"Our Days Are Numbered"

Slide in Lev Reyzin's JMM talk "Problems in AI and ML for Mathematicians" Reyzin is paraphrasing Telgarsky. Posted with permission.

Last week I attended the Joint Mathematics Meeting in Seattle with a theme of

We Decide Our Future: Mathematics in the Age of AI

With little computational complexity in the conference, I attended many of the AI talks. You could divide them into Math for AI and AI for Math. Mathematics of course plays critical roles in machine learning optimization, and several talks focused on provably good learning algorithms, though they overemphasized the importance of such. Do you get on a plane because you understand the physics or because air travel has a strong safety record?

But let's focus on AI for Math which generated the most discussion and angst. 

I started the conference sitting on a panel on the challenges of peer review in mathematics. Math doesn't have the replication crisis that dogs other scientific communities. But math does have papers with very specialized, long, detailed proofs and few qualified referees willing to check them over with care. 

We're not there yet, but in the "near future" we ought to have AI systems that can verify well-written proofs by compiling them using proof assistants like Lean. Referees could spend less time on checking proofs and more on deciding whether the model, theorem and/or techniques merit publication. We might get to the point that you couldn't even submit a paper to a journal until the proofs have been verified.

It's what comes next that really spooks mathematicians. While AI has made significant progress in solving competition problems with DeepMind's AlphaProof and Open AI's O3, it has only played minor roles in developing new ideas for theorems. Eventually AI systems will find critical steps for publishable theorems. Who do we give the credit to? When AI systems become stronger that typical PhD students, what kinds of problems to we give the students? 

We'll get plenty of AI slop, flooding journals with mathematical papers that technically prove new theorems but don't offer any particularly interesting or illuminating insights. But we'll also get mathematicians who can unleash an army of virtual grad students to find new connections between different subfields, or make significant progress on major open problems in the field. 

Some mathematicians don't want AI trained on their papers and using their techniques and theorems, even though they wouldn't have problems with other mathematicians doing so. In any case, they might not have much choice as many publishers are making deals with AI companies.

The future of mathematicians might follow that of artists and programmers. The best will do fine since they can find proofs beyond what an AI can do. Mathematicians who know how to ask the right questions can harness AI to make themselves far more productive. All mathematicians will have to navigate this brave new world.

"At least they'll still need us to teach the courses," one mathematician told me. Don't be so sure.

Random Thought on AI from someone in the REAL WORLD

Guest Post from Nick Sovich. 

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Bill Gasarch recently blogged on RANDOM THOUGHTS ON AI here . He is in the realm of theory. I am in the realm of applications so I asked if I could do a post on AI from that background. He agreed, and here's the post:

1) Words Matter

Today you tell a foundation model who to be and how to behave by giving it a system prompt and a chat prompt. The system prompt is the AI analog of 

 I’m a CS professor, I’m an expert in Ramsay theory, and my goal is to help students learn. 

The chat prompt is the AI analog of

“Hi CS Professor, can you please help me write a blog post to formulate my thoughts on AI”?

Words comprise these system prompts and chat prompts. Getting the right words is very important, in the same way writing the right lines of code is important.

So if you have an AI that *can* be any 157-level IQ expert in any field, you still haveto tell it what kind of expert to be, what its worldview is, and how to communicate. Because it can’t have infinite world views and communication styles all at once.

Example System Prompt and Chat Prompt is here.

 
2) If English becomes the #1 programming language, then what’s the difference between a poet and a computer scientist?

There has been talk in the industry of English eventually becoming the number 1 programming language. What does that mean? A poet has mastery over the English language. A computer scientist has mastery over programming languages. A poet can create works of art.  A poet can be creative, but so can a computer scientist. A computer scientist can be pragmatic, but so can a poet. Will poets become better programmers than computer scientists?  Will computer scientists become better artists than poets? Does it matter?

3) What is code and what is poetry?

Bad code doesn’t compile, or it compiles but doesn’t run, or it runs but has bugs. Bad poetry doesn’t express the author’s intent, or it doesn’t evoke the reader’s emotion, or it doesn’t convey a message to society. Bad code is still code, bad poetry is still poetry. Humans can produce good and bad code and good and bad poetry. AI can produce good and bad code and bad poetry. Can AI produce good poetry?


