AI Is Coming for Mathematics, Too?

Consider this.

AI Is Coming for Mathematics, Too

The New York Times

Siobhan Roberts, July 2, 2023

An artificial intelligence (AI)-driven transformation of mathematics looms, with former Google computer scientist Christian Szegedy forecasting computers will match or surpass human mathematicians' problem-solving ability by 2026. Terence Tao at the University of California, Los Angeles said mathematicians' concerns about AI potentially threatening mathematical aesthetics or their profession have emerged in the last several years. The University of Wisconsin-Madison's Jordan Ellenberg suggested AI gadgets could help optimize mathematicians' work. Microsoft's open source Lean proof assistant, which uses automated reasoning powered by AI, is drawing interest for its recent achievements, yet its frequent complaining of being unable to understand the mathematician's inputs makes research awkward. Geordie Williamson at Australia's University of Sydney said mathematicians and computer scientists should participate in discussions about AI's mathematical implications more aggressively.  ,,, ' 

Monumental Math Bubble Problem Solved

Solution further to some kinds of 'coverage' applications?

‘Monumental’ Math Proof Solves Triple Bubble Problem and More

The decades-old Sullivan’s conjecture, about the best way to minimize the surface area of a bubble cluster, was thought to be out of reach for three bubbles and up — until a new breakthrough result.

By Erica Klarreich, Contributing Correspondent October 6, 2022

The Prime Number Conspiracy - The Biggest Ideas in Math from Quanta – Available now!

When it comes to understanding the shape of bubble clusters, mathematicians have been playing catch-up to our physical intuitions for millennia. Soap bubble clusters in nature often seem to immediately snap into the lowest-energy state, the one that minimizes the total surface area of their walls (including the walls between bubbles). But checking whether soap bubbles are getting this task right — or just predicting what large bubble clusters should look like — is one of the hardest problems in geometry. It took mathematicians until the late 19th century to prove that the sphere is the best single bubble, even though the Greek mathematician Zenodorus had asserted this more than 2,000 years earlier.

The bubble problem is simple enough to state: You start with a list of numbers for the volumes, and then ask how to separately enclose those volumes of air using the least surface area. But to solve this problem, mathematicians must consider a wide range of different possible shapes for the bubble walls. And if the assignment is to enclose, say, five volumes, we don’t even have the luxury of limiting our attention to clusters of five bubbles — perhaps the best way to minimize surface area involves splitting one of the volumes across multiple bubbles.  ...   much more ...  (Math/Technical) 

Stories, Dice and Rocks that Think. A Look at the Historical Trajectory of AI and How it is Shaping the Future

 Just completed this excellent book by Byron Reese about how we got to thinking about 'artificial Intelligence', and were it can go.   Here is their overview:  

'Stories, Dice and Rocks That Think'.    Book by Byron Reese

 This fascinating tale explores the three leaps in human history that made us what we are today-and will change how you think about our future.

Look around. Clearly, we humans are radically different from the other creatures on this planet. But why? Where are the Bronze Age beavers? The Iron Age iguanas? 

In Stories, Dice, and Rocks That Think, Byron Reese argues that we owe our special status to our ability to imagine the future and recall the past, escaping the perpetual present that all other living creatures are trapped in.

Reese shows us how this escape enabled us to share knowledge on an unprecedented scale, and predict- -and eventually master--the future.

Thoughtful, witty, and compulsively readable, Stories, Dice, and Rocks That Think unravels our history as an intelligent species in three acts:

Act I (Stories): Ancient humans undergo "the awakening," developing the cognitive ability to mentally time-travel and share knowledge using language.

Act II (Dice): In 17th century France, the mathematical framework known as "probability theory" is born--a science for seeing into the future that we used to build the modern world.

Act III (Rocks That Think): Beginning with the invention of the computer chip, humanity creates machines to model the future with even more precision, and overcome the limits of our brains.

A fresh new look at the history and destiny of humanity, Stories, Dice, and Rocks That Think will give readers a new understanding of what they are- not just another animal, but a creature with a mastery of time itself. ... 

(Available in all the usual places)

Pushing the Frontiers of Mathematical Research

My earliest work did this kind of research

Pushing the Frontiers of Mathematical Research,   By Allyn Jackson

Commissioned by CACM Staff, September 8, 2022

For decades, mathematicians have turned to computers for help with tasks like big numerical calculations and visualizing complex geometric objects. Like a blackboard, the computer has been a handy tool that picks up where human capacity to juggle numbers, symbols, and pictures drops off.

