Sunday, March 27, 2016

218: Itching for the I Ching

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Recently I was reading a biography of John Cage, the quirky avant-garde 20th-century classical composer who I have mentioned in a few previous podcasts.    One of the most fascinating aspects of Cage’s composing was his attempt to introduce random elements into his music, starting in the 1950s, in order to free himself from preconceived patterns.   He experimented with numerous sources of randomness, including die rolls, ambient noise from the environment, and even imperfections in the paper he was writing on.   But one method that absorbed his interest for a long time was the ancient Chinese book of divination known as the I Ching.    Once Cage discovered the I Ching, it became his main guide in the selection of random numbers.   Some of his compositions required thousands of random numbers to be completed.   As a result, many of his visitors noted that anyone who stepped through Cage’s door was soon drafted into tossing coins for a few hours to generate I Ching trigrams for use in Cage’s music.

The I Ching,  or Book of Changes, is said to be one of the world’s oldest books, written around 3000 years ago.   It is based around interpreting the significance of various patterns of whole and broken lines, traditionally determined by tossing yarrow sticks, or by an equivalent method based on tossing coins.   The most fundamental set of patterns generated by the I Ching are the eight trigrams, patterns of three lines, which may each be solid or broken.   Each of the eight trigrams has several possible meanings, such as the mind, the spirit, emotions, or bodily sensations.   Pairs of these trigrams can be combined into one of 64 possible hexagrams, for an even richer set of possible meanings to interpret.    Being a listener of this podcast, you have probably realized by now that the combinations of three or six lines, each of which can be solid or broken, is precisely equivalent to a three- or six- digit binary number, if you interpret the solid lines as 1s and the broken ones as 0s.   So essentially, the I Ching is a divination system based on random numbers, expressed in binary, or base-2, notation, between 0 and 63.     Now I’m sure Chinese scholars will say I’m shortchanging the deep philosophy of the system, since these random divinations are accompanied by thousands of pages of interpretive text.   But it’s undeniable that these numbers are the basis.

Because of this numerical aspect, it’s actually not uncommon among historians of science to credit the ancient Chinese for first coming up with the idea of the binary number system, which is critical to modern computers.   Personally I’m a bit skeptical of this aspect of I Ching studies:  while the ancient book discussed many ways to combine and interpret the trigrams and hexagrams, they weren’t using these as a basis for a numerical system or for calculations of mathematical significance.   On the other hand, the legendary Gottfried Leibniz, co-inventor of calculus and early designer of ideas for calculating machines, did credit the I Ching for inspiring the idea of binary arithmetic in some 17th-century writings.    I think this may have been largely due to the fact that there were no other precedents for this idea in Leibniz’s time, though.   Most likely, he was astonished to see some basic ideas of his base-2 new arithmetic system in this ancient text, though he probably would still have developed the binary system if unaware of these writings.

As I read more about the I Ching online though, I was surprised to see that its description as a system of binary numbers is actually a bit of an oversimplification.   The reason is that the I Ching describes a complex procedure for generating the lines,  not the simple 0/1 coin toss you would have guessed.   When using the coin method to generate a solid or broken line, you are to toss three coins, with one side of each coin considered the “yin” side and the other the “yang” side.   Each yin toss has a value of 2, which each yang toss has a value of 3.   You then add the values together, to get a total between 6 and 9.   A 6 or 8 is a broken line, while a 7 or 9 is a solid line.    But there is more to it:  the less probable 6 or 9 values indicate that their line is “moving”, while the 7 or 8 lines are “stable”.   While the symbolic trigrams or hexagrams are still drawn with mainly solidness or brokenness visible, you need to note which are moving and which are stable, as this can make a major difference in the results of your divination.    Thus, one might say that the I Ching is really a base-4 divination system rather than binary.   In some of his writings, John Cage actually claimed to be using these stable and moving aspects to guide his randomly generated music.