4) AI is a tool, like auto-complete. 

This is both good and bad.

Under use the tool, and you waste a lot of time.

Over use the tool and you don't understand your own product.

5) Where should AI start and end in academia?

This is an important question worthy of a separate blog that I might do in the future.

6) AI can either replace people or empower people. 

We should look for ways that it empowers people. A nice 38 second clip about that here.

As an example:

Good: Giving the vet tech tools to record my intake questions through natural language instead of pen and paper.

Bad: Automating away all vet tech jobs and replacing them with humanoid robots that would scare my dog anyway. It’s traumatic already when a human pins her down to give her eye drops.

Example System Prompt and Chat Prompt here.

7) How many tokens and how much test-time computation would it take to fully capture my dog’s personality?

First of all, the real thing is better. To see what the latest models are capable of, I used the reasoning models to help generate prompts for the diffusion models, to see how closely I could replicate my dog’s appearance. The reasoning models capture the veterinary terminology very well, and increased specificity in prompting leads to better results from the diffusion models. The diffusion models get close, but don’t seem to have the level of expert veterinary knowledge that the reasoning models do (yet).

Example Prompt and Resulting Image here.

 
8) If spending an hour a day on social media can radicalize you politically, can spending an hour a day reasoning with an AI on technical topics make you more technical?

Spending an hour a day searching the internet for technical topics and reading them can certainly make you more technical. If AI helps you get that information more efficiently, and if the AI actually works (hallucinates less than doing a web search would lead you to incorrect technical information), then it follows that AI can be more efficient than a web search in making you more technical. AI needs to be “aligned”, just like the web search needs to be “unbiased”. And make sure that one of the topics you learn, either through web search or AI, is how to read a map if you lose internet access.

BILL's comment on point 8: While I agree that spending an hour a day (or more) reasoning with an AI on a technical topic is a good idea, make sure you don't fall into a rabbit hole.  For example, while using AI to help me with Ramsey Theory I, on a whim, asked it to write me a poem about Ramsey Theory. It wasn't that good but it was better than what I could do. I won't reproduce it for fear of sending my readers down a rabbit hole.

 

When DO Names Change? When SHOULD Names Change?

 BILL: Good news for Jimmy Carter! He won  The Betty White Award! (see here).

LANCE: That's not good news. He had to die to get it.

BILL: Call it a mixed bag. Good news for me, in that I have a famous person for The Betty White award. And I later found out that Manmohan Singh, a former prime minister of India passed away on Dec 26, 2024, so I have two famous people for The Betty White Award.

LANCE: You should change the name to The Jimmy Carter Award. Did Betty White start a Department of Education?

BILL: No can do. What if someone even more famous dies next year. I don't want to play musical chairs with the name. Past winners were Pele and The Pope Emeritus. Were they more famous than Betty White?

LANCE: YES!

BILL: And that's the problem. If I changed it to The Pele Award and later to The Jimmy Carter Award, that's three names in three years. Also,

If it was The Pele Award, people would think it has to do with Soccer.

If it was The Pope Benedict award people would think it has to do with the Catholic Church.

If it was The Jimmy Carter award, people would think it was about building houses.

If it was The Manmohan Singh award, people would think I am not an ignorant American who knows nothing about Indian Politics.

With Betty White there is nothing so striking about her to think its about something else. (Not quite true- she was involved with Animal Rights.)

LANCE: Yup, Carter got the Nobel prize for homebuilding. You over estimate how many people care about  The Betty White Award. But you named the award after Betty White only because she died shortly before you started the award. Hardly seems like a good criteria. 

BILL: Well, we do this all the time in computer science.

LANCE: Indeed. Some of them, like the Turing Award, survived the test of time. Others, which we shall not name, have not.

But anyway, all of this raises the question, should we change the name when we have more famous dead people? 

BILL: Rarely. Do you have examples of names being changed?

LANCE: Yes, though not replaced by other names.

The best student paper award at the Complexity Conference was named after Ronald Book from 1999 to 2104 and then quietly dropped when the conference left the IEEE. And of course when Rolf Nevanlinna got cancelled in 2019, changing his award to the Abacus Medal. Not a person named Abacus but the ancient computing device. Personally I'm holding out for the Slide Rule Statue.

