Calculus Made Easy, 1910: All the little bits

Calculus Made Easy is a book on introductory calculus originally written in 1910 by Silvanus P. Thompson. It was such a well-received text for teaching calculus, that it can still be found in print today (it's now in the public domain and also freely available online).

The first bit of text from the first chapter is fairly intriguing. So much so, that I thought you might like to read it here:

Thinking of the integral of dx as an easy way of saying "the sum of all the little bits of x" is a really intriguing approach to explaining some of the most basic ideas of calculus. 

If you've struggled at all with learning calculus, then you may want to give Calculus Made Easy a read. It may be over 100 years old, but much of the writing is still of great value for the interested student.

Writing Motivation - "writer s surf"

writer s surf, by Medi Belortaja

I most certainly need motivation right now to continue working on a current research article, which is also going to be part of my graduate dissertation. Beyond my habitual coffee, I like to jump into my writing by first reflecting on the work ahead and by considering the thoughts and creations of others.

Sometimes, writing seems to come naturally to me, while there are other times where it most certainly requires some personal taunting, teasing, inspiring, and just plain ol' "sucking-it-up-and-getting-it-done". In looking for a quick spot of motivation on my day's writing, I found this beautiful cartoon by Medi Belortaja. He titled the piece "writer s surf". 

"Writer s surf" brings to mind those feelings of the surge of a wave under my board as the ocean picks me up and thrusts me toward shore (a feeling I haven't experienced in reality in too many years). The piece also evokes the feeling of being sucked in to a work of writing or of reading. Some writings (be they short articles, fantastical stories, scientific research reports, or epic journeys into other worlds) really do take the author/reader on rides, much as surfing a wave.

Belotaja brings forward in his cartoon the feeling of riding the page, surfing the written word, and embracing the motions of the cosmos as we engage with a moving force of nature. Write on!

Worthy Reading: "Writing Your Dissertation in Fifteen Minutes a Day"

My friend and prior office mate has recently made a big transition in her life. After successfully completing her Ph.D. and becoming Dr. Kelsey, she packed up her stuff in the office and said good-bye to Boulder, Colorado as she moved on to the next step in her career. After she had left, I took a look at where her desk used to be and noticed that she had left something behind. A little, well-worn book titled "Writing Your Dissertation in Fifteen Minutes a Day". It was a very serendipitous moment for me, as I've been having troubles with getting myself into the mindset of writing my Ph.D. dissertation. I read the book through right away, and I have found it to be a great reassurance and a swift kick in the butt for motivating me to write my graduate dissertation.

"Writing Your Dissertation in Fifteen Minutes a Day" was published in 1998, but I found it to be as fitting to writing a graduate dissertation now as it likely was then. The author, Joan Bolker, is a long-time teacher of writing and a counselor of writers. The way that she writes in this book makes it feel like a personal coach's practical approach to writing a dissertation. 

Bolker takes the reader through all of the stages of graduate school life and suggests a great methodology for embracing graduate work by writing on a constant basis. She admits in the book that we obviously can't just write a dissertation in 15 minutes a day (a dissertation is practically a book-length report of our studies), but she advises starting with writing a small amount of time each day and then building to longer times. I've found this to be a tremendously helpful suggestion. I've already started working my way up to writing my dissertation for an hour each day.

Graduate school can be a hellacious roller coaster of happiness, guilt, discovery, feelings of imposterism, excitement for learning new things, and depression. The good and the bad seem to balance themselves most of the time, but sometimes the emotional impacts of grad life can teeter us one way or the other in great, or sometimes not-so-great, ways. We get paid for part-time work, but are expected to be overworked. We honestly have it better than many people and usually don't have as much to complain about as it sometimes feels, but one definite thing that seems worth complaining about is the dissertation. It looms over our heads like a vast storm system that we can see rolling in from our bedroom windows and we just know is going to keep us inside all day. A lot of people will almost never again read their dissertation once it's written, while some people will take their dissertation forward and publish it as a book. Since I'm looking at my final year(s) in graduate school, now is most definitely the time to be speeding ahead on writing my dissertation.

Quite honestly, I wish I had found Joan Bolker's book years ago. It would have been very helpful to build up a writing habit for my dissertation as soon as I started grad school. Indeed, I now highly recommend just that (as well as this book) for anyone who is looking into graduate school. If I had built up a pacing of writing for a short time each day over the past few years, I might actually have most of a couple of dissertations written by now. Still, I look forward to getting deeper into my writing, to letting it guide my future work as the writing helps me discover what needs to come next, and to having some decent drafts of my dissertation for revision in the coming months. I think the serendipitous action of discovering this left-behind book, "Writing Your Dissertation in Fifteen Minutes a Day", has been one of the best "kicks in the ass" that I've had in my time as a graduate student.