But on top of the base-4 complication, there is yet one more mathematical wrinkle.   While the totally random methods such as tossing sticks and coins are the most commonly used, one online scholar notes that the I Ching describes another, more complex, method for generating the next 6/7/8/9 line number based on the current one, using a series of mathematical calculations.    These calculations are actually pseudo-random, similar to the Linear Congruential Generation algorithms used by modern computers.  This means that the results are deterministic, though hard enough to predict that they appear random.   Furthermore, according to this online analysis, the official I Ching algorithm is somewhat biased:  while solid and broken lines are equally likely, the 9 is much more probable than the 6, meaning that solid lines are significantly more likely to be “moving” than broken ones.   I’m sure New Age mystics would say there is some deep meaning in this, and that Yang is more mobile than Yin, or something like that.   Being a bit more of a cynic, I would lean towards the interpretation that the ancient Chinese were just not mathematically advanced enough to notice the problem.

Anyway, I’m not sure how all this was supposed to lead to John Cage generating better music:  while I really enjoy reading about his bizarre random methods, trying to listen to the resulting music for more than a few seconds at a time is not a very pleasant experience.   It’s also amusing that Cage put so much energy into generating numbers using I Ching methods, when he could have bought books of pre-generated random numbers, which were available for engineering and cryptographic applications for decades before the advent of modern computers, and saved a lot of time.   But I wonder if Cage’s avant-garde admirers would have claimed to like his music as much, if he told them the source was the Rand Corporation rather than ancient Chinese mysticism.

And this has been your math mutation for today.


Friday, March 4, 2016

Where Are The Missing Episodes?

Hi everyone--  you may have noticed that a bunch of episodes recently disappeared from this blog & the corresponding podcast feed.    Each of the disappeared episodes is included in the new Math Mutation book, which will be released in Spring 2016, and is already available for pre-order from Amazon.

Apologies to those of you who  are searching for those episodes.   I really wanted to keep them up on the web, but the publisher's contract requires their removal, out of fear that their free availability would hurt book sales.   I tried to push back, but it's a standardized contract & there's not much wiggle room.

Note that the other episodes will remain on this site, and I will continue to release new ones (at least) monthly and post them as usual.

Thanks again for your interest, and please check out the book page on Amazon!

Wednesday, February 24, 2016

217: The Oxford Calculators

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Before we start, just a quick reminder that I’ll be taking down a bunch of old episodes from the web any day now, due to their inclusion in the upcoming Math Mutation book.   So be sure to download any old ones you’re interested in ASAP.    Now, on to today’s topic.

Often we think of the centuries before Copernicus and Galileo as a huge black hole of science, with little of note happening other than medieval knights spending all day jousting at dragons.   But actually, the pioneers of the Scientific Revolution who became household names, like Copernicus and Galileo, did not spontaneously arise from a vacuum like some quantum particle.   They owed a major debt to a number of earlier scholars who re-examined the knowledge passed down by the ancients, and explored the possibility of more carefully applying mathematical methods to understand the natural world.   One of the most important such groups were the cluster of philosophers from the 1300s, led by John Dumbleton, Richard Swineshead, Thomas Bradwardine, and William Heytesbury, who became known as the Oxford Calculators.   

The most critical contribution of the Oxford Calculators was probably the concept that nearly any physical attribute worth studying could be measured and quantified.   Nowadays we take this for granted from the beginning of our science education in school, but this was by no means obvious for most of human history.   Among the ancient Greeks, Aristotle had discussed measurements related to size and motion, but had not addressed the issue of whether other aspects of the physical world could be measured numerically.   The ancients seemed to have an implicit assumption that many aspects of reality, like heat and light, could only be discussed qualitatively.    The Oxford Calculators challenged this approach, and tried to quantify their discussions whenever possible, believing that it really should be possible to quantitatively specify nearly everything you could discuss.   They were primarily thinking of themselves as theologians and philosophers, so did not conduct the actual experiments that would be critical to the real advances of the later scientific revolution, but even thinking in these terms was a major step.

Another important contribution was their willingness to reopen discussion of matters that had been supposedly solved by the ancients, and question teachings that had been passed down since Aristotle.   One example is Artistotle’s belief that the velocity of an object would be proportional to the force exerted on it, and inversely proportional to its resistance.   In modern terms, we would write this as V = kF/R, with V = velocity, K = some constant, F = force, and R = resistance.    One of the Oxford Calculators, Thomas Bradwardine, pointed out that there was something fundamentally wrong with this formula:  if the force and resistance precisely balanced, we should expect an object not to move at all.   However, the Aristotelian formula would impart a constant velocity, since F/R would equal 1 in such a case.   Bradwardine proposed an alternate formula, which we would now write as V = k log (F/R).   This was also horribly wrong, but at least fixed a key flaw in Aristotle’s teachings, so we need to give him some credit.   Since the log of 1 is 0, Bradwardine’s approach would at least predict objects not to move when force and resistance are balanced, a pretty important characteristic for a basic theory of motion.   This wrong formula also helped advance the definition of logarithms, a major mathematical building block for further developments centuries later.

