Originally posted by VDO
...However, I am shocked. Really, utterly surprised at people's opinions. If someone wants to say that "philosophically" 0.9r!=1, they may do so, and I respect the opinions of Harley & Co. for restricting their arguments to the field of philosophy (where, unlike in Maths, there are valid opposing opinions and there can be debate), and memphisto for accepting the truth - but the people I rebutted were those claiming that, in essence, they were smarter than Stephen Hawking and every single other mathematician in history (without, of course, offering proof - perish the thought!), and people saying that, despite incontrovertible proof - even "God" himself telling them - they would refuse point blank to accept that 0.9r is mathematically equal to 1. AcidHell2 and Xenoxide, I mean you......
I think you're still missing what I'm saying. Look at your maths philosophically and ask questions, or at least, understand limits.
There are situations conventional maths systems have trouble handling. One of them, probably the biggest, is infinity and, by inference, infinitely small. Mathematicians therefore come up with a way to make those systems handle these awkward cases. Hence infinity. But infinity isn't a number. It isn't quantifiable. It's an abstract designed to stop those systems falling over when you hit the extreme case but, despite that, you're quite happy to stick an abstract like "infinity" into a system which is otherwise fully quantifiable.
And that's fine. Because it makes the system work and much of our daily lives derive (whether we realise it or not) from using those systems - like algebra.
So, all the proofs quoted (ad nauseum) work. And so they should, because the point which people have been trying to get at (that any difference betweem 0.9r and 1 is "infinitely small") is assumed out of the system used to provide those proofs, or the system wouldn't work.
Now, I have no problem with that. The system works, in the vast bulk of cases and provides us with immense benefits. But, it does nonetheless, has some incongruities. The cardinality of the sets of natural and even numbers, as I mentioned once before, is one of them.
I'm not convinced I'm getting through to the mathematicians here what I'm getting at, about challenging your systems. Maybe I'm just not wording it very well. I may be wrong, and apologise if I am , but I distinctly get the impression that the "philosophical" line of argument is being dismissed (as in VDO's quote above), perhaps as some sort of fluffy, artsy, non-scientific stuff not worthy of the attention of a scientist or mathematician. If I'm right in that, then
wake up. Questioning your assumptions, and even the foundations of maths, leads to a greater understanding, not a lesser one. Otherwise, all you're doing with even advanced maths is mechanistically applying the system. It might be a complex system to understand, but it would still make you a mechanic. Don't be put off by the word "philosophical". Philosophy in general might relate to a nebulous understanding "the meaning of life, the universe and everything" (\Hitchhikers Guide mode off) but the philosophy of mathematics is about using reason to understand the theories and ideas of maths, to question the limits and principles in order to further that knowledge.
Anyway, if so, I found a document that just might make you reassess that judgement. I only found it this morning, so I haven't been using it as a crib sheet for my comments. You will, however, have to take my word for it. In fact, I haven't finished reading it myself yet, but the line of argument it takes will, hopefully, make you realise what I've been getting at all this time. To add some weight, it is an article written by a member of the Mathematics Education Research Centre at Warwick University, so this is not just the opinion of some bloke on a forum, it's the work of a maths academic.
Once again, though, I've never said your proofs don't work. My point is that there's more than one way of looking at it. I'm not trying to undermine your methodology, or to dismiss the results. I'm trying to make you think more about what assumptions you make when you use these methods, and to adopt what I can only ascribe as being a more enlightened and less dogmatic view. That's what I've meant (and
maybe some others here have too) about another viewpoint, or 'thinking outside the box'.
The document is
here
Please, Alpha, VDO, xyphic, etc - take a look.
) - namely that nothing after the "0.9r" can ever exist. You can put "blue cheese" after the "r", it won't make an ounce of difference.
