Infinite Powers to explain

This post continues a series of post were I provide my thoughts on books that I deem worth reading.

This time it is the Infinite Powers book by Steven Strogatz that takes the reader into a realm of taming infinity to grasp nature’s secrets. The title of the book ambiguously plays on a method of using power series to approximate curves and powers which such a method, when exercised skillfully, brought to humankind. The book artfully describes how deferential and integral calculus was developed from Archimedes efforts to measure quadrature of curves through Descartes and Fermat, culminating in calculus invented by Isaac Newton in England and Gottfried Wilhelm Leibniz in Germany working independently. Lots of examples are provided showing how calculus is essential in many of inventions that are an important part of the modern civilization, be it GPS navigation, microwave ovens or the development of effective treatments to viruses induced diseases.

The power of insight

What I liked about the book is how the author is capable of explaining mathematical concepts, that usually require a solid mathematical background, mostly using analogies and down to earth explanations. Though, what I also liked that a mathematically inclined readers were not ignored, since what could, possibly, explained using proper mathematical notation was described as such. It is difficult to appreciate the beauty of calculus without using the mathematical symbols standing for derivative (differentials) and integral, that are so familiar to many people. I should say any person who studied at school should be familiar with them, and if not, the book provides a gentle and very intuitive explanation of what derivative stands for and why it is required, the same goes for integral.

As an example of a good explanation, I want to emphasize a Pizza Proof that is used to find an area of a circle. Since I recalled the formula for it being A = pi * R2, it was very interesting to see how the Pizza Proof showed clearly that A = R*C/2, where R is a radius of a circle and C its length. I think since school time I was curious where the power of two came in the area formula that I remembered. So, using the Pizza Proof result and substituting the C in it with the known formula for the length of the circle which is C = pi * 2R (which derivation was also explained in the book ) we get A = pi * R2. It was a nice insight, first one of many that the book provided.

I also liked how the method used by Fermat to find the maximum value of a curve, using a smart approach of double intersection, provides a correct result similar to what using derivatives would give. Another example, that was also insightful showed how the concept of derivative and curve interpolation could be used to find patterns in the seasonal changes of day length compared to the rate of change of day length, which both could be approximated by a sinusoidal function with a quarter cycle shift (pi/4 phase shift).

One can’t explain math without using it

Importantly, the concept of derivative was developed and shown very clearly using proper mathematical notation which should be clear even to readers coming from non-mathematical background, since the explanations gradually and systematically build up from simple to more advanced, as a reader progresses through the book (which means that the book should be read continuously). Then the concept of integral is shown quite remarkably well and the two concepts combined to showcase the Fundamental Theorem of Calculus about the duality of derivatives and integrals.

As I always mention, this book passes the test of providing references to other resources on the subject, like original papers of Newton or The Archimedes Palimpsest, which I was unaware of before reading the book.

Summary

It could be that reading Infinite Powers would provide your with appreciation of how calculus is essential in our day to day life and understanding of the world around us. And, maybe, show why mathematics could be beautiful in its own way and also applicable and useful, which could be an unexpected revelation to some.

Other resources

The books’ stack is changing

And so it continues…

This is a quick update on the status of the stack of the books I am reading. I am glad that it’s changing and worrying that its size remains the same overtime. The issue is, as I already mentioned in other posts, good books reference other good books and here we go. This time the culprit was Mind and the Cosmic Order book by Charles Pinter that mentioned the Selfish Gene and the ‘meme’ term created by Richard Dawkins. I have to say that I’ve heard about this book a long time ago, but never thought it was worth reading. But since a number of authors respected by me mentioned it I could no longer skip reading in. I should also mention that David Deutsch mentioned the meme term, coined by Dawkins, in his The Beginning of Infinity book. So did Jeff Hawkins in his recent A Thousand Brains book.

By the way, did you know that the preface to Jeff Hawkins’ book was written by Richard Dawkins?

