Ecco un estratto:
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\fancyhead[CE]{\footnotesize{\sc On the Formal Elements}}
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\author{Ernst Schr\”oder}
\title{\sc On the Formal Elements of the Absolute Algebra}
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{\sc Ernst Schr\”oder}\\
{\sc On the Formal Elements of the Absolute Algebra}\\
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{\sc davide bondoni\\
(Editor)}
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Author:\\
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Ernst Schr\”oder\\
(1842–1901)\\
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Editor:\\
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davide bondoni\\
independent scholar\\
via Bersaglio, 2\\
I-25070 — Anfo (BS)\\
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E-mail:davidebond[at]yahoo[dot]it\\
Homepage:www.davidebondoni.eu
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\noindent {\small 2010 Mathematical Subject Classification (primary; secondary): 01A55; 03-00,03G10, 06-00}\\
\noindent Key words: universal algebra, commutativity, connections, group of permutations, loop, groupoid, left/right-division, Galois, Cayley, Grassmann\\
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Front cover by\\
Lois Rottonara\\
via Planmurin, 10\\
I-39030 — La Villa (BZ)
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To my dear mom,\\
and to Francesca
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\chapter*{\sc Algebra, What Else?}
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\section{The Birth of a Masterwork}
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\footnotesize{The purely formal sciences,}\\
\footnotesize{logic and mathematics,}\\
\footnotesize{have to handle such relations,}\\
\footnotesize{which are or at least can be}\\
\footnotesize{independent from the determinate contents}\\
\footnotesize{and from the substance of the objects in issue}\footnote{Die rein formalen Wissenschaften, Logik und Mathematik, haben solche Relationen zu behandeln, welche unabh\”angig von dem bestimmten Inhalte, der Substanz der Objecte sind oder es wenigstens sein k\”onnen \cite[p.~1]{hankel}.}.
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\normalsize{In} the period between 1870 and 1874, Schr\”oder is teaching at the {\it Pro-} and {\it Real-} Gymnasium\footnote{In\label{ginnasio} order to explain to a non German reader what type of school the Gymnasium is, I quote from \cite[p.~333]{oxford2}: {\it The secondary school which prepares pupils for the} Abitur [the final exam at the Gymnasium]. {\it The Gymnasium is attended after the} Grundschule [primary school] {\it by the most accademically-inclined pupils. They spend nine years at this school, and during the last three years they have some choice as to which subjects they study}. In other words, the Gymnasium is attended by whose student willing to go at University.} in Baden-Baden, after having fought in the {\it French–Prussian War}. It is a lapse of time which Schr\”oder devotes to his favourite discipline, the {\it Algebra}. In 1873 he published the first volume of the {\it Handbook on Algebra}\footnote{\cite{schr73}.}, which remained unfinished, in 1874 a compendium of it\footnote{\cite{schr74}. Note that this differs from the preeceding, being adressed no more to {\it Teachers and Students}, but to students alone.} and the same year {\it On the Formal Elements of an Absolute Algebra}\footnote{\cite{schr74b}.}, which we translated here for the first time.\\
\indent The latter marks the closure of a phase focused on algebra and paths the way to the first logical investigations:
\begin{quotation}
It seems not impossible to bring to a completion the methods of logic in order that for any collection of premises we can derive its consequences with full completeness (\ldots)\footnote{See below, p.~\pageref{quoto1}.}.
\end{quotation}
\noindent Logic is seen as a consequence of the work in Algebra, as a tool to achieve what it seems impossible in the absolute algebra. For example, the question Schr\”oder puts forth near the end of this booklet is a sort of naive {\it decision problem}:\index{concepts}{Decision problem} given an algorithm $\xO_\xj$ and an equation $\alpha$ belonging to the set of those equations which were derived from $\xO_\xj$, is $\alpha$ a {\it consequence} of $\xO_\xj$ or does not? The problem Schr\”oder faces is such that in absolute algebra has not an answer; i.e. two arbitrary algebraic equations are {\it not} always confrontable, i.e. we are not able to say if the first implies the second or vice-versa:
\begin{quotation}
Until now we own no {\it directe} tool to determine with certainty if, given two arbitrary formulas (or set of formulas) belonging to absolute algebra, one implies the other, they are equivalent or they are mutually independent\footnote{See below, p.~\pageref{quoto2}. The italic is mine.}.
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