May 312023

I just finished uploading the latest release, 5.47.0, of our beautiful Maxima project.

It was a harder battle than I anticipated, lots of little build issues I had to fix before it was ready to go.

Maxima remains one of the three major computer algebra systems. Perhaps a bit (but only a bit!) less elegant-looking than Mathematica, and perhaps a bit (but only a bit!) less capable on some friends (exotic integrals, differential equations) than Maple, yet significantly more capable on other fronts (especially I believe abstract index tensor algebra and calculus), it also has a unique property: it’s free and open source.

It is also one of the oldest pieces of major software that remains in continuous use. Its roots go back to the 1960s. I occasionally edit 50-year old code in its LISP code base.

And it works. I use it every day. It is “finger memory”, my “go to” calculator, and of course there’s that tensor algebra bit.

Maxima also has a beautiful graphical interface, which I admit I don’t use much though. You might say that I am “old school” given my preference for the text UI, but that’s really not it: the main reason is that once you know what you’re doing, the text UI is simply more efficient.

I hope folks will welcome this latest release.

 Posted by at 8:54 pm
May 232023

In the last several years, we worked out most of the details about the Solar Gravitational Lens. How it forms images. How its optical qualities are affected by the inherent spherical aberration of a gravitational lens. How the images are further blurred by deviations of the lens from perfect spherical symmetry. How the solar corona contributes huge amounts of noise and how it can be controlled when the image is reconstructed. How the observing spacecraft would need to be navigated in order to maintain precise positions within the image projected by the SGL.

But one problem remained unaddressed: The target itself. Specifically, the fact that the target planet that we might be observing is not standing still. If it is like the Earth, it spins around its axis once every so many hours. And as it orbits its host star, its illumination changes as a result.

In other words, this is not what we are up against, much as we’d prefer the exoplanet to play nice and remain motionless and fully illuminated at all times.

Rather, what we are against is this:

Imaging such a moving target is hard. Integration times must be short in order to avoid motion blur. And image reconstruction must take into account how specific surface features are mapped onto the image plane. An image plane that, as we recall, we sample one “pixel” at a time, as the projected image of the exoplanet is several kilometers wide. It is traversed by the observing spacecraft that, looking back at the Sun, measures the brightness of the Einstein ring surrounding the Sun, and reconstructs the image from this information.

This is a hard problem. I think it is doable, but this may be the toughest challenge yet.

Oh, and did I mention that (not shown in the simulation) the exoplanet may also have varying cloud cover? Not to mention that, unlike this visual simulation, a real exoplanet may not be a Lambertian reflector, but rather, different parts (oceans vs. continents, mountain ranges vs. plains, deserts vs. forests) may have very different optical properties, varying values of specularity or even more complex optical behavior?

 Posted by at 12:06 am
May 052023

I got sent a link about an interesting, newly published book on the memoirs of Charles-Augustin de Coulomb. He was, of course, the French officer, engineer and physicist most famous for the Coulomb law that characterizes the electrostatic interaction.

As I occasionally receive e-mails from strangers about their self-published tomes or tomes published through vanity publishers of questionable credibility, I have come to the habit of dismissing such e-mails without paying them much attention. I am glad I paid more attention this time because this book is interesting, valuable, and genuine.

It is available as a matter of fact as a free PDF download from the authors but hey, I just bought the paperback. It was for some reason deeply discounted on Amazon Canada, so with free Prime shipping, all I am paying is the princely sum of $3.15. These days when even “cheap” paperback novels often cost 20 bucks if not more, how could I resist?

Of course it also helped that I looked at the PDF. I am sure the book has flaws (all books do) but it looks like a serious scholarly publication delivering real value to physicists and science historians both.

In fact, it is fascinating to see how modern, how advanced scientific thinking was already evident more than a quarter millennium ago. It makes me appreciate even more just how much of our collective human effort was needed to get from these early experiments to the present era of ubiquitous computer networks running amazing software that now mimics human intelligence, all powered by the same electricity that Coulomb was exploring.

