Recent New York Times Podcast

On October 13, the New York Times held a podcast interview with Dr. Steven Strogatz, faculty member in the Math Department at Cornell University. Dr. Strogatz is Susan and Barton Winokur Distinguished Professor for the Public Understanding of Science and Mathematics and a Stephen H. Weiss Presidential Fellow.

The title and description of the podcast were certainly provocative:

When A.I. Comes for Your Purpose in Life

In the span of just a few months, A.I. systems have solved some of the biggest outstanding problems in math — adding mathematicians to the list of those terrified of losing their job to the technology.

Artificial intelligence has gone from struggling with basic arithmetic to helping solve some of mathematics’ most famous open problems — and the field is reckoning with what that means.

Steven Strogatz, a professor of mathematics at Cornell University, describes the excitement and grief of watching A.I. take on work that gives people a sense of purpose — and asks what could be lost if machines deliver answers without the human struggle, insight and understanding that make solving problems meaningful.

Dr. Strogatz does a credible job of describing how quickly A.I. has progressed from being a laughingstock for mathematics just a couple years ago to tackling some of the great unsolved problems in the field. He also does an excellent job of explaining this to a lay audience and comparing these recent advances in A.I. with what had been the historical norms of the thrill of mathematical discovery. (Disclaimer: I would have chosen different metaphors in a couple of places. I don’t consider myself a hero for doing mathematics, and I would have likened proving a theorem to sex.)

The podcast ends with a description with an uncertain look toward the future, including a town hall of faculty and students at Cornell discussing, among other things, “What is the meaning of life?” In context: If A.I. can do the work of theorem-proving… if not now, then soon… then what is the point of studying mathematics in the future? I don’t begrudge any mathematician struggling with such existential questions in the face of what generative A.I. might produce in the very near future.

Dr. Strogatz appeals to the book of Ecclesiastes and Solomon’s similar musings about the meaning of life. I believe that is a more than appropriate source of wisdom, and I also recommend The Problem of Life with God: Living with a Perfect God in an Imperfect World, a discussion of Ecclesiastes by my former pastor (now semi-retired) Tommy Nelson.

Mathematics of the Future

I recently came across this very thoughtful opinion piece — leavened with appropriate cartoons — titled Mathematics of the Future, by Pavel Etingof and Slava Gerovitch of the MIT Department of Mathematics. The abstract:

Rapid advances in artificial intelligence are transforming mathematical research, raising critical questions about the role, training, evaluation, and social value of human mathematicians. We argue that, rather than making mathematicians obsolete, AI will prompt a reorientation of the profession away from an “ownership culture” centered on priority and publication records and toward intellectual agency: deep understanding, judgment, vision, communication, and stewardship of a shared mathematical culture. Mathematicians can use AI to explore a rapidly expanding mathematical landscape, interpret and synthesize AI-generated knowledge, repair and formalize the existing literature, and develop new forms of exposition and publication. We propose that hiring and graduate education rely more heavily on sustained mathematical interaction, including talks, discussion, oral examination, mentoring, and collaborative work, rather than treating papers alone as decisive evidence of ability. We also outline a model for AI-assisted research labs that combines collective human thinking with careful use and checking of AI output. Acknowledging serious ethical, social, and epistemic concerns and respecting those who choose not to use AI, we argue that the central challenge is not to compete with machines in producing theorems, but to preserve and expand humanity’s capacity to develop, understand, and communicate mathematics.

AI and the Riemann Hypothesis

Just a couple of months ago, major progress was made on the Riemann Hypothesis, one of the great unsolved problems in mathematics (and one of the seven Clay Millenial Problems worth $1 million to the first person who solves them). What’s especially noteworthy is that this progress was made by AI. From the opening paragraphs of an article in Forbes:

Number theorists spent 37 years moving one number by less than a single percentage point. The number is the proven share of the Riemann zeta function’s zeros that sit on the critical line, the place where one of the most famous unsolved problems in mathematics says all of them belong. Decades of human refinement had carried it to 41.6%. On August 10, Anthropic published a result from an unreleased research version of Claude that moved it to 67.2% in roughly a day and a half.

