This is a question regarding why the order of the real number subsets commonly used in the mathematics community is such:
$$ \mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R} $$
Here the concept of negative numbers is introduced before the concept of quotients. Theoretically, it...
This forum is not a discussion forum. Nor is it an opinion forum. So I think this question is off-topic here. A perhaps valid question on this would be: "Have there been published studies on whether a certain order for the topics is more effective for teaching? Or more desirable for some other purpose?"
@WillOrrick The significance is the (proposed) idea negative numbers were used on the Greek counting table at Salamis, known as the Salamis Tablet, dated to 300 BC. (arXiv:1206.4349 [math.HO]) The assumption is there were no previous uses of negative numbers which lasted in a way that contributed to our use of negative numbers today.
That's what I guessed you meant. Thanks for confirming. Is Stephenson really claiming the Greeks had the concept of negative numbers? I didn't find such a strong claim in the linked article after a quick perusal. (Wikipedia does cite the article for a somewhat less strong claim, but doesn't give a page reference.) Since I've never seen negative numbers attributed to the Greeks before, I would imagine this is controversial. I don't wish to derail the topic, but if you know of more literature on the significance of the Salamis tablet in the history of negative numbers, please share.
I assume you are also aware that the Chinese seem to have dealt with negative numbers in a natural way well before anyone else. Whether that discovery "stuck" is hard for me to say based on my limited knowledge, but the Indians were developing the arithmetic of negative numbers sometime later, followed by the Arabic algebraists. I am not sure to what extent Europeans picked up on negative numbers from the Arabic world-certainly their use was halting, but not nonexistent, in Europe for many centuries.
@WillOrrick Thank you for disputing this claim. I chose the Counting Table at Salamis as the inception of the concept of negative numbers because (1) due to my perceived popularity of the idea this was the first use of negative numbers; albeit crudely. (2) I chose the two advents as a starting point for a discussion. I used the tool of assumption so as to allow the dispute of said claim. I hope this clears my intentions up for you.
I have started a discussion on the topic on the historical dates of quotients and integers in the chat for this question.
@GeraldEdgar Are you suggesting that good ideas only come from published studies? If so, how did we develop the framework around published studies without good ideas extraneous to them? If not, why are you challenging me in seeking good answers by asking a question here, then providing a context as to why it could be a valid question? I would like to hear what you have to say as I respect your opinion enough to ask.
No, I am suggesting what I said. This is not a discussion forum, this is not an opinion forum. Good ideas may, indeed, arise from discussion and opinion. But this is not the place for that.
@GeraldEdgar I agree this is neither a discussion nor an opinion forum. I also can see how this question and its format could seem to be invalid. My question is, am I allowed to ask a question here, then give a (lengthy) justification as to why the question may be valid, on this Stack Exchange? In a way that could be useful. I think asking this question here is a better first step than going to Reddit, or writing a scientific paper.
I would, ideally, like to discuss this issue, but that is clearly not an option. I am trying to find a work-around.
This is a little beside the point, but I think there is a much stronger case that our present-day use of zero originated in Asia (specifically, India) than that our present-day use of negative numbers comes from the Salamis counting table, even if you could establish that the Salamis counting table somehow involved negative numbers.
More to the point, the case for following historical precedent relies on the notion that there is some fundamental mathematical truth that inevitably drives the direction of the development of mathematics throughout history. If we were to find that Chinese mathematics developed in a different sequence than European mathematics, in my opinion that would severely weaken the case that we should follow the European historical precedent in the development of real numbers for analysis.
Also, if a new historical discovery reveals that negative numbers have been lurking about long before true rational numbers, would we have to rework our account of the real numbers all over again?
My working assumption in approaching this topic would be that the history of mathematics is full of accidental discoveries as well as mistakes that are later corrected, and that its sequence of development was not at all inevitable, until proved otherwise. You would need a lot more evidence for this than we have seen in this discussion.
And if you could establish the inevitability of mathematical history, you would still need a non-question-begging argument that every account of modern mathematics should recapitulate the history of mathematics.
This is not to say that your opinion is illegitimate or cannot be discussed; merely that it is an opinion.
Discussion on question by Luke Nemeth…
Discussion on question by Luke Nemeth…