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What is the difference between the terms "analysis" and "synthesis" used in a mathematical context?
For example, Hawkins's Emergence of the Theory of Lie Groups p. 3 says that Klein and Lie
were self-styled "synthesists" in the midst of analysts and arithmeticians.
What does this mean?
There is a long history to analytic and synthetic. But after historical roots and philosophical battles were forgotten analytic and synthetic came to mean two different styles of reasoning, the more formal vs the more intuitive, the more discrete and reductive vs the more holistic, etc., and not even necessarily in mathematics. There was analytic and synthetic Cubism. In neuroscience the "analytic-synthetic theory" sometimes refers to the respective specializations of the left and the right hemispheres, the more "rational" and the more "creative" one.
In the 19th century mathematics the roughly Kantian usage was given a twist: "analytic" came to be associated with more formalization and arithmetization, while "synthetic" concerned the more intuitive and constructive approaches, especially in geometry (see e.g. synthetic geometry). When Frege argued against mathematics being synthetic he meant to rid it of "non-rigorous" Kantian intuition, and construction that required it. It is this loosened Kantian meaning that Hawking appears to have in mind. He follows Klein himself, who in Elementary Mathematics from a Higher Standpoint: Volume II: Geometry, pp.66-67 bemoaned the one-sidedness of the moderns:
"In higher mathematics, however, these words have, curiously, taken on an entirely different meaning. Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates. Rightly understood, there exists only a difference of gradation between these two kinds of geometry, according as one gives more prominence to the figures or to the formulas...
In mathematics, however, as everywhere else, men are inclined to form parties, so that there arose schools of pure synthesists and schools of pure analysts, who placed chief emphasis upon absolute "purity of method," and who were thus more one-sided than the nature of the subject demanded. Thus the analytic geometricians often lost themselves in blind calculations devoid of any geometric representation. The synthesists, on the other hand, saw salvation in an artificial avoidance of all formulas, and thus they accomplished nothing more, finally, than to develop their own peculiar language formulas, different from ordinary formulas."
The original use is derived from Pappus, who likely reports a consensus conception that emerged in late antiquity, see The Method of Analysis by Hintikka and Remes for modern commentary. And he clearly talks about two complementary aspects of solving a geometric problem:
"It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the Elder, and proceeds by the method of analysis and synthesis. Now analysis is the way from what is sought — as if it were admitted — through its concomitants [akolouthôn] in order to something admitted in synthesis. For in analysis we suppose that which is sought to be already done, and we inquire from what it results, and again what is the antecedent [proêgoumenon] of the latter, until we on our backward way light upon something already known and being first in order. And we call such a method analysis, as being a solution backwards [anapalin lysin].
In synthesis, on the other hand, we suppose that which was reached last in analysis to be already done, and arranging in their natural order as consequents [epomena] the former antecedents [proêgoumena] and linking them one with another, we in the end arrive at the construction of the thing sought. And this we call synthesis."
But the later usage was influenced by Kant's distinction between analytic and synthetic (anticipated by Locke), which altered the ancient meaning. Where analytic relied on formal logical (to Kant, syllogistic) derivations, synthetic relied on construction in intuition. Nonetheless, Kant's intuitive construction was explicitly modeled on the synthetic side of Euclid's demonstrations, see Friedman's Kant on Geometry and Spatial Intuition:
"It appears, in fact, that the proof-procedure of Euclid’s Elements is paradigmatic of construction in pure intuition throughout Kant’s discussion of mathematics in the first Critique — which includes a fairly complete presentation of the elementary Euclidean geometry of the triangle. In the Transcendental Aesthetic, for example, Kant presents the corresponding side-sum property of triangles — that two sides taken together are always greater than the third (Proposition I.20) — as an illustration of how geometrical propositions “are never derived from universal concepts of line and triangle, but rather from intuition, and, in fact, [are thereby derived] a priori with apodictic certainty” (A25/B39). And the Euclidean proof of this proposition proceeds, just like Proposition I.32, by auxiliary constructions and inferences starting from an arbitrary triangle ABC...".
As Francois Ziegler pointed out in his comment, the distinction of synthetic vs. analytic refers here to two approaches to geometry. In a nutshell, synthetic geometry is the study based solely on axioms, logical arguments and classical constructions, while analytic geometry allows the use of coordinates or calculus. The Wikipedia page gives a pretty good account: https://en.wikipedia.org/wiki/Synthetic_geometry
Klein did not completely ignore the analytic approach, as he wrote the following in his Erlangen Program:
``The distinction between modern synthesis and modern analytic geometry must no longer be re- garded as essential, inasmuch as both subject-matter and methods of reasoning have gradually taken a similar form in both. We choose therefore in the text as common designation of them both the term projective geometry . Although the synthetic method has more to do with space-perception and thereby imparts a rare charm to its first simple developments, the realm of space-perception is nevertheless not closed to the analytic method, and the formulae of analytic geometry can be looked upon as a precise and perspicuous statement of geometrical relations. On the other hand, the advantage to original research of a well formulated analysis should not be underestimated, - an advantage due to its moving, so to speak, in advance of the thought. But it should always be in- sisted that a mathematical subject is not to be considered exhausted until it has become intuitively evident, and the progress made by the aid of analysis is only a first, though a very important, step." (translated by M. W. Haskell; available at https://arxiv.org/abs/0807.3161)
I might be wrong on this, but it seems that the coordinate geometry is nowadays absent from the American high school curriculum; some elements are taught in multivariable calculus and/or linear algebra, which are considered ``advanced" courses (at the college level). Therefore the distinction between analytic and synthetic geometry appears mysterious. On the other hand, when I was in my junior high school grade in Poland in 1980s, I took a (compulsory) course on analytic geometry, from a text with a matching title. It was preceded by 2 years of axiomatic geometry, so the distinction was clear.
Grattan-Guinnesses's Convolutions in French Mathematics, 1800–1840 From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics §3.2.5 "'Analysis' and 'synthesis', and their mathematical connections", p. 135:
The basic distinction between analysis and synthesis in proof-methods in mathematics is that in analysis one starts out from the desired result and regresses until apparently impeccable principles are found, while synthesis begins with those principles and derives the result. However, especially during the 18th century mathematicians became accustomed to associate analysis with algebra and synthesis with geometry, although these connections were not clear, least of all in the development and use of the calculus.
