\documentclass[% version=last,% a5paper,% 12pt,% %headings=small,% bibliography=totoc,% index=totoc,% twoside,% open=any,% %parskip=half,% this option takes 2.5% more space than parskip draft=true,% titlepage=true,% DIV=12,% headinclude=false,% pagesize]% {scrbook} %{scrartcl} %\usepackage[notref,notcite]{showkeys} \usepackage{scrpage2} \pagestyle{scrheadings} \clearscrheadings \ofoot{\pagemark} \refoot{\leftmark} \lofoot{\rightmark} \usepackage{moredefs,lips} \usepackage{pstricks} \usepackage{subfig} \usepackage[neverdecrease]{paralist} \usepackage{hfoldsty,verbatim,verse} \usepackage{textcomp} % for \textlangle, \textrangle \newcommand{\insertion}[1]{\textlangle#1\textrangle} \usepackage{relsize} \usepackage[polutonikogreek,turkish,english]{babel} \usepackage{gfsporson} \newcommand{\gr}[1]{\foreignlanguage{polutonikogreek}% {\relscale{0.9}\textporson{#1}}} \newcommand{\grm}[1]{\ensuremath{\text{\gr{#1}}}} \renewenvironment{quote}{% \relscale{.90} \begin{list}{}{% \setlength{\leftmargin}{0.05\textwidth} %\setlength{\topsep}{0pt} \setlength{\topsep}{0.25\baselineskip} \setlength{\rightmargin}{\leftmargin} \setlength{\parsep}{\parskip} } \item\relax} {\end{list}} \renewenvironment{quotation}{% \begin{list}{}{% \relscale{.90} \setlength{\leftmargin}{0.05\textwidth} %\setlength{\topsep}{0\baselineskip} \setlength{\topsep}{0.25\baselineskip} \setlength{\rightmargin}{\leftmargin} \setlength{\listparindent}{1em} \setlength{\itemindent}{\listparindent} \setlength{\parsep}{\parskip} } \item\relax} {\end{list}} \newenvironment{mycomment}{% \smaller \begin{list}{}{% \setlength{\leftmargin}{0.0\textwidth} %\setlength{\topsep}{0pt} \setlength{\topsep}{0.0\baselineskip} \setlength{\rightmargin}{\leftmargin} \setlength{\parsep}{\parskip} } \item\relax} {\end{list}} %\newcommand{\mycomm}[1]{\textsmaller{#1}\mbox{}\par} \newcommand{\tur}[1]{\foreignlanguage{turkish}{\textsf{#1}}} \newcommand{\Kirketal}{Kirk \emph{\&\ al.}} \newcommand{\bce}{\textsc{b.c.e.}} \newcommand{\ce}{\textsc{c.e.}} \usepackage[all]{xy} \usepackage{amsmath,amsthm,amssymb,url,upgreek} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem*{porism}{Porism} \begin{document} %\frontmatter \title{Thales of Miletus\\ Sources and Interpretations} \subtitle{\slshape Miletli Thales\\ Kaynaklar ve Yorumlar} \author{David Pierce} \date{September 27, 2016} \publishers{Matematics Department\\ Mimar Sinan Fine Arts University\\ Istanbul\\ %\url{dpierce@msgsu.edu.tr}\\ \url{http://mat.msgsu.edu.tr/~dpierce/}} \lowertitleback{This is a collection of what I have been able to find or figure out about Thales of Miletus. It may be useful for anybody interested in Thales. I focus directly on the ancient sources that we have. \P\ I began collecting these notes in preparation to give one of several 20-minute talks at the Thales Meeting (\tur{Thales Bulu\c smas\i}) at the ruins of Miletus, now Milet, September 24, 2016. Talks at the meeting were in Turkish; the audience, members of the general population. I chose for my title ``Thales as the originator of the concept of proof'' (\tur{Kan\i t kavram\i n\i n \"onc\"us\"u olarak Thales}). \P\ The Thales Meeting was arranged by the office of the mayor of Didim. Part of Ayd\i n province, the district of Didim encompasses the ancient cities of Priene and Miletus, along with the temple of Didyma, which was linked to Miletus. Herodotus refers to Didyma under the name of the family of priests there, the Branchidae. \P\ One can visit all three of Priene, Didyma, and Miletus in a day. I did this in 2008, while teaching at the Nesin Mathematics Village in \c Sirince, in the district of Sel\c cuk, which contains also the ruins of Ephesus, home town of Heraclitus. My excellent guide was George Bean, \emph{Aegean Turkey} \cite{Bean}. %\P\ %The program for the Thales festival is in Appendix \ref{app:prog}, %page \pageref{app:prog}. } \maketitle \setcounter{tocdepth}3 \tableofcontents \listoffigures %\chapter{Introduction} \chapter{Sources} \section{A legend from Diogenes Laertius} In the \emph{Lives of Eminent Philosophers,} Diogenes Laertius (3rd century \ce) devotes to Thales the second chapter of Book I, the chapter comprising \P\P 22--44 of the book. In \P 24, Diogenes mentions Pamphila as attributing one theorem to Thales. I shall make use of Diogenes only for this attribution (see page \pageref{Pamphila} below) and for the story \cite[I.27--9]{Diogenes-I} to be quoted presently. I quote the story in part as an example of how Diogenes is not a critical historian, and because the story mentions all three of the ancient sites in Didim named above. The Wise Men mentioned at the beginning of the story are listed in \P13 as \begin{inparaenum}[(1)] \item Thales, \item Solon, \item Periander, \item Cleobulus, \item Chilon, \item Bias, and \item Pittacus; \end{inparaenum} but variations are given at \P\P40--2. \begin{quote} The well-known story of the tripod found by the fishermen and sent by the people of Miletus to all the Wise Men in succession runs as follows. Certain Ionian youths having purchased of the Milesian fishermen their catch of fish, a dispute arose over the tripod which had formed part of the catch. Finally the Milesians referred the question to Delphi, and the god gave an oracle in this form: \begin{quote} \settowidth{\versewidth}{Who shall possess the tripod? Thus replies} \begin{verse}[\versewidth] Who shall possess the tripod? Thus replies\\ Apollo: ``Whosoever is most wise.'' \end{verse} \end{quote} Accordingly they give it to Thales, and he to another, and so on till it comes to Solon, who, with the remark that the god was the most wise, sent it off to Delphi. Callimachus in his \emph{Iambics} has a different version of the story, which he took from Maeandrius of Miletus. It is that Bathycles, an Arcadian, left at his death a bowl with the solemn injunction that it ``should be given to him who had done most good by his wisdom.'' So it was given to Thales, went the round of all the sages, and came back to Thales again. And he sent it to Apollo at Didyma, with this dedication, according to Callimachus: \begin{quote} \settowidth{\versewidth}{Thales, of Greeks adjudged most wise,} \begin{verse}[\versewidth] Lord of the folk of Neleus' line,\\ Thales, of Greeks adjudged most wise,\\ Brings to thy Didymaean shrine\\ His offering, a twice-won prize. \end{verse} \end{quote} But the prose inscription is: \begin{quote} Thales the Milesian, son of Examyas [dedicates this] to Delphinian Apollo after twice winning the prize from all the Greeks. \end{quote} The bowl was carried from place to place by the son of Bathycles, whose name was Thyrion, so it is stated by Eleusis in his work \emph{On Achilles,} and Alexo the Myndian in the ninth book of his Legends. \end{quote} Diogenes writes in \P40, after describing the death of Thales, \begin{quote}\sloppy To him belongs the proverb ``Know thyself'' (\gr{Gn~wji saut'on}) which Antisthenes in his \emph{Successions of Philosophers} attributes to Phemono\"e, though admitting that it was appropriated by Chilon. \end{quote} There is little reason to lend credence to any of this, except perhaps insofar as it reflects what people really did believe about Thales. It may however be of interest to note also from \P25: \begin{quote} Thales is also credited with having given excellent advice on political matters. For instance, when Croesus sent to Miletus offering terms of alliance, he frustrated the plan; and this proved the salvation of the city when Cyrus obtained the victory. Heraclides makes Thales himself say [in a dialogue] that he had always lived in solitude as a private individual and kept aloof from State affairs. \end{quote} \section{Kirk, Raven, and Schofield} Thales wrote no book, except possibly one called \emph{Nautical Star-guide,} and even the authorship of this was disputed in ancient times. My source here is Chapter II, ``Thales of Miletus,'' of Kirk, Raven, and Schofield, \emph{The Presocratic Philosophers} \cite[pp.\ 76--99]{Kirk}. The whole book is based on 616 quotations of ancient authors. Given in the original language (Greek or Latin) and in English translation, the quotations are numbered in boldface, serially throughout (which is why I can say how many there are). Some of the quotations are given as part of the commentary by \Kirketal\ on the main quotations; some quotations are given in footnotes. Chapter I of the book is ``The Forerunners of Philosophical Cosmogony.'' The next five chapters concern ``The Ionian Thinkers,'' namely \begin{inparaenum}[(1)] \item Thales, \item Anaximander, and \item Anaximenes of Miletus, \item Xenophanes of Colophon, and \item Heraclitus of Ephesus. \end{inparaenum} (Colophon and Ephesus are now in \.Izmir province.) The chief sources on Thales are Herodotus, Aristotle, and Proclus (if he be considered ancient; he lived in the fifth century, after the founding of Constantinople). There are also passages of Diogenes Laertius and a few other authors, notably Plato. A summary of the Thales chapter of \Kirketal\ may be in order; I give it here by section (with abbreviated title) and number of quotation. Express quotations are from \Kirketal\ themselves or their translations, unless another reference is given. The parenthetical citations are those not the main ones. Citations \textbf{62--68} are \textbf{63--69} in the first edition \cite{Kirk-1}. Missing from the second edition is \textbf{70} in the first: \begin{quote} Plutarch \emph{de Is.\ et Osir.}\ 34, 364\textsc d. ``They think that Homer also, like Thales, made water principle and birth of all things through learning from the Egyptians'' \end{quote} Then citations \textbf{69--93} are \textbf{71--95} in the first edition. If \Kirketal\ give an explicit reference to one of the Diels--Kranz (DK) fragments listed in the next section, I give the reference too. \Kirketal\ have additional references to Diels and Kranz in their commentary. \begin{description} \item[Nationality.] Thales is said to be Phoenician, but was probably ``as Greek as most Milesians'' \begin{compactdesc} \item[62] Diogenes \textsc i, 22 (DK 11\textsc a1 \emph{init.}) \item[63] Herodotus \textsc i, 170 (from \textbf{65}) \item[64] (Herodotus \textsc i, 146) \end{compactdesc} \item[Activities.] See \S\ref{sect:Herodotus}, page \pageref{sect:Herodotus} \begin{compactdesc} \item[65] Herodotus \textsc i, 170. The bouleuterion at Teos \item[66] Herodotus \textsc i, 75. The crossing of the Halys \end{compactdesc} \item[Egypt.] He is said to have visited, and this is probably true \begin{compactdesc} \item[67] Aetius \textsc i, 3, 1. ``Thales\lips having practiced philosophy in Egypt came to Miletus when he was older'' \item[68] Proclus \emph{in Euclidem} p.\ 65 Friedl.\ (from Eudemus) (DK 11\textsc a11) \item[69] (Herodotus \textsc{ii}, 109, on the Egyptian origin of geometry \emph{qu\^a} surveying) \item[70] Herodotus \textsc{ii}, 20, on the Nile flooding; see \S\ref{H:170}, page \pageref{H:170} \item[71] Aetius \textsc{iv}, 1, 1. ``Thales thinks that the Etesian winds\lips,'' as in \textbf{70} \end{compactdesc} \item[Typical philosopher.] ``Neither of these stories is likely to be strictly historical'' \begin{compactdesc} \item[72] Plato, \emph{Theaetetus} 174\textsc a. He was mocked by a Thracian servant girl for falling into a well while looking at the stars \item[73] Aristotle, \emph{Politics} \gr A11, 1259a9. Mocked for being poor, he studied the heavens, predicted a bumper crop of olives, and rented all the olive presses, thus making a killing at harvest time. (Diogenes tells the story briefly at I.26, attributing it to Hieronymos of Rhodes) \end{compactdesc} \item[Astronomy.] See \S\ref{sect:Herodotus} \begin{compactdesc} \item[74] Herodotus \textsc i, 74. Thales predicted the solar eclipse of 585 \item[75] Diogenes \textsc i, 23. ``He seems by some accounts to have been the first to study astronomy, the first to predict eclipses of the sun and to fix the solstices; so Eudemus in his \emph{History of Astronomy}'' \cite{Diogenes-I} \item[76] Dercyllides \emph{ap.}\ Theon Smyrn.\ p.\ 198, 14 Hiller. ``Eudemus relates in the \emph{Astronomy} that Thales \insertion{first discovered} the eclipse of the sun and the variable period of its solstices'' \item[77] (Herodotus \textsc i, 29. Greek sages, including Solon, visited Sardis under Croesus) \item[78]\sloppy Callimachus \emph{Iambus} \textsc i, 52, fr.\ 191 Pfeiffer (DK 11\textsc a3a). Thales ``measured out the little stars of the Wain, by which the Phoenicians sail'' \end{compactdesc} \item[Mathematics.]\mbox{} \begin{compactdesc} \item[79] Diogenes \textsc i, 27. ``Hieronymus says that he [Thales] actually measured the pyramids by their shadow, having observed the time when our own shadow is equal to our height'' \item[80] Proclus \emph{in Euclidem} p.\ 352 Friedl.\ (DK 11\textsc a20). See \S\ref{sect:Proclus}, page \pageref{sect:Proclus} \end{compactdesc} \item[Writings.] See above \begin{compactdesc} \item[81] Simplicius \emph{Phys.}\ p.\ 23, 29 Diels \item[82] Diogenes \textsc i, 23 \item[83] Suda s.v.\ (from Hesychius) (DK 11\textsc a2) \end{compactdesc} \item[Cosmology.] See below (mainly \S\ref{sect:Aristo}, p.\ \pageref{sect:Aristo}). \begin{enumerate}[(i)] \item The earth floats on water, the source of all things. \begin{compactdesc} \item[84] Aristotle, \emph{De Caelo} \gr B13, 294a28. The earth rests on water \item[85] Aristotle, \emph{Met.}\ \gr A3, 983b6. The first principle is water \item[86] (Aristotle, \emph{De Anima} \gr A2, 405b1. Hippo said the soul was water) \item[87] (Heraclitus Homericus \emph{Quaest.\ Hom.}\ 22. ``\lips Thales declared that water, of the four elements, was the most active, as it were, as a cause'') \item[88] \sloppy Seneca \emph{Qu.\ Nat.}\ \textsc{iii}, 14 (presumably from Theophrastus, through a Posidonian source). ``For he [Thales] said that the world is held up by water and rides like a ship, and when it is said to `quake' it is actually rocking because of the water's movement'' \end{compactdesc} \item The inanimate can be alive; the world is full of gods \begin{compactdesc} \item[89] Aristotle, \emph{de an.}\ \gr A2, 405a19. The magnet has a soul \item[90] Diogenes \textsc i, 24 (similarly; see p.\ \pageref{90}) \item[91] Aristotle, \emph{de an.}\ \gr A5, 411a7: all things are full of gods \item[92] (Plato, \emph{Laws} 10, 899\textsc a. ``Is there anyone who will accept this and maintain that all things are \emph{not} full of gods?'') \item[93] (Aetius \textsc i, 7, 11. ``Thales said that the mind of the world is god, and that the sum of things is besouled, and full of daimons; right through the elemental moisture there penetrates a divine power that moves it'') \end{compactdesc} \end{enumerate} \end{description} \section{Diels and Kranz} Diels's \emph{Fragmente der Vorsokratiker} \cite{Diels} is apparently a comprehensive collection of everything said by or about everybody involved with Greek philosophy before Socrates. Here I give the citations about Thales, the transcription being mostly by cutting and pasting. As far as I can tell, all main quotations of \Kirketal\ appear here, except \textbf{67} and \textbf{71} of Aetius; and none of the Trying to find the quotations here caused me to realize, for example, that Diels's 22 really had \emph{two} quotations from \emph{De Anima.