Re: Calculus XOR Probability

cbrown@xxxxxxxxxxxxxxxxx said:
stephen@xxxxxxxxxx wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:
Tony Orlow wrote:

So, I feel justified, until I see a good counterexample, that
equality between quantitative expresseions proven inductively hold in the
infinite case.

So, your idea of a mathematical argument is: n*(n+1)/2 is the sum of
the naturals from 1 to n, because you haven't seen a good
counter-example?

I think you are misreading Tony here. Tony is claiming that
because 1 + 2 + .... n = n*(n+1)/2 when n is finite, it is also
true when n is infinite, and he will continue to believe in
general that any equality that holds for the finite case
also holds in the infinite case until you provide a counter example
where it is clear that the equality does not hold in the infinite case.


Well, one might then ask: why does he believe it in the finite case to
/start/ with? Because he read it in a book? Or because it can be proven
by a mathematical argument?

I thought about it in two ways.

First:
1. for n=1, sum(x=1->n: x)=1 = f(1)
2. n(n+1)/2 + (n+1) = n(n+1)/2 + 2(n+1)/2 = (n+2)(n+1)/2 = f(n+1)
3. Therefore, for all n in n, sum(x=1->n: n)=n(n+1)/2

Second, list the naturals, in unary.
1
11
111
1111
11111
111111
.......

See how it forms a diagonal half of a square? The side of that square is n, and
the area is n^2, so the 1's cover half of that, or n^2/2. But, the 1's also
cover the diagonal, half of which belongs to the other side, containing n 1's.
So, we subtract half of those n 1's from n^2/2 and get n^2/2-n/2 = (n^2-n)/2 =
n(n+1)/2.


Similarly, I ask: why should I believe him when he says that it holds
"when n is infinite" (whatever that may mean)? Because he read it in a
book? Or because it can be proven by a mathematical argument?

I can assert that inductive proof of equalities holds in the infinite case, but
you don't believe that's true. So, I offer this idea, as a visualization for
you. COnsider that each of those 1's is an infinitesimal square, and that you
have an infinite number of them covering half a square. If there are n rows of
such squares and n columns, there are n^2 overall, and the set covers half the
square. It's harder to see the additional n/2 elements in this picture, because
n/2 is infinitesimal compared to n^2/2, but you can see it in the example
above. The diagonal line, when included in the area of the triangle, increases
its are infinitesimally.

<snippetty doo dah>
Cheers - Chas



--
Smiles,

Tony
.

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