The Oxford Calculators’ most significant concrete contribution was probably the Mean Speed Theorem, an idea which is often incorrectly attributed to Galileo, but was actually first published by William Heytesbury in 1335.  This came out of attempts to understand accelerated motion, quite a challenge before the development of calculus, our key tool for understanding changing quantities.   You may remember the basic formula for the distance traveled under constant acceleration:  S = 1/2 A T squared, where S is the distance traveled, A is acceleration, and T is time.   Nowadays, we can trivially derive this using calculus.   But Heytesbury was able to essentially figure out the same formula centuries before calculus was available, reasoning that the total distance traveled by an object under constant acceleration would be equal to the average speed multiplied by the time.   In modern terms, the average speed is just 1/2 A T, and the time is T, so Haytesbury’s reasoning leads us to the same 1/2 A T squared formula, without the need for calculus.  Heytesbury didn’t write it precisely in the modern form, but managed to achieve the correct result, centuries before Galileo.    

Now, we should point out that while they provided critical stepping stones, the achievements of the Oxford Calculators do not diminish the significance of Galileo’s work or others involved in the later Scientific Revolution.   If their successors sometimes failed to recognize or mention them, it wasn’t out of malice or deceit, but mostly because the modern system of scientific citation and of publication credits had not yet been established.    With modern scholarship, we can give the Oxford Calculators the credit they deserve, without needing to take away credit from anyone else.   They played a small but critical role in laying the foundations for the rapid mathematical and scientific advances that continue to benefit and astonish us in our own time.

And this has been your math mutation for today.


Wednesday, February 3, 2016

216: Bowie Meets Escher

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Apologies that this podcast is a little late.   A bunch of my time has been taken up by a fun new project:  putting together a Math Mutation book.   Yes, I’ve actually found a publisher crazy enough to want one!  I’ll announce more details here as we get closer to the book’s release.    One slight downside though:  due to the realities of publishing contracts, any episodes that end up in the book will have to be taken down from the web.   Sorry about that; I pushed back a little, but they were firm on this aspect of the contract.   So, if you were planning to catch up on old episodes, be sure to download them ASAP, before they start disappearing.   Now, on to today’s topic.

With the sad passing of the musical pioneer David Bowie, it seems appropriate to try to create one more Math Mutation episode that focuses on him somehow.   You may recall that he has come up before, in our discussion on random song selection back in podcast 193.   But this time I thought it would be fun to talk about something a little different:  the climactic scene in the movie Labyrinth.   As you may recall, Labyrinth was a 1980s movie that starred Bowie, Jennifer Connelly, and lots of muppets.   Bowie played the Goblin King in this move, where the only other speaking human was Connelly, as a scared teenage girl trying to rescue her baby brother, who Bowie had kidnapped.   To catch up with him, she had to traverse a bizarre maze filled with strange traps and spooky muppet monsters.   When the girl finally catches up with the Goblin King, he is in a huge maze with staircases in every direction, clearly inspired by M.C.Escher’s classic 1953 lithograph Relativity.

Before we get into the movie, let’s talk about the original lithograph.   Relativity is one of Escher’s less absurd works, in that the 3-D structure he depicts is actually self-consistent, and can theoretically be built in three dimensions.   It centers around a triangular group of staircases, with various doorways, windows, and secondary staircases nearby, and faceless figures walking up and down in various locations.   Where the Escher mind-bending comes in is that there are multiple distinct sources of gravity in the picture, with each of the walking figures independently subscribing to one or the other, even if on the same staircase.   For example, in the staircase at the top, two figures seem to have their feet near the same stair, but the “tops” of the stairs to one of them are the “fronts” of the stairs to the other, so they are standing perpendicular to each other.   Similarly, the doors and windows each seem perfectly reasonable on their own, but all together don’t make much sense, creating multiple different impressions of which way is “up”.