Book Review: The Science of Avatar

I just finished reading The Science of Avatar by British scientist and science-fiction author Stephen Baxter. It was a most excellent read and definitely a book worth talking about.

Avatar

The film Avatar blew the socks off audiences worldwide back in 2009. Some people were even noted to have become depressed over not being able to personally experience the world that the film created. You've probably seen Avatar. If not, go watch it. You'll be happy you did, but here's a brief overview anyway:

Quick Summary: Avatar is a story about an ex-marine who travels to a world called Pandora to take control of an avatar, a genetically engineered organism that can be linked to a human "driver's" mind, so that he can interface with the indigenous, intelligent alien species (on whom the avatar is mostly based; though there are human components as well). This indigenous species are a humanoid group who call themselves the Na'vi (they are very reminiscent of various ancient Native American tribes). The ex-marine, Jake Sully, not only interfaces the Na'vi, but befriends (and eventually falls in love) with one of them. The conflict of the story is developed through the exploitation of Pandora's resources and disregard for the Na'vi and other wildlife by a company called RDA (Resources Development Administration) and their militarized force of employees (similar to our modern Private Military Contractors). The story develops as Jake Sully sides with the Na'vi in a battle against RDA and their "offense" forces. This is just a brief review of the story. If you haven't seen the film, go watch it. If you haven't seen it in a while, go watch it. I'll be here.

Seen it? Awesome!

I remember the first time I saw Avatar in the theater and thought to myself, "What a fantastical alien environment they've created here for the audience." I was stunned by the visual beauty and depth of the alien world, Pandora, and its similar-yet-alien biosphere (let's face it, most viewers would have a hard time watching the film if Pandora were truly alien - and so there are creatures that are very similar to life as we know it, and that's okay). I also really liked the story. 

It's a fairly typical story of the technologically-advanced bully wreaking destruction upon the indigenous ecology; we've seen this time and again. Many people made associations between the story of Avatar and that of stories such as Ferngully and Dances with Wolves. The blog Madatoms went as far as to create a rather humorous diagram showing some of the basic comparisons between Avatar with Dances with Wolves, Pocahontas, Dune, and Ferngully:

"Avatar Dances in Ferngully's Dune" from Madatoms

Sure, the story of Avatar was not exactly new, but I think James Cameron and the other creators of the film did an excellent job telling the story within the world they created. They gave us a thrilling look at a possible dystopian future for Earth (briefly), interstellar travel, a new biosphere on an alien world, genetically engineered surrogate organisms that we can literally control with our minds, a possible role of mercenaries in future warfare, and, probably coolest of all, a biosphere where organisms can neurologically link together (thus giving the entire biosphere a "brain/mind" of its own). 

Stephen Baxter did a fantastic job reviewing some of these key points in his book on the science of Avatar. I thought I'd share a few of his ideas as well as mine for each of the major sections of the book. 

Science fiction is often inspired by scientific discovery (and vice versa); The background is a painting by willroberts04 on Deviant Art

The Science of Avatar

Stephen Baxter is a fantastic author of hard sci-fi stories. He has a background in mathematics and engineering, but has made a name for himself by writing science fiction stories with alternate history and future history slants. His approach to The Science of Avatar comes from someone who understands sci-fi but who also has a great bearing on modern science. 

The primary sections of Baxter's book are "Earth", "RDA", "Venture Star", "Pandora", "Hell's Gate", "Living World", "Na'vi", and "Avatar"; these sections discuss the possible future of Earth, the resources extracted from Pandora, interstellar travel, the planetary science related to Pandora, the potential future of human colonization of other worlds, the biology developed for Pandora, the tribe of the Na'vi, and finally the nature of the avatars.

By breaking down the film into some key points, Baxter made it easier to tackle some of the more intriguing ideas that were developed by the film's creators. The book considers everything from the ecology of Pandora to the nature of the fictional resource "unobtainium" (which appears in various science fiction stories and in various forms; in Avatar, unobtainium appears to be a mineral that serves as a superconductor at room-temperature).

Credit: Dylan Glynn

We're only given a brief glimpse of the dystopian future of Earth in Avatar, although Jake Sully gives us an idea of what it might be like: "See, the world we come from, there's no green there. They've killed their mother..." Yes, that's a little vague, but it leads us to imagine a future Earth that has succumbed to global pollution, climate change, and sprawl such that ecosystems have been destroyed globally. 

