Areas of maths thats hand wavy?

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Certain areas of mathematics initially appear "hand wavy" as they often start with intuitive concepts before being rigorously defined. Popularized accounts of mathematical ideas tend to simplify complex topics, making them more accessible but less precise. Historical examples, such as Fourier series and Euler's solution to the Basel problem, illustrate how foundational concepts were initially viewed skeptically due to their lack of rigor. Over time, these areas evolve from informal to formalized frameworks as deeper understanding develops. This progression highlights the natural development of mathematical fields.
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Any areas in maths that's hand wavy? Is so where?
 
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Any area of math must be firmly grounded in order to be an area of math.

On the other hand, a "popularized" account of any area of math will probably need to be "hand wavy" not to be intelligible to the general public. It is not a matter of the "area" but of the depth to which an individual wants to study the area.
 
Many important areas of math begin handwavy and are only later made rigorous. For example, Fourier series were scoffed by mainstream mathematicians when Fourier first thought of them because he handwaved a few things.

Euler's big breakthrough at age 29 was the solution of the Basel problem... except his "proof" by "factoring" sin(x)/x wasnt made rigorous until much later!

This is the natural way new areas of math are developed.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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