Mathematics and Information
Dirk Bruere at NeoPax
asked...
Is here a minimum amount of information/entropy/energy associated with
the creation/discovery of a given mathematical theorem? I assume a
practical minimum is the amount of information required to store the
result, but its derivation?
--
Dirk
http://www.transcendence.me.uk/ - Transcendence UK
http://www.theconsensus.org/ - A UK political party
http://www.onetribe.me.uk/wordpress/?cat=5 - Our podcasts on weird stuff
Phillip Helbig---remove CLOTHES to reply
<dirk....@gmail.com> writes:
> Not quite sure how to phrase this since it's a question I have ever seen
> asked...
> Is here a minimum amount of information/entropy/energy associated with
> the creation/discovery of a given mathematical theorem? I assume a
> practical minimum is the amount of information required to store the
> result, but its derivation?
First, you have to define information. Murray Gell-Mann's THE QUARK AND
THE JAGUAR has a discussion of various definitions. For example, in one
sense a purely random number has a high information content, since there
is no encoding shorter than the length of the number itself. However,
we don't normally think of pure randomness as having a high information
content. The same goes for its opposite---to stick with numbers, say an
infinite number of the same digits.
Freeman Dyson has done some work on the relationship between entropy and
thought processes (with a view to working out how long life could exist
in various types of universe), as has Lawrence Krauss (in part in
response to Dyson and vice versa). I don't have any references on hand,
but this should be enough to get a search started.
Ian Parker
There are a few "pat" facts. One of them is that the minimum amount of
energy to perform a computation is nkT where n is a number which
guarantees accuracy. It should be noted that operating a computer at
room temperature is best since although we need less energy at reduced
temperature the second law of thermodynamics means that when you
include the energy required to run your refigerator you get nkT T
being the temperature of the hot end.
To find minimum energy you need a minimum number of computations. This
is hard to compute. We STILL don't even know whether
P = NP or P < NP
Until we do there is no well defined minimum number of computations.
One interesting sideline. The Quantum computer. If we write down
Shroedinger's equation in the form Hx = Ex and we solve this equation
by Cholsky's diagonalzation we get a well defined answer. Proportional
to the cube of matrix size. If we have a QM computer this is solved
intantaneously in an analogue way. Just a fly in the ointment!
- Ian Parker
Gerry Quinn
says...
> Not quite sure how to phrase this since it's a question I have ever seen
> asked...
> Is here a minimum amount of information/entropy/energy associated with
> the creation/discovery of a given mathematical theorem? I assume a
> practical minimum is the amount of information required to store the
> result, but its derivation?
It seems to me that if the theorem is proveable (which I suppose is
really part of the definition of a theorem), there is no new information
as it is embodied in the theory that is there already.
Bear in mind that theorems can be rather simple:
"36538756 + 3 = 36538759" is a perfectly good one.
However, an *unproveable* but true theorem (probably more correctly
described as an axiom) might have a real information content, I suppose.
- Gerry Quinn