An early draft of Cameron's Avatar screenplay had intended a bit more detail of future Earth: "Jake stares upward at the levels of the city. Maglev trains whoosh overhead on elevated tracks, against a sky of garish advertising... Most of the people wear filter masks to protect them from the toxic air... It is a marching torrent of anonymous, isolated souls." This sounds like a sad future for Earth, but it is by no means unheard of with regard to science fiction. There have been plenty of sci-fi stories that have considered dystopian futures for Earth (including the films Automata, Mad Max, Blade Runner, 12 Monkeys, Cloud Atlas, Elysium, Looper, and WALL-E). 

The future of Earth in Avatar appears to be one where industrial pollution and overpopulation have finally driven the Earth to the verge of global collapse. However, the real industrial trouble we see in the film is the extraction of resources (namely unobtainium) from Pandora without any regard for the ecosystems or intelligent species who live on that world. 

We currently extract an exceedingly large amount of raw material from the Earth each year to support our industrial, technological, and commercial infrastructure and to provide for the lives that many of us enjoy. It's hard to find good estimates of the actual amount of material we mine, but most numbers I've seen range in the billions to hundreds of billions of metric tons (around 60 billion metric tons of raw material each year sounds like a reasonable estimate). Although some nations have implemented various forms of regulation on mining operations to avoid industrial pollution or the devastation of ecosystems, mining across the globe still causes deforestation, acid mine drainage, and a range of other environmental impacts.

This is the Bagger 288. Like the other Bagger mining vehicles, this sucker is one of the largest land vehicles on the planet.

The Resources Development Administration (RDA) plays the "bad guy" in Avatar. They're developed to fit with the stereotypical image of a mining company or tech company that develops their product with little to no interest for the wellbeing of the ecosystem from which they take their goods. In The Science of Avatar, Stephen Baxter does a fantastic job of considering whether or not this is a potential future for resource development. Baxter mentions some of the current plans for resource development in our solar system, especially from asteroids. Given the potential for resource development organizations to act humanely and with vested interest in more than just profits, it's sad to admit that many times in the history of our species we have pushed forward for our own gain without regard for the rest of the natural world. Maybe our future of exploring new worlds will involve companies like RDA, though I hope by that point we'd learn to act better, especially when encountering new worlds.

The Hallelujah Mountains from Avatar

One of the cooler story points in Avatar is the nature of the Hallelujah Mountains. Much like the the floating island of Laputa in Jonathan Swift's Gullivers Travels, the Hallelujah Mountains fit with a common fantasy of islands floating in the sky, but in Avatar they give these floating mountains a scientific basis. The Hallelujah Mountains are composed mostly of the mineral unobtainium. The creators of Avatar built a backstory for the origination of unobtainium along with the formation of the moon Pandora and the planet it orbits (Polyphemus, in the Alpha Centauri system) - an object (either a planetary body or maybe a neutron star) impacted Pandora early in its formation causing an enrichment in the moon with the mineral unobtainium. As Stephen Baxter points out in the book, though, with a back-of-the-envelope calculation one could see that the magnetic field required to raise the Hallelujah Mountains into the air is thousands of times greater than the magnetic field here on Earth (not only that, but it would be far stronger than the magnetic fields produced by our Sun). Although the film's creators have built a Polyphemus and Pandora system with intense gravitational fields, it's still not likely to be enough to raise the Halleljuah Mountains. However, sometimes the fun of merging science fact with science fiction is having the creative license to push the boundaries of truth a bit. Which might be what the film does with the idea of the avatars.

Jake Sully checks out his avatar in its amino tank

The avatars are pretty sweet. It's not unthinkable that we may be able to port ourselves into created bodies in the future. We're already pushing into more immersive media with improved 3D films, steps towards better virtual reality with tech like the Oculus Rift, and developing CAVE systems (I had the chance to try one of these immersive systems at the Desert Research Institute some years ago). With increasing interest in virtual worlds (like Second Life), I can definitely see people in the future looking to find ways to take on new personae, possibly through the control of another body in the real world. In fact, a pretty fun movie came out not too long ago looking into just that: Surrogates is a 2009 film featuring Bruce Willis which portrays a future where nearly everyone interacts in the real world through surrogate robots. But will we get to the point where we can take control of a biological form like that of an avatar?

