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;
Duchamp in Maya Deren's Witch's Cradle (1943); 
Georg Cantor et les infinis, par Patrick Dehornoy (SMF/Youtube 2009);
Problems with Zero (Numberphile/Youtube);
Indeterminate: the hidden power of zero divided by zero (Mathologer/Youtube); 
L'étonnant puzzle fractal de von Koch (Mickaël Launey/Youtube); 
El Problema indemostrable (Mates Mike/Youtube); 
The Great Abyss Inframince (A/Z 2018, for more see here);

Foundation out of the common track:


"E pois, te digo, as estrelas,
no céu imenso espalhadas,
são a metade e outro tanto
das mesmas por Deus criadas;
e, se imaginas que minto
na quantidade que dei,
te desafio a contá-las...
para ver que não errei!"
Cancioneiro Guasca
"... but I'm prejudiced. I have loved Cantor's hierarchy of cardinals ever since I was exposed to it at a tender age."
Ian Hacking (Why Is There Philosophy of Mathematics At All?)

"... the objects of transfinite set theory... clearly do not belong to the physical world and even their indirect connection with physical experience is very loose (owing primarily to the fact that set-theoretical concepts play only a minor role in the physical theories of today)... The set-theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics... The mere psychological fact of the existence of an intuition which is sufficiently clear to produce the axioms of set theory and an open series of extensions of them suffices to give meaning to the question of the truth or falsity of propositions like Cantor's continuum hypothesis..."
Der Herr Warum (What is Cantor's Continuum Problem?/Supplement to the Second Edition)
"Es treten in uns fortwährend neue Vorstellungs-massen auf, welche sehr rasch aus unserm Bewusstsein wieder verschwinden. Wir beobachten eine stetige Thätigkeit unserer Seele. Jedem Act derselben liegt etwas Bleibendes zu Grunde, welches sich bei besonderen Anlässen (durch die Erinnerung) als solches kundgiebt, ohne einen dauernden Einfluss auf die Erscheinungen auszuüben. Es tritt also fortwährend (mit jedem Denkact) etwas Bleibendes in unsere Seele ein, welches aber auf die Erscheinungswelt keinen dauernden Einfluss ausübt. Jedem Act unserer Seele liegt also etwas Bleibendes zu Grunde, welches mit diesem Act in unsere Seele eintritt, aber in demselben Augenblick aus der Erscheinungswelt völlig verschwindet. Von dieser Thatsache geleitet, mache ich die Hypothese, dass der Weltraum mit einem Stoff erfüllt ist, welcher fortwährend in die ponderablen Atome strömt und dort aus der Erscheinungswelt (Körperwelt) verschwindet..."
Bernhard Riemann (Neue mathematische Prinzipien der Naturphilosophie)

"Considerer une infinité dénombrable d'objets à travers leur représentation finie idéale pouvait déjà paraître dépasser les possibilités humaines, mais Pythagore découvrit l'insuffisance de cette donnée, selon laquelle tout segment aurait dû être commensurable avec tout autre. Dès lors surgissaient sur la droite idéale, elle-même substitut d'une infinité dénombrable de droites, une infinité de points non rationnels, dont on apprit, avec Cantor, que l'ensemble cessait d'être dénombrable. En ce qui concerne les radicaux carrés et les expressions rationnelles d'un nombre fini de radicaux carrés, la doctrine des idées permit de résoudre entièrement le problème, dans la mesure où elle admit les constructions par la règle et le compass. Mais, résolu sur ce point, le scandale devait renaître avec les autres nombres algébriques, puis avec les nombres transcendants."
"[Dans la projection téréographique sur la sphère de Riemann] P peut être choisi comme représentant du nombre complexe x + iy... quand P se dirige vers N, Q tend à s'éloigner de l'origine et Ne peut être regardé comme le représentant unique de tous les point à l'infini du plan imaginaire, points qui deviennent uniques et perdent ainsi, dans la représentation intuitive de Riemann, le privilège qu'ils conservaient dans celle d'Argand-Cauchy. La conformité de la représentation est conservée quand on passe du plan à la sphère complexe."
Jules Vuillemin (La philosophie de l'algèbre) 

