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\begin{document}
%\frontmatter
%\titlehead{}
\subject{Milet'te 24 Eyl\"ul 2016 g\"un\"u verilmi\c s\\
bir konu\c sma}
\title{Thales}
\subtitle{Kan\i t kavram\i n\i n \"onc\"us\"u olarak}
\author{David Pierce}
\date{%24 Eyl\"ul 2016\\
\today\ g\"un\"u d\"uzeltilmi\c s}
\publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}

\maketitle
\tableofcontents
\listoffigures
%\begin{multicols}{2}
%\tableofcontents
%\end{multicols}


\nocite{Kirk,Diels}

\chapter{Bir kan\i t}

%\setcounter{section}{-1}
\section{Herodot}

\begin{sloppypar}
T\"urkiye'ye ilk geldi\u gimde
rehberim
\textbf{Halikarnasl\i\ Herodot'tu}
\cite{Herodotus-Penguin}.
Herodot Thales'ten bir y\"uzy\i l sonra ya\c sad\i\u g\i\ halde
Herodot'tan
Thales'in ya\c sam\i\ hakk\i nda
bir \c seyler
\"o\u grenebiliriz.
\end{sloppypar}

\"Orne\u gin
\"o\u gle yeme\u gimizde bize verilmi\c s bir ka\u g\i da g\"ore
Thales, Mil\^attan \"once 625 y\i l\i nda do\u gdu.
Herhalde bu y\i l Herodot'tan elde edilmi\c s.
Herodot'a g\"ore
bu topraklarda,
Lidyal\i lar ve Medler sava\c s\i yordu, 
ve alt\i\ y\i l sonra,
bir \textbf{g\"une\c s tutulmas\i} oldu.
Sava\c sanlar korktular,
sava\c s\i\ b\i rak\i p bar\i\c st\i lar.
Herodot'a g\"ore Thales, g\"une\c s tutulmas\i n\i\
\"onceden bildirmi\c sti
\cite[I.74]{Herodotus-Loeb}.
Bug\"un bu tutulman\i n
mil\^attan \"once 585 
y\i l\i nda oldu\u gunu
biliyoruz
\cite[sayfa 15, not 3]{Heath-Aristarchus}.
Thales o y\i l 40 ya\c s\i nda olsa,
625 y\i l\i nda do\u gmu\c stur.

\section{Pisagor Teoremi}

%\begin{sloppypar}
Bildi\u giniz gibi ba\c sl\i\u g\i m,
\textbf{Kan\i t kavram\i n\i n \"onc\"us\"u olarak Thales.}
\.Ilk olarak bir kan\i t \"orne\u gi verece\u gim,
ama Thales'in kan\i t\i\ de\u gil, \"Oklid'in.
Posterde
Pisagor Teoremi i\c cin
bir kan\i t var.
(\Sekle{fig:I.47} bak\i n.)
\begin{figure}
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\end{figure}
Resimde renkli \"u\c cgenler, d\"ortgenler,
ve be\c sgenler var,
ama bunlar 3 kareyi
ve bir b\"uy\"uk dik \"u\c cgeni olu\c sturur.
Kareler, 
b\"uy\"uk \"u\c cgenin kenarlar\i na
konulmu\c stur.
En b\"uy\"uk kare,
\"u\c c\-genin dik a\c c\i s\i n\i n kar\c s\i s\i nda,
ve iki dikd\"ortgene b\"ol\"un\"ur.
Soldaki dikd\"ortgen
4 par\c caya b\"ol\"un\"ur,
ve soldaki kare de
4 par\c caya b\"ol\"un\"ur,
ve bu par\c calar\i n her biri birine e\c sittir.
%benzer ve ayn\i\ boylu \c sekillere b\"ol\"un\"ur.
Bu durumda,
soldaki kare ve dikd\"ortgen birbirine e\c sittir.
Ayn\i\ nedenle
sa\u g\-daki kare ve dikd\"ortgen e\c sittir.
Sonu\c c olarak b\"uy\"uk kare,
k\"u\c c\"uk karelerin toplam\i na e\c sittir.
B\"oylece Pisagor Teoremi kan\i tlan\-m\i\c st\i r.
%\end{sloppypar}

