@fadesingh, While I do agree in part with your comment that the technical side of the math could have been explored a bit more, to me, part of what this story is highlighting (and to great benefit as well) is the "other" personal side of mathematics which is rarely seen by the broader public. Most of math and math history is full of seemingly brilliant solo (read: lone wolf) researchers developing mathematics de novo in dark, smoke and caffeine-filled rooms and emerging with iron clad proofs. This particular story shows the growing more collaborative and friendly side of math in addition to the years of slow development of friendships and theory which have culminated into something potentially interesting. I would suggest that in this case you not take them too hard to task on the subject, particularly as the article was being written contemporaneously with the publication of the journal article itself. (The article here was published <i>just</i> in advance of the arXiv post, such that it didn't even include the link to the paper itself, though it was added the following day.)

I love that Quanta is continually exploring the areas of math and science at the depth and level which they've become accustomed. They're filling a very important gap in science communication between technical journal articles and somewhat sophisticated outlets like National Geographic, Scientific American, and Wired (in which they are also distributed, but yet are still an editorial cut above comparatively. [Cross reference: <a href="http://stream.boffosocko.com/2015/evolution-of-a-scientific-journal-article-title-from-nature-to">Evolution of a Scientific Journal Article Title (from Nature to TMZ)</a>] I'm sure that C.P. Snow himself would praise them for helping to close the gap between the "Two Cultures."

RT @FrankWilczek Manuscript of Maxwell's "Dynamical Theory" paper: 150 years ago, the true beginning of the modern world. https://twitter.com/FrankWilczek/status/667696156051378176

My reply to @PeterCoffee "Information should be more ‘force’ than ‘mass’" | Diginomica

You're certainly asking the right types of questions here, but doing so in a somewhat flimsy framework in what is already a fairly solid mathematical theory. Mathematicians would call this a "hand-waving argument." Some of your conceptualization is clouded by mistaking the semantics of the commonly accepted definition of information with Claude Shannon's formal mathematical definition, something which he admonishes the reader about in the opening paragraphs of his seminal paper.

I would suggest that you take a look at Cesar Hidalgo's recent and very accessible book Why Information Grows: The Evolution of Order, from Atoms to Economies (MIT Press, 2015) [http://amzn.to/1MrA7Pf] which provides a more rigorous grounding of what information is and what it means, particularly within the context of economics and global business. In it, he re-frames your questions and provides a firmer theoretical platform for seeing farther and moving faster. The framework he provides will provide a more solid grounding for what is happening within the growing digital economy and proliferation of areas like big data.

For those who want to take the economics piece a step further, one can then delve into the broader topics at the intersection of fields like complexity theory and economics similar to those framed by the Santa Fe Institute over the past two decades. (W. Brian Arthur's Complexity and the Economy, (Oxford, 2014) [http://amzn.to/1MrB5ex] is a fairly good starting point without getting too deep too quickly for most).

From a business perspective, these theories underpin many of the ideas expounded by Judith Rodin's The Resilience Dividend: Being Strong in a World Where Things Go Wrong (Public Affairs, 2014) or many of the probabilistic arguments made in Nassim Nicholas Taleb's various works.

A general foothold into the overlap of maximum entropy methods and biology:

John Harte's work on applying the mathematical theory of maximum entropy to ecology is certainly one of the better known examples of the application of this area of mathematics to science, in part because he literally wrote the textbook: [Maximum Entropy and Ecology: A Theory of Abundance, Distribution, and Energetics (Oxford Series in Ecology and Evolution)][1]

To be clear, maximum entropy (also known as MaxEnt in some of the literature, though most/all researchers use the longer form in publications) is a mathematical tool stemming from the fields of probability theory, statistics, and information theory. It's use is classically most often seen in thermodynamics, statistical thermodynamics, physics, and information theory, primarily because these were the areas in which E.T. Jaynes was working when he posited the idea. [Wikipedia has links to his two seminal papers.][2]But because of it's mathematical form, it can be applied in a multitude of areas, typically where one can utilize probabilistic methods.

If you're looking for additional areas of application, simply google the phrase "applied maximum entropy" and you'll find a [wealth of areas][3] including: econometrics, natural language processing, nuclear medicine, queuing systems, mass spectrometry, image processing, machine learning, and many others.

For ecology related work, a cross search on maximum entropy and "genetics", "evolution", "species", and similar words will provide a wealth of papers like ["A maximum entropy approach to species distribution modeling"][4].

Given the generic nature of your question, I might suggest that you'll find E.T. Jaynes' paper ["On the Rationale of Maximum-Entropy Methods" (IEEE, 1982)][5] useful.

Those generally interested in the broader applications of information theoretic methods to biology will likely appreciate some of the work that came out of last year's [NIMBioS Workshop on Information and Entropy in Biological Systems][6] (which Harte both attended and presented at), the [BIRS Workshop Biological and Bio-Inspired Information Theory][7], and the 2014 [CECAM Entropy in Biomolecular Systems][8]. The NIMBios Workshop was organized by John Baez, a physicist, who has worked with MaxEnt methods and explored them on his blog "[Azimuth][9]".

Those with a more sophisticated mathematical background (including measure theory, functional analysis, etc.) may appreciate Henryk Gzyl's text [The Method of Maximum Entropy (World Scientific: Series on Advances in Mathematics for Applied Sciences, Vol 29, 1995)][10].


[1]: http://amzn.to/1SnNB2c
[2]: https://en.wikipedia.org/wiki/Principle_of_maximum_entropy
[3]: https://scholar.google.com/scholar?q=applied%20maximum%20entropy
[4]: http://dl.acm.org/citation.cfm?id=1015412
[5]: ftp://129.240.33.108/pub/outgoing/IMN/Prediction%20modelling%20artikler%20fra%20Anders%20K%20W/Jaynes%201982,%20On%20the%20rational%20of%20Maximum-Entropy%20models.pdf
[6]: http://boffosocko.com/2015/05/20/videos-from-nimbios-workshop-on-information-and-entropy-in-biological-systems/
[7]: http://www.birs.ca/events/2014/5-day-workshops/14w5170
[8]: http://www.cecam.org/workshop-1014.html
[9]: https://johncarlosbaez.wordpress.com/?s=maximum%20entropy
[10]: http://amzn.to/1MFrrIC

Wish I could have gone to Urban Design and Complexity talk. Looking forward to the full paper. #Complexity

Paul, thanks for the provocative piece, though the state of the art is certain much further along that your piece intimates. For the general reader, I would suggest reading MIT professor Cesar Hidalgo's recent book Why Information Grows (MIT Press, 2015) for some general structure and philosophy.

One of the best definitions and frameworks I've seen thus far has to be that of Christoph Adami. To start, and depending on your level of sophistication, take a look at his recent arXiv paper (Information-theoretic considerations concerning the origin of life - http://arxiv.org/abs/1409.0590 ) and then take a crack at this popular press article about it in Medium https://medium.com/the-physics-arxiv-blog/information-theory-and-the-origin-of-life-4cf6b93d156c). If it's something that blows your skirt up, then you can certainly begin to delve more deeply into some of his journal articles over the past decade or so.

For further references, I maintain a nice list of resources at Information Theory and Biology Resources [http://boffosocko.com/itbio/], as well as a "journal club" of sorts at Mendeley: ITBio: Information Theory, Microbiology, Evolution, and Complexity [https://www.mendeley.com/groups/2545131/itbio-information-theory-microbiology-evolution-and-complexity/].

For those who like to watch video material, I'll refer them to some videos from the NIMBioS Workshop on Information and Entropy in Biological Systems [http://boffosocko.com/2015/05/20/videos-from-nimbios-workshop-on-information-and-entropy-in-biological-systems/] organized by physicist John Carlos Baez. The Banff International Research Station also hosted a relatively recent week long workshop on Biological and Bio-Inspired Information Theory [http://www.birs.ca/events/2014/5-day-workshops/14w5170] which covered some interesting related ground with videos of many of the talks there as well.

It actually looks like you're missing one of the biggest -- and now, according to the Supreme Court, legal -- new loopholes for cheap textbooks.

One of the most important changes in the textbook market that every buyer should be aware of: last year in Kirtsaeng v. John Wiley & Sons, Inc. [http://www.supremecourt.gov/opinions/12pdf/11-697_d1o2.pdf] the US Supreme Court upheld the ability for US-based students to buy copies of textbooks printed in foreign countries (often at huge cut-rate prices) [see also Ars Technica: http://arstechnica.com/tech-policy/2013/03/thai-student-protected-by-first-sale-supreme-court-rules/]. This means that searching online bookstores in India, Indonesia, Pakistan, etc. will often find the EXACT same textbooks (usually with slightly different ISBNs, and slightly cheaper paper) for HUGE discounts in the 60-95% range.