Baxter points out in The Science of Avatar that the basic biological structure of the organisms of Pandora does not utilize DNA/RNA and proteins the way that Terran life does. This means that to make an avatar requires an understanding of Pandoran biology to the point of genetically engineering organisms, but, moreso, it requires that the scientists who create the avatars are capable of appropriately decoding human and na'vi heritable traits (not so hard for the human side these days) but then that they be able to logically code these two forms of life together to make a third form that becomes the avatar. It doesn't seem too outlandish, but it definitely would be a huge marvel of science to attain that level of biological engineering. It seems to me that if we were to get to that point then we could easily have fixed Jake Sully's paralysis or even may finally develop gene therapy to the point where we could make it so that human colonists could live naturally on the Pandoran surface.

Avatar ends with Jake Sully having his conscious self fully transitioned from his human body into his avatar body. It's an interesting idea, for sure. Although there really isn't a good reason to believe in dualism (the idea that our physical minds and our conscious selves are different), a lot of people have been wondering lately if we may be able to upload our brains to computers in the near future (neuroscientist Randal Koene wants to see this happen, though there are other who doubt it's even possible). The 2014 film Transcendence, starring Johnny Depp, is one of the more interesting takes on this idea in recent media. Not only does Avatar force us to question whether we could create a newly biological form like an avatar but also whether it would be possible to code everything from the structure of our brains into the brains of one of these organisms. A very intriguing consideration.

The Science of Avatar considers all of these points and more. For instance, Baxter goes into what we currently know about the possibilities for life beyond Earth, the relativistic effects of near speed-of-light travel, the history of rocketry, what we currently know about exoplanets, the potential for terraforming, and more. The Science of Avatar was definitely a fantastic read. It's inspired me to watch the film again (but this time paying a little bit closer attention to the world of the film). I think Baxter could have filled the book with so much material as to make it a tome of information, but he wrote it so that it's accessible to just about anyone and can be read in as little as one sitting. Now that I'm finished writing about the book, I think it's time to watch Avatar again. 

In case reading this hasn't inspired you to watch Avatar again yourself, maybe having a listen to the film's soundtrack will help:

Plutarch Dies at the End

Wallowing in my own self-pitty last night due to the continued presence of my runny nose, cough, and stuffed-up sinuses from this damned sickness I've had, I sought out a horror film to watch alone and in the dark.  I came upon John Dies in the End while searching through my Netflix queue.  I had added said film because it sounded promising, though I couldn't recall having ever seen a trailer or read any reviews.  So I jumped to the ol' Google and found the film's trailer to be enticing.  The film is definitely worth a watch for anyone who enjoys humorous comedy, but I'm not offering a review of the film here.  Rather, I'm writing this because of one interesting part of the film: the prologue.  

The opening of the film presents a simple thought experiment in a not-so-simple and enjoyably quirky way:

What do you think?  If you're anything like me, the first answer that comes to your mind is an obvious "no".  The axe has been completely re-constructed, so the original parts that were used to behead the now-rotting, corpsified zombie-dude are no longer in your possession and are most likely just adding to the mass of waste at some local landfill.  

However, that's not the reason that I think the answer of "no" is astoundingly obvious (you might not have caught the primary reason for the answer being "no" on your first watch of that video; if so, watch it again.  Good to go?  Awesome).  Hopefully you saw that the primary reason that the answer to the question is "no" is because what slew Swastika-Tongue in the first place was one, some, or all of the eight bullets that you had shot him with before using the axe to remove his head (like I said, it's a simple riddle).  However, if you take away the obvious answer and just allow yourself to assume that the real question of the riddle is whether or not the axe you now hold in your hand in the presence of Zombie-Swastika-Tongue is the same one that you had used the previous winter to remove the head of his former self, then you have another riddle that is really a re-hash of a much older thought experiment: The Ship of Theseus.

The Ship of Theseus is a thought experiment proposed by Plutarch in the first century C.E.  It goes something like this: the ship in which the hero Theseus and the young Athenian men returned from Crete (see the myth of Theseus and the Minotaur) was honored by Athenians and kept in good repair in the harbor of Athens for many centuries.  Over time, as parts of the ship would degrade, they were slowly replaced, so eventually there were not many of the original parts of the ship remaining.  The question then became "Is the ship, after replacing part for part over time, still the Ship of Theseus?"

The thought experiment, as it stands, really questions the value an object has based upon its parts.  It's a question of philosophical identity.  

There are other versions of this thought experiment.  Some of them predate Plutarch's Ship of Theseus.  For instance, there is a version in which Socrates and Plato each slowly exchanged the parts of their carriages such that the parts that once were in Plato's carriage have been completely replaced with parts from Socrates' carriage and vice versa and then the question is posited as to whether Plato is now using Socrates' carriage or if he's still in his own.  

