Posted by: Alexandre Borovik | September 29, 2020

What is it about mathematics that may change your thoughts?

My answer on Quora:

What is it about mathematics that may change your thoughts?

At the level of school mathematics, I am afraid, nothing is likely to change your mind. Changes start when you master proof (this does not happen in school mathematics nowadays, and less and less features in the undergraduate university mathematics). Mastering proofs means being able to solve problems of that kind:

given a statement you have never seen before, prove it or construct a counterexample.

An example of this problem:

Part 1

\(1^2=1\)

 

\(11^2 = 121\)

 

\(111^2 = 12321\)

 

\(1111^2 = 1234321\)

 

\(\dots\)

 

You see a distinctive symmetric pattern. Will this pattern continue forever? Prove this or give an example of when pattern breaks.

Part 2.

\(1^2 = 1\)

 

\((1+x)^2 = 1+2x+x^2\)

 

\((1+x+x^2)^2 = 1+2x+3x^2+2x^3+x^4\)

 

\((1+x+x^2+x^3)^2 = 1+2x+3x^2+4x^3+3x^4+2x^5 + x^6\)

 

\(\dots\)

 

You see a similar symmetric pattern. Will this pattern continue forever? Prove this or give an example of when pattern breaks.

Part 3. If you got (and justified by proof or counterexample) different answers in Part 1 and 2, explain why.

Mastering this kind of thinking means that you learn to look at the both alternatives: true / false without knowing in advance which one is correct. This is a skill that most people are lacking in their real life problem solving, they concentrate only on one option. This deficiency of thinking leads them to chose one option on the basis of … well, frequently they cannot explain, if asked, on the basis of what. Quite frequently, their choice was made on the basis of their prejudices and preconceptions.


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