Was the George Foreman Grill The Best Invention of the last 50 Years?

(This post was inspired by  George Foreman, who passed away March 21, 2025, at the age of 76.)

About 10 years ago I asked my class

What is the best invention or tech advance of the last 50 years?

Here are the answers I got NOT ranked.

1) The internet. We can look things up so easily! (see my post on one aspect of this here). Young people reading this blog have no idea what the world was like before the internet. Here is a short story that shows how much has changed.

At MIT in 1982 I saw Anil Nerode give a great talk about Recursive Mathematics: using recursion theory to show that some theorems whose proofs were not effective could not be made effective (e.g., there is a computable 2-coloring of pairs of naturals with no decidable infinite homogeneous set, so the proof of Ramsey theory on \(K_N\) has to be non-effective). The talk got me interested in the topic, so that day I went to the math library (ask your grandparents what a library is) and found the paper journals that had some of the articles he talked about (Journals used to be on paper? See my post here about that.) I then copied the articles on the copy machine (ask your grandparents what a copy machine is) and read them. A few years later Anil Nerode asked me to write a survey of recursive combinatorics for a collection of articles in recursive mathematics. He was one of the editors of a joint America/USSR project (ask your parents what the USSR was) which I was happy to do. The book was delayed when the USSR fell apart (ask your parents about that important historical era), but was eventually published. The survey is 131 pages and is here. The book, Handbook of Recursive Mathematics, is in two volumes. Its available on Amazon Volume 1 and Volume 2. I've also sometimes seen it available for free download though I don't know if it really is or if thats some kind of scam (if your grandparents try to download it warn them that it might be a scam).

2) Advances in medicine. You can have an operation and be home that night (this is partially medical advances and partially the insurance companies forcing you to have short stays). I asked the question pre-COVID. If I asked it again, I may have some people saying vaccines. 

3) VCR/DVD/DVR/Streaming so we have a lot more control over our entertainment (Is this really on a par with medical advances? Yes- you can watch TV of your choice while you recover.)

4) Easy Pass for Toll Booths. The students said that it was always a pain having to bring change to toss into a basket at the toll booth. And sometimes they missed. 

5) The George Foreman Lean Mean Fat-Reducing Grilling Machine. The student really liked using it and said that burgers were tastier and healthier. I asked him if he really thought George Foreman got to be the heavyweight champion of the world (twice!) because of the grill. He didn't know George Foreman had been a boxer, he thought that George Foreman was a pitchman.  Which is true but inomplete. Whether he is a pitchman or a boxer or an ex-boxer, does he really know enough about  healthy eating so that you should  take his advice? Note also that he has a financial incentive to  believe that the grill  produces tasty and healthy food.  See here for a blog post on the illogic of advertising. 

6) The Cell phone. It would be rude to, at a party, go into a corner and read a newspaper. But it is socially acceptable to go into a corner and read the news (or anything else) on your phone. I discussed this, though about laptops, as part of a post here. On the one hand, maybe it was good to be forced to talk to people. On the other hand, I can check my mail without being at home or the office!

7) THE CELL PHONE!!!!!! See Lance's post here.

8) Caller ID. The Cell Phone makes it possible to call anyone, anytime. Caller ID makes it possible to avoid talking to anyone, anytime. Symmetry!

9) ChatGPT was not on the list 10 years ago but might be now. And other AI things will be also.

But for now I ask: What do YOU think were the greatest tech advances of the last 50 years? If you prefer thinking on a full stomach then grill some burgers and eat them before answering. 

BILL to LANCE: Anything you want to add?

LANCE:

9) GPS - First launched in 1978 and now you never get lost, even at sea.

BILL: Some people trust their GPS to much, see here, though I think these stories happen less and less over time. 

10) CNN - Watching wars play out in real time was a game changer.

 BILL comment on CNN: In 1991 I was on a cruise. In the time I was gone there was an attempted coup against Gorbachev. I didn't hear anything about it until the cruise was over. CNN existed but it was not-a-thing to have it wherever you go. The next time I went on a cruise was 2022. While on that cruise, Gorbachev died and I heard about it right away. Two points here:(a) How fast we got news anytime-anywhere has changed a lot, and (b) For the sake of the Gorbachev family I should stop going on cruises. 

11) Cloud Computing - Enabled small startups to do big things.

BILL Sometimes very small, like one person. New word: Solopreneur

The Phone

I've heard this story from a few places. A father watches Back to the Future II with his kid. The 1989 movie view of 2015 looks entirely different when in fact not much has changed except for the fashion and the lack of mobile phones. This is supposed to be a parable about the lack of technological progress.

I disagree. It's about a lack of imagination of the future because of those phones. Phones that are hidden from sight, and have become so ubiquitous we take them for granted.

