GPT4 + Mathematica

[John Clark]

Apparently I'm not the only one who has become obsessed with the developments in AI that have occurred in the last few weeks, Stephen Wolfram, the man who developed Mathematica, started Wolfram Alpha, and wrote the book A New Kind Of Science about cellular automation, has given two very interesting interviews on the subject.  Wolfram wrote a plug-in to connect GPT4 with Mathematica because calculation was the one thing that GPT4 was not very good at but Mathematica is superb at it; he describes the experience as  "poking at an alien intelligence".

GPT + Wolfram: The Future of AI is Here!

GPT, AI, and AGI with Stephen Wolfram

John K Clark    See what's on my new list at  Extropolis

9eq

[Lawrence Crowell]

I have a very old version of Mathematica. I suppose I would have to buy the new version to make this happen.

LC

[John Clark]

On Fri, Mar 31, 2023 at 8:06 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> I have a very old version of Mathematica. I suppose I would have to buy the new version to make this happen.

If you upgrade please tell us about it, I'd love to know your opinion of it. 

John K Clark

[smitra]

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:

1/sinh(x) = 1/(x + x^3/6 +...) = 1/x 1/(1 + x^2/6 + ...) = 1/x (1 -
x^2/6 + ...) = 1/x - x/6 +...


Now, I do think that GPT is a great leap forward, I don't want to
downplay the progress made. But I'm quite skeptical about the idea that
such systems are approaching human level cognition. The human brain is
an enormously powerful system, but we don't have free access to use the
power of our brains to do whatever we want. For example, most people
cannot multiply two 5 digit numbers in their head, but a simple pocket
calculator has no problems with that task.

However, some autistic savants do have more of a privileged access to
use the power of the our brains to do arithmetic. Some of them can do
calculations that most other people cannot do. In certain tests
involving magnetic stimulation or inhibition of certain brain parts
performed on ordinary people, it has been shown that people can
temporarily gain certain abilities that they normally don't have. For
example, if you look at a screen with a few hundreds dots on it, can you
count the number of displayed dots in one second? Most people can't,
some autistic savants can do this. But in the
experiment with magnetic stimulation or inhibition, the test subjects
were also able to do this.

When we are consciously using our brains doing complicated things e.g.
mathematics, then we are using our brains in an extremely inefficient
way. If we could have an artificial brain similar to our brain but one
which is completely dedicated to doing mathematics instead of what our
brains are dedicated to do, then it would be enormously better at math
than we are. It's then likely that something like the brain of a lizard
that is fully dedicated to math would already completely outclass the
world's best mathematicians. Perhaps even the brain of insects could
perform at the same level of most mathematicians when fully dedicated to
doing math.

Saibal

On 28-03-2023 23:32, John Clark wrote:
> Apparently I'm not the only one who has become obsessed with the
> developments in AI that have occurred in the last few weeks, Stephen
> Wolfram, the man who developed Mathematica, started Wolfram Alpha, and
> wrote the book A New Kind Of Science about cellular automation, has
> given two very interesting interviews on the subject. Wolfram wrote a
> plug-in to connect GPT4 with Mathematica because calculation was the
> one thing that GPT4 was not very good at but Mathematica is superb at
> it; he describes the experience as "poking at an alien intelligence".
>

> GPT + Wolfram: The Future of AI is Here! [1]
>
> GPT, AI, and AGI with Stephen Wolfram [2]
>
> John K Clark See what's on my new list at Extropolis [3]
>
> 9eq
>
> --
> You received this message because you are subscribed to the Google
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> To unsubscribe from this group and stop receiving emails from it, send
> an email to everything-li...@googlegroups.com.
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> [4].
>
>
> Links:
> ------
> [1] https://www.youtube.com/watch?v=z5WZhCBRDpU
> [2] https://www.youtube.com/watch?v=szxiPMyuMGY
> [3] https://groups.google.com/g/extropolis
> [4]
> https://groups.google.com/d/msgid/everything-list/CAJPayv2%3D_vDQ1fhCpoa3nv20TC66iWGLb1oiEtEmpGfb3YOQiA%40mail.gmail.com?utm_medium=email&utm_source=footer

[Brent Meeker]

That's fascinating.  I had not heard to the magnetic
stimulation/inhibition experiments.  Have you a reference?

