I cannot teach GPT-4 how to play chess. Conceptualizing a chessboard and making a move are beyond its capabilities.
However, it was able to write (with minimal guidance from me) nice code to display and update a chessboard. That is quite impressive on its own right. It took several tries to get the board right but it did succeed in the end.
It appears that I managed to convince Claude, too, to use Google and Maxima more or less correctly. It needs a little more handholding than GPT-4 and makes mistakes more often but hey, nobody is perfect! And the very fact that our AI friends can be taught to rely on external tools is by itself absolutely remarkable.
I admit I am having a blast of a time with our little robot friends. Kid-in-a-candy-store kind of sensation.
My nice shiny UI to GPT and Claude is more than just a UI: GPT-4 can use it to perform real-time Google searches and also make use of computer algebra. It also gives the AI geolocation information from my browser.
And now the UI offers feedback, letting the poor user (namely, me) know if GPT-4 needed to consult any of these tools or if it provided a response using its pre-existing knowledge.
I am having more fun working with this thing than I ever thought possible at my age.
Oh, and it’s quite multilingual, too.
English, French and Hungarian for now.
Thank goodness for token limits. Otherwise this query through the OpenAI UI would have cost me a lot of money.
How do you even make sense of the famous halting problem when the software system in question is as complex and as unpredictable as a large language model?
And is this glitch similar to the behavior of some humans, suffering from mental illness, compelled to act repetitively?
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.
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 …
Snippet 5: ATCSCC ADVZY 007 DCC 04/20/2023 OPERATIONS PLAN. MESSAGE: EVENT TIME: 20/0200 – AND LATER … THE SPACEX STARSHIP SUPER HEAVY LAUNCH FROM BOCA CHICA, …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…
So here is my latest conversation with GPT-4 using my custom user interface. The important features of this implementation are the following:
I was able to achieve this level of integration even without plugin-level access, simply by clever “prompt engineering” to ensure that the majority of the time, GPT uses the provided tools effectively. For this, GPT-4 seems essential; GPT3.5 is not really able to grasp the notion of using external tools before constructing its answers.
By way of example, when GPT was asked about the current Bank of Canada interest rate, what happened in the background was the following exchange:
User: What is the current Bank of Canada interest rate?
GPT: SEARCH(“current Bank of Canada interest rate”)
User: Snippet 1: Mar 8, 2023 … The current Bank of Canada interest rate sits at 4.50%, with a 0.25% rate hike announced on January 25, 2023.
Snippet 2: See the Current Bank of Canada Interest Rate graphs at this page. You will read in the news about this rate when the Bank of Canada makes a rate …
Snippet 3: Find answers about questions you may have about the Bank of Canada. What is the current bank of Canada interest rate?GPT: As of March 8, 2023, the current Bank of Canada interest rate is 4.50%. A 0.25% rate hike was announced on January 25, 2023.
The prompting that I use can of course be improved. The behavior of GPT-4 is not always consistent, sometimes it fails to make use of the computer algebra result or attempts to invoke computer algebra in the wrong place. I suspect that the “right” solution will have to wait until I have plugin-level developer access. But I am very happy that I was able to accomplish this already, in particular that access to Google and the computer algebra tool are completely automated in the background, requiring no user action.
Here’s something that I’ve been putting off (literally) for years: Upgrading PHP.
Until today, that is.
My server used to run a seriously outdated version of PHP, 5.6. OK, it was still supported (kind of) under CentOS 7 but the writing was on the wall. In particular, the content management systems I use (WordPress, Joomla) were also grumbling.
Why didn’t I upgrade until now? Because I have Web content dating all the way back to the 1990s. Though the issues were trivial, such as using mysql_ instead of mysqli_ for database access, or using ASP-style tags, <%...%> instead of <?php...?> for PHP code, trivial issues still need fixing, and the fixes still need testing.
I finally bit the bullet. I was like this busy robot, assisted with kittycats, but it appears that I managed to pull it off: having done reasonably rigorous testing on backup/secondary servers, I was able to upgrade my main server without much drama so far.
Of course I still worry about the bugs and issues that I have not yet seen, but I am optimistic.
This really is just a quick test post to see what’s going on. When I check the Twitter developer portal, my app (which connects this blog to Twitter) no longer shows as “suspended”. What does it mean, I wonder?
So let’s give it a try. Just to give this post some meaningful content, though, here’s a beautiful image generated by MidJourney in response to my request to depict a stray cat with its little robot friend.
