Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: What is the "x" which occurs in "sin x"? Being an essay towards a
  conceptual Foundations of Mathematics   As Lebesgue implies, measure is the most fundamental concept of science. It
is not accidental that the principal branch of mathematics is that which deals
with measure: recently Analysis, before it Geometry. The transition of the
study of measure from geometry to analysis is a long process which is yet to be
investigated from both the logical and the conceptual points of view.
Trigonometry, thanks to Eudoxus' theory of ratio and proportion, was developed
within geometry. Now it involves analysis too. To analytically deal with the
trigonometric functions, several mathematicians answered Hardy's key question
"The whole difficulty lies in the question, What is the x which occurs in cos x
and sin x ?".
  Building on the works of Eudoxus and Lebesgue, it is shown that (to the best
of my knowledge) all of these answers are logically wrong or conceptually
unsound. In contrast, based on Eudoxus' theory, measure in general and
measuring angles in particular are investigated, and an answer to this
question, which is both logically correct and conceptually sound, is presented.
To comprehend the philosophical and historical background of the subject,
briefly several problems are considered which need further investigation and
several questions are raised which are yet to be answered. In an appendix a
correction of the proof of proposition XII.2 of Euclid's Elements, which
practically defines "pi", is given.

28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat