Chris Aldrich | Chris Aldrich

Popular Science Books on Information Theory, Biology, and Complexity

Previously, I had made a large and somewhat random list of books which lie in the intersection of the application of information theory, physics, and engineering practice to the area of biology. Below I’ll begin to do a somewhat better job of providing a finer gradation of technical level for both the hobbyist or the aspiring student who wishes to bring themselves to a higher level of understanding of these areas. In future posts, I’ll try to begin classifying other texts into graduated strata as well. The final list will be maintained here: Books at the Intersection of Information Theory and Biology.

Introductory / General Readership / Popular Science Books

These books are written on a generally non-technical level and give a broad overview of their topics with occasional forays into interesting or intriguing subtopics. They include little, if any, mathematical equations or conceptualization. Typically, any high school student should be able to read, follow, and understand the broad concepts behind these books. Though often non-technical, these texts can give some useful insight into the topics at hand, even for the most advanced researchers.

Complexity: A Guided Tour by Melanie Mitchell (review)

Possibly one of the best places to start, this text gives a great overview of most of the major areas of study related to these fields.

Entropy Demystified: The Second Law Reduced to Plain Common Sense by Arieh Ben-Naim

One of the best books on the concept of entropy out there. It can be read even by middle school students with no exposure to algebra and does a fantastic job of laying out the conceptualization of how entropy underlies large areas of the broader subject. Even those with Ph.D.’s in statistical thermodynamics can gain something useful from this lovely volume.

The Information: A History, a Theory, a Flood by James Gleick (review)

A relatively recent popular science volume covering various conceptualizations of what information is and how it’s been dealt with in science and engineering. Though it has its flaws, its certainly a good introduction to the beginner, particularly with regard to history.

The Origin of Species by Charles Darwin

One of the most influential pieces of writing known to man, this classical text is the basis from which major strides in biology have been made as a result. A must read for everyone on the planet.

Information, Entropy, Life and the Universe: What We Know and What We Do Not Know by Arieh Ben-Naim

Information Theory and Evolution by John Avery

The Touchstone of Life: Molecular Information, Cell Communication, and the Foundations of Life by Werner R. Loewenstein (review)

Information Theory, Evolution, and the Origin of Life by Hubert P. Yockey

The four books above have a significant amount of overlap. Though one could read all of them, I recommend that those pressed for time choose Ben-Naim first. As I write this I’ll note that Ben-Naim’s book is scheduled for release on May 30, 2015, but he’s been kind enough to allow me to read an advance copy while it was in process; it gets my highest recommendation in its class. Loewenstein covers a bit more than Avery who also has a more basic presentation. Most who continue with the subject will later come across Yockey’s Information Theory and Molecular Biology which is similar to his text here but written at a slightly higher level of sophistication. Those who finish at this level of sophistication might want to try Yockey third instead.

The Red Queen: Sex and the Evolution of Human Nature by Matt Ridley

Grammatical Man: Information, Entropy, Language, and Life by Jeremy Campbell

Life’s Ratchet: How Molecular Machines Extract Order from Chaos by Peter M. Hoffmann

Complexity: The Emerging Science at the Edge of Order and Chaos by M. Mitchell Waldrop

The Big Picture: On the Origins of Life, Meaning, and the Universe Itself (Dutton, May 10, 2016)

In the coming weeks/months, I’ll try to continue putting recommended books on the remainder of the rest of the spectrum, the balance of which follows in outline form below. As always, I welcome suggestions and recommendations based on others’ experiences as well. If you’d like to suggest additional resources in any of the sections below, please do so via our suggestion box. For those interested in additional resources, please take a look at the ITBio Resources page which includes information about related research groups; references and journal articles; academic, research institutes, societies, groups, and organizations; and conferences, workshops, and symposia.

Lower Level Undergraduate

These books are written at a level that can be grasped and understood by most with a freshmen or sophomore university level. Coursework in math, science, and engineering will usually presume knowledge of calculus, basic probability theory, introductory physics, chemistry, and basic biology.