} There could be such oversights still undetected; I cannot always understand Diels's notation. \centerline{\gr A. LEBEN UND LEHRE} \begin{compactdesc} \item[1]\sloppy \textsc{Diogenes Laertius} \textsc i 22--44. The whole chapter on Thales. \hfill KRS \textbf{62, 75, 79, 82, 90} \item[2] \textsc{Suidas} [\gr Z. 25--30 aus Hesychios Onomatologos\lips]\\ \mbox{} \hfill KRS \textbf{83} \item[3] \textsc{Schol.\ Platonis} in remp.\ 600\textsc a [aus Hesych] \item[3a] \textsc{Callimach.}\ Iamb.\ [fr.\ 94\lips]. \hfill KRS \textbf{78} \item[4] \textsc{Herodot.}\ \textsc i 170. \hfill KRS \textbf{63, 65} \item[5] --- \textsc i 74. \hfill KRS \textbf{74} \item[6] \textsc{Herod.}\ \textsc i 75. \hfill KRS \textbf{66} \item[7] \textsc{Euseb.}\ Chron. \item[8] \gr{>Eklog`h Aristot'elhs d`e ka`i ut`on ka`i to~is >y'uqois metadid'onai yuq~hs, tekmair'onenon >ek t~hs l'ijou t~hs magn'htidos ka`i to~u >hl'ektrou.} // Aristotle and Hippias affirm that, arguing from the magnet and from amber, he attributed a soul or life even to inanimate objects. \end{quote} \item Diogenes \textsc i, 27 \cite{Diogenes-I}: \begin{quote} \gr{>Arq`hn d`e t~wn p'antwn <'udwr 'emyuqon ka`i daim'onwn pl'hrh.} // His doctrine was that water is the universal primary substance, and that the world is animate and full of divinities. \end{quote} \item Aristotle. KRS \textbf{89} \item Aristotle. KRS \textbf{84} \item Diogenes \cite[\P35, pp.\ 36--7]{Diogenes-I}. (See also page \pageref{apophthegms2} below.) \begin{quote} Here too are certain current apophthegms\label{apophthegms} (\gr{>apo\-fj'egmata}) assigned to him: \begin{verse} Of all things that are, the most ancient is God, for he is uncreated.\\ \gr{presb'utaton t~wn >'ontwn je'os; >ag'enhton g'ar.}\\ The most beautiful is the universe, for it is God's workmanship.\\ \gr{k'alliston k'osmos; po'ihma g`ar jeo~u.} \end{verse} \end{quote} \item An extrapolation: \begin{quote}\sloppy That the earth `grazes' on water is not a doctrine anywhere expressed in the fragments of Thales or ascribed to him by any ancient writer, but I am not alone in thinking it implied in the recorded fragments and their context. `Le monde des choses est donc au milieu de l'eau et \emph{s'en nourrit'} (A. Rey, \emph{La jeunesse de la Science grecque,} Paris, 1933, p.\ 40: my italics). \end{quote} \end{compactitem} \end{compactdesc} \section{Herodotus}\label{sect:Herodotus} The quotations by \Kirketal\ of Herodotus that actually mention Thales are from Book I, chapters 74, 75, and 170. These are the only references to Thales in the index of Strassler's edition of Herodotus \cite{Strassler}, and they tell us the following. \subsection[Solar eclipse]{Solar eclipse (74)} Thales predicted the year of a solar eclipse, which occurred in the sixth year of war between the Lydians and the Medes; the two parties subsequently made peace. Strassler gives the date of eclipse as May 28, 585. This is apparently the Julian date; Guthrie in \emph{A History of Greek Philosophy} \cite[p.\ 46]{Guthrie-I} notes that the Gregorian date is May 22. One of Guthrie's references happens to be in my possession: Heath, \emph{Aristarchus} \cite{Heath-Aristarchus}, where the Julian date is given in note 3, page 15. As an example of historical detective work, I quote from this note. Heath mentions here references to the eclipse in Cicero and Pliny and also in \begin{quote}\sloppy Eusebius, \emph{Chron.\ (Hieron.),} under the year of Abraham 1433, `An eclipse of the sun, the occurrence of which Thales had predicted: a battle between Alyattes and Astyages'. The eclipse so foretold is now most generally taken to be that which took place on the (Julian) 28th May, 585. A difficulty formerly felt in regard to this date seems now to have been removed. Herodotus (followed by Clement) says that the eclipse took place during a battle between Alyattes and Cyaxares. Now, on the usual assumption, based on Herodotus's chronological data, that Cyaxares reigned from about 635 to 595, the eclipse of 585 \textsc{b.c.}\ must have taken place during the reign of his son; and perhaps it was the knowledge of this fact which made Eusebius say that the battle was between Alyattes and Astyages. But it appears that Herodotus's reckoning was affected by an error on his part in taking the fall of the Median kingdom to be coincident with Cyrus's accession to the throne of Persia, and that Cyaxares really reigned from 624 to 584, and Astyages from 584 to 550 \textsc{b.c.}\lips; hence the eclipse of 585 \textsc{b.c.}\ would after all come in Cyaxares's reign. Of two more solar eclipses which took place in the reign of Cyaxares one is ruled out, that of 597 \textsc{b.c.}, because it took place at sunrise, which would not agree with Herodotus's story. The other was on 30th September, 610, and, as regards this, Bailly and Oltmanns showed that it was not total on the presumed field of battle (in Cappadocia)\lips \end{quote} \Kirketal\ give only the year 585, and only at the beginning of their chapter, not where they quote Herodotus; but there they surmise that Thales used the records of the Babylonians, kept since 721 \bce. Citing \textbf{77,} as above, \Kirketal\ seem to suggest that the Babylonian records were available at Sardis. This does not make much sense, since the eclipse would have happened before the fall of Croesus, even before the rise, while Sardis was still Lydian. \subsection[Crossing of the Halys]{The crossing of the Halys (75)} Thales helped the army of Croesus cross the Halys River by diverting it around them, according to the Greeks; Herodotus thinks the army used the existing bridges. \subsection[Bouleuterion at Teos]{The bouleuterion at Teos (170)}\label{H:170} Thales of Miletus was Phoenician by descent, and he recommended that the Ionians have a single deliberative chamber \cite{Kirk} or council house \cite{Strassler}, that is, a bouleuterion (\gr{bouleut'hrion} \cite{Herodotus-Loeb}), centrally located in Teos. In fact Bean's account \cite[pp.\ 106--15]{Bean} of the ruins of Teos mentions no bouleuterion, although his index (under Council House) lists one for each of Heracleia, Miletus, Notium, and Priene. Nonetheless, what Bean calls the odeon of Teos is labelled as a bouleuterion at the site itself (according to my blog article%%%%% \footnote{\url{https://polytropy.wordpress.com/2015/05/19/teos/}} recording a visit in May of 2015, and another visit now in September, 2016). \Kirketal\ also quote Aetius (\textbf{71}) as attributing to Thales the theory that the flooding of the Nile is caused by the Etesian winds (``The regular N.W. winds which blow in summer from the Mediterranean'' \cite[p.\ 299, n.\ 1]{Herodotus-Loeb}); Herodotus states the theory (without naming Thales) at II.20 (quotation \textbf{70} above). Herodotus concludes his \emph{Histories} with the failed invasion of Greece by Xerxes begun in 480, more than a century after the 585 eclipse whose prediction Herodotus attributes to Thales. Presumably these two men were not alive at the same time. Strassler gives the years of Croesus's reign at Sardis as 560--547/6; probably Herodotus was not alive for this either. \section{Proclus}\label{sect:Proclus} \Kirketal\ are not elaborate in their use of Proclus. According to the index of Morrow's edition \cite{MR1200456}, there are five mentions of Thales in Proclus's \emph{Commentary on the First Book of Euclid's \emph{Elements.}} The lines of the Friedlein edition \cite{1873procli} are 65.7, 157.11, 250.20, 299.4, and 352.15. Morrow gives the last four in a footnote at the first place; he there also suggests as references \begin{compactitem} \item Heath, \emph{A History of Greek Mathematics} \cite[pp.\ 130--7]{MR654679}; \item Gow, \emph{History of Greek Mathematics} (Cambridge, 1884; reprinted New York, 1923), 138--45; and \item Van der Waerden, \emph{Science Awakening} (New York, 1961) 85--90. \end{compactitem} For now, I have only the first. Proclus's source on Thales seems to be the now-lost history of mathematics by Eudemus of Rhodes, pupil of Aristotle. Morrow refers to Heath, who weighs the evidence in his \emph{History} (pp.\ 118--20) and in the first volume of his edition of Euclid \cite[pp.\ 35--8]{MR17:814b}. What Proclus himself says is as follows; the descriptive headings are by me. \subsection[Origin of geometry]{Origin of geometry (65.7)} \begin{quote} Thales, who had travelled to Egypt, was the first to introduce this science into Greece. \end{quote} Proclus has discussed the origin of geometry in measuring lands after the Nile floods, as contrasted with the origin of arithmetic in the trading and exchange of the Pheonicians. In \cite[pp.\ 242--3]{MR3312989} I dispute the kind of materialistic account given by Proclus. The flooding of the Nile does not make you invent geometry; you invent geometry in order to deal with the flooding of the Nile. It is a question of responsibility: a river has none, but we do. \subsection[Bisection of circle]{Bisection of circle (157.11)} \begin{quote} The famous Thales is said to have been the first to demonstrate that the circle is bisected by the diameter. The cause of this bisection is the undeviating course of the straight line through the center; for since it moves through the middle and throughout all parts of its identical movement refrains from swerving to either side, it cuts off equal lengths of the circumference on both sides. If you wish to demonstrate this mathematically, imagine the diameter drawn and one part of the circle fitted upon the other. If it is not equal to the other, it will fall either inside or outside it, and in either case it will follow that a shorter line is equal to a longer\lips \end{quote} This is in the commentary on Definition XVII, which Pappus gives as follows. The Greek in Friedlein's edition agrees with that of Heiberg's edition \cite{Euclid-Heiberg} of Euclid, and Morrow's translation agrees with Heath's \cite{bones} except at the point indicated. (Also Heath italicizes ``diameter'' and has a comma where Morrow has a semicolon.) \begin{quote} A diameter of the circle is a straight line (\gr{e>uje~i'a tis}: Heath ``any straight line'') drawn through the center and terminated in both directions by the circumference of the circle; and such a straight line also bisects the circle. \end{quote} That the latter part of this is really a theorem is reason to think that it was not part of Euclid's original text. That Proclus elaborates at such length on a proof suggests that the theorem is not obvious. It is perhaps odd that a Platonist like Proclus would refer to the ``motion'' of the diameter through the circle, when in the \emph{Republic} \cite[527\textsc a]{Shorey-II} Socrates has ridiculed those who speak as if geometry were about \emph{doing} things, when in fact it is knowledge of something that always \emph{is.} However, Proclus has already addressed this issue, as Seidenberg \cite[pp.\ 265--6]{Seidenberg-Euclid} observes. Speusippus (Plato's nephew and successor at the Academy) thought all problems were really theorems; Menaechmus, all theorems were problems. both were right, says Proclus \cite[78.14--22]{MR1200456}: \begin{quote} The school of Speusippus are right because the problems of geometry are of a different sort from those of mechanics, for example, since the latter are concerned with perceptible objects that come to be and undergo all sorts of change. Likewise the followers of Menaechmus are right because the discovery of theorems does not occur without recourse to matter, that is, intelligible matter (\gr{<'ulhn t`hn noht'hn}). In going forth into this matter and shaping it, our ideas are plausibly said to resemble acts of production\lips \end{quote} \begin{sloppypar} Heath cites M. Cantor (not G. Cantor!)\ in saying that Thales's Theorem \begin{quote} may\lips have been suggested by the appearance of certain figures of circles divided into a number of equal sectors by 2, 4, or 6 diameters such as are found on Egyptian monuments or represented on vessels brought by Asiatic tributary kings in the time of the eighteenth dynasty. \end{quote} See Figure \ref{fig:circ}. \end{sloppypar} \begin{figure} %\centering \psset{unit=1.7cm} \begin{pspicture}(-1,-1)(1,1) \pscircle(0,0)1 \psline(-1,0)(1,0) \psline(0,-1)(0,1) \end{pspicture} \hfill \begin{pspicture}(-1,-1)(1,1) \pscircle(0,0)1 \psline(-1,0)(1,0) \psline(0,-1)(0,1) \psline(-0.7,-0.7)(0.7,0.7) \psline(-0.7,0.7)(0.7,-0.7) \end{pspicture} \hfill \begin{pspicture}(-1,-1)(1,1) \pscircle(0,0)1 \psline(-1,0)(1,0) \psline(0,-1)(0,1) \psline(-0.86,-0.5)(0.86,0.5) \psline(-0.5,-0.86)(0.5,0.86) \psline(-0.86,0.5)(0.86,-0.5) \psline(-0.5,0.86)(0.5,-0.86) \end{pspicture} \caption{The diameter divides the circle}\label{fig:circ} \end{figure} Now, if you pointed out to somebody that the sectors of the circle were all equal to one another, I should think the response would be, ``So what?'' I would expect the same response if you observed that the two halves of the circle made by any diameter were equal. But perhaps not; perhaps there are people who have no conception of comparing two things. The recent book \emph{Sapiens} \cite[p.\ 55]{Harari} points out how, since the Agricultural Revolution around 10,000 \bce, many of us need not know much about the world in order to survive: \begin{quote} The human collective knows far more today than did the ancient bands. But at the individual level, the ancient foragers were the most knowledgeable and skilful people in history. \end{quote} Alternatively, perhaps what is remarkable is Thales's recognition, not of the mere equality of the two halves of a circle, but of some kind of ``necessity'' in their equality. The necessity may have been something along the lines suggested by Proclus. The vertical sectors created by any two diameters are equal to one another; therefore, by adding up sectors and their opposites in one of the circles in Figure \ref{fig:circ}, we establish the equality of two semicircles. A sector of a circle resembles an isosceles triangle, considered in the next passage; and the equality of vertical sectors is related to the equality of vertical \emph{angles} in the passage after that. See page \pageref{ellipse} on generalizing the bisection of circles to ellipses. \subsection[Isosceles triangles]{Isosceles triangles (250.20)} \begin{quote} We are indebted to old Thales for the discovery of this and many other theorems. For he, it is said, was the first to notice and assert (\gr{>epist~hsai ka`i >eipe~in}) that in every isosceles the angles at the base are equal, though in somewhat archaic fashion he called the equal angles similar (\gr{t`as >'isas omo'ias gwn'ias fa'inetai fer'omenon} where \emph{equal} angles are meant. \end{quote} I should think the key to the present theorem would be that two angles $ABC$ and $CBA$ are equal, that is, congruent; and if $AB=BC$, this additional congruence establishes the congruence of the angles at $A$ and $C$. This is Pappus's proof, according to Proclus, who gives it just before the quotation above. Aristotle gave a proof different from Pappus's and Euclid's, but only for the sake of illustrating the syllogism. The relevant passage in the \emph{Prior Analytics} is quoted by Heath \cite[p.