Re: Kress's Probability Trilogy Q's

On Jun 9, 4:56 am, James A. Donald <jam...@xxxxxxxxxxx> wrote:
James A. Donald

: A human can see that the rules of simple
: arithmetic must be consistent. An algorithm
: cannot.

Wayne Throop

A human can belive it. Whether the human is right,

James A. Donald:

But we are right.
I think I understand now: We know it because it's
true; it's true because we we're right; we're right
because it's true.

We know our mathematics is true by direct insight, not
by experiment - but the value of that direct insight is
confirmed by an enormous number of experiments.

I think how we know a particular bit of mathematics is true depends on
the particular bit. Some experiments can be viewed as calculations. I
saw a Japanese TV clip on Yahoo, in which some people experimented to
see what would happen if you shot a tennis ball at 100 kph out of the
back of a truck moving at 100 kph. There was an element of empirical
testing involved (a change in speed might have - conceivably -
drastically altered the physics of the throwing mechanism). There was
also an element of calculation - this sort of thing could be used as a
very elaborate and expensive way to add numbers.

Any calculation has an experimental aspect to it (albeit one that is
usually neglected, because we just don't expect things to act funny).
And some things we know only by calculating them. It's only by
calculating the digits of pi that we actually know the nth digit of
pi. We do not know such things by direct insight. Nevertheless, the
identity of the nth digit of pi is a mathematical truth.

I submit that in many cases "direct insight" is nothing more than a
calculation so simple that we don't need a device to perform the
calculation, but can do the calculation in our heads. That 1+2=3 is a
fact not fundamentally different from the fact that the nth digit of
pi is y, but it is one that we know by what seems like "direct
insight".

We also know that 1+2 in some sense "must" equal 3, and does not
merely happen to come out to 3 in our current attempt to calculate it.
But I submit that this "knowledge" is more a matter of confidence
based on a lack of imagination - for which we are not to be faulted.
Indeed, in some sense, 1+2 must equal 3, but our inability to imagine
otherwise is not so much anything special about us, as it is an
inability of our simple mental calculator to come out with any result
other than 3 no matter how hard we try to force it. I would submit
that the raw inability of our simple mental calculator to produce a
result other than 1+2=3 even under willful attempts at self-delusion
is a consequence of the mathematical truth that 1+2=3, but that does
not mean that our minds are particularly in touch with a Platonic
realm of truth, because the exact same thing is true of our material
calculators, our abaci and so on. A calculator will continue to get
the result 1+2=3 even when placed under various forms of stress
(within limits - but it is the same with the brain, which also has
limits). The mathematical fact that 1+2=3 is in some fuzzy but real
sense constraining all calculators. Of course, the constraint is not
absolute - really poorly built calculators or calculators placed under
great strain of some sort can get wrong answers. But this is as true
of the human brain as it is of physical calculators. It is possible
for a person to get a calculation wrong. Even the feeling of certainty
that one has not gotten it wrong is not infallible (compare with the
feeling of deja vu, also fallible).

Humans of course can go back and catch their own past errors. But it
is important not to hastily conflate this phenomenon - the ability to
self-correct - with the phenomenon of that feeling of absolute
personal certainty that some calculation is correct (e.g. that 1+2=3).
The feeling of certainty, of "direct insight", is distinct from the
possibility of self-correction, which is something that occurs over
time rather than instantly, in a moment.

The ability to self-correct and the ability to construct calculators
is important, but not, I think, miraculous. I think there are certain
criteria which, if we apply to material things, forces them into
limited shapes. If we want to create a material object that is (a)
simple (e.g. minimize the number of lego pieces used), (b) predictable
(always produces the same output for the same input), (c) interesting
(doesn't always produces the same output whatever the input) - and
possibly a few other conditions - then at each level of complexity
there is only a finite number of possibilities. This is an objective
reality which we can gradually discover as we try to build gradually
more complicate devices that produce interesting output reliably.
We're not doing it. We're building the devices, but it is an objective
reality which is making it so that at any given level of complexity of
device, there is only a finite number of possibilities. In short, the
world's behavior suggests addition, subtraction, multiplication, and
so on, to us, because the simple non-trivial operations that we find
ourselves able to define on the basis of simple mechanisms (or what
amounts to the same thing, simple and learnable skills which we can
teach to each other while preserving agreement in our results) tend to
be the same ones over and over. Addition, multiplication, and so on.
To refer briefly to Kripke's Wittgenstein, the reason plus is favored
over quus is that a device that reliably implements the operation quus
is more complex than a device that reliably implements the operation
plus. (there are many caveats here, objections and answers which I'm
not going into. For example, there's the point that it is hard to
produce a device that adds arbitrary numbers and easy to produce a
device that loops back so that large number plus large number equals
negative number. But there are answers to these objections which
preserve the insight.)

We create devices (or skills or whatever) following certain simple
criteria, such as the need for repeatable output for any given input.
The world yields plus, minus, times, and a bunch of other things,
which we whittle down to the more useful things. Our "algorithm" is an
algorithm for finding mechanisms (or mental skills) that produce
extremely regular, repeatable behavior. This isn't the same thing as
an algorithm for writing down proofs given a set of axioms and rules
of inference - such an algorithm would be one of the outcomes, one of
the found things, of the former algorithm, which is an algorithm for
finding/building such things in the first place. Our algorithm is more
a combination of an ability to recognize regular behavior when we see
it, combined with an extreme affinity for it (see for example our
affinity for music), combined with a willingness to experiment until
we get what we're looking for (see for example a child's random
babbling followed by more regular speech, or a learning musician's
noise followed by a mastery of the skill of precisely producing sounds
which are clearly distinguished from each other in time and pitch).


.

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