BILL: So, we need more examples of name changes that are NOT because a person was cancelled or a conference changes organizations.

LANCE: It's hard to do because it's admitting you made a mistake to begin with. So instead we are getting gun shy on new award names. The ACM has made it harder to make a named award and the more recent SIGACT awards have been named after those who died long ago (Gödel) or still living (Knuth). An exception was made for the Danny Lewin Best Student Paper Award at STOC. Danny died fighting the terrorists on American Flight 11 on September 11, 2001 and the award was named for him starting in 2002.

BILL: So it's no longer worth dying early to get an award.

LANCE: I wouldn't recommend it.

The Betty White Award for 2024

In Jan of 2023 I estabalished the Betty White Award, see here which is given to people who died late in the prior year and hence won't be in the  those who we lost in year X articles. I also gave out a few from prior years. Here are past winners, some in retrospect.

2006: James Brown and Gerald Ford. See my post here. The post was not on the Betty White award since Betty White had not died yet. But the idea is there.

2021: Betty White and Bishop Tutu. See my post here. I didn't have the award yet but I do now. Can person X win the X award? I say yes. Its rare since usually the person an award is named after is dead, but in this case that's kind of the point.

2022: Pele (Soccor player), Barbara Walters (broadcast journalist), Pope Emeritus Benedict. See my post here. Three famous people! To quote my post: One was prominent in one of the worlds largest religions. The others were a broadcast journalist and a former Pope. 

2023: Tommy Smothers (Singer and Comedian). See my post here. Since I collect novelty songs I knew who he was, but I think most people did not. 

2024: I began writing this post on Dec 28. Bad idea- the whole point of the award is that we should WAIT until the year is OVER before doing retrospectives. In fact, the award was going to go to Ricky Henderson (famous baseball player), Greg Gumbel (sportscaster), and Olivia Hussey (Actress). The last two I had never head of until they died, but I didn't want to just give it to one person.

Then on Dec 29 Jimmy Carter died. Okay then. Greg and Olivia will still get Honorable Mention.  After that but before Jan 1, Linda Lavin died who will also get Honorable Mention. 

ADDED LATER: An alert commented that Manmohan Singh, who was prime minister of India 200 4-2014 (analogous to being president of America) passed away on Dec 26, 2024. Hence I have added him as well.

Here are the WINNERS of the Betty White Award for 2024:

Ricky Henderson A hall-of-fame baseball player who was truly a superstar. He died on Dec 20, 2024, at the age of 65. See his Wikipedia entry here. There are two criteria for the Betty White Award. 

a) Being famous enough so IF he had died earlier, he WOULD be in the those we lost in 2024 articles. On this criteria, Ricky Henderson is solid. Note that he holds the record for most stolen bases in a career by A LOT: RH has 1406, Lou Brock is second with  938.

b) Dying to late in the year to be on those lists. I checked- he is on some lists but not others. To NOT get the Betty White award because he died late but not late enough would be really sad. SO, even though on this criteria he is borderline, the judges have decided to give it to him.

Jimmy Carter A former president of the United States; however, he may be more known for his post-presidency work on charities. He won a Nobel Peace Prize in 2002. He died on Dec 29, 2024 at the age of 100. Providing a Wikipedia link sounds silly- you all know who he is. Here are some miscellany:

a) I was hoping he would last until Biden stepped down so there would be six living ex-presidents:

Carter, Clinton, Bush Jr, Obama, Trump, Biden.  I blogged about this here.

b) Carter is the prez who lived the longest. He also  had the longest marriage. Jimmy and Rosalyn Carter were married for 77 years. George and Barbara Bush were second with 73 years of marriage. 

c) Carter is the only president sworn into office by his nickname Jimmy (his `real' name is James). It annoyed me when BILL Clinton was sworn in as WILLIAM and when JOE Biden was sworn in as JOSEPH. If Jeb Bush had become prez he probably would have been sworn in by his nickname Jeb (his `real' name is John). Oh Well. 

Jimmy is Clearly famous enough and Clearly died late enough in the year. 