\ 253]{MR17:814b} and Thomas \cite[pp.\ 428--31]{MR13:419a}, but it may be useful to include also the preceding paragraph \cite[I.\textsc{xxiv}, 41\textsuperscript b7--22, pp.\ 322--5]{Aristotle-LI}. \begin{quote} Further, in every syllogism one of the terms must be positive, and universality must be involved. Without universality either there will be no syllogism, or the conclusion will be unrelated to the assumption, or there will be a \emph{petitio principii} (\gr{t`o >ex >arq~hs a>it'hsetai}). Suppose that we have to prove that musical enjoyment is commendable. Then if we postulate that enjoyment is commendable, unless `all' is prefixed to `enjoyment,' there will be no syllogism. If we postulate that some enjoyment is commendable, then if it is a different enjoyment, there is no reference to the original assumption; and if it is the same, there is a \emph{petitio principii} (\gr{t`o >ex >arq~hs lamb'anei}). \end{quote} It seems three arguments are contemplated: \begin{compactenum} \item Enjoying music is commendable, because some enjoyment is commendable. \item Enjoying music is commendable, because enjoyment of music is commendable. \item Enjoying music is commendable, because all enjoyment is commendable. \end{compactenum} The first is invalid; the second is begging the question; the third assumes even more than what is to be proved, but is nonetheless considered a valid syllogism. This does not make the syllogism unworthy of study. As ``negative evidence'' for his notion that Euclid does not employ the ``axiomatic method,'' Seidenberg \cite[p.\ 281--3]{Seidenberg-Euclid} notes Euclid's omission of the theorem that circumferences of circles are to one another as the diameters. Euclid omitted the theorem, because he could not prove it. It could be proved only with an axiom, such as Archimedes gives in \emph{On the Sphere and Cylinder I} \cite[p.\ 36]{MR2093668}: \begin{quote} That among lines which have the same limits, the straight \insertion{line} is the smallest. \end{quote} Says Seidenberg, \begin{quote} \textsc{Euclid} would have been thunderstruck! It would never have occurred to him that to prove a theorem (``the arc is greater than the chord''), it is all right to generalize it, and then assume the generalization. In fact, though with the Parallel Postulate he may have admitted he was stumped, there is no clear evidence that he thought it was all right to make any geometrical assumption whatever. \end{quote} Let us just note that, for the theorem omitted by Euclid, one needs the continuation of Archimedes's postulate, that if such lines are concave in the same direction, and one is contained between the other and the straight line, then that one is the smaller. Archimedes uses these implicitly in \emph{Measurement of a Circle} \cite[pp.\ 91--3]{Heath-Archimedes} to show that the circle is equal to the right triangle whose legs are respectively equal to the circumference and the radius. Meanwhile, Aristotle makes the same point about syllogisms with a mathematical example. I revert to Aristotle's letters (the Loeb translation uses Latin letters, the way Heath does). The diagram, apparently not found in the manuscripts, is supposed to be as in Figure \ref{fig:Aristo}, \begin{figure} \centering \psset{unit=3cm} \begin{pspicture}(-1,-1)(1,1) \pscircle(0,0)1 \pspolygon(0,0)(-0.86,-0.5)(0.86,-0.5) \uput[120](-0.43,-0.25){\gr A} \uput[60](0.43,-0.25){\gr B} \uput{4mm}[-30](-0.86,-0.5){\gr G} \uput{4mm}[210](0.86,-0.5){\gr D} \uput{8mm}[15](-0.86,-0.5){\gr E} \uput{8mm}[165](0.86,-0.5){\gr Z} \end{pspicture} \caption{Aristotle's proof of Euclid I.5}\label{fig:Aristo} \end{figure} where \gr{G, D, E,} and \gr Z are the angles indicated (the first two having an arc of the circle as a common side), while the angles \gr{AG} and \gr{BD} are the angles made with that arc by the radii \gr A and \gr B respectively. \begin{quote} The point can be seen more clearly in the case of geometrical theorems. \emph{E.g.,} take the proposition that the angles adjacent to the base of an isosceles triangle are equal. Let the lines \gr A and \gr B be drawn to the centre. Then if you assume that $\angle\grm{AG} = \angle\grm{BD}$ without postulating generally that the angles of semicircles are equal, and again if you assume that $\angle\grm G=\angle\grm D$ without also assuming that all angles of the same segment are equal, and further if you assume that\label{PA} when equal angles are subtracted from the whole angles the remaining angle \gr E and \gr Z are equal, unless you assume \insertion{the general principle} that when equals are subtracted from equals the remainders are equal, you will be guilty of a \emph{petitio principii.} \end{quote} The proof is bizarre, because the premises seem less clear than the conclusion. Nonetheless, perhaps Aristotle thought it a good proof, because the combination of a circle and a straight line is simpler than a triangle (the combination of three straight lines). Knowledge of the former kind of configuration perhaps ought to precede that of the latter. \subsection[Vertical angles]{Vertical angles (299.4)} \begin{quote} This theorem, then [namely XV], proves that, when two straight lines cut one another, their vertical angles are equal. It was first discovered by Thales, Eudemus says, but was thought worthy of a scientific demonstration only by the author of the \emph{Elements.} \end{quote} Proclus observes that Euclid's proof relies on Proposition XIII (that a straight line, stood on another straight line, makes either right angles or angles equal to two right angles) and two axioms (\gr{>axi'wmasi duo~in}, that equals to the same are equal to one another, and remainders are equal when equals are subtracted from equals). A note from Ian Mueller observes that Postulate IV (equality of all right angles) is also used. Seidenberg \cite[pp.\ 270--1]{Seidenberg-Euclid} also observes Proclus's failure to mention the fourth postulate and concludes ultimately that this postulate was an interpolation (possibly by Euclid himself). As I noted above, one might infer the equality of vertical angles from looking at Figure \ref{fig:circ}. Again one might verify it by symmetry. In Figure \ref{fig:vert}, \begin{figure} \centering \begin{pspicture}(5,3) \psline(0,1.5)(5,1.5) \psline(0.5,3)(4.5,0) \uput[l](0.5,3){\gr A} \uput[r](4.5,0){\gr B} \uput[r](5,1.5){\gr G} \uput[l](0,1.5){\gr D} \uput[ur](2.5,1.5){\gr E} \end{pspicture} \caption{Vertical angles are equal}\label{fig:vert} \end{figure} straight lines \gr{AB} and \gr{GD} meeting at \gr E, the angle \gr{AEG} being equal to \gr{GEA}, the demonstration of this by flipping the diagram over shows $\angle\grm{AED}=\angle\grm{GEB}$. Of course Euclid avoids this kind of proof. \subsection[Congruent triangles]{Congruent triangles (352.15)} \begin{quote} Eudemus in his history of geometry attributes the theorem itself [namely ASA and AAS, Proposition XVI] to Thales, saying that the method by which he is reported to have determined the distance of ships at sea shows that he must have used it. \end{quote} The last quotation from Proclus is number \textbf{80} of \Kirketal, who suggest that the method of measuring distances at sea was ``similar triangles,'' and that a ``primitive theodolite'' could have been used, ``two sticks (one as a sight-line, the other as an approximate level-line) pivoting on a nail.'' But could sufficient accuracy have been achieved with this method? I should think the theodolite would have to be high above the sea, on a hillside; and then we would be back to the question of how to measure this height. Perhaps two observation points along the shore, at a known distance from one another, were used instead. They do not make it an official quotation; but \Kirketal\ mention the account of Plutarch (\emph{Sept.\ Sap.\ Conv.}\ 2, 147 \textsc), \begin{quote} that the height of a pyramid is related to the length of its shadow exactly as the height of any mensurable vertical object is related to the length of its shadow at the same time of day. \end{quote} see \S\ref{sect:thm}, page \pageref{sect:thm}. According to \Kirketal, \textbf{80} shows that Thales may have used this more general method, not just the one in \textbf{79.} But \textbf{80} says nothing about similar triangles. The last four quotations above from Proclus are found also in Thomas \cite[pp.\ 164--7]{MR13:419a}, who in a note describes Heath's suggested method of measuring the distance of a ship \cite[pp.\ 132--3]{MR654679}. Climb a tower, note the angle of depression of the ship, then find an object on land at the same angle: the object's distance is that of the ship. This obviates any need to know the height of the tower, or to know proportions. Supposedly one of Napoleon's engineers measured the width of a river this way. \section{Diogenes Laertius: The angle in a semicircle} Thomas also quotes Diogenes Laertius, i.24--5: \begin{quote}\label{Pamphila} Pamphila says that, having learnt geometry from the Egyptians, he [Thales] was the first to inscribe in a circle a right-angled triangle, whereupon he sacrificed an ox. Others say it was Pythagoras, among them being Apollodorus the calculator. \end{quote} For some reason \Kirketal\ omit this quotation, although their \textbf{90} is part of i.24. (They have an index of all sources quoted in the book.) Heath cites the passage in his list of five theorems attributed to Thales \cite[pp.\ 130--1]{MR654679}. Thomas observes that Pamphila was a woman living during the reign of Nero. According to \emph{Wikipedia,} Nero's reign was 54--68; he was last in the Julio-Claudian line Augustus $\to$ Tiberius $\to$ Caligula $\to$ Claudius $\to$ Nero, Augustus, the first emperor, being adopted son of Julius Caesar. I have a photo of stone inscribed \gr{NERWN} from the lighthouse at Patara. The theorem in question is evidently that the angle in a semicircle is right. This is Euclid's III.31. Euclid's proof is based on his I.32, that the angles in a triangle are equal to two right angles; and according to Proclus at 379.2, \begin{quote} Eudemus the Peripatetic attributes to the Pythagoreans the discovery of this theorem, that every triangle has internal angles equal to two right angles, and says they demonstrated it as follows \end{quote} ---using Figure \ref{fig:I.32-alt}, as opposed to Euclid's Figure \ref{fig:I.32} (which is needed because Euclid also proves that the exterior angle is equal to the two opposite interior angles). \begin{figure} \mbox{}\hfill \subfloat[]{\label{fig:I.32-alt} \begin{pspicture}(0,-0.5)(4,3.5) \pspolygon(0,0)(4,0)(1,3) \psline(0,3)(4,3) \uput[u](1,3){\grm A} \uput[d](0,0){\grm B} \uput[d](4,0){\grm G} \uput[u](0,3){\grm D} \uput[u](4,3){\grm E} \end{pspicture} } \hfill \subfloat[]{\label{fig:I.32} \begin{pspicture}(0,-0.5)(5,3.5) \pspolygon(0,0)(4,0)(1,3) \psline(5,0)(4,0)(5,3) \uput[u](1,3){\grm A} \uput[d](0,0){\grm B} \uput[d](4,0){\grm G} \uput[d](5,0){\grm D} \uput[u](5,3){\grm E} \end{pspicture} } \hfill\mbox{} \caption{Euclid's I.32, by the Pythagoreans and himself} \end{figure} Nonetheless, Heath \cite[pp.\ 136--7]{MR654679} suggests how Thales might have recognized the theorem about semicircles without knowing the general theorem about the angles of a triangle. One might study a rectangle with diagonals as in Figure \ref{fig:rect}, \begin{figure} \centering \begin{pspicture}(0,-1)(4,4) \pspolygon(0,0)(4,0)(4,3)(0,3) \psline(0,0)(4,3) \psline(0,3)(4,0) \pscircle[linestyle=dashed](2,1.5){2.5} \psdots(0,0)(4,0)(4,3)(0,3)(2,1.5) \end{pspicture} \caption{Right angles inscribed in a circle}\label{fig:rect} \end{figure} and observe that the intersection point of the diagonals is equidistant from the four vertices. This assumes rectangles exist in the first place. Given a circle with two diameters drawn as in the figure, one may obtain the indicated quadrilateral, which is composed of two vertical pairs of congruent isosceles triangles. So the quadrilateral has equal opposite sides, and all four of its angles are equal. If we grant that these angles are right, we are done. \section{``Thales's Theorem''}\label{sect:thm} According to the English \emph{Wikipedia,} Thales's Theorem is precisely that the angle inscribed in a semicircle is right. In the Turkish \tur{Vikipedi,} this theorem is given in the article \tur{Thales teoremi (\c cember),} while the \tur{Thales teoremi} is basically Euclid's Proposition VI.2, that a straight line cutting two sides of a triangle cuts them proportionally if and only if it is parallel to the base. This naming is confirmed in a test preparation book \cite[p.\ 45]{YGS-LYS}: \begin{quote} \tur{\textbf{Thales Teoremi:} Paralel do\u grular\i n kendilerini kesen do\u grular \"uzerinde ay\i rd\i klar\i\ par\c calar kar\c s\i l\i kl\i\ olarak orant\i l\i d\i r.} \end{quote} The English \emph{Wikipedia} describes this under the title the Intercept Theorem, while acknowledging the term Thales's Theorem, which alludes to a passage from Plutarch's \emph{Dinner of the Seven Wise Men} \cite[\S2, pp.\ 351--3]{Plutarch-7}: \begin{quotation} Thales began to laugh, and said, ``If it is anything bad, go to Priene again! For Bias will have a solution for this, just as he had his own solution of the first problem.'' ``What,'' said I, ``was the first problem?'' ``The king,'' said he, ``sent to Bias an animal for sacrifice, with instructions to take out and send back to him the worst and best portion of the meat. And our friend's neat and clever solution was, to take out the tongue and send it to him, with the result that he is now manifestly in high repute and esteem.'' ``Not for this alone,'' said Neiloxenus, ``but he does not try to avoid, as the rest of you do, being a friend of kings and being called such. In your case, for instance, the king finds much to admire in you, and in particular he was immensely pleased with your method of measuring the pyramid, because, without making any ado or asking for any instrument, you simply set your walking-stick upright at the edge of the shadow which the pyramid cast, and, two triangles being formed by the intercepting of the sun's rays, \textbf{you demonstrated that the height of the pyramid bore the same relation to the length of the stick as the one shadow to the other.