\documentclass[% version=last,% a5paper, 10pt,% headings=small,% bibliography=totoc,% %abstract=true,% twoside,% titlepage=false,% leqno,% open=any,% cleardoublepage=empty,% %parskip=half,% draft=true,% DIV=classic,% %DIV=12,% headinclude=false,% %titlepage=true,% pagesize] %{scrreprt} {scrbook} %%%%%%%%%%%%%% TIME %%%%%%%%%%%%%%% \usepackage{calc} \newcounter{ampm}\newcounter{hours}\newcounter{minutes} \newcommand{\printtime}{% \setcounter{ampm}{\time/720}% \setcounter{hours}{(\time-\value{ampm}*720)/60}% \setcounter{minutes}{\time-\value{ampm}*720-\value{hours}*60}% \ifthenelse{\value{ampm}=0}% {\ifthenelse{\value{minutes}>9}% {\thehours:\theminutes~a.m.}% {\thehours:0\theminutes~a.m.}}% {\ifthenelse{\value{minutes}>9}% {\thehours:\theminutes~p.m.}% {\thehours:0\theminutes~p.m.}}} % % code adapted from the % LaTeX Companion (2d ed), p. 871 \usepackage{scrpage2} \pagestyle{scrheadings} \rehead{\leftmark} \lohead{\today, \printtime} \ohead{\pagemark} \ofoot{} %\renewcommand{\captionformat}{.} \usepackage{hfoldsty,url,multicol} \usepackage[neverdecrease]{paralist} \usepackage{pstricks} \usepackage{pst-plot,pst-3d} \usepackage{relsize} % Here \smaller scales by 1/1.2; \relscale{X} scales by X \renewenvironment{quote}{\begin{list}{} {\relscale{.90}\setlength{\leftmargin}{0.05\textwidth} \setlength{\rightmargin}{\leftmargin}} \item\relax} {\end{list}} \renewenvironment{quotation}{\begin{list}{}% {\relscale{.90}\setlength{\leftmargin}{0.05\textwidth} \setlength{\rightmargin}{\leftmargin} \setlength{\listparindent}{1em} \setlength{\itemindent}{\listparindent} \setlength{\parsep}{\parskip}} \item\relax} {\end{list}} %\usepackage{verbatim} %\let\oldquote\quote %\let\endoldquote\endquote %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage[polutonikogreek,english]{babel} % From the Greek Text Society \usepackage{gfsneohellenic} \newcommand{\gk}[1]{\foreignlanguage{polutonikogreek}{% \relscale{0.8}\textneohellenic{#1}}}% Greek text \newcommand{\gkm}[1]{\text{\gk{#1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\etal}{\emph{\&~al.}} \newcommand{\ce}{\textsc{c.e.}} \newcommand{\bce}{\textsc{b.c.e.}} \usepackage{amsmath,amssymb,bm} \newcommand{\dist}[1]{\operatorname{dist}(#1)} \newcommand{\stnd}[1]{\mathbb{#1}} \newcommand{\R}{\stnd R} \newcommand{\Z}{\stnd Z} \newcommand{\N}{\stnd N} \newcommand{\Q}{\stnd Q} \newcommand{\lto}{\Rightarrow} \newcommand{\liff}{\Leftrightarrow} \newcommand{\Forall}[1]{\forall{#1}\;} \newcommand{\Exists}[1]{\exists{#1}\;} \newcommand{\divides}{\mid} \newcommand{\size}[1]{\lvert#1\rvert} \usepackage{upgreek} \usepackage{mathrsfs} \newcommand{\included}{\subseteq} %\usepackage{verbatim} \renewcommand{\leq}{\leqslant} \renewcommand{\geq}{\geqslant} \renewcommand{\theequation}{\fnsymbol{equation}} \renewcommand{\phi}{\varphi} %\renewcommand{\emptyset}{\varnothing} \newcommand{\as}{\mathrel{:\;:}} \usepackage{amsthm} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} %\newenvironment{genprop}[1]{\textbf{#1.}}{} \begin{document} \title{Analytic geometry} \author{David Pierce} \date{\today, \printtime} \publishers{Mathematics Department\\ Mimar Sinan Fine Arts University\\ \url{dpierce@msgsu.edu.tr}\\ \url{http://mat.msgsu.edu.tr/~dpierce/}} \maketitle \tableofcontents \listoffigures \addchap{Introduction} The writing of this report was originally provoked both by frustration with the lack of rigor in analytic geometry texts and by a belief that this problem can be remedied by attention to mathematicians like Euclid and Descartes, who are the original sources of our collective understanding of geometry. Analytic geometry arose with the importing of algebraic notions and notations into geometry. Descartes, at least, justified the algebra geometrically. Now it is possible to go the other way, using algebra to justify geometry. Textbook writers of recent times do not make it clear which way they are going. This makes it impossible for a student of analytic geometry to get a correct sense of what a \emph{proof} is. I find this unacceptable. If it be said that analytic geometry is not concerned with proof, I would respond that in this case the subject pushes the student back to a time before Euclid, but armed with many more unexamined presuppositions. Students today have the idea that every line segment has a length, which is a positive real number of units, and conversely every positive real number is the length of some line segment. The latter presupposition is quite astounding, since the