## Schools of Thought in the Hard and Soft Sciences

A framework for determining the difference between 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.

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.

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?

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## String Theory, Black Holes, and Information

Amanda Peet presented the a lecture entitled "String Theory Legos for Black Holes" at the Perimeter Institute for Theoretical Physics.

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”

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

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.

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

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## NIMBioS Workshop: Information Theory and Entropy in Biological Systems

Web resources for participants in the NIMBioS Worshop on 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

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

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.

## Why Aren’t Math Textbooks More Straightforward?

Math textbooks often seem difficult, obtuse, and often incomprehensible. Here are some hints and tips for making the situation better for all students.

## Some General Advice for Math Students of All Ages

I recently saw the question “Why aren’t math textbooks more straightforward?” on Quora.

In fact, I would argue that most math textbooks are very straightforward!

The real issue most students are experiencing is one of relativity and experience.  Mathematics is an increasingly sophisticated, cumulative, and more complicated topic the longer you study it. Fortunately, over time, it also becomes easier, more interesting, and intriguingly more beautiful.

As an example of what we’re looking at and what most students are up against, let’s take the topic of algebra. Typically in the United States one might take introductory algebra in eighth grade before taking algebra II in ninth or tenth grade. (For our immediate purposes, here I’m discounting the potential existence of a common pre-algebra course that some middle schools, high schools, and even colleges offer.)  Later on in college, one will exercise one’s algebra muscles in calculus and may eventually get to a course called abstract algebra as an upper-level undergraduate (in their junior or senior years).  Most standard undergraduate abstract algebra textbooks will cover ALL of the material that was in your basic algebra I and algebra II texts in about four pages and simply assume you just know the rest! Naturally, if you started out with the abstract algebra textbook in eighth grade, you’d very likely be COMPLETELY lost. This is because the abstract algebra textbook is assuming that you’ve got some significant prior background in mathematics (what is often referred to in the introduction to far more than one mathematics textbook as “mathematical sophistication”, though this phrase also implicitly assumes knowledge of what a proof is, what it entails, how it works, and how to actually write one).

Following the undergraduate abstract algebra textbook there’s even an additional graduate level course (or four) on abstract algebra (or advanced subtopics like group theory, ring theory, field theory, and Galois theory) that goes into even more depth and subtlety than the undergraduate course; the book for this presumes you’ve mastered the undergraduate text and goes on faster and further.

## A Weightlifting Analogy

To analogize things to something more common, suppose you wanted to become an Olympic level weightlifter.  You’re not going to go into the gym on day one and snatch and then clean & jerk 473kg! You’re going to start out with a tiny fraction of that weight and practice repeatedly for years slowly building up your ability to lift bigger and bigger weights. More likely than not, you’ll also very likely do some cross-training by running, swimming, and even lifting other weights to strengthen your legs, shoulder, stomach, and back.  All of this work may eventually lead you to to win the gold medal in the Olympics, but sooner or later someone will come along and break your world record.

Mathematics is certainly no different: one starts out small and with lots of work and practice over time, one slowly but surely ascends the rigors of problems put before them to become better mathematicians. Often one takes other courses like physics, biology, and even engineering courses that provide “cross-training.”  Usually when one is having issues in a math class it’s because they’re either somehow missing something that should have come before or because they didn’t practice enough in their prior classwork to really understand all the concepts and their subtleties.  As an example, the new material in common calculus textbooks is actually very minimal – the first step in most problems is the only actual calculus and the following 10 steps are just practicing one’s algebra skills. It’s usually in carrying out the algebra that one makes more mistakes than in the actual calculus.

Often at the lower levels of grade-school mathematics, some students can manage to just read a few examples and just seem to “get” the answers without really doing a real “work out.” Eventually they’ll come to a point at which they hit a wall or begin having trouble, and usually it comes as the result of not actually practicing their craft.  One couldn’t become an Olympic weightlifter by reading books about weightlifting, they need to actually get in the gym and workout/practice. (Of course, one of the places this analogy breaks down is that weightlifting training is very linear and doesn’t allow one to “skip around” the way one could potentially in a mathematics curriculum.)

I’m reminded of a quote by mathematician Pierre Anton Grillet: “…algebra is like French pastry: wonderful, but cannot be learned without putting one’s hands to the dough.” It is one of the most beautiful expressions of the recurring sentiment written by almost every author into the preface of nearly every mathematics text at or above the level of calculus. They all exhort their students to actually put pencil to paper and work through the logic of their arguments and the exercises to learn the material and gain some valuable experience.  I’m sure that most mathematics professors will assure you that in the end, only a tiny fraction of their students actually do so. Some of the issue is that these exhortations only come in textbooks traditionally read at the advanced undergraduate level, when they should begin in the second grade.

