Editing Analysis (section)

===Mathematics===
{{Main|Mathematical analysis}}

Modern mathematical analysis is the study of infinite processes. It is the branch of mathematics that includes calculus.  It can be applied in the study of classical concepts of mathematics, such as [[real analysis|real numbers]], [[complex analysis|complex variables]], [[Fourier analysis|trigonometric functions]], and [[numerical analysis|algorithms]], or of [[non-classical analysis|non-classical]] concepts like [[constructivist analysis|constructivism]], [[harmonic analysis|harmonics]], [[non-standard analysis|infinity]], and [[functional analysis|vectors]].

[[Florian Cajori]] explains in [[wikiquote:A History of Mathematics|''A History of Mathematics'']] (1893) the difference between modern and ancient mathematical analysis, as distinct from logical analysis, as follows:
<blockquote>
The terms ''synthesis'' and ''analysis'' are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in [appended to] [[Euclid's Elements|Euclid]], XIII. 5, which in all probability was framed by [[Eudoxus of Cnidus|Eudoxus]]: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." 
</blockquote>

<blockquote>
The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule, added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions.
</blockquote>

James Gow uses a similar argument as Cajori, with the following clarification, in his [https://books.google.com/books?id=KSe_ZEmHaXEC& ''A Short History of Greek Mathematics''] (1884):

<blockquote>
The synthetic proof proceeds by shewing that the proposed new truth involves certain admitted truths. An analytic proof begins by an assumption, upon which a synthetic reasoning is founded. The Greeks distinguished ''theoretic'' from ''problematic'' analysis. A theoretic analysis is of the following kind. To ''prove'' that A is B, ''assume'' first that A is B. If so, then, since B is C and C is D and D is E, therefore A is E. If this be known a falsity, A is not B. But if this be a known truth and all the intermediate propositions be [[wiktionary:convertible#Adjective|convertible]], then the reverse process, A is E, E is D, D is C, C is B, therefore A is B, constitutes a synthetic proof of the original theorem. Problematic analysis is applied in all cases where it is proposed to construct a figure which is assumed to satisfy a given condition. The problem is then converted into some theorem which is involved in the condition and which is proved synthetically, and the steps of this synthetic proof taken backwards are a synthetic solution of the problem.
</blockquote>