On 17 Jul 2015, at 06:21, Pierz wrote:
On Thursday, July 16, 2015 at 7:07:50 PM UTC+10, Bruno Marchal wrote:
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.
And in string theory I believe there are two solutions to the size
of the universe relative to the Planck length - either we are much,
much bigger or we are much, much smaller. Once the universe
collapses below the size of the Planck length, it becomes
mathematically identical to expansion, so collapse becomes expansion
and vice versa. All very weird. Reminds me of a nightmare my brother
used to complain about as a kid which he called "macro-micro", in
which a boulder would become immense, and at the same time as it
expanded to infinity, it would become infinitesimal, like the cube
which can be seen in one of two perspectives. He found this
unpleasant, even terrifying.
I can understand. I made similar nightmare in my youth, notably
involving infinitely complex infinite knots.
At any rate, from the point of view of this question, infinitesimal
is as good as infinite, since computationally it involves the same
problem of unending iteration.
Sure, same difficulty. Now, is not like consciousness? I mean, it is
easy to do a machine which can grasp many notion of infinite and
reason with them. The mystery is more in the apparent instantaneous
qualitative understanding of infinity that we seem to have, when we
say that we understand that a program like "10-got-10" will never
stop. Simple machine can prove that, and know that in the Theaetetus'
sense, but how could they feel that in a finite time? That suggests to
me that consciousness is more on the side of truth (p) than
representation ([]p) in the Theaetetus definition of knower []p & p.
That would confirm the filter of consciousness that brain and
memories would impose in the relative way. It is very counter-
intuitive though, but *that* fact can be intutited when familiar with
machine's self-reference.
Bruno
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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