Saturday, March 31, 2018

239: The Shape Of Our Knowledge

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Recently I’ve been reading Umberto Eco’s essay collection titled “From the Tree to the Labyrinth”.   In it, he discusses the many attempts over history to cleanly organize and index the body of human knowledge.    We have a natural tendency to try to impose order on the large amount of miscellaneous stuff we know, for easy access and for later reference.   As typical with Eco, the book is equal parts fascinating insight, verbose pretentiousness, and meticulous historical detail.    But I do find it fun to think about the overall shape of human knowledge, and how our visions of it have changed over the years.

It seems like most people organizing a bunch of facts start out by trying to group them into a “tree”.   Mathematically, a tree is basically a structure that starts with a single node, which then links to sub-nodes, each of which links to sub-sub-nodes, and so on.   On paper, it looks more like a pyramid.   But essentially it’s the same concept as folders, subfolders, and sub-sub folders that you’re likely to use on your computer desktop.   For example, you might start with ‘living creatures’,   Under it you draw lines to ‘animals’, ‘plants’, and ‘fungi’.   Under the animals you might have nodes for ‘vertebrates’, ‘invertebrates’, etc.     Actually, living creatures are one of the few cases where nature provides a natural tree, corresponding to evolutionary history:  each species usually has a unique ancestor species that it evolved from, as well as possibly many descendants.

Attempts to create tree-like organizations date back at least as far as Aristotle, who tried to identify a set of rules for properly categorizing knowledge.   Later authors made numerous attempts to fully construct such catalogs.   In later times, Eco points out some truly hilarious (to modern eyes) attempts to create universal knowledge categories, such as Pedro Bermudo's 17th-century attempt to organize knowledge into exactly 44 categories.  While some, such as “elements”, “celestial entities”, and “intellectual entities” seem relatively reasonable to modern eyes, other categories include “jewels”, “army”, and “furnishings”.     Perhaps the inclusion of “furnishings” as a top-level category on par with “Celestial Entities” just shows us how limited human experience and knowledge typically was before modern times.

Of course, the more knowledge you have, the harder it is to cleanly fit into a tree, and the more logical connections you see that cut across the tree structure.   Thus our attempts to categorize knowledge have evolved more into what Eco calls a labyrinth, a huge collection with connections in every direction.  For example, wandering down the tree of species, you need to follow very different paths to reach a tarantula and a corn snake, one being an arachnid and the other a reptile.   Yet if you’re discussing possible caged parent-annoying pets with your 11-year old daughter, those two might actually be closely linked.    So our map of knowledge, or semantic network, would probably merit a dotted line between the two.     Thus, we don’t just traverse directly down the tree, but have many lateral links to follow, so Eco describes our real knowledge as more of a labyrinth.   He seems to prefer the vivid imagery of a medieval scholar wandering through a physical maze, but in a mathematical sense I think he is referring more to what we would call a ‘graph’, a huge collection of nodes with individual connections in arbitrary directions.

On the other hand, this labyrinthine nature of knowledge doesn’t negate the usefulness of tree structures— as humans, we have a natural need to organize into categories and subcategories to make sense of things.   Nowadays, we realize both the ‘tree’ and ‘labryrinth’ views of knowledge on the Internet.   As a tree, the internet consists of pages with subpages, sub-sub-pages, etc.   But a link on any page can lead to an arbitrary other page, not part of its own local hierarchy, whose knowledge is somehow related.   It’s almost too easy these days.   If you’re as old as me, you can probably recall your many hours poring through libraries researching papers back in high school and college.   You probably spent lots of time scanning vaguely related books to try to identify these labyrinth-like connections that were not directly visible through the ‘trees’ of the card catalog or Dewey Decimal system.

Although it’s very easy today to find lots of connections on the Internet, I think we still have a natural human fascination with discovering non-obvious cross connections between nodes of our knowledge trees.   A simple example is our amusement at puns, when we are suddenly surprised by an absurd connection due only to the coincidence of language.    Next time my daughter asks if she can get a tarantula for Christmas, I’ll tell her the restaurant only serves steak and turkey.    More seriously, finding fun and unexpected connections is one reason I enjoy researching this podcast, discussing obscure tangential links to the world of mathematics that are not often displayed in the usual trees of math knowledge.   Maybe that’s one of the reasons you like listening to this podcast, or at least consider it so absurd that it can be fun to mock.

And this has been your math mutation for today.


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Sunday, February 18, 2018

238: Programming Your Donkey

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You have probably heard some form of the famous philosophical conundrum known as Buridan’s Ass.   While the popular name comes from a 14th century philosopher, it actually goes back as far as Aristotle.   One popular form of the paradox goes like this:   Suppose there is a donkey that wants to eat some food.   There are equally spaced and identical apples visible ahead to its left and right.   Since they are precisely equivalent  in both distance and quality, the donkey has no rational reason to turn towards one and not the other, so it will remain in the middle and starve to death.   

It seems that medieval philosophers spent quite a bit of time debating whether this paradox is evidence of free will.   After all, without the tie-breaking power of a living mind, how could the animal make a decision one way or the other?   Even if the donkey is allowed to make a random choice, the argument goes, it must use its living intuition to decide to make such a choice, since there is no rational way to choose one alternative over the other.  

You can probably think of several flaws in this argument, if you stop and think about it for a while.   Aristotle didn’t really think it posed a real conundrum when he mentioned it— he was making fun of sophist arguments that the Earth must be stationary because it is round and has equal forces operating on it in every direction.   Ironically, the case of balanced forces is one of the rare situations where the donkey analogy might be kind of useful:  in Newtonian physics, it is indeed the case that if forces are equal in every direction an object will stay still.    But medieval philosophers seem to have taken it more seriously, as a dilemma that might force us to accept some form of free will or intuition.  