real numbers compose an uncountable set. Euclidean geometry can in fact be done in a countable space, as David Hilbert points out. I made notes on some of these matters. The notes grew into this report as I found more and more things that seemed worth saying. There are still many more avenues to explore. Some of the notes here are just indications of what can be investigated further, either in mathematics itself or in the existing literature about it. Meanwhile, the contents of the numbered chapters of this report might be summarized as follows. \begin{compactenum} \item The logical foundations of analytic geometry as it is often taught are unclear. The subject is not presented rigorously. \item What is rigor anyway? I consider some modern examples where rigor is lacking. Rigor is not an absolute notion, but must be defined in terms of the audience being addressed. \item Ancient mathematicians like Euclid and Archimedes still set the standard for rigor. \item I have suggested \emph{how} they are rigorous; but \emph{why} are they rigorous? I don't know. But we still expect rigor from our students, if only because we expect them to be able to justify their answers to the problems that we assign to them% ---or if we don't expect this, we ought to. \item I look at an old analytic geometry textbook that I learned something from as a child, but that I now find mathematically sloppy. \item Because that book uses the odd terms \emph{abscissa} and \emph{ordinate} without explaining their origin, I provide an explanation, \linebreak which involves the conic sections as presented by Apollonius. \item The material on conic sections spills over to another chapter. Here we start to look in more detail at the geometry of Descartes. How Apollonius himself works out his theorems remains mysterious: Descartes's methods do not seem to illuminate those theorems. \item I look at an analytic geometry textbook that I once taught from. It is more sophisticated than the textbook from my childhood. This makes its failures of rigor more frustrating to me and possibly more dangerous for the student. \item I spell out more details of the justification of the use of algebra in geometry. Descartes acknowledges the need to provide this justification. \item Finally, I review how the algebra of certain ordered fields can be used to obtain a Euclidean plane. \end{compactenum} My scope here is the whole history of mathematics. Obviously I cannot give this a thorough treatment. I am not prepared to \emph{try} to do this. To come to some understanding of a mathematician, one must \emph{read} him or her; but I think one must read with a sense of what it means to do mathematics, \emph{and} with an awareness that this sense may well differ from that of the mathematician whom one is reading. This awareness requires experience, in addition to the mere will to have it. I have been fortunate to read old mathematics, both as a student and as a teacher, in classrooms where \emph{everybody} is working through this mathematics and presenting it to the class. I am currently in the third year of seeing how new undergraduate mathematics students respond to Book I of Euclid's \emph{Elements.} I continue to be surprised by what the students have to say. Mostly what I learn from the students themselves is how strange the notion of proof can be to some of them. This impresses on me how amazing it is that the \emph{Elements} was produced in the first place. It may remind me that what Euclid even \emph{means} by a proof may be quite different from what we mean today. But the students alone may not be able to impress on me some things. Some students are given to writing down assertions whose correctness has not been established. These students' proofs may end up as a sequence of statements whose logical interconnections are unclear.%%%%% \footnote{Unfortunately some established mathematicians use the same style in their own lectures.} This is not the case with Euclid. And yet Euclid does begin each of his propositions with a bare assertion. He does not preface this enunciation or \emph{protasis} (\gk{pr'otasis}) with the word ``theorem'' or ``problem'' as we might today. He does not have the typographical means that Heiberg uses in his own edition of Euclid to distinguish the protasis from the rest of the proposition. No, the protasis just sits there, not even preceded by the ``I say that'' (\gk{l'egw <'oti}) that may be seen further down in the proof. For me to notice this, na\"{\i}ve students were apparently not enough, but I had also to read Fowler's \emph{Mathematics of Plato's Academy} \cite[10.4(e), pp.