Silhouette wrote:Not at all, it's possible from what I can tell from your words that you might actually have something to teach me. That will involve pointing out legitimate flaws in my argument, so feel no guilt for it. To be honest, I was tired of the repetitious illegitimate criticism that I was receiving. It frustrates me so when it's so clear to me that the respondent is so very misguided - and by contrast, I seem to be recognising you as somebody who actually has knowledge about that which you're talking.

Thaks again. But remember it's an online forum. What forums sometimes lack in sanity and/or knowledge they make up in free speech and alternate points of view. So rather than be frustrated, we should enjoy online forums for what they are, which is NOT the proceedings of the Royal Society!

Silhouette wrote:Wetness aside

I totally don't understand that.

Silhouette wrote:by all means let's challenge this assumption and see if we can't fall out as in all proper internet arguments

Looking ahead, I did not see a clear thesis or question articulated. I think we agree that multiplication was originally repeated addition, both at the level of mathematical development and individual student understanding. You seem to be emphasizing that but I perfectly well stipulate it. I could not get a grip on what point you are making or what my reponse should be.

Silhouette wrote:Some clarification though: my maths background actually extends much further than my computing one, though I won't deny that my computing has had an influence on it and given me some insight into maths that I did not have before. Actually though, I am mostly basing my arguments loosely on a history of mathematics: starting with natural numbers, through subsequent discoveries of other categories that did not accord with the previous ones.

Ok. Also paragraphs? That would help. I know, I'm a born critic. Still.

Silhouette wrote:It all started off with addition forwards and backwards (subtraction) - although it wasn't necessarily the case that it was thought of in this way, "backwards addition" is more of a reductive perspective as I've already said. More than likely it was just thought of as "this much more" vs "this much less": backwards addition is just a way to think of it that was inspired by my education in computing. With integers, multiplication and division ought to be fairly straight-forwardly a shorthand for multiple additions/subtractions. Of course this runs into problems are you start branching out past natural numbers through integers and fractions to irrationals and beyond (what I am referring to as "anomalies" compared to what preceded them) - I'm just explaining how we got there.

Ok. You and I are in perfect agreement that in the development of humanity, and mirrored in the development of the individual human, numeracy proceeds from abstracting numbers of things to numbers, and then taking successors and adding numbers, and then multiplication is repeated addition. Nobody's debating that. I hope that's clear.

Silhouette wrote:I assume that your post was to catch me out

No I didn't post to "catch you out." You answered a question about the mathematical real numbers by giving a detailed exposition of floating point arithmetic in computers. I thought that was such a large category error that I called it out.

Calling out an error is not "catching you out." Catching you out is trying to trick you. I'm not trying to trick you. I'm trying to explain to you that the mathematical real numbers are as far removed from IEEE-754 floating point as fine wine is to grape juice. Bad grape juice.

Fun fact: Determining whether a floating point number is zero is not computable. Now how lame is that? You can't even recognize zero.

Silhouette wrote:by shifting the foundation from the beginning of mathematics to something more contemporary where things are much more complicated.

Well it's not 1400 anymore. If you want to talk about the mathematical real numbers, the default is that we are talking about them as they are understood by contemporary mathematicians. If you want to talk history, say that.

But I'm confused. You answered a question about the real numbers by saying something about floating point numbers, a completely different topic. I don't understand why you did that. I still don't.

And now I don't understand why you are saying your approach is historical. I'm a little lost in your argument.

Silhouette wrote:Perhaps you recognised my approach and wanted to change the context to something more modern - and why not?

I'm not tracking your line of thought. You talked about floating point numbers. That's not historical, that's just a category error, responding to a question about apples with an answer about armadillos.

But sure, historically multiplication was repeated addition. Ok. If that's your point, I agree with it.

Silhouette wrote:I admit conceptions have moved on from my explanation of how maths started.

I don't feel that you have addressed my concern that you think IEEE-754 describes the mathematical real numbers. You said that you've studied math but you didn't address the point. Why did you talk about floating point? That doesn't tell us anything about the real numbers. And it's not historical.

See my dilemma? Am I at least making clear why I'm confused at what you wrote?

Silhouette wrote:My contribution to this thread, though, was in the certainty that at no point has it ever been appropriate to equivocate by equating "NOT" with "negative" - nor will it ever be.

I take no position on other aspects of this thread, nor am I responsible for any words other than my own. Why u tellin' me dis?

Silhouette wrote:That was the whole reason I brought computing into it: what was being described was the NOT function, as in computing, only it was erroneously being called the "negative" operator.

Ah ... well ... hmm ........ let me think about that. No. Not buying it. I asked you about the real numbers and you told me about floating point. So this wasn't anything to do with that.

Well ok I hope you found some of this entertaining. We didn't even get to talk about the real numbers yet.