Upper Level Undergraduate

These books are written at a level that can be grasped and understood by those at a junior or senor university level. Coursework in math, science, and engineering may presume knowledge of probability theory, differential equations, linear algebra, complex analysis, abstract algebra, signal processing, organic chemistry, molecular biology, evolutionary theory, thermodynamics, advanced physics, and basic information theory.

Graduate Level

These books are written at a level that can be grasped and understood by most working at the level of a master’s level at most universities. Coursework presumes all the previously mentioned classes, though may require a higher level of sub-specialization in one or more areas of mathematics, physics, biology, or engineering practice. Because of the depth and breadth of disciplines covered here, many may feel the need to delve into areas outside of their particular specialization.

Nicolas Perony: Puppies! Now that I’ve got your attention, complexity theory | TED

For those who are looking for a good, simple, and entertaining explanation of the concept of emergent properties and behavior within complexity theory (or Big History), I just came across a nice TED talk that simplifies complexity using a few animal examples including a cute puppy video as well as a bat and a meerkat example. The latter two also have implications for evolution and survival which are lovely examples as well.

[ted id=1916]

Schools of Thought in the Hard and Soft Sciences

A recent post in one of the blogs at Discover Magazine the other day had me thinking about the shape of science over time.

Neuroscientists don’t seem to disagree on the big issues. Why are there no big ideas in neuroscience?

Neuroskeptic, Where Are The Big Ideas in Neuroscience? (Part 1)

The article made me wonder about the divide between the ‘soft’ and ‘hard’ sciences, and how we might better define and delineate them. Perhaps in a particular field, the greater the proliferation of “schools of though,” the more likely something is to be a soft science? (Or mathematically speaking, there’s an inverse relationship in a field between how well supported it is and the number of schools of thought it has.) I consider a school of thought to be a hypothetical/theoretical proposed structure meant to potentially help advance the state of the art and adherents join one of many varying camps while evidence is built up (or not) until one side carries the day.

Firmness of Science vs. # of Schools of Thought — Simple linear approximation of the relationship, though honestly something more similar to y=1/x which is asymptotic to the x and y axes is far more realistic.

Theorem: The greater the proliferation of “schools of though,” the more likely something is to be a soft science.

Generally in most of the hard sciences like physics, biology, or microbiology, there don’t seem to be any opposing or differing schools of thought. While in areas like psychology or philosophy they abound, and often have long-running debates between schools without any hard data or evidence to truly allow one school to win out over another. Perhaps as the structure of a particular science becomes more sound, the concept of schools of thought become more difficult to establish?

For some of the hard sciences, it would seem that schools of thought only exist at the bleeding edge of the state-of-the-art where there isn’t yet enough evidence to swing the field one way or another to firmer ground.

Example: Evolutionary Biology

We might consider the area of evolutionary biology in which definitive evidence in the fossil record is difficult to come by, so there’s room for the opposing thoughts for gradualism versus punctuated equilibrium to be individual schools. Outside of this, most of evolutionary theory is so firmly grounded that there aren’t other schools.

Example: Theoretical Physics

The relatively new field of string theory might be considered a school of thought, though there don’t seem to be a lot of other opposing schools at the moment. If it does, such a school surely exists, in part, because there isn’t the ability to validate it with predictions and current data. However, because of the strong mathematical supporting structure, I’ve yet to hear anyone use the concept of school of thought to describe string theory, which sits in a community which seems to believe its a foregone conclusion that it or something very close to it represents reality. (Though for counterpoint, see Lee Smolin’s The Trouble with Physics.)

Example: Mathematics

To my knowledge, I can’t recall the concept of school of thought ever being applied to mathematics except in the case of the Pythagorean School which historically is considered to have been almost as much a religion as a science. Because of its theoretical footings, I suppose there may never be competing schools, for even in the case of problems like P vs. NP, individuals may have some gut reaction to which way things are leaning, everyone ultimately knows it’s going to be one or the other ( $P=NP$ or $P \neq NP$ ). Many mathematicians also know that it’s useful to try to prove a theorem during the day and then try to disprove it (or find a counterexample) by night, so even internally and individually they’re self-segregating against creating schools of thought right from the start.