As with many Escher prints, generations of college math majors have put this poster up on their walls, and enjoyed the absurd questioning of basic artistic and mathematical rules.   But is there a deeper meaning to the lithograph?   One blogger suggests that it is questioning the nature of who actually controls reality:  “Who controls the world, and reality, in this painting? It seems that the human-like figures do. By going about their everyday business they show no desire to change it. Perhaps Escher is trying to say something about human nature.  It seems as though as long as these beings can eat, walk, read, and go about their normal lives they are content to go along with the distorted world they live in, however ridiculous it is…  If we care enough to wake up and see what's going on, we will have the power to change it.”    This seems to be the most interesting analysis I can easily find on the web, and ties in nicely with some of the fan interpretations of Labyrinth.

Getting back to the movie:   as I mentioned, the climactic scene involves a chase through a Relativity-like maze, complete with inconsistent gravity from various angles.   This was before the days of cheap CGI effects, so the filmmakers actually built a large Relativity-like set, and used camera tricks to make it look like Bowie, Connelly, and the baby were subject to varying gravity in multiple directions.   Like most of the movie, this scene seemed to come out of nowhere, with nothing earlier specifically alluding to it.  Many critics panned the movie for basically that reason, just being an accelerating series of oddities with no underlying rules— initially it wasn’t much of a box office success, though it is now considered a cult classic.  In the years since it came out, legions of fans have tried to discover a deeper underlying meaning.   

The easiest interpretation is that this is just another in a long line of absurd children’s tales, with crazy magic and monsters that don’t really have much deeper meaning.   A slightly more convincing interpretation is that it’s a coming-of-age tale, where the girl learns to take on the maturity and responsibility to make her own decisions.   This would put it squarely in the typical space of many popular fantasy stories.   However, there are darker possibilities.   One website, “Vigilant Citizen”, claims that the entire movie is an allegory for mind control, with each of the obstacles in the labyrinth being somehow related to the internal world of a brainwashing victim.    This also ties in well with the resolution of the scene, where Connelly tells Bowie “You have no power over me”, and as a result finds herself safely teleported home with her baby brother.   But is an interpretation this dark really appropriate for what is largely regarded as a children’s movie?

Ultimately, I’m not sure which view is correct.  Was the final Relativity stairway chase in Labyrinth a metaphor for pulling free of mind control, or a gentler coming-of-age ritual?   Or was it just another case of the legendary Bowie choosing to be weird for weirdness’s sake?    We’ll never be able to answer those questions completely, but I have no doubt that future generations will continue to enjoy both Escher’s Relativity and David Bowie’s Labyrinth.

And this has been your math mutation for today.


Wednesday, December 23, 2015

215: It's Not A Conspiracy

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Often when we are watching science fiction shows or movies, they imagine fantastic devices that are just at the edge of possibility according to modern physics.    Ideas like death stars, faster-than-light travel, or teleportation are all things we are unlikely to see for many years, but can’t totally dismiss as potential inventions of the far future.   But every once in a while, science fiction posits something so totally absurd that I can’t help but laugh.     The other day, while watching an episode of Star Trek: Deep Space Nine, I saw an idea in this category:   a small device that would alter the laws of probability.

The plot of this episode, titled “Rivals”, involved a series of strange events on a space station.    Gamblers would win against impossible odds, the infirmary suddenly filled with victims of freak accidents, a computer search of an unsorted file list would instantly discover just the right data, and a crew member who stunk at racquetball suddenly started making impossible trick shots.    As expected for a Star Trek episode, this all boiled down to an alien technological device, in this case one that could alter the laws of probability.    Once the captain found and destroyed the device, everything could go back to normal.

Now at first you might just label this as another piece of random technobabble used to advance a sci-fi plot.    But I think this hits at a popular misconception about probability.   Many people think then when the odds of some event are low, it’s some kind of conspiracy of the universe against them.    So if you have a one-in-a-million chance of making a trick racquetball shot, why shouldn’t some gizmo be able to alter that probability and help your game?    But actually, the low probability just reflects the fact that there are a million different ways you can hit the ball, all of which are equally valid executions of the laws of physics.   The universe doesn’t really care about the one shot that we abstractly label the great trick shot:  it’s just another in a huge sea of possibilities.    Tiny variations in the angle of your aim and the force of your swing can make a major difference in where the ball goes.   You can imagine a million parallel universes in which you hit the ball, and only one of them involves you making the shot successfully.     Which one is it?   You might care, but it’s none of the universe’s business.  All are roughly equally likely, depending on your exact position and momentum when you hit the ball.