The phone in your pocket is more powerful than the most powerful computer of 1985 and it's not even close. But that's the least interesting thing about the phone.

You no longer need a printed newspaper, encyclopedia, atlas, almanac, dictionary, thesaurus, dining guides, programming manual or any other reference book. The printed sports almanac at the center of the plot of the movie doesn't exist anymore. 

You can read virtually any book ever published, listen to any song ever recorded, watch any movie ever filmed. You have access to nearly all publicly available information. You can watch major sporting events live. On that phone.

You can buy tickets to any event and save and use them on your phone. You can pay for just about anything with your phone, and if you wish ship it anywhere.

You no longer have to fold a map or ask for directions. The phone will give you directions, taking account traffic and transit delays, and guide you along the way.

When visiting a foreign country you can point your phone at a sign in Japanese and have the words magically replaced by English. Take a picture of a menu in German and have it describe the dishes in English. Instantly translate voice as well.

You can have a conversation with your phone about anything. It will understand your voice and respond likewise. 

You can take photos and videos on your phone and share them instantly with your friends or with everyone around the world. The quality is far superior to any consumer-level camera from 1985. Or you can have the phone create its own photos and videos based on your descriptions. 

It's also, of course, a phone. You can speak to anyone anywhere, with video if desired, for zero marginal cost. Back in 1985 it cost about a dollar a minute to call someone in a different state, and much more internationally, on a landline. You can have impromptu video meetings and you can send messages to anyone and share them with everyone. 

And I'm just scratching the surface. 

So between the phone and the flying hoverboards, I'll take the phone any day.

A few more notes about Tom L

Tom Lehrer passed away on July 26, 2025, at the age of 97. I wrote a blog-obit here. One of my readers read the post and went down a rabbit hole (or did he?), which lead to a blog post about rabbit holes here. 

A few more notes about Tom L.

1) When someone of interest to our readers dies, Lance calls me (he seems to always know before I do) to discuss who should do the obit (me, him, someone else, or perhaps no obit needed). This was a case where there was no need for a discussion.  To say I am a fan of novelty songs would be an understatement---I've been a fan of novelty songs before I was a fan of mathematics. A link to my website of novelty songs by topic, including Math, is here

2) Whenever Lance calls I answer with Who Died? 

3) Who was the oldest novelty song singer to die in the Summer of 2025? The summer is not over yet but I think the answer is known. And it's NOT Tom L!

Jane Morgan, a singer, died on August 4, 2025, at the age of 101.  She recorded a parody of  Johnny Cash's A Boy Named Sue titled A Girl Named Johnny Cash 

A boy named sue:  here

A girl named Johnny Cash: here

Wikipedia Page for Jane Morgan: here

To my knowledge, Jane Morgan only had one novelty song  (she had many non-novelty songs) so calling her a novelty singer is a stretch, but its a great song, so I will call her a novelty singer.  (I checked if the flipside of that song was a novely song. It was Charley, a love song, not a novelty song. Ask your grandparents what a flipside is.) 

4) Tom Lehrer and Jim Lehrer (news anchor on PBS, deceased) are not related.

I googled how many people have last name Lehrer

Google-AI:  FamilySearch.org estimates there are 211,503 records for the surname.

But this website says approximately 5982 bear this surname.

I'm surprised by the large gap there.

5) There have been more elements found since Tom L did The Elements. I wondered whether someone updated it. Of course they did:

The Original by Tom L: here

Updated version by Leonard  Lehrman that has the latest element, Number 118, Oganesson: here

I've heard that 118 is natural barrier- we probably won't create 119 or higher. Elements that high are created, not discovered. They also don't last long. 

6) There are many elements, so a list-song made sense. There are a lot of complexity classes, so a list-song makes sense for those as well. 

Dave Barrington sings Song of the Complexity Classes: here

7) How many people emailed me to tell me that Tom L passed away? R(5).

8) One of my favorite and less known Tom L songs, whose advice he did not take: Selling Out

9) Tom L wrote a paper with R.E. Fagen (Not Ronald Fagin) for the NSA. It's a real math paper, but note the third reference in the bibliography, the paper is here.

Total Pixel Space

Last month the New York Times highlighted some AI generated short movies, including Total Pixel Space, by Jacob Adler that gets philosophical about information, à la infinite monkeys. It imagines the finite number of images and video that contain all human history, both real and fake, along with all possible creatures, and even the rise and fall of alien civilizations.

Let's talk about the math. According to the video there are roughly 7.8 x 10^7,575,667 possible 1024x1024 images, and 9.3 x 10^{1,309,075,411,322} possible two hour films, incomprehensibly large numbers. The narrator acknowledges that most of the images are random noise.

But within this ocean of pixel possibility, natural images are but a drop. Recognizable scenes, faces and objects are extremely rare islands in a vast sea of noise.