"We know the difference that separates the Hilbertian conception of mathematics from that of Russell and Whitehead’s Principia Mathematica (1910). Hilbert has replaced the method of genetic definitions with that of axiomatic definitions, and far from claiming to reconstruct the whole of mathematics from logic, introduced on the contrary, by passing from logic to arithmetic and from arithmetic to analysis, new variables and new axioms which extend each time the domain of consequences. Here is, for example, according to Bernays, who in the complete works of Hilbert published a study of all his work on the foundations of mathematics, all that is necessary to be given to formalize arithmetic: the propositional calculus, the axioms of equality, the arithmetic axioms of the ‘successor’ function (a + 1), the recurrence equations for addition and multiplication, and finally some form of the axiom of choice. To formalize analysis, it is necessary to be able to apply the axiom of choice, not only to numeric variables, but to a higher category of variables, those in which the variables are functions of numbers. Mathematics thus presents itself as successive syntheses in which each step is irreducible to the previous step. Moreover, and this is crucial, a theory thus formalized is itself incapable of providing the proof of its internal coherence. It must be overlaid with a metamathematics that takes the formalized mathematics as an object and studies it from the dual point of view of consistency and completion. The duality of planes that Hilbert thus established between the formalized mathematics and the metamathematical study of this formalism has as a consequence that the notions of consistency and completion govern a formalism from the interior of which they are not figured as notions defined in this formalism."
"... it is impossible to consider a mathematical ‘whole’ as resulting from the juxtaposition of elements defined independently of any overall consideration relative to the structure of the whole in which these elements are integrated. There thus exists a descent from the whole towards the part, as a ascent from the part to the whole, and this dual movement, illuminated by the idea of completion, allows the observation of the first aspect of the internal organization of mathematical entities. If one claims to admit that the study of such structural connections is an essential task for mathematical philosophy, one cannot fail to notice the differences that separate mathematical philosophy thus conceived from the entire current of logicist thought that developed after Russell had discovered the paradoxes of set theory. The logicians have since always claimed to prohibit non-predicative definitions, that is, those in which the properties of an element are supportive of the set to which that element belongs. Mathematicians have never been willing to admit the legitimacy of this interdiction, rightly showing the necessity, to define certain elements of a set, to sometimes call upon the global properties of this set... We thus hope to make evident this idea that the true logic is not a priori in relation to mathematics but that for logic to exist a mathematics is necessary."
"Facts are... organized under the unity of the notion that generalizes them. The real ceases to be the pure discovery of the new and unforeseeable fact, in order to depend on the global intuition of a supra-sensible entity. [Pierre] Boutroux [L'Ideal scientifique des mathématiciens, 1920] takes as an example the reality of the ellipse. The ellipse is for him neither the locus of points such that the sum of their distances to the foci is constant, nor a curve defined by its algebraic equation, nor a curve to the projective properties of conics. It is all that and much more. It is, he says, 'a whole that does not include parts... a sort of Leibnizian monad. This monad is pregnant with the properties of the ellipse; I mean that these properties, even though they have not been explicitly formulated (and they cannot be since they are infinite in number) are contained in the notion of ellipse.'"
Albert Lautman (Essay on the Notions of Structure and Existence in Mathematics, Simon B. Duffy's translation [it is an ill-fated instance of the malignancy of this world that it happens I don't have the French original of Lautman's essays, & shall therefore by my ashes stand forever indebted to the heroine soul (chaste star!) who peradventure send me, with all her fraternity, the French original of just at least these choicest morsels so I could definitely pen them down here as a seasonable kindness to our whole parish!])

