arts, ethics and religion” burbles Gerald Holton on the back cover, and Jared Diamond proclaims the eminent ant-man “one of the twentieth century’s greatest thinkers”.

Well, a man is not on his oath, in jacket copy.   Let us see whether the accomplishment matches the hype.

Wilson borrows the word from an 1840 work by William Whewell, which states:

The consilience of inductions takes place when an induction, obtained from one class of facts, coincides with an induction, obtained from another different class.  This consilience is a test of the truth of the theory in which it occurs.

Wilson comments: “Consilience is the key to unification.  I prefer this word over “coherence” because its rarity has preserved its precision.”  (p.8)

That is a common and valid reason for philosophical neologism (since the word consilience had become obsolete, Wilson effectively re-introduced it).  The word is not, however, used with much precision in this book -- at least, it does not truly preserve whatever meaning Whewell may have had in mind (his own wording is none too precise.)

Consilience -- It is a lovely word, rhyming with brilliance and resilience.  Wilson trots it out from time to time, rather like those “station identification” messages required by the FCC;  but over the whole course of the book, it seems to do no actual work.   As he himself suggests, it is roughly a synonym of coherence -- which would seem to be setting the bar pretty low.    

As Wiki  puts it:  “Wilson's concept is a much broader notion of consilience than that of Whewell, who was merely pointing out that generalizations invented to account for one set of phenomena often account for others as well.”  But broader is a nice way of saying diluted, weakened. --  In terms of ambition, however, Wilson’s consilience is a stronger program, in that it aims to rope in the humanities as well; but it can hope to do this only at the expense of diluting the proposed commonality.

For sheer emptiness, it touches bottom on page 117:

But surely, skeptics will say, that is impossible.  Scientific fact and art can never be translated  one into the other.  Such a response is indeed the conventional wisdom.  But I believe it is wrong.  The crucial link exists:  The common property of science and art  is the transmission of information.

Omigosh,  we didn’t we think of that before ?!?  All-kina stuffs uses information!! -- Or rather, “the information”, in the unidiomatic jargon being peddled by James Gleick’s recent (and similarly overambitious) book. -- In point of fact, there is more commonality between a sermon and a sex pheromone, since both, in addition to transmitting information, exhort. --

Or again:  We may regard elephants and chimps as simply alternate aspects of a single superorganism, since both may be seen as machines for processing peanuts into poop.
Or again, p. 219.  He quotes with approval: “Edward Rothstein, a critic trained in both mathematics and music, compares their creative processes”:

We begin with objects that look dissimilar.  We compare, find patterns, analogies with what we already know.  We distance ourselves and create abstractions…

I’d be prepared to concede (and the suggestion is intriguing) that, at a high and abstract level, there are similarities between the appreciation of … well, certain types of music (not “Who Let the Dogs Out”) and mathematics.  But surely the creative processes are quite different.   The description above (which does rather sound like a mathematician’s work)  is itself already such an abstraction, that it rings less like an analysis, than like a really terrific pick-up line  for naive and impressionable co-eds.

(There is a wonderful app, by the bye, for mathifying, or at least patternizing, our musical appreciation:

An animated graphic showing interval type

http://www.youtube.com/watch?v=X1QM6YZC45g&src_vid=asDXpfFMKNA&feature=iv&annotation_id=annotation_587547

Disney did something similar at the beginning of “Fantasia”, but this display is more precise and structured, and ultimately just as evocative as the artistic impressionism.)

However, there is another way of reading Whewell’s definition, which, whether or not Whewell himself meant it this way, leads to a more interesting notion, which we may call strong consilience.   This thesis, I shall argue, is exciting, though false; whereas the watered-down weak consilience is little more than consistency, and thus is uninteresting even if true.

[Note:   I have only just this minute stumbled upon the fact that H. Allen Orr has already written a (typically incisive) review of Consilience as well. That man is hard to top, whenever he opens his mouth about something.  So out of respect for the reader’s time, I must in all candor now urge you to cease reading this one, and go read Orr instead:

http://bostonreview.net/BR23.5/Orr.html

If, after you have savored and assimilated what Orr has to say (and which, fwiw, I heartily endorse), you still have appetite left for some leftovers, you may return to the present on-going essay.

Orr himself, incidentally, writing with Jerry Coyne, downplays consilience even within biology, in that the essential species concept is itself not tightly unified:

Because we rejected ecological differentiation as part of the Biological Species Concept in sexually reproducing groups, we obviously endorse the use of different species concepts in different groups.  We do not consider this pluralism to be a weakness of the BSC.  Because the causes of discreteness may well differ among taxa, so may the concepts appropriate to addressing the species problem.

