Re: The Unreasonable Effectiveness of Mathematics
- From: Paul Grieg <pgrieg@xxxxxxxxx>
- Date: Thu, 23 Aug 2007 11:52:09 -0700
... the early Greeks, who doubted the existence of the real
number system, we have decided that there should be a number that
measures the length of the diagonal of a unit square (though we need
not do so), and that is more or less how we extended the rational number
system to include the algebraic numbers. It was the simple desire to
measure lengths that did it. How can anyone deny that there is a
number to measure the length of any straight line segment?
But we can we measure it accurately?
Very few of us in our saner moments believe that the
particular postulates that some logicians have dreamed up create the numbers -
no, most of us believe that the real numbers are simply there
Where?
But let us not confuse
ourselves-Zeno's paradoxes are still, even after 2,000 years, too
fresh in our minds to delude ourselves that we understand all that we wish
we did about the relationship between the discrete number system and the
continuous line we want to model.
Yup I'm stilled worried about how Achilles catches that tortoise.
Anyone have a good
explanation?
How do we know there is a continuous line?
We know, from nonstandard analysis...
... there are some
mathematicians who doubt the existence of the conventional real number
system. A few computer theoreticians admit the existense of only "the
computable numbers."
So some mathematicians actually do some philosophy! Do the computable
numbers exist only in the mind, or are they Platonic ideals, or are
they somehow in the real world?
"Putting aside the mental tortures involved multiply (5 + sqrt 15) by (5 - sqrt
-15) making 25-(-15) ...." Thus he clearly recognized that the same
formal operations on the symbols for complex numbers would give
meaningful results. In this way the real number system was gradually
extended to the complex number system, except that this time the
extension required giving up the property of ordering the numbers-the
complex numbers cannot be ordered in the usual sense.
This *is* incredibly weird. It's like postulating that the tooth fairy
actually exists and being able to predict (say) when Peter will lose
his teeth by postulating this. Of course tooth fairies are ridiculous,
but they work!
Cauchy was apparently led to the theory of complex variables by the
problem of integrating real functions along the real line. He found
that by bending the path of integration into the complex plane he could
solve real integration problems.
As if adding infinitely thin rectangles wasn't mind boggling
enough :-)
... came away with the feeling that "God made the universe out of
complex numbers."
Poor mad fool :-)
... Yet how does it happen that no theorem in
all the thirteen books is now false? Not one theorem has been found to
be false, though often the proofs given by Euclid seem now to be
false... over half of the new theorems published these days are
essentially true though the published proofs are false.
Interesting. Does this show that mathematical intuition is more
precise tool than formal proof?
...we simply will not abandon much of mathematics no
matter how illogical it is made to appear by research in the
foundations.
Does this mean pragmatism rules? If the best coherent philosophy leads
to foundations that are incongruent with mathematics what is the
honest mathematician to do? Give up his world view or his mathematics?
I guess he'll live inconsistently with both, but how can he be content
to do this? Mathematicians don't like inconsistency, especially if
they have a philosophical bent. Probably mathematics has to go, but
what if it pays the rent?
The dominant attitude in science is that we are not
the center of the universe, that we are not uniquely placed...
Is this still so dominant? This was written before the anthropic
principle took off...
... Thus there are many results in mathematics that
are independent of the assumptions and the proof.
Doesn't that make them postulates? You end up with a large web of
postulates, many that don't look axiomatic. But it might be more
honest to recognize this.
I picked the example of scientists in the earlier part for a good
reason. Pythagoras is to my mind the first great physicist. It was he
who found that we live in what the mathematicians call L2-the sum of
the squares of the two sides of a right triangle gives the square of the
hypotenuse. As I said before, this is not a result of the postulates
of geometry-this is one of the results that shaped the postulates.
But is it physics? In physics you, say, suggest the existence of new
particles and then design experiments to try to find them. How is this
like what Pythagoras was doing? If that's physics then isn't all maths
physics?
The more he thought about it-and the more you think about it-the more
unreasonable becomes the question of when two bodies are one. There is
simply no reasonable answer to the question of how a body knows how
heavy it is-if it is one piece, or two, or many. Since falling bodies
do something, the only possible thing is that they all fall at the same
speed-unless interfered with by other forces.
Great argument! But did Galileo really think this way? Maybe he just
thought "well they might just fall at the same speed" and scurried up
the tower to test them.
I know that the textbooks often present the falling body law as an
experimental observation; I am claiming that it is a logical law, a
consequence of how we tend to think.
When I was 14 and doing the ticker tape experiments I'm sure I took
the experimental observation route.
Thus my first answer to the implied question about the unreasonable
effectiveness of mathematics is that we approach the situations with
an intellectual apparatus so that we can only find what we do in many
cases.
What's all the fuss about? Isn't this obvious? We might ask why we are
so good at digging. But we don't, because the answer is obvious. We
are good at digging because we approach digging with a spade, arms,
and a good back.
It is both that simple, and that awful.
Why awful?
... what Truth, Beauty, and Justice are. But so far as I can see science has
contributed nothing to the answers...
He said earlier that some mathematical equations are beautiful. So
science at least provides examples of beauty (and truth!) Isn't that
contributing something? Maybe not an explanation, but (something
better) actual beautiful and true things.
We seem not to be able to think appropriately about
the extremes beyond normal size.
Who cares about 'appropriately'? We can and do think about the large
and the small effectively. QFT predicts activities on a very small
scale to incredible accuracy -- isn't that appropriate?
Just as there are odors that dogs can smell and we cannot, as well as
sounds that dogs can hear and we cannot...
We can with appropriate devices (or dogs!)
"Perhaps there are
thoughts we cannot think,"
Well we can think about them, isn't that grist enough for our mill?
There are an endless number of interesting thoughts we can have why
worry about those we can't?
*Conclusion.* From all of this I am forced to conclude both that
mathematics is unreasonably effective...
Just because he can't give reasons for it doesn't mean it is
unreasonable. That's like 18th century scientists saying it is
unreasonable that there could be a heavier than air flying machine.
...we-meaning you, mainly-must
continue to try to explain why the logical side of science-meaning
mathematics, mainly-is the proper tool for exploring the universe as
we perceive it at present.
Can you wait until next week for the answer :-)
I (Larry Frazier, who (with R. Hamming's permission) scanned this and
put it online) was pleased to note that 58 people visited this essay
in a recent 2-month period.
I find it sad - 5 million people watched big brother
It is the most profound essay I have seen regarding philosophy
of science; important, significant, in fact, for our whole understanding
of thought, of knowing, or reality.
It's interesting, as a kind of glossy magazine overview. But its too
broad and too short to cover the issues in anything like the depth
they deserve. He does have some good references though, which may help
give some of the required depth. More recent books (Hersh,
Lakoff, ...) are also worth reading.
.
- Prev by Date: Recent quotations
- Next by Date: Re: Recent quotations
- Previous by thread: Recent quotations
- Next by thread: Re: The Unreasonable Effectiveness of Mathematics
- Index(es):
Relevant Pages
|