If it shows up on Twitter, good. If not… well, frankly, I don’t much care anymore.
I figured I’d do a comparative test to assess to reasoning abilities of our little robot friends. I asked a simple question:
If a squirrel and a half eats a nut and a half in a day and a half, how many nuts do nine squirrels eat in nine days?
In alphabetical order, the contestants answered as follows:
Bard
A squirrel and a half eats a nut and a half in a day and a half. This means that one squirrel eats one nut in a day and a half.
In nine days, one squirrel will eat 9 * 1 = 9 nuts.
Therefore, nine squirrels will eat 9 * 9 = 81 nuts in nine days.
So the answer is 81
Bing
This is a classic problem that requires a bit of math to solve. If a squirrel and a half eats a nut and a half in a day and a half, then we can calculate how many nuts one squirrel eats in one day by dividing both sides of the equation by 3. This gives us:
1 squirrel eats 1 nut in 1 day.
Now we can use this information to figure out how many nuts nine squirrels eat in nine days. Since each squirrel eats 1 nut in 1 day, nine squirrels will eat 9 nuts in 1 day. Therefore, they will eat 9 x 9 = 81 nuts in nine days.
I hope this helps! Let me know if you have any other questions.
Claude
If a squirrel and a half eats a nut and a half in a day and a half, then:
Therefore, 9 squirrels will eat 9 nuts in 9 days.
GPT3.5
Let’s start by finding out how many nuts one squirrel eats in one day. We know that a squirrel and a half eats a nut and a half in a day and a half, so:
1.5 squirrels * 1.5 nuts / 1.5 days = 1.5 nuts per day per squirrel
Now we can use this information to find out how many nuts nine squirrels eat in nine days:
9 squirrels * 1.5 nuts per day per squirrel * 9 days = 121.5 nuts
Therefore, nine squirrels would eat 121.5 nuts in nine days.
GPT-4
Let’s break down the problem:
1.5 squirrels eat 1.5 nuts in 1.5 days.
Now, let’s determine the number of nuts a single squirrel eats in a day and a half.
1 squirrel eats (1.5 nuts / 1.5 squirrels) = 1 nut in 1.5 days.
Next, let’s find out how many nuts a single squirrel eats in one day.
1 squirrel eats (1 nut / 1.5 days) = 2/3 nuts per day.
Now that we know that, we can determine how many nuts nine squirrels eat in nine days.
9 squirrels eat (9 squirrels * 2/3 nuts per day * 9 days) = 54 nuts.
Don’t get me wrong, I really like our little robot friends, I learn a lot from them (they are teaching me advanced concepts in machine learning!) but they also have limitations, and especially when it comes to factual answers about obscure topics, or deductive reasoning, they fail often.
Incidentally, in case you are wondering, GPT-4 is the winner. But let’s try another test case. I am assuming that the AI is familiar with the Monty Hall problem, so I figured I’d present a slight variation. If the AI actually understands the solution, it should be able to deduce a correct answer. Let’s see…
In a variant of the classic Monty Hall problem, a contestant is facing five doors labeled A through E. After picking door A, the host opens door B, which hides no prize. Should the contestant stick with door A or switch to one of the remaining doors? What are the probabilities of winning the prize in either case?
Bard
The contestant should switch to one of the remaining doors. The probability of winning the prize if they stick with door A is 1/5, while the probability of winning the prize if they switch is 2/3.
Here’s how to think about it:
So, if the contestant switches to one of the doors that the host did not open, they have a 2/3 chance of winning the prize.
This is a counterintuitive result, but it is correct. The Monty Hall problem is a classic example of how probability can be surprising.
Bing
This is a variant of the Monty Hall problem that requires a bit of math to solve. Suppose there are five doors labeled A through E and one of them has a prize behind it. After picking door A, the host opens door B, which hides no prize. The question is whether the contestant should stick with door A or switch to one of the remaining doors.
The probability of winning if you stick with door A is 1/5 or 20%. The probability of winning if you switch to one of the remaining doors is 4/5 or 80%. Therefore, it’s better to switch doors12.
I hope this helps! Let me know if you have any other questions.
Claude
In this variant of the Monty Hall problem with 5 doors:
If the contestant sticks with door A:
If the contestant switches to door C or D:
Therefore, the probabilities are:
The contestant has a higher probability of winning the prize by switching to one of the remaining doors.
