The number 1 stupid person is Donald Trump. He micromanaged the budget of the National Science Foundation to realign it with his favorite topics: AI, computer science, and quantum computing, even offering a $200 million dollar prize for a working quantum computer. The Republican hate of DEI has caused drastic cuts in grants and brought about reductions in support for graduate students. Xenophobia has prevented a flow of talent from abroad. A friend of mine said this effort should be called Make China Great Again.

Now openAI has shown that it has no clue about how to conduct mathematical research by dropping the solution of 372 problems, the intellectual equivalent of Taylor Swift releasing 30 albums at once. I spent some time reading the article in Scientific American about the greatest hits (and you can too if you give them your email address). It would have made much more sense to concentrate on one or maybe two dozen achievements and write them up in a way that people can read and understand the achievements. If this was done then there would have been a sudden burst of progress. As it is we will have to sift through the wheat to find the chaff, and as openAI admits probably 10% of the solutions are simply wrong.

A mathematicians who work at Anthropic called it the greatest moment in the history of mathematics. Well maybe so, but I am pretty sure he has the sign wrong. Never has there been such a steaming heap of crap dumped into the literature that will set back mathematics for years. We all knew AI could kill us all but we didn’t expect how or when.

At this point we do not know which of 372 new theorems in openAI’s upload are breakthroughs or which ones were solved by putting together insights created by human minds and uploaded to the internet, where they were found by AI bots and assembled like legos. But I know that (i) These agents are the not the Ramanujans of the 21 century. (ii) It will take dozens of humans hundreds of hours to decode each new morsel of knowledge.

None of this really matters. Mathematicians are a community. In addition to working together to decipher the mysteries of mathematics, we teach undergraduate students to understand well-established methods for solving problems, guide them in navigating ther undergraduate careers  and initiate graduate students into the arts of research, teaching and mentoring.

It is possible that 20,000 AI agents could have collaborated to solve the mysteries of Navier and Stokes without spying on other people’s work, but they could not have done it without the recent published breakthough of Córdoba and Martínez-Zoroa,. That holds true for the recent progress on the problem of showing no percolation at the critical value in any dimension. There would be no solution without the conjectures of Gady Kozma and Shahaf Nitzan that set-up the framework for solving the problem. See Gil Kalai’s blog

Yes, the landscape of mathematical has suddenly changed, but humans are needed now more than ever. Robots may be able to run faster than Usain Bolt but they can’t figure out how to stop before crashing into the wall at the end of the course. For a laugh use google to find a video. Much of the progess AI has made in math was simply come  from mining the gold tand the less precious minerals hat has accumulated over the last 100 years in the mathematical literature. Large language models are talented aggregators of information, which offer mathematicians an unprecedented ability to retrieve that information, compared to crawling the graph of papers liked by citations. But if they really understood what they read then they would be able to write their proofs n a way that we can understand them.

The purpose of this note is to explain some simple math that is very relevant to current events.

Every disease has a basic reproduction number, R. It is the average number of secondary infections created by an infectious individual (in an otherwise susceptible population). Early in covid R was 2.5 but it grew to 5 or more as the disease evolves. Measles has an R that is much higher. According to Google’s AI overview “it is widely cited as being between 12 and 18.”

The other number we need is u, the fraction of the population that is unvaccinated. If Ru < 1 herd immunity has been achieved and the disease cannot spread. If we suppose R = 14.28 then this means 93% of the population has to be successfully vaccinated. The word successful refers to the fact that sometimes vaccinations do not produce the desired immunity. My wife Susan was vaccinated against whooping cough but caught it from her Christian Scientist neighbors.

As usual reality is more complicated than math. Even if Ru < 1 there can be groups of people who for religious, political, or other reasons do not believe in vaccination. This creates areas in which the disease can spread.  The 2025 Texas measles outbreak was centered in a close-knit and under-vaccinated Mennonite community in West Texas’ Gaines County, where low vaccination rates were a key factor in the rapid spread of the highly contagious virus. The outbreak started within this community but eventually spread beyond it to other Texas counties and into New Mexico. 

