A circle can be written as the solution set of an equation $x^2+y^2=1$.

I have heard that Fermat associated figures with equations in the 17th century.

Did people see figures as sets of points before set theory? Otherwise, what is the difference between the view of Fermat et al. and the view of set theorists?

  • 3
    $\begingroup$ No, the notion "solution set" did not exist in those days. A "line" was not considered a "set of points". Rather: there are points and there are lines, and in some cases a point lies on a line. $\endgroup$ May 14 at 0:04
  • $\begingroup$ As far as I know, Cantor was the first to give a formal analysis of the continuum as a point set. Prior to Cantor it was generally thought that a continuum could not contain any points. The reason being that any part of a continuum must be divisible while a point is not divisible. AFAIK, the arithmetization of geometry that began with Descartes' La Géométrie made use of infinitesimals and allowed one to talk of points lying on a line without making any explicit claim about the nature of the continuum. $\endgroup$
    – nwr
    May 14 at 2:17
  • 1
    $\begingroup$ That some lines associated to a curve stand in some relation ("equation") does not require assuming that the curve is made of points. People did not see figures as sets of points until 20th century, but they did associate figures to equations since middle ages, and to their predecessors called "symptoms" since antiquity. Such figures were called loci (plural of locus). Aristaeus wrote Five Books on Solid Loci c. 350 BC, Fermat's work was called Introduction to Loci. $\endgroup$
    – Conifold
    May 14 at 4:12
  • $\begingroup$ Even in the second half of the 20th century I was taught the notion of locus for things like circles. $\endgroup$
    – mdewey
    May 15 at 15:52


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