A Historical Introduction to the
Philosophy of Science
Ch. 3: The Ideal of Deductive Systematization
The following is a summary of the third chapter of John Losee's book, A Historical Introduction to the Philosophy of Science (fourth edition), with some ancillary notes.
Philosophers discussed in this chapter: Euclid (fl. 300 BCE); Archimedes (287–212 BCE)
[p. 20] Ancient writers considered science a deductive system of axioms, definitions and theorems, along the lines of Euclid's geometry and Archimedes' statics.
[p. 20–1] Deductive systematization requires:

axioms and theorems are deductively related

axioms are selfevident truths

theorems agree with observations
[p. 21] Some philosophers of science disagree with 2. and 3.
Euclid and Archimedes deduced theorems from axioms by:

reductio ad absurdum arguments – assume theorem to be proved is not true and then deduce a contradiction (e.g., Archimedes' proof that 'weights that balance at equal distances from a fulcrum are equal')

method of exhaustion – show consequences of contraries of a theorem inconsistent with axioms (e.g., Archimedes' proof that the area of a circle is equal to the area of a certain right triangle)
Euclid's geometry failed first requirement that theorems are deduced from axioms. This was fixed by Hilbert in the 19th Century.
[p. 21–2] Aristotle and Pythagorean tradition agreed with second requirement that axioms are selfevident truths.
[p. 22] Thinkers saving the appearances in mathematical astronomy disagreed with Aristotle and Pythagoreans—all that is required is that the deductive consequences agree with observations.
The third requirement that the theorems agree with observations poses problems not recognized by Euclid and Archimedes. Some of the terms in the deductive system apply to idealized entities that cannot occur in nature (e.g., 'rod' that does not bend and has perfectly uniform weight distribution).
[p. 23] Archimedes' focus on ideal entities reflects the Platonic view that the phenomenal realm (the observed world) is but an "imitation" or "reflection" of the "real world". This dualism was important for Galileo and Descartes.
Questions to Consider:

Must the axioms of mathematics and geometry be selfevident to be accepted? What of the postulates used in scientific theories?

What is the relation between the idealized items in a set of axioms and the real world?

Is our observed world simply a world of appearance?