So what would it mean for an alien device to change the laws of probability?    It is theoretically possible for some specific technical gizmo attached to your arm to bias it towards the successful racquetball shot.    But a generalized probability gizmo that could enhance your shot, improve computer data searches, enable victories at roulette, and cause freak accidents?   How would this device know how to bias the universe precisely in ways that we label as “unlikely” results, in all these diverse domains?    In the racquetball shot example, we’re estimating that there are a million possibilities, so *every* shot you take will be a one-in-a-million result:  no matter what ends up happening, there was only that same tiny chance of that specific shot occurring.    So a machine that caused an “unlikely” result for events would be useless at choosing the victory shot for you— every non-victory shot is equally as likely, and there are 999,999 of them.    On the outer fringe of possibility, perhaps if the machine had full artificial intelligence capability, there might be possibilities here.  But then it would be a techno-gremlin notable mainly for its intentional meddling in other’s lives, and nobody would describe this strange robot’s actions as a change in the laws of probability.   

We can see similar laws at work in many parts of our daily lives.   Is my daughter engaged in a specific effort to make her room messy?       If I don’t enter my daughter’s room for a few weeks, books, toys, and clothes are strewn about everywhere, with barely a path available to the bed.     But is she deliberately trying to make the room messy?   Though there is some doubt about her intentions here, I think it’s really a result of the fact that there are many more configurations of the room that are messy than non-messy.   You can put a sock in 1000 locations that are not the sock drawer, but need to spend some energy to intentionally put it in the sock drawer if that’s what you want.    Combine all the objects in the room, and it seems there are uncountably more ways for the room to be messy than clean.    Thus, without intentional action to drive it towards one of these clean configurations, small continuous changes will probabilistically lead to complete messiness.   So she doesn’t need to be trying, the messy room is just something that will happen with high probability unless someone invests energy to prevent it.   This is similar to the basic principles that the thermodynamic laws of entropy are built upon, though I won’t say much more about that right now, due to the large number of websites that seem to admonish us against the messy-room metaphor for this concept.   

So, in short, when some event we want has a low probability of occurring,  it is usually just a measure of the fact that there are many possibilities of what can occur, and only a small number have property that is significant to our human interpretations.    The universe doesn’t know which one we want, so it has no particular reason to deliver the desired outcome.   Imagining a technical device that can alter the laws of probability is like imagining a device that can make 2+2=5, or can cause triangles in a Euclidean plane to have angles totaling 190 degrees:   it simply violates the fundamental mathematics of the situation.     Over the next few millennia, humanity may see miraculous inventions such as laser pistols, teleportation, starships, or halfway decent William Shatner albums, but I think we can safely bet that no machine will ever alter the laws of probability.    If you’re frustrated sometimes by things not going your way, you should take comfort in the fact that there are many ways things can go, and figure out what you can do to reduce the number of possible bad outcomes.   The universe is not engaged in some kind of conspiracy against you that needs to be fixed with a magical gadget.

And this has been your math mutation for today.


Sunday, November 29, 2015

214: In Search Of The Ultimate Math Game

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With the holiday season upon us, many of you out there are probably giving or getting new tech gadgets as gifts.    Once you unwrap your fancy new iPad, iPhone, or Android tablet, you’re probably asking yourself, “Now what do I do with this?”    While you will probably be downloading lots of fun games and apps, you will somehow need to justify all the hours you spend in front of the screen to your family.    One of the best ways to do this is to install a few math-related games, and provide some educational value for your children.   But there is a bewildering array of supposedly educational games available for these systems.   How do you know which ones to get?     Today I will share some suggestions based on my experiences.

The first thing I should point out is that there are hundreds of games out there that are basically glamorized flashcards, presenting math problems directly and giving some kind of in-game reward for correct solutions.    For example, they will put up a math problem, like “What is 5 x 5?”, and if the answer is correct, the player gets a few points.   These points can then be traded in for virtual stickers, virtual ammunition against alien robots, or similar rewards.   While there is nothing wrong with this type of game, and they have the advantage of being able to easily draw on large libraries of problems for different skill levels, I don’t find them very exciting.    My daughter will play them if I tell her she needs to practice her math, but doesn’t usually come to me asking to play them.   What I really want are original games that both teach math and can stand on their own as fun games.   Fortunately, I have found two games that fit these requirements.