} But, as I said, you have been unjustly accused of having an animosity against kings, Band certain offensive pronouncements of yours regarding despots have been reported to him. For example, he was told that, when you were asked by Molpagoras the Ionian what was the most paradoxical thing you had ever seen, you replied, `A despot that lived to be old.' And again he was told that on a certain convivial occasion there was a discussion about animals, and you maintained that of the wild animals the worst was the despot, and of the tame the flatterer. \end{quotation} Diogenes \cite[1.36]{Diogenes-I} too mentions Thales's aged despot, without naming a source. \section{Aristotle}\label{sect:Aristo} \Kirketal\ observe that ``our knowledge of Thales' cosmology depends virtually completely'' on their \textbf{84} and \textbf{85}, from \emph{De Caelo} and the \emph{Metaphysics} respectively. I reserve commentary, except to observe that, if Aristotle recognizes for causes, but Thales and the other Ionians saw but one, then it will be misleading to think of that one as if it were just one of Aristotle's four. \subsection{\itshape De Caelo} Quotation \textbf{84} is the sentence that I have emboldened, with the two ensuing sentences, in the following passage from Book II of \emph{De Caelo} \cite[II.13, pp.\ 429--30]{Aristo-Heavens}. \begin{quotation} III. There are similar disputes about the \emph{shape} of the earth. Some think it is spherical, others that it is flat and drum-shaped. For evidence they bring the fact that, as the sun rises and sets, the part concealed by the earth shows a straight and not a curved edge, whereas if the earth were spherical the line of section would have to be circular. In this they leave out of account the great distance of the sun from the earth and the great size of the circumference, which, seen from a distance on these apparently small circles appears straight. Such an appearance ought not to make them doubt the circular shape of the earth. But they have another argument. They say that because it is at rest, the earth must necessarily have this shape. For there are many different ways in which the movement or rest of the earth has been conceived. The difficulty must have occurred to every one. It would indeed be a complacent mind that felt no surprise that, while a little bit of earth, let loose in mid-air moves and will not stay still, and the more there is of it the faster it moves, the whole earth, free in midair, should show no movement at all. Yet here is this great weight of earth, and it is at rest. And again, from beneath one of these moving fragments of earth, before it falls, take away the earth, and it will continue its downward movement with nothing to stop it. The difficulty then, has naturally passed into a common place of philosophy; and one may well wonder that the solutions offered are not seen to involve greater absurdities than the problem itself. By these considerations some have been led to assert that the earth below us is infinite, saying, with Xenophanes of Colophon, that it has `pushed its roots to infinity',---in order to save the trouble of seeking for the cause. Hence the sharp rebuke of Empedocles, in the words `if the deeps of the earth are endless and endless the ample ether---such is the vain tale told by many a tongue, poured from the mouths of those who have seen but little of the whole'. \textbf{Others say the earth rests upon water. This, indeed, is the oldest theory that has been preserved, and is attributed to Thales of Miletus.} It was supposed to stay still because it floated like wood and other similar substances, which are so constituted as to rest upon but not upon air. As if the same account had not to be given of the water which carries the earth as of the earth itself! It is not the nature of water, any more than of earth, to stay in mid-air: it must have something to rest upon. Again, as air is lighter than water, so is water than earth: how then can they think that the naturally lighter substance lies below the heavier? Again, if the earth as a whole is capable of floating upon water, that must obviously be the case with any part of it. But observation shows that this is not the case. Any piece of earth goes to the bottom, the quicker the larger it is. These thinkers seem to push their inquiries some way into the problem, but not so far as they might\lips \begin{comment} It is what we are all inclined to do, to direct our inquiry not by the matter itself, but by the views of our opponents: and even when interrogating oneself one pushes the inquiry only to the point at which one can no longer offer any opposition. Hence a good inquirer will be one who is ready in bringing forward the objections proper to the genus, and that he will be when he has gained an understanding of all the differences. \end{comment} \end{quotation} \subsection{\itshape Metaphysics} Quotation \textbf{85} of \Kirketal\ is the second and third paragraphs (with ellipsis of the example involving Socrates) from the \emph{Metaphysics} \cite[Book I, ch. 3, 983\textsuperscript a24, pp.\ 693--5]{Aristo-Metaphysics-Ross}. I quote also the first paragraph for its list of the four causes. I introduce their \emph{typographical} enumeration and have taken their Greek names from \cite{Aristotle-XVII}. \begin{quote} \noindent\textbf 3\quad Evidently we have to acquire knowledge of the original causes (\gr{t~wn >ex >arq~hs a>it'iwn}) (for we say we know each thing only when we think we recognize its first cause), and causes are spoken of in four senses. \begin{compactenum}[{[}1{]}] \item In one of these we mean the substance, i.e.\ the essence (\gr{t`hn o>us'ian ka`i t`o t'i >~hn e>~inai}) (for the `why' is reducible finally to the definition, and the ultimate `why' is a cause and principle); \item in another the matter or substratum (\gr{t`hn <'ulhn ka`i t`o arq`h t~hs kin'h\-sews}), and \item in a fourth the cause opposed to this, the purpose and the good (\gr{t`o o<~u <'eneka ka`i t>agaj'on}) (for this is the end of all generation and change). \end{compactenum} We have studied these causes sufficiently in our work on nature [\emph{Physics} II. 3, 7], but yet let us call to our aid those who have attacked the investigation of being and philosophized about reality before us. For obviously they too speak of certain principles and causes; to go over their views, then, will be of profit to the present inquiry, for we shall either find another kind of cause, or be more convinced of the correctness of those which we now maintain. \end{quote} Briefly, the causes are \begin{inparaenum}[(1)] formal, \item material, \item efficient or motor, and \item final. \end{inparaenum} After this first paragraph, I interject the account of the causes in a different order, and now \emph{with examples,} from the \emph{Physics} \cite[\gr B.3, pp.\ 29--30]{Aristo-Phy-Apost}; the typographical enumeration is by the translator Apostle: \begin{quotation} In one sense, a ``cause'' means (1) that from which, as a constituent, something is generated; for example, the bronze is a cause of the statue, and the silver, of the cup, and the genera of these [are also causes]. In another, it means (2) the form or the pattern, this being the formula or the essence, and also the genera of this; for example, in the case of the octave, the ratio $2:1$, and, in general, a number and the parts in the formula. In another, it means (3) that from which change or coming to rest first begins; for example, the adviser is a cause, and the father is the cause of the baby, and, in general, that which acts is a cause of that which is acted upon, and that which brings about a change is a cause of that which is being changed. Finally, it means (4) the end, and this is the final cause [that for the sake of which]; for example, walking is for the sake of health. Why does he walk? We answer, ``In order to be healthy''; and having spoken thus, we think that we have given the cause. And those things which, after that which started the motion, lie between the beginning and the end, such as reducing weight or purging or drugs or instruments in the case of health, all of them are for the sake of the end; and they differ in this, that some of them are operations while others are instruments. The term ``cause'', then, has about so many senses. And since they [the causes] are spoken of in so many ways, there may be many nonaccidental causes of the same thing; for example, in the case of a statue, not with respect to something else but qua a statue, both the art of sculpture and the bronze are causes of it, though not in the same manner; but the bronze as matter and the art as source of motion. There may be also causes of each other; for example, exercise is a cause of good physical condition, and good physical condition is a cause of exercise, although not in the same manner, but good physical condition as an end, while exercise as a principle of motion\lips \end{quotation} Now we continue where we left off, with the next few paragraphs from the \emph{Metaphysics,} I.3. \begin{quotation} \textbf{Of the first philosophers, then, most thought the principles which were of the nature of matter were the only principles of all things} (\gr{T~wn d`h pr'wtwn filosofhs'antwn oen <'ulhs e>'idei m'onas >w|'hjesan >arq`as e>~inai p'antwn}). That of which all things that are consist, the first from which they come to be, the last into which they are resolved (the substance remaining, but changing in its modifications), this they say is the element and this the principle of things%%%%% \footnote{Because of the use of gender, I wonder if a ``respectively'' is implied, as if the meaning is, ``That of which all things consist they call the element; that from which they come to be and into which they are resolved, the principle.''} %%%%%%%%%%%%%%%%%%%%%%%%%% (\gr{>ex o<~u g`ar >'estin <'apanta t`a >'onta, ka`i >ex o<~u g'ignetai pr'wtou ka`i e>is <`o fje'iretai teleuta~ion, t~hs m`en o>us'ias arq'hn fasin e>~inai t~wn >'ontwn}), and therefore they think nothing is either generated or destroyed, since this sort of entity is always conserved, as we say Socrates neither comes to be absolutely when he comes to be beautiful or musical, nor ceases to be when loses these characteristics, because the substratum, Socrates himself, remains. just so they say nothing else comes to be or ceases to be; for there must be some entity---either one or more than one---from which all other things come to be, it being conserved. Yet they do not all agree as to the number and the nature of these principles. \textbf{Thales, the founder of this type of philosophy, says the principle is water (for which reason he declared that the earth rests on water),} getting the notion perhaps from seeing that the nutriment of all things is moist, and that heat itself is generated from the moist and kept alive by it (and that from which they come to be is a principle of all things). He got his notion from this fact, and from the fact that the seeds (\gr{t`a sp'ermata}) of all things have a moist nature, and that water is the origin of the nature of moist things.\label{seed} Some think that even the ancients who lived long before the present generation, and first framed accounts of the gods, had a similar view of nature; for they made Ocean and Tethys the parents of creation, and described the oath of the gods as being by water, to which they give the name of Styx; for what is oldest is most honourable, and the most honourable thing is that by which one swears. \textbf{It may perhaps be uncertain whether this opinion about nature is primitive and ancient, but Thales at any rate is said to have declared himself thus about the first cause} (\gr{per`i t~hs pr'wths a>it'ias}). Hippo [of Samos] no one would think fit to include among these thinkers, because of the paltriness of his thought. Anaximenes and Diogenes [of Apollonia, contemporary with Hippo, latter half of 5th c.]\ make air prior to water, and the most primary of the simple bodies, while Hippasus of Metapontium and Heraclitus of Ephesus say this of fire, and Empedocles says it of the four elements (adding a fourth---earth---to those which have been named); for these, he says, always remain and do not come to be, except that they come to be more or fewer, being aggregated into one and segregated out of one. \begin{sloppypar} Anaxagoras of Clazomenae, who, though older than Empedocles, was later in his philosophical activity, says the principles are infinite in number; for he says almost all the things that are made of parts like themselves, in the manner of water or fire, are generated and destroyed in this way, only by aggregation and segregation, and are not in any other sense generated or destroyed, but remain eternally. \end{sloppypar} \end{quotation} \subsection{\itshape De Anima} Finally, the three quotations from \emph{De Anima} \cite{Aristo-DA,Aristotle-DA-OCT} are the following. \subsubsection[The magnet has a soul]{The magnet has a soul (405\textsuperscript a19)} I start earlier, at 404\textsuperscript b30. \begin{quotation} As to the nature and number of the first principles opinions differ. The difference is greatest between those who regard them as corporeal and those who regard them as incorporeal, and from both dissent those who make a blend and draw their principles from both sources. The number of principles is also in dispute; some admit one only, others assert several. There is a consequent diversity in their several accounts of soul; they assume, naturally enough, that what is in its own nature originative of movement must be among what is primordial. That has led some to regard it as fire, for fire is the subtlest of the elements and nearest to incorporeality; further, in the most primary sense, fire both is moved and originates movement in all the others. Democritus has expressed himself more ingeniously than the rest on the grounds for ascribing each of these two characters to soul; soul and mind are, he says, one and the same thing, and this thing must be one of the primary and indivisible bodies, and its power of originating movement must be due to its fineness of grain and the shape of its atoms; he says that of all the shapes the spherical is the most mobile, and that this is the shape of the particles of fire and mind. Anaxagoras, as we said above, seems to distinguish between soul and mind, but in practice he treats them as a single substance, except that it is mind that he specially posits as the principle of all things; at any rate what he says is that mind alone of all that is simple, unmixed, and pure. He assigns both characteristics, knowing and origination of movement, to the same principle, when he says that it was mind that set the whole in movement. \textbf{Thales, too, to judge from what is recorded about him, seems to have held soul to be a motive force, since he said that the magnet has a soul in it because it moves the iron.