By Keith Devlin @KeithDevlin@fediscience.org**, **@profkeithdevlin.bsky.social

There are, of course, a wide range of circumstances, character traits, and opportunities that lead us to choose the career path we do. The question is particularly intriguing for mathematicians, since the subject requires considerable effort to master, much of it necessarily involving long periods of unaided, solo effort, struggling to proceed in the face of errors. At the research level, a success rate of one correct solution after ten or more attempts is routine – you have to be persistent.

On the other hand, once you are into it, the world of mathematics provides a “safe haven” where you can shut yourself off from the rest of your world and are in full control. And there’s nothing like that adrenalin rush you get when you finally solve a problem you struggled with for hours (or days, months, even years).

A quick look through the biographies of famous mathematicians turns up a surprising number who came from impoverished, broken, or abusive homes; Sir Isaac Newton, Évartiste Galois, Alexander Grothendieck, Srivinasa Ramanujan, Nils Henrik Abel, and Mary Sommerville, to name just a few. Mathematics surely provided an escape from their circumstances. (In my case, being born right after World War II in a heavily bombed industrial city in the North of England, to working class parents with a strong opposition to education and intellectual pursuits – much like that of today’s Republican Party – I spent most evenings shut away in my room studying, particularly science and mathematics, about which my parents hadn’t a clue. It was my “world”, and they were not in it .)

Whatever your story is, the pleasure to be had from working on, and sometimes solving, challenging mathematical problems is addictive, provided you get to a stage where you can engage in such problems. A school with a traditional, “basics first,” instructional approach (often with timed tests) can be deadly for many students.

I asked Google’s AI engine for a summary of the reasons people give for doing mathematics. Here is what it came up with:

Q: Give me a summary of the reasons people give for doing mathematics

*People do mathematics to develop critical thinking, problem-solving, and analytical skills, which are useful in daily life and across many careers. They also engage with math for its practical applications in areas like cooking and finance, for the intellectual satisfaction of solving puzzles and understanding abstract concepts, and for career opportunities in fields like STEM and economics. *

Practical applications

• Daily life:* Math is used for everyday tasks like measuring ingredients for recipes, managing a budget, calculating distances, and understanding financial concepts like loans and interest.*
• Career opportunities:* A strong math background is valuable for many jobs, not just in STEM fields. It’s essential for careers in computing, engineering, accounting, economics, and many others. *

Intellectual and personal development

• Critical thinking and logic:* Math trains the brain to think logically and systematically, helping to solve problems by breaking them down into smaller, manageable steps.*
• Analytical skills:* It improves the ability to notice patterns, analyze information, and make informed decisions, which is useful in many real-life situations.*
• Abstract thinking and beauty:* Some people are drawn to the intellectual challenge and beauty of abstract concepts and the way math provides a framework to describe the world.*
• Brain function:* Regularly practicing math can help keep the brain healthy and improve cognitive skills. *

Career and future prospects

• Job market:* Strong math skills are highly valued by employers, and a background in mathematics can lead to high-earning and successful careers.*
• Understanding data:* It helps people understand, appreciate, and critically evaluate the math done by others in areas like research and media.*

Knowing how today’s AI system works, I had expected it would lean heavily towards applications and employment opportunities (since that’s how most people view mathematics). So then I repeated the question, slanted towards our community.

Q: What do mathematicians say they like about mathematics?

*Mathematicians are drawn to math for its beauty and elegance, its certainty and logic, and its power as a universal language for describing reality. They also find satisfaction in the intellectual challenge of problem-solving, the creativity involved in finding new proofs and approaches, and the sense of mastery and discovery that comes from overcoming difficult problems. *

Why mathematicians like mathematics

• Beauty and elegance:* Many find beauty in the structure, logic, and elegance of mathematical theorems, seeing it as a form of art that can also describe the physical world.*
• Certainty and logic:* They appreciate that math is not subjective; there is one right answer, and the results are not open to interpretation or empirical disproof. This provides a profound sense of certainty.*
• Intellectual challenge:* The pursuit of solving a difficult problem is a major draw. This process requires perseverance, and the gratification of finally finding a solution is a strong motivator.*
• Creativity and discovery:* Problem-solving is seen as a creative act, with many different approaches to a single problem possible. It is also a playground for the imagination to explore ideas and uncover universal truths.*
• A universal language:* Mathematics is a universal language that connects different fields and provides a way to express complex ideas concisely and deeply. It is seen as a way to understand the fundamental structure of the universe.*
• Sense of mastery:* The journey of learning and mastering mathematics can lead to a powerful sense of accomplishment and identity, motivating further study even as problems become harder.*
• Connections and patterns:* Mathematicians are fascinated by the connections between different areas of math and the patterns that are fundamental to the discipline and the universe itself. *

That answer is the one MAA members and other readers of “Devlin’s Angle” will surely recognize as essentially the one they would give. Hardly surprising, given the way Large Language Models operate. The system just gathered together all the stuff we have written about throughout our careers. (But don’t blame me for stating that in mathematics “there is one right answer.” If you want to say that, you need to add a caveat that you are referring to scholarly pure mathematics, based on axioms. Most (practical) problems for which mathematics is used do not have unique answers. See my Angle post for November 2019).

What the AI did not pick up on is the fact that for those of us who grew up and were schooled in the UK prior to the year 2000, A-level Mathematics was split into two subcategories: Pure Mathematics and Applied Mathematics.