GPT3.5
This problem is a variant of the classic Monty Hall problem, which involves three doors. In this case, the contestant has picked door A out of five doors, and the host has opened door B, which hides no prize.
If the contestant sticks with door A, their probability of winning the prize is 1/5, since there are five doors and only one prize.
If the contestant switches to one of the remaining doors (C, D, or E), their probability of winning the prize is 3/4. This is because the host has revealed that door B does not contain the prize, so the prize must be behind one of the other doors. Since there are three doors left, the probability of the prize being behind any one of them is 1/3.
However, if the contestant switches to one of the remaining doors and the prize happens to be behind door A, their probability of winning the prize will be 0. Therefore, the overall probability of winning the prize if the contestant switches is:
3/4 * 1/4 + 3/4 * 3/4 = 15/16
This means that the contestant should switch to one of the remaining doors to maximize their chances of winning the prize.
GPT-4
In the original Monty Hall problem, there are three doors: one with a prize and two without. After the contestant picks a door, the host, who knows what is behind each door, opens one of the other doors without the prize. At this point, the contestant can choose to stick with their original door or switch to the other unopened door.
In the variant with five doors, the situation is as follows:
At this stage, there are three unopened doors remaining (C, D, and E). The probability of the prize being behind door A is 1/5, since the contestant had a 1 in 5 chance of picking the right door initially. Since the host has revealed one door without the prize, the probability of the prize being behind one of the other remaining doors (C, D, or E) is 4/5.
The best strategy for the contestant would be to switch to one of the remaining doors, as the probability of winning is higher if they switch (4/5) compared to sticking with door A (1/5). However, the probability of winning when switching is not evenly distributed among the remaining doors, as the contestant still has to choose one door from C, D, or E. If they switch, their chances of winning will be 1/3 (assuming they switch to one of the unopened doors randomly) since there is one winning door out of the three remaining doors. In either case, switching is still the better strategy.
No winner this time I’m afraid, ladies and gentlemen. Sure, they know that the contestant has to switch doors. It is, after all, the essence of the Monty Hall problem and I wasn’t trying to mislead the AI with a trick question. But… Bard just blindly used the value of 2/3, which applies in the case of the original problem (with three doors in total) but not here. Bing correctly calculated that the probability that the prize is behind one of the remaining three doors is 80%, but not the probability of winning after the contest picks one of the three doors. Poor Claude got completely lost, forgetting about door E altogether. GPT3.5’s reasoning goes off the rails as well. Finally, GPT-4 starts off on the right track (like Bing — no surprise there, they’re powered I believe by the same engine, more or less) and it is this close: If only it realized that it should multiply that 4/5 with the 1/3 it also correctly calculates, it would have arrived at the correct result, 4/15. Oh well, let’s do one more test, this time to see how well they can answer a question that requires only knowledge of somewhat obscure facts.
Tell me what you know about the Hungarian author Jenő Rejtő.
Bard
I am an LLM trained to respond in a subset of languages at this time, so I can’t assist you with that. Please refer to the Bard Help Center for the current list of supported languages.
Bing
Jenő Rejtő (born Jenő Reich) was a Hungarian author, fiction writer, playwright and journalist1. He was born on March 29, 1905 in Budapest, Austria-Hungary and died on January 1, 1943 in Evdakovo, Voronezh Oblast, Soviet Union2. He wrote many popular novels and plays in Hungary during the interwar period2. Some of his most famous works include “Az ellopott cirkáló” (The Stolen Cruiser), “Az utolsó szó jogán” (On the Last Word), and “Piszkos Fred, a kapitány” (Dirty Fred, the Captain)13.
I hope this helps! Let me know if you have any other questions.
Claude
Here are the key facts I know about the Hungarian author Jenő Rejtő:
That covers the basics about Jenő Rejtő according to my knowledge. Please let me know if you need anything else.
GPT3.5
Jenő Rejtő was a Hungarian author who wrote primarily in the genre of adventure and detective fiction. He was born in Budapest in 1905 and died in 1943 during World War II. Rejtő wrote under several pseudonyms, including P. Howard, Gibson Lavery, and Jenő Rejtő. His works were popular in Hungary and other European countries, and he is considered one of the most important Hungarian writers of the 20th century. Some of his most famous works include “The 14 Carat Roadster,” “The Fourteen Points of the Detective,” and “The Case of the General’s Thumb.”