Example: Religion

Looking at the furthest end of the other side of the spectrum, because there is no verifiable way to prove that God exists, there has been an efflorescence of religions of nearly every size and shape since the beginning of humankind. Might we then presume that this is the softest of the ‘sciences’?

What examples or counter examples can you think of?

String Theory, Black Holes, and Information

Four decades ago, Stephen Hawking posed the black hole information paradox about black holes and quantum theory. It still challenges the imaginations of theoretical physicists today. Yesterday, Amanda Peet (University of Toronto) presented the a lecture entitled “String Theory Legos for Black Holes” yesterday at the Perimeter Institute for Theoretical Physics. A quick overview/teaser trailer for the lecture follows along with some additional information and the video of the lecture itself.

The “Information Paradox” with Amanda Peet (teaser trailer)

“Black holes are the ‘thought experiment’ par excellence, where the big three of physics – quantum mechanics, general relativity and thermodynamics – meet and fight it out, dragging in brash newcomers such as information theory and strings for support. Though a unification of gravity and quantum field theory still evades string theorists, many of the mathematical tools and ideas they have developed find applications elsewhere.

One of the most promising approaches to resolving the “information paradox” (the notion that nothing, not even information itself, survives beyond a black hole’s point-of-no-return event horizon) is string theory, a part of modern physics that has wiggled its way into the popular consciousness.

On May 6, 2015, Dr. Amanda Peet, a physicist at the University of Toronto, will describe how the string toolbox allows study of the extreme physics of black holes in new and fruitful ways. Dr. Peet will unpack that toolbox to reveal the versatility of strings and (mem)branes, and will explore the intriguing notion that the world may be a hologram.

Amanda Peet is an Associate Professor of Physics at the University of Toronto. She grew up in the South Pacific island nation of Aotearoa/New Zealand, and earned a B.Sc.(Hons) from the University of Canterbury in NZ and a Ph.D. from Stanford University in the USA. Her awards include a Radcliffe Fellowship from Harvard and an Alfred P. Sloan Foundation Research Fellowship. She was one of the string theorists interviewed in the three-part NOVA PBS TV documentary “Elegant Universe”.

Web site: http://ap.io/home/.

Dr. Amanda Peet’s Lecture “String Theory Legos for Black Holes”

Richard Dawkins Interview: This Is My Vision Of “Life” | Edge.org

Bookmarked This Is My Vision Of "Life" by John Brockman (edge.org)

The Edge.org's interview with Richard Dawkins.

From “A Conversation With Richard Dawkins“ [4.30.15]

“My vision of life is that everything extends from replicators, which are in practice DNA molecules on this planet. The replicators reach out into the world to influence their own probability of being passed on. Mostly they don’t reach further than the individual body in which they sit, but that’s a matter of practice, not a matter of principle. The individual organism can be defined as that set of phenotypic products which have a single route of exit of the genes into the future. That’s not true of the cuckoo/reed warbler case, but it is true of ordinary animal bodies. So the organism, the individual organism, is a deeply salient unit. It’s a unit of selection in the sense that I call a “vehicle”. There are two kinds of unit of selection. The difference is a semantic one. They’re both units of selection, but one is the replicator, and what it does is get itself copied. So more and more copies of itself go into the world. The other kind of unit is the vehicle. It doesn’t get itself copied. What it does is work to copy the replicators which have come down to it through the generations, and which it’s going to pass on to future generations. So we have this individual replicator dichotomy. They’re both units of selection, but in different senses. It’s important to understand that they are different senses.”