## “It’s Easy to See”

A common phrase in almost every advanced math textbook on the planet is the justification, “It’s easy to see.” The phrase, and those like it, should be a watchword for students to immediately be on their guard!  The phrase is commonly used in proofs, discussions, conversations, and lectures in which an author or teacher may skip one or more steps which she feels should be obvious to her audience, but which, in fact, are far more commonly not obvious.

It’s become so cliche that some authors actually mention specifically in their prefaces that they vow not to use the phrase, but if they do so, they usually let slip some other euphemism that is its equivalent.

The problem with the phrase is that everyone, by force of their own circumstances and history, will view it completely differently.  A step that is easy for someone with a Ph.D. who specialized in field theory to “see” may be totally incomprehensible for a beginning student of algebra I in the same way that steps that were easy for Girgory Perelman to see in his proof of the Poincaré conjecture were likewise completely incomprehensible for teams of multiple tenured research professors of mathematics to see. (cross reference: The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O’Shea (Walker & Co., 2007))

## How to Actively Read a Math Text

### The Problem

So how are students to proceed? It will certainly help to see a broader road map of what lies ahead and what the expected changes in terrain will look like. It will also help greatly if students have a better idea how to approach mathematics for themselves and even by themselves in many cases.

In my opinion, the most common disconnect occurs somewhere between high school mathematics and early college mathematics (usually a calculus sequence, linear algebra) and then again between linear algebra/differential equations (areas which usually have discussion followed by examples and then crank-out problems) and higher abstract mathematical areas like analysis, abstract algebra, topology (areas in which the definition-theorem-proof cycle of writing is more common and seemingly more incomprehensible to many).

The first big issue in early college mathematics is the increased speed at which college courses move. Students used to a slower high school pace where the teachers are usually teaching to the middle or lower end of the class get caught unaware as their college professors teach to the higher ability students and aren’t as afraid to leave the lower end of the spectrum behind. Just like high school athletes are expected to step up their game when they make the transition to college and similarly college athletes who go pro, mathematics students should realize they’re expected to step up their game at the appropriate times.

Often math students (and really any student of any subject) relies on the teacher assigning readings or problems from their book rather than excersizing their curiosity to more avidly and broadly explore the material on their own.  If they can take the guidance of their teacher as well as that of the individual authors of books, they may make it much further on their own.  High school teachers often skip sections of textbooks for time, but students should realize that there is profitable and interesting material that they’re skipping.  Why not go and read it on their own?

Earlier I mentioned that an average undergraduate abstract algebra textbook might cover the totality of a high school algebra textbook in about three pages.  What does this mean for upper level mathematics students?  It almost always means that the density of material in these books is far greater than that of their earlier textbooks.  How is this density arrived at?  Authors of advanced textbooks leave out far more than they’re able to put in, otherwise their 300 page textbooks, if written at the same basic level as those that came before would be much more ponderous 1000+ page textbooks.  What are they leaving out?  Often they’re leaving out lots of what might be useful discussion, but more often, they’re leaving out lots of worked out examples. For example, a high school text will present a definition or concept and then give three or more illustrations or examples of problems relating to the concept.  The exercises will then give dozens of additional drill problems to beat the concept to death.  This type of presentation usually continues up to the level of calculus where one often sees massive tomes in the 800+ page length. Math texts after this point generally don’t go much over 300 pages as a rule, and it’s primarily because they’re leaving the examples out of the proverbial equation.

### The Solution

How does one combat this issue?  Students need to more actively think back to the math they’ve taken previously and come up with their own simple examples of problems, and work though them on their own.  Just because the book doesn’t give lots of examples doesn’t mean that they don’t exist.

In fact, many textbooks are actually presenting examples, they’re just hiding them with very subtle textual hints. Often in the presentation of a concept, the author will leave out one or more steps in a proof or example and hint to the student that they should work through the steps themselves. (Phrases like: “we leave it to the reader to verify” or “see example 2.”) Sometimes this hint comes in the form of that dreaded phrase, “It’s easy to see.” When presented with these hints, it is incumbent (or some students may prefer the word encumbering) on the student to think through the missing steps or provide the missing material themselves.

While reading mathematics, students should not only be reading the words and following the steps, but they should actively be working their way through all of the steps (missing or not) in each of the examples or proofs provided. They must read their math books with pencil and paper in hand instead of the usual format of reading their math book and then picking up paper and pencil to work out problems afterwards.  Most advanced math texts suggest half a dozen or more problems to work out within the text itself before presenting a dozen or more additional problems usually in a formal section entitled “Exercises”. Students have to train themselves to be thinking about and working out the “hidden” problems within the actual textual discussion sections.

Additionally, students need to consider themselves “researchers” or think of their work as discovery or play.  Can they come up with their own questions or exercises that relate the concepts they’ve read about to things they’ve done in the past? Often asking the open ended question, “What happens if I…” can be very useful.  One has to imagine that this is the type of “play” that early mathematicians like Euclid, Gauss, and Euler did, and I have to say, this is also the reason that they discovered so many interesting properties within mathematics.  (I always like to think that they were the beneficiaries of “picking the lowest hanging fruit” within mathematics – though certainly they discovered some things that took some time to puzzle out; we take some of our knowledge for granted as sitting on the shoulders of giants does allow us to see much further than we could before.)