Jerry Coyne and H. Allen Orr, Speciation  (2004), p. 52 ]

~

~  Posthumous Endorsement ~

"If I were alive today, and in the mood for a mystery,

this is what I'd be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(I am William Whewell, and I approved this message.)

~         ~

~

(To resume...)  ]

So, let us see if we can force an interpretation on Whewell’s words which will lead to something interesting.

An “induction”, whatever exactly that may be, is not a fact, but an orderly pattern of facts.   So let us by fiat define  strong consilience thus: 

actual isomorphism (or at least homomorphism) between established aspects of different disciplines. 

The model for what we have in mind  is the collection of surprising and delightful structural coincidences within and between mathematics and various parts of (originally) mostly physics, known under the slogan of the unreasonable effectiveness of mathematics.  (You can consult Wigner’s original essay of that title here.)  And since Wigner wrote, the essence of the history of mathematics over this past half-century or more, has been an efflorescing of consilience in just this neo-Whewellian sense:  subfield after subfield  turn out to be deeply interlinked.  It is a breathtaking tale (and has, incidentally -- see below -- nothing whatever to do with the thesis of Reductionism).

There have also been some remarkable recent examples of consilience between math and physics:  cases in which mathematicians, burrowing from (as it were) the east coast, and physicists, burrowing from (so to speak) the west, suddenly met in the middle.  Celebrated cases include:  fibre bundles (math) and gauge theories (physics); zeros of the Riemann zeta-function (math) and atomic energy levels  (physics).  These results are, I suspect, among the most important of the past century (as much as to say:  of the past two millennia), and the best thing would be for us all to quit our day-jobs and join mountaintop monasteries set up specially to study these matters.  (While not omitting matins;  first things first.)

Mathematics itself might be defined as the science of matching patterns;  so there is a certain appropriateness that it serves as a link between such superficially disparate phenomena as the pattern of spots on a leopard and the subject of reaction-diffusion equations (pioneered by none other than Alan Turing), or floret patterns and Fibonacci series.  An absorbing book on this subject is Life’s other secret (bad title, good book), by Ian Stewart, 1998.

“Strong Program” publications, that go way beyond such vacuous ecumenism, include the Encyclopedia of Unified Science (Vienna Circle).

[A more careful consideration of this kind of consilience  is available here.]

 

~     ~     ~

There are times when Wilson does seem to be talking about something like strong consilience  (p. 125):

The natural sciences have constructed a webwork of causal explanation  that runs all the way from quantum physics  to the brain sciences and evolutionary biology.

Yet at other times  (p. 229), consilient seems no more than a ten-dollar synonym for consistent:

The biological origin of the arts is a working hypothesis, dependent on the reality of the epigenetic rules … It is meant to be testable, vulnerable, and consilient with the rest of biology.

Which is to say:  When attempting to demonstrate that various foul-smelling chemicals lead directly to the Goldberg Variations  and  Paradise Lost, you’re not allowed to actually falsify the basic biological facts.

~      ~      ~

Wilson never explicitly defines the characteristics of consilience once and for all -- which is fine;  it’s more a term to guide the thoughts along.   I use “minimalism” in a similar way, in these essays:  connotation accretes to the term in the course of investigation.  And a connotation that accretes to Wilson’s term  is that of hierarchy:  which has indeed characterized biology ever since Linnaeus, and in ever richer ways since, but which seems a very odd fit for, say, mathematics or art.  P. 182:

The crucial difference between the two domains is consilience:  the medical sciences have it and the social sciences do not.  Medical scientists build upon a coherent foundation of molecular and cell biology.  They pursue elements of health and illness  all the way down to the level of biophysical chemistry. … Social scientists  by and large  spurn the idea of the hierarchical ordering of knowledge…

Sidenote:

Steven Pinker (The Blank Slate,p. 26) quotes from anthropologist Leslie White  the sort of statement that apparently drove the worthy Wilson u-batz:

The culture process [is] sui generis;  culture is explainable in terms of culture.

Not, note, even in terms of psychology;  let alone physics.

Anyhow:

If being hierarchical is a requirement for being consilient, then indeed the term in this sense is more substantive than it seemed when the author proclaimed that simply using “information” would suffice.  And accordingly it becomes more falsifiable (philosophically, this is a good thing), but also (though I won’t argue the point further) in the case of many quite respectable spheres of activity, false (not so good).