Richard Dawkins

RICHARD DAWKINS is an evolutionary biologist; Emeritus Charles Simonyi Professor of the Public Understanding of Science, Oxford; Author, The Selfish Gene; The Extended Phenotype; Climbing Mount Improbable; The God Delusion; An Appetite For Wonder; and (forthcoming) A Brief Candle In The Dark.

Watch the entire video interview and read the transcript at Edge.org.

Category Theory – the bedrock of mathematics? via Ilyas Khan | LinkedIn

Bookmarked Category Theory - the bedrock of mathematics ? by

Ilyas Khan (LinkedIn Pulse)

Category theory looks set to become the dominant foundational basis for all mathematics. It could, in fact, already have achieved that status through stealth.

Beauty, even in Maths, can exist in the eye of the beholder. That might sound a little surprising, when, after all, what could be more objective than mathematics when thinking about truth, and what, therefore, could be more natural than for beauty and goodness, the twin accomplices to truth, to be co-joined ?

In the 70 odd years since Samuel Eilenberg and Saunders Mac Lane published their now infamous paper “A General Theory of Natural Equivalences“, the pursuit of maths by professionals (I use here the reference point definition of Michael Harris – see his recent publication “Mathematics without Apologies“) has become ever more specialised. I, for one, don’t doubt cross disciplinary excellence is alive and sometimes robustly so, but the industrially specialised silos that now create, produce and then sustain academic tenure are formidable within the community of mathematicians.

Beauty, in the purest sense, does not need to be captured in a definition but recognised through intuition. Whether we take our inspiration from Hardy or Dirac, or whether we experience a gorgeous thrill when encountering an austere proof that may have been confronted thousands of times before, the confluence of simplicity and beauty in maths may well be one of the few remaining places where the commonality of the “eye” across a spectrum of different beholders remains at its strongest.

Neither Eilenberg nor Mac Lane could have thought that Category theory, which was their attempt to link topology and algebra, would become so pervasive or so foundational in its influence when they completed and submitted their paper in those dark days of WW 2. But then neither could Cantor, have dreamt about his work on Set theory being adopted as the central pillar of “modern” mathematics so soon after his death. Under attack from establishment figures such as Kronecker during his lifetime, Cantor would not have believed that set theory would become the central edifice around which so much would be constructed.

Of course that is exactly what has happened. Set theory and the ascending magnitude of infinities that were unleashed through the crack in the door that was represented by Cantor’s diagonal conquered all before them.

Until now, that is.

In an article in Science News, Julie Rehmeyer describes Category Theory as “perhaps the most abstract area of all mathematics” and “where math is the abstraction of the real world, category theory is an abstraction of mathematics”.

Slowly, without fanfare, and with an alliance built with the emergent post transistor age discipline of computer science, Category theory looks set to become the dominant foundational basis for all mathematics. It could, in fact, already have achieved that status through stealth. After all, if sets are merely an example of a category, they become suborned without question or query. One might even use the description ‘subsumed’.

There is, in parallel, a wide ranging discussion in mathematics about the so called Univalent Foundation that is most widely associated with Voevodsky which is not the same. The text book produced for the year long univalence programme iniated at the IAS that was completed in 2013 Homotopy type theory – Univalent Foundations Programme states:

“The univalence ax-iom implies, in particular, that isomorphic structures can be identified, a principle that mathematicians have been happily using on workdays, despite its incompatibility with the “official”doctrines of conventional foundations..”

before going on to present the revelatory exposition that Univalent Foundations are the real unifying binding agent around mathematics.

I prefer to think of Voevodsky’s agenda as being narrower in many crucial respects than Category Theory, although both owe a huge amount to the over-arching reach of computational advances made through the mechanical aid proffered through the development of computers, particularly if one shares Voevodsky’s view that proofs will eventually have to be subject to mechanical confirmation.

In contrast, the journey, post Russell, for type theory based clarificatory approaches to formal logic continues in various ways, but Category theory brings a unifying effort to the whole of mathematics that had to wait almost two decades after Eilenberg and Mac Lane’s paper when a then virtually unknown mathematician, William Lawvere published his now much vaunted “An Elementary Theory of the Category of Sets” in 1964. This paper, and the revolutionary work of Grothendieck (see below) brought about a depth and breadth of work which created the environment from which Category Theory emerged through the subsequent decades until the early 2000’s.

