Sunday, June 24, 2018

Dedekind on ontology

[A footnote to this essay.]

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).

More from the Mindscape

A footnote to this essay:

Dedekind … allowed his philosophy of mind  much reign, with a ‘proof’ that “there are infinite systems”;  for he gave  as evidence “the totality S of all things, which may be objects of my thought”, since  as well as any of its elements s,  it contained also “the thought s’ that can be the object of my thought …This ‘proof’ did not gain a good reception.”
-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 105

Tuesday, June 19, 2018

Summer Solstice

The sun at apogee, an empire all of light.
The sweet corn basks and ripens in its rays.
All memory of winter  has melted from our minds.

Yet even now the worm lies in the bud, one day to blast it.
The long limbs of daylight   that had stretched and stretched
now imperceptibly  begin to shrink:
even as the cosmos  attains its outmost limit,
and sighs back  to collapse.

Alas, the Elves of Ice  lurk still in northern forests,
plotting their return.

Monday, June 18, 2018

The Beasts: a post-Benthamite assessment (Take Two)

The Beasts:  a post-Benthamite  assessment (Take Two  Three)

[The Sage of Houndsditch, updated]

The question about animals is not, “Can they reason?”,
but  “Can they handle algebraic geometry  Hodge theory  motivic cohomology?”

[Note:  We have further updated this post, in an attempt to outpace the rise in cleverness  among the squirrels, which has been growing at an alarming pace.]

Sunday, June 17, 2018

Ex-Word of the Day: Toplinear

In her diligent book of epistemology, Evidence and Inquiry (1993), Susan Haack steers her bark between the twin cliffs of Foundationalism and Coherentism, before christening her own proposed juste milieu with the,  alas, ungainly name of :


There was something comical about the coinage which I could not quite put my finger on.

Then this morning, happening to dip into one of Serge Lang’s textbooks on Differential Geometry, discussing topological vector spaces[**], I re-encountered his own stillborn coinage:

“… a Toplinear isomorphism, referring to the topology  and the linearity”

It is gauche in much the same way.

Neither term has much caught on, outside the readership-circle of its coiner.

[**  A TVS is a uniform structure on a linear space.]

Saturday, June 16, 2018

Contra Constructivism

“Only an atheist
 need fear infinity.”

Note:   In this place,
an unafraid theologian presents an Orthodox account of the doctrine of absolution and the remission of sins,  conscientiously noting  along the way  a possible mathematical objection to the Church’s handling of infinity in this connection, which he is not personally qualified to resolve.

Fortunately, I was able to reply in complete reassurance on this point:  the Church doctrine is fully consonant with set-theory in its post-Cantorian understanding.

For more:

Friday, June 15, 2018

A Dactylic Pentamer ad Differential Forms

Thése are the thíngs which occúr under íntegral sígns.”
-- Bárret O’Néill, Elementary Differential Geometry

[Note:  The author’s name is prosodically a choriamb.]

[Note:  In an alternative terminology, a quantity at home beneath the integral sign  is called a density.]

Wednesday, June 6, 2018

Apophthegms of Physics

Hertz: “Maxwell’s theory is Maxwell’s system of equations.”
By itself, this is a witty, eminently quotable, and meaningless comment.
-- Abraham Pais, Subtle is the Lord (1982), p. 119


There may be such a thing as habitual luck.  It is like hidden parameters in physics.
-- Stanislaw Ulam, Adventures of a Mathematician (1976), p. 119

Or as Yogi Berra once put it:  That is too much of a coincidence to be coincidental.

Saturday, June 2, 2018

The Riddle of the Sphinx (mis à jour)

Contemporary Sphinx:  What walks on four legs in the morning, two legs at noon, and six legs at evening?

Modern Oedipus:  A person:   a crawling baby;  an adult with grown-up gait; and an oldster with a walker (which itself is a quadruped, like a rug-rat).