As a result of this newly discovered rule, students will readily find that while they could read a dozen pages of their high school textbooks in just a few minutes, it may take them between a half an hour to two hours to properly read even a single page of an advanced math text. Without putting in this extra time and effort they’re going to quickly find themselves within the tall grass (or, more appropriately weeds).

Another trick of advanced textbooks is that, because they don’t have enough time or space within the primary text itself, authors often “hide” important concepts, definitions, and theorems within the “exercises” sections of their books.  Just because a concept doesn’t appear in the primary text doesn’t mean it isn’t generally important.  As a result, students should always go out of their way to at least read through all of the exercises in the text even if they don’t spend the time to work through them all.

One of the difficult things about advanced abstract mathematics is that it is most often very cumulative and even intertwined, so when one doesn’t understand the initial or early portions of a textbook, it doesn’t bode well for the later sections which require one to have mastered the previous work.  This is even worse when some courses build upon the work of earlier courses, so for example, doing well in calculus III requires that one completely mastered calculus I. At some of the highest levels like courses in Lie groups and Lie algebras requires that one mastered the material in multiple other prior courses like analysis, linear algebra, topology, and abstract algebra. Authors of textbooks like these will often state at the outset what material they expect students to have mastered to do well, and even then, they’ll often spend some time giving overviews of relevant material and even notation of these areas in appendices of their books.

As a result of this, we can take it as a general rule: “Don’t ever skip anything in a math textbook that you don’t understand.” Keep working on the concepts and examples until they become second nature to you.

Finally, more students should think of mathematics as a new language.  I’ve referenced the following Galileo quote before, but it bears repeating (emphasis is mine):

Though mathematical notation has changed drastically (for the better, in my opinion) since Galileo’s time, it certainly has its own jargon, definitions, and special notations.  Students should be sure to spend some time familiarizing themselves with current modern notation, and especially the notation in the book that they choose.  Often math textbooks will have a list of symbols and their meanings somewhere in the end-papers or the appendices. Authors usually go out of their way to introduce notation somewhere in either the introduction, preface, appendices, or often even in an introductory review chapter in which they assume most of their students are very familiar with, but they write it anyway to acclimate students to the particular notation they use in their text.  This notation can often seem excessive or even obtuse, but generally it’s very consistent across disciplines within mathematics, but it’s incredibly useful and necessary in making often complex concepts simple to think about and communicate to others. For those who are lost, or who want help delving into areas of math seemingly above their heads, I highly recommend the text Mathematical Notation: A Guide for Engineers and Scientists by my friend Edward R. Scheinerman as a useful guide.

A high school student may pick up a textbook on Lie Groups and be astounded at the incomprehensibility of the subject, but most of the disconnect is in knowing and understanding the actual language in which the text is written.  A neophyte student of Latin would no sooner pick up a copy of Cicero and expect to be able to revel in the beauty and joy of the words or their meaning without first spending some time studying the vocabulary, grammar, and syntax of the language.  Fortunately, like Latin, once one has learned a good bit of math, the notations and definitions are all very similar, so once you can read one text, you’ll be able to appreciate a broad variety of others.

#### Actively Reading a Mathematics Text Review:

• Work through the steps of everything within the text
• Come up with your own examples
• Work through the exercises
• Read through all the exercises, especially the ones that you don’t do
• Don’t ever skip anything you don’t fully understand
• Math is a language: spend some time learning (memorizing) notation

## Exceptions

Naturally there are exceptions to the rule. Not all mathematics textbooks are great, good, or even passable.  There is certainly a spectrum of textbooks out there, and there are even more options at the simpler (more elementary) end, in part because of there is more demand.  For the most part, however, most textbooks are at least functional.  Still one can occasionally come across a very bad apple of a textbook.

Because of the economics of textbook publishing, it is often very difficult for a textbook to even get published if it doesn’t at least meet a minimum threshold of quality. The track record of a publisher can be a good indicator of reasonable texts. Authors of well-vetted texts will often thank professors who have taught their books at other universities or even provide a list of universities and colleges that have adopted their texts. Naturally, just because 50 colleges have adopted a particular text doesn’t necessarily mean that that it is necessarily of high quality.

One of the major issues to watch out for is using the textbook written by one’s own professor.  While this may not be an issue if your professor is someone like Serge Lang, Gilbert Strang, James Munkres, Michael Spivak, or the late Walter Rudin, if your particular professor isn’t supremely well known in his or her field, is an adjunct or associate faculty member, or is a professor at a community college, then: caveat emptor.

