Can math be non-algorithmic? (and Quantum Gravity)
Warren
Are all mathematics algorithmic? Is there such thing as non-
algorithmic mathematics? What if quantum gravity is based on non-
algorithmic mathematics? Then we can no longer model it using math.
Does this means physics has to end (since physics involve mathematics)?
Sam Wormley
rotchm
> Are all mathematics algorithmic? Is there such thing as non-
> algorithmic mathematic
It depends on what is meant by 'algorithmic' (and math).
Math is not 'algorithmic' per se. The 'algorithmic' part is more about
HOW we humans (or computers) figure out the sought answers. It is the
'step-by step plan' used to attain the goal.
For instance, there is no algorithm in the fact that 1+2+3+...+n =n(n
+1)/2. But *we* develop an algorithm to find or prove such identities,
and there are many different ways (algorithms) to prove this identity.
Aatu Koskensilta
> Are all mathematics algorithmic? Is there such thing as non-
> algorithmic mathematics?
This depends on what you mean by "algorithmic". Mathematics in general
is not algorithmic in the sense that it does not deal solely with
algorithms or objects presented in terms of algorithms, and we don't
usually do mathematics by applying any explicit algorithms.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
artful
> Are all mathematics algorithmic?
Define what you mean by 'algorithmic'
> Is there such thing as non-
> algorithmic mathematics?
> What if quantum gravity is based on non-
> algorithmic mathematics?
You mean on the mathematics that you don't even know whether or not
exists? Why not ask if it is based on fairies ... it would make more
sense
> Then we can no longer model it using math.
But you just said it was based on mathematics .. so you can.
> Does this means physics has to end (since physics involve mathematics)?
What if the moon was made of green cheese? What a stupid question
Warren
I heard it from Roger Penrose about non-algorithmic thing. He said the
brain operates on non-algorithmic mechanisms meaning mathematics can't
model it. So I wonder how to describe non-algorithmic mathematics
which can't be solved. What if quantum gravity is like this. See "The
Emperor New Mind"
Daryl McCullough
>I heard it from Roger Penrose about non-algorithmic thing. He said the
>brain operates on non-algorithmic mechanisms meaning mathematics can't
>model it. So I wonder how to describe non-algorithmic mathematics
>which can't be solved. What if quantum gravity is like this. See "The
>Emperor New Mind"
I have to preface what I am about to say by saying that Roger Penrose
is a brilliant man, and I am not. But that doesn't mean that everything
Penrose says is brilliant. In particular, I think his speculations that
Godel's theorem implies that the human brain is non-algorithmic, and
that quantum gravity is somehow involved in human intelligence are
almost certainly wrong.
--
Daryl McCullough
Ithaca, NY
rotchm
the
> brain operates on non-algorithmic mechanisms meaning mathematics can't
> model it. So I wonder how to describe non-algorithmic mathematics
> which can't be solved. What if quantum gravity is like this. See "The
> Emperor New Mind"
Again, one needs a concise definition of 'algorithmic'. But lets go
along with its usual/vague concept... There are some math, some
concepts or sought results that can not be found via an algorithm...
the only way to get the result is to let the system (equations) evolve
till the answer comes out. For instance the 3-body problem ( general
PDE's). There is no way and no algorithm to find the solution; We can
approximate the solution or just wait till the system evolve and
look for the answer when it comes out. This is perhaps what is
generally happening for most processes in our universe (including in
our brains). If that is the case, then we cannot predict all that is
possible in our universe.... This is also related to computable/non-
computable algorithms.
There is also the concept of 'randomness' or 'quantum'. This also
produces a universe that can not be pre-calculated for at every
instant there can be a change in the parameters. There is also the
Goedel 'effect' in that some questions can not be answered at all,
hence algorithm or not, it cant be solved. This too can produce
'randomness' in the universe; such universe can have answer '1' or
answer '2' w/o violating any other principle (equations) of the
universe.
All these are big subjects. Consult more recent books by Penrose.
Also, for a deep discussion on algorithmic-calculable-randomness-etc,
check out Wolframs "A new kind of science". Read all those and come
back for your non-answered questions ;)
Edward Green
> On Mar 20, 10:11 pm, Warren <warrenco...@hotmail.com> wrote:
>
> > Are all mathematics algorithmic? Is there such thing as non-
> > algorithmic mathematic
>
> It depends on what is meant by 'algorithmic' (and math).
Good point. It sounds like buzz-word city to me. Algorithm has a
fairly well defined meaning: a step by step recipe for finding a given
answer, which removes the necessity of thought. A model is the
computer algorithm, which simply accepts inputs and generates outputs.
(I know you know all this already. I just find the non-algorithmic
ability to stir up discussion dwindling at the moment).
Huang
Wow !! Wasnt aware of that - really slick proof if it's any good.
JohnF
See the wikipedia entry
http://en.wikipedia.org/wiki/Digital_physics
that states, "the universe is, at heart, describable by information,
and is therefore computable".
However, that doesn't jibe with the (GHZ-like) gedanken experiment
described by Mermin in
http://www.iafe.uba.ar/e2e/phys230/history/moon.pdf
(which is a reprint of his article in Physics Today,
April 1985, pages 38-47) where, on page 9 of the pdf reprint
(page 44 of the print article), he concludes "there can be no
instruction sets", meaning the observed behavior of particles
in the gedanken experiment he describes can't be explained by
any possible programming (of the particles' behavior).
So Mermin appears to undermine wikipedia's "digital physics" premise.
You figure it out. Mermin's counterexample seems right to me.
In either case, even if we can't mathematically model specific
behavior, that doesn't necessarily mean we can't calculate
meaningful information about it, like probabilities. Turns out
we can (or seem to be able to, so far). So that's useful -- you
can build lots of devices based on such information that people
will pay money for.
--
John Forkosh ( mailto: j...@f.com where j=john and f=forkosh )