The Riemann hypothesis has been open since 1859. It predicts the hidden structure behind how prime numbers are distributed, and it carries a million-dollar Clay Institute bounty. The hypothesis itself remains open. What Claude established is that at least two-thirds of the zeros behave the way it predicts, the largest single advance in the record’s history…

The tempting story is that an AI out-thought the mathematicians. The evidence points somewhere more specific. Number theory is organized into specializations deep enough that a career fits inside one of them, and the researchers behind the two lines Claude combined publish in the same journals without needing each other’s machinery. Claude read both lines at once and found that they fit. The advance was sitting in the published literature, waiting for a reader with no specialization.

Ken Ono feared AI. Now he trains it

Physics Today recently published an interesting and thought-provoking Q&A with eminent mathematician Ken Ono, who recently took a leave of absence from his tenure-track position at the University of Virgina to join Axiom Math, an company developing AI tools to verify mathematical proofs. From the article’s opening paragraphs:

“Am I doing a better service to science by writing 20 more papers that only people in my field will read or by participating in this movement?” Ken Ono asked himself last year. At the time, Ono was an endowed chair at the University of Virginia. In the following months, he’d take a leave of absence from the university, move to Silicon Valley, and become the founding mathematician at an AI startup, Axiom Math.

Axiom is one of several companies focused on developing AI tools to verify mathematical proofs. In the future, similar tools could be applied to check AI-generated computer code, a potentially lucrative service now that AI tools can write lines of code faster than any human can debug them. Startups Harmonic and Math Inc as well as Google’s DeepMind are targeting similar AI-powered verification tools.

Axiom has attracted investor attention, reaching a valuation of $1.6 billion earlier this year. (Ono says that he has shares in the company and that he receives a salary that is similar to his former compensation at Virginia.) His interests, he says, are in helping researchers use AI to think in more-creative ways and across more disciplines.

News flash: ChatGPT sometimes asserts false statements are true (Part 2)

I feel like I’ve done my good deed for the day by uncovering another instance when ChatGPT claimed a “fact” from the secondary mathematics curriculum that is simply incorrect. This one’s a doozy: it claimed that the binomial coefficent $\displaystyle {23 \choose 6} = 100,947$ is equal to $\displaystyle {22 \choose 5} = 26,334$ . (The first identity in the first line below is correct; the second one is not.)

Lesson: ChatGPT is a nice tool but you get what you pay for.

News flash: ChatGPT sometimes asserts false statements are true (Part 1)

I feel like I’ve done my good deed for the day by uncovering an instance when ChatGPT claimed a “fact” from the secondary mathematics curriculum that is simply incorrect.

While using ChatGPT to do some brainstorming on a research project, I received the following response (as part of a bigger response):

Sadly, whenever ChatGPT asserts a fact without documentation, it behooves the user to double-check the “fact.” In this case, there is a pretty blatant sign error.

As taught in high school AP calculus, the Taylor series expansion for $\ln(1+y)$ is

$\ln(1+y) = \displaystyle \sum_{q=1}^\infty \frac{(-1)^{q-1}}{q} y^q$ ,

and so

$-ln(1-y) = \displaystyle -\sum_{q=1}^\infty \frac{(-1)^{q-1}}{q} (-1)^q y^q$

$= \displaystyle\sum_{q=1}^\infty \frac{(-1)^{2q}}{q} y^q$

$= \displaystyle\sum_{q=1}^\infty \frac{y^q}{q}$

Said another way,

$\displaystyle \sum_{q=1}^{A-1} \frac{y^q}{q} + \sum_{q=A}^\infty \frac{y^q}{q} = -\ln(1-y)$

$\displaystyle \sum_{q=A}^\infty \frac{y^q}{q} = -\ln(1-y) - \sum_{q=1}^{A-1} \frac{y^q}{q}$

$\displaystyle \sum_{r=0}^\infty \frac{y^{r+A}}{r+A} = -\ln(1-y) - \sum_{q=1}^{A-1} \frac{y^q}{q}$

$\displaystyle \sum_{r=0}^\infty \frac{y^{r}}{r+A} = \frac{1}{y^A} \left[ -\ln(1-y) - \sum_{q=1}^{A-1} \frac{y^q}{q} \right]$

So that’s the correct answer. Notice that there is a minus sign in front of the sum on the right-hand side, not a plus sign.

So I asked ChatGPT to double-check. Here’s the first part of the response.

Lesson: You get what you pay for.

