Sunday, February 10, 2019

the only three types of ingenuity when the parameter is infinity: the limited, the false & the genuine

Bernhard Bolzano, picture taken from the Internet;

"... la majorité est travaillé par une minorité proliférante et non dénombrable qui risque de détruire la majorité dans son concept même, c'est-à-dire en tant qu'axiome... le étrange concept de non-blanc ne constitue pas un ensemble dénombrable... Le propre de la minorité, c'est de faire valoir la puissance du non-dénombrable, même quand elle est composée d'un seul membre. C'est la formule des multiplicités. Femme, nous avons tous à le devenir, que nous soyons masculins ou féminins. Non-blanc, nous avons tous à le devenir, que nous soyons blancs, jaunes ou noirs."
Deleuze & Guattari
"Note the parallels between ordinary awareness, classical physics, and the natural and counting integers..."
Dean Radin (Real Magic)

In his The Mystery of the Aleph, Amir D. Aczel writes: 
"Infinity is an intimidating concept—one where our everyday intuition no longer servers to guide us."
We start by finding our three types by identifying two more general types: the ones who stick to everyday intuition, the ones who dare to face infinity's paradoxes. 
I argue that, in any case, the paradoxes related to infinity are real
Then to stick to everyday intuition is limited or false ingenuity. To face infinity, genuine ingenuity.
The most clear and distinct example of limited ingenuity is to be found in Descartes. He preferred to stick to everyday intuition, but acknowledged that infinity is (and has to be) real.
False ingenuity, which should rather be called bêtise or deliberate ingenuousness is the case of a person who not only sticks to everyday intuition but denies that the paradoxes of infinity can be real and really hates them. There are so many examples of this lamentable case, we fortunately won't need to give any names. It might just as well be the rule. Choose your own favourite member of whatever academic Kroneckian priesthood you happen to be acquainted with, reverenced scientists (several might do the job), inveterate analytic philosophers, hard-nosed economists, bureaucrats &/or famous CEOs.
Very ordinary people don't count because they are all geniuses in a very Warholian peculiar way: by never raising such questions on their own. Thanks gosh, they definitely don't belong to any of our three categories.

Examples of genuine ingenuity are the followings (under construction list) [it was hard to agree on a truly certified list, so we left a few ghostly holes in the form of question marks]:
- Nicolas of Cusa;
- Giordano Bruno;
- Galileo Galilei (?);
- G. W. Leibniz (?);
- Bernhard Bolzano;
- Bernhard Riemann (?);
- Kingdon Clifford (?);
- Karl Weierstrass;
- Nietzsche (?);
- Sonja Kowalewski (?);
- Gösta Mittag-Leffler (?);
- Richard Dedekind;
- Georg Cantor;
- Raymond Roussel (?);
- Henri Bergson (?);
- Der Herr Warum; 
- Claude Lévi-Strauss (?);
- Jacques Lacan (?);
- Georges Bataille (?);
- Maurice Blanchot (?);
- Pierre Klossowski (?);
- Gilles Deleuze (?);
- Jacques Derrida (?);
- Melanie Klein (?);
- Julia Kristeva (?);
- me;

Amir Aczel also provides a parable about the fate of genuine ingenuity by giving an account of Bernhard Bolzano's life:
"In 1805, Bolzano was ordained a priest and nominated to the chair of the department of the philosophy of religion at the University of Prague. Bolzano had wanted the position for several years but had been passed over for promotion by lesser-qualified but better-connected individuals... A mere decade and a half after his installment as chair, Bolzano was summarily fired and stripped of his priestly rank... One B. Frint had written a textbook which he had hoped would be used by Bolzano in his courses. But Bolzano, in his new position, resisted the pressure and did not adopt the book. Frint successfully turned people against the new chair of the philosophy of religion department. The slow but systematic case against Bolzano was built in a series of state papers documenting what officials considered objectionable elements in Bolzano's sermons. The most offensive infraction was Bolzano's preaching peace to the students... When the first attacks on him occurred, Bolzano had the support of the Archbishop of Prague, and this helped him evade any serious consequences" The Mystery of the Aleph (WSP, 2000).

Poincaré & Wittgenstein:

On Wittgenstein and finitism &/or on the difference between the infinite and the huge (see also here): 
"Wittgenstein’s famous matching of finitism and behaviourism, united by their denial of the existence of something (infinite sets and inner states, respectively), in the correct but badly executed attempt to avoid confusion (that between the infinite and a very large quantity, and that between an inner state and a private entity), shows, on this point, the agreement and, at the same time, the distance between the Austrian philosopher’s position and finitism. The denial of the existence of infinite sets is a mistaken way to draw a grammatical distinction which, though it may be opportune, should be done differently: by showing that the grammar of the word “infinite” cannot in the slightest be clarified by taking into account only the picture of something huge, a picture which usually accompanies the use of the word. As Wittgenstien affirms in one of his lectures in 1939: “If one were to justify a finitist position in mathematics, one should say just that in mathematics ‘infinite’ does not mean anything huge. To say ‘There’s nothing infinite’ is in a sense nonsensical and ridiculous. But it does make sense to say we are not talking of anything huge here”... Wittgenstein moves some criticisms against the platonistic interpretation of the true import of Cantor’s proof; nevertheless they do not originate in any way from a presupposed identification of legitimate mathematics with finitist mathematics and, even less so, from the violation, by Cantor’s proof, of the requirements imposed by strict finitism. Once the appropriate clarifications have been made about what, in his opinion, it really demonstrates, Cantor’s proof is more than good enough for Wittgenstein, in spite of the certainly non-finite nature of the “objects” it deals with... proofs which are finitistically (not only strict finitistically) unacceptable are actually accepted by Wittgenstein or are not questioned on the basis of the restriction of admissible mathematical procedures to the finitary ones," Pasquale Frascolla, Wittgenstein's Philosophy of Mathematics (Routledge 1994).
Russel & Quine with irony: "[they] were born to be nominalists even if the hard knocks of mathematical and philosophical experience shattered childhood complacency. (Russell actually began as an idealist in the Hegelian manner of late Victorian England, but that was his infancy, not his childhood)," Ian Hacking, Why is there Philosophy of Mathematics at all (Cambridge 2014).

See also:
And also:

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