To stick with your math example: I recently bought an international edition of Walter Rudin's Principles of Mathematical Analysis (Amazon $121 http://amzn.to/1U74CD7) for $5 (and it even happened to ship from within the US for $3). Not only was this 96% off of the cover price, but it was 78% off of Amazon's rental price! How amazing is it to spend almost as much to purchase a book as it is to ship it to yourself!? I'll also note here that the first edition of this book appeared in 1964 and this very popular third edition is from 1976, so it isn't an example of "edition creep", but it's still got a tremendous mark up in relation to other common analysis texts [http://amzn.to/1V3QVlm] which list on Amazon for $35-50.

Hint: Abe Books (a subsidiary of Amazon http://affiliates.abebooks.com/c/133691/77416/2029?u=http%3A%2F%2Fwww.abebooks.com%2Fbooks%2FTextbooks%2Finternational-editions.shtml) is better than most at finding/sourcing international editions of textbooks.

The same Calculus textbook you mentioned as a Sixth Edition can be easily found in paperback for $55.44 (I only did one search to find it, but I'm sure a little elbow grease might cut the price further) and ships within the US for less than $10. And because many students are apt to take the multi-variable calculus course as a follow up, this textbook also includes that material as an added bonus as well. http://www.abebooks.com/servlet/BookDetailsPL?bi=14224082117&searchurl=sts%3Dt%26sortby%3D20%26kn%3D%22International+Edition%22%26an%3DDeborah+Hughes-Hallett

The other option you leave out is purchasing the fifth edition, which isn't substantially different from the 6th edition and which can be easily found for less than $6 in good condition and including shipping.

Of course all this belies the true discussion of the how's and why's for why the textbook market is overpriced in the first place. For some of those ideas and suggestions of how we can fix them, I've written a short essay: "To Purchase, Rent, or Pirate? The Broken Economics of Textbooks in the Digital Age" http://boffosocko.com/2015/08/24/to-purchase-rent-or-pirate-the-broken-economics-of-textbooks-in-the-digital-age/

My reply to: Daniel Tung's answer to What do people think of Cesar Hidalgo's thesis in Why Information Grows? - Quora https://www.quora.com/What-do-people-think-of-Cesar-Hidalgos-thesis-in-Why-Information-Grows/answer/Daniel-Tung

I've read his book and think his views are more nuanced than you posit. Shortly, I hope to write a longer review, so for now I'll add some more nuance and some facts to the comments above.

I think Hidalgo would disagree with the criticism of his view in part A. I suggest you read his first few chapters to better tease apart how Hidalgo (and even Shannon himself) viewed information. In Shannon's original paper (in the second paragraph!) he totally divorces the semantic meaning of the message and only considers the mathematical portion of the information. Hidalgo follows along with this premise (page xvi in his introduction), though he doesn't go as deeply into the semantics of the example you've given, primarily as the target audience for the book is the mass market rather than specialists. From a thermodynamics perspective, the system under consideration was never just the Bugatti (which is the view Daniel is taking), but the system is the larger universe which includes the Bugatti, and all of the potential people in the societies which might encounter that Bugatti, including the New Guinean Highlanders who might view it as relatively worthless.

As for the criticism in part B, Hidalgo on page 105 queries us: "Are iPhones Californian, Chinese, or Korean? With the dismemberment of production, the nationality of products no longer makes much sense." Here Hidalgo takes an external view of everything, which is much more sophisticated than the simpler viewpoints he takes earlier in the text in which he looks at individual economies with respect to each other. Here Daniel's criticism simply suffers from not having all the facts or examples Hidalgo provides.

@collabchem Happy to help you along. I'm hoping to have a blog post or paper about science and social media that includes it as a reference shortly.

I'm on page 25 of 256 of Why Information Grows
I love that his definitions of information, knowledge, and knowhow are more concrete than most(all?) authors. He gives W. Weaver a bit too much priority in Shannon's work; also doesn't mention the original title of Shannon's Bell Labs paper or its significance. I love his concrete examples of the Bugati, Rubic's cube, and DNA. His philosophy seems to fit in with my own so far; can't wait to get to the economics.