Other variants of the thought experiment have come since the time of Plutarch (with some interesting additions).  The version that appears most similar to the prologue from John Dies at the End is the one known as "My Grandfather's Axe": my grandfather had an axe which he gave to my father.  My father replaced the haft before giving the axe to me.  I had to replace the head.  Do I still have my grandfather's axe?

One of the more interesting variants of this thought experiment was proposed by Thomas Hobbes, the 1700's English philosopher and author of Leviathan.  Hobbes' addition to the thought experiment works this way: you have the Ship of Theseus.  You slowly take one piece of the ship off and replace it with a new piece.  The old piece you keep.  You continue in this manner, replacing pieces of the ship and saving the removed pieces.  As this is happening over time, you take the pieces that had been removed and use those pieces to build a new ship, of exactly the same structure and design.  By the time you have replaced the final original piece of the ship, you now have two identical ships.  Which one is the Ship of Theseus?

Here's a fantastic breakdown of the original Ship of Theseus thought experiment and the Hobbes version from Wireless Philosophy:

The thesis of Joseph Butler, as reviewed in that video and suggesting that "objects persist in only a loose and popular sense", seems like a nice way to shrug off the problem as not being a problem in the first place.  This is usually a fun approach to a lot of philosophy problems since a lot of the time it seems like there's no resolution to a lot of philosophy problems.  

The reason I like this thought experiment, be it after replacing axe parts following your unexplained need to slay and behead some dude with a swastika tattooed on his tongue or replacing pieces of Theseus' ship, is because it questions identity.  We are constantly shedding cells and gaining new ones, so are we ever identical with who we were previously?  Darth Vader was almost fully replaced by mechanical parts, so was he still Anakin Skywalker?  The philosopher Wittgenstein might have thought these questions were balderdash ("Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing") and, if that were the case, he might have been right.

It would be interesting if we could just say that something is such because people agree to call it such.  Maybe the Ship of Theseus is really just whatever anyone decides to call "the Ship of Theseus".  Maybe Darth Vader is Anakin Skywalker because someone calls him so.  If that were the case, then the answer to the thought experiment as proposed in John Dies at the End might be that the axe you're holding in your hand is the same axe as the one that beheaded Swastika-Tongue because his zombie has now said that it is the same axe (it might be a good conclusion since chances are you should be more worried about dealing with said zombie before considering philosophical puzzles anyway).  

However, I still feel like the answer would be "no".  Even if you had killed the dude with an axe in the first place (and not with one, some, or all of those eight bullets), the original axe has been completely replaced.  The answer feels like "no" because none of the original axe remains and there are only two major parts of the axe to replace.  When the problem is introduced as in the case of the Ship of Theseus, where the object is replaced a small amount at a time, that's when it gets harder to decide when to even consider the ship to no longer be the original.

Maybe one of the more interesting answers comes from those who like to add the temporal dimension to the consideration, such as in the Worm Theory as presented in the video above.  When we question the temporal aspect of an object along with it's identity, we start hinting at a possible answer to the question (see Temporal Parts at Stanford Encyclopedia of Philosophy).  We can say that any one thing is only ever fully identical to itself at one point in time, but then at other points in time it can only be similar to itself.  Then using a name to define something falls back to the "loose and popular" context that Butler suggested.  That sounds just about right, honestly.  The answer that would suggest then is that the Ship of Theseus was only the same ship in the sense that it bore similarity to itself over time and that people still called it the Ship of Theseus is the only thing that made it the Ship of Theseus.  In that case, the axe that hewed the head laden with a swastika-marked tongue and the axe that you now have to defend yourself against the zombie at the door are only similar, and maybe you would call it a different axe since you know you've replaced the parts but the zombie calls it the same axe since it looks similar to the original.  Not a very rewarding answer, but an answer nonetheless (and now you can get on with hacking down the zombie as he is more than likely about to come at you).

There's rich food for thought there.  Maybe Wittgenstein is right and it's nonsense to even worry about two things being identical.  That seems to fit well with the answer that considers the temporal aspect to mean that an object has unique temporal parts during its existence (look up perdurantism).  Whatever anyone's consideration of this little thought experiment may be, I think we can all agree that it's a lucky thing we don't truly live in a world where a guy can get shot 8 times and have his head chopped off with an axe but then still find a way to come back from death and then sew his head back on before coming to find us with the likely intent of exacting revenge.

Update (22 October 2017): I re-shared this recently and have had several people ask if I've read the book John Dies at the End. Happily, I can say, yes, I have, and I've also read the sequel, This Book is Full of Spiders. A third book in the series, What the Hell Did I Just Read: A Novel of Cosmic Horror, just came out this month. Looking forward to reading that as well!