Since mathematics is a subject about clear thinking, analysis, and application of knowledge, I recommend that students who feel they’re being sold a bill of goods in their required/recommended textbook(s), take the time to look at alternate textbooks and choose one that is right for themselves.  For those interested in more on this particular sub-topic I’ve written about it before: On Choosing Your own Textbooks.

Often, even with the best intentions, some authors can get ahead of themselves or the area at hand is so advanced that it is difficult to find a way into it. As an example, we might consider Lie groups and algebras, which is a fascinating area to delve into. Unfortunately it can take several years of advanced work to get to a sufficient level to even make a small dent into any of the textbooks in the area, though some research will uncover a handful of four textbooks that will get one quite a way into the subject with a reasonable background in just analysis and linear algebra.

When one feels like they’ve hit a wall, but still want to struggle to succeed, I’m reminded of the advice of revered mathematical communicator Paul Halmos, whose book Measure Theory needed so much additional background material, that instead of beginning with the traditional Chapter 1, he felt it necessary to include a Chapter 0 (he actually called his chapters “sections” in the book) and even then it had enough issueshewas cornered into writing the statement:

This is essentially the mathematician’s equivalent of the colloquialism “Fake it ’til you make it.”

When all else fails, use this adage, and don’t become discouraged. You’ll get there eventually!

Syndicated copies to:

## The Postdoctoral Experience (Revisited)

The Postdoctoral Experience Revisited (2014) (The National Academies Press)
The Postdoctoral Experience Revisited builds on the 2000 report Enhancing the Postdoctoral Experience for Scientists and Engineers. That ground-breaking report assessed the postdoctoral experience and provided principles, action points, and recommendations to enhance that experience. Since the publication of the 2000 report, the postdoctoral landscape has changed considerably. The percentage of PhDs who pursue postdoctoral training is growing steadily and spreading from the biomedical and physical sciences to engineering and the social sciences. The average length of time spent in postdoctoral positions seems to be increasing. The Postdoctoral Experience Revisited reexamines postdoctoral programs in the United States, focusing on how postdocs are being guided and managed, how institutional practices have changed, and what happens to postdocs after they complete their programs. This book explores important changes that have occurred in postdoctoral practices and the research ecosystem and assesses how well current practices meet the needs of these fledgling scientists and engineers and of the research enterprise. The Postdoctoral Experience Revisited takes a fresh look at current postdoctoral fellows - how many there are, where they are working, in what fields, and for how many years. This book makes recommendations to improve aspects of programs - postdoctoral period of service, title and role, career development, compensation and benefits, and mentoring. Current data on demographics, career aspirations, and career outcomes for postdocs are limited. This report makes the case for better data collection by research institution and data sharing. A larger goal of this study is not only to propose ways to make the postdoctoral system better for the postdoctoral researchers themselves but also to better understand the role that postdoctoral training plays in the research enterprise. It is also to ask whether there are alternative ways to satisfy some of the research and career development needs of postdoctoral researchers that are now being met with several years of advanced training. Postdoctoral researchers are the future of the research enterprise. The discussion and recommendations of The Postdoctoral Experience Revisited will stimulate action toward clarifying the role of postdoctoral researchers and improving their status and experience.

The National Academy of Sciences has published a (free) book: The Postdoctoral Experience (Revisited) discussing where we’re at and some ideas for a way forward.

Most might agree that our educational system is far less than ideal, but few pay attention to significant problems at the highest levels of academia which are holding back a great deal of our national “innovation machinery”. The National Academy of Sciences has published a (free) book: The Postdoctoral Experience (Revisited) discussing where we’re at and some ideas for a way forward. There are some interesting ideas here, but we’ve still got a long way to go.

## Introduction to Lie Groups and Lie Algebras (Part 2) | UCLA Extension

Dr. Mike Miller has just opened up registration for the second course in the series. His courses are always clear, entertaining, and invigorating, and I highly recommend them to anyone who is interested in math, science, or engineering.

Dr. Mike Miller, who had previously announced a two quarter sequence of classes on Lie Groups at UCLA, has just opened up registration for the second course in the series. His courses are always clear, entertaining, and invigorating, and I highly recommend them to anyone who is interested in math, science, or engineering.

Prior to the first part of the course, I’d written some thoughts about the timbre and tempo of his lecture style and philosophy and commend those interested to take a peek. I also mentioned some additional resources for the course there as well.  For those who missed the first portion, I’m happy to help fill you in and share some of my notes if necessary. The recommended minimum prerequisites for this class are linear algebra and some calculus.

## Introduction to Lie Groups and Lie Algebras (Part 2)

Math X 450.7 / 3.00 units / Reg. # 251580W
Professor: Michael Miller, Ph.D.
Start Date: January 13, 2015
Location: UCLA, 5137 Math Sciences Building
Tuesday, 7-10pm
January 13 – March 24
11 meetings total
Class will not meet on one Tuesday to be annouced.

