Brief reply to: Is majoring in liberal arts a mistake for students? https:/
What magisterial sounding pontification! Sadly, it’s not much different than the early philosophies of Socrates and Plato or many of the other early progenitors of the humanities and liberal arts. I get the impression that the author hasn’t read much philosophy and has not much grounding in the liberal arts. While I agree with the spirit in which the piece is written, I find it deplorable that there aren’t what should be obligatory mentions of words like trivium, quadrivium, or philosophy, but rather the corpus of work in which the author seems steeped is that of only modern day authors of popular science (Pinker, Gladwell, Kahneman, et. al.) who have some interesting viewpoints, but ones which require at least a grounding in the liberal arts to pick apart. Several times Khosla demeans the liberal arts and uses the repeated example that a reader should be able to pick apart and think critically about articles in The Economist. To do this requires a knowledge of logic and rhetoric which are two of the pillars of what? — yes, the liberal arts!
He also seems unaware of big movements within the humanities and sciences like Bill Gates and David Christian’s Big History Project which are going a long way towards providing a more balanced education in history, economics, physics, chemistry, biology and evolution. I find here, no prima facie evidence of his knowledge of Thomas Kuhn or Karl Popper, which might help win me to his argument. In all, aside from the passing references to one or two recent works, this entire argument is not much different from many that could have been written at the beginning of the industrial revolution. How blind so many must be to seemingly think there’s something new here.
Most appalling to me here is that the author doesn’t seem to give even a passing nod or small wink to C.P. Snow or “The Two Cultures” [http:/
Yes, we certainly need more emphasis on the quadrivium portion of the liberal arts, and in particular mathematics and critical thinking which seem to have been left by the wayside. It is deplorable that the highest extent of mathematics that 99% of college students are exposed to terminates in the 17th century for the most part. Sadly, many college students are left without the ability to think critically and deeply, not to mention the hordes of students in America who barely make it through high school and don’t attend college. One also only needs to skim through recent issues of Nature [http:/
Yes, we need far, far more, but alas, this poor article only touches the tip of the issue and it sadly only does so with less than half of the picture.
The physics of life http:/
Paradox at the heart of mathematics makes physics problem unanswerable http:/
<i>Manny wrote: "I see Jacobson was an expert on Lie algebras... yes, that might certainly be useful! Thanks..."</i>
Many/most graduate math texts on Lie Groups/Algebras utilize manifold theory as their basis, which can make the field more daunting for physicists. As a result, I recommend looking at books that take a more linear algebraic bent to the subject, which can make some of the quantum mechanics related areas more transparent. I've used [book:Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|1051566], [book:Matrix Groups for Undergraduates.|1162374], and [book:Lie Groups, Lie Algebras, and Some of Their Applications|424103] for these types of applications and viewpoints.
About a year ago, I took a two quarter sequence at UCLA on Lie Groups from an ("easier") matrix group perspective which I imagine you may find somewhat useful in your reading on quantum mechanics. Several engineers, programmers, amateur mathematicians, physicists, and quantum mechanics enthusiasts had spent several years coaxing the professor into teaching it from this perspective. It was geared toward the advanced undergraduate level, and based on your comments here and your reading of Shankar, should be relatively easily followable.
We loosely followed Hall's textbook and portions of [book:Matrix Analysis|647523], which will give you some of the advanced linear algebra you could possibly be missing depending on your background. If it helps, here's a link to a downloadable pdf copy of the notes for the first class with the audio of the lecture embedded (using Livescribe.com digital pen technology which should let you click on the notes and jump to the audio portion related to where you've clicked): http:/
Having this background may make Weyl and Woit's developing textbook more easily manageable. My guess is that Woit is doing a more thorough job of developing the math than typical physics-oriented texts like [book:Geometry, Topology and Physics|439357] which do a lot of hand-waving at the math in an effort to get to the physics more quickly, but at the detriment of understanding what is happening mathematically.
I've dipped into some of Weyl's work in the past, but also keep in mind that some of his notation and definitions can be dated in relation to more modern presentations.
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].
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Reminder Los Angeles area mathematicians: Algebraic Number Theory starts next week @UCLAExtension http:/
Jacob Bekenstein, Physicist Who Revolutionized Theory of Black Holes, Dies at 68 #Obit #Physics http:/
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:/
For further references, I maintain a nice list of resources at Information Theory and Biology Resources [http:/
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:/
Big History specifically and purposefully looks at history from a much broader perspective, knowing full well that it's going to miss the complexity of large portions of what is going on at the much smaller scales. But it does this because different structures and ideas emerge at these larger scales. We see different types of things going on that we don't see at the smaller time scales.
If you analogize it to a chess game, there are some very simple and straightforward rules which are responsible for a terribly complex potential number of moves and games to be played. Most history has been studied at the level of "let's take a look at the first three moves of a game of chess" or let's look at the last three moves. This type of close examination doesn't allow for the broader sweeping view of entire games. Both Boris Spassky and a typical four year old could play the same 4 opening chess moves, but the complexity of competing against them diverges rapidly as the game progresses. By looking at the bigger picture, one can better appreciate what is happening later in the game. This also gives one a better platform for analyzing smaller portions of the mid-game as well.