The books that were on the stack physically or virtually since the last time

  • Mind and the Cosmic Order by Charles Pinter (A Book of Abstract Algebra brought me here)
  • The Right Kind Of Crazy by Adam Steltzner (well, have you heard about Curiosity and Perseverance?)
  • The Interstellar Age : Inside the Forty-Year Voyager Mission by Jim Bell (courtesy of watching a documentary on YouTube)
  • A Thousand Brains: A New Theory of Intelligence by Jeff Hawkins (following him since 2004)
  • Extraterrestrial: The First Sign of Intelligent Life Beyond Earth by Avi Loeb (a Google suggestion)
  • Unknown Quantity by John Derbyshire (I read the Prime Obsession so it was a natural continuation)
  • All Things Being Equal by John Mighton (was referenced by Anders Ericsson in his book on expertise )

Currently in progress

Infinite Powers by Steven Strogatz (well his text book on dynamic systems is a culprit). With regard to Steven Strogatz I want to mention his article he wrote for the Notices of the American Mathematical Society in 2014. In this article he descried tips on successful popular-science writing.

Writing about Math for the Perplexed and the Traumatized

Next to be popped from the stack

Selfish Gene by Richard Dawkins (mentioned by David Deutsch, Jeff Hawkins, Charles Pinter and others)

Rest of the stack

  • A Book of Abstract Algebra by Charles Pinter (Unknown Quantity sparked an interested in abstract algebra in me)
  • Number-Crunching by Paul Nahin (Paul Nahin’s book about Oliver Heaviside brought me here)
  • Discreet Mathematics by Lovasz, Pelikan and Vesztergombi (bought as a used book in 2014 while roaming in US, New Hampshire)
  • Applied mathematics by J. David Logan (who can resists math?)
  • Mathematical Modeling by Mark M. Meerschaert (the same as above)
  • The Mathematical Experience by Davis Hersh ( bought as a used book in 2014 while roaming in US, New Hampshire )

Books that make you think… differently

This post isn’t about the book by Adam Steltzner, I promised to write, but on an entirely different topic altogether. And this time it’s a book that is a unique and one of a kind.

There are books that you may find interesting, there are books that are easy to read and there are books that make you think… differently. I think I’ve read only a couple of books that caused me to think deeply about the reality and perception. One of them was On Intelligence by Jeff Hawkins, the other was Sapiens: A Brief History of Humankind by Yuval Noah Harari, but the last one that was most impressive, in my opinion, was Mind and the Cosmic Order by Charles Pinter Professor of Mathematics, Emeritus at Bucknell University.

What do I mean by the books that cause you rethink a world we live in, are books that try to answer deep questions about how the brains might work, how we function as a society, how we came to be in this Universe and more. Usually, these books require mindful reading, attention and are not that easy or fun to read. I should say these books require working with them, working through them. The more you think about the content of such books, the more you play with the concepts they describe, the more you get out of them. One additional thing that such books provide a reader is references to other useful books that could help to understand information better.

Mind and the Cosmic Order is exactly such a book I described above. This book contains a number of very well thought ideas that are systematically elaborated to convey a very peculiar point of view on how what we call as a perception emerges in a mind, be it humans or animals one. The books makes use of the evolutionary approach to animals behavior that emphasizes that evolution provided animals with such a system for perception that fits the animals environment and most efficient for animals survival. Based on this assumption, the author then introduces the concept of Gestalt perception

The essence of a Gestalt is that it’s “different from the sum of its parts.”

Pinter, Charles. Mind and the Cosmic Order. Springer Nature, 2020.

which is defined as a perception of an outside world as a coherent whole, which is what suits us humans from the evolutionary point of view, and makes us active agents in this world.

One of the interesting points that I haven’t thought previously was that the mind not only don’t get sensory input directly from the senses, but it also projects the internal world-model that it creates, outside making us, essentially, feel that we see a world outside, which feels perfectly real, but being a construct of the mind. Further elaborating on this point, the author explains that our whole perception is Gestalt based and is the only way we perceive outside world.

Additional point was, that outside world has no form and structure, which are creations of the outside observer, in our case the perceiving mind. The structure and the form of objects we think are outside there are only for us to be able to function in the world efficiently. They occur only for us, but are not required by the world outside and, are not inherent in the physics of the outside world.