~      ~      ~

Wilson’s penultimate chapter is titled “Ethics and Religion”.   Its motto (p. 238) is:

Moral reasoning, I believe, is  at every level  intrinsically consilient with the natural sciences.

Now -- those of you who have followed this blog (in particular my intemperate trashing of morally reductionist neuroscientists), no doubt expect, that at this point I shall undergo the Incredible Hulk transformation and start blasting-out one of my patented rants  against atheists (for Wilson manfully avows himself, not really an atheist, but an a-theist, a non-theist), Nominalists (Wilson is, in general, not that), and Donald Trump (so far as I know, Wilson has never even met the man).  Well, not a bit of it.

First of all, because I myself do not pretend to really understand moral reasoning.  Logic, yes, given the premises;  but ahh, the premises !  But more important, because Edward O. Wilson is a remarkably genial and congenial companion;  and one who, given his deeply Christian upbringing, truly understands what it’s all about, even if he has reached different conclusions from those of his Protestant evangelical tutors (as have I).  Furthermore, it would seem that both the present and the previous Pope would agree with that statement.

He clarifies:

The split is not, as popularly supposed, between religious believers and secularists.  It is between transcendentalists [dbj:  cf. Realists], those who think that moral guidelines exist outside the human mind,  and empiricists [dbj: cf. Nominalists], who think them contrivances of the mind.  The choice between religious or nonreligious conviction  and the choice between ethically transcendentalist or empiricst conviction  are cross-cutting decisions made in metaphysical thought.

Absolutely true.  There is, of course, a question about how non-theistic and indeed non-deistic transcendental moral schemes are grounded, but that does not invalidate the logical point made here.

As Charles C. Gillispie puts it in his generous review:

Put baldly, this sounds like arid monistic materialism. Wilson does not put it baldly. He brings to his subject a disarming mixture of personal modesty and intellectual rigor. His reading is wide and his learning extensive. He writes as well of arts and letters as he does of science. It is difficult to think of a finer evocation of Milton's genius than Wilson’s passage on Satan’s invasion of the garden of Eden.

(Um, well, actually, not so much:  C.S. Lewis, A Preface to Paradise Lost.  Still, Wilson’s own tribute is very creditable indeed, for a myrmecologist.)

He adds:

There is no question in his mind but that conveying the essence of truth and beauty pertains to the arts.

You (as Sarah Palin would put it) betcha !  But -- contrary to the effusions of certain journalistic science-pornographers, “conveying the essence of beauty” is not, in fact, the brief of science.

~      ~      ~

The latter part of the book contains rather general sociopolitical discussions:  which, at his own assessment that the social sciences are lamentably unconsilient, means that we stray fairly far from genuinely scientific or Whewellian concerns. 

Despite the fact that his overarching project is ambitious (or made to sound so), Wilson himself is a becomingly modest man.  He even cops to a probably spurious potential charge, granting a “whiff of brimstone in the consilient world view, and a seeming touch of Faust to those committed to its humanistic core.”  Specifically, he points to the spectre of human genetic engineering:  “How much should people be allowed to mutate themselves and their desendants?”  The spectre is quite real, the charge being spurious only if laid at the door of a Whewellian philosophy of science.   What eventually -- or not-so-eventually -- comes to pass, will depend, neither on philosophy nor on theoretical science, but on what is fashionable and funded in places like Singapore and Hong Kong.  It is quite possible that, within the lifetime of some of you, you will witness a Wellsian world of Eloi and Morlocks.   Institutionally, not much stands in the way of this but the Historical Church.

~     ~      ~     ~     ~

This may be a tad off-topic but... The Atlantic Monthly has just published a very nice article on Professor Wilson's current activities.  Of particular interest to myself was reference to an article that Wilson recently co-authored with Martin Nowak, a splendid mathematical biologist whose acquaintance I had the pleasure of making at a conference he organized at the Princeton IAS, described in "Follia Linguistica" (in: Princeton Follies;  to appear).

~      ~      ~

Russell Standish, a mathematician, cites the work of his chatmate, one Bruno:

His conclusion is, instead of psychology being reducible to, or indeed emerging from the laws of physics, the fundamental laws of physics are in fact a conseuence of the properties of machine psychology.  This is indeed a revolutionary reversal of the traditional ontology of these subjects.