The first game I want to highlight is called “DragonBox Elements”.    This game is designed to teach the basics of geometric proofs, a seemingly advanced topic, but they present it in a very accessible and intuitive way.    Each basic shape, triangles and quadrilaterals, can summon a basic type of monster related to that shape.    So if you can identify a quadrilateral among the shapes on the screen, you trace it out and summon a quadrilateral-monster.   Line segments and angles are also marked with colors, such that any two objects with the same color have equal length, and you can upgrade the monsters to “special” ones using these.   So, for example, if you notice a triangle-monster has two equal sides, you can click on them to upgrade to the slightly more powerful isoceles-monster.   The monsters also have powers, which essentially invert this process:  so if you have been given an isosceles-monster and its two equal sides are not yet colored, you can click on the monster and the two sides to mark them as equal.   They also introduce other powers related to ideas like opposite angles, radii of circles, and parallel lines.   So the basic Euclidean concepts of definitions, axioms, and theorems have been transformed into monsters and powers.    I’m not totally sure how this will translate to actual proof skills when my daughter reaches that level of math class, but laying the foundations at such a young age can’t hurt.   And more importantly, she loves this game, even asking to replay all the levels at “hard” difficulty after beating it once.

The second truly engaging iPad math game I have discovered is called “Calculords”.   This is a card game, where each turn you have a bunch of cards in hand that you can use to summon creatures for battle.   There are two types of cards, number cards and creature cards.   It’s not a simple energy system like in most popular collectible card games though:   in order to summon a creature for battle, you need to add, subtract, and multiply number cards to reach the creature’s number.   The creatures are then placed on a lane-based battlefield, where they fight the evil monsters summoned by an alien enemy.    For example, suppose you have a Hungry Blob card, a monster with a summoning cost of 15, and your number cards are 3, 3, 4, and 1.   You can form a 15 using 3 x 4 + 3, so you can play those cards to summon your blob.   But an additional wrinkle, adding to the mathematical challenge, is that you also gain extra bonuses if you precisely use up all your number or creature cards.   So a better move would be to play 3 x 4 x 1 + 3, which still reaches your 15, but uses up your numbers.    Since you have 9 creature cards and 9 number cards on each turn, the number of potential choices and calculations is quite large, and the strategy to summon the best set of monsters while trying to use up cards to get the bonus can get very involved.    But the game offers many enemies at a variety of difficulty levels;  my daughter has been playing at the easier levels since she was in 2nd grade.   This is another game that she and I have found quite addictive, and an amazing way to get her to eagerly practice her basic arithmetic.   And at the top difficulty levels, even I find it challenging, when I sneak in a chance to play on my own.

So, in short, these are the two truly original smartphone/tablet math games I currently recommend for elementary-age students:   DragonBox Elements and Calculords.   Naturally, these are heavily influenced by my 4th-grade daughter’s tastes, and their effectiveness probably varies a lot at older and younger ages.   DragonBox elements provides the amusing and engaging transformation of Euclidean definitions, axioms, and theorems into monsters and powers.   And Calculords provides a strategic challenge involving arithmetic calculations that is accessible to young children at lower levels, and fun even for adult math geeks at the hardest settings.   If you have kids at the upper elementary level who could use some extra math practice, be sure to take a look at these excellent games.   Also be sure to post reviews on iTunes or similar sites if you like them, as this will increase the chance of further games appearing from these talented authors..   And as always, I’ll be interested to hear from you on this topic:  with such an overwhelming number of smartphone and tablet games out there, I’m sure there are a few great ones that I haven’t discovered yet.   

And this has been your math mutation for today.