} \end{quotation} \subsubsection[Hippo on the soul as water]{Hippo on the soul as water (405\textsuperscript b1)} \Kirketal\ give the following in a footnote to their commentary on the \emph{Metaphysics} quotation, speculating that Hippo might be the source of Aristotle's notion of why Thales thought water was the principle. \begin{quote} Of more superficial writers, some, e.g.\ Hippo, have pronounced it [the soul] to be water; they seem to have argued from the fact that the seed (\gr{gon'h}) of all animals is fluid, for Hippo tries to refute those who say that the soul is blood, on the ground that the seed, which is the primordial soul (\gr{t`hn pr'wthn yuq'hn}), is not blood. \end{quote} \subsubsection[All things are full of gods]{All things are full of gods (411\textsuperscript a7)} I go back to 411\textsuperscript a2. \begin{quotation} If we must construct the soul out of the elements, there is no necessity to suppose that all the elements enter into its construction; one element in each pair of contraries will suffice to enable it to know both that element itself and its contrary. By means of the straight line we know both itself and the curved-the carpenter's rule enables us to test both-but what is curved does not enable us to distinguish either itself or the straight. \textbf{Certain thinkers say that soul is intermingled in the whole universe, and it is perhaps for that reason that Thales came to the opinion that all things are full of gods} (\gr{p'anta pl'hrh je~wn e>~inai}). This presents some difficulties: Why does the soul when it resides in air or fire not form an animal, while it does so when it resides in mixtures of the elements, and that although it is held to be of higher quality when contained in the former? \end{quotation} \chapter{Interpretations} Several modern books in my possession comment on the Ionian philosophers. The books may have a better understanding of Thales than Aristotle does, though we remain indebted to Aristotle (and the scribes who copied him over the centuries) for giving us any idea of what Thales thought in the first place. \section{Collingwood}\label{sect:Collingwood} \emph{The Idea of Nature} \cite{Collingwood-IN} examines the absolute presuppositions about nature made in \begin{inparaenum}[(1)] \item ancient Greece, \item Renaissance Europe, and \item modern times. \end{inparaenum} Collingwood is providing an example of the work of the metaphysician as described in \emph{An Essay on Metaphysics} \cite{Collingwood-EM}. After an Introduction, \emph{The Idea of Nature} starts in with the Ionians: \begin{quote} According to Aristotle, the characteristic of this Ionian cosmology is the fact that whenever its devotees ask the question: \textbf{`What is nature?'}\ they at once convert it \textbf{into the question: `What are things made of?'}\ or `What is the original, unchanging substance which underlies all the changes of the natural world with which we are acquainted?' \end{quote} It appears Collingwood takes Aristotle seriously, even though, in a footnote, he acknowledges a warning about this; his response for now is that the critic himself follows Aristotle tacitly: \begin{quote} Monsieur E. Brehier (\emph{Histoire de la Philosophie,} Paris, 1928, vol.\ i, p.\ 42) says that the question `What are things made of?' is not Thales' question but Aristotle's question. There is certainly force in his warning that \textbf{our traditional view of the Ionian physicists through the spectacles of Aristotle places us in danger of ascribing exaggerated importance in the minds of these men to what may in fact have been little more than \emph{obiter dicta,}} and thus projecting fourth century problems back into the sixth century or even the late seventh. Yet Monsieur Brehier himself says `Le ph\'enom\`ene fondamental dans cette physique mil\'esienne est bien l'\'evaporation de l'eau de la mer sous I'influence de la chaleur' (p.\ 44). In other words, Monsieur Brehier in spite of his own warning continues to accept Aristotle's view that the fundamental concept of Ionian physics was the concept of transformation. \end{quote} Meanwhile, Collingwood continues in the main text: \begin{quotation} People who could ask this question must have already settled in their minds a large number of preliminary points\lips I will mention three of them. 1. \emph{That there are `natural' things}\lips 2. \emph{That `natural' things constitute a single `world of nature'}\lips 3. \emph{That \textbf{what is common to all `natural' things is their being made of a single `substance' or material.}} This was the special or peculiar presupposition of Ionian physics; and the school of Miletus may be regarded as a group of thinkers who made it their special business to take this as their `working hypothesis' and see what could be made of it: asking in particular the question: `That being so, what can we say about this single substance?' They did not consciously treat it as a `working hypothesis': it cannot be doubted that \textbf{they accepted it as an absolute and unquestioned presupposition of all their thinking;} but the historian of thought, looking back on their achievement, cannot fail to see that \textbf{what they really did was to test this idea of a single universal substance and to find it wanting.} \end{quotation} I would take issue with saying what somebody is ``really'' doing while denying that the person is capable of recognizing it. Freud speaks this way, saying somewhere that the reason for a neurosis cannot just be told to the person suffering it. Here though there is the chance that the patient will discover the reason through analysis. In \emph{An Essay on Metaphysics} \cite[p.\ 40]{Collingwood-EM}, Collingwood describes ``metaphysical analysis,'' which is ``the analysis which detects absolute presuppositions.'' \begin{quotation}\noindent Such analysis [says Collingwood on page 43 of the \emph{Essay}] may in certain cases proceed in the following manner. If the enquirer can find a person to experiment on who is well trained in a certain kind of scientific work, intelligent and earnest in his devotion to it, and unaccustomed to metaphysics, let him probe into various presuppositions that his `subject' has been taught to make in the course of scientific education, and invite him to justify each or alternatively to abandon it\lips when an absolute presupposition is touched, the invitation will be rejected, even with a certain degree of violence\lips This is a precarious method, because the qualifications it demands in the subject are too delicate\lips Perhaps there was a kind of justice in the allegation that Socrates, the great master of this method, `corrupted the young men'\lips The only altogether satisfactory method is for the analyst to experiment on himself\lips \end{quotation} It is hard to get somebody to recognize his absolute presuppositions; but it is not impossible. Could Thales have been induced to recognize the absolute presuppositions that Collingwood attributes to him? A passage in \emph{The Principles of History} \cite[p.\ 30]{Collingwood-PH} suggests that Collingwood may not think this is important: \begin{quote} Confronted with a ready-made statement about the subject he is studying, the scientific historian never asks himself: `Is this statement true or false?', in other words `Shall I incorporate it in my history of that subject or not?' The question he asks himself is: `What does this statement mean?' And this is not equivalent to the question `What did the person who made it mean by it?', although that is doubtless a question that the historian must ask, and must be able to answer. It is equivalent, rather, to the question `What light is thrown on the subject in which I am interested by the fact that this person made this statement, meaning by it what he did mean?' \end{quote} Thus, in the last quotation from \emph{The Idea of Nature,} Thales meant that what is common to all natural things is their being made of water; but this ultimately shows that what is common to all natural things cannot be a ``substance'' at all. Concerning Thales, Collingwood goes on to say, ``He held, as everyone knows, that the universal substance out of which things are made is water.'' This gives us two questions: \begin{compactenum} \item Why water? \item How does ``a thing made of water, such as a stone or a fish,'' differ from the water itself? \end{compactenum} \begin{quotation}\noindent On the second question we have no light at all. On the first, Aristotle himself has no information, but he has put forward two suggestions which are admittedly guesses. The first is that moisture is necessary for the nourishment of every organism; the second, that every animal's life begins in seminal fluid. The point to be noticed here is not what Aristotle says but what it presupposes, namely that \textbf{Thales conceived the world of nature as an organism: in fact, as an animal.} This is confirmed by the fragments which have come down to us of Thales' own utterances. According to these fragments, Thales regarded the world (the earth \emph{plus} the heavens, that is to say; what later Greek thinkers called \gr{k'osmos}, but the Milesians called \gr{o>uran'os},) as something `ensouled', \gr{>'emyuqon}, a living organism or animal, within which are lesser organisms having souls of their own\lips \textbf{he may possibly have conceived the earth as grazing, so to speak, on the water in which it floats,} thus repairing its own tissues and the tissues of everything in it by taking in water from this ocean and transforming it, by processes akin to respiration and digestion, into the various parts of its own body. We are told, moreover, that \textbf{he described the world as \gr{po'ihma jeo~u}, something made by God.} That is to say, the vital processes of this cosmic organism were not conceived by him as self-existent or eternal (for he said that God is `older' than the world) but as depending for their existence on an agency prior to them and transcending them. It is evident from these scanty records that the ideas of Thales were enormously remote from the Renaissance conception of the natural world as a cosmic machine made by a divine engineer in order to serve his purposes. He regarded it as a cosmic animal whose movements, therefore, served purposes of its own. \textbf{This animal lived in the medium out of which it was made, as a cow lives in a meadow. But now the question arose, How did the cow get there?} What transformed the undifferentiated water into that mass of differentiated and ensouled water which we call the world? Here the analogy between the world and a cow breaks down. The cosmic cow did not begin its life as a calf. The life of the world-animal does not include anything analogous to reproduction. \textbf{The world was not born, it was made; made by the only maker that dare frame its fearful symmetry: God.} But what kind of a making was this? It was very unlike that making which Renaissance cosmology attributed to the 'great architect of the universe'. For Renaissance thought, as that phrase indicates, the creative activity of God in its relation to the world of nature is in all points except one a scaled-up version of the activity by which a man builds a house or a machine; the one exception being that God is an architect or engineer who has no need of materials but can make His world out of nothing. If the divine activity of which Thales spoke in his phrase \gr{po'ihma jeo~u} is a scaled-up version of any human activity, this human activity is not the activity of an architect or engineer but the activity of a magician. God, in the cosmology of Thales, makes a cosmic animal out of water \textbf{as magically as Aaron made a snake out of a walking-stick, or as the Arunta in their \emph{inchitiuma} ceremonies make a supply of emus or witchetty grubs.} \end{quotation} This ends Collingwood's specific treatment of Thales. He continues with Anaximander and Anaximenes, then considers ``Limits of Ionian natural science'' and ``Meaning of the word ``nature.'' \section{Frankfort and Frankfort} I bought \emph{Before Philosophy} \cite{Frankfort} in a used bookshop somewhere; possibly this was in Annapolis when I was a freshman at St John's, but I do not recall clearly. In the Conclusion of the book, the Frankfort couple write movingly of the Ionians. I note two themes: \begin{compactenum} \item Greek thought is mathematical in the sense of being elaborated by deductive reasoning. See the end of my selection on this. (The Frankforts do not seem to mention mathematics as such, and it is not in their index.) \item Thales's water is not to be understood like the elements on the Periodic Table; but one should recall that the land is not always green, but becomes so when rain comes; and Homer \cite[14.201, 245--6]{Iliad-Loeb-II} refers to \begin{quote}%\centering \begin{verse} Oceanus, from whom the gods are sprung\\ the streams of the river Oceanus from whom they [the gods] all are sprung \end{verse} \end{quote} \end{compactenum} Here now the Frankforts, from their pages 248--61. \begin{quotation} In the sixth century \textsc{b.c.