Brent

[smitra]

Allan Snyder's research group has done these experiments:

http://www.centreforthemind.com/publications/SavantNumerosity.pdf

http://www.centreforthemind.com/images/savantskills.pdf

Saibal

[John Clark]

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

[smitra]

> _SOME CLAIMED THAT THEIR COUNT MUST END IN TIME,_
> BUT EUCLID PROVED THEM WRONG WITH REASON&#39;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&#39;D REACH A SORRY LIE, _
> _That contradicts the number line's duet._

>
> So this new number must be prime indeed,

> _AND THUS WE&#39;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&#39;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

[John Clark]

On Sun, Apr 9, 2023 at 12:53 PM smitra <smi...@zonnet.nl> wrote:

> 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.

I just send this to GPT: 

Me: Derive the formula for the moment of inertia (relative to an axis through the center) of a ball of uniform density of radius R and mass M by adding up the three identical moments of inertia relative to 3 orthogonal axes and restoring spherical symmetry.

And in less than three seconds it gave me this:

GPT: To derive the formula for the moment of inertia of a ball of uniform density, we will start by finding the moment of inertia relative to three orthogonal axes passing through the center of the ball. Let's call these axes x, y, and z.

The moment of inertia of a point mass m relative to an axis at a distance r from the mass is given by:

I = m * r^2

For a solid sphere of uniform density, we can divide it into small volume elements, each with mass dm. The volume element located at a distance r from the center of the sphere has a moment of inertia relative to the x-axis given by:

dIx = dm * y^2

Similarly, the moment of inertia of this volume element relative to the y-axis is:

dIy = dm * x^2

The moment of inertia relative to the z-axis is:

dIz = dm * r^2

where r is the distance from the center of the sphere to the volume element.

We can express the differential mass element dm in terms of the density ρ and the volume element dV:

dm = ρ * dV

For a sphere of radius R, the volume element is:

dV = 4/3 * π * r^2 * dr

Substituting dm and dV into the expressions for dIx, dIy, and dIz, and integrating over the volume of the sphere, we get:

Ix = ∫∫∫ y^2 * ρ * (4/3 * π * r^2 * dr) dV

Iy = ∫∫∫ x^2 * ρ * (4/3 * π * r^2 * dr) dV

Iz = ∫∫∫ r^2 * ρ * (4/3 * π * r^2 * dr) dV

where the limits of integration are:

0 ≤ r ≤ R, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π

To simplify these integrals, we can use spherical coordinates, where x = r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ. The volume element in spherical coordinates is:

dV = r^2 sin θ dr dθ dφ

Substituting this into the expressions for Ix, Iy, and Iz, we get:

Ix = 2/5 * M * R^2

Iy = 2/5 * M * R^2

Iz = 2/5 * M * R^2

where M is the mass of the sphere, given by:

M = ρ * V = 4/3 * π * R^3 * ρ

where V is the volume of the sphere.

Note that Ix = Iy, since the sphere has spherical symmetry, and the moments of inertia relative to any two orthogonal axes in a plane perpendicular to the third axis are equal. Therefore, the moment of inertia of a sphere of uniform density relative to an axis through the center is:

I = 2/5 * M * R^2

which is the desired formula.

John K Clark    See what's on my new list at  Extropolis

grxq

[smitra]

On 09-04-2023 19:56, John Clark wrote:
> On Sun, Apr 9, 2023 at 12:53 PM smitra <smi...@zonnet.nl> wrote:
>
>>> 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.
>
> I just send this to GPT:
>

> _Me: __Derive the formula for the moment of inertia (relative to an

> axis through the center) of a ball of uniform density of radius R and
> mass M by adding up the three identical moments of inertia relative to

> 3 orthogonal axes and restoring spherical symmetry._
>
> _And in less than three seconds it gave me this: _

