Thumbnails of mathematical terms

The following essay treats of the art of mathematical definition:

https://worldofdrjustice.blogspot.com/2014/01/what-is-mathematics-expanded.html

Here are some recent additions:

Ergodicity means, roughly, that … very long sample paths … end up resembling each other.

-- Nassim Taleb, Fooled by Randomness (2004), p. 59

Ergodicity, … that time will eliminate the annoying effects of randomness.

-- Nassim Taleb, Fooled by Randomness (2004), p. 144

Holonomy is a measure of how tangent vectors on a particular surface  get twisted up as you attempt to parallel-transport them on a loop.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 129

The Lie Bracket of X and Y … informally … represents the net direction of motion if one first moves an infinitesimal amount in the X direction, then in the Y direction, then back in the XS direction and back in the Y direction, in that order.

-- Mark Ronan, “Lie Theory”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 231

Logic

… [C. S.] Pierce’s last papers on logic, a subject which he defined rather surprisingly as ‘the stable establishment of beliefs’.

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 174

Logic is the infancy of mathematics, or conversely, mathematics is the maturity of logic.

-- Bertrand Russell, 1957, quoted in I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 315

Spinors are analogues of tangent vectors.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 131

"real (or mathematically viable)"

We cannot say for sure that such manifolds -- the non-Kähler ones -- are even real (or mathematically viable).  There is no broad existence proof, such as that pertaining to Calabi-Yau manifolds, and existence, so far, has only been established in a few isolated cases.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 243

(I quoted that passage for its nice mathcentric gloss “real (or mathematically viable) “, but the paragraph is actually paradoxical.  Up to the word “manifolds”, it makes sense and should mean: non-Kähler manifolds have been defined in principle, but so far we have seen no examples; not “mathematically viable” would mean that no object could actually meet the definitional criteria, hence we shall never come across any.  But then he adds, that some finite number of instances  have indeed been proved to exist (albeit perhaps only in the shadowy form that you get from a non-constructive “existence proof”).

For essays on the subject “real in mathematics” (in the ontological sense, not in the retronymic sense of “not containing the square root of negative one as a factor”), try these:

https://worldofdrjustice.blogspot.com/search/label/Platonism

SimCity: the foundations

Background:   Topology is familiarly, informally characterized as “rubber-sheet geometry”.  That is, unlike the Euclidean geometry that we learned in school, which applies to flat rigid surfaces,  you’re allowed to stretch and bend the space, so long as you don’t tear it or let it intersect itself.

But at some point, we might like -- without returning quite to the simplicities and rigidities of the Euclidean picture -- to make our space… a bit less rubbery.   As the godfather of Calabi-Yau manifolds puts it:

We start with some raw topological space, which is like a bare patch of land that’s been razed for construction.  On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 77

[The above is an update to this post:

https://worldofdrjustice.blogspot.com/2011/08/urysohn-metrization-theorem-for-real.html  ]

On Abstraction, Abstractly Considered

 

Arriving in California at the age of twenty… I had no idea of what direction to pursue.  I was initially inclined toward operator algebra, one of the more abstract areas of algebra, owing to my vague sense that  the more abstract a theory was, the better.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 35

[Update to this essay:
https://worldofdrjustice.blogspot.com/2011/06/on-vulgar-numbers.html ]

Doffing the Robe of Winter

grown men, weary with cares; 

old men, burthened with winters; --

they had mused, and slumbered, and departed

to a yet profounder sleep.

--  Nathaniel Hawthorne, The House of Seven Gables (1851)

Midwest Morning monostich

hawks

hung

in the air

-- Hamlin Garland, A Son of the Middle Border (1917)

Deux fois n’est pas coutume

(1)  Ever happen to you?

The noted Catholic philosopher and ethicist Alasdair MacIntyre, writes in  After Virtue (1981; ²1984), p.  152:

Although someone who possesses the virtues to a high degree will be far less apt than others  to commit grave offences, a brave and modest man  may  on occasion  commit murder…

(2)  An actual case from France

Swann la regardait, avec ce regard paisible et poli mais clair d’un médecin qui jauge un malade.

— Odette, lui dit-il, mon chéri, je sais bien que je suis odieux, mais il faut que je te demande des choses. Tu te souviens de l’idée que j’avais eue que tu aurais, jadis, dans le temps,  assassiné un certain gentleman?  Dis-moi si c’était vrai, lui ou bien un autre.

Elle secoua la tête en fronçant la bouche.

Dis-donc, chérie, n’aurais-tu pas par hasard,

ou par mégarde, tué  juste un petit homme ?

--Tu as bien tort de te figurer que je t'en voudrais le moins du monde, Odette, lui dit-il avec une douceur persuasive et menteuse.  Je ne te parle jamais que de ce que je sais, et j'en sais toujours bien plus long que je ne dis.  Mais toi seule peux adoucir par ton aveu ce qui me fait te haïr tant que cela ne m'a été dénoncée que par d'autres.  Ma colère contre toi ne vient pas de tes actions, je te pardonne tout puisque je t'aime, mais de ta fausseté, de ta fausseté absurde qui te fait persévérer à nier des choses que je sais.  Mais comment veux-tu que je puisse continuer à t'aimer, quand je te vois me soutenir, me jurer une chose que je sais fausse.  Odette, ne prolonge pas cet instant  qui est une torture pour nous deux.  Si tu le veux, ce sera fini dans une seconde, tu seras pour toujours délivrée.  Dis-moi sur ta médaille, si oui ou non, tu as jamais fais ces choses.

--Mais je n'en sais rien, moi, s'ecria-t-elle avec colère, peut-etre il y a tres longtemps, sans me rendre compte de ce que je faisais, peut-etre deux ou trois fois.

Du coup  il sentit un grand soulagement.

-- Tiens! se dit-il, « deux ou trois fois » :  ce n’est quand-même pas un massacre !

~

[Sponsored content]

The Foreign Excellent Trenchcoat Company:

Clothiers to gentlemen since 1917

~

Frontier Days monostichs

  new worlds of virgin land   lay ever just beyond

-- Frederick Jackson Turner, “Pioneer Ideals”,  Commencement Address at the University of Indiana, 1910

  the solitary backwoodsman, wielding his ax
  at the edge of a measureless forest

-- id.

[For a different perspective -- a slanting glance -- at those early days of our great nation, try this:]

https://worldofdrjustice.blogspot.com/2011/02/tales-of-frontier-times.html