Manmohan Singh A former prime minister of India; however, some of my American readers may not know that (indeed- I did not). He served as prime minister from 2004 to 2014, serving two 5-year terms. He was the first Sikh prime minister of India. See his Wikipedia entry here. 

Manmohan is not famous in America but he is very famous in India and in countries that pay more attention to world events than Americans do. And he died late enough in the year.

AND three Honorable Mentions:

Greg Gumbel A sportscaster, died on December 27, 2024 at the age of 78. See his Wikipedia entry here. SO, does he deserve it? 

a) Being famous. I've read that he is famous but frankly, I had never heard of him until I saw the obit and said  Betty White Award Contender.? That he had an obit on some news site IS an indicator of fame. And more to the point of the award, had he died a a few days later, in 2025, he would have been on the those who we lost in 2025 lists. If Jimmy Carter hadn't died he would have won the award (with Ricky Henderson and Olivia Hussey and Linda Lavin) so I decided to give him Honorable Mention.  My award, My rules. 

b) Dying late. OH YEAH! Dec 27 is very late. 

Olivia Hussey An actress, died December 27, 2024 at the age of 73. See her Wikipedia entry here. SO, does she deserve the award? 

The Being Famous and Dying late comments are identical to those for Greg Gumbel. 

So she also gets an Honorable Mention. 

Linda Lavin An actress, died December 29, 2024 at the age of 87. See her Wikipedia entry here. SO does she deserve the award? 

The Being Famous and Dying late comments are identical to those for Greg Gumbel. 

So she also gets an Honorable Mention. Also she gets credit for dying really late in the year.

My Drunken Theorem

Bill's SIGACT Open Problems Column remembering Luca Trevisan is out. I chose the problem of whether Promise-ZPP in P implies Promise-BPP in P, an extension of an earlier theorem by Luca and his co-authors, which showed that Promise-RP in P implies Promise-BPP in P. But now, let me share a story that I didn’t include in print.

In the mid-1990s, I receive an email from Luca saying that Noam Nisan had told him I’d come up with an easier proof of one of his theorems. Luca asked if he could use it in his upcoming paper. I had no idea what he was talking about.

Then, I vaguely remembered…

I was in Dagstuhl, back when we’d hang out in a room meant for drinking beer and wine. I had, shall we say, more than one good German Pilsner, when Noam came by and asked if I knew how to show that Promise-RP in P implies P = BPP. I mumbled something about how it might follow from Lautemann's proof that BPP is in the second level of the polynomial-time hierarchy. Lautemann’s proof uses the probabilistic method in both directions, which I thought might fit nicely into Promise-RP.

Now to all you kids out there: you should never drink and derive. A theorem you prove after a couple of beers usually falls apart when you sober up. But this time it turns out I was right—and I totally forgot about it until I got Luca’s email.

I never admitted this to Luca but did give him permission to include it in his paper with Andreev, Clementi, and Rolim. And they did.

However, Lautemann’s proof doesn’t tell us anything about Promise-ZPP, so that problem remains open. Go ahead, read Bill’s column, and give it a try. If you drink a couple of Warsteiners along the way, it may not help you prove the theorem—but at least you’ll enjoy some good beer.

Complexity Year in Review

Back in the day (circa 1989) we studied locally random reductions which would lead to all those exciting interactive proof results. Somehow locally random reductions got rebranded as locally correctable codes and this year's result of the year settled a long-standing open question. 

An Exponential Lower Bound for Linear 3-Query Locally Correctable Codes

Pravesh Kothari and Peter Manohar

Roughly if you want a code where each bit is a linear combination of three other appropriately-chosen random bits with constant error, you're going to need a very long code. More in Quanta. 

Things Bill wanted me to mention in this post: R(5), new Mersenne prime, Busy Beaver, Vazirani's delayed proof, formal verification of the sum-check protocol and AI song generation.

2024 was quite a year, we saw a computational complexity theorist, Avi Wigderson, win the Turing Award and computer scientists win Nobel Prizes in both chemistry and physics. Also some elections, wars and college protests. It's all a prelude to a perfect storm for US higher education with the oncoming trains of the new administration, artificial intelligence, fiscal challenges and the demographic cliff. Hang on tight, it's going to be a bumpy ride.