### Course Description

A Lie group is a differentiable manifold that is also a group for which the product and inverse maps are differentiable. A Lie algebra is a vector space endowed with a binary operation that is bilinear, alternating, and satisfies the so-called Jacobi identity. This course is the second in a 2-quarter sequence that offers an introductory survey of Lie groups, their associated Lie algebras, and their representations. Its focus is split between continuing last quarter’s study of matrix Lie groups and their representations and reconciling this theory with that for the more general manifold setting. Topics to be discussed include the Weyl group, complete reducibility, semisimple Lie algebras, root systems, and Cartan subalgebras. This is an advanced course, requiring a solid understanding of linear algebra, basic analysis, and, ideally, the material from the previous quarter.Internet access required to retrieve course materials.

### Recommended Textbook

Hall, Brian. Lie Groups, Lie Algebras, & Representations (Springer, 2004) ISBN: 9781441923134

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## BIRS Workshop on Biological and Bio-Inspired Information Theory | Storify Stream

Over the span of the coming week, I'll be updating (and archiving) the stream of information coming out of the BIRS Workshop on Biological and Bio-Inspired Information Theory.

Over the span of the coming week, I’ll be updating (and archiving) the stream of information coming out of the BIRS Workshop on Biological and Bio-Inspired Information Theory.

Editor’s note: On 12/12/17 Storify announced they would be shutting down. As a result, I’m changing the embedded version of the original data served by Storify for an HTML copy which can be found below:

# BIRS: Biological and Bio-Inspired Information Theory

A 5 Day workshop on Biology and Information Theory hosted by the Banff International Research Station