 Posted by at 9:46 pm
May 022023

Not exactly the greatest discovery, I know, but GPT-4 still managed to offer an impressive demonstration of its understanding of gravitational physics when I asked it to build a Newtonian homogeneous universe:

What distinguishes GPT-4 from its predecessor is not that its training dataset is larger, but that it has significantly improved reasoning capabilities, which is well demonstrated by this answer. GPT 3.5 and Claude have the same knowledge. But they cannot put the pieces together quite like this (although they, too, can do impressive things with appropriate human guidance, one step at a time.)

 Posted by at 12:37 pm
Apr 202023

Another paper of ours, in all its 36-page glory, was accepted by Physical Review D and I am delighted.

One of the things we do in this paper is the use of symmetric trace-free tensors (STF) that allow us, among other things, to effect a rotation of spherical harmonic coefficients. This approach likely has many uses as a bit of practical math, beyond modeling gravitational fields.

 Posted by at 7:13 pm
Apr 202023

I just asked GPT-4 using my freshly developed UI about Starship.

In the background, GPT-4 correctly utilized a facility to perform Google searches and make use of the resulting snippets (I’m sure I’ll find a way to prevent it from referencing those snippets, which are not visible to the user, in the actual answer). Here is the critical part of the conversation that took place in the background:

User: What happened to Starship this morning?

GPT: SEARCH(Starship event on 2023/04/20)

System: Snippet 1: 3 hours ago … The rocket got off the launchpad before exploding above the Gulf of Mexico. This segment airs on April 20, 2023. Audio will be available after …
Snippet 2: UPCOMING. April 19, 2023. STARLINK MISSION · April 20, 2023. Starship Flight Test.
Snippet 3: 6 hours ago … April 20, 2023 … SpaceX launches the inaugural test flight of the integrated Starship and Super Heavy rocket from the company’s Starbase …
Snippet 4: 5 hours ago … April 20, 2023. Elon Musk has called it the “holy grail” for space technology. Starship, a new SpaceX rocket system that launched for the …

GPT: This morning, SpaceX launched the inaugural test flight of the integrated Starship and Super Heavy rocket from the company’s Starbase. Unfortunately, the rocket got off the launchpad but exploded above the Gulf of Mexico during the flight test (Snippet 1, Snippet 3).

In case you are wondering, yes, it knows the current date, too, thanks to the way my UI sets it up.

And yes, Starship blew up. But it blew up after the job of the first stage was nearly complete, at separation, at an altitude of some 40 km. Which means that despite the spectacular RUD (rapid unscheduled disassembly) event, this launch is still a huge success.

Now if we could only convince Elon to spend more time building rockets and less time honing his skills at corporate value destruction at Twitter, pardon me, X Corp…

 Posted by at 3:22 pm
Apr 132023

Once again, we have summer in April.

This picture shows the melting pile of dirty snow in our visitor parking area, photographed from our upstairs bedroom window.

The date and the temperature widget speak for themselves. (Yes, I still use Windows widgets, through a third-party tool. What can I say? I like them.)

 Posted by at 5:14 pm
Apr 032023

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:


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:


where a prime denotes differentiation with respect to \(r\).

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


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)$
(%t7)                          mcs        = ---
                                  1, 1, 2   2 A

(%t8)                          mcs        = ---
                                  1, 2, 1   2 B

(%t9)                          mcs        = ---
                                  2, 2, 2   2 A

(%t10)                          mcs        = -
                                   2, 3, 3   r

(%t11)                          mcs        = -
                                   2, 4, 4   r

(%t12)                         mcs        = - -
                                  3, 3, 2     A

(%t13)                        mcs        = ------
                                 3, 4, 4   sin(u)

                                          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

                        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:

 Posted by at 2:06 pm
Mar 222023

Students at the California State University, Northridge, are currently working on a short documentary film about our work on the Solar Gravitational Lens.

The project hopes to attract modest crowdfunding. They also produced a teaser trailer.

Of course I hope they succeed; our SGL work could use some good publicity.

 Posted by at 9:39 pm
Feb 272023

More than 20 years ago, I wrote a program that showed what Mars would look like if it had oceans.

I wrote it originally in C++, but then ported it to Java and turned it into a Web application.

That was a long time ago. Unfortunately, browsers no longer support Java applets for security reasons. So my beautiful applet was not working anymore.

