Adivi
Regular Member
dad said:Now, watch out, the highest triangular number you can form, of the same digits only goes as high as the number of a man, that is 666!
"666 is the largest triangular number which you can form of the same digits "
How limited is that!!?
Not limited at all. I mean, there aren't that many numbers with the same digits; there's only 42 of them less than a million, 84 less than a trillion, and in general, 6n+6 of them less than 10^n. The unusual sequences of numbers are the ones that don't have such an upper bound.
No. Real numbers are strictly defined: they are the numbers r so that r*r is a positive number. The positive numbers are defined as those that are greater than zero, and the operaion of 'greater than' can be put on a rigorous footing.Well, real is relative to the universe it exists in. More specifically, the state it exists in. If we add the spiritual, what you call positive may be affected, as in the loaves and fishes example.
There you go misunderstanding numbers again. Rational in this context means 'representable as the ratio of two integers', not 'logical'. For example, 1/2, 5948, -3, and -6/11 are all rational, but the square root of 2, pi, and Euler's constant are not.dad said:Rational is determined not only by how you imagine the new heavens math to be. It would be determined by the realities of the day.
I don't need to know the fundamentals of it to know that it works; it's a lot more than you know (oh, and I messed up orderd pairs; the ordered pair (a, b) is represented as {{a, b}, {a}}, but that's irrelevant). I can guarantee you can't explain to me the exact workings of the computer you're typing all of your replies out on, but that doesn't stop you from using it.dad said:Wow. So you really don't know the nitty gritty fishbowl fundamentals. OK. Our concept of negative/positive is something that might need a tweak on the other side anyhow.
Oh, and after a quick Google search, I found the general construction mechanism for the real numbers from the naturals: let <a, b> denote the ordered pair {{a, b}, a}. Then the non-negative fractions are defined as classes (which is similar to, but not the same as, a set) of ordered pairs of non-negative integers, where <m, n> and <p, q> are in the same class if and only if there is some x so that mq=np. So the non-negative rational number [<3,7>] is the class of all ordered pairs <m, n> with 7m=3n.
The general rational numbers are defined as equivalence classes of ordered pairs of non-negative rational numbers, where <r, s> and <t, u> are equivalent if r+u=s+t. So we can say that -3/7 is the equivalence class [<[<0, 1>], [<3, 7>]>]. In other words, it is the class of all ordered pairs <a, b> with a-b=3/7. So aside from <[<0, 1>], [<3, 7>]>, there is also <[<4, 7>], [<1, 1>]> and many many others. We can then construct the reals in the way I mentioned above.
I wouldn't make the assumption that y=4 and x=2 in the first place. If you said that "2x = y and x = 2", then I would assume that y=4 and x=2. But if you didn't say anything about x, I would only assume that whatever it is, y has twice its value. I know that x represents a quantity and y represents twice that quantity.Well, then they must represent something. Otherwise, they are unknown. Like if I say 2x = y
You might assume that y = 4, and x = 2. But I might assign 1,000,000,008 as the value for x. You would be wrong. Now if I wrote the actual numbers, rather than little letters that you do not know what they really are, why, you would have known!
It doesn't get there, though. Because there's no 'there' to get to. If you're saying that it doesn't actually get arbitrarily large, than could you show me a number that it doesn't get larger than? Or could you prove mathematically the existence of one?And large it gets indeed, especially in your head, as it gets anywhere near actual infinity. To actually get there is another matter altogether.
When I write "2x=y", I mean that 'no matter what x is, y has twice its value'. If someone else writes "2x = y", then that means the same thing. The specific values for x and y are not important; what is important is their relation.Only if you cook up the problem. This universe you never cooked up. If someone else wanted the little letters to mean something else, and they wrote the equation, you would wrong. See, it is relative.
But if we go back to the Zermelo-Fraenkel definition, then I can essentially construct mathematics out of the ten axioms and nothing else. What's there for you to change? If you change the axioms, you're no longer talking about the same thing as you were before.What all is there and what exists is actually a foundational reality, like a carpet upon which your maths sit. A carpet I can yank out at will, with all justification, when you misuse the concepts, by trying to apply them above where they can be applied. Making a fancy equation from them, or based on them is smoke and mirrors.
Well, if you have one pile of hay and one pile of hay, and you smash them together, you get one pile of hay. But that doesn't mean that 1+1=1. Even if God can make 40,000 loaves appear from five, that doesn't mean 40,000 = 5. Because those concepts transcend material objects; I can define the concept of '5' without referring to 5 loaves of bread, or fish, or anything: I say that it is represented by a certain set: {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}}. The number 40,000 can similarly be represented by a set, whose precise definition is too long to write here. Equality has a precise definition: two sets are equivalent if they contain the same elements. The set 5 and the set 40000 do not contain the same elements, so they are not equal.if we refer to the loaves and fishes again, and the first family takes (1plus another member takes 2, and the dad takes 3) and then the teen brother takes 4, and the loaves replicated as they either looked at them, or took them, they got 10 loaves from 5. Yet, we might assume there is now a dozen loaves in there for the next family to take, thereby making even more. So triangular numbers don't seem to help and drawing them as a square doesn't either.
When the dust settles, the numbers represent something, when we change what that is, the old math cannot apply. Whether you stack it, put it in a pyramid, or in a pie plate.
That's ultimately my entire argument; all of mathematics emerges from the ZF axioms. If you change them, you're no longer talking about the usual set of mathematics. To use an analogy you might be familiar with, how would Christianity be different if the Ten Commandments were replaced by the nearist Shinto or Buddhist or Islamic equivalent? It wouldn't be Christianity anymore, it'd be something different.
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