‘Hello world’ with mathematical proofs – why care?

“Prove that root 2 is irrational.”
If you read the sentence above and thought “Why in the world do mathematical proofs exist?” then welcome to my world! I’ve always believed that mathematical proofs exist so professors can torment students. If you are in Engineering or precisely Computer Science or Mathematics, then chances are you have an idea of what I mean. Mindless proofs that you can’t seem to make a connection with the physical world. For instance, it is easy to see that knowing how to proving root 2 is irrational will not suddenly make you see the light in life. Or maybe it will?

Do you think proofs matter?

It is always easy to make up reasons about why proofs are pointless especially if because many proofs are hard to connect with the physical world. If you are on the extreme side of disliking proofs, then before you think up your endless reasons to avoid them, consider the few counter-points for why they matter:

Yes, proofs matter because:

  • With a proof, you are certain a claim always works.
  • Without proofs, it is sometimes difficult to find counter-examples to claims. For example, the equation n^2 + n + 41 = prime number (for all positive integers). This seems true but it is false. There’s an example that contradicts the claim. Hint: the number is less than 45.
  • Not all test-cases for a claim can be checked with a computer. For instance, 313(x^3 + y^3) = z^3 (for all positive integers), has a contradiction. This is a 1000-digit number!
  • Intuition gained from proofs can help you debug your programs if you are a programmer.

There are many more reasons. If you are interested in beginner proof cases then take a look at the book in reference section.

References

  1. http://www.cs.princeton.edu/courses/archive/spr10/cos433/mathcs.pdf
  2. http://www.faradayschools.com/favourite-bits/quick-reads/what-is-proof/
  3. http://faradayschools.com/wp-content/uploads/proof.jpg
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