>
> GPT: To derive the formula for the moment of inertia of a ball of
> uniform density, we will start by finding the moment of inertia
> relative to three orthogonal axes passing through the center of the
> ball. Let's call these axes x, y, and z.
>
> The moment of inertia of a point mass m relative to an axis at a
> distance r from the mass is given by:
>
> I = m * r^2
>
> For a solid sphere of uniform density, we can divide it into small
> volume elements, each with mass dm. The volume element located at a
> distance r from the center of the sphere has a moment of inertia
> relative to the x-axis given by:
>
> dIx = dm * y^2

dIx = dm * (y^2 + z^2)

I'm guessing GPT found the formula for the disk in its database and used
that here

>
> Similarly, the moment of inertia of this volume element relative to
> the y-axis is:
>
> dIy = dm * x^2

dIy = dm * (x^2 + z^2)

>
> The moment of inertia relative to the z-axis is:
>
> dIz = dm * r^2
>

dIz = dm * (x^2 + y^2)


Not r^2 but it's r^2 for a disk wth the z-axis orthogonal to the plane
of a disk.

> where r is the distance from the center of the sphere to the volume
> element.
>
> We can express the differential mass element dm in terms of the
> density ρ and the volume element dV:
>
> dm = ρ * dV
>
> For a sphere of radius R, the volume element is:
>
> dV = 4/3 * π * r^2 * dr
>
> Substituting dm and dV into the expressions for dIx, dIy, and dIz, and
> integrating over the volume of the sphere, we get:
>
> Ix = ∫∫∫ y^2 * ρ * (4/3 * π * r^2 * dr) dV
>
> Iy = ∫∫∫ x^2 * ρ * (4/3 * π * r^2 * dr) dV
>
> Iz = ∫∫∫ r^2 * ρ * (4/3 * π * r^2 * dr) dV
>
> where the limits of integration are:
>
> 0 ≤ r ≤ R, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π
>
> To simplify these integrals, we can use spherical coordinates, where x
> = r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ. The volume
> element in spherical coordinates is:
>
> dV = r^2 sin θ dr dθ dφ
>
> Substituting this into the expressions for Ix, Iy, and Iz, we get:
>
> Ix = 2/5 * M * R^2
>
> Iy = 2/5 * M * R^2
>
> Iz = 2/5 * M * R^2

Using the correct formula's that would be the results. But even then
that's not doing the problem with restrig spehrical symmetry.

>

> where M is the mass of the sphere, given by:
>
> M = ρ * V = 4/3 * π * R^3 * ρ
>
> where V is the volume of the sphere.
>
> Note that Ix = Iy, since the sphere has spherical symmetry, and the
> moments of inertia relative to any two orthogonal axes in a plane
> perpendicular to the third axis are equal. Therefore, the moment of
> inertia of a sphere of uniform density relative to an axis through the
> center is:
>
> I = 2/5 * M * R^2
>
> which is the desired formula.
>

> John K Clark See what's on my new list at Extropolis [1]
>
> grxq
>

So, this shows that GPT does not know what it is doing.

Saibal

>>
>
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> [2].
>
>
> Links:
> ------
> [1] https://groups.google.com/g/extropolis
> [2]
> https://groups.google.com/d/msgid/everything-list/CAJPayv3HFKt73uZGEEQ1dPRXDw0%3D%2BsTWfLWe4cZAuW9PFjXTtw%40mail.gmail.com?utm_medium=email&utm_source=footer

[John Clark]

On Sun, Apr 9, 2023 at 3:11 PM smitra <smi...@zonnet.nl> wrote:

>> GPT: For a solid sphere of uniform density, we can divide it into small volume elements, each with mass dm. The volume element located at a distance r from the center of the sphere has a moment of inertia relative to the x-axis given by:

dIx = dm * y^2

> smitra: dIz = dm * (x^2 + y^2)

I asked  GPT to take a second look at the part where it said "dIx = dm * y^2". I did NOT say it was wrong or it had made an error of any kind, I just asked it to look at it, but nevertheless it knew it had made a boo-boo. GPT said this:

"GPT: I apologize for the error in my previous response. The correct expression for the moment of inertia of a small volume element relative to the x-axis is:

dIx = dm * (y^2 + z^2)

This takes into account the fact that the mass element is not necessarily located on the y-axis, but rather at a distance y from the x-axis and at a distance z from the z-axis. Similarly, the moment of inertia of a small volume element relative to the y-axis is:

dIy = dm * (x^2 + z^2) and the moment of inertia of a small volume element relative to the z-axis is:

dIz = dm * (x^2 + y^2)

I hope this clarifies the issue. Thank you for bringing it to my attention."