I would suggest that Christian's audio lectures in [book:Big History: The Big Bang, Life On Earth, And The Rise Of Humanity|7956320] do a far better job of showing this viewpoint than his text [book:Maps of Time: An Introduction to Big History|745703].
I would say that big history is anything but "new age-y pseudo-science", and because it is rooted more heavily in physics, cosmology, biology, chemistry, microbiology and various other hard science disciplines, it is actually on more solid grounding than most other history-based courses. Those who have read C.P. Snow will recognize that big history has a grounding in both of the "two cultures" rather than being solely entrenched in just one. It's somewhat akin to taking a complexity theory-based approach to economics compared to the older 18th and 19th century simplified approaches followed by Adam Smith, Keynes, et al well into the 1980's. With approaches to economics exemplified by Kahneman/Tversky or more modern schools like the Santa Fe Institute, there is more being uncovered by shifting paradigms and taking more nuanced approaches to the subject than using the simplifications of the past.
Taking new approaches doesn't necessarily invalidate prior work in these fields, it just provides a broader basis for how all the various parts interact to create a larger whole.
Christian is without a doubt a historian through and through, and is quite upfront about his general lack of scientific expertise and background. He has however spent quite a bit of time working with and consulting physicists, chemists, biologists, and other scientists to supplement the appropriate portions of his bigger thesis. I would say though, that he's got firm footing in both of C.P. Snow's "Two Cultures."
Christian references Prigogine only once, though includes two Prigogine related footnotes in the last quarter of the text. He's not as Prigogine-centric as [author:César Hidalgo|13831217] is in his recent [book:Why Information Grows: The Evolution of Order, from Atoms to Economies|25472587], which touches on some of the related physics of information theory and entropy (and general complexity theory - although I don't recall him using this specific term) as they relate to economics. I'd classify Why Information Grows as a "big history" book, though Hidalgo wasn't aware of the conceptualization of "big history" when he wrote it.
I wrote a slightly longer review of Christian's book(s) on my blog: http:/
#bighistory
Over the past several years, there's been a growing movement of "citizen science" and a handful of related games which send data back to scientists to assist in various areas of work, including primarily genetics. Googling for "games" and "citizen science" will bring back some interesting possibilities for you. Many should be integrateable into a big history program, particularly the genetics related ones which explore some of the evolutionary related space along with curricula in biology, chemistry, and physics. In particular, students may be able to experience first hand how physics influences evolution in the mid-level thresholds from the start of life onwards.
Here's a particular example that was recently in Scientific American: http:/
#BigHistory
@preskill @Bill_Gross @seanmcarroll For the Los Angeles area Math/Physics/Tech community, would you mind RTing: https:/
@charlestonchoo To dip your toe back in, try @cesifoti's new book Why Information Grows: about math, physics, economics, HR, & leadership
This seems like an extension of Colin McNamara's post a while back, which Eric Waldstein is quietly referencing by copying the two of us: https:/
I'm also reminded of some of the simple examples that Warren Weaver and John Robinson Pierce presented shortly after the publication of Claude Shannnon's seminal work "The Mathematical Theory of Communication" which, because it underpins ALL of the modern day communications revolution (digital communication, computers, satellites, cell phones, etc.) since the 1940's, is often cited (along with religious texts like the Bible, the Koran, etc.) as one of humanity's most influential written works. [http:/
Shannon had previously taken George Boole's new "algebra" from the late 1800's (now known as Boolean Algebra, which underpins most of modern logic) and applied it to electronic circuits. Though somewhat technical most advanced high school algebra students should be able to read and understand most of Claude Shannon's MIT master's thesis, which itself is often cited as one of the most influential theses ever written. [http:/
In any case, Pierce in particular does some simple exercises relating to language in his book "An Introduction to Information Theory: Symbols, Signals and Noise" [http:/
This also to some extent, underpins our ability to look at the following paragraph and understand what it says. All the letters have been jumbled (mixed). Only the first and last letter of ecah word is in the right place:
I cnduo't bvleiee taht I culod aulaclty uesdtannrd waht I was rdnaieg. Unisg the icndeblire pweor of the hmuan mnid, aocdcrnig to rseecrah at Cmabrigde Uinervtisy, it dseno't mttaer in waht oderr the lterets in a wrod are, the olny irpoamtnt tihng is taht the frsit and lsat ltteer be in the rhgit pclae. The rset can be a taotl mses and you can sitll raed it whoutit a pboerlm. Tihs is bucseae the huamn mnid deos not raed ervey ltteer by istlef, but the wrod as a wlohe. Aaznmig, huh? Yaeh and I awlyas tghhuot slelinpg was ipmorantt! See if yuor fdreins can raed tihs too.
Pierce's book, while it does have some small amount of math, should be generally readable by most high school students. He also provides a simple model of language in his chapter "A Mathematical Model" which many may find interesting, or be able to pull some interesting examples out of. This particular book is also an excellent example of the intertwining of the ideas of physics, mathematics, engineering, psychology, thermodynamics as well as the ideas of emergence, convergence, and allows for the increased complexity and collective learning that underpins the philosophy of Big History.