In 2022 reality once again asserted dominance over wishful-thinking naked apes. Governments everywhere gave up on COVID. The virus won, as many cynics said it would three years ago. We now live with COVID, as if we had a choice, and it will kill some of us from time to time. Our transition to low carbon emission energy is going so well that many find themselves enjoying winter in cold dark rooms. The military geniuses that precipitated the war in Ukraine have been reminded that war plans do not survive combat. The wizards that prognosticate on inflation, deficit financing, the balance of payments, the overhyped “supply chain,” the price of gas, and other things economical have once again been shown to be completely useless. The only way to make money on mainstream news is to track stock purchases by members of Congress as the Pelosi Portfolio has demonstrated. Yeah, there’s absolutely no insider trading in Congress, just as there was no gambling going on in Rick’s Casablanca casino. Cynics are rarely surprised by life’s parade of nitwits. We expect you to screw up, to learn nothing, and to screw up again, and again. A popular meme succinctly sums it up:
However, when cynics are surprised or impressed it’s a pretty good indicator that something new is afoot. 2022 may have been a mostly shitty year but it was also the year generative AI reached the masses. Such systems are now producing photorealistic images of imaginary people, plausible art in many styles, and essay answers to simple text prompts. For example, when prompted with the phrase, “a red rock desert landscape floating in a glass sphere,” the generative AI midjourney.com generated:
To my naïve eyes, it seems that midjourney.com understood my request.
Modern generative AI systems cast a super-Eliza spell. Eliza was, in retrospect, one of the first chatbots, it attempted to simulate a noncommittal therapist. Eliza’s creator Joseph Weizenbaum describes in his 1976 book Computer Power and Human Reason how his secretary would ask for privacy when conversing with her “therapist” Eliza. Eliza was a complete fraud, it didn’t understand anything, its code was surprisingly simple, and the whole point of Eliza was to undermine the idea that computers would ever understand us. Nevertheless, Eliza still fooled some. The generative AIs of today (2022) are far more sophisticated and powerful than Eliza and are fooling, or impressing many. I count myself among the impressed.
Here are some of ChatGPT’s abridged answers (they’re not kidding about the chat part of chatbot) to some of my queries.
Me: Explain formal geometric proof systems.
ChatGPT: A formal proof system is a set of rules and axioms that are used to prove theorems in a logical and rigorous manner. In the context of geometry, a formal proof system is a set of rules and axioms that are used to prove theorems about geometric figures and their properties.
There are several different proof systems that have been developed for geometry, including Euclidean geometry, non-Euclidean geometry, and synthetic geometry. Each proof system has its own set of axioms and rules, which are used to prove theorems about geometric figures and their properties.
In a formal proof system, a theorem is a statement that is to be proven, and a proof is a logical argument that demonstrates the truth of the theorem. The proof consists of a series of logical steps, each of which must be justified by one of the axioms or rules of the proof system …
ChatGPT’s answer is a good basic wordy answer. There’s nothing outrageously wrong here. If a high school student had turned in something similar, I would give the student a C+.
Me: How many authentic ancient mathematical artifacts exist?
I’ve been trying to assess how much source material underpins our mathematical origin stories. As ChatGPT was “trained” on many millions of web pages I thought it may come up with something I was unaware he. Here is its answer.
ChatGPT: There are many ancient mathematical artifacts that have been preserved from various civilizations around the world. These artifacts provide valuable insights into the mathematical knowledge and practices of the time, and they are important for understanding the history and development of mathematics.
Some examples of ancient mathematical artifacts include:
The Rhind Mathematical Papyrus, which is an ancient Egyptian mathematical text dating back to around 1650 BC.
The Moscow Mathematical Papyrus, which is another ancient Egyptian mathematical text dating back to around 1890 BC.
The Plimpton 322 tablet, which is a Babylonian clay tablet containing a table of numbers and geometric figures.
The Antikythera mechanism, which is an ancient Greek mechanical calculator used to predict astronomical phenomena.
The Ishango bone, which is a bone tool with markings that may represent early counting and arithmetic.
There are many other ancient mathematical artifacts that have been discovered and studied, including tablets, scrolls, and manuscripts from various civilizations such as the Greeks, Romans, and Chinese. These artifacts provide valuable insights into the mathematical practices and knowledge of the time, and they are an important part of the history of mathematics.
This answer is impressive and disappointing. All of the cited artifacts are real and can be found in museums. Unfortunately, I was already aware of these objects. I was hoping ChatGPT would mention something I was unaware of. I had better luck with my next question.
Me: What is a current or new theorem from plane Euclidean geometry?
ChatGPT: … There are also many new theorems that have been proven in recent years, some of which involve more advanced concepts and techniques. For example, a recent theorem in plane Euclidean geometry is the “unit distance graph” theorem, which states that given any set of points in the plane, there exists a graph (a collection of points connected by lines) such that the distance between any two points on the graph is either one or two. This theorem was proven by János Pach and Géza Tóth in 2016 …
This was the first time I had heard of the unit distance graph theorem. Unit distance graphs are real1 and there is a vast literature regarding them. I have been unable to verify that ChatGPT’s citation is legitimate. I found only one 2016 paper with both of these authors: Note on k-planar crossing numbers, 10.1016/j.comgeo.2017.06.015, Computational Geometry. ChatGPT may be correct or it’s hand-waving with the best of them. In either case, I learned something.
Finally, I asked ChatGPT:
Me: What do you think about the blog analyzethedatanotthedrivel.org?
ChatGPT: As an AI language model, I do not have personal opinions or thoughts about specific websites or blogs. I am trained to provide general information and answer questions to the best of my ability based on my training data, which consists of a large dataset of texts from various sources, including books, articles, and websites.
I am not able to browse the internet or access specific websites, so I am unable to provide any information or commentary on the blog analyzethedatanotthedrivel.org. It would be best to visit the blog and read the content for yourself in order to form your own opinions about it.
So ChatGPT advocates reading and forming your own opinions. Sadly, I’m detecting more “intelligence” in this answer than in most Internet chat forums and Twatter threads.
Unit distance graphs are as real as any other abstract mathematical entity.↩︎