So how do we capture which images actually matter? Let's visit our friend Kolmogorov Complexity, which measures information by the amount we can compress. A JPEG image typically compresses (lossily) to about 200 KB, which reduces the number of images to 8.7 x 10^1,061,650. Still enormous.

All the images and videos were generated by short prompts. JPEG compression reduces raw pixel space by discarding details our eyes don’t notice. Prompts do something similar at a higher level: they compress ideas, not pixels.

Consider prompts of length 100 over a 10,000 word alphabet typically used in prompting and we're down to 10⁴⁰⁰ possibilities, a mere quintic of the number of atoms in the universe. And you could cut that down by considering grammatical structure.

Now from a prompt you won't get the same image each time, based on the randomness of machine learning. But that's the whole power of AI, the randomness just gives us examples of the structure embedded in the prompt. It's the structure that matters.

And perhaps those monkeys would be more efficient if they entered prompts into AI, instead of generating random text. What images they could produce!

My Tom L post inspired a mathematical definition of Rabbithole

NICK: I read and enjoyed your blog post on Tom L (see here). I then spent 40 minutes down a rabbithole listening to his music on YouTube.

BILL: You call that a rabbit hole?!  A while back I spent 3 hours reading questions and answers on quora ranging from is Michelle Obama actually a man? to Is Donald Trump the Anti-Christ? (The answer to both is no.) THATS a rabbithole. Listening to Tom L is not.

NICK: SO... what is and isn't a rabbit hole? Also, is it rabbithole or rabbit hole?

BILL: Spellcheck is happy with rabbithole, hence so am I. As to your question,  we need a function \(f\) and a threshold \(T\) such that

if you spend \(x\) minutes on A, and 

you get y enjoyment out of it, where \(0\le y\le 10\), and

 \(f(x,y)\ge T\)  

then A is a  rabbithole.

(Note- I was kidding. There can't possibly be a function f that works.)

ONE DAY LATER

NICK: I asked Gemini  what \(f\) and \(T\) are  and it told me:

 \(f(x,y) = \frac{x}{y+1} \) and  

\(T=20\).

  It also gave me some examples:

1) The Tom L. Example: 40 mins, 8/10 enjoyment. Score 4.44. NOT a Rabbit Hole!

2) Doomscrolling: 90 mins, 1/10 enjoyment. Score  45. Rabbit Hole!

3) Binge-Watching Mediocre TV: 3 hours (180 mins), 4/10 enjoyment. Score = 36. Rabbit Hole!

4) Incredible Documentary: 3 hours (180 mins), 9/10 enjoyment.  18. Worthwhile but its close.

Gemini also output a heatmap, see here.

BILL: Uh- I was only kidding.

NICK: Well, the jokes on you. 

AI and ...

AI and Vacation

I'm back from my German vacation. This was my first AI vacation, by which I mean how I used AI to navigate a foreign country. Taking a picture of a hand-written menu board, not just to translate the dishes, but to get a description of each. Visiting a palace with a German-speaking tour guide and translating in real time. Even the more mundane taking pictures of buildings to learn about them.

On the TV I saw something about Künstliche Intelligenz, and Chatty told me it was German for Artificial Intelligence. The Germans use KI instead of AI partly because AI can be confused with Ei (egg). At least that's what AI tells me, so it must be true.

AI and Math

At the beginning of my vacation, Google announced that they achieved gold medal status in the International Mathematical Olympiad. An impressive achievement though Terry Tao makes a good point that comparing an AI system to a time-constrained high-school students is not an apples-to-apples comparison.

I already find talking to AI about math topics quite useful, though it's like talking to an early PhD student. Sometimes they just say things that aren't correct, but usually they are. The reasoning models are particularly good at finding holes in P v NP proofs. For example here's the conclusion of ChatGPT o3-pro's review of the paper from Eric's guest post.

The paper is a reminder that lower‑bound proofs live or die on the exact breadth of the algorithmic model they exclude—too narrow and the result is unsurprising, too broad and the proof tends to break. At present this work sits in the first category.

What I want to see is AI come up with a solution to an open math problem, a true new insight beyond just some optimized search. I'm not looking for P ≠ NP, just some result that would be publishable in a respectable conference or journal, even just a new completeness result. We haven't really seen that yet, but I suspect we will soon and then we can figure out where math goes from there.

AI and Bill

In his presidents question and solution, Bill states that AI had failed to find the right answer to his problem. Back in June, I saw Bill's draft post and tried AI to solve it.

AI initially failed the test but for a good reason. Bill's initial draft post had Ford and Dole in Group Two because they received LLBs instead of JDs. In the past the LLB was the professional law degree. Yale didn't change to JD until 1971. Ford got his LLB from Yale in 1941.

When I removed Ford and Dole, ChatGPT o3-pro correctly gave the reason for the partition, though it did take over 13 minutes.

