Woeful word from the front

The version

(∃x) (x is a sloop . I want x)

is suitable insofar only as there may be said to be a certain sloop that I want.  If what I seek is mere relief from slooplessness, [this expression] gives the wrong idea.

-- Quine, “Quantifiers and Propositional Attitudes” (1956)

Grave news :  Dr Justice has been diagnosed with a severe case of hypoyachtopoenia -- a not so rare condition in these troubled times.    His berth at the Millionaires Marina  cries out for very yachtlessness.   His lounge-chair at the Mariners Club sits empty;  and his cut-glass tumbler, which by rights should brim with the rarest of single-malt (topped off again and again  as he regales his fellow salty-sea-dogs  with tales of the bounding main), stands forlorn and empty.

(“Is there anything we can do is there anything we can dooo?!??” the readers interrupt, in anguish.)

-- Why, yes, as a matter of fact;  glad you asked.

Simply get your tush on over to this handsome site,

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

and BUY MY BOOKS.

[“My name is Dr J, and I have approved this message.”]

On “Rounding Out”

The shortest and best way between two truths of the real domain  often passes through the imaginary one.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 123

We have noticed Quine’s grudging acceptance of the irrationals given his unquestioning acceptance of the rationals, a process we alluded to as “rounding out”.

But it is really not so much rounding out as filling out -- or rather, filling in: filling in the gaps between the rationals.  And the result is, unfortunately, not so rational as the rationals themselves.  The rationals -- that is, fractions -- are forced upon you by Nature already in nursery school:  How shall we divide these two cupcakes among the three children? (Answer:  Each gets two-thirds.)  But the Reals are (we admit this, despite our Realism) a bit unreal.  Full of all manner of set-theoretic paradox.  Inscrutable.  You can still work with them in practical terms, because the rationals, which are well understood, are, though no more numerous than the integers, dense in R, providing a sort of well-defined ladder or footbridge along which we may proceed.

Here in any event  is the testimony of a first-rate mathematician,  to the effect that the transition to the full reals  is essentially a forced move:

We shall show how to construct a complete ordered field  from a simple chain [Think:  the natural numbers].  This … proves that any contradiction inherent in the postulates for a complete ordered field -- that is, the real number system -- is latent in the postulates for a simple chain, which is a far less complicated structure  whose consistency is almost guaranteed by our intuition.

Note that we do not discuss the existence of the simple chain.  In spite of its intuitive simplicity, a simple chain carries within itself  the germs of all the difficulties in logic and mathematics;  we are obliged to take its existence as axiomatic.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p.  112

Such is the very axiom -- we need but two -- which we seized upon to begin this whole series of essays.   We quoted it in the form made famous by the Nominalist (and our otherwise-foe) Kronecker, which invokes the deity -- possibly casually or ironically, but perhaps more pertinantly than he knew:  for whence this “intuition” of which we are so sure?

~

The next step beyond the reals were named the “imaginary” numbers, and the name stuck.  Adjoin these, and you generate the Complex Plane.   And, unforced and arbitrary as this move might initially feel, it is on the Complex Plane  that you at last truly get a notion of a Natural Setting.  Everything just suddenly… works, and works better than you ever thought it could.  You have, all of a sudden, a Circle of Convergence -- sounds like something out of Lord of the Rings, and it is just as good.  Differentiable functions turn out to be perfectly smooth, and with a natural notion of their own domain.  (Try to define them on too small a region, and they will propagate themselves by analytic extension till they are nice and fat.)

On the complex plane, things are rounder.  (Note:  Round is good.) In R, a ‘ball’ is a line-segment, and a ‘sphere’ (the surface of a ball) is two points.  In C, they’re a disk and a circle respectively.  And you can round out or rather round off the complex plane yet further, by adjoining a single ‘point at infinity’, which is the limit of any ray pointing in any direction.  The plump, rotund result:  the Riemann Sphere.  This is homeomorphic to the surface of a penguin,  the world’s most perfect shape.

As Penrose puts it:

It is as though Nature had herself entrusted to these numbers  the operation of her universe.

(Again, note the theistic language which, all unbidden, surges forth at such a time, from even the driest of nibs.   It is a very early and natural theology, such as is depicted in that fine chapter of The Wind in the Willows, "The Piper at the Gates of Dawn".)

Another indication of the greater naturalness of the complex plane as a nursery for functions:  A real function may be C-infinity (infinitely differentiable) at a point, yet somehow “off” at this point, a fact revealed by the fact that its complex analogue is not there analytic.   Thus, as one writer put it, (complex) analytic functions are “smoother” than real functions.

[Example:  exp(-1/x), for x > 0; 0 at x = 0.  That last point is artificially “tacked on”, and in the complex picture, it shows.]

