It's been rumoured that Brian D Harper said:
>
> >> 2) Were natural laws invented or discovered?
>
> I tend to waffle somewhat (alternate tuesdays) on the
> first question but not on the second. It is true that
> mathematical expressions of laws are often approximations
> but the approximations tend to get better and better
> and, IMHO, are approaching some real law of nature that
> exists independently of our representations of it.
> In other words, there is a real law that is discovered
> despite the imperfection of our representation of the
> law.
OK, is a painting of a still life an "invention" or
a "discovery"? As the painter gets better in thier
representation, aren't they both discovering new things
in the still life, and inventing new painting techniques?
I guess I have to ammend my claim, and say that a physical
law is more like a painting: an invented representtion of
a discovered reality.
> >> 3) Why are natural laws mathematical?
> >
> >Because they cannot be anything else. Mathematics is a kind of a
> >language, a shorthand for the english language that allows you to say
> >more with fewer marks on the page. Thus, if you can say it
> >in english, you can always recast it as a formula. Conversly,
> >if it cannot be said, then it is not truly understood, and therefore,
> >not expressible mathematically.
>
> Well, I think you should have inserted "Contrary to popular
> wisdom ..." for this one as well :), since the peculiarity
> that nature is described by mathematics has been a
> puzzlement for many great physicists and philosophers.
Hmm, well, being a much dimmer thinker than Wigner or Feynman,
I can only suggest that they failed to propose "what else it
could have been, if not mathematics".
> Surely its possible to imagine a universe in
> which there were a great many laws and facts for
> many situations but with those laws and facts being
> disjoint and unconnected. Just having laws in themselves
> is somewhat amazing, but having them all connected
> by logic is startling, for me anyway. Perhaps I'm
> simple minded ;-).
Ah, well, this is the crux. I think the surprise is not
so much that things are mathematical, but that there
are so few fundamental laws, and that all else seems to
derive from them.
Naively, I guess this is a surprise, but let me counter
with a mathematical example: John Conways construction
of the integers, rationals, reals, infinitessimals, and
infinities, and all two-player games, all from a few axioms.
1) There exists something called {|}
Its like the empty set, but it has a left and a right side.
2) rule of addition:
define {x|y} == {a|b} + {c|d} by the following.
x = { {a|b} + c , a + {c|d} }
y = { {a|b} + d , b + {c|d} }
where the outer braces & comma mean "set" in the ordinary sense,
and p+q is undefined (doesn't exist) if either p or q don't exist.
equivalent to 0. In particular, 0+0 = 0
Next, {0|} is mathematically equivalent to 1. One can discover
that 0+1 = 1+0 = 1
Similarly, {|0} is -1, and -1+1 = 1 + (-1) = 0
Next, one finds out that {|0,1} == {|1} == 2, etc.
One can quickly find all of the integers. Turns out {0 | 1} == 1/2,
which one can see as true, because 1/2 + 1/2 == 1
Mental hint: everything on the left hand side of the bar is "less than"
the number, and everything on the right hand side is greater.
BTW, note that 0 == {|} == {0|0} == {1,0 | 1,0}, etc.
It's not hard to come up with the rules for subtraction, multiplication,
division, sqrts, etc. Also, the rules for "less than" and "greater
than". One can then see that
0<1 is true, but
0<0 is false
that -1 < 0 < 1/2 < 1, etc.
One can also see that 0 x 1 = 1 x 0 = 0, but 1x1 = 1 etc.
---------
To get the infinities, one uses w = {|0,1,2,3,4,...n, ...}
and the rules, you can see that w > any finite number.
You can also add, subtract, mutlple and divide w. Turns out
2w > w and that 1/2 of w is < w. Turns out that 1/2 w is still
greater than any finite number; almost every expression involving w is
greater than any finite number (except for w-w which is zero, and
likewise...).
Turns out that w-n is greater than 1/2 w, for any finite number n.
One can then contemplate w, w^2 (w squared), w^3, ... w^n ... w^w
each being greater than the last, and manipulable just fine with
addition, subtraction, multiplication, etc. They are less than or
greter than to each other pretty much as one might expect.
One can contemplate E = w^w^w^w^... i.e. w itself raised to the power
of w to the power of w ... which is indeed a difficult number to
understand ... and E^2, E^3, ...E^w, ... E^E, ... E^E^E, ...
Somewhere in here, the tools at Conway's disposal break down, and
he is no longer able to talk much about these objects.
--------
In a similar way, one can find a galaxy of infinitesimals, which,
by using less than and greater than, you can discover that they are
greater than zero, but smaller than any positive number.
-----
Finally, you have odd thigs like {1|} ... which turns out to be a very
simple two-player game, where there is a strategy for the player
on the left to win. A trans-infinite collection of two-player games
evolves soon thereafter ...
------
So, to respond to feynmann's surprise, I offer a different surprise:
how is it possible, that merely by assuming that the empty-set
exists, and defining a construction for something called addition,
all numbers as we know them, and a few that we didn't, with all
of thier richness and relationships, just pop into being?
We are in a situation that all of mathemtics seems to arise
out of only a few basic constructions; the surprise is that
all physical fact seems to arise from a few physical laws.
That, perhaps is the grand mystery: and at this point, I feel
forced to become mystical: the physical universe is just math
acquired mass and solidity. In the moment before the big-bang,
there existed the empty set, from which exploded all of math;
and that was too much to bear, and thus the universe was born
in a big bang.
> "... we have learned from much experience that all
> philosophical intuitions about what nature is going
> to do fail." -- Richard Feynman
Well, then again, I could be wrong ...
--linas