"The supermultitudinous character of Peirce’s continuum shows, according to Peirce, that the Cantorian real line is just 'the first embryo of continuity', 'an incipient cohesiveness, a germinal of continuity.' Nevertheless, the cardinal indetermination of Cantor’s continuum inside ZF—a profound discovery of 20th century mathematical logic that Peirce could not imagine—shows that the Cantorian model can also be considered as a valid generic candidate to capture the supermultitudeness of the continuum... The indetermination of 2ˆAleph0 is not well appreciated by the specialists, who consider that the incompleteness of ZF must be repaired. In any of the 'normative' responses given to the cardinal size of the continuum, it ceases immediately to be supermultitudinous since the additional axioms force it to adjust within a determinate level of a hierarchy."
"... while Cantor and, systematically, most of his followers, try to bound (and bind) the continuum, Peirce tries to unbound it: to approach a supermultitudinous continuum, not restricted in size, truly generic in the transfinite, never totally determined. It comes then, as a most remarkable fact, that many indications of the indeterminacy of the continuum found at the core of contemporary Cantorian set theory (free analysis of the set theoretic universe through disparate filters, using forcing techniques, with many phenomena possibly coexistent) seem to assure in retrospect the appropriateness of Peirce’s vision. The generality of Peirce’s continuum implies, as we shall now see, that it cannot be reconstructed from the 'particular' or the 'existent', and that it must be thought in the true general realm of possibilia."

"Nous connaissons qu'il y a un infini, et ignorons sa nature..."
Pascal (Pensées, 233)
"Ainsi il y a des propriétés communes à toutes choses, dont la connaissance ouvre l'esprit aux plus grandes merveilles de la nature. La principale comprend les deux infinites qui se rencontrent dans toutes: l'une de grandeur, l'autre de petitesse... De même, quelque grande que soit un nombre... Et au contraire, quelque petit que soit un nombre..."
"Voilà l'admirable rapport que la nature a mis entre ces choses, et les deux merveilleuses infinites qu'elle a proposées aux hommes, non pas à concevoir, mais à admirer..."
Pascal (Fragments de l'esprit géométrique)
"... la conception d’une quantité finie porte toujours sur l'objet du savoir, sur l’être qui est opposé au savoir et qui constitue l’objet de la connaissance; tandis que l’idée de l’infini... ne peut porter que sur les fonctions mêmes du savoir, où elle introduit la plus haute unité intellectuelle ou la plus haute signification dans la production même de la connaissance de la quantité."
Höené Wronski (Philosophie de l'Infini)
"Quis neget, naturam instinctu solo, sine etiam ratiocinatione, docere geometriam?"
Johannes Kepler (Stereometria Dolii Austriaci in Specie, Theorema V)

"... 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 (Mille plateaux)
"Ce qui caractérise le chaos, en effet, c'est moins l'absence de déterminations que la vitesse infinie à laquelle elles s'ébauchent et s'évanouissent: ce n'est pas un mouvement de l'une à l'autre, mais au contraire l'impossibilité d'un rapport entre deux déterminations, puisque l'une n'apparaît pas sans que l'autre ait déjà disparu, et que l'une apparaît comme évanouissante quand l' autre disparaît comme ébauche."
Deleuze  & Guattari (Qu'est-ce que la philosophie)
"Et je reçus de lui de merveilleuses explications et des éclaircissements extrêmement précis sur la façon dont le Peyotl ressuscite dans le trajet entier du moi nerveux, la mémoire de telles vérités souveraines, par lesquelles la conscience humaine, me fut-il dit, ne perd plus, mais au contraire, retrouve la perception de l'Infini."
"Et d'après des vieux Chinois le foie est le filtre de l'inconscient mais la rate est le répondant physique de l'infini."
A. Artaud (Le Rite du Peyotl chez les Tarahumaras)
"This relative freedom of a hero does not violate the strict specificity of the construction, just as the specificity of a mathematical formula is not violated by the presence of irrational or transfinite quantities."
Mikhail Bakhtin (Problems of Dostoevsky's Poetics, Caryl Emerson translation)

