4

I am reading about mathematical reasoning in proofs, and the author, Terence Tao, speaks about usefulness and efficiency in proofs. He states that

Being true is different from being efficient. For instance the statement $2 = 2$ is true, but unlikely to be very useful. The statement $4 \le 4$ is also true but not very efficient (the statement $4 = 4$ is more precise). It may also be that a statement may be false yet still be useful, for instance $\pi = \frac{22}{7}$ is false, but is still useful as a first approximation. In mathematical reasoning, we only concern ourselves with truth rather than usefulness or efficiency.

So what I am trying to understand are a couple things here even if the author is saying usefulness and efficiency are not even considered in mathematical reasoning.

  1. When making a proof what does this usefulness even mean? Is this being used like I would say "A hammer is useful." or is there some mathematical evaluation for it?
  2. Does efficiency mean "precision" here? Is there a more rigorous definition for what efficiency means?

Edit: Not exactly sure how to Cite here. I enter the information on the Cite button popup but it just disappears. Quote is from Analysis 1 by Terence Tao published in 2006. Quote is from page 353.

GirnarTea
  • 51
  • Please add a citation for the quotation in your question. I formatted it as a block quote and added MathJax. – Greg Nisbet Jan 22 '23 at 18:06
  • Basically, no, there are no mathematical definitions for those words, they mean what they usually mean in natural language, and I think the quoted section illustrates that meaning pretty usefully and efficiently. – Captain Lama Jan 22 '23 at 18:08
  • Mathematicians use truths (well, provable results) to prove others; Tao is identifying, if non-rigorously, a respect or two in which some such statements can better facilitate this project than others. – J.G. Jan 22 '23 at 18:12
  • Ok so these terms are informally used. Thank you for your comments Greg Nisbet, Captain Lama and J.G.. I see no way to upvote comments, probably because I am a new user. – GirnarTea Jan 22 '23 at 18:25
  • You get that ability when your reputation is 50 or more. I've upvoted your question, which brings you nearer. – J.G. Jan 22 '23 at 18:32

1 Answers1

6

That last sentence - "In mathematical reasoning, we only concern ourselves with truth rather than usefulness or efficiency" - is saying "usefulness and efficiency, while natural ideas, are not concepts used in mathematical reasoning". In other words, there can't be "rigorous" definitions for these qualities, because they aren't rigorous concepts!

My view is that Tao's intent here is not to tell you how to make proofs, but what not to expect from proofs. Because we care more about truth than usefulness, a proof might reasonably include a statement like "2 = 2", which as Tao points out is true but does not seem helpful. Because we care more about truth than efficiency, a proof might reasonably include a statement like "$4 \leq 4$", which is true but "seems weird" because there's a more precise version that's easy to say.

I'm glad Tao brings this up, because these are issues which often confuse students. In ordinary English (and any other language, really) sentences like these often carry connotations. For example, "the sky is either blue or neon green" would generally be considered a weird or untrue thing to say, because it seems to allow the possibility that the sky might be neon green. But from a mathematical perspective, it's a true statement - the sky is blue, so it's either blue or neon green. The same thing is going on with that $4 \leq 4$ example; a student might feel like that's saying "4 might be equal to 4, but it's also possible that 4 is less than 4". Tao's point that "we only concern ourselves with truth" means that, in mathematical reasoning, we ignore these connotations and look only at what's literally true.

Reese Johnston
  • 17,988