On 03-04-2023 22:23, John Clark wrote:
On Sun, Apr 2, 2023 at 5:58 PM smitra <smi...@zonnet.nl> wrote:
While GPT hooked up to Mathematica may greatly improve its
performance,
I would not be impressed when judging that in terms of approaching
human level cognition.
A long time ago I was teaching a statistical mechanics class, the
students had to do a certain calculation for homework that involved
doing the series expansion of 1/sinh(x) to order x. I was shocked to
see
that none of the students could do it. But some did come up with the
right answer, they had shown some failed attempts and then wrote
that
they used Mathematica and Mathematica says that the answer is 1/x -
x/6
...
The students couldn't do it because they could not use the Taylor
expansion formula due to the singularity at x = 0. Most students
had
not bothered to think about how to bypass that problem, otherwise at
least a few would have written down the correct derivation
Forget algebra, trigonometry and calculus, even when a human performs
something as basic as long division he's usually just blindly
following an algorithm that he learned in grade school and is not
consciously thinking about why it works. But GPT4 is certainly capable
of providing mathematical proofs if necessary and can show it
understands them by correctly answering questions about them and even
giving the proof in a very different way. I just asked GPT3.5 to give
me Euclid's proof of the infinity of the primes but do so in the form
of a poem, and in less than two seconds this is what it produced, no
doubt GPT4 could do even better. And this is before Mr.GPT got hooked
up with Mathematica:
"In Euclid's time, a great debate was rife,
About the primes that fill the number line,
_SOME CLAIMED THAT THEIR COUNT MUST END IN TIME,_
BUT EUCLID PROVED THEM WRONG WITH REASON'S MIGHT.
SUPPOSE WE HAVE A FINITE SET OF PRIMES,
_AND MULTIPLY THEM ALL TO GET A LINE, _
_WE ADD ONE TO THIS PRODUCT, QUITE SUBLIME, _
A new number appears, that's not confined.
This number is not divisible
_BY ANY OF THE PRIMES IN OUR FINITE SET, _
_FOR IF IT WERE, WE'D REACH A SORRY LIE, _
_That contradicts the number line's duet._
So this new number must be prime indeed,
_AND THUS WE'VE FOUND A NEW ONE TO CONCEDE, _
_ADDING IT TO OUR SET, WE THEN PROCEED, _
_To find another prime with the same breed._
_And thus we prove there's infinite primes to find, _
_A TRUTH THAT STANDS THE TEST OF SPACE AND TIME, _
_THE PRIMES ARE INFINITE, IN NUMBER AND KIND, _
_THANKS TO EUCLID'S PROOF, SO CLEAR AND DIVINE."_
John K Clark
GPT used its language skills to morph the standard proof into a poem. It
is good enough on language to reformulate the text so that its meaning
doesn't change, but that doesn't imply that it has a good understanding
of the text.
To test GPT's math skills you need to get it do produce a result that is
not in its database. The fact that GPT fail at simple arithmetic betrays
that GPT doesn't understand math. It doesn't have the answers to all
simple sums in its database and therefore it cannot reproduce such
results.
I don't have a lot of time to test GPT myself, but you should try the
following. For some simple physics or math result consider different
ways of getting to that result where one of these ways is not widely
published and is likely not in GPT's database. Take e.g. different ways
of computing the moment of inertia of a ball of uniform density of
radius R and mass M (relative to an axis through the center). There are
many ways to do this, but I've not seen my favorite way of doing this on
any webpage, which is to restore spherical symmetry by adding up the
three identical moments of inertia relative to 3 orthogonal axes.
Because you are integrating over the square of the distance to each axis
which is the square of the distance to the origin minus the square of
the coordinate along that axis, the sum becomes the integral of 2 times
the squared distance to the center. So, the moment of inertia of a ball
is given by:
2/3 M integral from 0 to R of 4 pi r^4 dr / (4/3 pi R^3) = 2/5 M R^2
So, we get to a simple one-line derivation that's so simple that I can
easily do it in my head. But this may not be in GPT's database, and if
it's not then GPT will fail to reproduce this very simple result while
it will have no difficulties spewing out the more complex derivations,
formulate those as poems etc. etc.
There are quite a few of such cases where you have a widely published
result which is more complicated than the most efficient way of getting
to that result but with that more efficient way not being widely
published, so many tests like this van be done.
Saibal
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