![\"borel.jpg\"](\"https:\/\/www.joostrekveld.net\/wp\/wp-content\/uploads\/2010\/03\/borel.jpg\")
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A while ago I finished reading this book:
\nEmile Borel, ” Les Nombres Inaccessibles”, Gauthier-Villars, Paris, 1952.<\/p>\n
I found out about Borel via Gregory Chaitins ‘Meta Math !’, and earlier on I posted<\/a> a passage of Borel quoted by Chaitin. What I find great about Borel is the Borgesian twist many of his thought experiments have. Also it’s a kind of mathematics that is very philosophical and relatively accessible since Borel writes in French rather than formulas. His “Nombres Inaccessibles” traces the surprising consequences of the idea that the amount of numbers we tend to used is dwarfed by the infinitely higher amount of numbers that are so huge that we can not even formulate them. In this way he unfolds a view on mathematics as something man-made and with a constructivist tinge sometimes (or is that in the eye of the beholder ?).<\/p>\n Below a longish quote (and translation) to give an idea of the poetry in parts of this book; Borel has many examples like these, another favourite of mine is when he talks about the number that is formed by regarding the outcomes of all games ever played in the history of humanity as a sequence of digits forming one very long binary number..<\/p>\n ” ” A while ago I finished reading this book: Emile Borel, ” Les Nombres Inaccessibles”, Gauthier-Villars, Paris, 1952. I found out about Borel via Gregory Chaitins ‘Meta Math !’, and earlier on I posted a passage of Borel quoted by Chaitin. What I find great about Borel is the Borgesian twist many of his thought experiments … Continue reading nombres inaccessibles<\/span>
\n… un page d’un livre fran\u00e7ais (ou anglais, etc.) se pr\u00e9sente ainsi comme la suite d’un certain nombre de 26 lettres a, b, c, …, x, y, z. On peut donc consid\u00e9rer cette page comme un nombre \u00e9crit dans le syst\u00e8me de de num\u00e9ration de base 26, a correspondant au chiffre 0, b au chiffre 1, etc., z au chiffre 25.
\nUn volume d’un certain nombre de pages pourra de m\u00e9me \u00e9tre regard\u00e9 comme un seul nombre; … \u00c9tablir les conventions [pour cela] … est chose simple et facile si l’on ne consid\u00e8re qu’un seul volume; il en serait autrement si l’on voulait, pour d\u00e9finir un nombre tr\u00e8s grand, classer les uns apr\u00e8s les autres tous les volumes d’une grande biblioth\u00e8que. …
\nSi nous nous bornons \u00e0 un seul volume, nous savons que l’\u00e9dition que nous avons entre les mains a g\u00e9n\u00e9ralement \u00e9t\u00e9 imprim\u00e9e \u00e0 de nombreux exemplaires, dont plusieurs subsistent, si la publication n’est pas trop ancienne. … Les volumes de la m\u00eame \u00e9dition d\u00e9finissent … un m\u00eame nombre pouvant comporter environ un million de chiffres (dans le syst\u00e8me de base 26).
\nEst-il besoin de dire qu’un tel nombre ne pr\u00e9sente aucun int\u00e9r\u00eat pour le math\u00e9maticien, qui sera toujours incapable d’en d\u00e9montrer la moindre propri\u00e9t\u00e9 ? Mais personne n’aurait jamais song\u00e9 \u00e0 \u00e9crire dans le syst\u00e8me d\u00e9cimal un nombre aussi grand et nous aurions beaucoup de peine \u00e0 concevioir combien est grande la vari\u00e9t\u00e9 de nombres aussi \u00e9lev\u00e9s.
\nL’exemple du volume \u00e9crit en fran\u00e7ais parle bien davantage \u00e0 notre imagination, car nous savons quelle complexit\u00e9 et quelle richesse il y a dans une seule page d’un livre. La vari\u00e9t\u00e9 possible nous appara\u00eet comme d’une richess inouie, si nous songeons que, dans les millions de volumes d’une biblioth\u00e8que, comportant des milliards de lignes, il n’y a pas deux lignes identiques, sauf dans le cas o\u00f9 l’on fait une citation.
\nPour celui qui conna\u00eetrait toutes les propri\u00e9t\u00e9s des nombres, il est probable que deux nombres de 50 chiffres appara\u00eetraient comme ayant chacun une personnalit\u00e9 propre, tout comme deux lignes diff\u00e9rentes en langue fran\u00e7aise.
\nLes nombres qui jouent en analyse un r\u00f4le important, tels que e ou pi, dont les chiffres d\u00e9cimaux nous apparaissent comme une masse confuse d’o\u00f9 ne se d\u00e9gage aucune loi, devraient appara\u00eetre, \u00e0 celui qui saurait en d\u00e9gager les lois complexes, comme aussi int\u00e9ressants et aussi beaux qu’un c\u00e9l\u00e8bre sonnet ; …
\n”<\/p>\n
\n.. in this way a page of a French book (or English, etc.) presents itself as a sequence of a certain number of 26 characters a, b, c, …, x, y, z. We can therefore consider that page as a number written in a numbering system with base 26, a corresponding to the number 0, b to the number 1, etc, z to the number 25.
\nA volume of a certain number of pages can similarly be considered as one number; … To establish conventions [for this] … is a simple matter if we only look at one volume; it would be different if we would like to order one after another all the volumes of a large library, in order to define a very large number. …
\nIf we limit ourselves to one volume, we know that the edition we have in our hands has generally been printed in many copies, of which several still exist if the publication is not too ancient. … The volumes of the same edition define … the same number, made up of about a million characters (in the base 26 system).
\nIs it necessary to say that such a number is of no interest whatever to the mathematician, who will always be incapable of showing the slightest property this number has ? But nobody would have ever thought of writing such a large number in the decimal system and we would have a lot of trouble to imagine how great the variety of such high numbers is.
\nThe example of a volume written in French speaks much more to our imagination, because we know what complexity and abundance there is in one page of a book. The variety that is possible to us seems like an unheard of wealth, if we think that in the millions of volumes in a library, containing billions of lines, there are no two lines that are identical, except in the case of a quote.
\nTo him who would know all properties of all numbers, it is likely that two numbers of 50 characters would each appear with its own personality, just like two different lines in the French language.
\nThe digits of numbers that play an important role in analysis, like e or pi, appear to us like a confused mass from which we can derive no law. To him who would be able to derive its complex laws, they would seem as interesting and beautiful as a famous sonnet ; …
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