Sunday, October 25, 2015

213: Proof of the Fourth Dimension

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Rudolf Steiner was a prolific Austrian author and philosopher of the late 19th and early 20th centuries.    He felt a strong connection to mysticism and spiritualism, ever since he supposedly communicated with the ghost of a recently deceased aunt at the age of 9.   Steiner is well-known for having led a group that split off from the popular circle of European mystics known as the Theosophical Society, which seemed heavily inclined to regard the religions of East Asia as somehow providing the keys to understanding spirituality.   Steiner called his new group the Anthroposophical Society, and this competing group believed that Western science and culture were just as strongly connected to the spiritual-- it was just a matter of intepreting them properly.    One particular Western idea that Steiner was fond of was the concept of a fourth physical dimension, another mathematically-defined direction that we cannot percieve but is just as real as length, width, and height.     Steiner believed that our consciousness extended into this fourth dimension, and that phenomena like ghosts and ESP resulted from activity in this hidden dimension.      And most interestingly, he believed he had a simple philosophical proof that this fourth dimension really does exist, and our human minds really do extend into this additional dimension.

Here's how Steiner's proof goes.   We all know that a creature of a particular dimension, if it looks out at its world, really only sees a view that is one dimension smaller.   For example, a one-dimensional creature living in Lineland, a universe that exists entirely on a single straight line, can only perceive a single point on either side of himself:   a zero-dimensional view.    Similarly, a two-dimensional Flatlander, living on a plane, really only sees a line;  it is only us three-dimensional creatures, looking down on the plane from above, who can truly comprehend its full two-dimensional world.    And in real life, when we look out with our eyes, we are only seeing a plane.   Yet somehow we do believe we fully understand and perceive the three dimensions of our world.   Steiner draws what he believes is a natural conclusion from this:   "The fact that we can delineate external beings in three dimensions and manipulate three-dimensional spaces means that we ourselves must be four-dimensional...  We float in a sea of the fourth dimension just like ice cubes on water."   In other words, our ability to fully perceive our three-dimensional space shows that our minds must extend beyond those three dimensions.  

It's a fun thought, but you can see something fishy there right away, if you think about the world of modern computing.    I can think of all sorts of situations in which an object in three dimensions is represented by a model in fewer dimensions.     For example, most computer memories and circuits that power modern three-dimensional computer games are essentially stored in flat two-dimensional circuit boards.   While these are technically 3-D like all physical objects, the memory storage can be thought of as truly two-dimensional in some sense, as each (x,y) coordinate on the circuit board only stores one encoded value at any given time.    More basically, you may recall the concept of a Turing Machine discussed in some earlier podcasts:  this is a theoretical model of computing, based on writing and reading values from a long, essentially one-dimensional, tape.   It has been shown that any modern computer can be modelled by a very slow, but 100% accurate, Turing machine equivalent.    So even the 3-D models in a modern computer game could, with enough work, be represented in one dimension.

I think the main flaw in Steiner's argument is his fundamental premise, that a creature of n dimensions can only perceive n-1 dimensions.   It is true that through the sense of sight, a creature can only see one dimension lower, but our senses are not limited to sight.   Think about a blind man, who perceives the world mainly by walking around and tapping items with his cane to understand their form:  he can walk forward, back, right, or left, and even climb ladders up and down.    He is truly perceiving the full three dimensions of his world, travelling within all three of those dimensions and building a mental model based on his real experiences.    This applies to the lower-dimensional examples as well:   the flatlander can move around and perceive his full plane, and even the poor Linelander can move back and forth on his line.    Thus, the idea that perceiving your full dimensionality requires capabilities from a greater dimensionality does not really seem to ring true.    You need to think of perception much more generally than simple line-of-sight.

Naturally, this does not fundamentally prove that Steiner was wrong about our minds extending into the fourth dimension; it just means that the proof of such an idea is not so simple.   So it's still entirely possible that the concept of our mystical four-dimensional minds is correct but unproven, and the rest of Steiner's Anthroposophical Society ideas might still be valid.    This philosophy of the fourth dimension was just a launching point for a variety of mystic concepts, related to traveling along this fourth dimension to the astral plane where you could encounter ghosts, life after death, etc.   Some of Steiner's lectures get amusingly specific on details of the astral plane-- apparently he believed that his meditation and similar activities had actually taken him to this place, so he could talk about how astral dimensions mirrored our own, and writing there would appear backwards.   Personally, I'm a bit of a skeptic on this topic, but these kinds of ideas do seem to have a lasting appeal, as shown by the New Age sections you can find in many modern bookstores.   If you're into that stuff, try meditating hard enough, and maybe you too can follow Steiner's path into the astral plane through the fourth dimension.   While you're there. see if you can track down Steiner's spirit, to discuss the flaws in his philosophical proofs.

And this has been your math mutation for today.