Solving Problems Submitted to MAA Journals (Part 6c)

The following problem appeared in Volume 97, Issue 3 (2024) of Mathematics Magazine.

Two points $P$ and $Q$ are chosen at random (uniformly) from the interior of a unit circle. What is the probability that the circle whose diameter is segment $\overline{PQ}$ lies entirely in the interior of the unit circle?

As discussed in the previous post, I guessed from simulation that the answer is $2/3$ . Naturally, simulation is not a proof, and so I started thinking about how to prove this.

My first thought was to make the problem simpler by letting only one point be chosen at random instead of two. Suppose that the point $P$ is fixed at a distance $t$ from the origin. What is the probability that the point $Q$ , chosen at random, uniformly, from the interior of the unit circle, has the desired property?

My second thought is that, by radial symmetry, I could rotate the figure so that the point $P$ is located at $(t,0)$ . In this way, the probability in question is ultimately going to be a function of $t$ .

There is a very nice way to compute such probabilities since $Q$ is chosen at uniformly from the unit circle. Let $A_t$ be the probability that the point $Q$ has the desired property. Since the area of the unit circle is $\pi(1)^2 = \pi$ , the probability of desired property happening is

$\displaystyle \frac{\hbox{area}(A_t)}{\pi}$ .

So, if I could figure out the shape of $A_t$ , I could compute this conditional probability given the location of the point $P$ .

But, once again, I initially had no idea of what this shape would look like. So, once again, I turned to simulation with Mathematica. As noted earlier in this series, the circle with diameter $\overline{PQ}$ will lie within the unit circle exactly when $MO+MP < 1$ , where $M$ is the midpoint of $\overline{PQ}$ . For my initial simulation, I chose $P$ to have coordinates $(0.5,0)$ .

To my surprise, I immediately recognized that the points had the shape of an ellipse centered at the origin. Indeed, with a little playing around, it looked like this ellipse had a semimajor axis of $1$ and a semiminor axis of about $0.87$ .

My next thought was to attempt to find the relationship between the length of the semiminor axis at the distance $t$ of $P$ from the origin. I thought I’d draw of few of these simulations for different values of $t$ and then try to see if there was some natural function connecting $t$ to my guesses. My next attempt was $t = 0.6$ ; as it turned out, it looked like the semiminor axis now had a length of $0.8$ .

At this point, something clicked: $(6,8,10)$ is a Pythagorean triple, meaning that

$6^2 + 8^2 = 10^2$

$(0.6)^2 + (0.8)^2 = 1^2$

$(0.8)^2 = 1 - (0.6)^2$

$0.8 = \sqrt{1 - (0.6)^2}$

Also, $0.87$ is very close to $\sqrt{3}/2$ , a very familiar number from trigonometry:

$\displaystyle \frac{\sqrt{3}}{2} = \sqrt{1 - (0.5)^2}$

So I had a guess: the semiminor axis has length $\sqrt{1-t^2}$ . A few more simulations with different values of $t$ confirmed this guess. For instance, here’s the picture with $t = 0.9$ .

Now that I was psychologically certain of the answer for $A_t$ , all that remain was proving that this guess actually worked. That’ll be the subject of the next post.

Solving Problems Submitted to MAA Journals (Part 6b)

The following problem appeared in Volume 97, Issue 3 (2024) of Mathematics Magazine.

Two points $P$ and $Q$ are chosen at random (uniformly) from the interior of a unit circle. What is the probability that the circle whose diameter is segment $\overline{PQ}$ lies entirely in the interior of the unit circle?

As discussed in the previous post, I guessed from simulation that the answer is $2/3$ . Naturally, simulation is not a proof, and so I started thinking about how to prove this.

$\displaystyle \frac{\hbox{area}(A_t)}{\pi}$ .

So, if I could figure out the shape of $A_t$ , I could compute this conditional probability given the location of the point $P$ .

But, once again, I initially had no idea of what this shape would look like. So, once again, I turned to simulation with Mathematica.

First, a technical detail that I ignored in the previous post. To generate points $(x,y)$ at random inside the unit circle, one might think to let $x = r \cos \theta$ and $y = r \sin \theta$ , where the distance from the origin $r$ is chosen at random between 0 and 1 and the angle $\theta$ is chosen at random from $0 to 2\pi$ . Unfortunately, this simple simulation generates too many points that are close to the origin and not enough that are close to the circle:

To see why this happened, let $R$ denote the distance of a randomly chosen point from the origin. Then the event $R < r$ is the same as saying that the point lies inside the circle centered at the origin with radius $r$ , so that the probability of this event should be

$F(r) = P(R < r) = \displaystyle \frac{\pi r^2}{\pi (1)^2} = r^2$ .

