posted January 13, 2007 12:21 PM            

See also: http://www.bluemoon.net/~watson/proof.htm
It is supposed to be funny but it is certainly not silly.

"What we believe but cannot prove", Harper Perennial, 2006 --
by John Brockman (editor).

In this book, 100 scientists give their personal answers.
If you can't find the book, go to http://www.edge.org/
A sample quote from there:
Noam Chomsky: On the ordinary problems of human life, science
tells us very little, and scientists as people are surely no
guide. In fact they are often the worst guide, because they
often tend to focus, laser-like, on their professional interests
and know very little about the world. ...

posted January 13, 2007 05:00 PM            

What was IT ?

Perhaps we should adopt the red queen attitude:

Proof by red queen logic
"I want it to be true so it IS true"

posted January 13, 2007 05:29 PM            

That doesn't sit right with me. What is impossible doubt and how do we show that?

'beyond all doubt', IMO, would be better.

However, that would be impossible to achieve.

If someone doubts something we should ask for their reason which leads us to the legal phrase 'beyond all reasonable doubt'.

Added: Of course, something may be beyond reasonable doubt today but not tomorrow which is why I am against the death penalty in practice; I don't have a problem with it in principle.

posted January 13, 2007 06:20 PM            

They are so fundamental that they cannot possibly be doubted
(as long as you understand what they mean of course).
http://en.wikipedia.org/wiki/Probability_axioms

My attempt to translate into plain English:

1. An event has on non-negative probability of happening.
2. All events taken together have a 100% probability of happening.
3. Independent sets of events combined have a probability that is the sum of their probabilities.

All probability theory stems from these axioms.

posted January 13, 2007 09:53 PM            

David: I was vague & did cite a math web site, so anything is fair.

------------------

posted January 14, 2007 01:18 AM            

I didn't go there. In that case I stand corrected. Apologies, Charles.

posted January 14, 2007 04:48 AM            

'Lets pretend' is about the best thing you can do with axioms.
They are rules, and as such only have to prove themselves to be
useful in a theorem.

Given that methematicsl entities do not exist in nature, may I
suggest that mathematics is no more than an instrument of cognition,
and should be considered a branch of psychology

Which makes me wonder what is the state of the art in chimp mathematics?

posted January 14, 2007 08:19 AM            

As a retired psychologist (my specialty was monkeys & apes) I
hope that that is not so. Psychologists have argued and conducted
experiments for about 100 years on whether any nonhuman can even
count, and they are still arguing about it. I'm not. On that
issue I'd think it reasonable to say, Let's assume it, and get on
to more interesting problems.

The status of "chimpanzee mathematics" is nowhere near as advanced
as that of "turtle geometry" but that is only because the former
has not yet attracted the attention of such able (human) mathematicians
as Harold Abelson and Andrea di Sessa. I refer here to their book
on turtle geometry (MIT Press, 1980 or 81). I loved that book when
I read it and wish that I knew enough math to develop a formal
"chimp geometry" in much the same spirit.

>'Lets pretend' is about the best thing you can do with axioms.

If so, "Let's pretend" is a pretty darned good strategy, since
it (arguably) produced all of human mathematics. But of course
that's pragmatism, which a good Platonist might reject -- and
some of my best friends are Platonists.

P.S. I am fascinated by www.Pegge.net and also by the website on
Kolomogorov, to which I was led by the Wikipedia reference. I'll
have to tell my primatologist son & his artist wife about the
former.
------------------

posted January 14, 2007 11:00 AM            

quote:

---

In fact they are often the worst guide, because they
often tend to focus, laser-like, on their professional interests
and know very little about the world. ...

---

...over...

quote:

---

In fact they are often the worst guide, because they
often tend to broaden, sprinkler-like, on their lobbyist's interests
and know very little about the world. ...

---

...any day. Now start guessing whom I described with the later paragraph

posted January 14, 2007 05:23 PM            

web based turtle graphics:
http://sonic.net/~nbs/webturtle/

logo foundation:
http://el.media.mit.edu/logo-foundation/products/index.html

Re: Chimp mathematics

I imagine that chimps are not very symbolic in their thinking, but
have highly developed skills related to their rain-forest habitat.
Can Chimps recognise familiar objects or places within pictures?