Today, however, computers are playing an entirely new role: they are learning modern mathematics.

A loose-knit international group is using computer proof assistants, originally developed to check formal software correctness, to create online libraries of mathematical theorems and proofs. The theorems housed in these libraries can then be called upon as building blocks for proofs of new mathematical results. The hope is that the libraries one day will encompass the entirety of mathematical knowledge.

"It's a completely new way to do mathematics that is very satisfying," said mathematician Kevin Buzzard of the U.K.'s Imperial College London.

Buzzard discussed this work in one of the most-watched lectures at the 2022 International Congress of Mathematicians in July this year. (The Congress, originally scheduled to be held in Saint Petersburg, Russia, was transformed into an entirely online event after that country's invasion of Ukraine.) Around the same time, a paper appeared on the arXiv containing what might be called a "proof assistant manifesto," laying out progress achieved and describing challenges ahead. Buzzard is one of the paper's 10 authors, along with ACM A.M. Turing Award recipient Leslie Lamport.

During the course of his career in number theory and algebraic geometry, Buzzard has worked on pure mathematics with no obvious applications. He describes his latest work with computer proof assistants as also "100% blue-sky research," explaining, "It's interesting, important, and beautiful. But I can't see the future. I don't know what killer app might come out of this work."

Verifying Proofs Down to the Axioms

Computer proof assistants, also called interactive theorem provers, began to have a significant impact in mathematics starting with the work of Thomas Hales of the University of Pittsburgh.  In the late 1990s, Hales announced a solution to a venerable sphere-packing problem called the Kepler conjecture. While his approach was generally accepted as correct, Hales' use of a computer program to sort through a huge number of possible packings left many skeptical.

Hales then embarked on a multi-year project to validate his solution by recasting it as a "formal proof," a thoroughgoing version of the proof in which every logical inference is checked, down to the fundamental axioms of mathematics. Formal proofs, far too verbose and tedious for humans to read, are tailor-made for computer proof assistants. In 2014, Hales and 20 collaborators completed a computer verification of the Kepler proof. 

In parallel with this work, during the early 2000s, a few mathematicians began to use proof assistants to formalize proofs of several classic mathematical results. The work was slow and painstaking, and the technology rather off-putting. As a result, formal proofs remained something of a niche area, far from the frontier of mathematics.

Buzzard got into the act after listening to a 2017 lecture in which Hales described his experiences using proof assistants. "That lecture changed my life," said Buzzard, as he realized that proof assistants could handle the theorems that mathematicians are working on at the frontier of research today. "Computer scientists were figuring out how to write the software, to make it useful," Buzzard said. "They've done that now. Now it's our turn."

The True Value of Proof Assistants

As the name suggests, a computer proof assistant is very good at verifying proofs, but its true value in mathematics lies elsewhere, in its ability to store mathematical results and the fundamental logic supporting them. Every theorem, lemma, and proposition that was used or proved in the course of verifying, for example, Hales's solution of the Kepler conjecture, now sits in the memory of a proof assistant, ready to be called upon as raw material to build proofs of new theorems.  ... ' 

Stories, Dice, and Rocks That Think:

Just  Reading, very interesting, will review further as I progress.  by a correspondent I have often mentioned here.  

Stories, Dice, and Rocks That Think: How Humans Learned to See the Future--and Shape It  ...    by Byron Reese 

". . . Byron Reese gets to the heart of what makes humans different from all others." —Midwest Book Review

What makes the human mind so unique? And how did we get this way?   Amazon Description:

This fascinating tale explores the three leaps in our history that made us what we are—and will change how you think about our future.

Look around. Clearly, we humans are radically different from the other creatures on this planet. But why? Where are the Bronze Age beavers? The Iron Age iguanas? In Stories, Dice, and Rocks That Think, Byron Reese argues that we owe our special status to our ability to imagine the future and recall the past, escaping the perpetual present that all other living creatures are trapped in. 

Envisioning human history as the development of a societal superorganism he names Agora, Reese shows us how this escape enabled us to share knowledge on an unprecedented scale, and predict—and eventually master—the future.