Today, I finally spent some time reimplementing the applet in plain JavaScript. It’s working again! (Apart from some minor cosmetics.) And it has data sets for Mars, Earth, Venus and the Moon as well.

 Posted by at 2:47 am
Jan 262023

“Stop the presses! The Earth’s core stopped spinning! In fact it is now spinning backwards!”

Well, that’s pretty much how much of the popular press handled a recent article, published in Nature Geoscience, under the far less pretentious title, “Multidecadal variation of the Earth’s inner-core rotation”.

And indeed, the first half-sentence in the abstract says it all (emphasis mine): “Differential rotation of Earth’s inner core relative to the mantle“.

It’s not like the core stopped spinning. It’s just that the core is sometimes spinning slightly faster, sometimes spinning slightly slower than the mantle, an oscillatory pattern that has to do with the complex interaction between the two.

How much faster/slower? Don’t expect anything dramatic. At most a few degrees a year, but more likely, just a small fraction of a degree a year. So even if the ~70-year cycle (deduced by the authors of the recent article — there are other estimates) is valid, the core would only get ahead, or behind, the mantle by just a few degrees before it slows down or catches up again.

And this is what supposedly happened: the core was slowing down until a few years ago, its rotation came to be in sync with that of the mantle. Slowing down further, it’s now falling ever so slightly behind, only to catch up again, presumably, a few decades from now.

The way it is misleadingly presented in the media and the degree to which it is sensationalized demonstrate that we live in the era of hype.

 Posted by at 6:21 pm
Dec 122022

The National Ignition Facility has achieved a net power gain in its experimental fusion reactor. This is heralded as a major breakthrough.

Does this mean that in 50 years, we will have practical nuclear fusion power our world?

Oh wait. We were told exactly that some 50 years ago:

At the beginning of the 1950s, it seemed that success is not far away. But later, difficulties arose one after another […]

Unfortunately today there are still gigantic difficulties in the path towards utilizing this fabulously rich supply of energy […]

In fourteen countries of the world, more than two thousand engineers and scientists are laboring on working out different types of fusion devices.

To date, more than a hundred different models have been devised […]

Let us introduce only one group of these: the Soviet Tokamak devices, because around the world, these are the ones in which researchers have the most faith, viewing them as prototypes of future fusion power plants.

A year and a half ago, in an experiment carried out in collaboration between Soviet and English physicists, they directly measured the temperature and density of the plasma of Tokamak-3, and it became clear that the results were even better than indicated by prior measurements. To date, no other device could produce plasma of such quality.

When will the first fusion power plants be realized, when will the investigation of controlled nuclear fusion exit the constraints of laboratory experiments? According to Professor Igor Golovin, the world-renowned expert on thermonuclear research, it will be possible to develop Tokamak devices into electricity-producing equipment by the last decade of our century. L. Hirsch, one of the leading physicists of the American Atomic Energy Commission is a little more cautious. According to him the path from the first experiments to the worldwide spread of fusion power plants is longer, and we’re lucky if they will enter the world’s energy production market in fifty years.

These are all quotes (my translations) from a 1972 Hungarian-language educational children’s publication, “Boys’ Almanac 1973”.

As I express my (probably uninformed) skepticism concerning practical fusion power generation, I note that in the deep interior of the Sun, under gravitational confinement due to the combined mass of more than 300,000 Earths, fusion progresses at the leisurely rate of a few hundred watts per cubic meter. (The power output of a well-maintained industrial compost pile.) For practical power generation, we need something that is at least a million times that, a few hundred megawatts per cubic meter… and we don’t have 300,000 Earths for gravitational confinement.

Of course I’d be delighted if they proved me wrong.

 Posted by at 1:09 pm
Nov 072022

Every so often, I am presented with questions about physics that go beyond physics: philosophical questions of an existential nature, such as the reasons why the universe has certain properties, or the meaning of existence in light of the far future.

I usually evade such questions by pointing out that they represent the domain of priests or philosophers, not physicists. I do not mean this disparagingly; rather, it is a recognition of the fact that physics is about how the universe works, not why, nor what it all means for us humans.

