Mouse in English means a mouse ... :) The computer mouse is translated into several languages, 'souris' in French, 'raton' in English, 'maus' in German. In Italian, the mouse is left mouse. They did not know the Americans of IBM when they published an instruction manual.
"Balls of mice are now available as spare parts. If your mouse has difficult to operate properly, or does it in spurts, it is possible that it will need a ball replacement. Because of the delicate nature of the replacement process of the balls, it is recommended that it be performed by a qualified expert. Before proceeding, determine what kind of balls he needs your mouse. To do this, just examine its underside. Balls of American mice are usually larger and harder than those of mice overseas. The procedure for removal of a ball depends on the kind of rat. The protection of the balls of the mice from overseas can be simply blown away with a paperweight on the protection of American mice must first be exercised a clockwise or counterclockwise. Normally, the balls of mice are charged with static electricity, but it is better to treat them with caution, to avoid unplanned. Once you complete the replacement, the mouse can be used immediately. It is recommended to staff trained to carry constantly with them a couple of balls in reserve, so as to ensure maximum customer satisfaction. In the event that the scarce spare balls, you can send a request to the central distribution using the following code ... "
numbers and succession: metamathematical reflections, historical and educational activities on a song Leopardi (4) 4. Looking to the historic teaching explicit attention the operational concept, then, is the basis of some important attempts formalization of arithmetic. Such an attitude is constant in the history of mathematics, as stated by F. Arzarello, L. Bazzini and G. Chiappini: "The development of the concept of numbers you can see how the conduct of a chain of transitions from the operational and structural conceptions. On the other hand , even before the process of generating new numbers were regarded as objects , mathematicians used them and combines them into operations (Arzarello, Bazzini & Chiappini, 1994, p. 9). Concludiamo osservando che l'annotazione storica secondo la quale molto spesso l'aspetto operativo precede quello strutturale, assume una netta rilevanza in numerose questioni di didattica della matematica (8). A. Sfard, in una nota ricerca (1991), dopo avere sottolineato la sostanziale astrazione che caratterizza la matematica (9), sottolinea la possibilità di concepire (e di presentare) parallelamente i contenuti matematici in termini strutturali (interpretandoli, dunque, come "oggetti") ed in termini operativi (interpretation, therefore, as "processes") "Being able to see a mathematical entity as an object means being able to refer to it as a real thing, a static ... and manipulate as a whole ... Interpret the notion as a process means considering as potential rather than actual entities, which comes to light in the face of a sequence of actions. So while the structural conception is static ... itananea and overall the operation is dynamic, sequential and detailed "(Sfard, 1991: 4). The Sfard also extends this distinction to coding (and the author seems here again ideally records Leopardi mentioned above): 'verbal encodings can not be captured' at a glance 'and must be processed sequentially, so they seem more suited to present calculation procedures. Thus, the internal representation is not iconic relevant to operational thinking "(Sfard, 1991, p. 7, re: Hadamard, 1949, p. 77). Without claim to exhaust a subject very deep and sensitive, even from epistemological point of view, we can therefore conclude that the introduction of operational many fundamental concepts of mathematics (and, these, the elements of arithmetic ) is a particularly important and debated in a teaching environment.