}\ the Greeks, in their great cities on the coast of Asia Minor, were in touch with all the leading centres of the civilized world: Egypt and Phoenicia; Lydia, Persia, and Babylon. There can be no doubt that this contact played some part in the meteoric development of Greek culture\lips And yet Hesiod was without oriental precedent in one respect: the gods and the universe were described by him as a a matter of private interest. Such freedom was unheard of in the Near East, except among the Hebrews, where Amos, for instance, was a herdsman\lips \begin{sloppypar} The same freedom, the same unconcern as regards special function and hierarchy, is characteristic for the Ionian philosophers who lived a century or more after Hesiod. Thales seems to have been an engineer and statesman; Anaximander, a map-maker\lips \end{sloppypar} \lips Like Hesiod, the Ionian philosophers gave their attention to the problem of origins; but for them it assumed an entirely new character. The origin, the \gr{>arq'h}, which they sought was not understood in the terms of myth\lips Yet the doctrines of the early Greek philosophers are not couched in the language of detached and systematic reflection. Their sayings sound rather like inspired oracles. And no wonder, for these men proceeded, with preposterous boldness, on an entirely unproved assumption. They held that the universe is an intelligible whole\lips The speculative courage of the Ionians is often overlooked. Their teachings were, in fact, predestined to to be misunderstood by modern---% or rather, nineteenth-century---scholars. When Thales proclaims water to be the first cause, or Anaxi\-menes air; when Anaximander speaks of the `boundless', and Heraclitus of fire; when, moreover, Democritus' theory of atoms can be considered the outcome of these earlier speculations; then we need not be astonished that commentators in a positivistic age unwittingly read familiar connotations into the quasi-materialist doctrines of the Ionians and regard these earliest philosophers as the first scientists. No bias could more insidiously disfigure the greatness of the Ionian achievement. The materialist interpretation of their teachings takes for granted what was to be discovered only as a result of the labours of these ancient thinkers---% the distinction between the objective and the subjective. And only on the basis of this distinction is scientific thought possible. In actual fact the Ionians moved in a curious borderland. They forefelt the possibility of establishing an intelligible coherence in the phenomenal world; yet they were still under the spell of an undissolved relationship between man and nature\lips \begin{sloppypar} \lips Anaximenes recognized in air something variable enough to make it seem possible to interpret the most diverse phenomena as its manifestations. Thales had preferred water, but he, too, did not consider his first cause merely as a neutral, colourless liquid. We must remember that \textbf{seeds and bulbs and the eggs of insects lie lifeless in the rich soil of Eastern Mediterranean lands until the rains come}---% remember, also, the preponderant role of watery substances in the processes of conception and birth in the animal kingdom. It is possible that the ancient oriental view of water as a fertilizing agent had retained its validity for Thales. It is equally possible that he endorsed the oriental conception of a primeval ocean from which all life came forth. Homer, as we have seen, called Okeanos the origin of gods and men\lips \end{sloppypar} \lips In the first place, early Greek philosophy (in Cornford's words) `ignored with astonishing boldness the prescriptive sanctities of religious representation' [\emph{Cambridge Ancient History,} IV, 532]. Its second characteristic is a passionate consistency. Once a theory is adopted, it is followed up to its ultimate conclusion irrespective of conflicts with observed facts or probabilities. Both of these characteristics indicate an implicit recognition of the autonomy of thought; they also emphasize the intermediate position of early Greek philosophy\lips Its disregard for the data of experience in its pursuit of consistency distinguishes it from later thought\lips \lips With conviction they propounded theories which resulted from intuitive insight and which were elaborated by deductive reasoning\lips \end{quotation} \section{Guthrie} W. K. C. Guthrie, \emph{The Greek Philosophers} \cite{Guthrie} was used in an ancient Greek history course the year \emph{after} I took it as a freshman in high school. When I expressed interest, the teacher gave me a copy. In the course, we had read some of Cornford's version of the \emph{Republic} (perhaps only on the Divided Line and the Cave). Guthrie suggests in a note on his page 2 that the job of his essay ``has been done, as well as it is ever likely to be, by F. M. Cornford in \emph{Before and After Socrates} (Cambridge University Press, 1932).'' For Guthrie (p.\ 23), the Milesian School \begin{quote} [1] looked for something permanent, persisting through the chaos of apparent change; and they [2] thought that they would find it by asking the question: `What is the world made of?' \end{quote} I would question this two-part analysis. As Guthrie notes, one could alternatively suppose ``that the permanent and comprehensible element lies in its structure or form'' (p.\ 25). But I do not know any reason to think that the Ionians had Aristotle's distinction between form and matter. It's not as if the Ionians were faced with a choice of one or the other. Thales said the world was made of water. Why? ``The explanation which occurs most readily to modern scholars'' (p.\ 25) is that water is seen to exist in three phases: solid, liquid, and gas. It seems to me that this brings us back to the original question. What is it that persists when ice melts and water boils? After considering Anaximander and Anaximenes, Guthrie says (pp.\ 31--2, first ellipsis in original), \begin{quotation} As Cornford put it, `If we would understand the sixth-century philosophers, we must disabuse our minds of the atomistic conception of dead matter in mechanical motion and of the\lips dualism of matter and mind.' Aristotle, who was already criticizing the Ionians for (as it appeared to him) `lazily shelving' the question of the motive cause, remarks in one place, without comment, that none of them made earth the primary substance. There was surely a good reason for this. They wanted a substance which would \emph{explain its own movement,} as in those early days it was still possible to imagine it doing. One thought of the ceaseless tossing of the sea, another of the rushing of the wind\lips \end{quotation} Presently Guthrie quotes the end of passage \textbf{85} of \Kirketal, that is, the end of the paragraph ending on page \pageref{seed} above, as having more to be said for it ``than modern commentators are inclined to allow.'' \chapter{Proof} Here are some notes concerning the origin of mathematical proof as we know it. \section{Plato}\label{sect:Plato} Plato's dialogues provide examples of proofs of assertions not normally considered mathematical. For example, in Book X of the \emph{Republic,} Socrates offers a deductive proof of the immortality of the soul as follows. \begin{compactenum} \item Everything has its own badness, evil, disease: the eyes have ophthalmia; grain, mildew. \item Nothing is destroyed, except by its own badness. \item The badness of the soul is injustice, ignorance, \&c. \item These do not kill the soul. \item Therefore the soul cannot die. \end{compactenum} The whole argument is in Appendix \ref{app:Rep}. According to Diogenes \cite[I.24]{Diogenes-I}, \begin{quote} some, including Choerilus the poet, declare that he [Thales] was the first to maintain the immortality of the soul. \end{quote} Do we find arguments for such propositions to be valid and worth making, if we think proofs of \emph{mathematical} assertions are worth making? \section{Autolycus and Aristoxenus} Autolycus and Aristoxenus wrote astronomy and music, respectively, around the time of Euclid, possibly earlier, in somewhat of Euclid's style. In \emph{The Forgotten Revolution} \cite[pp.\ 48--9]{MR2038833}, Lucio Russo argues that even if, as is said, the results presented in the \emph{Elements} were known before Euclid, \begin{quote} the main feature of Euclid's work is not the set of results presented, but the way in which these results connect together, forming infinitely extensible ``networks'' of theorems, drawn out from a small number of distinguished statements. To judge the originality of the \emph{Elements}, therefore, one must ask whether a similar structure (without which one cannot extend the the\-% ory by doing ``exercises'': that is the whole point!) had been achieved prior to Euclid. \end{quote} As Russo has explained on his page 17 (though without explicit mention of students), in a deductive science, one can assign the exercise of \emph{proving} a particular theorem. This is possible, because there is universal agreement on what constitutes a proof, once the foundations are laid down. Russo thinks Euclid must have been the first to lay down such foundations. There are earlier proofs, but none based on such foundations as Euclid's postulates. However, Russo's search for evidence does not go explicitly beyond Plato and Aristotle, as the continuation of the last quotation shows. Here the Plato reference is to \emph{Republic} VI, 510\textsc c; the Aristotle, to \emph{Analytica posteriora} I.\textsc x, 7a\textsuperscript a40; the common notion from Euclid was used in the example from the \emph{Prior Analytics} (page \pageref{PA}). \begin{quotation} In the surviving fragments on pre-Euclidean mathematics there is no evidence for sets of postulates similar to Euclid's. The works of Plato and Aristotle, moreover, offer an explicit description of what the ``principles'' accepted by mathematicians as the initial assumptions of their science were like at the time. Plato writes that ``those who work with geometry, arithmetic, and the like lay as `hypotheses' evenness and oddness, fig\-% ures, the three kinds of angles and similar things.'' Aristotle, in a passage where he discusses the role of principles in the deductive sciences, makes a distinction between the principles common to all sciences and those par\-% ticular to each. As an example of the first type he mentions the assertion ``Subtract equals from equals and equals remain'', which appears in the \emph{Elements} exactly as one of the ``common notions''. Immediately before that he had written: ``Particular [principles] are `The line is such-and-such', and likewise for straightness.'' There is an obvious difference between the type of ``geometric princi\-% ples'' exemplified by Plato and Aristotle, which surely could not serve as the basis for proving theorems, and the postulates contained in the Ele\-% ments. As to the premises actually used in the demonstration of geometric theorems, several passages from Plato and Aristotle attest to a deductive method much more fluid in the choice of initial assumptions than that transmitted by the \emph{Elements} and later works. The logical unity of the \emph{Elements}, or of a large portion of it, is clearly not due to chance; it is the result of conscious work on the part of the same mathematician to whom we owe the postulates. There is no reason to suppose that this unity is not an innovation due to Euclid, and a very important one at that. \end{quotation} Autolycus is not in Russo's index (nor does he come up in a search of the \url{djvu} file from which I have cut and pasted the quotations above). His work \cite{Autolycus-Hultsch,Autolycus-Mogenet} is in the ``\emph{protasis}-style'' recognized by Fowler in \emph{The Mathematics of Plato's Academy} \cite[p.\ 386--7]{MR2000c:01004}, with statements (\emph{protases}) followed by their proofs. Fowler takes up the question of whether mathematics before Euclid ``was striving towards the content and style of Euclid's \emph{Elements.}'' Taking for the \emph{Elements} the conventional date of 300 \bce, Fowler says that, before then, \begin{quote} We shall find that the \emph{only} evidence for \emph{protasis}-style in this period will be in Aristotle, and most of that\lips will not be in mathematics but in logic, in his \emph{Prior Analytics.} Outside Aristotle, the evidence will be in music, with Aristoxenus' \emph{Elements of Harmony,} Book III; or in contexts which may not actually be pre-Euclidean, like Autolycus, \emph{On Risings and Settings and On the Heavenly Spheres.} \end{quote} Concerning Aristoxenus, Autolycus, and Euclid, Fowler says on pages 392--3: \begin{quote} There are even problems with all of these authors: the name of \emph{Elements of Harmonics} might have been given to Aristoxenus' treatise after the tradition of naming books \emph{Elements of}\lips had been established; the evidence that Autolycus pre-dates Euclid's \emph{Elements} is no more convincing than the other way round; and we know nothing of the composition---place, date, and author or authors---of Euclid's \emph{Elements.} And I do not know whether the use in mathematics of \emph{protasis}-style could be an importation of its use in logic by Aristotle, and know of no discussion of this. \end{quote} Perhaps we cannot say categorically that Euclid is the originator of \emph{protasis}-style for mathematics; but this style did arise around when Euclid himself (whoever he was) was working, and nobody else used postulates as he did. Thomas \cite[p.\ 490, n.\ \textit a]{MR13:419a} thinks Euclid's \emph{Phaenomena} is based on Autolycus, though he does not quote him; he quotes Euclid here only to the extent of: \begin{quote} If a cone or cylinder be cut by a plane not parallel to the base, the resulting section is a section of an acute-angled cone which is similar to a shield. \end{quote}\label{ellipse} This theorem allows a generalization of Thales's supposed theorem that the circle is bisected by the diameter. When an ellipse is understood in the original sense of a ``section of an acute-angled cone,'' then its symmetry is not obvious. Fowler discounts Heath's argument \cite[pp.\ 348--53]{MR654679} that Euclid's \emph{Phaenomena} is cribbed from Autolycus: the similarities could mean the cribbing was the other way. Moreover, we do not know whether the \emph{Phaenomena} was written before the \emph{Elements,} or even whether it is by the same person. Nonetheless, in \emph{The Shaping of Deduction in Greek Mathematics} \cite[p.\ 275]{MR1683176}, Reviel Netz says, \begin{quote} The solid starting-point for Euclidean-style geometry is neither Euclid nor Autolycus, but Aristotle. \end{quote} Consulting Barker, \emph{Greek Musical Writings} \cite[p.\ 170]{Barker-writings-2}, I see that Book III of Aristoxenus is indeed in \emph{protasis}-style, the first proposition being, ``Successive tetrachords are either conjunct or disjunct.'' Barker explains the style himself in a note: \begin{quote} The third book is quite unlike the others. It consists of a set of theorems deriving propositions from principles already adopted. The theorems are what Aristoxenus, following Aristotle, calls \emph{apodeixeis,} `demonstrations', and are thought of as explaining why the propositions are true, as well as proving them (see the introductions to chapter 3 and to this chapter). Each proposition is first stated, then demonstrated (I follow Macran in italicising the initial statements, which I have also numbered, in order to bring out this pattern; it parallels that of Euclid in 8 \emph{Sect.\ Can.