 

This Complex Plane  is a real find; it is not just a waystation to something better yet.  (David Berlinski calls complex numbers "instruments that providence had provided for the recovery of lost symmetries," a neatly postlapsarian formulation.)  There is very little beyond this, by way of fields suitable for the calculus -- certainly nothing that approaches the leap that the complex numbers represented beyond the reals.   There are the quaternions, which have their points, but are a very poor cousin indeed: the theory is poorer, not richer, for the extra generating elements, since the field of quaternions offers no analogue of holomorphic functions. ( “Quaternions have more or less dropped by the wayside.” -- Thomas Hankins, Sir William Rowan Hamilton (1980), p. 325)

Then there are the octonions, for which no-one has ever found much of a use.  And there’s an end to it.

~

Other mathematical instances of “rounding out”:

*  The adjunction of zero to the natural numbers, and of the empty-set to the world of sets.  Both function exactly like their less spectral congeners.

And a nice aesthetic note -- both are represented by a round symbol: respectively, a goose egg, and a goose egg barre sinistre.

* There are various elaborate ways of constructing things out of other things, like a Stone-Cech compactification.  But in “taking the power set”, we just stand back and let it happen.  Again and again.  Yielding the “beth numbers”, and more infinities than most folks know what to do with.

* The mathematics of string theory adds extra tiny “compactified” spatial dimensions to the three of everyday experience; in these, you just go round and round.  But this isn’t rounding-out, really, since the large spatial dimensions may themselves be compact, in which any sufficiently long journey circles back on itself.  (“Compact” doesn’t mean “tiny”;  it’s a topological, not a metrical notion.)  Space could even be flat, yet finite -- thus having the topology of a three-torus.

~

Footnote:

It is well-known that it took mankind a long time to recognize zero as itself a number.  Less well known is that “not until modern times was unity considered a number” (D.E. Smith, History of Mathematics, vol. II, p. 26.)  Or that the negative numbers were long qualified as "false".

Compare the uncertainty over whether white qualifies as a “color”.  (And if it does, what about black, or grey?)

~

So where is Minimalism in all this?  Are we just tacking on turrets and wing-additions to some increasingly sprawling McMansion?

Not a bit of it.  The operative word here really is round.  For, round things are minimal surfaces -- indeed, the very simplest class of these -- in the sense of using-up a minimal area to enclose a prescribed volume.   Our purpose is, indeed, to group like with like, and to enclose them in some stable structure.  This is no multiplication of entities for their own sake -- the itchy-clutching witchfingers of insensately proliferating fractals, which is the very architecture of the dungeons of Hell.   In rounding out, the mathematician is seeking a coherent minimal structure to regiment and account for what he has hitherto seen:  one which, upon acquaintance, may become more intuitive than the partial structures initially encountered.  (The “upon acquaintance” part may of course require a bunch of Ph.D.’s and several hundred years.)

            And the things which we have seen, and which need explanation -- or at least for agencement into some larger and more natural whole -- do keep arising.  Connections are detected among them which cry out for elucidation.  So we ascend to a yet loftier bird’s-eye -- eagle-eye -- phoenix-eye view.  To arrive, it may be, eventually at Topos Theory, or the Lord of Hosts.

(For the latter, though note:  that ladder reaches only so high.  We quote the saint:

Remaneret igitur humanum genus, si sola rationis via ad Deum cognoscendum pateret, in maximis ignorantiae tenebris.
-- Thomas Aquinas,  Contra Gentiles, lib. 1 cap. 4 n. 4 )

~

The examples we gave were mathematical, merely for clarity.  But the principle of Rounding Out  applies to any field with structure.

These vague words ‘capable’ and ‘normal’  allow the grammarian scope for shaping his task to suit his convenience.  Seeking simplicity, he will round out and round off.

-- Quine, “Reply to Harmon”, in The Philosophy of W.V.O. Quine (1986)

~

Footnote re the irrationals:

Dedekind sttressed the distinction of category  between cut and number  in 1888; against the view of his friend Heinrich Weber  that “the irrational number is nothing other than the cut itself”, he explained that “as I prefer it, to creat something New distinct from the cut, to which the cut corresponds.  We have the right to grant ourselves such power of creation”,  and cuts corresponding to both rational and irrational numbers were examples.

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 87

A seemingly slight, even pedantic distinction;  but like many another such, it might have its point.   Cf. my astonished delight in junior high-school, upon meeting the distinction between  x (the thing itself) and ‘x’ (the name of x) -- already adequately foreshadowed in Alice in Wonderland, but encountered now in a new context.  Likewise the difference between  x and {x} (the singleton-set of x).