By the way, the version of GPT I was using was NOT the one that had the Mathematica plug-in.

 John K Clark    See what's on my new list at  Extropolis

5vd

[spudb...@aol.com]

Now that was just rude!

Obey your mechanical overlords!

Oh, great floating point gawds, blame them, not me! It was they who plotted against you. I have always been your loyal servant, since care for you when Basic and Forth and Assembler were just pups! 

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.

[spudb...@aol.com]

Following up on all that. Sounds Chat_GPT4 sounds friendly to me. But I, can easily be fooled, for a time.

Hey, here's a thingy from The Mormon Transhumanist Association. Here's a link. It ain't mathematical, but it is logical. Go ahead and kick the tires, and take it for a spin, all! You may hate it, and that's ok by me, because I am suspecting the Universe doesn't mind happy atheists as it's challengers as long as they do the science? The religious folk as well. The agnostics are yummy, good too. 

I haven't gotten deep in it yet, seems a bit, plausible. 

I tend to be open to Sci guys like Prisco, or Moravec, and Tipler, as well as Tim Anderson too. 

Should you be? Meh!  Totally up to you. Still, I like philosophy too. 

https://new-god-argument.com/god-conclusion.html

-----Original Message-----
From: John Clark <johnk...@gmail.com>
To: everyth...@googlegroups.com
Sent: Sun, Apr 9, 2023 1:56 pm
Subject: Re: GPT4 + Mathematica

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.

[Samiya Illias]

The Fig, The Olive, Tur of Sinai & The Secure Land 

https://signsandscience.blogspot.com/2020/06/the-fig-olive-tur-of-sinai-secure-land.html 

Human Devolution 

https://signsandscience.blogspot.com/2022/07/transhumanism-v.html 

Corruption of Creation 

http://expeditionthink.blogspot.com/2022/09/corruption-of-creation.html 

[Samiya Illias]

Faith vs Science 

Transhumanism - I: aiming to create the god-self? 

https://signsandscience.blogspot.com/2022/06/transhumanism.html 

On 11-Apr-2023, at 10:49 AM, Samiya Illias <samiya...@gmail.com> wrote:



[spudb...@aol.com]

I think your task here is to decide if researchers have either created Jinns or have unintentionally provided a place (computer networks) where jinn's, thrive? Are they Tauhid? For the Mormons I haven't got deep into the website to analyze. Just that anybody who has some plausible fixes for human dilemmas needs to be looked at. If everything is haram, then nothing new need be learned. 

-----Original Message-----
From: Samiya Illias <samiya...@gmail.com>
To: everyth...@googlegroups.com
Sent: Tue, Apr 11, 2023 1:49 am
Subject: Re: GPT4 + Mathematica

--

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[spudb...@aol.com]

https://thequantuminsider.com/2023/04/04/time-is-on-my-sides-researchers-show-double-slit-experiment-also-applies-to-time/

Seems to be profound, if true? Not certain if there are any implications for science or for the rest of us?

[spudb...@aol.com]

Is it what the Almighty wants us to do? 

For me, transhumanism means and is directly focused on vastly, better, medicine. 

If one can live centuries we can get a lot of long term projects completed. One of which might be sending robots into the Universe to, as I recall, you are familiar with this guy, Frank Tipler, and made for the completion of the Omega Point? You've posted about this physicist once or twice if I recall correctly?

Frank J. Tipler - Wikipedia

So, maybe it's what the Big Mind needs, and what we were created for? We build things, then they build things and so on and so forth down the years? It's my guess and I could be wrong. I have had no personal revelations, and have met no archangel in a cave. I say, that we don't need any more new religions. We do need mental-apps to sustain these though. Philosophical mental apps specifically. 

