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Princess_Stargirl comments on Open thread, Dec. 8 - Dec. 15, 2014 - Less Wrong Discussion
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I don't really like these questions. "What exactly is a number?" doesn't really have an answer. I can give the standard answer about representing integers as certain sets. And I give the details of constructing the real numbers either as cuts or Cauchy sequences of rationals. But neither answer is very satisfying imo. Saying that integers "are" certain kinds of sets seems wrong to me (as it does to Tim Gowers). My feeling is I don't know what numbers exactly are.
I understand you probably are going to attack someone's expertise if they blank and can't say anything. But people react to things differently. I could imagine a version of myself who was didn't realize she needed to spout information even if she couldn't answer the question fully.
The other problem is my best friend studied computer science not mathematics. She is however much more intelligent than myself. Her knowledge of math is really quite good. She can give the "standard answer" to "what is an integer" but cannot give the details of a construction of the real numbers (I just asked her).
So I really think we should be careful about these gotcha questions.
I might edit the OP. It's not about there being a right or wrong answer. The whole point of asking the question is to discriminate between "'what exactly is a number?' is a deep, nebulous, philosophical question we don't have a satisfactory answer for" and "'what exactly is a number?' is a trivial question I simply haven't asked myself yet".
It's also not necessarily about posing awkward questions to other people, but about mechanically assembling these questions for any given topic, which we can then ask ourselves.
A number is anything that acts numbery under certain operations.
Integers are very numbery. Reals are pretty numbery. Polynomials and matrices are still pretty numbery. Strings and graphs are somewhat less numbery. Rubber chickens are scarcely numbery at all.
"What is a mathematical operation?" is maybe a better question.
Inspired by this, I am going to come up with an amazing new philosophical theory of truth based on Perl.