~385--6]{MR2000c:01004}. \chapter{The problem} Textbooks of analytic geometry do not make their logical foundations clear. Of course I can speak only of the books that I have been able to consult: these are from the last century or so. Descartes's original presentation \cite{Descartes-Geometry} in the 17th century \emph{is} clear enough. In an abstract sense, Descartes may be no more rigorous than his successors. He does get credit for actually inventing his subject, or at least for introducing the notation we use today: minuscule letters for lengths, with letters from the beginning of the alphabet used for known lengths, and letters from the end for unknown lengths. As for his mathematics itself, Descartes explicitly bases it on an ancient tradition that culminates in the 4th century with Pappus of Alexandria. More recent analytic geometry books start in the middle of things, but they do not make it clear what those things are. I think this is a problem. The chief aim of these notes is to identify this problem and its solution. How can analytic geometry be presented rigorously? Rigor is not a fixed standard, but depends on the audience. Still, it puts some requirements on any work of mathematics, as I shall discuss in Chapter~\ref{ch:failures}. In my own mathematics department, students of analytic geometry have had a semester of calculus, and a semester of synthetic geometry from its own original source, Book I of Euclid's \emph{Elements} \cite{MR1932864,bones}. They are the audience that I especially have in mind in my considerations of rigor. But I would suggest that any students of analytic geometry ought to come to the subject similarly prepared, at least on the geometric side. Plane analytic geometry can be seen as the study of the Euclidean plane with the aid of a sort of rectangular grid that can be laid over the plane as desired. Alternatively, the subject can be seen as a discovery of geometric properties in the set of ordered pairs of real numbers. I propose to call these two approaches the \emph{geometric} and the \emph{algebraic,} respectively. Either approach can be made rigorous. But a course ought to be clear \emph{which} approach is being taken. Probably most courses of analytic geometry take the geometric approach, relying on students to know something of synthetic geometry already. Then the so-called Distance Formula can be justified by appeal to the Pythagorean Theorem. However, even in such a course, students might be asked to use algebraic methods to prove, for example, that \emph{the base angles of an isosceles triangle are equal.}%%%%% \footnote{I had a memory that this problem was assigned in an analytic geometry course that I was once involved with. However, I cannot find the problem in my files. I do find similar problems, such as to prove that the line segment bisecting two sides of triangle is parallel to the third side and is half its length, or to prove that, in an isosceles triangle, the median drawn to the third side is just its perpendicular bisector. In each case, the student is explicitly required to use analytic methods.} Perhaps what would be expected is something like the following. \begin{proof}[Proof 1.] Suppose the vertices of a triangle are $\bm a$, $\bm b$, and $\bm c$, and the angles at $\bm b$ and $\bm c$ are $\beta$ and $\gamma$ respectively as in Figure~\ref{fig:abc}. \begin{figure}[ht] \centering \begin{pspicture}(2,2.5) \pspolygon(0,0)(2,0)(1,2) \uput[dl](0,0){$\bm b$} \uput[32](0,0){$\beta$} \uput[dr](2,0){$\bm c$} \uput[148](2,0){$\gamma$} \uput[u](1,2){$\bm a$} \end{pspicture} \caption{An isosceles triangle in a vector space}\label{fig:abc} \end{figure} These angles are given by \begin{gather*} (\bm a-\bm b)\cdot(\bm c-\bm b) =\size{\bm a-\bm b}\cdot\size{\bm c-\bm b}\cdot\cos\beta,\\ (\bm a-\bm c)\cdot(\bm b-\bm c) =\size{\bm a-\bm c}\cdot\size{\bm b-\bm c}\cdot\cos\gamma, \end{gather*} We assume the triangle is isosceles, and in particular \begin{equation*} \size{\bm a-\bm b}=\size{\bm a-\bm c}. \end{equation*} Then \begin{align*} (\bm a-\bm c)\cdot(\bm b-\bm c) &=(\bm a-\bm c)\cdot(\bm b-\bm a+\bm a-\bm c)\\ &=(\bm a-\bm c)\cdot(\bm b-\bm a)+(\bm a-\bm c)\cdot(\bm a-\bm c)\\ &=(\bm a-\bm c)\cdot(\bm b-\bm a)+(\bm a-\bm b)\cdot(\bm a-\bm b)\\ &=(\bm c-\bm a)\cdot(\bm a-\bm b)+(\bm a-\bm b)\cdot(\bm a-\bm b)\\ &=(\bm