Mathematics

Yes, this Science! group includes math, and also statistics and biostatistics. The more, the merrier. I've not see too much so far, though, of math books for the general public. Any suggestions, please?

Here is one forthcoming math book, bring published a few months from now:

https://www.amazon.com/Why-Machines-Learn-Elegant-Behind/dp/0593185749/ref=sr_1_...

Wednesday, March 13, 2024 --- CNN website published a mostly text-based report via the Apple iPhone news app explaining what their reporter Will Ripley just learned via an in-person visit about Taiwan's approach to its global manufacturing of advanced computer chips. The following report from the CNN website today is that same data, reported in video format:

https://www.cnn.com/videos/tv/2024/03/12/tsmc-chips-intl-spc-hnk.cnn

Key to that manufacturing process are microscopes, which is where the math comes in. From the included photos, those are tabletop microscopes, not electron microscopes.

Please feel free to attach your further comments and questions to this thread about this topic.

Newer and new books on these topics:

Chip War: the fight for the world's most critical technology; Understanding Semiconductors : a technical guide for non-technical people; Essential Math for AI : next-level mathematics for efficient and successful AI systems; published on March 4, 2024 via W.W. Norton---> The Heart and the Chip : our bright future with robots.

Finding predictability in randomness? There's a prize for that:

https://www.newscientist.com/article/2423192-mathematician-wins-2024-abel-prize-...

Not being a math person myself, the last time I thought about randomness was when I read Michael Crighton's superb novel, Jurassic Park. And the only other time I remember thinking about randomness was during a biostatistics class at UCLA.

In a video clip from Taiwan, CNN reporter Will Ripley has more to say about TSMC, the world's leading manufacturer of semiconductor chips:

https://www.cnn.com/videos/business/2024/03/22/taiwan-semiconductor-manufacturin...

Benoit Mandelbrot's most widely read book for the lay public on financial markets, The Misbehavior of Markets, gives a popular understanding of the role of fractals in shaping financial markets. He was a member of the US National Academy of Sciences before his death in 2010. That book is published by Basic Books:

https://www.hachettebookgroup.com/titles/benoit-mandelbrot/the-misbehavior-of-ma...

Turing Prize awarded for studies on randomness:

https://www.msn.com/en-us/money/companies/for-turing-award-winner-everything-is-...

A new approach on Pi:

https://phys.org/news/2024-06-physicists.html

Declining star formation in this universe:

https://www.naoj.org/en/results/2012/11/05/2386.html

Scientific American (SCIAM) also has a paywalled journal article on that topic. SCIAM claims that these research findings can be expressed in one formula.

A few that come to mind would be

Mathematics From the Birth of Numbers - it is a very big book but the nice thing is the chapters are essentially self contained so you can pick something that strikes your fancy and just jump in anywhere.

If you want to gain a basic understanding of the actual mechanics of statistics The Cartoon Guide to Statistics is an excellent place to begin.

The book A History of Pi is a good history of the evolution of the computation and understanding of that number. The author takes the time to give the reader a sense of what was going on in the world when various advances were made.

There's the old standby How to Lie with Statistics by Huff. The only problem is it is too funny. A much better book which covers the same ground and which highlights when and where these mistakes happen is Flaws and Fallacies in Statistical Thinking by Campbell.

For some of these books you as a reader will have to take some time to understand some of the mathematical expressions but if you are interested in this subject you will find the time well spent.

Here's another report that's way above my pay grade to opine on:

https://www.wired.com/story/sensational-proof-delivers-new-insights-into-prime-n...

Wavefield mathematics is used to develop an understanding of the geodynamo mechanism in the Earth's core:

https://www.sciencedaily.com/releases/2024/08/240830164142.htm

Forthcoming on October 1, 2024:

https://www.hachettebookgroup.com/titles/malcolm-gladwell/revenge-of-the-tipping...

Revenge of the Tipping Point

There are some free math books digitized for reading online:

https://www.openculture.com/free-math-textbooks

My favorites are the statistics books. Biostatistics is a fundamental part of public health college coursework at the undergraduate and graduate school levels.

A new math theory:

https://www.wired.com/story/groups-underpin-modern-math-heres-how-they-work/

The mathematics of US economics, or the Federal Reserve as a depressive factor on the US middle class? Holy hand grenade. Link:

https://www.cnbc.com/2024/10/04/middle-class-wages-policy-suppressing-growth.htm...

The dumbing-down continues for the recycling of drinking water:

https://laist.com/news/climate-environment/california-water-recycling-rules-soca...

Almost all reporters in the USA are unfamiliar with trigonometry, specifically the formula for common exponential decay. That formula demonstrates why contaminants in a solution can never be reduced to a zero level.