numbers and succession: metamathematical reflections, historical and educational activities on a song Leopardi (3) 3. The intuitionistic setting: numbers and counting
We therefore refer in this paragraph, setting intuitionistic arithmetic (5) by RL Goodstein (1957). Use the following symbols: counting operator N a, b, ..., l, ... objects a & b, a & b & c, ..., L, L & a, ... sets of objects recursively define the act of counting (6): N (l) = 1 N (L & l) = N (l) +1 Let S be the operator leads to the conclusion that the successor (denoted by Sx ie the successor of x). The usual addition operation: (a, b) → a + b è definita ricorsivamente da: x+0 = x x+Sy = S(x+y) Ad esempio, per determinare 6+3 si procede nel modo seguente: 6+0 = 6 6+1 = 7 6+2 = (6+1)+1 = 7+1 = 8 6+3 = (6+2)+1 = 8+1 = 9 La moltiplicazione: (a; b) → F1(a; b) = a·b è definita ricorsivamente da: F1(x; 0) = 0
F1(x; Sy) = x+F1(x; y) È quindi possibile una definizione ricorsiva di altre operazioni. Per ogni n naturale, n≥2, definiamo ricorsivamente le operazioni: Fn(x; 0) = 1 Fn(x; Sy) = Fn–1[x; Fn(x; y)] Ad esempio, per n = 2 si ottiene la definizione ricorsiva di F2(a; b) = ab (operazione di esponenziazione) (7): F2(x; 0) = 1 F2(x; Sy) = F1[x; F2(x; y)] = x·xy Per n = 3 si ottiene la definizione ricorsiva di un'operazione non usuale, che possiamo indicare mediante la posizione F3(a; b) = ba: F3 (x, 0) = 1 F3 (x, Sy) = F2 [x, F2 (x, y)] = (yx) x It
eg F3 (x 1) = 1x = x F3 (x, 2) = 2x = x ^ x F3 (x 1) = 3x = x ^ x ^ x
numbers and succession: metamathematical reflections, historical and educational track on a leopard ( 2) 2. A look at history:
set of Peano axioms From a historical perspective, says N. Bourbaki: "Before the nineteenth century, it seems there has been no attempt to define the addition and multiplication of natural numbers if relying direct intuition; Leibniz is the only one who, true to its principles, is expressly noted that the "obvious truth" as 2 +2 = 4 are also likely demonstration if we reflect on the definitions of numbers that will appear , he did not consider it as a given the commutativity of addition and multiplication. But no further than his reflections on this regard, and in the mid-nineteenth century, no progress had yet been reached "(Bourbaki, 1963). With the work on the concept of number (1891), Giuseppe Peano (1858-1932), rielaborando alcune idee introdotte da Wilhelm Richard Dedekind (1831-1916) nello scritto Was sind und was sollen die Zahlen? (1888), propose un'introduzione assiomatica dell'aritmetica basata su tre concetti primitivi (l'unità, che in una seconda stesura fu sostituita con lo zero; il numero; il successivo) e su sei assiomi (definitivamente enunciati nel 1898 in Aritmetica, la II parte del II vol. del Formulaire de mathematiques: Peano, 1908, p. 27; Kennedy, 1983): Assioma zero. I numeri formano una classe (2). Assioma I. Lo zero è un numero. Assioma II. Se a è un numero, il suo successivo a+ è un numero. Assioma III. Se s è una classe contenente lo zero e, per ogni a, se a appartiene a s, il successivo a+ appartiene a s,; allora ogni numero naturale è in s ("principio di induzione": si tratta in effetti di uno schema di assiomi: Chang & Keisler, 1973) (3). Assioma IV. Se a e b sono due numeri e se i loro successivi a+, b+ sono uguali, allora a e b sono uguali. Assioma V. Se a è a number, its following + is not zero. The second Peano addition is based on the following two conditions, given in the original symbolism (Peano, 1908, p. 29): Addition I. If a is a number, a +0 = a. Addition II. If a and b are two numbers, a + (b +) = (a + b) +. is clear the close analogy that links the introduction of Goodstein (presented in the previous paragraph) to this one. By implication, therefore, Peano shows that if a, b are numbers, even a + b is a number (see: Peano, 1908; demonstrations sono riportate in: Carruccio, 1972). La relazione introdotta da Peano è un'applicazione: a→a+ avente per dominio l'insieme dei numeri naturali e per codominio l'insieme dei numeri naturali non nulli, e che è una biiezione. Si può inoltre dimostrare che Peano introduce nell'insieme dei numeri naturali un ordine stretto. Possiamo dunque concludere che dall'impostazione peaniana, basata sull'applicazione che ad ogni numero naturale associa il suo successivo, emerge il ruolo essenziale del concetto di successione.
