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

Category Theory - the bedrock of mathematics ? by Ilyas KhanIlyas 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

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

 

Mathematics Without Apologies by Michael Harris
Mathematics Without Apologies by Michael Harris

 

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

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.
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.

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

A series of articles by producer Gavin Polone can serve as an excellent introduction to the business of Hollywood.

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

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