Russell Standish, Theory of Nothing (2006; 2^nd edn. 2011), p. 129

Such a view is certainly consilient, though it stands Wilsonian consilience on its head, along with common sense.  And Standish embraces it thus (solecistically):  “I agree with this fundamental tenant.”  By that he means tenet;  but “machine psychology” is no typo -- sic, as it stands -- albeit it would make about as much sense to derive the laws of physics rather from marine psychology :  We are all but the dreams of porpoises …

[Update 16 III 2012] A striven-for consilience of art criticism and neuroscience:
http://chronicle.com/article/Eric-Kandels-Visions/131095/

The Ladder of Abstraction (with added rungs)

The following  logically belongs in the “Abstraction” section of our essay Consilience in Mathematics.  But as that effort is growing overlong,  we begin to cultivate here a particular idea  building upon that of abstraction simpliciter :  namely, the tendency, in modern mathematics -- and indeed this may serve virtually as the defining characteristic of modern (even: modernist) mathematics -- to abstract from any given abstraction, layer upon layer, rise upon rise, to a virtual (topless/cloud-topped) Babel, reaching to the Beyond.

(Oh, and here again we have a term from the arts, Modernism, which, as it includes “abstract art”, metaphorically applies to mathematics.  Compare our earlier essay on Minimalism in Mathematics.)

In normal practice, mathematicians mostly talk to one another -- and indeed, mostly just to those within their own hyperspecialized neck of the woods.  But occasionally, one writes an undergraduate textbook, and thus must descend to earth, if only for the nonce, and address the laity.  Thus:

This “intrinsic” formulation of Calculus, due to its greater “abstraction”, and in particular  to the fact that, again and again, one has to leave the initial spaces, and to climb  higher and higher  to new “function spaces” (especially when dealing with the theory of higher derivatives), certainly requires some mental effort, contrasting with the comfortable routine of the classical formulas.  But we believe that the result is well worth the labor, as it will prepare the student to the still more general idea of Calculus on a differentiable manifold.

-- Jean Dieudonné, Foundations of Modern Analysis (1960), p. 141

We dub this the “ladder of abstraction”, taking the phrase from our teacher of yore. Referring likewise to ascent into functions-of-functions, and function spaces, and functions from one function space to another, and to the duals of all that:

Detached from any context, this construction is a pointless formality.  But as we move up the ladder of abstraction, we find that constructions such as this  become commonplace …

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 43

The metaphor of ascent is well attested.  Gödel speaks of

the infinite series of ever stronger axioms of infinity, each of which expresses a new idea or insight.

-- quoted in Hao Wang, From Mathematics to Philosophy  (1974), p. 325

A mathematician writes of

the inferential staircase  leading from the laws of physics  to the world that lies about us …

-- David Berlinski, “The End of Materialist Science”, collected in:  The Deniable Darwin (2009), p. 160

~

Saunders MacLane,  in his book Mathematics:  Form and Function (1986), p. 36ff, has a section called “Mathematical Activities”, structured somewhat like our own in the Consilience essay.  Some of the topics are the same (analogy, abstraction, generalization), while others, not relating to consilience especially, differ (conundrums, axiomatization, proof).  One, intrinsic structure, seems to relate to consilience, but is only briefly developed; and the last, completion, we have treated under the more Quinean label of rounding out.

Now, abstraction and generalization are related notions, but neither entails the other.  MacLane acutely adduces the example of group theory.  Originally, this grew out of the concrete examples known as groups of transformations.  Later, algebraists abstracted into abstract groups.  Whether a generalization has thereby been achieved, is (as Chomsky likes to put it) “an empirical question”;  and in this case, it turns out, it has not. “No new groups turn up in this process, in view of the famous theorem of Cayley, which asserts that every (abstract) group is isomorphic to a group of transformations.”  Thus, in this case, the ladder of abstraction has only one rung.  (By contrast, abstract rings do turn out to generalize upon their original model, rings of integers.)

~

Seeking analogues of the Ladder of Abstraction  outside of mathematics proper, I happened upon this:

Quine suggests that levels of abstractness, modeled on Russell’s Theory of Types, might be established.  “In the beginning  there are only concrete objects.”  These constitute type zero and are the values of bound individual variables.  “To be is to be a value of a variable.”  Next comes first-order classes and relations:  they constitute entities of type 1 and are the values of bound predicate variables.  Classes of classes, and relations, constitute entities of type 2;  and so on.