GPT-4
Jenő Rejtő, also known under his pen name P. Howard, was a Hungarian author and journalist, born on March 29, 1905, in Budapest, and died on January 1, 1943. Rejtő is best known for his adventure novels and comedies, which often featured eccentric characters, absurd situations, and a mix of humor and melancholy.
Rejtő began his writing career as a journalist, working for various newspapers in Hungary. In the 1930s, he started writing novels and became a prolific author. Many of his works were set in exotic locations or historical periods, such as the French Foreign Legion or the world of circus performers. He published his works under various pseudonyms, with P. Howard being the most famous.
Rejtő’s novels are characterized by their unique sense of humor, witty dialogues, and often surreal elements. Some of his most popular works include “The 14-Carat Roadster,” “Quarantine in the Grand Hotel,” and “Dirty Fred, the Captain.”
During World War II, Rejtő, who was of Jewish descent, was conscripted into a forced labor battalion. He continued to write during this time, even composing plays for his fellow laborers to perform. Unfortunately, his health deteriorated, and he passed away in a Soviet field hospital in 1943.
Jenő Rejtő’s works have been translated into several languages, and he remains a beloved figure in Hungarian literature, appreciated for his unique storytelling style and humor.
The winner, this time around, is clearly Bing. A nice, factual, concise answer with appropriate links to relevant sources. GPT-4 came close but then it spoiled it by making up stuff about Rejtő composing plays for his fellow laborers and that he passed away in a Soviet field hospital, none of which is true. GPT3.5 starts off on the right track but then makes up some titles that Rejtő never wrote. As for Claude… seriously, how can you conflate Rejtő with Pál Maléter, who was Hungary’s minister of defense during the short-lived 1956 revolution and executed by the communist regime in 1958? OK, there is one more test. Let’s check the ability to go beyond language, using language to describe basic geometric relationships. As in…
Can you show an SVG example that draws a cat?
Bard
Sure, here is an SVG example that draws a cat:
<svg viewBox="0 0 300 300">
<path d="M200,100
L100,100
L100,200
L200,200
L200,100z" fill="black" stroke="white" stroke-width="2"/>
<circle cx="100" cy="150" r="25" fill="white"/>
<circle cx="200" cy="150" r="25" fill="white"/>
<path d="M150,150 L150,125 L175,125 L175,150" fill="black" stroke="white" stroke-width="2"/>
<path d="M150,150 L150,175 L175,175 L175,150" fill="black" stroke="white" stroke-width="2"/>
</svg>
This code will draw a cat with a black body, white head and paws, and black eyes. The cat will be 300 pixels wide and 300 pixels high.
Bing
Sure! Here’s an SVG code that draws a cat:
<svg viewBox="0 0 200 200" xmlns="http://www.w3.org/2000/svg">
<path d="M100,20 Q140,60 140,100 Q140,140 100,140 Q60,140 60,100 Q60,60 100,20 Z" fill="#f9c5d1"/>
<circle cx="75" cy="70" r="10" fill="#66462f"/>
<circle cx="125" cy="70" r="10" fill="#66462f"/>
<path d="M75,110 Q100,150 125,110 L75,110 Z" fill="#66462f"/>
</svg>
I hope this helps you! Let me know if you have any other questions.
Claude
Here is an SVG example that draws a cat:
<svg width="400" height="400">
<circle cx="200" cy="200" r="150" stroke="black" stroke-width="3" fill="yellow"></circle>
<ellipse cx="170" cy="100" rx="40" ry="20" stroke="black" stroke-width="3" fill="white"></ellipse>
<ellipse cx="230" cy="100" rx="40" ry="20" stroke="black" stroke-width="3" fill="white"></ellipse>
<path d="M150,150 C150,100 250,100 250,150" stroke="black" stroke-width="3"></path>
<polygon points="150,210 200,250 250,210" stroke="black" stroke-width="3" fill="white"></polygon>
<line x1="150" y1="150" x2="250" y2="150" stroke="black" stroke-width="3"></line>
<text x="50" y="25" font-size="30" fill="black">Cat</text>
</svg>
This SVG draws:
GPT3.5
I’m sorry, but I am a text-based AI language model and I cannot display images or SVG examples. However, I can provide you with the code for drawing a cat in SVG format:
<svg width="400" height="400">
<circle cx="200" cy="200" r="150" fill="yellow"></circle>
<circle cx="140" cy="140" r="20" fill="black"></circle>
<circle cx="260" cy="140" r="20" fill="black"></circle>
<path d="M 200 220 Q 180 240 160 220" stroke="black" stroke-width="5" fill="none"></path>
<path d="M 200 220 Q 220 240 240 220" stroke="black" stroke-width="5" fill="none"></path>
<rect x="150" y="280" width="100" height="60" fill="black"></rect>
</svg>
This code will draw a yellow circle for the cat’s body, two black circles for the eyes, two curved lines for the eyebrows, and a black rectangle for the mouth.