We remember Rance Cleaveland, Peter Higgs, Thomas Kurtz, Phil Lewis, Steven Rudich, Frank Ryan, Jim Simons, Luca Trevisan, Dominic Welsh and Niklaus Wirth.

We thank all our guest posters and collaborators Eric Allender, Martin Bullinger, Max Burkes, James De Santis, Mohammad Hajiaghayi, Neil Immerman, Valentine Kabanets, Harry Lewis and Larry Washington.

Enjoy the holidays and we'll see you in January. 

Information is Physical?

I've heard a few times recently the phrase "Information only exists in a physical state". It come from the quantum computing world where they claim quantum changes the game when it comes to representing information.

As one who has spent his career studying theoretical information that has never and never will exist in a physical state, how can we reckon with such a statement? For starters let's consider the set of all primes--how does that infinite set exist in our finite world? 

Information is physical but not directly, but rather as its description. We can discuss a computational process or more generally a mathematical model that captures the set of all primes and we can and have store that description physically.

Let's consider a single prime, the recently discovered Mersenne prime \(2^{136279841}-1\). Note how we must describe the number in a very compressed format, certainly not as a collection of \(2^{136279841}-1\) ping pong balls or even \(2^{136279841}-1\) atoms, far more than the roughly \(2^{365}\) atoms in the observable universe.

In a similar fashion, a large language model stores information through its weights--not a direct encoding of the sentences it can generate. 

Now let's think of quantum computing. The quantum algorithm is always classically defined. All the information in quantum states has a classical description. An entangled quantum state may require an exponentially large explicit description, but the algorithm generating it provides a short classical physical description. So if we allow information to only physically represented by its description then it's hard to argue that quantum is somehow special. There are differences to how quantum works but when we try to simplify the message, it can confuse people into thinking quantum is more powerful than it really is.

Random Thoughts on AI (Human Generated)

 (I wrote this post without any AI help. OH- maybe not- I used spellcheck. Does that count? Lance claims he proofread it and found some typos to correct without any AI help.)

Random Thought on AI

I saw a great talk on AI recently by Bill Regli, who works in the field. 

Announcement of the talk: here

Video of the talk:  here

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1) One item Bill R mentioned was that AI requires lots of Energy so
3-mile Island is being reopened. See here.

Later I recalled the song

        The Girl from 3-Mile Island

to the tune of

        The Girl from Ipanema.

The song is in my audio tape collection but that is not useful so I looked for it on the web. The copy on YouTube doesn't work; however, this website of songs about 3-mile island here included it.

In the 1990's I was in charge of the Dept Holiday Entertainment since I have an immense knowledge of, and collection of, novelty songs- many in CS and Math.

Today- My talents are no longer needed as anyone can Google Search and find stuff. I did a blog on that here. I still have SOME advantage since I know what's out there, but not as much. Indeed, AI can even write and sing songs. I blogged about that and pointed to one such song here.

SO, some people's talents and knowledge are becoming obsolete.  On the level of novelty songs I am actually HAPPY that things change- I can access so much stuff I could not before. But humans becoming obsolete is a serious issue of employment and self worth. Far more serious then MACHINES TAKE OVER THE WORLD scenarios.

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2) When technology made farming jobs go away, manufacturing jobs took their place. That was true in the LONG run, but in the SHORT run there were starving ex-farmers. The same may happen now.

(ADDED LATER; someone emailed me that Machines taking over farming and other things has caused standards of living to go up. YES, I agree- in the LONG run very good, but in the short run people did lose their livelihoods.)

Truck Drivers and Nurses may do better than Accountants and Lawyers:

Self Driving trucks are 10 years away and always will be.
Nurses need to have a bedside manner that AI doesn't (for now?).

One ADVANTAGE of AI is that if it makes white collar workers lose jobs the government might get serious about

Guaranteed Basic Income, and

Univ. Health care

(ADDED LATER: someone emailed me that there GBI is not the way to go. Okay, then I should rephase as when white collar workers lose their jobs then the problem of a social saftey net will suddently become important.) 

Similar: If global warming makes the Cayman Island sink then suddenly Global Warming will be an important problem to solve.