Lawvere’s work has, at times, been seen as an attempt to simply re-work set theory in Category theoretic terms. This limitation is no longer prevalent, indeed the most recent biographical reviews of Grothendieck, following his death, assume that the unificatory expedient that is the essential feature of Category theory (and I should say here not just ETCS) is taken for granted, axiomatic, even. Grothendieck eventually went much further than defining Category theory in set theoretic terms, with both Algebraic Topology and Mathematical Physics being fields that now could not be approached without a foundational setting that is Category theory. The early language and notation of Category Theory where categories ‘C’ are described essentially as sets whose members satisfy the conditions of composition, morphism and identity eventually gave way post Lawvere and then Lambek to a systematic adoption of the approach we now see where any and all deductive systems can be turned into categories. Most standard histories give due credit to Eilenberg and Mac Lane as well as Lawvere (and sometimes Cartan), but it is Grothendieck’s ‘Sur quelques points d’algebre homologique’ in 1957 that is now seen as the real ground breaker.

My own pathway to Category theory has been via my interest in Lie Groups, and more broadly, in Quantum Computing, and it was only by accident (the best things really are those that come about by accident !) that I decided I had better learn the language of Category theory when I found Lawvere’s paper misleadingly familiar but annoyingly distant when, in common with most people, I assumed that my working knowledge of notation in logic and in set theory would map smoothly across to Category theory. That, of course, is not the case, and it was only after I gained some grounding in this new language that I realised just how and why Category theory has an impact far beyond computer science. It is this journey that also brings me face to face with a growing appreciation of the natural intersection between Category theory and a Wittgensteinian approach to the Philosophy of Mathematics. Wittgenstein’s disdain for Cantor is well documented (this short note is not an attempt to justify, using Category theory, a Wittgensteinian criticism of set theory). More specifically however, it was Abramsky and Coecke’s “Categorical Quantum Mechanics” that helped me to discern more carefully the links between Category Theory and Quantum Computing. They describe Category Theory as the ‘language of modern structural mathematics’ and use it as the tool for building a mathematical representation of quantum processes, and their paper is a thought provoking nudge in the ribs for anyone who is trying to make sense of the current noise that surrounds Quantum mechanics.

Awodey and Spivak are the two most impressive contemporary mathematicians currently working on Category Theory in my view, and whilst it is asking for trouble to choose one or two selected works as exemplars of their approach, I would have to say that Spivak’s book on Category Theory for the Sciences is the standout work of recent times (incidentally the section in this book on ‘aspects’ bears close scrutiny with Wittgenstein’s well known work on ‘family resemblances’).

Awodey’s 2003 paper is as good a recent balance between a mathematical and philosophical exposition of the importance of category theory as exists whilst his textbook is often referred to as the standard entry point for working mathematicians.

Going back to beauty, which is how I started this short note. Barry Mazur wrote an article in memory of Saunders Mac Lane titled ‘When is one thing equal to another‘ which is a gem of rare beauty, and the actual catalyst for this short note. If you read only one document in the links from this article, then I hope it is Mazur’s paper.

Don’t get the impression that I actually read more than a few pages

In the second place, the notes are there to convince the reader that I didn’t make things up. But please don’t get the impression that I actually read more than a few pages of most of the references quoted.

The notes are also a convenient hiding place for the author’s true opinions. But what do they matter?

Michael Harris, number theorist,
on why his book Mathematics Without Apologies has so many footnotes.

Mathematics Without Apologies by Michael Harris

Visiting ancestors…

Instagram filter used: Normal

Photo taken at: Natural History Museum of Los Angeles County

An Asterisk Does Not Denote a Hard Problem†

It should be noted that an asterisk does not denote a hard problem: we have reserved the dagger for this.