Thoughtful, witty, and compulsively readable, Reese unravels our history as an intelligent species in three acts: 

Act I: Ancient humans undergo “the awakening,” developing the cognitive ability to mentally time-travel using language

Act II: In 17th century France, the mathematical framework known as 'probability theory' is born—a science for seeing into the future that we used to build the modern world

Act III: Beginning with the invention of the computer chip, humanity creates machines to gaze into the future with even more precision, overcoming the limits of our brains

A fresh new look at the history and destiny of humanity, readers will come away from Stories, Dice, and Rocks that Think with a new understanding of what they are—not just another animal, but a creature with a mastery of time itself.  ... ' 

Can Computers Be Mathematicians?

Would make sense, hypotheses could be automatically spun off into tests as needed. 

Can Computers Be Mathematicians?    By The Joy of Why  in ACM Opinion, July 5, 2022

"I think that one of the things that's happening in this collaboration is that computer scientists are beginning to learn more about the nature of what modern mathematics actually looks like." -Kevin Buzzard

For the last few years, researchers and amateurs all over the world have worked together to translate the essential axioms of mathematics into a programming language called Lean. Armed with this knowledge, theorem-proving programs that understand Lean have begun helping some of the world's greatest mathematicians verify their work.

In an interview, Kevin Buzzard, professor of pure mathematics at Imperial College London, talks about the effort to "teach" math to Lean—and how projects like this one could shape the future of mathematics.

From The Joy of Why   

View Full Article

AI Translates Math Problems into Code

 Useful Direction.  

AI Translates Math Problems into Code to Make Them Easier to Solve

New Scientist, Alex Wilkins, June 6, 2022

Google's Yuhuai Wu and colleagues used the Codex neural network of artificial intelligence (AI) research company OpenAI to translate mathematical problems from plain English into formal code. Codex correctly translated 25% of 12,500 secondary-school math competition problems into a format compatible with a formal proof-solver program called Isabelle. Wu said the system's inability to understand certain mathematical concepts was responsible for many of the unsuccessful translations. The team then tested the process by applying Codex to problems pre-formalized by humans. The network produced its own formal versions, and the researchers used the MiniF2F AI to solve both versions; the auto-formalized versions raised MiniF2F's success rate from 29% to 35%, suggesting Codex's formalization was superior to that of humans.... '

The Now Much Older Pythagorean Theorem

 The Pythagorean theorem, you know the very handy rule about the square of the hypotenuse.   Invented by Pythagoras. ...   Well no.    We now have just recently learned, it has been around, and identified in archeology tablets much,  much earlier.  in common use in Babylonian times.     This is a quite remarkable finding, pass on to your math friends.  

Mathematical mystery of ancient Babylonian clay tablet revealed by Daniel Mansfield & Norman Wildberger   Date:    Friday, 25th August 2017

UNSW scientists have discovered the purpose of a famous 3700-year-old Babylonian clay tablet, revealing it is the world’s oldest and most accurate trigonometric table, possibly used by ancient mathematical scribes to calculate how to construct palaces and temples and build canals.

The new research shows the Babylonians, not the Greeks, were the first to study trigonometry – the study of triangles – and reveals an ancient mathematical sophistication that had been hidden until now.

Known as Plimpton 322, the small tablet was discovered in the early 1900s in what is now southern Iraq by archaeologist, academic, diplomat and antiquities dealer Edgar Banks, the person on whom the fictional character Indiana Jones was based.

It has four columns and 15 rows of numbers written on it in the cuneiform script of the time using a base 60, or sexagesimal, system.

“Plimpton 322 has puzzled mathematicians for more than 70 years, since it was realised it contains a special pattern of numbers called Pythagorean triples,” says Dr Daniel Mansfield of the School of Mathematics and Statistics in the UNSW Faculty of Science.

“The huge mystery, until now, was its purpose - why the ancient scribes carried out the complex task of generating and sorting the numbers on the tablet.    .. 

“Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles. It is a fascinating mathematical work that demonstrates undoubted genius.  ..... '

Action at a Distance Works

An approach that's already being used for some kinds of encrypted transmission.  Here is a largely non technical explanation using Bell's Theorem.  

How Bell’s Theorem Proved ‘Spooky Action at a Distance’ Is Real

The root of today’s quantum revolution was John Stewart Bell’s 1964 theorem showing that quantum mechanics really permits instantaneous connections between far-apart locations. .Spookiness indeed.  ... '

Matrix Multiplication Inches Closer to Mythic Goal

Mentioned previously, here a little more succinctly.   Technical. 