The islet of Gavrinis, just off the coast of Brittany, France not far from the megalithic site at Carnac, includes many examples of drawings and shapes which are still undergoing much study. It was actually connected by causeway to the mainland until sea level rise at the end of the last ice age. Links:

https://sacred.numbersciences.org/2020/02/02/gavrinis-1-its-dimensions-and-geome...

https://en.wikipedia.org/wiki/Gavrinis

Helpful photos displayed in the online site Pinterest:

https://www.pinterest.com/rafgoblin2/gavrinis/

In recent years, scientists described it as a passage tomb, to describe its general shape. Some have found mathematical values coded into some of those shapes, reputedly the number Pi.

Recent discoveries about mathematical constants uncovered with the aid of A.I.:

https://www.pnas.org/doi/10.1073/pnas.2321440121

Another longstanding problem solved:

https://www.popularmechanics.com/science/math/a62945966/math-brauer-problem-solv...

>6 MaureenRoy: I met Dr. Mandelbrot during a lecture at UCLA when I was not a math major but attending department events anyway. He had to be one of the nicest "celebrities" that I met. The sad part of his lecture was following along nicely and then just going off a cliff of understanding.

Good morning,

I happened upon an interesting article about a new mathematical object: the Alena Tensor.

There is a little bit of "enthusiasm" in the piece regarding unifying curved and flat spacetime but it is measured enthusiasm.

https://modernsciences.org/new-mathematical-discovery-unite-quantum-mechanics-ge...

I have not had a change to read the underlying technical papers but they do look interesting.

A new provable mathematical theory of time:

https://phys.org/news/2025-06-theory-dimensions-space-secondary-effect.html

A pyramid shape that always lands on the same face:

https://gizmodo.com/this-weird-pyramid-always-lands-on-the-same-face-confirming-...

A more extensive discussion of that phenomenon plus its mathematical underpinning:

https://www.quantamagazine.org/a-new-pyramid-like-shape-always-lands-the-same-si...

>1 MaureenRoy: I found Why Machines Learn to be an excellent book and wrote a review of it:
https://www.librarything.com/work/31081521/book/277171301
I looked at the ratings and people either loved it (4 or 5 star) or hated it (1 or 2 star)
No 3 star ratings.

Did you read it?

I have not read it, but it's the most popular science book on Amazon. At 480 pages it sounds to be authoritative. I would suggest caution, however, regarding AI ability to "diagnose cancer." At least in the USA, human radiologists still read X-rays, CAT scans, MRI scans, etc.

A better counting method for automated systems:

https://washingtonweeklytimes.com/science-2/this-number-system-beats-binary-but-...

A new book:

https://maa.org/book-reviews/proof-the-art-and-science-of-certainty/
Adam Kucharski

Know your authoritative math sources:

https://phys.org/news/2025-09-systematic-fraud-uncovered-mathematics.html

Wolfram Alpha online now includes a statistics section:

https://www.wolframalpha.com/examples/mathematics/statistics

An overview of the traceability of bitcoin payments in kidnapping ransom demands:

https://www.forbes.com/sites/martinadilicosa/2026/02/06/the-kidnappers-of-savann...

On Tuesday, February 10, 2026, an international blockchain expert confirmed to CNN reporter Jake Tapper that even a bitcoin payment of one penny can be traced through blockchain systems. He then noted that US sheriffs typically discuss bitcoin payments as untraceable (as has been stated so far by the US local Sheriff in the Nancy Guthrie kidnapping in Tuscon, Arizona) despite evidence to the contrary on which the US sheriffs have previously been briefed.

In the United States to date, national law enforcement, investigative and clandestine branches of the Federal government are knowledgeable about tracing bitcoin money transfers while many local law enforcement groups are not.

About the math problems explored in the Hollywood film “Good Will Hunting”:

https://mathworld.wolfram.com/GoodWillHuntingProblems.html

A more efficient Earth-Moon route has been calculated:

https://phys.org/news/2026-05-mathematical-method-efficient-earth-moon.html

I've just been reading Paul Lockhart's A Mathematician's Lament.
{ https://www.amazon.com/Mathematicians-Lament-School-Fascinating-Imaginative/dp/1... }
He says that current mathematics teaching is like teaching music by teaching children everything about musical notation, without letting them hear or make music itself.
I can sympathyze with a lot of what he says, but ... how should mathematics be taught?
If anyone else has read this book, I'd be interested in your thoughts.

>34 Doug1943: How about giving us a brief summary of exactly what he means when he says, "current mathematics teaching is like teaching music by teaching children everything about musical notation, without letting them hear or make music itself."

>34 Doug1943: I read this some time ago and still find it incredibly "current" and "distressing". I have lived in several countries and often spoke with colleagues about how their children learned, not just math but many topics.