-- Harold Lee, “Discourse and Event”, in: Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 297

The resemblance to Russell’s Theory of Types had not escaped me, but I rejected mention of it, since, rather than leading -- as the Ladder does -- to ever greater depth (the metaphor is here in distress -- maybe think of it as a ladder down a mineshaft), it seems to lead mostly to More of the Same.  In other words, forming those strata, as described above, is less like the dizzying and ethereal Abstract Ascent of mathematics, than simply forming new sets via the Power Set operation (a new and larger set consisting of all the subsets of the original set).  Now this, if we start with a finite set, leads absolutely nowhere.  It’s just like counting.  If you start with the whole of the Natural Numbers, now the Power Set operation does become more powerful, leading to new and incomparable levels of infinity.    Whether this leads to true new depth, or is rather a mere formal exercise, I do not know, since I lack all intuition of any infinities beyond the countable, let alone the Power of the Continuum or Measurable Cardinals.  Perhaps it does;  espresso-sodden Berkeley conversations about Quality emerging out of Quantity, return to mind.

Still, I am inclined to doubt it.  The very fact that the fellow can say “and so on”  virtually proves as much.  For there is no “and so on” to true mathematical abstraction.   There is nothing mechanical about such ascent -- it is more like a miracle.  You can proceed only one step -- nay rather, one leap at a time;  and the interval between leaps may take decades or even centuries.   Above the calculus lies Function Theory;  above that, Topology.  Above that, Algebraic Geometry. Far, far above us, hovers Category Theory, unreachably aloft.  And far, far above and beyond that, soars Topos Theory.  What comes next  is known only to angels.

To vary Nestroy’s celebrated epigram -- “Bis die Topologie gehts noch, aber von da bis sheaf theory  zieht sich der Weg.”

Additionally, Quine introduced the term semantic ascent.  There is some similarity to Gleason’s ladder of abstraction, but the ascent doesn’t go very high, and Quine himself -- perhaps surprisingly for a logician -- is wary of the upper reaches, preferring basic-level entities  behaviourally grounded.

Here the Russian author A. D. Aleksandrov, instead of envisaging a ladder,  uses the metaphor of layers  or (appropriately enough) of nesting, like Russian dolls, in the procession to affine or projective geometry and on to topology:

The properties of space are stratified … with respect to their depth and stability.  The ordinary Euclidean geometry was created by disregarding all properties of real bodies other than the geometrical;  here we perform yet another abstraction within geometry.

-- Aleksandrov et al, eds, Mathematics: Its Content, Methods, and Meaning (publication in the original Russian: 1956;  Eng. tr. publ. 1963), vol. III, p. 133

~

The more I think about it, the more this Ladder of Abstraction idea seems possibly fruitful.  Not so much as in the Theory of Types, but as in the scala naturae, which encompasses angelology.  (Compare also graded algebras.)

By contrast, mere ungraded “abstractness” in itself is of little interest. Thus, to take MacLane’s Group Theory example:  the so-called “abstract” groups (MacLane himself uses the sneer-quotes here) mean to lift aloft from Groups of Transformations, in that they retain the laws (associativity, inverses, and all that) while becoming agnostic as to the nature of the elements of the group.  But, first of all, groups of transformations are, compared with, say, pickles, already quite Abstract;  so the word adds, really, nothing.  Indeed, as soon as you say that two apples plus two apples are four apples, and that in the same sense  two penguins plus two penguins make four penguins (well, and a few more, after a while, if the sex mix is right), you are already indulging in such abstraction.

~

I tried looking up “abstraction” in the index of the various math textbooks and philosophy treatises on my shelves, and basically came up with  bupkes.  Thus, in Dummett’s omnibus volume, Truth and Other Enigmas (1978), we find no reference to abstraction per se, let alone to the Ladder of Abstraction, but only to “abstract objects” -- i.e., pickles versus the Meaning of ‘Pickle”,  the Idea of a Pickle, the set-containing-a-pickle, the… sandwich containing a pickle, the -- but enough.  Mathematics is so far beyond this, no comment is required.

~

The ethic -- even, the aesthetic -- of abstraction for its own sake, sociologically chronicaled here (“On Vulgar Numbers”), eventually evoked a backlash.

The Bourbaki group sought to present the entire abstract structure of all mathematical concepts in one set of volumes, the Eléments de Mathématique. In that treatise, the real numbers, which most of us regard as a starting point, only appeared midway into the series, as a special “locally compact topological group”.

An opposing idea, promoted especially in the Russian school, is that a few well-chosen examples can illuminate an entire field.