Matrix Multiplication Inches Closer to Mythic Goal  By Quanta Magazine,  March 24, 2021

A paper posted in October describes the fastest-ever method for multiplying two matrices together.

For computer scientists and mathematicians, opinions about "exponent two" boil down to a sense of how the world should be.

"It's hard to distinguish scientific thinking from wishful thinking," said Chris Umans of the California Institute of Technology. "I want the exponent to be two because it's beautiful."

"Exponent two" refers to the ideal speed — in terms of number of steps required — of performing one of the most fundamental operations in math: matrix multiplication. If exponent two is achievable, then it's possible to carry out matrix multiplication as fast as physically possible. If it's not, then we're stuck in a world misfit to our dreams.

Matrices are arrays of numbers. When you have two matrices of compatible sizes, it's possible to multiply them to produce a third matrix. For example, if you start with a pair of two-by-two matrices, their product will also be a two-by-two matrix, containing four entries. More generally, the product of a pair of n-by-n matrices is another n-by-n matrix with n2 entries.

For this reason, the fastest one could possibly hope to multiply pairs of matrices is in n2 steps — that is, in the number of steps it takes merely to write down the answer. This is where "exponent two" comes from.

And while no one knows for sure whether it can be reached, researchers continue to make progress in that direction....

From Quanta magazine

Topology and Homology Address the Form of Data

Nice short piece on Topology.   I was introduced to it at an early age, and liked aspects of how it linked to the real world in a special ways.   But I never saw very much direct use for it.   Except, as the article suggests, for categorizing shapes in an interesting way. But now something called Homology, which lets you think about shapes in data. Now thinking of how it might be used.    In Quanta Mag, a largely, but subtly non-technical description: 

How Mathematicians Use Homology to Make Sense of Topology

Originally devised as a rigorous means of counting holes, homology provides a scaffolding for mathematical ideas, allowing for a new way to analyze the shapes within data.

In Quanta Magazine By Kelsey Houston-Edwards    Contributing Writer  ..... '

Defining Epidemics

Late to this, but interesting and well done. . Mostly non technical and visually presented.

Chasing the Elusive Numbers That Define Epidemics

By Jordana Cepelewicz, Staff Writer   in Quanta Magazine

Most modeling efforts during the COVID-19 pandemic have sought to address urgent practical concerns. But some groups aim to bolster the theoretical underpinnings of that work instead.

Researchers can’t directly observe many key features of disease transmission. As a result, they rely on statistical models to translate what they can see to what they want to know. But they’re finding that for COVID-19 in particular, some of these methods have been giving them the wrong answers.  ... " 

Intro to Bayesian

Good intro to the idea  of Bayesian computation and modeling,   But ultimately does include math.  We successfully used the method for simulating relatively complex process.   Or 'computing' probabilistically the real world based on available data.  This is an intro, and you are best advised to not implement it directly, but use a package as the basis for use.    But it is also good to use Bayesian methods to think about your problem and map it out usefully ... 

The ABCs of Approximate Bayesian Computation

An introduction into parameter inference using Approximate Bayesian Computational methods.  By Tom Leyshon in TowardsdataScience

What is Bayesian statistics?

Bayesian statistics are methods that allow for the systematic updating of prior beliefs in the evidence of new data [1]. The fundamental theorem that these methods are built upon is known as Bayes’ theorem.... " 

Faster Linear Equations

 Like the aspect of guessing to solve, implies the useful introduction  of randomness. 

Algorithm Breaks Speed Limit for Solving Linear Equations

By Quanta Magazine

Grade school math teachers admonish students not to just guess the answer to a problem. But a new proof establishes that, in fact, the right kind of guessing is sometimes the best way to solve systems of linear equations, one of the bedrock calculations in math.

As a result, the proof establishes the first method capable of surpassing what had previously been a hard limit on just how quickly some of these types of problems can be solved.

The new method, by Richard Peng and Santosh Vempala of the Georgia Institute of Technology, is decribed in "Solving Sparse Linear Systems Faster Than Matrix Multiplication,"    which was presented at SODA21, the ACM-SIAM Symposium on Discrete Algorithms, where it won the best-paper award.  ... "

Article in Quanta

Imaginary Numbers Essential for a Quantum Reality

Good piece, math-technical, worth a skim regardless.   Intro below.  

Imaginary Numbers May Be Essential for Describing Quantum Reality

A new thought experiment indicates that quantum mechanics doesn’t work without strange numbers that turn negative when squared.