"... the world burst into a brilliant but complex series of geometric patterns so elaborate that no mere mortal could even BEGIN to comprehend their true significance. Only  God and I understood..."
James St. James
"The vistas echoed endlessly, paralleling the way that drum hits, guitar chords, and horn licks were turned into reverb trails by dub producers like King Tubby..."
Simon Reynolds
"Note the parallels between ordinary awareness, classical physics, and the natural and counting integers..."
Dean Radin (Real Magic)
"... interrompidas não porque terminassem, mas porque ninguém podia levá-las a um fim."
Clarice Lispector
"So Art is limited to infinite, and beginning there cannot progress."
Whistler

"Of course, not everyone in postpunk attended art school, or even college. Self-educated in a scattered, omnivorous fashion, figures like John Lydon or Mark E. Smith of the Fall fit the syndrome of the anti-intellectual intellectual, ravenously well read but scornful of academia and suspicious of art in its institutionalized forms."
Simon Reynolds
"I had been occluded from space-time like an eel's ass occludes when he stops eating on the way to Sargasso... Locked out... Never again would I have a Key, a Point of Intersection... The Heat was off me from here on out... relegated with Hauser and O'Brien to a landlocked junk past where heroin is always twenty-eight dollars an ounce and you can score for yen pox in the Chink laundry of Sioux Falls... Your plan was unworkable then and useless now... Like da Vinci's flying machine plans..."
William S. Burroughs

"Denken wir uns nun einen äußersten Fall: daß ein Buch von lauter Erlebnissen redet, die gänzlich außerhalb der Möglichkeit einer häufigen oder auch nur seltneren Erfahrung liegen, — daß es die erste Sprache für eine neue Reihe von Erfahrungen ist. In diesem Falle wird einfach Nichts gehört, mit der akustischen Täuschung, daß, wo Nichts gehört wird, auch Nichts da ist... Dies ist zuletzt meine durchschnittliche Erfahrung und, wenn man will, die Originalität meiner Erfahrung."
"Ich komme aus Höhen, die kein Vogel je erflog, ich kenne Abgründe, in die noch kein Fuß sich verirrt hat..."
"Man büßt es teuer, unsterblich zu sein..."
Nietzsche

"Car toutes ces femmes étaient des actrices du monde, et il est vrai que même considérée à ce point de vue, la comtesse Molé n'était pas égale à l'extraordinaire réputation d'intelligence qu'on lui faisait, et qui donnait à penser à ces acteurs ou à ces romanciers médiocres qui à certaines époques ont une situation de génie, soit à cause de la médiocrité de leurs confrères, parmi lesquels aucun artiste supérieur n'est capable de montrer ce qu'est le vrai talent, ou de la médiocrité du public, qui, existât-il une individualité extraordinaire, serait incapable de la comprendre."
Marcel Proust (le narrateur, La Prisonnière)

"... one of the two professors assigned to assess his thesis, doubtless bewildered by the extent of its expertise and scholarship, declared it incomprehensible and threatened not to approve it."
Malcolm Hayes (Anton von Webern)
"N'ayant jamais consenti à vivre dans notre univers salarié, le célèbre linguiste n'avait pas cotisé à la M.G.E.N. Résultat: impossible de le faire soigner dans une clinique sérieuse. N'était-ce pas inadmissible?"
Olga (Les Samouraïs)
"On n'élimine pas, on neutralize en marginalisant..."
Edward (Les Samouraïs)
"Au-delà de l'erreur, au-delà de la bêtise elle-même: une certaine bassesse de l'âme..."
Deleuze/Nietzsche
"Dans le pain et le vin destinés à sa bouche
Ils mêlent de la cendre avec d'impurs crachats..."
Charles Baudelaire
"One will know well if ignorance is removed. Those who know well always desire to act."
Myoe (Mark Unno, Shingon Refractions)