Richard Millman & George Parker in Elements of Differential Geometry

Book Review of Dominic O’Brien’s “Quantum Memory Power”

Quantum Memory Power by Dominic O'Brien — Quantum Memory Power by Dominic O’Brien

I’ve read many of the biggest memory related books over the past three decades and certainly have my favorites among them. I’ve long heard that Dominic O’Brien’s Quantum Memory Power: Learn to Improve Your Memory with the World Memory Champion! audiobook was fairly good, and decided that I’d finally take a peek having known for a while about O’Brien and his eponymous Dominic System.

General Methods

Overall, I was fairly impressed with his layout and positive teaching style, though I don’t particularly need some of the treacly motivation that he provided and which is primarily aimed at the complete novice. While I appreciate that for some, hearing this material may be the most beneficial, I would have preferred to have some of it presented visually. In general, I wouldn’t recommend this as a something to listen to on a commute as he frequently admonishes against doing some of the exercises he outlines while driving or operating heavy machinery.

Given the prevalence of and growth of memory systems from the mid-20th century onwards, I personally find it difficult to believe all of his personal story about “rediscovering” many of the memory methods he outlines, or at least to the extent to which he tempts the reader to believe.

Differences from Other Systems

Based on past experience, I really appreciate his methods for better remembering names with faces as his conceptualizations for doing this seemed better to me than the methods outlined by Bruno Furst. I do however, much prefer the major mnemonic system’s method for numbers over the Dominic system for it’s more logical and complete conversion of consonant sounds for most languages. The links between the letters and numbers in the major system are also much easier to remember and don’t require as much work to remember them. I also appreciate the major system for its deeper historical roots as well as for its precise overlap with the Gregg Shorthand method. The poorer structure of the Dominic system is the only evidence I can find to indicate that he seems to have separately re-discovered some of his memory methods.

I appreciated that most of his focus was on practical tasks like to do lists, personal appointments, names and faces, but wish he’d spent some additional time walking through general knowledge examples like he did for the list of the world’s oceans and seas.

While I appreciated his outlining the ability to calculate what day of the week any particular date falls on (something that most memory books don’t touch upon), he failed to completely specify the entire method. He also used a somewhat non-standard method for coding both the days of the week and the months of the year, though mathematically all of these systems are equivalent. I did appreciate his trying to encode a set up for individual years, which will certainly help many cut down on the mental mathematics, particularly as it relates to the dread many have for long division. Unfortunately, he didn’t go far enough and this is where he also failed to finish supplying the full details for all of the special cases for the years. He also failed to mention the discontinuities with the Gregorian versus the Julian calendar making his method more historically universal. For those interested, Wikipedia outlines some of the more familiar mathematical methods for determining the day of the week that a particular date would fall on.

Instead of having spent the time outlining the calendar, which is inherently difficult to do in audio format compared to printed format, he may have been better off having spent the time going into more depth memorizing poetry or prose as an extension of his small aside on memorizing quotes and presenting speeches.

I could have done without the bulk of the final disk which comprised mostly of tests for the material previously presented. The complete beginner may get more out of these exercises however. The final portion of the disk was more interesting as he did provide some philosophy on how memory systems engage both lobes of the brain within the right-brained/left-brained conceptualizations from neuropsychology.

While O’Brien doesn’t completely draw out his entire system, to many this may be a strong benefit as it forces individuals to create their own system within his framework. This is bound to help many to create stronger personalized links between their numbers and their images. The drawback the beginner may find for this is that they may find themselves ever tinkering with their own customized system, or even more likely rebuilding things from scratch when they discover the list of online resources from others that rely on people having a more standardized system.

O’Brien also provides more emphasis on creativity and visualization than some books, which will be very beneficial to many beginners.

Overall, while I’d generally recommend this to the average mnemonist, I’d recommend they approach it after having delved in a bit and learned the major system from somewhere else.