Charlie Wood   Contributing Writer  Quanta Magazine

Mathematicians were disturbed, centuries ago, to find that calculating the properties of certain curves demanded the seemingly impossible: numbers that, when multiplied by themselves, turn negative.

All the numbers on the number line, when squared, yield a positive number; 22 = 4, and (-2)2 = 4. Mathematicians started calling those familiar numbers “real” and the apparently impossible breed of numbers “imaginary.”

Imaginary numbers, labeled with units of i (where, for instance, (2i)2 = -4), gradually became fixtures in the abstract realm of mathematics. For physicists, however, real numbers sufficed to quantify reality. Sometimes, so-called complex numbers, with both real and imaginary parts, such as 2 + 3i, have streamlined calculations, but in apparently optional ways. No instrument has ever returned a reading with an i.

Yet physicists may have just shown for the first time that imaginary numbers are, in a sense, real.

A group of quantum theorists designed an experiment whose outcome depends on whether nature has an imaginary side. Provided that quantum mechanics is correct — an assumption few would quibble with — the team’s argument essentially guarantees that complex numbers are an unavoidable part of our description of the physical universe.

“These complex numbers, usually they’re just a convenient tool, but here it turns out that they really have some physical meaning,” said Tamás Vértesi, a physicist at the Institute for Nuclear Research at the Hungarian Academy of Sciences who, years ago, argued the opposite. “The world is such that it really requires these complex” numbers, he said.

In quantum mechanics, the behavior of a particle or group of particles is encapsulated by a wavelike entity known as the wave function, or ψ. The wave function forecasts possible outcomes of measurements, such as an electron’s possible position or momentum. The so-called Schrödinger equation describes how the wave function changes in time — and this equation features an i. ... ' 

Computers for Charisma, Empowerment, Learning

I was briefly involved with an OLPC effort.  And have always been intrigued by the idea of how it is different to teach coding, than it is to teach mathematics, though they are closely connected.   I know many people who were truly expert coders, but avoided any math.   Coding has become 'charismatic' in a sense, but I still don't think it helps to have everyone learn to code.  Coding is more an extreme in paying attention to detail,  than the esoterica of math.  Giving out laptops empowers people to make things happen, but it rarely makes them coders or mathematicians.  That makes a difference.  Aiming to read these books. 

Computers for Learning: Charisma that Fails to Disrupt?

January 5, 2021  By Jeremy Roschelle

Learning technology often bursts into our awareness with powerful promises to personalize learning, to accelerate progress, to scale powerful learning to everyone. Too often, learning technology fails to deliver. Why? Recently, I read two books addressing this dilemma, both of which are grounded in strong empirical research traditions.

The first book is "The Charisma Machine" by Morgan G. Ames, who conducted an ethnography of One Laptop Per Child (OLPC) in South America in 2009-2010. Dr. Ames spent extensive time in schools and towns in Paraguay as the OLPC "XO" laptop rolled out and slowly fell into disuse. Some of the reasons for declining use were mundane: the XO laptops were underpowered, the trackpad was hard to use, and necessary software wasn’t available. The laptops were designed to survive a fall when closed, but students often carried them open in order to use the video camera. Once broken, the laptops could not be repaired locally. The mesh networking rarely worked and internet connections were poor. 

Despite these mundane problems, the choice to feature "charisma" in the book title points to the deeper lessons that can be learned by reading this book. These lessons generalize beyond the particular missteps of OLPC program. Charisma connotes unusual attractiveness that commands devotion, and also suggests the ability to distort reality such that obvious downsides become hidden from view.  Reading this book made me realize how many technologies for education are constructed as charismatic, implying "don’t anticipate difficulties based on what you already know, because this new technology will be a complete game changer!"  This  charismatic portrayal can facilitate rapid initial adoption but rarely translates into lasting change.   ...  (Much more follows)

Automated Math Reasoning

In our earliest AI courses,we learned about theorem proving using AI. And yes, it was not automated math reasoning.  But it gave you the hope that it could be done, if only you could state the problem at hand as purely mathematical.   Or even parts of it.  But it was never so.  Like the article says, it rarely intersects exactly with the real world, except for elements of the real world that are also approximations within contexts.  Bottom line, its still hard.    Good article explains it, with hopes for the next generation.