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3) An example of AI taking away jobs is the Writers Strike.

OLD WAY: There were 10 people writing Murder She Wrote Scripts.

NEW WAY: AN AI generates a first draft and only needs 2 people to polish it.

KEY: In a murder mystery the guilty person is an innocuous character you saw in the first 10 minutes or a celebrity guest star. Sometimes the innocuous character is the celebrity guest star.

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4) ChatGPT and school and cheating.

Calculator Scenario: We will allow students to use Chat GPT as we now allow calculators. Students are not as good at arithmetic, but we don't care.  Is Chat GPT similar?

Losing battle scenario: Ban Chat GPT

My solution which works--- for now: Ask questions that Chat GPT is not good at, allow chat GPT, insist the students understand their own work, and admit they used it. Works well in Grad courses and even senior courses. Might be hard in a Freshman courses.

Lance's Solution--- Stop giving out grades. See here

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5) Bill R said that we will always need humans who are better at judgment.

Maybe a computer has better judgment. I blogged on this here

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6) I asked two AI people at lunch if the AI revolution is just because of faster computers and hence is somewhat limited. They both said YES.

SO- could it be that we are worrying about nothing?

This also may be an issue with academia: if we hire lots of AI people because it's a hot area, it may cool off soon. Actually I thought the same thing about Quantum Computing, but I was wrong there.

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7) LLM's use LOTS of energy. If you get to ask one How do we solve global warming? they might say

First step: Turn me off!

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8) Scott did a great  blog post about the ways AI could go. See here.

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9) I recently emailed Lance a math question.

He emailed me the answer 5 minutes later.

I emailed that I was impressed

He emailed that he just asked  Chat GPT. He had not meant to fool me, he just assumed I would assume that. Like if you asked me what 13498*11991 was and I answered quickly you would assume I used a calculator. And if there is a complicated word in this post that is spelled correctly then you would assume I used spellcheck - and there is no embarrassment in that.

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10) If a painting is done with AI does any human get credit for it?

I always thought that people who forge paintings that look JUST LIKE (say) a van Gogh should be able to be honest about what they do and get good money since it LOOKS like a van Gogh who cares that it is NOT a van Gogh.  Same with AI- we should not care that a human was not involved.

IF an AI finds a cure for cancer, Great!

If an AI can write a TV series better than the human writers, Great!

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11) AI will force us to make moral choices. Here is a horrifying scenario:

Alice buys a self-driving car and is given some options, essentially the trolley problem:

If your car has to choose who to run over, what do you choose?

You have the option of picking by race, gender, age, who is better dressed, anything you want.

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12) Climate Change has become a political problem in that

Democrats think it IS a problem
Rep think it is NOT a problem

Which is a shame since free-market solutions that would normally appeal to Reps are not being done (e.g., a Carbon Tax). Indeed, we are doing the opposite- some states impose a tax on Hybrid cars

SO- how will AI go with politics? Scenarios

a) Dems are for regulation, Reps are against it. Elon Musk worries about AI and he is a powerful Rep so this might not happen.  Then again, he supports Reps, many of whom have made E-cars in their states harder to get or own.

(ADDED LATER: I originally said states had BANNED e-cars. A commenter inquired which states did this so I looked it up. NONE so I ammended the post. Some states are making it harder to have an E-car:

Extra fees for owning an E-car. See here. The reasonaing given is that E-Cars don't pay gas taxes. While that is true, I don't really buy that- republicans seem to be agains ALL taxes EXCEPT thoseon E-cars (and lately tarrifs).

Want to phasa out E-cars: see here

Right now it is illegal to sell cars directly to customers- must go to dealers. This has nothing to do with e-cars but is clearly an idiotic law. E-cars are trying to get around it, and there is pushback on that, see here.

)

 
b) AI-doomsayers want more regulation, AI-awesomers do not, and this cuts across party lines.

c) We will ignore the issue until it's too late.

If I was a betting man ...

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13) International cooperation on being careful with AI. Good luck with that.

My cynical view: International Treaties only work when there is nothing at stake

The Chem Weapons ban works because they are hard to use anyway.

The treaty on exploring Antarctica was working until people found stuff there they wanted. It is now falling apart