Yesterday, I came across a wonderful 1915 painting by Russian avant-garde painter Lyubov Popova, entitled Portrait of a Philosopher:

What can I say? This painting sums up how I feel perfectly.

 Posted by at 1:19 am
Nov 052022

Today is November 5, 2022. Here we are in Ottawa, supposedly the second coldest capital city in the world after Ulan Bator. It is 9:15 PM.

And it is 21 degrees (centigrade — 70 F, for my American friends.)

Our A/C ran several times today, especially while we were baking something.

This is beyond incredible. I’ve seen snowstorms in this town in October. I’ve never seen summer-like weather in November.

If this is global warming… well, if folks who will likely be swept away by the sea in places like Florida don’t care, who am I to complain?

Still… weather like this in November is a bit creepy.

[And yes, I still use Windows gadgets, with the help of third-party software. What can I say? I like them.]

 Posted by at 9:23 pm
Oct 242022

There are only about six days left of the month of October and I have not yet written anything in this blog of mine this month. I wonder why.

Ran out of topics? Not really, but…

… When it comes to politics, what can I say that hasn’t been said before? That the murderous mess in Ukraine remains as horrifying as ever, carrying with it the threat of escalation each and every day? That it may already be the opening battle of WW3?

Or should I lament how the new American radical right — masquerading as conservatives, but in reality anti-democratic, illiberal authoritarianists who are busy dismantling the core institutions of the American republic — is on the verge of gaining control of both houses of Congress?

Do I feel like commenting on what has been a foregone conclusion for months, Xi “Winnie-the-pooh” Jinping anointing himself dictator for life in the Middle Kingdom, ruining the chances of continuing liberalization in that great country, also gravely harming their flourishing economy?

Or should I comment on the fact that prevalent climate denialism notwithstanding, for the first time in the 35 years that I’ve lived in Ottawa, Canada, our air conditioner came online in the last week of October because the house was getting too hot in this near summerlike heat wave?

Naw. I should stick to physics. Trouble is, apart from the fact that I still feel quite unproductive, having battled a cold/flu/COVID (frankly, I don’t care what it was, I just want to recover fully) my physics time is still consumed with wrapping up a few lose ends of our Solar Gravitational Lens study, now that the NIAC Phase III effort has formally come to a close.

Still, there are a few physics topics that I am eager to revisit. And it’s a nice form of escapism from the “real” world, which is becoming more surreal each and every day.

 Posted by at 7:41 pm
Sep 282022

I don’t always agree with Sabine Hossenfelder but every once in a while, she hits the nail spot on.

Case in point: Her article, published in The Guardian on September 26, about the state of particle physics.

Imagine going to a zoology conference, she says, where a researcher discusses a hypothesis (complete with a computer-generated 3D model) of a 12-legged purple spider living in the Arctic. Probably doesn’t exist but still, how about proposing a mission to the Arctic to search for one? After all, a null result also contains valuable data. Or how about a flying earthworm that lives in caves? Martian octopuses, anyone?

Zoology conferences do not usually discuss such imaginary monsters but, Sabine argues (and she is spot on) this is pretty much what particle physics conferences are like: “invent new particles for which there is no evidence, publish papers about them, write more papers about these particles’ properties, and demand the hypothesis be experimentally tested”. Worse yet, real money is being spent (wasted might be a better word) on carrying out such experiments.

She points out that while it is true that good science is falsifiable, the opposite isn’t always the case: Just because something is falsifiable does not make it good science.

And not just particle physics, I hasten to add. How about cosmology and gravitation? Discussions about what may or may not have happened during the Planck epoch? Exploring exotic spacetime topologies, often in dimensions other than four? And let me not even mention quantum computing or fusion energy…

Perhaps I am a born skeptic lacking imagination, but to me, these are all 12-legged purple Arctic spiders. The science we actually know and have the ability to confirm are general relativity in a spacetime that is by and large the perturbed Minkowski metric; and the Standard Model of particle physics, extended with a neutrino mass mixing matrix. These are the things that work. Not perfectly, mind you. General relativity needs “dark matter” (name aside, we don’t know what it is except that it has a dust equation of state) and “dark energy” (again, it has a name but beyond that, we don’t know what it is beyond its equation of state) to account for galaxy dynamics and cosmic evolution. The Hubble tension, the discrepancy between values of the Hubble parameter measured using different methods, is real. Observations by the James Webb space telescope suggest that we do not understand the “dark ages”, the first few hundred million years after the surface of last scattering (i.e., the epoch when the cosmic microwave background radiation was produced), well. Massive neutrinos invite the question about the apparent absence right-handed neutrinos.