How Close Are Computers to Automating Mathematical Reasoning? in Quanta Mag.  Stephen Ornes
Contributing Writer 

AI tools are shaping next-generation theorem provers, and with them the relationship between math and machine.

n the 1970s, the late mathematician Paul Cohen, the only person to ever win a Fields Medal for work in mathematical logic, reportedly made a sweeping prediction that continues to excite and irritate mathematicians — that “at some unspecified future time, mathematicians would be replaced by computers.” Cohen, legendary for his daring methods in set theory, predicted that all of mathematics could be automated, including the writing of proofs.

A proof is a step-by-step logical argument that verifies the truth of a conjecture, or a mathematical proposition. (Once it’s proved, a conjecture becomes a theorem.) It both establishes the validity of a statement and explains why it’s true. A proof is strange, though. It’s abstract and untethered to material experience. “They’re this crazy contact between an imaginary, nonphysical world and biologically evolved creatures,” said the cognitive scientist Simon DeDeo of Carnegie Mellon University, who studies mathematical certainty by analyzing the structure of proofs. “We did not evolve to do this.”

Computers are useful for big calculations, but proofs require something different. Conjectures arise from inductive reasoning — a kind of intuition about an interesting problem — and proofs generally follow deductive, step-by-step logic. They often require complicated creative thinking as well as the more laborious work of filling in the gaps, and machines can’t achieve this combination.

Computerized theorem provers can be broken down into two categories. Automated theorem provers, or ATPs, typically use brute-force methods to crunch through big calculations. Interactive theorem provers, or ITPs, act as proof assistants that can verify the accuracy of an argument and check existing proofs for errors. But these two strategies, even when combined (as is the case with newer theorem provers), don’t add up to automated reasoning.  .... "

Numbers, Mathematics and the Reality of Science

Why can numbers do such a good job of describing reality?   Can they describe all of reality?

Recently republished:  https://medium.com/@ruth.ym.ng/do-numbers-exist-251e9b61508

Do Numbers Exist?  by Ruth Ng
November 2nd 2018

In 1960, Eugene Wigner began the closing paragraph of his paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences with a beautiful summary of the problem philosophers face when it comes to the existence of numbers. He said:

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

He’s talking about the sheer power, the disproportionate usefulness and beauty of mathematics. Its ability to seemingly describe reality in a way our ordinary language never could is uncanny.

The Collatz Conjecture is one such curiosity, as is the Fibonacci sequence. (If these sorts of things interest you, my favourite books on this kind of thing are Ian Stewart’s Incredible Numbers, Freiberger & Thomas’ Numericon and David Acheson’s 1089.) ... 

Tool Turns Math into Pictures

Lovely thought, you can browse the images, and edit or choose the best.  To produce a best explanation.   A means to communicate math concepts with management, decision makers?   Good examples at the link.

Carnegie Mellon Tool Automatically Turns Math Into Pictures
By Byron Spice

A tool created by Carnegie Mellon University (CMU) researchers allows anyone to render mathematical abstractions as illustrations. The Penrose tool enables diagram-drawing experts to encode their math-into-diagram methods; users simply type in an ordinary mathematical expression, and Penrose produces the drawing. Once the computer learns how the user wants to visualize mathematical objects, it uses the encoded rules to draw several candidate diagrams, which the user can from choose and edit. The researchers created a special programming language for this purpose, which CMU’s Keenan Crane said mathematicians should have no trouble learning. Said Crane, "Our vision is to be able to dust off an old math textbook from the library, drop it into the computer, and get a beautifully illustrated book—that way, more people understand."  .. '

Math Aware Search

Interesting  idea, and in some cases could be useful.  Perhaps search for similar math stated algorithms as starting points for analysis?

RIT Researchers Create Easy-to-Use Math-Aware Search Interface
Rochester Institute of Technology
By Scott Bureau
June 23, 2020

Rochester Institute of Technology (RIT) researchers have developed MathDeck, an online search interface that allows anyone to easily create, edit, or look up complex mathematical formulas. MathDeck users can enter and edit formulas in multiple ways using the scientific markup language LaTeX, including handwriting, uploading a typeset formula image, and text input. The math-aware interface can identify formula images and hand-drawn symbols via image processing and machine learning. MathDeck also features an auto-complete function for formulas and keywords; users looking for a popular symbol or formula will likely find an entity card displaying the formula, the name of its associated concept, and a brief description. MathDeck is a component of the multi-institutional MathSeer project, which RIT's Richard Zanibbi said aims "to produce new technologies to provide 'math search for the masses.'"   ... '