The main and the most interesting point in the book that builds upon previous ones is that Quantum Mechanics paradoxes, namely, The Measurement Problem, can be resolved easily if we accept previous points, that human mind is incapable of sensing outside world directly, it has a bespoke world-model that serves a mind carrier survival and that an observer could only make conclusion based on an explanation of the world-model inferred from senses. In essence, what Charles Pinter proposes is an inversion of what the Copenhagen interpretation of quantum mechanics states. This means, that there is no wave function collapse during the measurement, instead it is the perception of the observer that is changing, while the world outside remains in quantum superposition. What is interesting in this explanation that is radically different from the multiverse point of view, where an observer is split with each measurement along with the universe where this measurements happen.

All in all, Mind and the Cosmic order provides you with many ideas to think about, It’s an interesting, but not an easy read. It requires your attention, it requires you to think and work through what is explained. In addition, the author mentions great many books and papers that worth reading to get a wider picture of what is mind, how it emerges and creates a world that we are so used to think of as being the only real one possible.

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These are the voyages of Voyager 1 and 2

Space, the final frontier. These are the voyages of the Starship Enterprise...

Probably, most of you can recall these opening lines from the Star Trek sci-fi franchise. But in this post I want to write about real space voyagers which are Voyager 1 and Voyager 2 that where launched in 1977.

Not long ago friends visited us at home and while having a lovely conversation we talked about space exploration. Eventually, we mentioned the Voyager 1 and 2 robotic spacecrafts. When our friends left I was curious to check what YouTube had with regard to interesting documentaries about Voyagers space mission. As you might guess, there were a lot on the YouTube, but one particular documentary called The Farthest produced by HHMI Tangled Bank Studios I liked the most. This documentary has a reasonable coverage of Voyagers mission from its inception until present and it features interviews with key scientists who were responsible for making Voyagers a successful endeavor.

As usually happens to me, while watching this documentary I’ve noticed that one of the people who was interviewed for this documentary was Jim Bell, a professor of Astronomy who wrote a book called The Interstellar Age : Inside the Forty-Year Voyager Mission which was the basis for the documentary. Again, as usual, I bought that same book and, boy, wasn’t I disappointed. The book was also very interesting and it covered robotic space program including early Mariner missions, but mostly focused on Voyager 1 and Voyager 2 journeys beginning in 1977 and until these days when they plow interstellar space continuing ad infinitum and, some may say, beyond. The books mentions interesting discoveries that where made by each Voyager in their encounters with giant planets following on a Grand Tour of Planets. It also covers what will be the fait of the Voyagers when we as a civilization will be long gone. I particularly, liked when Jim Bell described how the Golden Record was prepared to be carried by the Voyagers and also it was interesting to know that some moons of Jupiter and Saturn could be potential harbors of life besides Earth.

All in all, I recommend you to check out the documentary and if you like it you also may consider reading the book. The next book that I have in queue is The Right Kind Of Crazy by Adam Steltzner, the chief engineer of Curiosity and Perseverance Mars rovers, but this and more in my next post.

Rendezvous with Rama… sorry ʻOumuamua interstellar object. Are We Alone?

It all started when I, as usual, was skimming through the Google digest on a mobile phone before I went to sleep. There was this article about Avi Loeb a theoretical physicist and Professor of Science at Harvard University who hypothesized that an interstellar object that zipped through the Solar System in December 2017 could have been an artifact or a spaceship from an alien civilization.

This is how I ended up reading unexpectedly Extraterrestrial: The First Sign of Intelligent Life Beyond Earth book written by Avi Loeb about possible origins of ʻOumuamua. The book itself is not that long and it has a mix of factual information about interstellar object ʻOumuamua and Avi’s thoughts on Scientific Method, philosophy and his childhood in Israel. The book felt too repetitive at times and could have been much shorter. It also could have had more than Drake equation. In a number of places it could have benefited from using math notations instead of describing numbers in words.