\section{E\c sitlik}

Buradaki e\c sitlik, ayn\i l\i k de\u gildir.
Resimdeki gibi
bir kare ve bir dikd\"ortgen, 
birbiriyle ayn\i\ olmadan 
birbirine e\c sit olabilir.
Bu durumda \c sekillerin \emph{alanlar\i} ayn\i\ say\i labilir;
ama alan bir say\i\ de\u gil, daha soyut bir \c seydir.

Bug\"unk\"u $=$ ``e\c sittir'' i\c saretinde,
iki farkl\i\ ama e\c sit (ve paralel) do\u gru vard\i r
\cite{Recorde}.
\emph{\.Insan haklar\i\ evrensel beyannamesi}'ne g\"ore
\begin{quote}
\textbf{Kanun \"on\"unde herkes e\c sittir}
ve farks\i z olarak kanunun e\c sit korumas\i ndan istifade hakk\i n\i\ haizdir.
\end{quote}
Dilin eski oldu\u gunu biliyorum,
ama Birle\c smi\c s Milletlerin web sayfas\i ndan ald\i m.
Hepimiz birbirimizden farkl\i y\i z,
ama bir \c sekilde birbirimize e\c sitiz.

\section{Ger\c cek kan\i t}

\begin{sloppypar}
Pisagor Teoreminin ger\c cek kan\i t\i\
resimde de\u gil, akl\i m\i zdad\i r.
Resmin kendisini sanat olarak g\"orebiliriz.
Bu durumda,
birbirimizle ayn\i\ fikirde olmayabiliriz.
Bir ki\c si resmi be\u gene\-bilir,
ba\c ska bir ki\c si be\u genmeyebilir.
Ama bir kan\i t\i n do\u grulu\u gu konusunda
fikrimiz varsa,
hepimiz ayn\i\ fikirde olmal\i y\i z.
Bir teorem anlamak istedi\u gimiz durumda,
e\u ger teoremin bir kan\i t\i n\i\ size \"onersem,
ve bunu kabul etmezseniz,
ben tekrar d\"u\c s\"unmeliyim.
\end{sloppypar}

\chapter{\"Oklid}

\section{\.Iki \"onerme}

\begin{sloppypar}
Bug\"une kalan en eski matematiksel kan\i tlar\i n \c co\u gu,
\"Oklid'in mil\^attan \"once 300 y\i l civar\i nda yazd\i\u g\i\
\emph{\"O\u geler} adl\i\ eserin 13 kitab\i ndad\i r.
Bug\"un,
%Mimar Sinan G\"uzel Sanatlar \"Universitesi'ndeki
\c cal\i\c st\i\u g\i m matematik b\"ol\"um\"une giren \"o\u grenciler,
\emph{\"O\u geler}'in ilk kitab\i n\i n \"onermelerini okuyup 
kan\i tlar\i n\i\ birbirine anlat\i rlar
\cite{Oklid-2014-T}.
Bu \"onermelerin en \"onemlilerinden biri,
Pisagor Teoremidir.
Di\u gerine g\"ore,
kenarlar\i\ do\u gru olan her tarlan\i n alan\i\ \"ol\c c\"ulebilir.
Dedi\u gimiz gibi \"Oklid i\c cin bu alan bir say\i\ de\u gil;
burada bir dikd\"ortgendir.
K\i saca \textbf{her \c cokgen bir dikd\"ortgene e\c sittir.}
(\Sekle{fig:4-gen} bak\i n.)
Ayr\i ca bu dikd\"ortgen
verilmi\c s bir do\u gru par\c cas\i nda in\c sa edilebilir.
\end{sloppypar}
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\caption{Her \c cokgen bir dikd\"ortgene e\c sittir}\label{fig:4-gen}
\end{figure}