1. I know where I’ll be in Oct 2014! Let’s hear it for Biology & Information Theory!  https://www.birs.ca/events/2014/5-day-workshops/14w5170  #ITBio #Banff @andreweckford
2. . @andreweckford You might be interested in this grouping of research papers:  http://www.mendeley.com/groups/2545131/itbio/  #ITBio #Banff
3. Wishing I was at the Gene Regulation and Information Theory meeting starting tomorrow  http://bit.ly/XnHRZs  #ITBio
4. #ITBio: @andreweckford has a new book on Molecular Communication available Oct 31.  http://bit.ly/15uEUzF
5. Mathematical and Statistical Models for Genetic Coding starts today.  http://www.am.hs-mannheim.de/genetic_code_2013.php?id=1  @andreweckford might borrow attendees for BIRS
6. Mathematical Foundations for Information Theory in Diffusion-Based Molecular Communications  http://bit.ly/1aTVR2c  #ITBio
7. Bill Bialek giving plenary talk “Information flow & order in real biological networks” at Feb 2014 workshop  http://mnd.ly/19LQH8f  #ITBio
8. Workshop on Information Theoretic Incentives for Artificial Life  http://jhu.md/1lM8tAn  #ITBio #ALife14 @alifeofficial @14thALIFE @cxdig
9. Researchers working in information theory & biology  http://jhu.md/1gieQGR  #ITBio @andreweckford @ChristophAdami @wbialek @johnhawks
10. CECAM Workshop: “Entropy in Biomolecular Systems” starts May 14 in Vienna.  http://jhu.md/1faLR8t  #ITBio pic.twitter.com/Ty8dEIXQUT
11. Currently organizing my Banff workshop on bio-information theory …  https://www.birs.ca/events/2014/5-day-workshops/14w5170
12. Last RT: wonder what the weather is going to be like at the end of October for my @BIRS_Math workshop
13. @JoVanEvery I’m organizing a workshop in Banff in October … hopefully this isn’t a sign of weather to come!
14. How information theory could hold the key to quantifying nature.  http://wrd.cm/1uy1xdX  by @vero_greenwood pic.twitter.com/ek5DUb2Ul9
15. Good morning from Banff. Current temp: -1 C
16. Banff takes its name from the town of Banff, Scotland, not to be confused with Bamff, also Scotland.
17. Good morning from beautiful Banff. How can you not love the mountains? pic.twitter.com/mxYBNz7yzl
18. “Not an obvious connection between utility and information, just as there is no obvious connection between energy and entropy” @BIRS_Math
19. Peter Thomas (Case Western Reserve University), Signal Transduction and Information Theory  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410270940-Thomas.mp4
20. Last RT: a lot of discussion of my signal transduction work with Peter Thomas.
21. Live now: Nicolo Michelusi of @USCViterbi on Stochastic Model for Electron Transfer in Bacterial Cables  http://www.birs.ca/live  #ITBio
22. Nicolo Michelusi (University of Southern California), A Stochastic Model for Electron Transfer in Bacterial Cables  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410271450-Michelusi.mp4
23. Listening to the always awesome @cnmirose talk about the ultimate limits of molecular communication.
24. “Timing is fundamental … subsumes time-varying concentration channel” @cnmirose @BIRS_Math
25. Chris Rose (Rutgers University), Molecular Communication Channels: timing vs. payload  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410271538-Rose.mp4
26. Standard opening quote of these talks: “I’m not a biologist, but …” @BIRS_Math
27. Stefan Moser (ETH Zurich), Capacity Bounds of the Memoryless AIGN Channel – a Toy-Model for Molecular Communicat…  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410271610-Moser.mp4
28. Weisi Guo (University of Warwick), Communication Envelopes for Molecular Diffusion and Electromagnetic Wave Propag…  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410271643-Guo.mp4
29. Terrific introduction of Canada/Banff by Andrew Eckford (York)The Landscape  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410270858-Eckford.mp4
30. Biological and Bio-Inspired Information Theory workshop videos!  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos  @BIRS_Math
31. .@ChrisAldrich @andreweckford @Storify @BIRS_Math Sounds like a fascinating workshop on bioinformation theory in Banff.
32. Toby Berger, winner of the 2002 Shannon award, speaking right now. @BIRS_Math
33. Naftali Tishby (Hebrew University of Jerusalem), Sensing and acting under information constraints – a principled a…  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410281032-Tishby.mp4
34. “…places such as BIRS and the Banff Centre exist to facilitate the exchange and pursuit of knowledge.” S. Sundaram  http://www.birs.ca/testimonials/#testimonial-1454
35. We’re going for a hike tomorrow. Many thanks to Lukas at the @ParksCanada info centre in Banff for helpful advice! @BIRS_Math
36. Behnaam Aazhang (Rice University), Real-Time Network Modulation for Intractable Epilepsy  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410281337-Aazhang.mp4
37. Alexander Dimitrov (Washington State University), Invariant signal processing in auditory biological systems  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410281416-Dimitrov.mp4
38. Joel Zylberberg (University of Washington), Communicating with noisy signals: lessons learned from the mammalian v…  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410281450-Zylberberg.mp4
39. Robert Schober (Universitat Erlangen-Nurnberg), Intersymbol interference mitigation in diffusive molecular communi…  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410281549-Schober.mp4
40. Rudolf Rabenstein (Friedrich-Alexander-Universitat Erlangen-Nurnberg (FAU)), Modelling Molecular Communication Cha…  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410281627-Rabenstein.mp4
42. THis week @BIRS_Math ” Biological and Bio-Inspired Information Theory ” @thebanffcentre #biology #math @NSF
43. “Your theory might match the data, but the data might be wrong” – Crick @BIRS_Math
44. So information theory seems to be a big deal in ecology. @BIRS_Math
45. Tom Schneider (National Institutes of Health), Three Principles of Biological States: Ecology and Cancer  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410290904-Schneider.mp4
46. “In biodiversity, the entropy of an ecosystem is the expected … information we gain about an organism by learning its species” @BIRS_Math
47. Seriously, I’m blown away by this work in information theory in ecology. Huge body of work; I had no idea. @BIRS_Math
48. .@andreweckford @BIRS_Math Harte’s book Maximum Entropy & Ecology is excellent in this area  http://amzn.to/1DwIl3V  pic.twitter.com/EIBDpM35uf
49. .@andreweckford @QuantaMagazine had a nice overview of some of John Harte’s work in September  http://bit.ly/1DwIWCD  @BIRS_Math
50. Chan-Byoung Chae (Yonsei University), Molecular MIMO: From Theory to Practice  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410281705-Chae.mp4
51. John Baez (University of California, Riverside), Biodiversity, entropy and thermodynamics  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410291038-Baez.mp4
52. I encourage @BIRS_Math attendees at Biological & Bio-Inspired Information Theory to contribute references here:  http://bit.ly/1jQwObk
53. Christoph Adami (Michigan State University), Some Information-Theoretic Musings Concerning the Origin and Evolutio…  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410291114-Adami.mp4
54. .@ChristophAdami talk Some Information-Theoretic Musings Concerning the Origin of Life @BIRS_Math this morning #ITBio pic.twitter.com/VA8komuuSW
55. ICYMI @ChristophAdami had great paper: Information-theoretic Considerations on Origin of Life on arXiv  http://bit.ly/1yIhK2Q  @BIRS_Math
56. Johnston Canyon selfie @BIRS_Math pic.twitter.com/MEKeY5To5s
57. Baez has a post on Tishby’s talk “Sensing & Acting Under Information Constraints”  http://bit.ly/1yIDonR  @BIRS_Math pic.twitter.com/dFuiVLFSGC
58. INFORMATION THEORY is the new central …
59. I’m listening to a talk on the origin of life at a workshop on Biological and Bio-Inspired Information Theory. …  https://plus.google.com/117562920675666983007/posts/gqFL7XY3quF
60. Ilya Nemenman @EmoryUniversity on Predictive information  http://bit.ly/1titfOw
61. Now accepting applications for the #Research Collaboration Workshop for Women in #MathBio at NIMBioS  http://ow.ly/DzeZ7
62. Inkpots selfie from yesterday’s hike. @BIRS_Math pic.twitter.com/0A6ZQsQVwE
63. On the way home from Inkpots. @BIRS_Math pic.twitter.com/1XhO8mLOkq
64. Toby Berger (University of Virginia), Neruoscience Applications of GIG Distributions  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410280914-Berger.mp4
65. We removed a faulty microphone from our lecture room this morning. We’re now fixing the audio buzz in this week’s videos, and reposting.
66. Daniel Polani (University of Hertfordshire), Informational Principles in Perception-Action Loops  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410301038-Polani.mp4
67. Didn’t get enough information theory & biology this week @BIRS_Math? Apply for NIMBioS workshop in April 2015  http://bit.ly/1yIeiWe  #ITBio
68. Amin Emad (University of Illinois at Urbana-Champaign), Applications of Discrete Mathematics in Bioinformatics  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410301329-Emad.mp4
69. Paul Bogdan (University of Southern California), Multiscale Analysis Reveals Complex Behavior in Bacteria Populati…  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410301401-Bogdan.mp4
70. Robert Shaw (ProtoLife Inc.), Information and Causality in a Reaction-Diffusion System  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410301434-Shaw.mp4
71. Lubomir Kostal (Institute of Physiology, Academy of Sciences of the Czech Republic), Efficient information transmi…  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410301534-Kostal.mp4
72. Nima Soltani (Stanford University), Applications of Directed Information to Neuroscience  http://www.birs.ca/events/2014/5-day-workshops/14w5170/videos/watch/201410301647-Soltani.mp4
73. Banff ☀️❄️🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲🌲❤️
74. @lrvarshney I shoulda invited you to this BIRS workshop …
75. @conservativelez I’m a big fan of your dad’s research & was reminded of much of it via a workshop on Biological Information Theory
76. @conservativelez Though he may not have been able to attend, he can catch most of the talks online if he’d like  https://www.birs.ca/events/2014/5-day-workshops/14w5170
77. Depressed that @BIRS_Math Workshop on Biological & Bio-Inspired Information Theory is over? Relive it here:  http://bit.ly/1rF3G4B  #ITBio
78. Kudos @andreweckford, Toby Berger, Peter Thomas, @NGhoussoub, @BIRS_Math & friends on a fantastic workshop!  http://bit.ly/1ckttZq
79. This @BIRS_Math Workshop was biggest thing in #informationtheory & #biology since the Gatlinburg Symposium in 1956.  http://bit.ly/1rF4RRr
80. See you later Calgary. pic.twitter.com/mkmU6yrmVz
81. A few thoughts about that workshop while I wait for my flight back to Toronto.
82. 1/ Everyone I talked to said it was the best workshop they’d ever been to, and they’d like to do a follow-up workshop @BIRS_Math
83. 2/ There is an amazing diversity of work under the umbrella of “information theory”. @BIRS_Math
84. 3/ Much of this work is outside the IT mainstream, and an issue is that people use different terms for related concepts. @BIRS_Math
85. 4/ Some community building is in order. I think this workshop was a good first step. @BIRS_Math
86. 5/ Many many thanks to @BIRS_Math and huge kudos to @NGhoussoub for excellent service to the Canadian scientific community. BIRS is a gem.
87. 6/ Also many thanks to the participants for their excellent talks, and to @ChrisAldrich for maintaining a Storify.