In the case of an algebraic number like √2, a simple number staring you in the face out of a hypotenuse  versus the infinite train of rational pilgrims (never quite arriving at their destination) of a Dedekind cut,  one is reminded of the variety of definitions of something so familiar as a tangent:  the slope of a curve (at a point); the closest linear approximation to the curve (at that point); versus the distressing definition in Loomis & Sternberg as an infinite equivalence-class of curves (through that point).

Ask Dr. Science !

  (1)  The Dawood Illusion

<==========================>

  

 

       >======<    

Admit it -- the line at the top looks longer to you.  Yet scientists assure us they are exactly the same length!   (The illusion is produced by the relative happiness of the adjoining hamsters.)

(2)  Turing Test

Q:  Is there a God?

A:  No.

Q:  Is there any basis for morality?

A:  No.

Q:  Is consciousness real?

A:  No.

Q:  Do we have Free Will?

A:  No.

Q:  Am I, mm, speaking to a robot?

A:  …. No.

[Note:  All the responses were false except the last.  The responder was in fact not a robot, but a neuroscientist.  They can be hard to tell apart.]


(3)   Questions that stumped the experts, knocked over the left-field fence by Dr.  J.

Q:  What time is it on the sun?

A:  Five o’clock.

Q:  What color is God?

A:  Blue.

Q:  Why don’t atoms have individual names?

A:  Some do, some don’t.

“Science You Can Trust”  ®

The Psychology of Mathematical Invention

In the introduction to his classic study The Psychology of Invention in the Mathematical Field (1945), Jacques Hadamard writes, with great acuity:

Our title is “Psychology of Invention in the Mathematical Field” and not “Psychology of Mathematical Invention”. … Mathematical invention is but a case of invention in general…

We do not aim at any such generality, and thus choose the more circumspect title.

The other aspect of Hadamard’s title we are tempted to change, is “invention”, vice discovery; for we have argued throughout these essays that, just as in physics, we discover mathematical truths that exist antecedently and independently of the fumblings of any particular forked-radishes -- though our characterizations and purported “proofs” of these truths, are  to be sure  fallible and contingent.

It turns out, though, that Hadamard himself  is a proper Platonist, and we disagree not at all:  “We speak of invention;  it would be more correct to speak of discovery.”  He goes on to make several acute remarks about the distinction in general between discovery and invention, which we hope, in good time, to treat in another place.   (Of course, by the time I get around to it, I might myself be in Another Place, in which case the practicalities of publication might become problematic.)

[Note, by the way, that this distinction between discovery and invention is, for a philosopher, more than a niggling nicety;  for indeed it turns the significance of his opening statement, “Mathematical invention is but a case of invention in general”, quite upon its head.  As stated, and as it would normally be understood, it assimilates the -- let us call it, neurtrally,  attainment of mathematical truths, to the invention of, say, the cuckoo-clock, or the beer bong.  Whereas, substitute discovery, and now we have the Platonist position in full:  you discover Riemann surfaces they way you discover the origins of the Nile.]

The final thing I wish I could change about Hadamard’s title, is “psychology”.   For, at this level, the mental foibles of hominids or turtles  is of very little interest in itself.   But -- we are incarnated.  (I almost wrote, “Alas!”, save that Our Lord, Himself, did not disdain to don a human frame.)   And at least we are treating of cognitive psychology, and indeed that of creative mathematicians, rather than the psychology of nihilists or Donald Trump.

~

In addition to Hadamard’s well-known booklet, the combinatoric mathematician George Pólya wrote a whole series of volumes  on mathematical heuristics.  One of them, How to Solve It, is quite widely known;  I myself  for some reason  could never really get into it, but many consider it a classic.

~

Andrew Gleason offers some stray comments,  which might illuminate the creative process, which I have gleaned from his one layman-accessible book.

Mathematical research  is largely a process of winnowing theorems from a melange of hunches, vague analogies, and geometrical images.

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

Gleason is here  speaking for himself and for some others, but not all.  Hadamard mentions Hermite’s actual hatred for geometrical images.

Mathematicians continue to rely on what is ultimately a subjective process for evaluating proofs.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 7

Since mathematics exceeds all other activities whatsoever in intellectual rigor, this statement should not be taken as some kind of fatal confession.   Indeed, since the defeat (at the hands of Gödel and others) of Hilbert’s well-intentioned but doomed Formalist program, mathematics itself has abounded in deep subtleties, illuminating this “subjectivity” in ways that show it to be far beyond any simple falling-short of objectivity, any de gustibus.   (Cf. startling results such that something may be uncountable within a model, but countable ‘from outside’.)

It may happen that a conditional and its contrapositive have a distinctly different intuitive flavor…

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 10

(Language as mediator  between math and the mere brain.)