\section{M\i s\i rl\i lar}

\begin{sloppypar}
\"Oklid'den \c cok y\"uzy\i l \"once M\i s\i rl\i lar,
vergiler i\c cin tarlalar\i n\i\ \"ol\c c\"uyorlard\i.
Herodot'a g\"ore Yunanlar,
geometriyi M\i s\i rl\i lar\-dan \"o\u grenmi\c sler
\cite[II.109]{Herodotus-Loeb}.
Asl\i nda
Yunanca'da \textbf{geometri, yeri \"ol\c cmek demektir.}
Ama M\i s\i rl\i lar\i n kulland\i\u g\i\ kurala g\"ore 
d\"ort kenarl\i\ bir tarlan\i n alan\i,
kar\c s\i t kenarlar\i n ortalamalar\i n\i n \c carp\i m\i yd\i\
\cite[sayfalar 232, 281]{MR2000c:01004}.
Yani
(\Sekilde{fig:4-gen}ki gibi)
kenarlar
s\i ras\i yla $a$, $b$, $c$, ve $d$ ise,
o zaman d\"ortgenin alan\i,
\begin{equation*}
\frac{a+c}2\cdot\frac{b+d}2.
\end{equation*}
%$a$ art\i\ $c$ b\"ol\"u $2$ \c carp\i\ $b$ art\i\ $d$ b\"ol\"u $2$.
Bu kural, tamamen do\u gru de\u gil.
Ama kural kanundu, ve M\i s\i rl\i lar kanuna uydular.
\end{sloppypar}

\section{Matematik}

\begin{sloppypar}
Bug\"unk\"u matematik \"o\u grencileri ayn\i\ \c sekilde d\"u\c s\"unebilirler.
Hoca bir kural verir, 
ve \"o\u grenciler kanun olarak kurala sayg\i\ g\"osterir.
Bu matematik de\u gil.
K\i saca 
\begin{compactenum}[1)]
\item
\"ozerk ve
\item
evrenseldir.
\end{compactenum}
A\c c\i klamak gerekirse
\begin{compactenum}[1)]
\item
bir \"onermenin do\u grulu\u gunu kendim i\c cin bilmeden
ve 
\item
\"onermenin neden do\u gru oldu\u gunu a\c c\i klayamadan
\end{compactenum}
\"onermenin matematik oldu\u gunu s\"oyleyemem.
Benzer \c sekilde
\begin{compactenum}[1)]
\item
Bir \"onermenin do\u grulu\u gunu
kabul etmenizi emredemem;
\item
\"onermenin do\u grulu\u gu,
kavga etmekten de\u gil,
konu\c smak ve d\"u\c s\"unmekten gelir.
Dedi\u gimiz gibi
ilgilenen matematik\c ciler
sonunda ayn\i\ fikirde olmal\i lar.
\end{compactenum}
\end{sloppypar}

\chapter{Thales}

\section{D\"ort teorem}

G\"ord\"u\u g\"um\"uz gibi Thales,
\"Oklid'den
a\c sa\u g\i\ yukar\i\
\"u\c c y\"uz y\i l \"once ya\c sam\i\c st\i r.
Herodot de\u gil,
ama ba\c ska eski yazarlara g\"ore Thales,
\"Oklid'in \emph{\"O\u geler}'inde bulunan birka\c c \"onermeyi biliyordu.
Ay\c se zaten biri hakk\i nda konu\c stu.
Ben Thales'in d\"ort \"onermesi daha hakk\i nda konu\c smak isterim.
\begin{description}\sloppy
\item[1.si,]
iki do\u gru kesi\c sti\u ginde ters a\c c\i lar
birbirine e\c sittir
\cite[299.4]{MR1200456}.
\item[2.si,]
ikizkenar bir \"u\c cgende
tabandaki a\c c\i lar birbirine e\c sittir
\cite[250.20]{MR1200456}.
\item[3.s\"u,]
bir daire, \c cap\i, taraf\i ndan
iki e\c sit par\c caya b\"ol\"un\"ur
\cite[157.11]{MR1200456}.
\item[4.s\"u,]
Yar\i\c cemberdeki a\c c\i\ diktir
\cite[I.24--5]{Diogenes-I}.
\end{description}
(\Sekle{fig:4-thm} bak\i n.)
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\psline(-2,0)(2,0)
\psline(2;60)(2;240)
\pspolygon(2;0)(2;60)(2;180)(2;240)
\end{pspicture}}
\hfill\mbox{}
\caption{Thales'in teoremlerinden d\"ord\"u}\label{fig:4-thm}
\end{figure}