http://blog.blogosfere.it/mte/mt-tb.php?tb_id=232196 For
4 friends who read my "box of things" that have been reviewed as an example "what not to do a blogger." As my reviewer is right I hope to make it known right ( and alleviate the pain), stating his conviction that I post: http://sciencebackstage.blogosfere.it/2010/07/le-buone-pratiche-del-blogging.html and adding SCIENCEBACKSTAGE ( http://sciencebackstage.blogosfere.it ) to the blog rool on the side.
Luca Pacioli After 500 Years And here's the press release:
On 21 October 2010, with the presentation of the Conference Proceedings of study 500 years after Pacioli (Sansepolcro, 22 and May 23, 2009 ) was Pacioli review of the achievements of the Project and were presented the results of two years of study and research, which involved students, teachers and students of secondary schools of the Tuscan and Umbrian Tiber Valley, with the cooperation of local authorities, and of individuals and companies operating in the Territory. The speakers were Professors James Banker, Argante Ciocci, Roberto Manescalchi, Enzo Mattesini, Pier Daniele Napolitani, Fausto Casi (Mostra A scuola di scienza e tecnica) e Giovanni Cangi (Quaderno Pacioli fra Arte e Geometria). Ha introdotto la Prof.ssa Paola Refice, Presidente della Fondazione Piero della Francesca. Ha condotto Matteo Martelli, Presidente del Centro Studi Mario Pancrazi. Nella Sala Conferenze della Fondazione Piero della Francesca, in Viale Aggiunti a Sansepolcro, il folto pubblico ha partecipato alla ricostruzione della figura intellettuale di Luca Pacioli ed ha avuto modo di constatare la validità scientifica del volume degli Atti. Dagli interventi degli studiosi e dei docenti è emerso che il volume del Centro Studi non solo rappresenta un aggiornamento importante delle conoscenze biografiche del frate del Borgo, ma raccoglie gli studi most up to date on the works of Francis, on behalf of its recognition to the language of science in late fifteenth and early sixteenth century, and illuminates the relationship between Piero and Luca Luca between Leonardo and, in the context of the statement of the mathematical sciences, Pacioli places the protagonist of the great revolution of printing with movable type. The book will be presented by the President of the Centre, November 10, 2010 in Leon in Spain, during the proceedings of the VII Encuentro Internacional AEC. For an audience of experts in accounting and business management, the development program will be shown the work of Pacioli promoted by local authorities, da Aboca Museum e dal Centro Studi “Mario Pancrazi”. Partendo dal Convegno del 1994 (Luca Pacioli e la Matematica del Rinascimento) sarà illustrata la bibliografia pacioliana fino al volume degli Atti. Un’attenzione particolare sarà rivolta alle pubblicazione delle Edizioni Aboca, dalla monografia pacioliana di Argante Ciocci (2009) all’edizione in facsimile di tre manoscritti del minorita di Sansepolcro: il De ludo scachorum (2007), il De viribus quantitatis (2009), il De divina proportione (manoscritto della Biblioteca Universitaria di Ginevra) del 2010. Il Centro Studi “Mario Pancrazi” nei prossimi mesi pubblicherà il terzo quaderno della Serie R&D diretta da Francesca Giovagnoli, the result of the seminar devoted to astronomy January 31, 2010 (Conference Room of the Diocesan Museum of Città di Castello) and will contribute to the realization, in cooperation with the Central Apennines Business - Review of Territory and boarding INPDAP "Regina Elena", the seminar entitled economy after the crisis - Labor / Business / Company (Theatre Hall boarding INPDAP, 3 and 4 December 2010). They are in the works, finally, two initiatives: the Leonardo Project and the Tiber Valley, which includes a seminar and to publish the papers by the end of 2011 and an International Meeting on June 19 in Sansepolcro that will take stock on the biography and on pacioliana 'importance Brother of mathematics in the history of accounting and business economics.