"We do not confine ourselves any more to using infinity as a figure of speech, or as shorthand for the statement that no matter how great a number there is one greater: the act of becoming invokes the infinite as the generating principle for any number; any number is now regarded as the ultra-ultimate step of an infinite process; the concept of infinity has been woven into the very fabric of our generalized number concept."
"... while Galileo dodged the issue by declaring that the attributes of equal, greater, and less are not applicable to infinite, but only to finite quantities, Cantor takes the issue as a point of departure for his theory of aggregates. And Dedekind goes even further: to him it is characteristic of all infinite collections that they possess parts which may be matched with the whole... The reader will remember Liouville's discovery of transcendentals..."
Tobias Dantzig, Number: the language of science

"In contrast to the possibility of eliminating infinity as just described stand a number of results that show that some finitary statements can only be proved through infinitary considerations. These results originally emerge with Gödel’s incompleteness theorems (1931) but have been recently refined by displaying statements of mathematical interest... establishing the truth of the Gödel sentence and of the new incompleteness results requires appeal to some “infinitary” principles (when the truth of the Gödel sentence G is established through appeal to the statement expressing the consistency of Peano Arithmetic, it is establishing the latter that requires some portion of infinitary reasoning, such as induction up to an infinite ordinal called ε0)."

"I should advise that by number... there be understood not the class (the system of all mutually similar finite systems), but rather something new (corresponding to this class) which the mind creates. We are of divine species and without doubt possess creative power not merely in material things (railroads, telegraphs), but quite specially in intellectual things... you say that the irrational number is nothing else than the cut itself, whereas I prefer to create something new (different from the cut), which corresponds to the cut..."
Richard Dedekind (to Heinrich Weber, as quoted in Ian Hacking's Why Is There Philosophy of Mathematics At All?)

"More surprising still is the fact that dimensionality is not the arbiter of the power of a set. The power of the set of points in a unit line segment is just the same as that of the points in a unit area or in a unit volume—or, for that matter, all the three-dimensional space. (Dimensionality, however, retains some measure of authority in that any one-to-one mapping of points in a space of unlike dimensionality is necessarily a discontinuous mapping.) So paradoxical were some results in point-set theory that Cantor himself on one occasion in 1877 wrote to Dedekind, 'I see it, but I don't believe it'; and he asked his friend to check the proof."
"Dedekind and Cantor were among the most capable mathematicians, and certainly the most original, of their day; yet neither man secured a top-ranking professional position. Dedekind spent almost a lifetime teaching on the secondary level..."

Carl B. Boyer (A History of Mathematics)

"Expressions like 1/0, 3/0, 0/0 etc. [that is, divisions by zero] will be for us meaningless symbols. For if division by zero were permitted, we could deduce from the true equation 0.1 = 0.2 the absurd consequence 1 = 2. It is however sometimes useful to denote such expressions by the symbol ∞ (read 'infinity'), provided that one does not attempt to operate with the symbol ∞ as though it were subject to the ordinary rules of calculation with numbers."
"We cannot include the symbol ∞ in the real number system and at the same time preserve the fundamental rules of arithmetic. Nevertheless the concept of the infinite pervades all mathematics..."
Richard Courant & Herbert Robbins (What is Mathematics)
********************************************************************

Memoire presenté à Messieurs les Docteurs de Sorbonne (if to be detected):


In his The Mystery of the Aleph (a book written to the literal edification of this scurvy and disasterous world of ours & in spight of all gentlemen reviewers in England & the Continent), Amir D. Aczel, person (if I mistake not) of no small note and consequence, writes: "Infinity is an intimidating concept—one where our everyday intuition no longer servers to guide us."
As you (known by the name of "reader") and I are in a manner perfect strangers to each other, before finding our three types, we shall start 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 (your worship) 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 (wether she was deceiving or deceived in this manner) really hates them. There are so many examples of this dismally lamentable case (trifling in nature, tedious in telling), we fortunately won't need even to give any names (I'm positive). It might just as well be the rule (though in no way becomes me to decide). And since alle Schweiger sind dyspeptisch, 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 (whose number has been lately worn down to a thread), hard-nosed economists, bureaucrats &/or famous CEOs (who presently are legion). Choose and vituperate it out & aloud—with abhorrence.
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. Blessed be the Lord, they definitely don't belong to any of our three categories, remaining (with great candor & modesty) baffled & overthrown.