c-\bm b)\cdot(\bm a-\bm b), \end{align*} and so $\cos\beta=\cos\gamma$. \end{proof} If one has the Law of Cosines, the argument is simpler: \begin{proof}[Proof 2.] Suppose the vertices of a triangle are $\bm a$, $\bm b$, and $\bm c$, and the angles at $\bm b$ and $\bm c$ are $\beta$ and $\gamma$ respectively, again as in Figure~\ref{fig:abc}. By the Law of Cosines, \begin{gather*} \size{\bm a-\bm c}^2 =\size{\bm a-\bm b}^2+\size{\bm c-\bm b}^2 -2\cdot\size{\bm a-\bm b}\cdot\size{\bm c-\bm b}\cdot\cos\beta,\\ \cos\beta =\frac{\size{\bm c-\bm b}}{2\cdot\size{\bm a-\bm b}}, \end{gather*} and similarly \begin{equation*} \cos\gamma =\frac{\size{\bm b-\bm c}}{2\cdot\size{\bm a-\bm c}}. \end{equation*} If $\size{\bm a-\bm b}=\size{\bm a-\bm c}$, then $\cos\beta=\cos\gamma$, so $\beta=\gamma$. \end{proof} In this last argument though, the vector notation is a needless complication. We can streamline things as follows. \begin{proof}[Proof 3.] In a triangle $ABC$, let the sides opposite $A$, $B$, and $C$ have lengths $a$, $b$, and $c$ respectively, and let the angles as $B$ and $C$ be $\beta$ and $\gamma$ respectively, as in Figure~\ref{fig:ABC}. \begin{figure}[ht] \centering \begin{pspicture}(2,2.5) \pspolygon(0,0)(2,0)(1,2) \uput[dl](0,0){$B$} \uput[32](0,0){$\beta$} \uput[dr](2,0){$C$} \uput[148](2,0){$\gamma$} \uput[u](1,2){$A$} \uput[ul](0.5,1){$c$} \uput[ur](1.5,1){$b$} \uput[d](1,0){$a$} \end{pspicture} \caption{An isosceles triangle}\label{fig:ABC} \end{figure} If $b=c$, then \begin{gather*} b^2=c^2+a^2-2ca\cos\beta,\\ \cos\beta=\frac a{2c}=\frac a{2b}=\cos\gamma.\qedhere \end{gather*} \end{proof} Possibly this is not considered \emph{analytic} geometry though, since coordinates are not used, even implicitly. We can use coordinates explicitly, laying down our grid conveniently: \begin{proof}[Proof 4.] Suppose a triangle has vertices $(0,a)$, $(b,0)$, and $(c,0)$, as in Figure~\ref{fig:abc0}. \begin{figure}[ht] \centering \begin{pspicture}(0,-0.5)(2,3) \psline(0,0)(1,2)(2,0) \psline{->}(-1,0)(3,0) \psline{->}(1,-0.5)(1,3) \uput[d](0,0){$b$} \uput[d](2,0){$c$} \uput[ur](1,2){$a$} \end{pspicture} \caption{An isosceles triangle in a coordinate plane}\label{fig:abc0} \end{figure} We assume $a^2+b^2=a^2+c^2$, and so $b=-c$. In this case the cosines of the angles at $(b,0)$ and $(c,0)$ must be the same, namely $\size b/\sqrt{a^2+b^2}$. \end{proof} In any case, as a proof of what is actually Euclid's Proposition I.5, this whole exercise is logically worthless, assuming we have taken the geometric approach to analytic geometry. By this approach, we shall have had to show how to erect perpendiculars to given straight lines, as in Euclid's Proposition I.11, whose proof relies ultimately on I.5. One could perhaps develop analytic geometry on Euclidean principles without proving Euclid's I.5 as an independent proposition. For, the equality of angles that it establishes can be established immediately by means of I.4, the Side-Angle-Side theorem, by the method attributed to Pappus by Proclus \cite[pp.~249--50]{1873procli}. In this case, one can still see clearly that I.4 is true, without needing to resort to any of the analytic methods suggested above. \chapter{Failures of rigor}\label{ch:failures} The root meaning of the word \emph{rigor} is stiffness. Rigor in a piece of mathematics is what makes it able to stand up to questioning. Rigor in mathematics \emph{education} requires helping students to see what questions might be asked about a piece of mathematics. An education in mathematics will take the student through several passes over the same subjects. With each pass, the student's understanding should deepen.%%%%% \footnote{It might be counted as a defect in my own education that I did not have undergraduate courses in algebra and topology before taking graduate versions of these courses. Graduate analysis was for me a continuation of my high school course, which had been based on Spivak's \emph{Calculus} \cite{0458.26001} and (in small part) Apostol's \emph{Mathematical Analysis} \cite{MR49:9123}.