Abstract
Reading older mathematical texts always involves a hermeneutical dilemma: in order to make sense of the mathematical content of a given old text one wants to interpret it in modern terms; in order to see the difference between the modern mathematical thinking and older ways of mathematical thinking one wants to avoid anachronisms and understand the old text in its own terms (Unguru 1975).
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Notes
- 1.
Being between Scylla and Charybdis is an idiom deriving from Greek mythology. Scylla and Charybdis were mythical sea monsters noted by Homer. Scylla was rationalized as a rock shoal (described as a six-headed sea monster) on the Italian side of the strait and Charybdis was a whirlpool off the coast of Sicily. They were regarded as a sea hazard located close enough to each other that they posed an inescapable threat to passing sailors; avoiding Charybdis meant passing too close to Scylla and vice versa. (after WikipediA)
- 2.
See the full quote from Friedman in the end of Sect. 3.5.
- 3.
As far as mutual influences are concerned two things are certain: (i) Proclus read Aristotle and (ii) Aristotle had at least a basic knowledge of the mathematical tradition, on which Euclid later elaborated in his Elements (as Aristotle’s mathematical examples clearly show Heath 1949). It remains unclear whether Aristotle’s work could influence Euclid. In my view this is unlikely. However Aristotle’s logic certainly played an important role in later interpretations and revisions of Euclid’s Elements. I leave this interesting issue outside of the scope of this book.
- 4.
The difference A − B of two figures A, B is a figure obtained through “cutting” B out of A; the sum A + B is the result of concatenation of A and B. These operations are not defined up to congruence of figures (for there are, generally speaking, many possible ways, in which one may cut out one figure from another) but, according to Euclid’s Axioms, these operations are defined up to Euclid’s equality. This shows that Euclid’s equality is weaker than congruence: according to Axiom 4 congruent objects are equal but, generally, the converse does not hold. In the case of (plane) figures Euclid’s equality is equivalent to the equality (in the modern sense) of their air.
- 5.
Here are some quotes:
By first principles of proof [as distinguished from first principles in general] I mean the common opinions on which all men base their demonstrations, e.g. that one of two contradictories must be true, that it is impossible for the same thing both be and not to be, and all other propositions of this kind. (Met. 996b27-32, Heath’s translation, corrected)
Here Aristotle refers to a logical principle as “common opinion”. In the next quote he compares mathematical and logical axioms:
We have now to say whether it is up to the same science or to different sciences to inquire into what in mathematics is called axioms and into [the general issue of] essence. Clearly the inquiry into these things is up to the same science, namely, to the science of the philosopher. For axioms hold of everything that [there] is but not of some particular genus apart from others. Everyone makes use of them because they concern being qua being, and each genus is. But men use them just so far as is sufficient for their purpose, that is, within the limits of the genus relevant to their proofs. Since axioms clearly hold for all things qua being (for being is what all things share in common) one who studies being qua being also inquires into the axioms. This is why one who observes things partly [=who inquires into a special domain] like a geometer or a arithmetician never tries to say whether the axioms are true or false. (Met. 1005a19-28, my translation)
Here is the last quote where Aristotle refers to Axiom 3 explicitly:
Since the mathematician too uses common [axioms] only on the case-by-case basis, it must be the business of the first philosophy to investigate their fundamentals. For that, when equals are subtracted from equals, the remainders are equal is common to all quantities, but mathematics singles out and investigates some portion of its proper matter, as e.g. lines or angles or numbers, or some other sort of quantity, not however qua being, but as […] continuous. (Met. 1061b, my translation)
The “science of philosopher” otherwise called the “first philosophy” is Aristotle’s logic, which in his understanding is closely related to (if not indistinguishable from) what we call today ontology. After Alexandrian librarians we call today the relevant collection of Aristotle’s texts by the name of metaphysics and also use this name for a relevant philosophical discipline.
- 6.
Although the choice of letters in Euclid’s notation is arbitrary the system of this notation is not. This traditional geometrical notation has a relatively stable and rather sophisticated syntax, which I briefly describe in what follows.
- 7.
I reproduce here Fitzpatrick’s footnote about Euclid’s expression “let it be postulated”:
The Greek present perfect tense indicates a past action with present significance. Hence, the 3rd-person present perfect imperative Hitesthw could be translated as “let it be postulated”, in the sense “let it stand as postulated”, but not “let the postulate be now brought forward”. The literal translation “let it have been postulated” sounds awkward in English, but more accurately captures the meaning of the Greek.
- 8.
Remind that the concepts of infinite straight line and infinite half-line (ray) are absent from Euclid’s geometry; thus the result of OP2 is always a finite straight segment. However the result of OP2 is obviously not fully determined by its single operand. This shows that OP2 doesn’t really fit the today’s usual notion of algebraic operation.
- 9.
Problem 1.1 involves a difficulty that has been widely discussed in the literature: Euclid does not provide any principle that may allow him to construct a point of intersection of the two circles involved into the construction of this Problem. This flaw is usually described as a logical flow. In my view it is more appropriate to describe this flow as properlygeometrical and fill the gap in the reasoning by the following additional postulate (rather than an additional axiom):
Let it have been postulated to produce a point of intersection of two circles with a common radius.
Even if this additional postulates is introduced here purely ad hoc, the way in which it is introduced gives an idea of how Euclid’s Postulates could emerge in the real history.
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Rodin, A. (2014). Euclid: Doing and Showing. In: Axiomatic Method and Category Theory. Synthese Library, vol 364. Springer, Cham. https://doi.org/10.1007/978-3-319-00404-4_2
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