Florida has recently made a dangerous decision to no longer require children be vaccinated for measles and other diseases in order to attend school. While it will take several years until herd immunity is lost, the fact that school populations are stratified by age it will almost immediately create subpopulations in which diseases can spread. The problem will grow worse over time. When it gets bad it will be very hard to raise vaccination levels in the population to stop it. .

Vaccine policy needs to be informed by science. It is not enough for RFKJr to say “I am not taking away anyone’s vaccines.” It is important to have people in HHS and CDC who advocate for vaccination, and educate the public about the need for them and their safety and effectiveness. Last but not least we need to remove barriers to getting vaccinations such as the current need for a prescription to get a covid vaccination. Fortunately that virus has evolved to be less deadly but it can still cause serious disease in people with “high risk health conditions.” These people can get vaccinated but they will be much less safe if there is a significant amount of  covid infection  in the population as a whole.

This was written Sunday October 27, 2024 and will not be revised.

1. The president is not elected by popular vote.

I remember watching Headline News on CNN a few days before the 2012 election and hearing that the election between Obama and Romney was a statistical dead heat. Each candidate was predicted to have 47% of the vote. However, the results in the Electoral College were not close Obama 332, Romney 206. People who forecasted based on a state by state basis (Nate Silver and a talented amateur at Princeton) concluded on the days before the election that the probability Obama would win was 99%.

2. Even when the difference between the candidates is less than the margin of error of the poll it still contains information.

Let’s start with the question: what is the margin of error? It is often reported, but rarely with the sample size of the poll which is needed to be able to check the calculation. Some examples with both pieces of data are Franklin-Marshall for Penn 794/4.3; Emerson College for North Carolina 1226/3.6, and for Wisconsin 800/3.4; Marist for Arizona and Georgia 1193/3.9.

If Ɵ is the estimated frequency of votes for Harris then using standard formulas from an intro states class the radius of the 95% confidence interval the margin of error is 1.96(Ɵ(1-Ɵ)/n)^0.5.  Since Ɵ is usually in (0.4,0.6) then 0.24 < Ɵ(1-Ɵ) < 0.25 and replacing 1.96 by 2 we can simplify this to 1/n^0.5 and our data points become

n,error               800,  0.0353       1200, 0.288

somewhat smaller than the reported margins of error.

3. Frequencies should be reported to one decimal place and should be recalculated to remove responses that are not one of the two major candidates to make it easier to compare predcitions.

As some readers have noticed I have not gotten to #2 yet. That is coming soon. Bloomberg, the one poll that I have seen which reports frequencies with one decimal place had on October 23:

Harris/Trump    H/(H+T)           P(H win)   538 ave.

Michigan            49.6/46.9           51.4                     79.2         H 0.6

Penn.                  50/48.2               50.9                     67.7         H 0.3

Nevada               48.8/48.3           50.3                     59.2         H 0.2

Wisconsin          48.3/48               50.2                     51.6         H 0.1

Arizona               49.1/46               50.2                     51.6         T 1.8

48. Carolina 48.5/48.8 49.4                     43.9         T 1.2

Georgia               48.4/49.9           49.3                     35.2         T 1.5

If the uncertainties look large, it is because they are based on one poll. Once one has the average of several polls then the uncertainties decrease. Of course comparing Bloomberg with 538 average shows that there are sampling biases. The Washington Post is even more discordant. They have Harris leading in all states except Arizona. I got these figures from an article in Forbes but when I went to check them the Post wanted $2 to read the article.

Combining these projections to calculate is difficult without a computer and a good programmer. 538 uses simulations to conclude that, when this was written: Trump wills with probability 0.54 and Harris with probability 0.45.

https://projects.fivethirtyeight.com/2024-election-forecast/

One final selling point for people who publish polls, if you make predictions like this you can’t be wrong.