I think that there will always be a tension between, "practical and useful" knowledge for survival and beauty or understanding. As one example, look at the Kumon method or some of the Indian JEE materials. You work problems until you know all of the tricks and solutions, but the why is never really part of the learning. You solve the problems. However, the magic is missing. The "look at this, what do you think" is missing. Don't think, solve. I am being a bit hyperbolic in my description to be fair.

I am not a professional teacher so I can't speak to the challenges of the classroom. I can say that real learning in many topics appears to require more time than we want to give them. Process and efficiency moves things along, understanding will be less efficient in the short term (my proposition).

Now when the parents are stuck on their cell phones or not engaged when they can be...

>36 SRB5729: Since >34 Doug1943: doesn't appear to want to elaborate with respect to what has been quoted could I impose on you and ask the same thing?

This is my humble take. Very early on in the essay version, there is discussion of a triangle in a box where the vertices of the triangle touch a side. The question is posed, how much space of the box does the triangle take? Through reference to older Greek mathematics, we learn that this particular one takes up half of the box.

The process to discover this answer was one originated by curiosity, not some desired outcome. Trial and error led to a discovery.

In today's classrooms we learn geometry with certain givens, before and after that we learn algebraic rules. Then we learn about triangles. According to Israel Gelfand, Trigonometry gives us the precision that the Greeks could not originally achieve with geometry.

There is a beauty in the abstract. A search for equations that govern the universe but also, why and how they work. According to the book/essay, it was abstract thinking mathematicians who proposed the existence of a black hole, not physicists.

Just memorizing formulas as discrete objects leaves out the beauty. Many things are all connected. This connection is the search we are in now in many areas. For example the Langlands Program. But even at a lower level, watch any yet to be crushed child marvel when you explain how multiplication is many "adds" all done at once rather than one at a time. Or ponder why a straight line is the shortest distance. Memorization is our selling point in too many places. Memorize this trick, repeat it, don't waste my time on non practical questions like why, or even worse, why only sometimes.

Without the why and the pondering of what is going on, there is no room for curiosity. Curiosity is the very thing that drives advancement.

Here is an example that leaves me furious. There is a very solid YouTube channel called Black Pen Red Pen. He teaches lots of types of math and provides great introductions or help with respect to a variety of mathematical items. In one video he explains how his child's math teacher said that zero divided by zero is one because both the numerator and denominator were the same. The teacher could not conceptually understand that zero represents nothing. As a numerator, you have none of whatever your denominator represents. All good there. However as a denominator, there is nothing to be addressed or evaluated. therefore, such a fraction is undefined. You cannot have something of nothing. I would have definitely taken this to the school board, but so far have not encountered it for my child.

I have stepped down from my soapbox now.

Edited to add example.

>38 SRB5729: Thank you. If rote teaching without any kind of discussion as to why mathematics matters and what it does (many adds at once) is the point of the quote and it is a summary of the way things are done today then that is a sad state of affairs and I guess things have changed quite a bit since I was in grade/high school.

I did have some teachers who went in for rote memorization but most of them took the time to show you the mechanism and then show you what it meant and how it allowed you to view things in a completely different way. It's interesting you mentioned introducing multiplication as doing many adds at once. That was exactly the way my 4th grade teacher introduced it to the class. I can't speak for the rest of the class but I found that idea fascinating. There were other teachers along the way who did similar things such as introducing us to quadratic equations, showing us the graph that resulted when we plugged numbers into one, and then showing us where such curves showed up in the real world and why they mattered.

I do know if you introduce mathematics using a combination of experimentation, explanation, and practice with a focus on how it matters and what it tells you it is possible to take students a very long way in a very short time. I taught 3rd and 4th graders statistics for 6 one hour periods (two years in a row) and I got them all the way from understanding the concepts of means and standard deviations to actually using (and understanding) a two sample t-test. They had a lot of fun along the way and, two years later, I received some unexpected feedback with respect to what they had taken away from the courses from a 6th grade teacher who told me about an incident in her class concerning means comparisons and the response of one of her (formerly my) students.

>10 alco261: I have recommended Gullberg to well over 30 people of varying ages.

>35 alco261: This statement, using musical education as an analogy, is about as 'summarized' as it can be, already. I can't make it any more simple than it is. Sorry.

Lockhart's view is that mathematics should be presented to students as something that is fun, period. Not as something that is practical, or useful. I suppose he's "bending the stick" a bit.

The real question is: given the current education system's approach to teaching mathematics -- which I suspect is the same in all countries -- what could, in real life, be done to address his concern?

The only thing I can think of is: once a month, have a 'Lockhart hour', in which the teacher leads the students in a 'discovery session' of interesting mathematical facts.