NIMBioS Workshop: Information Theory and Entropy in Biological Systems

Over the next few days, I’ll be maintaining a Storify story covering information related to and coming out of the Information Theory and Entropy Workshop being sponsored by NIMBios at the Unviersity of Tennessee, Knoxville.

For those in attendance or participating by watching the live streaming video (or even watching the video after-the-fact), please feel free to use the official hashtag #entropyWS, and I’ll do my best to include your tweets, posts, and material into the story stream for future reference.

For journal articles and papers mentioned in/at the workshop, I encourage everyone to join the Mendeley.com group ITBio: Information Theory, Microbiology, Evolution, and Complexity and add them to the group’s list of papers. Think of it as a collaborative online journal club of sorts.

Those participating in the workshop are also encouraged to take a look at a growing collection of researchers and materials I maintain here. If you have materials or resources you’d like to contribute to the list, please send me an email or include them via the suggestions/submission form or include them in the comments section below.

Resources for Information Theory and Biology

RSS Feed for BoffoSocko posts tagged with #ITBio

Machiavelli in Hollywood | Gavin Polone’s ‘Textbook’ on the Entertainment Industry

Dearth of (Great) Textbooks on The Entertainment Business

In having previously taught several classes on the business of the entertainment industry, I was never quite able to pick out even a mediocre textbook for such a class. There are a handful that will give one an overview of the nuts and bolts and one or two that will provide some generally useful numbers (see the syllabi from those classes), but none comes close to providing the philosophy of how the business works in a short period of time.

A Short Term Solution

To remedy this problem, I was always a fan of producer and ex-agent Gavin Polone, who had a series of articles in New York Magazine/Vulture. I’ve recently gone through and linked to all of the forty-four articles, in chronological order, he produced in that series from 9/21/11 to 5/7/14.

I’ve aggregated the series via Readlists.com, so one can click on each of the articles individually. Better yet, for students and teachers alike, one can click on the “export” link and very easily download them all in most ebook formats (including Kindle, iPad, etc.) for your reading/studying convenience.

My hope is that for others, they may create an excellent starter textbook on how the entertainment business works and, more importantly: how successful people in the business think. For those who need more, Gavin is also an occasional contributor to the Hollywood Reporter. (And, as a note for those not trained in the classics and prone to modern-day stereotypes, I’ll make the caveat that I use the title “Machiavelli” above with the utmost reverence and honor.)

I’m still slowly, but surely making progress on my own all-encompassing textbook, but, until then, I hope others find this series of articles as interesting and useful as I have.

Gavin Polone is an agent turned manager turned producer. His production company, Pariah, has brought you such movies and TV shows as Panic Room, Zombieland, Gilmore Girls, and Curb Your Enthusiasm. Follow him on Twitter @gavinpolone.

👓 How our 1,000-year-old math curriculum cheats America’s kids | LA Times

Read How our 1,000-year-old math curriculum cheats America's kids – LA Times by Edward Frenkel (Los Angeles Times)

Imagine you had to take an art class in which you were taught how to paint a fence or a wall, but you were never shown the paintings of the great masters, and you weren't even told that such paintings existed. Pretty soon you'd be asking, why study art? That's absurd, of course, but it's surprisingly close to the way we teach children mathematics. In elementary and middle school and even into high school, we hide math's great masterpieces from students' view. The arithmetic, algebraic equations and geometric proofs we do teach are important, but they are to mathematics what whitewashing a fence is to Picasso — so reductive it's almost a lie. Most of us never get to see the real mathematics because our current math curriculum is more than 1,000 years old. For example, the formula for solutions of quadratic equations was in al-Khwarizmi's book published in 830, and Euclid laid the foundations of Euclidean geometry around 300 BC. If the same time warp were true in physics or biology, we wouldn't know about the solar system, the atom and DNA. This creates an extraordinary educational gap for our kids, schools and society.

An interesting train of thought to be sure. I should post in response to this, or at least think about how it could be structured. I definitely want to come back to write more about this topic.