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I am quite happy with that "definition", not least because it's obviously not intended to be formal. But anyway, a very standard thing for a mathematics lecture

I am quite happy with that "definition", not least because it's obviously not intended to be formal. But anyway, a very standard thing for a mathematics lecturer to say is "This is the only thing it could be", or "you know what this concept is already; we are just making some formal symbols to capture it", or words to that effect. It's pedagogically important to distinguish between genuinely new things that the reader is going to have to put some effort into rearchitecting their worldview around, versus things the reader already knows but might not have seen in this form before. Since everybody in this target audience already knows how to count, the treatment of the natural numbers is surely going to be in the latter category.

lisper 5y ago
It's worse than non-formal, it's circular, and hence vacuous. It has the same information content as "A foo is an object that you build to act like the concept of a foo."
- Smaug123 5y ago
  No! But I am not going to re-explain it; I have already said precisely why the information content is different, in the comment to which you replied. I shall merely quote it again. > It's pedagogically important to distinguish between genuinely new things that the reader is going to have to put some effort into rearchitecting their worldview around, versus things the reader already knows but might not have seen in this form before. Since it is common knowledge that nobody already knows what a "foo" is, but it is common knowledge that everyone already has a pretty uniform idea of what a "number" means in everyday life (really, how many ways are there you can count from 1 to 10), my assertion applies.
  - lisper 5y ago
    > everyone already has a pretty uniform idea of what a "number" means in everyday life Well, that's true, but that common notion is wrong. So if that is what you are appealing to then you have different problems. For example, on the common notion of number, there is no number whose square is negative one. (And indeed mathematicians got hung up on that for centuries before they figure out that they were wrong!)
    - Smaug123 5y ago
      Oh come now, it's not wrong. I shall quote the article. > # The natural numbers > Numbers are objects you build to act like the concept of a number. Here we begin to construct the natural number object. This is obviously referring to the natural numbers. It says so both before and after the statement you're talking about. There is no natural number whose square is negative one.
      - lisper 5y ago
        Let's not lose the plot here. I was critiquing a piece called "Math Foundations from Scratch". It's a little hard to tell whether it was intended to be pedagogical, targeted toward people who didn't already understand the foundations of math, or a sort of review or reference, targeted at those who did. But either way I think it fails, notwithstanding the fact that the specific flaw that I cited, a circular definition of the word "number", appeared in the context of a discussion of natural numbers. That's my opinion. I've explained at some length why I hold this opinion, but it's still just my opinion.
AnimalMuppet 5y ago
At best, it's a badly-written sentence. If he meant what you mean, he could have said "Everyone has an intuitive understanding of what 'numbers' are. In mathematics, we make a precise definition for the same concept." And that would have been fine. But with the sentence the author actually wrote, you're left wondering whether that's what he actually meant. Does the first "number" mean the same thing as the second "number" (in which case it's circular), or are they subtly different (in which case, is the reader sure that they understand the distinction)? And that reveals the bigger problem. Mathematics has been described as the study of things that have precise definitions. If the writer is this fuzzy in the introduction, can I trust him to be correctly precise?

lisper 5y ago

It's worse than non-formal, it's circular, and hence vacuous. It has the same information content as "A foo is an object that you build to act like the concept of a foo."

Smaug123 5y ago
No! But I am not going to re-explain it; I have already said precisely why the information content is different, in the comment to which you replied. I shall merely quote it again. > It's pedagogically important to distinguish between genuinely new things that the reader is going to have to put some effort into rearchitecting their worldview around, versus things the reader already knows but might not have seen in this form before. Since it is common knowledge that nobody already knows what a "foo" is, but it is common knowledge that everyone already has a pretty uniform idea of what a "number" means in everyday life (really, how many ways are there you can count from 1 to 10), my assertion applies.
- lisper 5y ago
  > everyone already has a pretty uniform idea of what a "number" means in everyday life Well, that's true, but that common notion is wrong. So if that is what you are appealing to then you have different problems. For example, on the common notion of number, there is no number whose square is negative one. (And indeed mathematicians got hung up on that for centuries before they figure out that they were wrong!)
  - Smaug123 5y ago
    Oh come now, it's not wrong. I shall quote the article. > # The natural numbers > Numbers are objects you build to act like the concept of a number. Here we begin to construct the natural number object. This is obviously referring to the natural numbers. It says so both before and after the statement you're talking about. There is no natural number whose square is negative one.
    - lisper 5y ago
      Let's not lose the plot here. I was critiquing a piece called "Math Foundations from Scratch". It's a little hard to tell whether it was intended to be pedagogical, targeted toward people who didn't already understand the foundations of math, or a sort of review or reference, targeted at those who did. But either way I think it fails, notwithstanding the fact that the specific flaw that I cited, a circular definition of the word "number", appeared in the context of a discussion of natural numbers. That's my opinion. I've explained at some length why I hold this opinion, but it's still just my opinion.

AnimalMuppet 5y ago

At best, it's a badly-written sentence. If he meant what you mean, he could have said "Everyone has an intuitive understanding of what 'numbers' are. In mathematics, we make a precise definition for the same concept." And that would have been fine. But with the sentence the author actually wrote, you're left wondering whether that's what he actually meant. Does the first "number" mean the same thing as the second "number" (in which case it's circular), or are they subtly different (in which case, is the reader sure that they understand the distinction)? And that reveals the bigger problem. Mathematics has been described as the study of things that have precise definitions. If the writer is this fuzzy in the introduction, can I trust him to be correctly precise?