What I liked

Avi’s thoughts on science where he suggested that science should be preoccupied with practical theories that try to explain existing evidence sounds similar to the thoughts expressed in Sabine Hossenfelder’s Lost in Math book. But this view is not shared by David Deutsch which he expressed in The Beginning of Infinity book and with which I tend to agree more. David Deutsch supports

Karl Popper’s epistemology, especially its anti-inductivism and requiring a realist (non-instrumental) interpretation of scientific theories. (a quote from Wikipedia)

The hypothesis that ʻOumuamua could be an alien interstellar visitor, based on its anomalies is an interesting one. Since Avi was participating in the Breakthrough Starshot project, which proposed to use a solar sail to travel to nearest star, he suggested that ʻOumuamua could have been similar object. Then he described how an advanced civilization could surround a star, about to explode, with such solar sails that would be blasted away in all directions, serving as a probes, into space. The only issue with this approach is that somehow it assumes that an advanced civilization, that is capable of surrounding a star with millions of solar sails, would use such technology, can’t we assume that such a civilization could have discovered laws of physics that we cannot imagine yet and could use other means to traverse space.

An advanced ancient civilization here on Earth

There are ample evidence, here on Earth, that shows that in ancient Egypt, and in South America (namely Machu Picchu and other locations) we find hundreds of granite blocks that have marks of being cut by machining tools, like large diameter disk saws and wire saws, that required a power supply and an infrastructure similar to what we have in the factories that use CNC machines to produce granite blocks. The 7 great pyramids and Osirion in Egypt are the best examples of the machining tools applied to granite and limestone blocks.
More details can be found in a good documentary produced by the Laboratory of Alternative History (LAH)

The Film “Mysteries Of Ancient Egypt (2005)

The filmmakers decided not to rely on a particular theory, but on real facts, logic and common sense. This approach leads inexorably to the conclusion that in the land of Egypt for thousands of years before the first pharaohs a highly developed civilization existed that was superior in their knowledge and technology not only primitive society of the ancient Egyptians, but modern humanity.

Additionally, in Israel (Western Wall Tunnel) and in Lebanon (Temples of Baalbek) there are building blocks of some 100 tons that cannot be extracted and moved easily taking into consideration what we know about the level of human technology in ancient times.

Here on Earth and there in space

So not only we should start looking more carefully at what is in the space, we have the evidence of an advanced ancient civilization (interstellar or not) here on Earth.

Why random reading could be useful

Random thoughts on a reading process

I have to confess I am an obsessive reader. I like books, I like to read them a lot, I like to read them daily. It seems like the most efficient way of reading books or doing other tasks is doing it in a sequential way, where each book completed before the next one is read. The issue is that I cannot help, but defy this approach. I can read a number of books in parallel, jumping from one to another and returning back again. I also can be distracted by a reference to a different book, and so it goes.

Now, we may ask is there any point is such haphazard reading, where the focus is constantly lost, things and thoughts are getting mixed? Personally, I do not find this confusing or disorganizing, but actually, I see some merit in this approach. First, you do not get bored and have some fresh point of view when you return to a book (if you remember where you’ve left last time). Second, since good books are just like candies, it’s difficult to decide where to start, what have next and when to finish.

Stack overflow of the books

Having described my non-linear approach to reading I should mention that nevertheless, on average, I usually able to read 1.5 books a month. This is nice, but there are a couple of books that are still in the stack and they tend to overflow it. There is this book What’s Math Got Do With It by Jo Boaler, then there is The Fabric Of Reality by David Deutsch, underneath is the Unknown Quantity by John Derbyshire. Further below is All Things Being Equal by John Mighton, traveling by US post is a Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction by Paul Nahin, and last but not least is infinite powers by Steven Strogatz.

Speaking of Steven Strogatz book. Last year I was looking for a good book on applied mathematics and stumbled upon Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering which is very good. Then I checked other books Steven wrote and found that in addition to textbooks he also wrote a number of popular science books. One of them was infinite powers. It is enough to read just a few pages of this book to understand that it’s a pure gem in the world of popular science books and if you’d like to really get a good understanding of what differentiation and integration is without going directly to a calculus course then this book may be of interest to you. When I finish reading it, I’ll write a more extensive review of the book.

Random walking

For now, keep reading and try reading sequentially, otherwise start a random walk. Who knows what you stumble upon and where it take you.

Good books come in tuples

This post continues a number of post were I kind of reviewed books that I read and thought it would be helpful to share them with other readers. All Things Being Equal: Why Math Is the Key to a Better World by John Mighton is such a book that deserves to be shared and read by people who care about math education, their children’s math education and math in general. The title of this post is not a mere gimmick, but it means that good books always mention or reference other authors or books that worth reading. This is what exactly happened when I read the book by Anders Ericsson that I mentioned in the previous post. In the Peak Ericsson mentioned John Mighton a Canadian mathematician that incorporated elements of deliberate practice with clear goals and problems that had increasing level of difficulty to teach math to children. This approach is now known as JUMP Math and it is taught to thousands of kids helping them master mathematics while enjoying the subject, unlike in the usual way math is taught in schools.