%\section{Ters a\c c\i lar}

\begin{enumerate}[(a)]
\sloppy
\item
Neden ters a\c c\i lar birbirine e\c sittir?
Bir a\c c\i y\i\ \c cevirirsek,
kendisiyle \c cak\i\c sacak.
Yani
\begin{equation*}
\angle ABC=\angle CBA.
\end{equation*}
\.Iki do\u gru bu \c sekilde kesi\c sirse,
ve solumdaki a\c c\i y\i\ \c cevirirsem,
alt a\c c\i\
\"ust a\c c\i yla \c cak\i\c sacak.
Bu nedenle \"ust ve alt a\c c\i lar
e\c sittir.
K\i saca \emph{simetri} sayesinde teorem do\u grudur.

Thales'in kan\i t\i n\i\ bilmiyoruz.
Belki bir kan\i t vermedi.
Ama simetriyle Thales'in di\u ger teoremlerini
kan\i tlayabiliriz.
\item
\.Ikizkenar \"u\c cgeni bu \c sekilde \c cevirirsek,
tabandaki a\c c\i lar birbiriyle \c cak\i\c s\i r.
\item
Daireyi
\c cap\i n\i n etraf\i nda \c cevirirsek,
kendisiyle \c cak\i\c s\i r.
\item
A\c c\i\ i\c ceren yar\i\c cemberi merkezinden \c cevirirsek,
d\"ortgen i\c ceren bir \c cember elde ederiz,
ve bu d\"ortgenin t\"um a\c c\i lar\i\ birbirine e\c sittir,
dolay\i s\i yla diktir.
\end{enumerate}

\section{Deneme}

Baz\i\ \"onermeler,
deneme ile kan\i tlanabilir.
\"Orne\u gin
denememe g\"ore,
ba\c s a\u gr\i m varsa,
aspirin a\u gr\i y\i\ keser.
Bu durumde denemem, aspirinin etkili oldu\u gunun bir kan\i t\i d\i r.
Ama bu kan\i t,
matematiksel bir kan\i t de\u gildir.

Ters a\c c\i lar\i n her zaman e\c sit olmas\i,
birka\c c tane \"ornekten elde edilemez.
G\"orebildi\u gimiz bir prensipten elde edilmeli.
Bu prensibe ``simetri'' diyorum.

\section{D\"unyan\i n ilkesi}

Aristo'ya g\"ore Thales,
t\"um d\"unyan\i n arkas\i nda,
t\"um d\"unyan\i n alt\i nda,
tek bir taban\i n, tek bir ilkenin oldu\u gunu d\"u\c s\"unm\"u\c s.
Bu ilke i\c cin Thales'in verdi\u gi isim suydu
\cite[I.3]{Aristotle-XVII}.
Bildi\u gimiz gibi su olmadan hi\c c bir \c sey ya\c sayam\i yor
\cite[sayfa 31]{Collingwood-IN}.

Thales'in g\"ord\"u\u g\"um\"uz geometri \"onermelerinin arkas\i nda
tek bir ilke veya prensip vard\i r.
Buna ``simetri'' dedim.
B\"oyle prensipler bulmak, 
bir matematik\c cinin i\c sidir.
Bu \c sekilde Thales,
ilk matematik\c cilerden biri say\i labilir.

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\end{document}