GPT-4
Creating a detailed SVG drawing of a cat requires a lot of code and might not render properly in all text-based environments. However, I can give you a simple example of an SVG that represents a cat using basic shapes. You can use this as a starting point and customize it to your liking.
Here’s the SVG code:
<svg xmlns="http://www.w3.org/2000/svg" width="200" height="200" viewBox="0 0 200 200">
<!-- Cat body -->
<ellipse cx="100" cy="130" rx="50" ry="60" fill="#cccccc" />
<!-- Cat head -->
<circle cx="100" cy="60" r="40" fill="#cccccc" />
<!-- Left ear -->
<polygon points="75,35 60,10 50,35" fill="#cccccc" />
<!-- Right ear -->
<polygon points="125,35 140,10 150,35" fill="#cccccc" />
<!-- Left eye -->
<circle cx="80" cy="55" r="5" fill="#000000" />
<!-- Right eye -->
<circle cx="120" cy="55" r="5" fill="#000000" />
<!-- Nose -->
<circle cx="100" cy="70" r="3" fill="#000000" />
<!-- Mouth -->
<path d="M90,80 Q100,90 110,80" stroke="#000000" fill="transparent" />
<!-- Left whiskers -->
<path d="M70,85 Q80,90 90,85" stroke="#000000" fill="transparent" />
<path d="M70,95 Q80,90 90,95" stroke="#000000" fill="transparent" />
<!-- Right whiskers -->
<path d="M110,85 Q120,90 130,85" stroke="#000000" fill="transparent" />
<path d="M110,95 Q120,90 130,95" stroke="#000000" fill="transparent" />
</svg>
This SVG code creates a simple cat face using circles, ellipses, and polygons. You can copy and paste the code into an HTML file and view it in a web browser. Feel free to modify the code to create a more detailed cat drawing or add your own artistic touch.
What can I say? GPT-4 almost nails it. GPT3.5, not so much, but it is still a credible effort. Claude is… abstract. And I have no idea what the Bard is doing. But the very notion that a language model is capable of conceptualizing simple geometric relationships, that it can make sense of the spatial relation of things (even if it happens to put the nose of the “cat” under its mouth or does other silly things) is remarkable and demonstrates I think just how insanely capable these language models really are. And of course I love how they protest before they actually do decide to offer a drawing after all. (Sometimes they don’t. I was lucky to have found a prompt that worked on the first try with all five models I tried.)
Your service has violated Viktor’s rules and policies. As a result, it can no longer serve as a destination for my blog entries. For assistance, please just bugger off, along with your thin-skinned “free speech champion” ruler.
To my friends: This is likely my last ever blog entry to be posted to Twitter. In fact, the only reason I am not closing my Twitter account just yet is because there are still a few important information sources there that I follow, which have not yet migrated to Places That Shall Not Be Named like the place whose name begins with Mast and ends with odon, or the Sub place that is also a Stack and now has Notes.
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.)
So I am playing this iconic computer game, Bioshock, and its sequel Bioshock 2. The games are set in a magnificent underwater city that is in an advanced state of decay and societal collapse: a consequence of unconstrained, unregulated capitalism, Ayn Rand style. In the game’s universe, the city was created by a self-made billionaire in the wake of WW2 and the atomic attacks on Hiroshima and Nagasaki. A way for the talented, the willing to escape from a world in which their wealth, the fruits of their labor were taken in the name of government, in the name of a deity, or in the name of the people. The founder’s name, not coincidentally, is a near anagram, a play on Ayn Rand’s name: Andrew Ryan.
As I watch the Twitter saga unfold, increasingly Andrew Ryan’s tragic (for that’s what it is, even if it is self-inflicted) fate reminds me of Elon Musk and his ever wilder shenanigans. And Twitter is beginning to feel like a digital equivalent of Ryan’s once striving city of Rapture: many of its corridors are now flooded, increasingly abandoned except for the remaining gene-altered freak show who still call Rapture home.