Today’s mathematics is based on set theory. … Mathematics is thus reduced, in a sense, to glorified combinatorial problems.  While this approach is decried by some  for making mathematics nonintuitive, it does, in fact, lead to a new kind of intuition  which is indispensable in modern algebra …

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 55

(Psycho)logical note:   The purported foundation of mathematics upon set theory  does not subtract one whit from all that went before, but simply adds an understory.  This rhetoric of “reduced … glorified…” is therefore an example of mathematical humility, a topic whereof I hope to treat  in another place, at another time.

~

There follow some glimpses of the mathematician at his workbench -- not presenting his finished results to the world, but puzzling over how to make progress.

*     *     *

~ Commercial break ~

Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

*     *     *

A pioneer of Hilbert space theory, expounding the then-contemporary state-of-the-art for a nonspecialist mathematical audience, particularly as regards dilations and extensions of operators:

There do not seem to be any conspicuous and challenging yes-or-no questions that serve to indicate the direction in which the search for new results might begin,  but I have faith.  There is depth in the subject;  the trouble is that the surface has not been explored enough  to show where the deepest parts lie.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

He goes on:

From a certain point of view, the main problem of group theory is to decide when two groups are isomorphic, and the main problem of topology is to decide when two spaces are homeomorophic.

--  Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 18

In practice,  algebraists and topologists  usually focus on a different problematics;  but this analogy does suggest a research program for Hilbert space -- which, however turns out to be difficult to pursue in practice, since such questions are “usually too broad (and too vague) ever to arrive at a satisfactory solution."  We can picture the researcher sitting puzzled at his desk:

Special cases of the problem of unitary equivalence  can sometimes turn out to be rewarding;  it is usually hard to tell in advance  whether they will or not.

-- id., p. 19

Heuristic, rule-of-thumb, seat-of-the-pants  research programs:

The hope is that, if we know all about all invariant subspaces of many operators, we might get an insight into either the construction of an operator without any, or the proof that all operators have one.

--  Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 12

Topology in its full glory being intractable,

We associate with topological spaces and with continuous maps  certain algebraic objects, called topological invariants, under the hopeful assumption that algebra is easier than topology.  These invariants, in order to be useful, must be computable, which often is also just a hopeful assumption.

--Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 101.

~

James Newman (The World of Mathematics, p. 2039) calls Hadamard's essay "entertaining ... but not very enlightening."


Weiteres zum Thema:
Above, we treated of the psychology of successful mathematical endeavor.   Another psychological problem is why, for humans, including professional mathematicians, mathematics is so darn hard.
For a fairly funny essay on the subject, click here:
Oligophrenia mathematica

 

From outside mathematics:

Aus den Mitteilungen einiger höchstproduktiven Menschen, wie Goethe und Helmholtz, erfahren wir doch eher, da das Wesentliche und Neue ihrer Schöpfungen  ihnen einfallsartig gegeben wurde, und fast fertig  zu ihrer Wahrnehmung kam.

-- Sigmund Freud, Die Traumdeutung (1900)

.

Tourism vs. Travel

 

The old voyagers risked life and limb to get where they were going.  Every journey was one of discovery.  For long, there were no practical maps, just traditional itineraries.

Many have commented on the decline of travel into mere tourism -- fatsos clad in flowered shorts,  having arrived by plane, snapshotting everything  and seeing nothing.

But better than either  is … pilgrimage:

http://blog.myocn.com/travel-blogs/imagining-the-holy.html


For more from Drs.  Massey & Justice:
  http://linguasacrapublishing.com/massey.html
  http://linguasacrapublishing.com/justice.html

On Angels

I don’t think about them much -- just take them for granted, like woodchucks --  delighting when I happen to run into one, is all.   I can’t recall having ever set down to write about them, since I know so little.  But it turns out that several posts do touch upon the subject:

http://worldofdrjustice.blogspot.com/search/label/angelology

Something to Kindle your interest

Well how about that....  Amazon has just debuted a "Read the first chapter for free" feature.
Here, from the collection of new and previously-published short stories, interspersed with philosophico-sociological "entrefilets", starring Michael Xavier Murphy, the wise-cracking, gun-toting, two-fisted postlapsarian preConciliar  private eye:

http://www.amazon.com/dp/0984343229/ref=as_li_qf_sp_asin_til?tag=keitmassintea-20&camp=0&creative=0&linkCode=as1&creativeASIN=0984343229&adid=0PY17FNEY2TXVF74T0N0


For more about this wise & wacky character, click here:
http://murphybros.blogspot.com/


Note:  Equal time for aficionades of the Nook:
http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

Sermons from the Mitt

The poor ye have always with you.

Let them eat cake.