% (find-LATEX "2020badiou-low.tex")
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% «.title»	(to "title")
% «.2020aug04»	(to "2020aug04")

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\catcode`\^^J=10
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\def\Id{\mathbf{Id}}


% «title»  (to ".title")

{\setlength{\parindent}{0em}
\footnotesize

Notes on Alain Badiou's

``Logics of Worlds: Being and Event, 2'' (2006, translation 2009):

\url{https://www.bloomsbury.com/uk/logics-of-worlds-9781441172969/}

\ssk

These notes are at:

\url{http://angg.twu.net/LATEX/2020badiou-low.pdf}

\ssk

See:

\url{http://angg.twu.net/LATEX/2020favorite-conventions.pdf}

\url{http://angg.twu.net/math-b.html\#favorite-conventions}

I wrote these notes mostly to test if the conventions above
are good enough.


}



% (find-books "__cats/__cats.el" "badiou-low")
% (find-badioulowpage (+ 18 153) "II.3. Algebra of the Transcendental")
% (find-badioulowpage (+ 18 155) "II.3.2. Function of Appearing and Formal Definition")
% (find-badioulowtext (+ 18 155)      "2. Function of Appearing and Formal Definition")
% (find-badioulowpage (+ 18 156)         "The minimal requirement")
% (find-badioulowtext (+ 18 156)         "The minimal requirement")
% (find-badioulowpage (+ 18 157) "II.3.3. Equivalence-Structure and Order-Structure")
% (find-badioulowtext (+ 18 157)         "Equivalence-Structure and Order-Structure")

% (find-badioulowpage (+ 18 191) "Book III Greater Logic, 2. The Object")
% (find-badioulowpage (+ 18 193) "Introduction")
% (find-badioulowpage (+ 18 199) "Section 1 For a New Thinking of the Object")
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\newpage

Book II: Greater Logic

II.3. Algebra of the Transcendental

\subsection*{II.3.2. Function of Appearing and Formal Definition of the Transcendental}

(Page 157):

% (find-badioulowpage (+ 18 157) "The idea--a very simple one--")
% (find-badioulowtext (+ 18 157) "The idea--a very simple one--")

The idea --- a very simple one --- is that in every world, given two
beings $α$ and $β$ which are there, there exists a value $p$ of
$\Id(α,β)$. To say that $\Id(α,β)=p$ means that, with regard to their
appearing in that world, the beings $α$ and $β$ --- which remain
perfectly and univocally determined in their multiple composition ---
are identical `to the $p$ degree', or are $p$-identical. The essential
requirement then is that the degrees $p$ are held in an
order-structure, so that for instance it can make sense to say that in
a fixed referential world, $α$ is more identical to $β$ than to $γ$.
In formal terms, if $\Id(α,β) = p$ and $\Id(α,γ) = q$, this means that
$p > q$.

\newpage

% (find-latexgimp-links "GIMP/badiou-low-p157")
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% (find-latexgimp-links "GIMP/badiou-low-p157-2")
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% (ph1p 5 "positional")
% (ph1    "positional")
% (favp 50 "functors-as-objects")
% (fav     "functors-as-objects")

The positional notations are explained in \cite[Section 1]{PH1} and
\cite[Section 7.12]{FavC}.


\newpage

% «2020aug04»  (to ".2020aug04")

Oi Caron! Tou tentando traduzir algumas definições da seção ``II.3.2.
Function of Appearing and Formal Definition of the Transcendental'' do
LoW pra uma terminologia mais padrão...

Eu deixei a câmera do celular aberta o tempo todo? Caramba...

Vou escrever umas duvidas aqui, aé quando você tiver tempo você ou me
responde ou me diz pra onde eu devo mandar...

Os ``degress of identity'' vão ser os elementos da álgebra de Heyting
dos valores de verdade do topos

Num dos exemplos que eu discuti com você e com o Gabriel a gente
começava com o ``house-shaped DAG'' $H$ que aparece aqui na pagina 27,

\url{http://angg.twu.net/LATEX/2017planar-has-1.pdf\#page=27}

E aí quando a gente montava o topos $\Set^H$ esse topos tinha 10
valores de verdade - a figura no topo da página 27.

Seja 1 o objeto terminal do topos $\Set^H$. Os valores de verdade
desse topos podem tanto ser vistos como os subobjetos desse 1 - lembra
que a gente pode usar a notacao $\Sub(A)$ pra falar do conjunto dos
subobjetos de um objeto $A$

quanto podem ser vistos como os morfismos do objeto 1 pro objeto
$\Omega$, onde $\Omega$ é o classificador.

Eu acho o $\Sub(1)$ mais fácil de visualizar.

Se a gente tem um objeto $A$ num topos os pontos de $A$ são os
morfismos do objeto 1 pro objeto $A$

Eu tou com a impressão de que quando o Badiou define $\Id(\alpha,
\beta)$ esses $\alpha$ e $\beta$ (que na terminologia dele são
``multiples'', se não me engano) são uma coisa um pouco mais
complicada que "pontos" do topos...

...porque tanto $\alpha$ quando $\beta$ podem ter um ``extent'' que é
um subobjeto do 1 que não é o próprio 1.

Na pagina 246 do PDF do LoW que eu tenho o Badiou define $𝐛E x :=
\Id(x,x)$

e um ``multiple'' $\alpha$ não é um morfismo de 1 para $A$, e sim um
morfismo de $𝐛E\alpha$ para $A$. Nao lembro a terminologia usual em
topos theory pra isso... acho que a gente chama de ``partial points''
ao inves de ``points''.

Se for isso eu posso fazer uns desenhos e mandar pro pessoal do seminário

Na verdade eu já tenho vários desses desenhos, é só reciclá-los...



\printbibliography

\end{document}

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