This blog post is based on the previous one The Arithmetic of Presidential Elections. There we used data from the last three presidential elections to argue that there are Red States with 170 electoral vote, Blue States with 215, and Swing States listed below with 153.

Turning to https://projects.fivethirtyeight.com/polls/president-general/2024/

for data, we see that Biden trails in every swing state

States               Votes                 Biden                 Trump

Ohio                   18                        36.8                    46.1

Georgia             16                        38.3                    44.8

Arizona              11                        38.9                    43.9

Iowa                   6                          32                        50.4

Carolina           15                       38.0                    45.2

Florida               29                        36.5                    46.1

*Penn.               20                       41.6                    43.6

*Wisc.                10                        41.8                    42.8

Nevada              6                          38.7                    42.6

*Michigan        16                        41.2                    43.0

*New Hamp.   4                          42                        44

Since this predicts Biden will only carry the Blue States this gives the prediction above.

Flipping the four closest states marked with * would result in 273 – 265 but likely the polls do not yet fully reflect his abysmal debate performance.

Here’s a little problem to test your skills at combinatorial probability.

You have a set of six cups and saucers. Two are NC State red R, two are UNC light blue b, and two are Duke dark blue B. You place the saucers in a line on the table RRbbBB. Then a blind man comes in and puts the saucers on the cups in random order. Let M be the number of cups that match the color of the saucer they are on. Your job is to compute the distribution of M.

To get you started I will specify a probability space which is the first step in solving any problem of this type. I once thought it was good to number the cups but a student in my class this year taught me it was better to treat the two cups of a given color as indistinguishable so we have 6!/(2!2!2!) = 90 outcomes instead of 720. To help check the solution note that not only should the probabilities sum to 1, but we must have EM = 6(1/3)=2. In the next paragraph I will start to reveal the solution starting at 6 and working down, so if you want to discover it on your on you should stop scrolling.

P(M=6)=1/90. In our probability space there is only one outcome where the cups all match, which is better than the situation when the cups are numbered and there are 2 x 2 x 2.

P(M=5)=0. If say the 2R match and 2b match then the 2B must match so 5 is impossible

P(M=4)=12/90. Matching 2-2-0 is impossible by the reasoning for 5, so we must have 2-1-1. There are 3 ways to pick the color with two matches, and for each color with only one match 2 choices of where the matching cup is. The rest of the outcome is now forced, e.g., RRbBBb.

P(M=3)=16/90. Matching 2-1-0 is impossible so we must have 1-1-1. We can pick the locations of the matching cups in 2 x 2 x 2 ways. The other three nonmatching cups must be either BRb or bBR

P(M=2)=27/90. We can have 2-0-0. Once we pick the double matching color in 3 ways the rest is forced, e.g. RRBBbb. We can have 1-1-0. We can pick the color with no match in 3 ways and 2 x 2 ways for the location of the matching cups. Suppose dark blue has no match. Then the two B cups must be on R and b, but there are 2 ways to put R and b on the B sauces for a total of 24 + 3 = 27.

P(M=1) =24/90. We can pick the location of the matching cup in 6 ways. Suppose it is the first R saucer. The second Red saucer can be B or b (2 ways). If it is B then we have bb on the Blue saucers, and we can have RB or BR on the b sauces (x2). If it is B then we have BB on the blue saucers and we have two possibilities on the B saucers, but autocorrect in Word will not let me type them.

P(M=0) =10/90. We can have BB on the Red saucers and then must have RR on b and bb on B. The situation is similar for bb on Red. This gives 2 outcomes. If we have {Bb} on the red saucers then we must have {Rb} on b and {Rb} on B where the set braces indicate we have not specified the order, so there are 2 x 2 x 2 = 8 outcomes.

1+12+16+27+24+10=90, 6 x 1+12 x 4+3 x16 +2 x 27+1 x 24 = 180 (so the mean is 2).