Examples of genuine ingenuity are the following (under construction now made public list) [it was hard to agree on a truly certified census beyond all possibility of doubt, so we left a few ghostly holes in the form of question (& others totally hidden) marks]:
- Anaximander;
- Melissos;
- Zeno; 
- Democritus; 
- Plato (?); 
- Epicuros (?);
- Archimedes (?); 
- Lucretius (?);
- Hasdai Crescas (?);  
- Nicolas of Cusa;
- Girolamo Cardano (?);
- Giordano Bruno;
- Johannes Kepler (?); 
- Galileo Galilei (?);
- Blaise Pascal; 
- Torricelli (?); 
- Girard Desargues (?); 
- John Wallis (?);
- G. W. Leibniz (?);
- Newton (?); 
- Euler (?);
- Salomon Maimon;
- Jean Baptiste Bordas-Demoulin; 
- Höené Wronski;
- Bernhard Bolzano;
- Bernhard Riemann (?);
- Kingdon Clifford (?);
- Karl Weierstrass (?);
- Charles Sanders Peirce (?);
- Sonja Kowalewski (?);
- Gösta Mittag-Leffler (?);
- Richard Dedekind;
- Georg Cantor;
- David Hilbert (?); 
- Henri Bergson (?);
- Raymond Roussel (?);
- Melanie Klein (?);
- Jacques Lacan (?);
- Georges Bataille (?);
- Maurice Blanchot (?);
- Pierre Klossowski (?);
- Andre Weil (?); 
- Albert Lautman;
- Jules Vuillemin (?);
- Claude Lévi-Strauss (?);
- Gilles Deleuze;
- Jacques Derrida ["... l'axiome fondamental de tout ce qu'il dit partout, c'est la divisibilité du point. Tout ce qu'il écrit, tout ce qu'il pense est une protestation contre le point comme indivisible..." Hélène Cixous];
- Julia Kristeva (?);
- Sylvia Leclercq ["Jamais comblée, une différence demeure (cf. Hekhalot, Demeures Célestes, Moradas) toujours entre l'Être et l'ensemble ouvert des 'sujets', des 'singularités', des 'nombres' susceptibles de l'exprimer, et dans lesquels je me dissémine en écrivant et en agissant. J'appartiens à une géométrie qui n'est plus algébrique, mais analytique. Je suis un site du signifiant illimité..."];
- Alexander Grothendieck (?);
- Paul Cohen (?);
- Hugh Woodin (?);
- Alain Connes (?);
- Fernando Zalamea (?);
- me (which I own);

Amir Aczel provides us also with a parable about the fate of genuine ingenuity by giving an account of a set of pitiful misadventures and cross accidents sustained by Bernhard Bolzano throughout his 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).
To this it may be interesting to juxtapose the following francisé passage from Nietzsche (no muddle-headed): "Il y a des vies où les difficultés touchent au prodige; ce sont les vies des penseurs. Et il faut prêter l'oreille à ce qui nous est raconté à leur sujet, car on y découvre des possibilités de vie, dont le seul récit nous donne de la joie et de la force, et verse une lumière sur la vie de leurs successeurs. Il y a là autant d'invention, de réflexion, de hardiesse, de désespoir et d'espérance que dans les voyages des grands navigateurs; et, à vrai dire, ce sont aussi des voyages d'exploration dans les domaines les plus reculés et les plus périlleux de la vie..." (as quoted by Deleuze in Nietzsche et la philosophie).
Also, from Ecce Homo: "Wenn trotzdem an mir manche kleine und große Missethat verübt worden ist, so war nicht der 'Wille', am wenigsten der böse Wille Grund davon: eher schon hätte ich mich —ich deutete es eben an — über den guten Willen zu beklagen, der keinen kleinen Unfug in meinem Leben angerichtet hat."

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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