According to this site Zeno of Sidon argued that

even if we admit the fundamental principles of geometry, the deductions from them cannot be proved without the admission of something else as well which has not been included in the said principles, and he intended by means of these criticisms to destroy the whole of geometry

I am pretty confident that I saw some other author that wrote that Zeno just stated that more axioms were needed to deduce certain theorems in Euclid and didn't state the above position in any generality. What's the consensus on the matter if there is one?

  • @MauroALLEGRANZA The first link is certainly not a good source for answering my question since it doesn't seem to have anything with Zeno of Sidon.As for the second I can't see the content of the paper, why are you torturing me with references that may have little to nothing to do with my questions? – GEP Jan 11 at 17:00
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    The paragraph is quoted from here based on Proclus here. I suspect that the general statement of the argument and the part about trying to destroy geometry is just Heath's commentary. – b a Jan 11 at 20:05
  • @ba That's what I think too since the general version of the argument seems too much of a conceptual leap for any of the ancients to have been able to make. – GEP Jan 11 at 22:04

We do not have too many original sources for Zeno of Sidon, most are cited in Sedley, Epicurus and the mathematicians of Cyzicus. But Proclus gives an extensive response to his critiques of Euclidean geometry in A commentary on the first book of Euclid's Elements, 214-218. Heath's comment is a close paraphrase of Proclus's:

"Since some persons have raised objections to the construction of the equilateral triangle with the thought that they were refuting the whole of geometry, we shall also briefly answer them. The Zeno whom we mentioned above asserts that, even if we accept the principles of the geometers, the later consequences do not stand unless we allow that two straight lines cannot have a common segment."

There is nothing in either Heath or Proclus about "any geometric system", nor is such a concept even hinted at until 18th century. Nor do they imply that the unstated assumptions must be there. If anything, Zeno criticiszes geometers for not stating them, and using wrong unstated assumptions at that. As Sedley discusses, Zeno's objection likely stems from the Epicurean concept of elahiston, the smallest conceivable length, existence of which would contradict what Zeno says geometers need to allow. But both Euclid and Epicureans like Zeno believed in just one "true" geometry, not multiple "geometric systems", they just disagreed as to what its correct principles were.

It is interesting that Proclus's response assigns to definitions part of the role played by axioms in modern axiomatic systems, further indicating that Euclid's "system" was structurally different:

"Therefore, says Zeno, even if the principles be granted, the consequences do not follow unless we also presuppose that neither circumferences nor straight lines can have a common segment. To this we must reply first that in a sense it is presupposed in our first principles that two straight lines cannot have a common segment. For the definition of a straight line contains it, if a straight line is a line that lies evenly with all the points on itself."

One could say that Zeno anticipated the movement of 16-17th century geometers to extract such definitional presuppositions into separate postulates. The closest Proclus comes to something like modern pluralism of "systems" is when discussing Posidonius's response to Zeno:

"Besides, he adds, Epicurus himself, and all other philosophers, admit that they have proposed many possible as well as many impossible hypotheses for the sake of examining their consequences."

But the "hypotheses" proposed do not form alternative systems. They are temporary assumptions within "the system", like assumptions about the figure that lead to different cases to be considered in demonstrations until a contradiction is reached (or the desired conclusion). Hypotheses function similarly to conditional assumptions discharged in natural deduction proofs, only sometimes discharging can not be done and they remain as additional postulates. The parallel postulate is conjectured to have emerged in this way (Euclid takes pains to avoid using it for as long as he can in book I of Elements).

On the role of definitions as axioms characteristic of pre-modern approach to geometry, and mathematics generally, and on how even Euclidean geometry only slowly emerges as a "geometric system" in Leibniz's work see De Risi, Leibniz on the Parallel Postulate and the Foundations of Geometry.

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