Oh I see. Well this fits in with something I keep saying. If we defined the symbol .999... to mean 47, then since 47 is different than 1, it must be that .999... ≠ 1. Of course all these alternate systems are inherently interesting, so if this discussion has turned people on to the hyperreals or the p-adics or the surreals, then it's valuable that way.Arising_uk wrote: I was interested in the later bits on skepticism and the alternative number systems where it doesn't equal 1.
But my thesis is that none of these alternate number systems bear on the question of whether .999... = 1 or not. It is unquestionably the case that if you give the symbols their standard mathematical interpretation, .999... = 1 just like the knight in chess moves the way it moves. There is no more meaning to one than the other. If someone asked, "But how does the night REALLY move?" or "How do we know the knight really moves that way," the question would be a category error. Chess is a game played according to formal rules and so is math.
All the problems come from trying to imagine .999... = 1 "means' something. It's not a statement about physics, or philosophy, or computer science. It's just a legal position in the game of formal math.
And even if one is not actually a mathematical formalist, when we are doing math, we should think like formalists!
That's my thesis in a nutshell, for what it's worth. So by that logic, even though the alternate number systems are interesting in their own right, they don't actually bear on the question.
Now if I'm missing some philosophical point, I'd like to know. Someone could say, "Oh, it's meaningful in physics because math reveals the truth about physics," well that's simply not true. Math is quite independent of physics.
That's my story and I'm stickin' to it!
Oh. Well I'm glad. But I'm also sad. If you find this thread of interest I must not have made my point. Because my thesis is that this subject is as pointless as arguing over how the knight "really" moves. In other words not only is .999... = 1 in standard math; and not only is this simply irrefutable; but it is also, and especially, a pointless conversationArising_uk wrote:Sorry unclear, I meant I found the discussion between the two of you interesting.
Thanks arising_uk for giving me this opportunity to be as clear as I could possibly be about my own opinion of this. But tell me, am I wrong? Am I missing some deeper meaning of all this? Is math required to mean something? Just because the physicists like to use math to build things, why is that math's problem?