}\ and in the \emph{Elements of Geometry}). \end{quote} There is no preamble to Book III of Aristoxenus: he starts right in with propositions. Autolycus begins \emph{On the Heavenly Spheres} with two definitions, the second bracketed by Mogenet, of \begin{compactenum}[1)] \item \gr{'axwn sfa'iras} ``axis of a sphere.'' \end{compactenum} The LSJ \cite{LSJ} does not cite Autolycus for the meaning of the first, although he is on the list of sources for the lexicon. For my awareness of Autolycus in the first place, I thank Ay\c se Berkman for having called to my attention an article \cite{Autolykos} in the ``Science Technology'' (\tur{Bilim Teknoloji}) supplement of \tur{\emph{Cumhuriyet}} newspaper, concerning the publication in Turkish of the oldest book of science, namely Autolycus of Pitane. \begin{sloppypar} The \emph{protasis}-style may not be original to Euclid; but the postulates---the very idea of using such postulates---% would seem to be original, or at least there is no evidence otherwise. \end{sloppypar} \section{Hypsicles} Euclid makes the postulate that all right angles are equal to one another. It should be obvious to everybody that they \emph{are} equal. At least it is tacitly accepted by anybody who uses a set square. On the other hand, would everybody give Euclid's definition? It is one of the few now bound with the \emph{Elements} that are needed: \begin{quote} When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the angles is \emph{right} (\gr{>orj'h}), and the straight line standing on the other is called a \emph{perpendicular} (\gr{k'ajetos}) to that on which it stands. \end{quote} One could define a right angle to be any of the angles in an equiangular quadrilateral; but then a postulate that all of these are equal would yield the Parallel Postulate. Some people may prove that all right angles are equal by observing that every right angle measures 90 degrees; but this begs the question of whether all degrees are the same! Inspired by the thought of mountains, one might conceive of the surface of every right cone (bounded by the circumference of its base) as a circle. If a degree is a 360th of the ``way around,'' then different cones will yield different degrees. In any case, the first division of the circle into 360 degrees by a writer in Greek---% even the first indication of any influence of Babylonian sexagesimal arithmetic---% is said to be by Hypsicles. One source here is Fowler, \emph{The Mathematics of Plato's Academy} \cite[p.\ 219, n.\ 52]{MR2000c:01004}, who observes (pp.\ 223, 83) that Hypsicles flourished around 150 \bce\ and wrote the so-called Book XIV of the \emph{Elements.} For Fowler it is important that Babylonian arithmetic did not appear in Greek mathematics till late (p.\ 399). \section{Equality}\label{sect:=} Equality in Euclid is not sameness, but congruence, as discussed in my paper \cite[pp.\ 238--40]{MR3312989}. Neither is equality sameness in ordinary life. Here is Article 7 of the Universal Declaration of Human Rights:%%%%% \footnote{\url{http://www.un.org/en/universal-declaration-human-rights/}, accessed September 2, 2016}%%%%% %proclaimed by the United Nations General Assembly in Paris on December 10, 1948, \begin{quote} All are equal before the law and are entitled without any discrimination to equal protection of the law. All are entitled to equal protection against any discrimination in violation of this Declaration and against any incitement to such discrimination. \end{quote} In Turkish, this is \tur{Madde~7} of the \tur{\.Insan haklar\i\ evrensel beyannamesi:}%%%%% \footnote{\url{http://www.ohchr.org/EN/UDHR/Pages/Language.aspx?LangID=trk}, accessed September 2, 2016} \begin{quote} \tur{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. Herkesin i\c sbu Beyannameye ayk{\i}r{\i} her t\"url\"u ay{\i}rdedici mualeleye kar\c s{\i} ve b\"oyle bir ay{\i}rdedici muamele i\c cin yap{\i}lacak her t\"url\"u k{\i}\c sk{\i}rtmaya kar\c s{\i} e\c sit korunma hakk{\i} vard{\i}r.} \end{quote} The notion of equality before the law is traced to the Funeral Oration of Pericles in Athens, 431/0, as recounted by Thucydides \cite[II.37, p.\ 145]{Thucydides}: \begin{quotation} Let me say that our system of government does not copy the institutions of our neighbours. It is more the case of our being a model to others, than of our imitating anyone else. Our constitution is called a democracy because power is in the hands not of a minority but of the whole people %καὶ ὄνομα μὲν διὰ τὸ μὴ ἐς ὀλίγους ἀλλ᾽ ἐς πλείονας οἰκεῖν δημοκρατία κέκληται: (\gr{ka`i >'onoma m`en di`a t`o m`h >es >ol'igous >all'' >es ple'ionas o>ike~in dhmokrat'ia k'eklhtai}). \textbf{When it is a question of settling private disputes, everyone is equal before the law;} when it is a question of putting one person before another in positions of public responsibility, what counts is not membership of a particular class, but the actual ability which the man possesses. No one, so long as he has it in him to be of service to the state, is kept in political obscurity because of poverty. And, just as our political life is free and open, so is our day-to-day life in our relations with each other. We do not get into a state with our next-door neighbour if he enjoys himself in his own way, nor do we give him the kind of black looks which, though they do no real harm, still do hurt people's feelings. We are free and tolerant in our private lives; but in public affairs we keep to the law. This is because is commands our deep respect. We give our obedience to those whom we put in positions of authority, and we obey the laws themselves, especially those which are for the protection of the oppressed, and those unwritten laws which it is an acknowledged shame to break. \end{quotation} This is, in the Turkish of Furkan Akderin \cite[p.\ 82]{Thucydides-Akderin}: \begin{quote} \tur{Siyasi yap\i m\i z\i n kom\c sular\i m\i zdan bir fark\i\ yok. Hatta onlardan \"ust\"un oldu\u gumuzu bile s\"oyleyebiliriz. \c C\"unk\"u biz onlara g\"ore de\u gil, onlar bize g\"ore yasalar\i n\i\ yap\i yorlar. Bizim devletimiz az\i nl\i\u g\i n de\u gil \c co\u gunlu\u gun \c c\i karlar\i n\i\ g\"ozetmektedir. Bu nedenle de ismi demokrasidir. Herhangi bir anla\c smazl\i k an\i nda herkes yasalar kar\c s\i s\i nda e\c sittir. Ancak konu kamu ya\c sam\i na kat\i lmak oldu\u gunda kim di\u gerlerinden daha \"ust\"unse y\"onetimde o bulunur. Atina'ya hizmet eden hi\c c kimse kendisinin fakir olmas\i ndan ya da o an i\c cin devletin bulundu\u gu g\"u\c c durumdan dolay\i\ utan\c c duymaz. Devlet i\c cinde esas olan \c sey \"ozg\"url\"ukt\"ur. Fakat g\"undelik ya\c sam\i m\i zda \"ozg\"url\"ukten kavram\i ndan yola \c c\i karak \"on\"um\"uze gelen herkese k\"ot\"u davranmay\i z. Ya da bir insan maddi durumundan dolay\i\ ba\c skalar\i\ taraf\i ndan a\c sa\u g\i lanmaz. G\"undelik i\c slerimizde kar\c s\i m\i zdaki insanlar\i\ zorlayamay\i z. Di\u ger yandan memurlara, yasalara \"ozellikle de zor durumda olanlara kar\c s\i\ su\c c i\c slenmesini engelleyen yasalara kar\c s\i\ sayg\i\ g\"osterilmesi ve sayg\i\ g\"ostermeyenlerin d\i\c slanmas\i\ \c cok \"onemlidir.} \end{quote} In mathematics, the sign $=$ of equality was introduced by Robert Recorde in 1557 \cite{Recorde} as an icon of two parallel lines having the same length, \begin{quote}\centering bicause no. 2. thynges, can be moare equalle. \end{quote} The equals sign is an icon in the strict sense of Charles Peirce \cite{Peirce-signs}, because it \begin{quote} would possess the character which renders it significant, even though its object had no existence; such as a lead-pencil streak as representing a geometrical line. \end{quote} \section{The Pythagorean Theorem} I have tried to come up with a visual proof of the Pythagorean Theorem based on Euclid's proof. The point is to be able to sketch a proof of a nontrivial theorem using only a diagram (which can be prepared in advance). See Figure \ref{fig:I.47-big}. \begin{figure} \centering \psset{unit=1.6mm} \begin{pspicture}(-12,-25)(37,28) %\psset{linestyle=none,linewidth=0} %\psset{fillstyle=solid} \pspolygon[fillcolor=yellow](37,16)(21,16)(21,28) \pspolygon[fillcolor=yellow](25,0)(25,-16)(13,-16) \pspolygon[fillcolor=white](0,0)(0,9)(6.75,9) \pspolygon[fillcolor=white](0,0)(9,0)(9,-6.75) \pspolygon[fillcolor=yellow](25,0)(37,16)(25,16) \pspolygon[fillcolor=yellow](25,0)(9,12)(9,0) \pspolygon[fillcolor=yellow](25,-25)(9,-13)(9,-25) \pspolygon[fillstyle=vlines,hatchangle=53](25,0)(9,0)(9,-13)(13,-16) \pspolygon[fillstyle=hlines,hatchangle=53](25,0)(25,16)(12,16)(9,12) \psset{fillstyle=crosshatch,hatchangle=0,hatchsep=6pt} \pspolygon(-12,9)(-5.25,18)(1,18)(9,12)(6.75,9) \pspolygon(0,0)(9,-6.75)(9,-13)(3,-21)(0,-18.75) \psset{fillstyle=solid} \pspolygon[fillcolor=black](-3,21)(-5.25,18)(1,18) \pspolygon[fillcolor=black](0,-25)(0,-18.75)(3,-21) \psset{fillcolor=lightgray} \pspolygon(21,16)(12,16)(21,28) \pspolygon(25,-16)(25,-25)(13,-16) \pspolygon(0,0)(-12,9)(0,9) \pspolygon(0,0)(9,12)(9,0) \pspolygon(0,-25)(9,-13)(9,-25) \end{pspicture} \caption{The Pythagorean Theorem}\label{fig:I.47-big} \end{figure} \section{Thales} The \emph{Wikipedia} article on Thales needs thorough editing, at least as far as Thales's mathematics is concerned, to temper dogmatic assertions about what Thales could prove. However, the article does provide a useful reference, to the apophthegm of Thales quoted by Diogenes just after the two cited by Collingwood (page \pageref{apophthegms}). Perhaps all of them should be given: \begin{verse}\label{apophthegms2} Of all things that are, the most ancient is God, for he is uncreated.\\ The most beautiful is the universe, for it is God's workmanship.\\ The greatest is space, for it holds all things.\\ The swiftest is mind, for it speeds everywhere.\\ The strongest, necessity, for it masters all.\\ The wisest, time, for it brings everything to light.\\ \gr{presb'utaton t~wn >'ontwn je'os; >ag'enhton g'ar.}\\ \gr{k'alliston k'osmos; po'ihma g`ar jeo~u.}\\ \gr{m'egiston t'opos; >'apanta g`ar qwre~i.}\\ \gr{t'aqiston no~us; di`a pant`os g`ar tr'eqei.}\\ \gr{>isqur'otaton >an'agkh; krate~i g`ar p'antwon.}\\ \gr{sof'wtaton qr'onos; >aneur'iskei g1ar p'anta.} \end{verse} The claim in \emph{Wikipedia} (as of September 27, 2016) is, \begin{quote} Topos is in Newtonian-style space, since the verb, chorei, has the connotation of yielding before things, or spreading out to make room for them, which is extension. Within this extension, things have a position. Points, lines, planes and solids related by distances and angles follow from this presumption. \end{quote} Beyond the bare apophthegm about space, no justification is offered for this claim; thus I think it is original research, unfit for \emph{Wikipedia.} It is suggested that Thales knew about the Egyptian \emph{seked,} the inverse of our notion of slope, except run is measured in palms; and rise, cubits; there being seven palms in the cubit. The textbook William G. Shute, William W. Shirk, George F. Porter, \emph{Plane and Solid Geometry,} American Book Company (1960), pp.\ 25-27, is cited for the claim: \begin{quote}\sloppy when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: all straight angles are equal, equals added to equals are equal, and equals subtracted from equals are equal. \end{quote} Apparently the textbook offers a proof in the style of Euclid. My idea is that the proof would be by symmetry. This seems to be the idea of ``Lockhart's Lament,'' pages 18--9, available through ``Devlin's Angle,'' March 2008.%%%%% \footnote{\url{www.maa.org/external_archive/devlin/devlin_03_08.html}} \appendix \chapter{Program}\label{app:prog} The invitation to speak at the Thales festival on September 24, 2016, came by email as follows. \begin{quotation}\tur{% TURAD'{\i}n (Turizm Ara\c st{\i}rmalar{\i} Derne\u gi) Didim Belediyesi ad{\i}na y\"ur\"utt\"u\u g\"u ``Markala\c sma ve Turizmde S\"urd\"ur\"ulebilirlik Projesi'' kapsam{\i}nda d\"uzenlenen bir dizi etkinlikle, Didim'in tarihi ve k\"ult\"urel de\u gerlerini \"ulkemiz ve d\"unyada tan{\i}tarak uzun y{\i}llar boyunca ya\c satmay{\i} hedefliyoruz. Bu ba\u glamda d\"uzenlenen I. Thales Bulu\c smas{\i} 24 Eyl\"ul'de Thales'in memleketi Milet'te ger\c cekle\c stirilecektir.} \tur{Amac{\i}m{\i}z, \"on\"um\"uzdeki y{\i}llarda Thales etkinli\u gini uluslararas{\i} alanda bir konferansa d\"on\"u\c st\"urmek ve de\u gerli akademisyenleri her y{\i}l ayn{\i} tarihlerde Didim'de misafir etmektir.} \tur{Konferans{\i}n yan{\i}s{\i}ra, Belediye Ba\c skanl{\i}\u g{\i} Didim'de matemati\u gin g\"uncel ya\c samdaki yerini benimsetecek, analitik d\"u\c s\"unme yetkinli\u gi kazand{\i}racak, soran, sorgulayan bireyler yeti\c stirecek Thales Bilim Akademisi'ni kurmay{\i} da hedeflemektedir. T\"urk Matematik Derne\u gi Y\"onetim Kurulu, projemize destek vermeyi kabul etmi\c stir. Dernek, ilk y{\i}l organizasyonunun bilimsel ve akademik aya\u g{\i} olmay{\i} \"ustlenmi\c s ve sizi davetli konu\c smac{\i} olarak \"onermi\c stir. Milet Anfitiyatro'da, bir konu\c sma yapmak \"uzere sizi de aram{\i}zda g\"ormekten onur duyar{\i}z.} \tur{Konu\c sma, siz de\u gerli misafirlerimizin tercih edece\u gi bir konuda, yakla\c s{\i}k 20 dakikal{\i}k s\"urede tamamlanacak \c sekilde planlanm{\i}\c st{\i}r.} \tur{PROGRAM} \begin{compactitem} \item \tur{24 Eyl\"ul Saat 12:00'de \"o\u gle yeme\u ginde bulu\c sma} \begin{compactitem} \item \tur{Yemekten sonra Apollon Tap{\i}na\u g{\i} gezisi ve Milet'e gidi\c s} \item \tur{Milet Antik Kenti gezisi, anfitiyatroda konu\c smalar} \item \tur{Bas{\i}n a\c c{\i}klamas{\i}} \end{compactitem} \item \tur{Milet M\"uzesi bah\c cesinde kokteyl} \item \tur{20:00 Ak\c sam kapan{\i}\c s yeme\u gi (D-Marin)} \item \tur{25 Eyl\"ul Kahvalt{\i} ve d\"on\"u\c s} \end{compactitem} \tur{(Konaklama : Venosa Beach Resort Hotel)} \tur{Didim Belediyesinin evsahipli\u ginde ger\c cekle\c secek bu program\-da, kat{\i}l{\i}m teyidinizi takiben seyahat ve konaklama organizasyonunuz ekibimiz taraf{\i}ndan ger\c cekle\c stirilecektir.} \tur{\.Ilginize \c simdiden te\c sekk\"ur eder, geri d\"on\"u\c s\"un\"uz\"u rica ederiz.} % %\tur{24 Eyl\"ul'de Didim'de g\"or\"u\c smek dile\u giyle,} \end{quotation} The list of speakers and titles was later announced as: \begin{compactdesc} \item[Bet\"ul Tanbay] \tur{A\c c\i l\i\c s Konu\c smas\i} \item[Ay\c se Berkman] \tur{Tales'in Hesaplar\i} \item[David Pierce] \tur{Kan\i t Kavram\i n\i n \"Onc\"us\"u Olarak Thales} \item[Attila A\c skar] \tur{Finans Piyasalar\i nda Futures Yat\i r\i m Sistemi} \item[Ali Karatay] \tur{Tales ilk matematik felsefecisi miydi?} \item[Alp Eden] \tur{Cumhuriyet D\"onemi Matematik\c cileri} \end{compactdesc} All speakers had the title \tur{Profes\"or Doktor.