} At an early stage, the student need not and cannot be told all of the questions that might be raised at a later stage. But if the mathematics of an early course resembles that of a different later course, it ought to be equally rigorous. Otherwise the older student might assume, wrongly, that the earlier mathematics could in fact stand up to the same scrutiny that the later mathematics stands up to. Concepts in an earlier course must not be presented in such a way that they will be misunderstood in a later course. By this standard, students of calculus need not master the epsilon-delta definition of limit. If the students later take an analysis course, then they will fill in the logical gaps from the calculus course. The students are not going to think that everything was already proved in calculus class, so that epsilons and deltas are a needless complication. They may think there is no \emph{reason} to prove everything, but that is another matter. If students of calculus never study analysis, but become engineers perhaps, or teachers of school mathematics, they are not likely to have false beliefs about what theorems can be proved in mathematics; they just will not have a highly developed notion of proof. By introducing and using the epsilon-delta definition of limit at the very beginning of calculus, the teacher might actually violate the requirements of rigor, if he or she instills the false notion that there is no rigorous alternative definition of limits. How many calculus teachers, ignorant of Robinson's ``nonstandard'' analysis \cite{MR1373196}, will try to give their students some notion of epsilons and deltas, out of a misguided notion of rigor, when the intuitive approach by means of infinitesimals can be given full logical justification? On the other hand, in mathematical circles, I have encountered disbelief that the real numbers constitute the unique complete ordered field. Since every valued field has a completion, it is possible to suppose wrongly that every ordered field has a completion. This confusion might be due to a lack of rigor in education, somewhere along the way.\footnote{I may have been saved from this confusion by Spivak's final chapter, ``Uniqueness of the real numbers'' \cite[ch.~29]{0458.26001}.} There are (at least) two ways to obtain $\R$ from $\Q$. One way is to take the quotient of the ring of Cauchy sequences of rational numbers by the ideal of such sequences that converge to $0$. Another way is to complete the ordering of $\Q$, as by taking Dedekind cuts, and then to show that this completion can be made into a field. The first construction can be generalized to apply to non-Archimedean ordered fields; the second cannot. More precisely, the second construction can be applied to a non-Archimedean field, but the result is not a field. It might be better to say that while the first construction achieves the completeness of $\R$ as a metric space, the second achieves the \emph{continuity} of $\R$ as an ordered field. At any rate, continuity is the word Dedekind uses for what his construction is supposed to accomplish \cite{MR0159773}. In an elementary course, the student may learn a theorem according to which certain conditions on certain structures are logically equivalent. But the theorem may use assumptions that are not spelled out. This is a failure of rigor. In later courses, the student learns logical equivalences whose assumptions \emph{are} spelled out. The student may then assume that the earlier theorem is like the later ones. It may not be, and failure to appreciate this may cause the student to overlook some lovely pieces of mathematics. The word \emph{student} here may encompass all of us. The supposed theorem that I have in mind is that, in number theory, the principles of induction and well ordering are equivalent.\footnote{``Either principle may be considered as a basic assumption about the natural numbers'' \cite[ch.~2, p.~23]{MR1373196}. I use this book as an example because it is otherwise so admirable.} Proofs of two implications may be offered to back up this claim, though one of the proofs may be left as an exercise. The proofs will be of the standard form. They will look like other proofs. And yet, strictly speaking, they will make no logical sense, because: \begin{compactitem} \item Induction is a property of algebraic structures in a signature with a constant, such as $1$, and a singulary function symbol such as ${}'$ (``prime'') for the operation of adding $1$. \item Well ordering is a property of ordered structures. \end{compactitem} When well ordering is used to prove induction, a set $A$ is taken that contains $1$ and is closed under adding $1$, and it is shown that the complement of $A$ cannot have a least element. For, the least element cannot be $1$, and if the least element is $n+1$, then $n\in A$, so $n+1\in A$, contradicting $n+1\notin A$. It is assumed here that $n