But what exactly is “Applied Mathematics”? Most people would surely say it is the use of mathematics to solve problems in the world, in domains such as design, engineering, the sciences, and so on. My recollection of the GCE Applied Mathematics curriculum in place when I was in my last two years of high school was that it consisted predominantly of Classical Mechanics and Three-Dimensional Geometry, and amounted to applications of calculus. It definitely had a very different flavor than the Pure Mathematics curriculum.

Like many high school students, I was attracted to (pure) mathematics by its inner (logical, structural) beauty and the fact that in that kind of mathematics, problems do have unique solutions. But I came to it via Applied Mathematics, with Physics being my favorite subject and intended university major. (I took Physics and the two mathematics courses for my A-levels.) But by the time I was applying to university, Pure Mathematics was the one I wanted to pursue. I had been smitten by the inner beauty, and after my bachelors degree in Pure Mathematics I selected Mathematical Logic for my Ph.D., which is essentially mathematics applied to mathematics itself.

But after ten years of post-doctoral research in Axiomatic Set Theory, my interests took a much more applied bent, applying mathematical thinking to linguistics and communication, a hot topic in the era when the Internet was starting to be adopted. In that world, applying mathematics and mathematical thinking had a very different success criterion than pure mathematics. In pure mathematics, proof from axioms (a.k.a. mathematical truth) is the defining objective; in “applied mathematics”, utility is the objective. Indeed, none of the problems I worked on for the remainder of my career had unique solutions.

It was that shift in focus that led directly to my invitation to spend a year at Stanford in 1987, a transatlantic move that became permanent, with the last eighteen years of my career spent at that institution. It was at that university that, in the mid 1990s, one of the most consequential pieces of “applied mathematics” was created: the Google search algorithm. That was not about a “right answer”, rather an effective one. It’s a great example of mathematics pursued not for a right answer, but one that worked better than all the other attempted solutions that had been developed to find information on the newly introduced World Wide Web.

I suspect many readers of the *Angle *have at most a vague sense of how Google search works, so I’ll end this essay by summarizing the original idea. (These days, the algorithm is way more complicated, the details of which are a trade secret.)

The company Google was founded in 1996 by two graduate students in computer science at Stanford, Larry Page and Sergei Brin. With the World Wide Web emerging, the research topic Page chose for his masters dissertation was to examine mathematically how people used the new computer network.

The Web was built for people to search for information using the key idea of a link from one site to another. Page wondered what mathematical properties the Web graph would exhibit as more and more people started to use it. With Web-links providing the means to search forward from one site to another, he decided to investigate paths going backward. What can you say about the paths that lead to a site?

Can you say which sites a surfer would be more likely to visit than others? If you trace backwards and find you get to a site that gets lots of visitors (such as a reputable news site), then the site you are on may also be popular. (Why did that news site lead to yours?) Page called his investigation “backrub”.

Page and Brin developed a formula to provide, for each Webpage u on the WWW, a ranking of its popularity that they called PageRank (a play on the two uses of “page”), typically denoted by PR(u). Formally, PR(u) is the probability that** **a surfer would eventually visit u. The intuition is that the more popular sites there are that link to u, the more likely the surfer will end up at u. Among those more popular sites linking to u, the ones that themselves have popular sites linking to them will likely play a bigger role. And so on (for potentially many steps). So, tracing back, what gives a site higher rank is that lots of sites link to it that themselves have higher rank, and so on. This means that they had to specify an algorithm for computing PageRank by an iteration following the Weblinks backwards.

The problem was, that this sounds like the computation might go on forever. But mathematicians have ways to ensure an iteration converges to a fixed limiting value. Introduce an element of randomness, to take advantage of the fact that random activity generates long term patterns. (When you stir the milk into your coffee, eventually every sip is the same temperature. You don’t find hot spots or cold ones; the heat is distributed evenly.)

Suppose the number of pages in the network is N. Let L(v) denote the number of links going out from the page v. At first pass, you could set PR(u) to be the sum of all the numbers PR(v)/L(v) for all those v that link to u; the intuition being that each v’s rank PR(v) is spread across all those links that go out from v. So we could set

PR(u) = \sum_{\textit{v links to u}} PR(v)/L(v)

as a way to calculate PR(u) from the ranks of all the sites linking to u. This provides the basis for iterating in order to compute ranks for all sites. But there is no guarantee the process of iteration will be convergent, i.e., close in on a particular value. The way Brin and Page got past that possibility was to introduce a random element (like stirring your coffee).

So, they said, let’s assume the surfer uses the above formula d percent of the time but jumps to a random site the other (100-d) percent of the time, for some d. Except they didn’t use percentages in their formula, rather probabilities, so d was between 0 and 1. The probability of ending up on a site u from a random jump is 1/N (since each of the N pages in the network is equally likely). This led to their original definition of PageRank.

PR(u) = \frac{1-d}{N} + d \sum_{\textit{v links to u}} PR(v)/L(v)

The value for d they chose was around 0.85. (Sometimes math is like baking; you just see what proportion works best.) That means they have 0.15 parts random jumps and 0.85 parts formula. With that modification, the algorithm computes PR values iteratively until they converge. The random term ensures that will eventually occur. (If you continued long enough, eventually every site would be visited.)

When Page and Brin first tested it on Page’s Stanford home page in the campus network, it converged after 30 to 50 iterations. On the full Web, you could need a lot more. A whole lot more. The theory behind iterating processes with a random element simply says you will get convergence “eventually”. In practice, search-folk let it run until the difference in values from one iteration to the next is sufficiently small, perhaps 0.001. The point is, the math Page and Brin needed was all known (and Brin knew about it).

Why was Google so dramatically successful? Not because it gave the “right” answer. Rather it gave a really useful one!