Comments on: Math Proofs vs. Explanations (aka Nutrition vs. Taste)

By: Shirley

Shirley — Tue, 08 Nov 2016 03:22:31 +0000

Hey Kalid, your points inspired me thinking about how i want to learn, how we should learn, and why i was struggling before. What i found most difficult is the inability to phrase a proper questions that can help myself. Your great article get me into thinking where things go wrong.
As i have been reading more and more science textbooks, suddenly i realized that most authors tend to present math formulas rather than talking about ideas. The issue is, quite often those math formulas are just correlations rather than causality. I was troubled before with lots of topics such as physics, and now i see why: i got confused by math correlations and causality. Why mathematical correlations give us the ability to predict one thing based on others, they offer no intuition as to why. And without understanding the causality, i constantly found myself unable to connect things, explain things. Unfortunately nowadays people confuse the two as well.

By: v s s sastry

v s s sastry — Mon, 07 Nov 2016 08:53:03 +0000

beautifully explained very soothing to read

By: Hitoshi

Hitoshi — Thu, 03 Nov 2016 12:53:10 +0000

Nice article. I also agree about 4-color map theorem. I hope someday I can see the "understandable" proof of it.
- Hitoshi

By: kalid

kalid — Thu, 03 Nov 2016 01:14:53 +0000

In reply to Amit Patel.

Great question. The missing step (I should have mentioned it explicitly, it's in the article https://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/) is that Area is proportional to (any line segment)^2.

In a square, we usually write Area = side^2. But we could also say because perimeter=4s, then

Area = 1/16 p^2

Or, we could measure the diagonal (s * sqrt(2)) and get

Area = 1/2 diagonal^2

The constant factor F (1, 1/16, 1/2, etc.) just depends on the shape and the line segment we pick. That gives us:

Area(big) = Area(part1) + Area(part2)

Fc^2 = Fa^2 + Fb^2 [assume c, a and b are the same line segment in each shape]

c^2 = a^2 + b^2 [divide out scaling factor: the line segments in each shape are related using this formula]

Yep, in the circle diagram, I could have picked circles with circumference 3, 4, 5 instead of radius 3, 4, 5.