However, in the above simulation, $R$ was chosen uniformly from $[0,1]$ , so that $P(R < r) = r$ . All this to say, the above simulation did not produce points uniformly chosen from the unit circle.

To remedy this, we employ the standard technique of using the inverse of the above function $F(r)$ , which is clearly $F^{-1}(r) = \sqrt{r}$ . In other words, we will chose randomly chosen radius to have the form $R= \sqrt{U}$ , where $U$ is chosen uniformly on $[0,1]$ . In this way,

$P(R < r) = P( \sqrt{U} < r) = P(U < r^2) = r^2$ ,

as required. Making this modification (highlighted in yellow) produces points that are more evenly distributed in the unit circle; any bunching of points or empty spaces are simply due to the luck of the draw.

In the next post, I’ll turn to the simulation of $A_t$ .

Solving Problems Submitted to MAA Journals (Part 6a)

The following problem appeared in Volume 97, Issue 3 (2024) of Mathematics Magazine.

Two points $P$ and $Q$ are chosen at random (uniformly) from the interior of a unit circle. What is the probability that the circle whose diameter is segment $\overline{PQ}$ lies entirely in the interior of the unit circle?

It took me a while to wrap my head around the statement of the problem. In the figure, the points $P$ and $Q$ are chosen from inside the unit circle (blue). Then the circle (pink) with diameter $\overline{PQ}$ has center $M$ , the midpoint of $\overline{PQ}$ . Also, the radius of the pink circle is $MP=MQ$ .

The pink circle will lie entirely the blue circle exactly when the green line containing the origin $O$ , the point $M$ , and a radius of the pink circle lies within the blue circle. Said another way, the condition is that the distance $MO$ plus the radius of the pink circle is less than 1, or

$MO + MP < 1$ .

As a first step toward wrapping my head around this problem, I programmed a simple simulation in Mathematica to count the number of times that $MO + MP < 1$ when points $P$ and $Q$ were chosen at random from the unit circle.

In the above simulation, out of about 61,000,000 attempts, 66.6644% of the attempts were successful. This leads to the natural guess that the true probability is $2/3$ . Indeed, the 95% confidence confidence interval $(0.666524, 0.666764)$ contains $2/3$ , so that the difference of $0.666644$ from $2/3$ can be plausibly attributed to chance.

I end with a quick programming note. This certainly isn’t the ideal way to perform the simulation. First, for a fast simulation, I should have programmed in C++ or Python instead of Mathematica. Second, the coordinates of $P$ and $Q$ are chosen from the unit square, so it’s quite possible for $P$ or $Q$ or both to lie outside the unit circle. Indeed, the chance that both $P$ and $Q$ lie in the unit disk in this simulation is $(\pi/4)^2 \approx 0.617$ , meaning that about $38.3\%$ of the simulations were simply wasted. So the only sense that this was a quick simulation was that I could type it quickly in Mathematica and then let the computer churn out a result. (I’ll talk about a better way to perform the simulation in the next post.)

Square roots and Logarithms Without a Calculator: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on computing square roots and logarithms without a calculator (with the latest post added).

Part 1: Method #1: Trial and error.

Part 2: Method #2: An algorithm comparable to long division.

Part 3: Method #3: Introduction to logarithmic tables.

Part 4: Finding antilogarithms with a table.

Part 5: Pedagogical and historical thoughts on log tables.

Part 6: Computation of square roots using a log table.

Part 7: Method #4: Slide rules

Part 8: Method #5: By hand, using a couple of known logarithms base 10, the change of base formula, and the Taylor approximation $\ln(1+x) \approx x$ .

Part 9: An in-class activity for getting students comfortable with logarithms when seen for the first time.

Part 10: Method #6: Mentally… anecdotes from Nobel Prize-winning physicist Richard P. Feynman and me.

Part 11: Method #7: Newton’s Method.

Part 12: Method #8: The formula $\sqrt{b} \approx \displaystyle \frac{a+b}{2\sqrt{a}}$