Geometry as made rigorous by Euclid and Descartes

Seminar at Mimar Sinan

David Pierce

Seminar at Mimar Sinan

David Pierce

For Immanuel Kant (born 1724), the discovery of mathematical proof by Thales of Miletus (born around 624 B.C.E) is a revolution in human thought. Modern textbooks of analytic geometry often seem to represent a return to prerevolutionary times. The counterrevolution is attributed to René Descartes (born 1596). But Descartes understands ancient Greek geometry and adds to it. He makes algebra rigorous by interpreting its operations geometrically.

The definition of the real numbers by Richard Dedekind (born 1831) makes a rigorous converse possible. David Hilbert (born 1862) spells it out:

geometry can be interpreted in the ordered field of real numbers, and even in certain countable ordered fields.

Geometry

Menaechmus's solutionin fact one of his two solutions to the problem above can be understood as follows.

We obtain two equations\begin{align*}xy&=2,& 2x&=y^2.\end{align*}It is known that these are the equations of certain

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Figure 1: Menaechmus’s finding of two mean proportionals

Figure 1: Menaechmus’s finding of two mean proportionals

Then the axis of the parabola will be the horizontal asymptote of the hyperbola. The

Was Menaechmus doing analytic geometry as we understand it? Perhaps not. Today we would just calculate the solution to the original system as\begin{equation*}(x,y)=(\sqrt[3]2,\sqrt[3]4).\end{equation*}But a point with these coordinates cannot be found with the usual tools of straightedge and compass. Menaechmus gives us reason to believe that this point exists anyway. The reason he gives is geometric.

Two thousand years later, René Descartes seems to share the view that solutions to equations should be understood geometrically. For example, in figure below,

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Figure 2: Descartes’s locus problem

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Figure 2: Descartes’s locus problem

assuming $GE=EA=AI=a$, suppose we want the locus of points $C$ such that\begin{equation*}CF\cdot CD\cdot CH=CB\cdot CM\cdot AI,

\end{equation*}that is,\begin{equation*}(2a-y)(a-y)(a+y)=yxa.\end{equation*}Given any value of $y$, we can compute $x$ and thus sketch the curve as in figure below.

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Figure 3: The locus itself

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Figure 3: The locus itself

But Descartes finds it worthwhile to do more. He shows that the point $C$ lies on the intersection, shown in figure below,

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Figure 4: Descartes’s geometrical solution

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Figure 4: Descartes’s geometrical solution

• a parabola with axis $AB$ and

• the straight line through $GL$, where $KL=a$.

Thus the curve given by the cubic equation above becomes geometrically meaningful.

Again, we think the problem of duplicating the cube is solved simply by taking the cube root of $2$.

But how is this taken? There is an algorithm for finding

But why do we think these approximations have a limit? We can just declare that $\sqrt[3]2$ is some infinite decimal expansion.

But why do we think that infinite decimal expansions like this compose a field?

Richard Dedekind claims that, before he gave a rigorous definition of the rational numbers, the theorem\begin{equation*}\sqrt2\cdot\sqrt3=\sqrt6\end{equation*}had not been proved. David Fowler (author of

There is

3.1415926535\dots+0.8584073464\dots

\end{equation*}It is either $3.9\dots$ or $4.0\dots$, but we cannot specify a number of digits that are sufficient to tell us which. For example\begin{equation*}

1.222\dots\times0.818181\dots\end{equation*}which is\begin{gather*}\left(1+\frac29\right)\times\frac{81}{99}=\frac{11}9\times\frac{81}{99}=1;\end{gather*}but no amount of multiplying finite decimal approximations tells us that the product is not required to begin as $0.9$.

Dedekind's definition of the real numbers explicitly avoids making use of geometric notions. Therefore we can use the set of ordered pairs of real numbers as a

Thus there are two complementary approaches to analytic geometry. Either geometry or algebra can be taken as fundamental. But textbooks assume both of these foundations. I think this is a defect of rigor.