## Information Theory is the New Central Discipline

Information Theory is the new central discipline. by Nassim Nicholas Taleb (facebook.com)

I’m coming to this post a bit late as I’m playing a bit of catch up, but agree with it wholeheartedly.

In particular, applications to molecular biology and medicine are really beginning to come to a heavy boil in just the past five years. This particular year is the progenitor of what appears to be the biggest renaissance for the application of information theory to the area of biology since Hubert Yockey, Henry Quastler, and Robert L. Platzman’s “Symposium on Information Theory in Biology at Gatlinburg, Tennessee” in 1956.

Upcoming/recent conferences/workshops on information theory in biology include:

At the beginning of September, Christoph Adami posted an awesome and very sound paper on arXiv entitled “Information-theoretic considerations concerning the origin of life”  which truly portends to turn the science of the origin of life on its head.

I’ll note in passing, for those interested, that Claude Shannon’s infamous master’s thesis at MIT (in which he applied Boolean Algebra to electric circuits allowing the digital revolution to occur) and his subsequent “The Theory of Mathematical Communication” were so revolutionary, nearly everyone forgets his MIT Ph.D. Thesis “An Algebra for Theoretical Genetics” which presaged the areas of cybernetics and the current applications of information theory to microbiology and are probably as seminal as Sir R.A Fisher’s applications of statistics to science in general and biology in particular.

