As for the 20th century discussion of the matter, Hasse und Scholz
coined the title of their article in 1928. As far as I am aware,
"Grundlagenkrisis" was a term invented during the Hilbert-Weyl
discussion between 1919 and 1922, occurring e.g. in Weyl's
U"ber die neue Grundlagenkrise der Mathematik.
Math.Z. 10 (1921) 39-79 .
This debate found wide attention, and that also in the Steinitz-
Fraenkel seminar in Kiel which both Hasse (then a young number
theorist) and Scholz (then a professor of philosophy of religion)
attended. Their title no doubt was chosen to draw attention to
their claim that besides the Grundlagenkrisis in everybody's
mouth there also had been a Grundlagenkrisis already with the Ancients.
And in distinction to the 19th century, to them the Archimedian
comments in "Ad Eratothenem", discovered only about 1904, were
known and become an important part of their argumentation.
This was centered around the methodological remark that proofs
must be finite and that the Theory of Proportions TP in book V was
the only tool (before "Weierstrass") which made such proofs possible.
Their emphasis was that TP was an axiomatic description of 'domains
of magnitudes, connected by the relation of equality between pro-
portions', and although such formulation could not have been stated
before Peano's 1891 "cutting the umbilical chord between geometry
and reality" (Freudenthal), today's mathematician has to admit that
this precisely expresses the way TP was used. [I should add though that
O.Becker in his "Eudoxos-Studien I ", Quellen und Studien 2 (1933)
311-333, observed that the older, antiphairetic theory of proportions
does suffice for the theorems up to Euclid's book X ]
I also might add that Wolfgang Krull in
U"ber die Endomorphismen von total geordneten Archimedischen
Abelschen Gruppen. Math.Z. 74 (1960) 81-90 and
Automorphismen und Spiegelungen eudoxischer Halbgruppen
Math.Z 79 (1962) 53-68
has shown that, for a totally ordered, dense Archimedian group G it
is equivalent that
(1) G is the additive group of an ordered field
(2) for any three positive a,b,c in G there exists the fourth
proportional d such that [a,b]=[c,d]
where this 'proportional equality' means that for all positive
integers m, n : ma > nb iff mc > nd ;
groups satisfying (2) Krull calls Eudoxian. In the second article he
shows that the ordered field in (1) quadratically closed if, and only
if, for any two positive a,d there exists the middle proportional x
such that [a,x] = [x,d] .
W.F.