On first principles, Wikipedia says:

A first principle is an axiom that cannot be deduced from any other within that system. The classic example is that of Euclid's Elements; its hundreds of geometric propositions can be deduced from a set of definitions, postulates, and common notions: all three types constitute first principles.

See: https://en.wikipedia.org/wiki/First_principle#:~:text=In%20formal%20logic,-In%20a%20formal&text=A%20first%20principle%20is%20an,three%20types%20constitute%20first%20principles.

I had been under the impression that first principles in philosophy are analogous to axioms in mathematics; for example, the five celebrated axioms of Euclidean geometry.

However, the above citation suggests that first principles also include definitions and common notions. If so, then which definitions? all of them?. Also, in particular, which common notions?

It is well known that Euclid defined a point as "A point is that which has no part."---which, of course, is not very clear. Hence, we may leave "point" as an undefined term and presume that anyone who reads or hears that word will have the same ``common notion'' of it as anyone else. Otherwise, we must define "part," which will produce another term in need of clarification ... and so on.

Hence, terms like ``point'' are probably best left undefined.

So, I ask:

Besides the famous 5 axioms of Euclidean geometry, what else comprises all of the FIRST PRINCIPLES of Euclidean geometry? Are all generally accepted definitions included? If not, which ones? And, specifically, what are all of the common notions that are included in those principles?

  • 1
    This source lists 7 axioms, 4 definitions, and 5 postulates. Dec 10, 2023 at 23:00
  • @KristianBerry Thank you for providing the link. It seems that the "5 postulates" are what I have referred to as axioms. Unless I am mistaken, it appears to me that the "7 axioms" are taken to be fundamental logical principles. However, I'm not sure about the four definitions which you allude to. I didn't see them succinctly listed. Perhaps I should check again.
    – DDS
    Dec 10, 2023 at 23:42
  • The four definitions are sandwiched between the things they refer to as axioms and the postulates. I'm not versed in The Elements so I can't corroborate the webpage's listing myself... Dec 11, 2023 at 0:07
  • 2
    The seven axioms set out a general theory of magnitudes which is largely separable from geometry proper. Here is a translation with some very good commentary: mathcs.clarku.edu/~djoyce/java/elements Dec 11, 2023 at 2:23
  • See Hilbert's axioms for modern axiomatization. Compared to Euclid's original one (linked above) it filled some gaps. The original one was based on 5 postulates, i.e. specific assumption regarding geometrical entities, and 5 common notions, i.e. assumptions regarding equality. Dec 11, 2023 at 8:26


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