On 15 Jul 2015, at 20:54, John Mikes wrote:
I think JC resoinded to Brent:
"I don't have a visceral grasp of the true immensity of infinity.
Do you?"
I wonder if 'immensity' means - B I G - ? in which case I cannot
refrain from thinking about the infinite SMALL as well.
The infinitely small is infinitely large, in the relative way.
The Mandelbrot set illustrates this well:
https://www.youtube.com/watch?v=bo-MB1QPZ7E
The more you zoom, the more the mini-mandelbrot set are small, with
ever bigger filaments around them.
Perhaps it is even clearer in this zoom where we go near 8 mini-brots:
https://www.youtube.com/watch?v=nkNNrfZz7dg
Just like I may think for 'eternal' as
being momentary and timeless.
Sometimes we can distinguish eternity from timelessness, depending on
the context.
We like to imagine meanings for concepts as
we like.
That is why we have to be careful to not introduce wishful thinking in
the picture.
We must be able to not deny the logical consequences of our beliefs.
Bruno
JM
On Wed, Jul 15, 2015 at 4:26 AM, Bruno Marchal <[email protected]>
wrote:
On 14 Jul 2015, at 20:25, [email protected] wrote:
On 07/14/15, John Clark wrote:
On Tuesday, July 14, 2015 , Brent wrote:
> Just ask yourself how you grasp the notion of infinity.
I don't have a visceral grasp of the true immensity of infinity.
Do you?
No, I don't, which was more or less my point. What we think of as
our "grasp of infinity" is an ability to consistently manipulate and
use some symbol that just means "bigger than anything else we're
concerned with". In mathematics it mostly comes up in proofs by
induction. There's an interesting book available online,
http://www.cs.utexas.edu/users/moore/publications/moore-
wirth-2014a.pdf
which describes one somewhat successful effort to have a computer do
automatic proof by induction; which is what I would regard as one
kind of 'grasping infinity'.
Another one is a theorem prover for a formal and effective (the
theorems are recursively enumerable) set theory. It has the axiom
that there is an infinite set, but soon or later the theorem prover
will prove Cantor theorem that all sets have a smaller cardinal than
their power set. This is proved by diagonalization instead of
induction. In fact diagonalization is very often effective or
computable, and that is why machine can be aware of their own
limitation.
Bruno
Brent
On Mon, Jul 13, 2015 at 9:26 PM, Pierz <[email protected]> wrote:
> Sure. It's a concept even very young children can understand
Have you actually tried this experiment? I think if you ask a very
young child for the largest number there is he will say something
like a million zillion, if you counter with a million zillion +1 he
will look puzzled for a second and then with a note of triumph in
his voice will say a million zillion +2 and it will take some time
to convince him that still isn't quite right.
> Computers just iterate until told or forced to stop, they
cannot reason about their own iterative processes.
Actually they can.
The computer program Mathematica
uses iteration to calculate the numerical value of PI, if you
tell it to calculate the first 500 digits to the right of the
decimal point it can do it in about half a second, if you tell it to
calculate the first
10,000 digits to the right of the decimal point it can do it in
about
3 second
s, but if you ask it to calculate an infinite number of digits to
the right of the decimal point it won't even start the iteration
procedure, instead it will tell you that is an impossible task and
you're being a idiot for asking it to do such a thing. Well OK,...
the program is more polite than that and its language more
diplomatic but I have a hunch that's what it's thinking.
> infinity and zero are about equally easy mathematical
concepts to grasp - historically both appeared in Indian mathematics
around the same time.
And yet the idea that there was more than one sort of infinity and
some infinite things were bigger than others wasn't
discovered until about 1880, not because the proof was so
technically difficult it isn't (the ancient Greeks could have
discovered it), but because before Georg Cantor nobody had even
tried; before Cantor everybody thought it was obvious that nothing
could be larger than infinity and that was that. Everybody thought
they understood infinity but they did not.
John K Clark
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