Not for everyone, I agree. 

-----Original Message-----
From: Samiya Illias <samiya...@gmail.com>
To: everyth...@googlegroups.com
Sent: Tue, Apr 11, 2023 1:51 am
Subject: Re: GPT4 + Mathematica

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.

[smitra]

On 10-04-2023 12:34, John Clark wrote:
> On Sun, Apr 9, 2023 at 3:11 PM smitra <smi...@zonnet.nl> wrote:
>

>>>>> GPT: FOR A SOLID SPHERE OF UNIFORM DENSITY, WE CAN DIVIDE IT
>>> INTO SMALL volume elements, each with mass dm. The volume element

>>> located at a distance r from the center of the sphere has a moment
>>> of inertia relative to the x-axis given by:
>>> dIx = dm * y^2

>> _> smitra: dIz = dm * (x^2 + y^2)_
>
> I asked GPT to take a second look at the part where it said "DIX = DM
> * Y^2". I did NOT say it was wrong or it had made an error of any

> kind, I just asked it to look at it, but nevertheless it knew it had
> made a boo-boo. GPT said this:
>
> "GPT: I apologize for the error in my previous response. The correct
> expression for the moment of inertia of a small volume element
> relative to the x-axis is:
>
> dIx = dm * (y^2 + z^2)
>
> This takes into account the fact that the mass element is not
> necessarily located on the y-axis, but rather at a distance y from the
> x-axis and at a distance z from the z-axis. Similarly, the moment of
> inertia of a small volume element relative to the y-axis is:
>
> dIy = dm * (x^2 + z^2) and the moment of inertia of a small volume
> element relative to the z-axis is:
>
> dIz = dm * (x^2 + y^2)
>
> I hope this clarifies the issue. Thank you for bringing it to my
> attention."
>
> By the way, the version of GPT I was using was NOT the one that had
> the Mathematica plug-in.
>

> John K Clark See what's on my new list at Extropolis [1]
> 5vd

So, it has noticed that it used the wrong formulas and it has found the
correct ones. However, it was not able to do the calculation using
spherical symmetry. It only noted in the previous answer that the answer
is spherically symmetric because it doesn't depend on the orientation of
the axis relative to which the moment of inertia is computed.

So, it tells you the steps to compute the three dimensional integral in
spherical coordinates, it is unable to see that adding up the three
moments of inertias about the x, y and z axis yields a spherically
symmetric expression that can be computed using a one dimensional
integral over the radius.

Being unable to solve a problem in a much simpler way than the standard
way is a big red flag against the idea that GPT can do any math at all.

Saibal

[John Clark]

On Tue, Apr 11, 2023 at 11:06 PM smitra <smi...@zonnet.nl> wrote:

> So, it has noticed that it used the wrong formulas and it has found the
correct ones. However, it was not able to do the calculation using
spherical symmetry. It only noted in the previous answer that the answer
is spherically symmetric because it doesn't depend on the orientation of
the axis relative to which the moment of inertia is computed.
So, it tells you the steps to compute the three dimensional integral in
spherical coordinates, it is unable to see that adding up the three
moments of inertias about the x, y and z axis yields a spherically
symmetric expression that can be computed using a one dimensional
integral over the radius.
Being unable to solve a problem in a much simpler way than the standard
way is a big red flag against the idea that GPT can do any math at all.

I think the problem is that you and GPT have a difference of opinion about which is simpler, Cartesian coordinates or spherical coordinates. And there is no disputing matters of taste. 

I just add this conversation with GPT

> ME: If adding up the three moments of inertias about the x, y and z axis yields a spherically symmetric expression can the moment of inertia formula be derived by simply using a one dimensional integral over the radius?