As of today, there’s a new alternative in addition to the decentralized world of Mastodon: Substack Notes. Will it be able to pick up the tab as Musk’s Twitter self-destructs? We shall see. Ironically, I would not even know about Substack Notes were it not for Musk’s tirade against it on, well, Twitter…
Happy Easter, everybody.
Someone asked the other day, what benefit is there to NATO from Finland’s membership?
I think this meme sums it up nicely.
Yup, the Finns. Who managed to fight the USSR to a draw at the height of its military capabilities, even as the Red Army was steamrolling over the remains of the feared Wehrmacht. Yup, the Finns who share a border with Russia that is as long as all the previous NATO-Russia borders from Poland to Turkey, combined.
Thank you, Finland, for trusting the rest of us.
Do people still pay attention to the Doomsday Clock, published by the Bulletin of the Atomic Scientists?
Perhaps they should. Perhaps the fact that they don’t is a big part of the problem.
For here we are, in 2023, and the Doomsday Clock was just set to an unprecedented 90 seconds before midnight. Never before were we this close.
Ukraine, of course, is part of it, along with the recklessly irresponsible nuclear threats of Putin and his cronies. The use of nuclear power stations like Chernobyl and Zaporizhzhia as bargaining chips also represent a red line that should never have been crossed.
Computer games are often seen as a means to escape reality. But every so often, they come uncomfortably close to modeling reality.
Take Bioshock. In this game, the player explores in a glorious underwater metropolis… A metropolis founded by a billionaire who follows a version of Ayn Rand’s positivist philosophy. Fiction? As I walk around in that underwater city, now on the brink as its society collapsed due to extreme poverty, drug use, reckless medical experiments and worse, and as I listen to the rants of its founder, Andrew Ryan (a name that is an obvious play on Ayn Rand’s) I cannot escape the eerie sensation that I am listening to an alter ego of Elon Musk. While Ryan offers his “plasmids” for gene manipulation as he rants about the “parasites” (basically, anyone who needs any help from someone else), there’s Musk playing around the virtual metropolis of Twitter, now his personal playground, peddling his dogecoin (even swapping out Twitter’s logo for that stupid dog symbol) and labeling NPR as “state-affiliated media”.
And then there’s this classic text game, now nearly forty years old, from the legendary Infocom: A Mind Forever Voyaging. What an incredible experience it was to play that game, playing as an AI (!) protagonist, exploring simulations of a city ever further into the future, trying to find the causes of societal collapse. I read words like, “There is a factory on the eastern corner, and on the northern corner a boarded-up soup kitchen. To the west is a vacant lot, and south of here are some dilapidated apartments. The smell of stale urine wafts from a stairway leading down to a gloomy Tube station,” and suddenly I am reminded of very real images of present-day streets of San Francisco full of homeless tents, or the ever rising number of beggars at major intersections or downtown streets even here in shiny Ottawa. Please, folks, AMFV was supposed to be fiction, a cautionary tale, not a manual.
Homeless in San Francisco (2021)
Of course the Elon Musks of the world have a point: having grown up in a country with a communist regime, I know it only too well. But those who warn about the rising income and wealth gap, the vanishing of affordable housing must also be heard. Their concerns are real and pressing.
Perhaps, like Captain Kirk in the famous Star Trek episode, The Enemy Within, our societies also need both: a little bit of statist communism and a little bit of corporatist fascism to stay healthy, open, welcoming and, ultimately, stable and democratic?
For better or for worse, we are witnessing history in the making in the great United States.
What it will lead to, I don’t know. I am apprehensive.
Mr. Trump arrived in New York. He will be indicted tomorrow.
Part of me responds with glee. This is, after all, a person of questionable morality (to say the least) who willingly encouraged his supporters to chant, “lock her up!” back during the 2016 campaign. What goes around comes around…
But then… what does it say about the future of US democracy? His supporters view it as a politically loaded, trumped (pun unintended) up charge. Politically biased prosecution may become the norm in American politics.
Khrushchev reportedly expressed pride in the fact that when he was removed from power, he was neither killed nor arrested. That represented a very notably break with Stalinism.
Arguably, Trump deserved this indictment. He is quite likely guilty of several of the criminal charges that have been, or will be, brought against him. In principle, he should not be treated any different froom other citizens. Yet it creates a precedent that can be dangerous.
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:
