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.