numbers and succession: metamathematical reflections, historical and educational activities on a song Leopardi (1) GIORGIO TOMASO BATHROOMS
1. Medley of thoughts, November 28, 1820 wrote the twenty-two Giacomo Leopardi: "The man without the knowledge of a speech, can not conceive the idea of \u200b\u200ba fixed number . Imagine you have thirty or forty stones, senz'avere a name to be given to each one, that is, one, two, three, to the last name, ie thirty or forty, which contiene la somma di tutte le pietre, e desta un'idea che può essere abbracciata tutta in uno stesso tempo dall'intelletto e dalla memoria, essendo complessiva ma definita ed intera. Voi nel detto caso, non mi saprete dire, né concepirete in nessun modo fra voi stesso la quantità precisa delle dette pietre; perché quando siete arrivato all'ultima, per sapere e concepire detta quantità, bisogna che l'intelletto concepisca, e la memoria abbia presenti in uno stesso momento tutti gl'individui di essa quantità, la qual cosa è impossibile all'uomo. Neanche it should help the eye, wanting to know why the number of certain objects present, and not knowing how to count, you need the same operation and simultaneous individual memory. So if you do not know except one numerical designation, and counting could not say more than one, one, one; to how much attention you would put, akin to collect progressively with the soul and memory The precise amount of these units until the last, you'd in the same case. So if I did not know that two other names etc.. Except a very small amount, like five or six, which memory and intellect can not conceive of speech, because you get to simultaneously present all it amounts to a few individuals ... Typically the idea of \u200b\u200bthe precise number, or with the help of speech or not, has never instantaneous, but composed of succession, more or less long, more or less difficult, according to the measure the amount (by Zibaldone of thoughts, 28 November 1820 : Leopardi, 1969). This track offers some reflections on the concept of natural number. First, it is clear the central importance that the author attributes to the language
(1). Early onset ("The man without the knowledge of a speech,
can not conceive the idea of \u200b\u200ba fixed number), in fact, Leopardi
explicitly refers to the essential role of the name of the individual numbers
natural: it is thanks to 'name' that we can enumerate the elements of a finite
(that we can identify, gradually, his
subsets of increasing cardinality) until you hear the whole
(to fix "an idea that can all be embraced in the same time
intellect and memory, but defined as total and complete ").
This concept is therefore closely linked to the numerical count, when
to enumerate: the role of the 'speech' is to be decisive in its
as it allows the unfolding of the proper implementation of such a measure ("And so if you do not
I knew except a single numerical designation, and counting
could not say no more than one, one, one ...»). Leopardi, towards the end of the passage quoted,
clearly recognizes that the very concept of numbers through (and beyond)
the succession of 'Name', is linked inextricably
to count ("The idea of \u200b\u200bthe exact number, or with the help of speech or not, is not
never instantaneous, but composed of succession) (Borg & Pepe, 1998).
This consideration can be modern and thorough recovery
by examining some of the settings of arithmetic, as we shall see that based on the introduction of recursive
their roots.
The Abraham Lincoln connection From site http://www.mathopenref.com/euclid.html
At age forty, Abraham Lincoln Studied Euclid for training in reasoning, and as a traveling lawyer on horseback, kept a copy of Euclid's Elements in his saddlebag. In his biography of Lincoln, his law partner Billy Herndon tells how late at night Lincoln would lie on the floor studying Euclid's geometry by lamplight. Lincoln's logical speeches and some of his phrases such as "dedicated to the proposition" in the Gettysburg address are attributed to his reading of Euclid.
Lincoln explains why he was motivated to read Euclid: "In the course of my law reading I constantly came upon the word "demonstrate". I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?
I consulted Webster's Dictionary. They told of 'certain proof,' 'proof beyond the possibility of doubt'; but I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man.
At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies."
