Why is Euclid's SAS proof considered not valid?

MegaMoh

New member
It seems reasonable. Why wouldn't it.

pka

Elite Member
To what proof of Euclid does it refer?
Can you refer us to text material that is the basis for you quarry?

JeffM

Elite Member
In the 19th century, mathematicians began to demand far more rigor in proofs than previously. In part this was because of the development of non-Euclidean geometries early in the century, and in part because of concerns about the logical validity of calculus, a branch of mathematics that was then historically recent and far from rigorous in its development.

When these much higher standards of rigor were applied to Euclid, it was found that a number of his early theorems did not meet the new standards. (This is not really a criticism of Euclid; his standards were considered good enough for thousands of years. And the average high school student is likely to consider the deficiencies mere pettifogging.) So Euclid was done over. The first answer in the link below gives more details with respect to the specific issue of SAS, but I point out that the very first proof in Euclid is also considered to not meet modern standards of rigor though common sense will not even notice the defect.

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MegaMoh

New member
In the 19th century, mathematicians began to demand far more rigor in proofs than previously. In part this was because of the development of non-Euclidean geometries early in the century, and in part because of concerns about the logical validity of calculus, a branch of mathematics that was then historically recent and far from rigorous in its development.

When these much higher standards of rigor were applied to Euclid, it was found that a number of his early theorems did not meet the new standards. (This is not really a criticism of Euclid; his standards were considered good enough for thousands of years. And the average high school student is likely to consider the deficiencies mere pettifogging.) So Euclid was done over. The first answer in the link below gives more details with respect to the specific issue of SAS, but I point out that the very first proof in Euclid is also considered to not meet modern standards of rigor though common sense will not even notice the defect.

JeffM

Elite Member

I also found this paper, whose introduction discusses various flaws in Euclid's proofs, without singling out I.4: https://link.springer.com/content/pdf/10.1007/s10472-018-9606-x.pdf
Wow. The second link is a powerhouse. It would take me a week to grasp it fully. Thank you for the link.

By the way, I have an historical supposition about the logical deficiencies in Euclid. He was idealizing compass and straight-edge geometry and may never have asked himself about things that are obviously true about physical reality. That is, his geometry may have been about idealized points, lines, circles, and polygons, but he missed axioms needed to translate from the physical to the ideal. Platonism makes it easy to make that mistake. If reality is the shadow of the ideal, then the physical fact that two approximate circles scratched in the sand intersect means that the corresponding ideal circles must intersect in the ideal world of the imagination.

pka

Elite Member
It seems reasonable. Why wouldn't it.
Wow. The second link is a powerhouse. It would take me a week to grasp it fully. Thank you for the link.
By the way, I have an historical supposition about the logical deficiencies in Euclid. He was idealizing compass and straight-edge geometry and may never have asked himself about things that are obviously true about physical reality. That is, his geometry may have been about idealized points, lines, circles, and polygons, but he missed axioms needed to translate from the physical to the ideal. Platonism makes it easy to make that mistake. If reality is the shadow of the ideal, then the physical fact that two approximate circles scratched in the sand intersect means that the corresponding ideal circles must intersect in the ideal world of the imagination.
Here is a small & cheap used paperback worth your consideration.