And yes, we are very much in the blind concerning these issues. Nature has not yet provided hints and we are not smart enough to figure out the answers entirely on our own. But how is that an excuse for inventing 12-legged spiders?

I think it isn’t.

 Posted by at 6:40 pm
Sep 262022

Putting aside Trumpism, woke-ism, the politics of the day, populism, the whole kaboodle, here’s something to contemplate.

Tonight, Russia is continuing its efforts to subjugate the independent nation of Ukraine, not refraining from committing serious, intentional, criminal acts against the country’s civilian population to further its goals.

Also tonight, the space agency of the United States, NASA, is conducting a ground-breaking experiment, the first of its kind testing a method that might one day avert a global disaster, protecting the entire planet from an asteroid impact.

DART: View of the asteroid Dimophos 20 minutes to impact

I think it speaks volumes about the different ways in which these countries see their respective roles in the world.

I almost long for Soviet times. The regime was assuredly brutal, but at least it professed to seek noble goals. Not anymore, I guess.

 Posted by at 7:18 pm
Sep 132022

Oops. It’s past midnight already, so technically it was yesterday but to me it is still today, September 12.

The sixtieth anniversary of John F. Kennedy’s famous “we choose to go to the Moon” speech. How many more years before another human sets foot on the Moon?

Oh, and it was thirty years ago that Ildiko and I became married.

Yup, that’s us; 1979 vs. 2019.

 Posted by at 1:54 am
Sep 012022

A few days ago I had a silly thought about the metric tensor of general relativity.

This tensor is usually assumed to be symmetric, on account of the fact that even if it has an antisymmetric part, \(g_{[\mu\nu]}dx^\mu dx^\nu\) will be identically zero anyway.

But then, nothing constrains \(g_{\mu\nu}\) to be symmetric. Such a constraint should normally appear, in the Lagrangian formalism of the theory, as a Lagrange-multiplier. What if we add just such a Lagrange-multiplier to the Einstein-Hilbert Lagrangian of general relativity?

That is, let’s write the action of general relativity in the form,

$$S_{\rm G} = \int~d^4x\sqrt{-g}(R – 2\Lambda + \lambda^{[\mu\nu]}g_{\mu\nu}),$$

where we introduced the Lagrange-multiplier \(\lambda^{[\mu\nu]}\) in the form of a fully antisymmetric tensor. We know that


since the product of an antisymmetric and a symmetric tensor is identically zero. Therefore, variation with respect to \(\lambda^{[\mu\nu]}\) yields \(g_{[\mu\nu]}=0,\) which is what we want.

But what about variation with respect to \(g_{\mu\nu}?\) The Lagrange-multipliers represent new (non-dynamic) degrees of freedom. Indeed, in the corresponding Euler-Lagrange equation, we end up with new terms:

$$\frac{\partial}{\partial g_{\alpha\beta}}(\sqrt{-g}\lambda^{[\mu\nu]}g_{[\mu\nu]})=

But this just leads to the trivial equation, \(\lambda^{[\mu\nu]}=0,\) for the Lagrange-multipliers. In other words, we get back General Relativity, just the way we were supposed to.

So in the end, we gain nothing. My silly thought was just that, a silly exercise in pedantry that added nothing to the theory, just showed what we already knew, namely that the antisymmetric part of the metric tensor contributes nothing.

Now if we were to add a dynamical term involving the antisymmetric part, that would be different of course. Then we’d end up with either Einstein’s attempt at a unified field theory (with the antisymmetric part corresponding to electromagnetism) or Moffat’s nonsymmetric gravitational theory. But that’s a whole different game.

 Posted by at 11:40 pm
Aug 202022

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?

 Posted by at 10:08 pm