For those commenting on the post who were interested in a layman’s introduction to information theory, I recommend John Robinson Pierce’s An Introduction to Information Theory: Symbols, Signals and Noise (Dover has a very inexpensive edition.) After this, one should take a look at Claude Shannon’s original paper. (The MIT Press printing includes some excellent overview by Warren Weaver along with the paper itself.) The mathematics in the paper really aren’t too technical, and most of it should be comprehensible by most advanced high school students.

For those that don’t understand the concept of entropy, I HIGHLY recommend Arieh Ben-Naim’s book Entropy Demystified The Second Law Reduced to Plain Common Sense with Seven Simulated Games. He really does tear the concept down into its most basic form in a way I haven’t seen others come remotely close to and which even my mother can comprehend (with no mathematics at all).  (I recommend this presentation to even those with Ph.D.’s in physics because it is so truly fundamental.)

For the more advanced mathematicians, physicists, and engineers Arieh Ben-Naim does a truly spectacular job of extending ET Jaynes’ work on information theory and statistical mechanics and comes up with a more coherent mathematical theory to conjoin the entropy of physics/statistical mechanics with that of Shannon’s information theory in A Farewell to Entropy: Statistical Thermodynamics Based on Information.

For the advanced readers/researchers interested in more at the intersection of information theory and biology, I’ll also mention that I maintain a list of references, books, and journal articles in a Mendeley group entitled “ITBio: Information Theory, Microbiology, Evolution, and Complexity.”

Syndicated copies to:

## Announcement: Special issue of Entropy: “Information Theoretic Incentives for Cognitive Systems”

Dr. Christoph Salge asked me to cross-post this notice from the Entropy site here.

Editor’s Note: Dr. Christoph Salge asked me to cross-post the following notice from the Entropy site here. I’m sure many of the readers of the blog will find the topic to be of interest.

Dear Colleagues,

In recent years, ideas such as “life is information processing” or “information holds the key to understanding life” have become more common. However, how can information, or more formally Information Theory, increase our understanding of life, or life-like systems?

Information Theory not only has a profound mathematical basis, but also typically provides an intuitive understanding of processes, such as learning, behavior and evolution terms of information processing.

In this special issue, we are interested in both:

1. the information-theoretic formalization and quantification of different aspects of life, such as driving forces of learning and behavior generation, information flows between neurons, swarm members and social agents, and information theoretic aspects of evolution and adaptation, and
2. the simulation and creation of life-like systems with previously identified principles and incentives.

Topics with relation to artificial and natural systems:

• information theoretic intrinsic motivations
• information theoretic quantification of behavior
• information theoretic guidance of artificial evolution
• information theoretic guidance of self-organization
• information theoretic driving forces behind learning
• information theoretic driving forces behind behavior
• information theory in swarms
• information theory in social behavior
• information theory in evolution
• information theory in the brain
• information theory in system-environment distinction
• information theory in the perception action loop
• information theoretic definitions of life

## Submission

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed Open Access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs).

Deadline for manuscript submissions: 28 February 2015

## Special Issue Editors

Guest Editor
Dr. Christoph Salge
Adaptive Systems Research Group,University of Hertfordshire, College Lane, AL10 9AB Hatfield, UK
Website: http://homepages.stca.herts.ac.uk/~cs08abi
E-Mail: c.salge@herts.ac.uk
Phone: +44 1707 28 4490
Interests: Intrinsic Motivation (Empowerment); Self-Organization; Guided Self-Organization; Information-Theoretic Incentives for Social Interaction; Information-Theoretic Incentives for Swarms; Information Theory and Computer Game AI

Guest Editor
Dr. Georg Martius
Cognition and Neurosciences, Max Planck Institute for Mathematics in the Sciences Inselstrasse 22, 04103 Leipzig, Germany
Website: http://www.mis.mpg.de/jjost/members/georg-martius.html
E-Mail: martius@mis.mpg.de
Phone: +49 341 9959 545
Interests: Autonomous Robots; Self-Organization; Guided Self-Organization; Information Theory; Dynamical Systems; Machine Learning; Neuroscience of Learning; Optimal Control

Guest Editor
Dr. Keyan Ghazi-Zahedi
Information Theory of Cognitive Systems, Max Planck Institute for Mathematics in the Sciences Inselstrasse 22, 04103 Leipzig, Germany
Website: http://personal-homepages.mis.mpg.de/zahedi
E-Mail: zahedi@mis.mpg.de
Phone: +49 341 9959 535
Interests: Embodied Artificial Intelligence; Information Theory of the Sensorimotor Loop; Dynamical Systems; Cybernetics; Self-organisation; Synaptic plasticity; Evolutionary Robotics

Guest Editor
Dr. Daniel Polani
Department of Computer Science, University of Hertfordshire, Hatfield AL10 9AB, UK
Website: http://homepages.feis.herts.ac.uk/~comqdp1/
E-Mail: d.polani@herts.ac.uk
Interests: artificial intelligence; artificial life; information theory for intelligent information processing; sensor Evolution; collective and multiagent systems

Syndicated copies to: