We all know that GPT is very competent, very knowledgeable. So let me go ahead and ask a technical question. It’s not trivial but not terribly complicated either; any student studying general relativity should be able to show the correct derivation. Here is my question:

Derive the geodesic equations for $$d\tau^2=Bdt^2-Adr^2-r^2d\theta^2-r^2\sin^2\theta d\phi^2$$, where $$A$$ and $$B$$ are functions of $$r$$ only.

Let’s see how GPT3.5 responds:

The geodesic equations for the given metric are given by:

$$\frac{d^2x^\alpha}{d\tau^2}+\Gamma^\alpha_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=0$$

where $$\tau$$ is the proper time, $$x^\alpha$$ are the coordinates $$(t,r,\theta,\phi)$$, and $$\Gamma^\alpha_{\mu\nu}$$ are the Christoffel symbols of the second kind, given by:

$$\Gamma^\alpha_{\mu\nu}=\frac{1}{2}g^{\alpha\beta}(\partial_\mu g_{\beta\nu}+\partial_\nu g_{\beta\mu}-\partial_\beta g_{\mu\nu})$$

where $$g_{\alpha\beta}$$ is the metric tensor.

For the given metric, the non-zero Christoffel symbols are:

\begin{align*}
\Gamma^t_{rr}&=\frac{1}{2}B’\\
\Gamma^t_{tt}&=\frac{1}{2}B\frac{A’}{B}\\
\Gamma^r_{rt}&=\frac{1}{2}\frac{A’}{B}\\
\Gamma^r_{rr}&=-\frac{1}{2}\frac{A’}{B}\\
\Gamma^\theta_{r\theta}&=\frac{1}{r}\\
\Gamma^\theta_{\theta\phi}&=-\sin\theta\cos\theta\\
\Gamma^\phi_{r\phi}&=\frac{1}{r}\\
\Gamma^\phi_{\theta\phi}&=\cot\theta
\end{align*}

where a prime denotes differentiation with respect to $$r$$.

Using these, we can write down the geodesic equations explicitly:

\begin{align*}
\frac{d^2t}{d\tau^2}+\frac{B’}{2B}\left(\frac{dt}{d\tau}\right)^2+\frac{A’}{2B}\left(\frac{dr}{d\tau}\right)\left(\frac{dt}{d\tau}\right)&=0\\
\frac{d^2r}{d\tau^2}-\frac{A’}{2B}\left(\frac{dt}{d\tau}\right)^2+\frac{A’}{2B}\left(\frac{dr}{d\tau}\right)^2+r\left(\left(\frac{d\theta}{d\tau}\right)^2+\sin^2\theta\left(\frac{d\phi}{d\tau}\right)^2\right)&=0\\
\frac{d^2\theta}{d\tau^2}+2\frac{dr}{d\tau}\frac{d\theta}{d\tau}-\sin\theta\cos\theta\left(\frac{d\phi}{d\tau}\right)^2&=0\\
\frac{d^2\phi}{d\tau^2}+2\frac{dr}{d\tau}\frac{d\phi}{d\tau}+2\cot\theta\frac{d\theta}{d\tau}\frac{d\phi}{d\tau}&=0
\end{align*}

Hmmm… sounds eminently knowledgeable, does it not?

Well… maybe not. How about we ask an actual computer algebra system:

(%i1) load(ctensor)$(%i2) derivabbrev:true$
(%i3) ct_coords:[t,r,u,v]$(%i4) depends([A,B],[r])$
(%i5) lg:matrix([B,0,0,0],[0,-A,0,0],[0,0,-r^2,0],[0,0,0,-r^2*sin(u)^2])$(%i6) cmetric(false)$
(%i7) christof(mcs)$B r (%t7) mcs = --- 1, 1, 2 2 A B r (%t8) mcs = --- 1, 2, 1 2 B A r (%t9) mcs = --- 2, 2, 2 2 A 1 (%t10) mcs = - 2, 3, 3 r 1 (%t11) mcs = - 2, 4, 4 r r (%t12) mcs = - - 3, 3, 2 A cos(u) (%t13) mcs = ------ 3, 4, 4 sin(u) 2 r sin (u) (%t14) mcs = - --------- 4, 4, 2 A (%t15) mcs = - cos(u) sin(u) 4, 4, 3 (%i16) geod:[0,0,0,0]$
(%i17) cgeodesic(true)\$
B t    + B  r  t
s s    r  s  s
(%t17)                     geod  = -----------------
1           B

2        2           2          2                     2
2 r sin (u) (v )  + 2 r (u )  - B  (t )  - 2 A r    - A  (r )
s           s      r   s          s s    r   s
(%t18) geod  = - --------------------------------------------------------------
2                                  2 A

2
r cos(u) sin(u) (v )  - r u    - 2 r  u
s        s s      s  s
(%t19)        geod  = - ----------------------------------------
3                        r

r sin(u) v    + 2 r cos(u) u  v  + 2 r  sin(u) v
s s               s  s      s         s
(%t20)     geod  = -------------------------------------------------
4                       r sin(u)



Looks different, doesn’t it. And no, I don’t mean LaTeX vs. the fixed pitch character representations of equations in a text terminal. Rather, the content.

The thing is, what GPT produces looks plausible. It has the right idea. The equations seem to make sense. Unless you know what to expect, you are likely to accept the result as correct, since it appears correct. But GPT sucks at math. It gets easily confused. It is a text model that is optimized to write equations that look right… but only has a superficial understanding of what it produces. Kind of like a student who is trying hard to remember, produces something that resembles the right thing, but without a perfect memory (and keep in mind, trained neural nets are not like any other software we are used to using, as they have no perfect memory!) and without in-depth understanding, fails.

I am sure over time this will improve. GPT-4 is already better at it than 3.5 (which was used to produce this outcome). And future versions may likely interface with computer algebra subsystems (among other things) to augment the neural net with specific capabilities. But for now, perhaps I can be forgiven for asking GPT’s cousin, DALL-E, to draw me a cat, exasperated by the bad math GPT produces:

I’ve been wanting to write about this all the way back in April, when folks became rather upset after Florida rejected some school math textbooks. A variety of reasons were cited, including references to critical race theory and things like social-emotional learning.

Many were aghast: Has the political right gone bonkers, seeing shadows even in math textbooks? And to a significant extent, they were correct: when a textbook is rejected because it uses, as an example, racial statistics in a math problem, or heaven forbid, mentions climate change as established observational fact, you can tell that it’s conservative denialism, not genuine concern about children’s education that is at work.

But was there more to these rejections than ludicrous conservative ideology? Having in the past read essays arguing that mathematics education is “white supremacist”, I certainly could not exclude the possibility. Still, it seemed unlikely. That is, until I came across pages like Mrs. Beattie’s Classroom, explaining “How to spark social-emotional learning in your math classroom“.

Holy freaking macaroni! I thought this nonsense exists only in satire, like a famous past Simpsons episode. But no. These good people think the best way to teach children how to do basic math is through questions like “How did today’s math make you feel?” — “What can you do when you feel stressed out in math class?” — “What self-talk can you use to help you persevere?” or even “How can you be a good group member?” The line between reality and satire does not seem to exist anymore.

In light of this, I cannot exactly blame Florida anymore. Conservatives may be living in a deep state of denial when it comes to certain subjects (way too many of them, from women’s health the climate change) but frankly, this nonsense is almost as freakishly crazy. If I were a parent of a school age child in the United States today, I’d be deeply concerned: Does it really boil down to a choice between schools governed by some form of Christian Taliban or wokeism gone berserk?

A few hours ago, my phone rang and my friend David told me the sad news: His father, Peter Ada-Winter passed away*.

I’ve known Peter since the late 1970s. He was an educator who played a pioneering role in the introduction of computer programming and computer science into the Hungarian school curriculum.

Peter was a true scholar. His home, a cramped apartment in Budapest’s historic Castle District, was full of books. The walls were lined with tall bookshelves. His large desk was also covered with teetering piles of textbooks, newspapers, and computer printouts.

Peter was the son of Ernő Winter, engineer, physicist and inventor, a towering figure in the early development of vacuum tube technology in the 1920s.

Born in 1923, Peter survived the Holocaust that wiped much of his extended family off the face of the Earth.

He was in his mid-40s when, in 1968, he was asked to organize a regular computer programming course for high school students in the same high school where I studied a decade later.

I met Peter when I became friends with his son David in high school. I always looked up to him. I learned quite a few things from him. Not just technical matters, basic human values as well.

Peter’s interest in information technology never diminished. In the mid-1980s, together with his son David they published a book on the 8-bit ZX Spectrum microcomputer. In the 1990s and early 2000s, despite his advancing age, Peter became well acquainted with the Internet; surfing the Web for news became part of his daily life.

Even after he passed 90, Peter remained in good health and mentally active. Only in the last few years did his health begin to gradually decline. Nonetheless, David remained hopeful that they would be able to celebrate Peter’s 100th birthday in 2023 with a ginormous birthday cake. These hopes were squashed by the news David received today.

Though I share David’s sense of grief, I reminded him that instead of grieving, we should remember the long, productive life of a very good man. Someone that I feel privileged to have known.

*No, not a COVID-19 statistic. Simply old age.

OK, so we’ve had Trump for nearly two years now, and we know that the White House has become a combination of kindergarten and insane asylum. My conservative friends still support Trump because he “delivers”, and are willing to completely overlook the fact that this president is not only a bumbling dilettante, an offensive excuse of a human being (waste of skin, to borrow a phrase from Lexx, a science-fiction series from a few years ago) but quite possibly a traitor to his nation, too, working for Putin’s Russia.

But if I hoped that Trump’s opposition is any better, they bitterly disappoint each and every day.

Take, for instance, the made-up controversy of a Kavanaugh aide presumably flashing “white power” hand signs while sitting behind Kavanaugh during his Supreme Court hearing, visible to cameras. Never mind that the hand sign was, in fact, a perfectly ordinary OK sign. Never mind that it was a well-documented Internet hoax from last year that suggested that this OK sign is, in fact, a secret hand gesture used by white supremacists. None of that stops many of my liberal friends from tweeting and retweeting the meme, complete with obscenities and death threats. Fact checking is for wimps, I guess.

And now I am reading about the bitter fate of a paper exploring the mathematics behind a controversial hypothesis dating back to Darwin’s times, called the “Greater Male Variability Hypothesis” (GMVH). The GMVH basically asserts that there are more idiots and more geniuses among men than women. It was Darwin who first noted that such greater variability is prevalent across many species in the animal kingdom. But politically correct guardians of science publishing would have none of that. Poor Darwin… the right hates him because he dares to challenge the idea that the world was created 6,000 years ago, but now the left hates him, too, because he dares to offer us politically incorrect science. The paper by Theodore P. Hill was first accepted and then rejected by journals, including a journal that already published the paper online, only to replace it with another a few days later. Never even mind the attack on academic freedom that this represents, but how about blatant sexism? You know, those impressionable young female scientists, fragile little flowers that they are, who cannot handle scientific truth and must be protected at all costs, unlike their ever so manly male colleagues…

One of the guests on Fareed Zakaria’s show today on CNN was Jonathan Haidt, one of the authors of the book, The Coddling of the American Mind. The authors explore the consequences of what they dub “safetyism”: Keeping children away from danger, real or perceived, at all costs, thus denying them a chance to become independent human beings. The result, according to the book, is that rates of anxiety, depression, even suicide are rising at an alarming rate, even as both students and professors on college campuses walk on eggshells, less they offend someone with a careless word, or heaven forbid, a hand gesture…

All in all, I am ready to conclude that the world is going bonkers, and those who seek salvation from Trump’s political opposition on the left (or seek salvation left-wing political opposition to right-wing populism and nativism elsewhere in the world) are deluding themselves.

Warning: the language of this post is offensive. I find ordinary words insufficient to express how I feel about Iran’s latest, a broad ban on female attendance at Iran’s universities.

When my wife asked the obvious question, “why?” I could only offer a very crude answer: because the ayatollahs had trouble jerking off last night and needed to find another way to get a hard-on. There really is no sensible way to describe this medieval theocracy that has been Iran’s curse for the past several decades.

There is a silver lining though: by depriving themselves of a capable, educated work force (quite capable, in fact, as women routinely outperformed men), perhaps they are hastening their moronic regime’s imminent demise. Cannot happen too soon, if you ask me.

Congratulations to Mariam Sultana, reportedly Pakistan’s first PhD in astrophysics. (Or in the subfield of extragalactic astrophysics, according to another news site. Either way, it’s a laudable achievement.)

I knew women scientists have an especially difficult time in very conservative Muslim countries.

I didn’t know astrophysicists (presumably, both male and female) had to pass an extra hurdle: apparently, illiterate Islamists don’t know the difference between astrophysics and astrology. The practice of astrology, like other forms of fortune telling, is considered haraam, a sin against Allah.

Am I ever so glad that I live in an enlightened, secular country.

One of Dr. Sultana’s (I am boldly assuming that Sultana is her last name, though I am well aware that Pakistani naming conventions do not necessarily follow Western traditions) examiners was James Binney, whose name is well known to anyone involved with galactic astrophysics; the book colloquially known as “Binney and Tremaine” (the real title is Galactic Dynamics) is considered one of the field’s “bibles”. (Darn, I hope no religious fanatic misconstrues the meaning of “bible” in the preceding sentence!)

I wish Dr. Sultana the brightest career. Who knows, maybe I’ll run into her one day somewhere, perhaps at the Perimeter Institute.

Neutrinos recently observed by CERN’s OPERA experiment may have been traveling faster than light. Or may have not. I have been discussing with physicists a number of possibilities: the role of statistics, errors in time or distance measurements, comparisons to SN 1987A, Cherenkov radiation, or the necessity for a Lorentz-violating theoretical framework.

Fortunately, there is one thing I did not need to discuss: How faster-than-light neutrinos relate to the Koran. Physics educators in Pakistan, such as Pervez Hoodbhoy writing for the Express Tribune, are not this lucky: they regularly face criticisms from fundamentalists, and if they choose to confront these head-on, they provoke ominous reader comments that call on all Muslims to “reject this evil experiment”.

Yet, there is a glimpse of hope: a Pakistani reader mentions Carl Sagan’s The Demon-Haunted World, one of Sagan’s last books, and a superb one about rational thinking versus superstition. I don’t know how popular Sagan’s book is in Pakistan, but I am glad it’s not forgotten.

So a few days ago, I wrote a blog entry about Ontario’s new grade school curriculum. The one that has since been withdrawn due to objections by conservative groups. I have to concede: they may have a point. I used no words in my blog post that were not used in the curriculum itself, yet the result was apparently too strong for Facebook; their automated software did not pick up and paste the entry onto my Facebook page.

Still, I stand by what I said: after I looked at the actual curriculum (as opposed to the sensationalized headlines about it) there really was nothing in it that a sane person could possibly object to. It’s not about sanity, of course, it’s about politics, which is why Ontario Liberals decided to abandon the updated curriculum after all. They can only fight one battle at a time, they say, according to the Toronto Star. I just wish that the battle they chose to keep fighting was this one, as opposed to the astonishingly braindead idea of messing up pharmacies by blocking payments to them by generic drug companies. Or the HST… which would have been a good idea back when the GST was introduced, but now, it’s just a badly disguised tax grab.

If you listened to Canadian news recently, like I did, you could be forgiven if you got the impression that Ontario’s liberal government released a 208 page curriculum that will teach first graders about masturbation and anal intercourse.

I actually took the trouble this morning and looked at the document in question. I was curious, has Ontario really become Liberal Hell?

Yes, the document does mention masturbation. Once. (Grade 6).

And yes, it does mention anal intercourse, not once, but three times (Grade 7). The context? Delaying sexual activity and avoiding sexually transmitted diseases.

While sexual education is an important subject in these 208 pages (Human Development and Sexual Health being one of the topic titles) the curriculum is not about sex. It is titled Health and Physical Education, and that’s precisely what it is about, with sexuality being a prominent, but certainly not dominant, part.