"Analysis" vs. "Synthesis" terms

Question

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?

Answer 1

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.