Re: Calculus XOR Probability

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?

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?

Of course your stairstep example was meant to be exactly
that, but Tony does not see it that way, especially since
he has now decided that points have a "direction", although
he did admit that some points have multiple directions.

I wonder what direction the isolated point (0,0) is supposed to have.
All directions? No directions? An uncountable number of directions?

Is the curve "x^2 + y^2 = 0" the "same as" the point (0,0), or does it
have a direction as well? Is that direction the same as the curve "x^20
+ y^20 = 0"?

Is the limit of circles with radius 1/n and tangent to both the x and y
axes the "same" as either of the two previous "(0,0)" points?

Is that in turn different than the limit of squares centered on the
origin, with side length 1/n, with the nth square rotated about the
origin by n radians?

One can only guess; because he provides no mathematical argument for
his positions; aside from the fact that no one has provided a
"believable counter-example" to his undefined assertions.

Cheers - Chas

.

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