Every name in Group One spent time in a law school—most completed a J.D. or LL.B., and the two exceptions (Al Gore and, in some accounts, Lloyd Bentsen) still enrolled in law studies.

• examples – Walter Mondale (University of Minnesota Law) en.wikipedia.org,
  JD Vance (Yale Law) en.wikipedia.org,
  Al Gore (attended Vanderbilt Law) en.wikipedia.org.

No one in Group Two ever attended a law school; their highest formal education is in fields such as engineering (Jimmy Carter), economics (George H. W. Bush), business (Donald Trump), political science (Paul Ryan), or acting (Ronald Reagan) en.wikipedia.orgen.wikipedia.org.

So the distinguishing property is legal education: every Group One figure went to law school, while none in Group Two did.

Another commentor got a similar result for ChatGPT o4-mini-high. I just tried it on Gemini 2.5-Pro and it also gave the correct response, this time in seconds.

On the other hand, E tried several base models and none of them succeeded. The lesson: You want to solve a tricky problem, pony up the $20/month and use a reasoning model.

Some thoughts on journals, refereeing, and the P vs NP problem

A guest post by Eric Allender prompted by an (incorrect) P ≠ NP proof recently published in Springer Nature's Frontiers of Computer Science.

For a time, I served as Editor-in-Chief of ACM Transactions on Computation Theory, and in this role I had to deal regularly with submissions that claimed to resolve the P vs NP problem. Finding referees for these papers was sometimes challenging, so I frequently ended up reviewing them myself.  Dealing with such submissions involves enough overhead that ToCT, J.ACM and ACM Transactions on Algorithms limit the frequency with which authors can submit work of this sort.  But for every submission, on any topic, the overall process was the same: Find someone on the Editorial Board who has sufficient expertise to find referees and make knowledgeable use of their recommendations.  If there was no such person on the Editorial Board, then it was always the case that the submission was out of scope for the journal.

These thoughts are brought to mind by a recent case, where it seems to me that the editorial process broke down.

Springer publishes several high-quality academic journals that deal with Theoretical Computer Science.  One Springer journal, Frontiers of Computer Science, recently published an article entitled SAT Requires Exhaustive Search, where one of the authors is a Deputy Editor-in-Chief of the journal.  The abstract of the article states that it proves a result that is stronger than P not equal to NP.  The Editorial Board of the journal has some members who are expert in computational complexity theory.  However, all the ones whom I know personally have asserted that they had no knowledge of this paper, and that they were not involved at all in handling the paper.

When Ryan Williams and I learned about the publication of this article, we drafted a comment, which we sent to the Editor-in-Chief.  We recommended that the paper be retracted, in which case there would be no need to publish our comment.  However, the Editor-in-Chief declined to retract the article, saying that he could find no evidence of misconduct, and thus we have been assured that an edited version of our comment will appear in the journal.

Our comment calls attention to some shortcomings in the proof of the main theorem (similar to shortcomings in several other failed attempts to separate P from NP).  But we were able to say more.  Often, when one is looking for bugs in a purported proof, one has to deal with the situation where the claimed theorem is probably true, and the only problem is that the proof is not convincing.  However, the main theorem in their paper (Theorem 3.2) states that a particular constraint satisfaction problem requires time more than \(d^{cn}\) for any constant \(c<1\) (where \(d\) is the domain size, and \(n\) is the number of variables).  In particular, their purported proof claims that this holds even when \(k=2\) (meaning that each constraint has at most 2 variables).  However, Ryan Williams presented an algorithm more than two decades ago that runs in time \(O(d^{(0.8)n})\) in this special case, contradicting the lower bound claimed in the article.

The article contains an appendix, with glowing testimonies from various researchers; the lead-in to the appendix also contains enthusiastic comments from Gregory Chaitin.  I contacted Gregory Chaitin, and he asserts that he did not read the paper, and that he was quoted out of context.

The edited version of the comment that Ryan Williams and I wrote, which will supposedly appear in Frontiers of Computer Science soon, differs from the version linked here (our original submission) primarily in one respect:  Our closing paragraph was removed.  Here is that paragraph:

Finally, it is our opinion that the publication of this article is a complete embarrassment to this journal and its publisher. We believe that, at the very least, the paper should be withdrawn, and Springer should conduct an investigation to understand how such a paper could have made it through the peer review process.

Update (Lance, 8/5/25): The authors of the paper in question have asked me to link to their reply to the Allender-Williams comment. I do so with no endorsement. 

Eric Allender asked me to note that their claim about Ryan’s algorithm requiring exponential space is misleading; the amount of space is not more than the runtime of the algorithm, which is less than the lower bound that they claim. (Their theorem does not state that \(d^{cn}\) time is required under the additional assumption that the algorithm use only \(n^{O(1)}\) space.)