} We had been told that one talk would be called \tur{Thales'in bug\"un\"u etkileyen \c cal\i\c smalar\i}; this was apparently replaced with the talk whose title referred to finance. \chapter{My talk} %\begin{mycomment} In June 2015 in Istanbul, at the 5th World Congress and School on Universal Logic, I gave course of three lectures on the Compactness Theorem. After the course, I typed up my handwritten lecture-notes, correcting them according to my memory of what I had actually written on the whiteboards and said out loud. I needed about six typeset pages (size A5, 12 point type, text body covering nine sixteenths of the area of a page) to cover each lecture; I had covered about four handwritten pages in each lecture. Each lecture had lasted one hour. For the 2016 Thales Meeting, I had to prepare to speak for a third of an hour. Should I distill my thoughts on Thales into two pages? I did not expect to be writing on boards, though I planned to use some models to explain some theorems. In the event, practice showed that 20 minutes were enough for six typeset pages (of the dimensions described, but including some diagrams). Including front page and contents page, I had eight pages (printed on either side of two sheets of A4 paper) to take to the Meeting. Below, typeset as a quotation, is the English text that I had started with. In preparing the Turkish version from this, I ended up dropping \begin{compactenum}[1)] \item the paragraph about Teos; \item the second sentence in the quotation from the Universal Declaration of Human Rights, and all of the ensuing quotation from Pericles; and \item any mention of the rule $A=\uppi r^2$. \end{compactenum} I didn't sketch the simple Figure \ref{fig:I.47}, since the Meeting organizers had been able to fulfill my request to print out, on a foam board of size A1, the color version of Figure \ref{fig:I.47-big}. Since Thales's birth year had been given as 625 in an earlier talk, I pointed out in my own talk that this year was based on the assumption that Thales had been 40 years old in the year of the solar eclipse. I had planned to use two chopsticks to talk about vertical angles. I had also prepared some cardboard triangles and circles. In the event, I forgot these in the hotel. At the Didim restaurant where we lunched, I was able to pick up two drinking straws to use in place of the chopsticks. At the theater of Miletus, I folded a sheet of paper into an isosceles triangle. Since a whiteboard was available, I drew something like Figure \ref{fig:rect}. I had assumed that the voices of speakers would be unamplified, as they would have been in ancient times. But we had to speak into a microphone: either the microphone fixed to the podium, which to my feeling was too far from the audience, or the handheld microphone, which kept my from using props (the drinking straws, the paper triangle) as I wished. I had practiced my talk, but not memorized it. I ended up referred to my printed text more than I had hoped. Sometimes my words did not flow. I do not know what to do about this but continue to practice speaking Turkish. %\end{mycomment} \begin{quotation} When I first came to Turkey, I used Herodotus as a travel guide. With Ay\c se's family, when I visited the ruins of Sardis, east of Izmir, I knew that it had been the capital of Lydia. The Lydians had once been at war with the Medians for six years, when a solar eclipse occurred. This scared the warring parties into making peace. According to Herodotus, Thales of Miletus had predicted the eclipse. We know now that the eclipse happened in 585 \bce Two weeks ago, Ay\c se and I were in Teos, which is today S\i\u gac\i k. In Teos there is a well-preserved bouleuterion, a council chamber. According to Herodotus, Thales recommended that the Ionians build this. I say in my title that I want to talk about Thales as the originator of the concept of \emph{proof.} A proof shows how a certain proposition follows from previously accepted principles. The earliest \emph{mathematical} proofs that we still have are (mostly) those in the thirteen books of Euclid's \emph{Elements.} These books were written around 300 \bce{} In the mathematics department of Mimar Sinan Fine Arts University, our first-year students read the first book of the \emph{Elements.} This culminates in two important theorems: \begin{compactenum} \item The Pythagorean Theorem: the square on the longest side (the hypotenuse) of a right triangle is equal to the squares on the other two sides. \item Every plot of land with straight borders can be measured, in the sense of being shown equal to some rectangle on a given base. \end{compactenum} In Figure \ref{fig:I.47}, \begin{figure} \centering \psset{unit=1.25mm} \begin{pspicture}(-12,-25)(37,28) \pspolygon[fillstyle=solid,fillcolor=lightgray](9,12)(0,0)(9,0)(9,12)(25,0)(9,0)(9,-25)(0,-25)(0,0)(-12,9)(-3,21) \psline(9,12)(21,28)(37,16)(25,0)(25,-25)(9,-25) \uput[dl](0,0){$A$} \uput[u](9,12){$B$} \uput[dr](25,0){$C$} \end{pspicture} \caption{The Pythagorean Theorem}\label{fig:I.47} \end{figure} angle $ABC$ is right. Euclid shows that the shaded square on the left is equal to the shaded rectangle, and the blank square on the right is equal to the blank rectangle. The two rectangles together are a square. Thus the Pythagorean Theorem is proved. Equality here is not \emph{sameness.} That the shaded square is equal to the shaded rectangle means the two figures have the same \emph{area}; but they are different figures. That equality is not sameness is often forgotten in mathematics today. However, according to the Universal Declaration of Human Rights, \begin{quote} All are equal before the law and are entitled without any discrimination to equal protection of the law. All are entitled to equal protection against any discrimination in violation of this Declaration and against any incitement to such discrimination. \end{quote} The notion of equality before the law is traced to the Funeral Oration of Pericles in Athens, around 430 \bce: \begin{quote} Our constitution is called a democracy because power is in the hands not of a minority but of the whole people. When it is a question of settling private disputes, everyone is equal before the law; when it is a question of putting one person before another in positions of public responsibility, what counts is not membership of a particular class, but the actual ability which the man possesses. \end{quote} In mathematics, the modern sign $=$ of equality consists of two \emph{equal} parallel straight lines: they are two lines, not one. The Egyptians had already been measuring land for centuries before Euclid. Herodotus observed that the Greeks had learned mathematics from the Egyptians. And yet the Egyptians computed the area of a four-sided field by taking the product of the averages of the opposite sides. This formula is not exact, unless the field is a rectangle. Euclid provides an exact measurement. Perhaps Egyptian tax law defined how fields were to be measured. Our students may be given the impression that mathematics works this way. Somebody tells them a rule---say, that the area of a circle is given by \begin{equation*} A=\uppi r^2 \end{equation*} ---and they have to learn it. But this is not mathematics. An assertion that I make is not mathematics unless \begin{compactenum}[1)] \item I know it is true, and \item I can explain why it is true. \end{compactenum} In fact the equation $A=\uppi r^2$ is a modern formulation of the most difficult theorem in Euclid's \emph{Elements.} It is a theorem of what we now call calculus. Several propositions in the \emph{Elements} are said to have been known to Thales. Here are four of them: \begin{compactenum} \item A diameter of a circle bisects the circle. \item The base angles of an isosceles triangle are equal to one another. \item When two straight lines intersect, the vertical angles that are created are equal, each to its opposite. \item The angle inscribed in a semicircle is right. \end{compactenum} These are theorems about \emph{every} circle, \emph{every} isosceles triangle, \emph{every} pair of intersecting straight lines. They cannot be proved by measuring specific examples. They can be understood by means of a single principle: symmetry. [Now demonstrate, using props.] Thales is also said to have thought the whole world was unified by a single underlying principle, which could be identified with water. In the same way, perhaps, Thales recognized that a single principle could account for every instance of the four propositions above. Identifying such principles is what mathematics is about; and in this sense, Thales may have been the first mathematician. \end{quotation} \chapter{Proof of the immortality on the soul}\label{app:Rep} Referred to in \S\ref{sect:Plato} (page \pageref{sect:Plato}), the following is from Book X of the \emph{Republic} \cite{Shorey-II}, with formatting and highlighting by me. The \emph{Republic} is told in the first person by Socrates himself. The reason for including Socrates's argument here is that it is an example of a mathematical proof of a non-mathematical assertion. \begin{quotation} ``Have you never perceived,'' said I, ``that \textbf{our soul is immortal} (\gr{>aj'anatos agaj`on l'egeis}), [609a] as for example for the eyes ophthalmia, for the entire body disease, for grain mildew, rotting for wood, rust for bronze and iron, and, as I say, for practically everything its congenital evil and disease?'' ``I do,'' he said. ``Then when one of these evils comes to anything does it not make the thing to which it attaches itself bad, and finally disintegrate and destroy it?'' ``Of course.'' ``Then the congenital evil of each thing and its own vice destroys it, or \textbf{if that is not going to destroy it, nothing else} [609b] \textbf{remains that could} (\gr{e>i m`h to~uto >apole~i, o>uk >`an >'allo ge a>ut`o >'eti diafje'ireien}); for obviously the good will never destroy anything, nor yet again will that which is neutral and neither good nor evil.'' ``How could it?'' he said. ``If, then, we discover anything that has an evil which vitiates it, yet is not able to dissolve and destroy it, shall we not thereupon know that of a thing so constituted there can be no destruction?'' ``That seems likely,'' he said. ``Well, then,'' said I, ``\textbf{has not the soul something that makes it evil} (\gr{yuq~h| >~ar'' o>uk >'estin <`o poie~i a>ut`hn kak'hn})?'' ``Indeed it has,'' he said, ``all the things that we were just now enumerating, [609c] \textbf{injustice and licentiousness and cowardice and ignorance} (\gr{>adik'ia te ka`i >akolas'ia ka`i deil'ia ka`i >amaj'ia}).'' ``\textbf{Does any one of these things dissolve and destroy it} (\gr{>~H o>~un ti to'utwn a>ut`hn dial'uei te ka`i >ap'ollusi})? And reflect, lest we be misled by supposing that when an unjust and foolish man is taken in his injustice he is then destroyed by the injustice, which is the vice of soul. But conceive it thus: Just as the vice of body which is disease wastes and destroys it so that it no longer is a body at all, in like manner in all the examples of which we spoke it is the specific evil which, [609d] by attaching itself to the thing and dwelling in it with power to corrupt, reduces it to nonentity. Is not that so?'' ``Yes.'' ``Come, then, and consider the soul in the same way. Do injustice and other wickedness dwelling in it, by their indwelling and attachment to it, corrupt and wither it till they bring it to death and separate it from the body?'' ``They certainly do not do that,'' he said. ``But surely,'' said I, ``it is unreasonable to suppose that the vice of something else destroys a thing while its own does not.'' ``Yes, unreasonable.'' ``For observe, Glaucon,'' [609e] said I, ``that we do not think it proper to say of the body either that it is destroyed by the badness of foods themselves, whether it be staleness or rottenness or whatever it is; but when the badness of the foods themselves engenders in the body the defect of body, then we shall say that it is destroyed owing to these foods, but by its own vice, which is disease. [610a] But the body being one thing and the foods something else, we shall never expect the body to be destroyed by their badness, that is by an alien evil that has not produced in it the evil that belongs to it by nature.'' ``You are entirely right,'' he replied. ``On the same principle,'' said I, ``\textbf{if the badness of the body does not produce in the soul the soul's badness we shall never expect the soul to be destroyed by an alien evil apart from its own defect} (\gr{>e`an m`h s'wmatos ponhr'ia yuq~h| yuq~hs ponhr'ian >empoi~h|, m'h pote >axi~wmen allotr'iou kako~u >'aneu t~hs >id'ias ponhr'ias yuq`hn >ap'ollusjia})---one thing, that is, by the evil of another.'' ``That is reasonable,'' he said. ``Either, then, we must refute this [610b] and show that we are mistaken, or, so long as it remains unrefuted, we must never say that by fever or any other disease, or yet by the knife at the throat or the chopping to bits of the entire body, there is any more likelihood of the soul perishing because of these things, until it is proved that owing to these affections of the body the soul itself becomes more unjust and unholy. But when an evil of something else occurs in a different thing and the evil that belongs to the thing is not engendered in it, [610c] we must not suffer it to be said that the soul or anything else is in this way destroyed.'' ``But you may be sure,'' he said, ``that nobody will ever prove this, that the souls of the dying are made more unjust by death.'' ``But if anyone,'' said I, ``dares to come to grips with the argument and say, in order to avoid being forced to admit the soul's immortality, that a dying man does become more wicked and unjust, we will postulate that, if what he says is true, injustice must be fatal [610d] to its possessor as if it were a disease, and that those who catch it die because it kills them by its own inherent nature, those who have most of it quickest, and those who have less more slowly, and not, as now in fact happens, that the unjust die owing to this but by the action of others who inflict the penalty.'' ``Nay, by Zeus,'' he said, ``injustice will not appear a very terrible thing after all if it is going to be fatal to its possessor, for that would be a release from all troubles. But I rather think it will prove to be quite the contrary, [610e] something that kills others when it can, but renders its possessor very lively indeed, and not only lively but wakeful, so far, I ween, does it dwell from deadliness.'' ``You say well,'' I replied; ``for when the natural vice and the evil proper to it cannot kill and destroy the soul, still less will the evil appointed for the destruction of another thing destroy the soul or anything else, except that for which it is appointed.'' ``Still less indeed,'' he said, ``in all probability.'' ``Then since it is not destroyed by any evil whatever, [611a] either its own or alien, it is evident that it must necessarily exist always, and that if it always exists it is immortal.'' ``Necessarily,'' he said. \end{quotation} \AfterBibliographyPreamble{\smaller} %\bibliographystyle{plain} %\bibliography{../../references} \begin{thebibliography}{10} \bibitem{MR2093668} Archimedes. \newblock {\em The Two Books \emph{{O}n the Sphere and the Cylinder}}, volume~I of {\em The Works of {A}rchimedes}. \newblock Cambridge University Press, Cambridge, 2004. \newblock Translated into English, together with Eutocius' commentaries, with commentary, and critical edition of the diagrams, by Reviel Netz. \bibitem{Aristotle-Basic} Aristotle. \newblock {\em The Basic Works of {A}ristotle}. \newblock Random House, New York, 1941. \newblock Edited and with an Introduction by Richard McKeon. \bibitem{Aristo-DA} Aristotle. \newblock De anima. \newblock In {\em The Basic Works of {A}ristotle\/} \cite{Aristotle-Basic}, pages 533--603. \newblock Translated by J. 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