-- David Mumford, Forward to Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. xv

~

For the latest in fine reading, check this out:

http://www.linguasacrapublishing.com/justice.html

For more about abstraction, here:
         http://worldofdrjustice.blogspot.com/search/label/abstraction

Rome by Different Roads

[In our basic essay on Consilience in Mathematics [which is under reconstruction, following a blogspot glitch; temporarily redirected to this post]  one topic was the way in which a given mathematical topic or object can be approached from quite different angles, or defined in superficially quite different ways, so that it is finally surprising that (after a lot of work and lucubration) you wind up at the same mathematical metropolis.   And just as with the phenomenon of parallax, or triangulation (which we learned in Boy Scouts), the result is a more multidimensional appreciation than we began with.]

We have discussed the notions of abstraction and of generalization as regards the overall ‘topology’ (or geography) of the  mathematical landscape.   Not the same as either of these are cases where two intuitively quite distinct notions turn out to imply each other -- to be provably logically equivalent.  As, the (growing) menagerie of propositions that are equivalent or weakly equivalent to the Axiom of Choice.

The closest notion to this that we have previously touched on is that of  “Rome by Different Roads”, which in the simplest case means merely that the same proposition, say of number theory, can be reached (proven) with distinct sets of tools, e.g. an heavy-machinery analytic proof versus an ‘unplugged’ proof (generally using quite different ideas) which ascetically restricts itself to using only elementary methods.   The proposition you arrive at remains unchanged, the variety of proofs  something of a curiosity.  (As, the many different ways of proving the Pythagorean Theorem, most elegantly involving no more than a single diagram -- a small square tiltedly inscribed within a larger.)   It’s like -- tiens!  this road leads to Rome as well.

Whereas in this deeper phenomenon, the different equivalences -- the different perspective views of the same central Thing -- actually wind up changing or rather sublating what one’s concept of Rome is.

It’s a bit like spooky action-at-a-distance in quantum physics, in which you can’t consider the particles to be quite independently existing, even though they are distinct.   There seems to be looming some supernatent notion, some eka-conception, of which each individual proposition from the pool of those interderivable, are like the sundry avatars of Vishnu, and not Vishnu himself.

The study of computability came to be known as recursion theory, because early formalizations by Gödel and Kleene relied on recursive definitions of functions.When these definitions were shown equivalent to Turing's formalization involving Turing machines, it became clear that a new concept – the computable function – had been discovered, and that this definition was robust enough to admit numerous independent characterizations.

-- Wiki, “Mathematical Logic”

Specifically, equivalent formulations of the intuitive notions of computability, independently arrived-at by different thinkers, include:  Church’s lambda-calculus; Kleene’s general recursive functions; Post’s automata; and Markov algorithms.

For the notion of “independent characterizations”, compare the Heisenberg formalism vs. the Schroedinger formalism  in quantum mechanics.   That one is a cause célèbre, and with few comparable situations in the history of physics;  whereas mathematics is chock-full of people discovering the same mountain via different slopes.

The Wikipedia article on “Representation theory” highlights “the diversity of approaches to representation theory.  The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatoriecs  and topology.”

~     ~     ~

For an application of this theme to the topic of uniform spaces,  consult this essay.

 

Dr J hearkens to Vox Populi

After a careful review of the blogstats, we must reluctantly conclude that the themes for which this blog was founded -- to include Cantorian Realism, Trinitarian Minimalism, Mathematical Consilience, and suchlike worthy fare -- is not what most folks are hankering for, after a long day at the lab.  Even the Humble Woodchuck receives fewer visitors than would be his due, for all his furry butt.  No, what the broad masses seem to have a yen for, is T&A -- and I don’t mean Theorems & Axioms.

Accordingly, in reverence to the inevitable, we offer this:

David Does Dallas

[you must be 21 to view]

And this:

Fifty Shades of Jack Bauer

(We asked his Jackness, if he had any regrets.)

JB:  I do.  When, in Season Seven, Renee Walker slapped me hard across the face, not once but twice, while asking me (patronizingly enough), in an acid voice, whether I “felt that” -- I just walked away, without so much as a parting shot.

I realize now that -- so obvious in retrospect -- I should have…have…

pulled down her fawn lace panties then and there,

and striped her buttocks with my studded belt,

till ridges rose upon that quivering skin

like as the moldwarp’s tunnels ‘cross the mold.

[Lines quoted from the “B” manuscript of Shakespeare’s Taming of the Shrew, omitted from the “A” recension.]