Re: Platonics

I used to be keen on Platonic ideas but reading Buckminster Fuller
persuaded me otherwise. He maintained that abstract geometric
objects such as points and lines do not exist and also that chemistry
is really engineering on a small scale.

I would say that Platonic entities only embody the subset of
properties we happen to be interested in. They could only exist in
our imagination (and inside computers). This is inverted Platonism:
physical objects are real and platonic objects are their shadows.

Having said that. Platonic space is very useful, when you run out
of physical space. (I am thinking of my garden here.)

posted January 14, 2007 07:11 PM            

That reminds me of an (unpublished) test I once did on two chimps
who were quite sophisticated about video pictures. They could see
and hear me only via live closed-circuit TV. I pointed to any particular
spot on a map of the world and said, "Touch here" -- Miami,
New York, London, Hokkaido, etc... They had an identical map of their
own. It could be in various places in their cage, and rotated to various
angles, including upside-down. They very accurately pointed to
almost any "place" that I picked -- right from the first trial of the
test.

>Symbols(?)
The same chimps had learned many arbitrary "lexigrams" or icons
that represented various objects, people, and activities such
as play, chasing or grooming. I could also point to any one
of 20 or more of these lexigrams and if the chimps saw me do
this on a video monitor and I "asked" them to point to the corresponding
real-world lexigram on a keyboard in their own cage, they did so
virtually without error.

More impressive stuff has been published, by Duane Rumbaugh and
Sue Savage-Rumbaugh, who raised these two chimps, and by
numerous other people besides. Some of the tests more clearly
related to mathematics.

>Plato.
I used to be anti-Plato, first as an undergraduate after reading
Lord Byron's hilarious satire on Platonic love in his "Don Juan",
and much later after studying Karl Popper and Ernst Mayr. I changed
my mind after studying Martin Gardner and William James. But I
will not say that I'll never change my mind again. And that
reminds me of my first class in philosophy. Our professor told
us that everyone was basically and fundamentally either a Platonist
or an Aristotlean. One cannot (he said) be both, and one cannot
equivocate, at least if one is a real philosopher. I cracked my
head on that for a couple of years and decided I was not a real
philosopher -- albeit I was still a good bit less of a mathematician.
Alternatively, James' "pluralism" sooner or later made sense. As Niels
Bohr himself said, James was not nearly so confused or so self-
contradictory as some of his critics claim; he was preaching the
strict analog of "complementarity" ala Bohr.

------------------

posted January 15, 2007 10:32 AM            

Object Oriented programming is very close to Platonism. This article
expresses it well:

The Metaphysics of Object Oriented Programming
http://news.softpedia.com/news/The-Metaphysics-of-Object-Oriented-Programming-24906.s html

With software you can stretch a philosophical paradigm to its
breaking point, which in the case of OOP is its rigid taxonomy of
classes (corresponding to Platonic forms), consequently bloating
OOP software, trying to make it all work together. I like the idea of
a Wittgensteinian based computer language.

Chimp Cognition:

Computer games for chimps (or parrots even). Would be a fun way
to explore their cognitive abilities.

This is getting close:
http://www.units.muohio.edu/dragonfly/family/chimp.shtml

>Dr Sue Rumbaugh: very impressive video clip on Kanzi understanding
'novel sentences'.
http://www.greatapetrust.org/research/srumbaugh/rumbaugh.php#

------------------
www.pegge.net

posted January 15, 2007 06:06 PM            

I agree. A fair amount of such work has been; I done a bit myself,
& with home-made PB-DOS programs. Source code available on request for
nonprofit research or recreational purposes.

Commercial games are of little use for research purposes: They are
mostly "eye candy"; you cannot record or capture the user's
performance in much detail; and the games are usually too complex,
either for the animal or for purposes of detailed analysis. Generally
speaking, simpler is better.

My ex-colleagues & co-workers still think of (e.g.) mazes in terms
of intelligence, planning, foresight, cognition & so forth. I think
of them more often in terms of shortest-path estimation &
optimization, especially when talking to those who understand
such terms. Relatively few researchers really know both languages,
or how to program for themselves. So if you want to volunteer or
advertise your programming services I'd be happy to tell you a
few places to write. Some researchers can even pay, albeit
generally speaking federal funding is very tight right now.