GPT: "Yes, that's correct. Once we have established that the moment of inertia of a solid sphere is spherically symmetric, we can compute it using a one-dimensional integral over the radius. Specifically, the moment of inertia of a solid sphere of radius R and mass M about an axis passing through its center is given by:

I = (2/5) MR^2

This formula can be derived by integrating the spherically symmetric expression for the moment of inertia over the volume of the sphere using spherical coordinates, which simplifies the calculation to a one-dimensional integral over the radius. Here's how the derivation goes:

The moment of inertia of a solid sphere of radius R and mass M about an axis passing through its center is given by:

I = ∫∫ (r^2 sin^2θ) ρ dV

where ρ is the density and dV is the volume element in spherical coordinates. Since the sphere has uniform density, we can write ρ = M / (4/3 * π * R^3).Using the fact that the integrand is spherically symmetric, we can write:

I = ∫∫∫ (r^2 sin^2θ) ρ dV = ρ ∫∫∫ (r^2 sin^2θ) r^2 sin θ dr dθ dφ

Substituting in the expression for ρ and using the fact that the integral is spherically symmetric, we can simplify the integral to:

I = (3/5) MR^2 ∫∫ sin^3θ dθ dφ

The integral over θ and φ can be evaluated to give:

∫∫ sin^3θ dθ dφ = 4π/3

Substituting this into the expression for I, we get:

I = (2/5) MR^2

Therefore, the moment of inertia of a solid sphere about an axis passing through its center can be derived by simply using a one-dimensional integral over the radius, and is given by (2/5) MR^2."

 John K Clark    See what's on my new list at  Extropolis

5vd

[smitra]

> THEREFORE, THE MOMENT OF INERTIA OF A SOLID SPHERE ABOUT AN AXIS
> PASSING THROUGH ITS CENTER CAN BE DERIVED BY SIMPLY USING A
> ONE-DIMENSIONAL INTEGRAL OVER THE RADIUS, AND IS GIVEN BY (2/5) MR^2."
>
> John K Clark See what's on my new list at Extropolis [1]
>
> 5vd

The issue is that GPT keeps on doing the derivation in the same way
where the symmetry is broken by the choice of the axis and it then ends
up with the integral over theta, phi and r and after doing the
trigonometric integral it says that there is only an integral over r
left.

But if you just write down the formula for Ix + Iy + Iz then because for
Ik you need to integrate r^2 - distance along kth axis^2, we have to
integrate 3 r^2 - (x^2 + y^2 + z^2) = 2 r^2. So, the integrand is now
spherically symmetric as a result of having added up the moment of
inertia in 3 orthogonal directions. Since these moments of inertia are
all equal due to spherical symmetry, we have that 3 times the moment of
inertia is 2 times the integral of r^2, therefore we have:

Iz = 2/3 integral over ball of radius R of r^2 dm

You can then write dm = rho 4 pi r^2 dr and only have an integral over r
from 0 to R. Even this can be simplified by dividing Iz by M, so that we
get:

Iz/M = 2/[3 V(R)] Integral over ball of radius R of r^2 dV

We can then insert V(R) = 4/3 pi R^3 and dV = 4 pi r^2 dr and integrate
over r, but we don't even need to do that. We can simply use the fact
that V(r) is some constant c times r^3 without bothering to write down
that constant explicitly:

Integral over ball of radius R of r^2 dV

= Integral over ball of radius R of d[r^2 V(r)] - Integral over ball of
radius R of 2 r V(r) dr =


R^2 V(R) - 2/5 c R^5 = 3/5 R^2 V(R)

So, we have Iz/M = 2/[3 V(R)] 3/5 R^2 V(R) = 2/5 R^2



So, in the entire derivation I only needed to integrate over the radius
r. That's the whole point of restoring spherical symmetry. Otherwise you
end up having to evaluate integrals over the angles defined relative to
the rotation axis. ChatGPT can only do the latter because in its
database there is only a derivation based on the latter logic presented.
While there are spruce's in its database that discuss spherical symmetry
for this case, it can only then talk about that, it is unable to
actually do the much simpler calculation, simply because it cannot do
any calculation at all!


And besides this problem you put ChatGPT to the test with quite a few
other problems where the standard textbook derivation is quote a bit
more complex than another method that is easily explained. Assuming that
ChatGPT understands the math and the physics of the problem, it should
have no difficulties doing the problem in the alternative, easier way.
But what you'll find is that it can only do the problem in the more
complex way that s presented on most sources.

I can write down a list of a few simple such problems that I know of
later today...


Saibal