Lockhart uses the fact that the sum of any sequence of odd numbers starting with 1 is always a square number to illustrate the magic of mathematical discovery. This is a particularly good example to use in class, since it can be illustrated with a visual example which makes it clear. (As can the sum of a sequence of whole numbers, starting with 1.)

Another book with Lockhart's attitude to mathematics is David Berlinski's, A Tour of the Calculus.
https://www.amazon.com/Tour-Calculus-David-Berlinski/dp/0679747885

>42 Doug1943: Thanks for the additional information. My experience with teaching mathematics is Lockhart's view is just about as dismal as the idea of rote memorization. Fun without any kind of an objective in mind doesn't do much of anything - we are amused and the amusement is very likely to be filed as fun to forget. My experience is to introduce something novel about mathematics and then have the students discover why it is novel and what one can do to address/understand/utilize the knowledge of that novel concept.

For example:

In preparation for teaching 3rd and 4th graders statistics I first purchase a full box of standard size M&Ms. I do this to insure all of the packets are from the same lot. I do this twice - once about a year before I teach and a second time about a week before the sessions are to begin.

When I start the first session I write down 2 and 3 on the blackboard and I ask the students to tell me what 2 x 3 is. Their looks tell me they think I've grossly under estimated their skills but someone finally says "6". My response is, "how do you know?" This usually elicits a number of proofs such as "two threes will add to 6". I then ask - how do you know those are actually 2 threes? In fairly short order one or more of the students will say something like, "A 3 is just a 3 and 2 of them make 6." My response is - "OK, fine you have defined 3 to be exactly 3. That's great and there is nothing wrong with your statement."

Then I will say - "The problem is that in life, whatever it is that corresponds to 3 is something you have to measure and when you do that you will not get exactly 3 every time." Then I write down a few numbers on either side of 2 and the same for 3 and then ask - "OK, now what is 2 x 3." This leads into a short discussion about measurement. Sometimes one of the kids will look at the distribution of numbers and recommend we take the average and multiply. When this happens it is always fun and I get to introduce a few other ideas.

At the end of the discussion session I pull out one of the boxes of M&M's, open it, remove a single packet and ask, "So how do I determine the most likely number of red M&M's in this packet?" Usually several people will say, "Just open it and count them." And I will admit I could do that but I don't care about the actual number I want to know about the most likely number." From there I briefly discuss scientific practice with respect to log entries and witnessing and I tell the students we will need to gather some data so everyone gets a packet, everyone is paired with someone else, one member of the team opens their packet, counts the red M&M's, the count is witnessed, and the value is plotted on a histogram I have up on the blackboard. Once all of this is done I can talk about the idea of a distribution and the concept of a mean. I point to the distribution they just finished building and I say, "Based on the research you just did what can you tell me NOW about the most probable number of red M&M's in the mystery packet?"

The usual response to this is the students pick the largest count. I respond by reminding them of what they just did and point out they got a number of values around the largest number but not everyone got the largest count so maybe we ought to estimate a range with a thought as to which values would be more or less likely again based on the research they just completed.

I write down the estimated ranges on the blackboard, pick a random student (usually who has a birthday closest to today) have them open the bag and do the count, I act as the witness, and then we plot the new data point and discuss it relative to the range estimates.
Usually the mystery bag count will fall well within most of the range estimates. Once in a great while we get an outlier and I get to introduce that idea as well.

The rest of the 5 sessions build on and extend the ideas of the first session. Lots of hands on graphing, actually using the pocket calculators and some of those weird symbols on the buttons at the top, etc. As I mentioned in my previous post, when we are finished the kids have a good understanding of many mathematical/statistical concepts as well as some understanding of how these things apply to the real world. It is also obvious from their responses during the 6 sessions that they are having a lot of fun while they are gaining that understanding.

Oh yes, at the end we repeat the M&M's experiment with the second box. Depending on how things have gone on the national economic scene we may find a distribution of red M&M's that is smaller, larger, about the same, shifted, etc. relative to the first distribution we built. When I post the first distribution next to the second the kids find it very interesting and questions about many aspects of the two distributions are raised - typically more than I have time to address and discuss before the last session ends.

I realize this is very long winded but I wanted to provide enough information about the details of the approach I use to give some understanding concerning my initial statement about just fun.

>42 Doug1943: Do we compare to Hardy and where his impractical became rather so? Albeit not right away.

Alco261: Whoa!!!! We need to video your sessions, incorporate these videos into an AI, and make them available to everyone.
SRB5729: It's been decades since I read Hardy. Can you elaborate?

>45 Doug1943: Sorry, have been traveling for work. I will try to reply soon.

I’m not sure yet if this finding can now be applied to a random number generator:

https://www.livescience.com/physics-mathematics/quantum-physics/physicists-achie...