What is so interesting about this book?

I am past 1/3 of the book, and so far I wasn’t disappointed. The book itself is not only about teaching math to kids. John Mighton discussed also psychological approaches, such as a research into Expertise that plays important role in education in general and in math in particular. He also provides us with an interesting observation that usual math education results in the same distribution of grades among pupils of public schools and among pupils in private schools. It worth mentioning, intellectual poverty a term he coined to emphasize that even though there is a research in to expertise that resulted in clear guidance on how to effectively approach teaching, we as a society still do not incorporate this approach, and what we get is a suboptimal outcome, where kids dislike math, since they think they are not good at it, they have no innate ability or inclination towards it.

Apart from this, John Mighton incorporates a number of examples from math lessons at schools, where he shed light on some of the arithmetic operations that are usually taught as a mere algorithms, without explaining how they work and why. For example, he provides a neat explanation why one could substitute a division of a number by a fraction, by a multiplication of the inverse of that fraction.

Overall

The book is worth reading, since it provides a fresh approach to teaching math to kids and adults alike, in an engaging and exciting way, where kids are gently guided by discovering math step by step, building on the knowledge they gain at a previous step, facing gradually increasing challenges along the way.

Critical Thinking and Scientific Approach

Critical Thinking is more important than ever before

Due to the January 6th, 2021 events in the USA where the was an attempt to overthrow the democratic process I decided that there is an urgent need to take a hard stand and defend democratic values. To this end this blog from now on will be focused on promoting Critical and Skeptical Thinking and Scientific Approach for people to use in their lives. People who do not want to use Critical Thinking are easily manipulated and can be used by others to achieve their goals.

To start on the right foot I advise people who read this post to check The Deamon-Haunted World a book by Carl Sagan from 1995 about Critical Thinking and Scientific approach.

There are other books on Critical Thinking and Scientific Approach in general that I will recommend and start reviewing and discussing in future posts.

Remember

Equipped with critical thinking the probability of someone taking advantage of you by manipulating you becomes very low.

Unknown Quantity is a math book to work through

This post is similar to my other posts on books I read or am in the process of reading. This time it is second book by John Derbyshire I read on mathematics. The previous book Prime Obsession was an inspiring, interesting and a pleasure to read, since it was all about the Riemann hypothesis. It took me though a little effort to not only read it, but also work through author’s explanations.

So this second book is called Unknown Quantity and it is as captivating as the Prime Obsession was. What is different about the Unknown Quantity that it has more of a historical context on how algebra developed from ancient Mesopotamia to our days.

What I like the most about how John explains mathematical topics in his books is the way he is capable of explaining mathematics the way I never experienced in a school or later in a college. Most of the time math was taught as a given, without trying to convey the essence of the subject, why this formula such and such, how it was conceived and developed. In my opinion, these are very important questions, if not the most important in mathematics. Questioning and curiosity are crucial in mathematical research.

For example, in the Unknown Quantity John shows with enough details how general solutions to second, third and forth degree equations were developed. Why determinant is useful in solving systems of linear equations and why it is important in matrices. These are only some examples, since I haven’t yet finished reading the book.

In short, if you are curious about algebra, and want to know how it evolved historically, and also get some new insight about math you were taught, but never really understood, then the Unknown Quantity is the book for you.

An exceptional book. A quick update

This post is a super quick and short. From time to time I stumble upon an exceptional subject to study or an insightful book. Peak: Secrets from the New Science of Expertise by Anders Ericsson and Robert Pool is just like that. This book is exceptional in its usefulness and practical advise that can help you become an expert or at least progress in that direction.

The book has a solid foundation in psychology of expertise and is based on the research results achieved in that field. A more detailed information about psychology of expertise can be found in The Cambridge Handbook of Expertise and Expert Performance.

I’ll write a more detailed post as I’ll near the end of the book.