I think my biggest problem with the whole idea of Buridan’s Ass as a philosophical conundrum is that it rests on a horribly restrictive concept of what is allowed in an algorithm.  By an algorithm, I mean a precise mathematical specification of a procedure to solve a problem.   There seems to be an implicit assumption in the so-called paradox that in any decision algorithm, if multiple choices are judged to be equally valid, the procedure must grind to a halt and wait for some form of biological intelligence to tell it what to do next.   But that’s totally wrong— anyone who has programmed modern computers knows that we have lots of flexibility in what we can specify.   Thus any conclusion about free will or intuition, from this paradox at least, is completely unjustified.   Perhaps philosophers in an age of primitive mathematics, centuries before computers were even conceived, can be forgiven for this oversight.

To make this clearer, let’s imagine that the donkey is robotic, and think about how we might program it.   For example, maybe the donkey is programmed to, whenever two decisions about movement are judged equal, simply choose the one on the right.   Alternatively, randomized algorithms, where an action is taken based on a random number, essentially flipping a virtual coin, are also perfectly fine in modern computing.    So another alternative is just to have the donkey choose a random number to break any ties in its decision process.    The important thing to realize here is that these are both basic, easily specifiable methods fully within the capabilities of any computers created over the past half century, not requiring any sort of free will.  They are fully rational and deterministic algorithms, but are far simpler than any human-like intelligence.   These procedures could certainly have evolved within the minds of any advanced  animal.

Famous computer scientist Leslie Lamport has an interesting take on this paradox, but I think he makes a similar mistake to the medieval philosophers, artificially restricting the possible algorithms allowed in our donkey’s programming.   For this model, assume the apples and donkey are on a number line, with one apple at position 0 and one at position 1, and the donkey in an arbitrary starting position s.   Let’s define a function F that describes the donkey’s position an hour from now, in terms of s.  F(0) is 0, since if he starts right at apple 0, there’s no reason to move.   Similarly, F(1) is 1.  Now, Lamport adds a premise:  the function the donkey uses to decide his final location must be continuous, corresponding to how he thinks naturally evolved algorithms should operate.   It’s well understood that if you have a continuous function where F(0) is 0, and F(1) is 1, then for any value v between them, there must be a point x where F(x) is v.   So, in other words, there must be points v where F(v) is not 0 or 1, indicating a way for the donkey to still be stuck between 0 and 1 and hour from now.      Since the choice of one hour was arbitrary, a similar argument works for any amount of time, and we are guaranteed to be infinitely stuck from certain starting points.   It’s an interesting take, and perhaps I’m not doing Lamport justice, but it seems to me that this is just a consequence of the unfair restriction that the function must be continuous.   I would expect precisely the opposite:   the function should have a discontinuous jump from 0 to 1 at the midpoint, with the value there determined by one of the donkey-programming methods I discussed before.

I did find one article online that described a scenario where this paradox might provide some food for thought though.   Think about a medical doctor, who is attempting to diagnose a patient based on a huge list of weighted factors, and is at a point where two different diseases are equally likely by all possible measurements.   Maybe the patient has virus 1, and maybe he has virus 2— but the medicines that would cure each one are fatal to those without that infection.   How can he make a decision on how to treat the patient?   I don’t think a patient would be too happy with either of the methods we suggested for the robot donkey:  arbitrarily biasing towards one decision, or flipping a coin.     On the other hand, we don’t know what goes on behind the closed doors after doctors leave the examining room to confer.   Based on TV, we might think they are always carrying on office romances, confronting racism, and consulting autistic colleagues, but maybe they are using some of our suggested algorithms as well.     In any case, if we assume the patient is guaranteed to die if untreated, is there really a better option?  In practice, doctors resolve such dilemmas by continually developing more and better tests, so the chance of truly being stuck becomes negligible.   But I’m glad I’m not in that line of work. 



And this has been your math mutation for today.

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Monday, January 15, 2018

237: A Skewed Perspective

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If you’re a listener of this podcast, you’re probably aware of Einstein’s Theory of Relativity, and its strange consequences for objects traveling close to the speed of light.   In particular, such an object will appear to have its length shortened in the direction of motion, as measured from its rest frame.    It’s not a huge factor— where v is the object’s velocity and c is the speed of light, it’s the square root of 1 minus v squared over c squared.    At ordinary speeds we observe while traveling on Earth, the effect is so close to zero as to be invisible.    But for objects near the speed of light, it can get significant.    

A question we might ask is:  if some object traveling close to the speed of light passed you by, what would it look like?    To make this more concrete, let’s assume you’re standing at the side of the Autobahn with a souped-up camera that can take an instantaneous photo, and a Nissan Cube rushing down the road at .99c, 99% of the speed of light, is approaching from your left.   You take a photo as it passes by.   What would you see in the photo?   Due to length contraction, you might predict a side view of a somewhat shortened Cube.   But surprisingly, that expectation is wrong— what you would actually see is weirder than you think.   The length would be shorter, but the Cube would also appear to have rotated, as if it has started to turn left.

This is actually an optical illusion:   the Cube is still facing forward and traveling in its original direction.   The reason for this skewed appearance is a phenomenon known as Terrelll Rotation.    To understand this, we need to think carefully about the path a beam of light would take from each part of the Cube to the observer.   For example, let’s look at the left rear tail light.    At ordinary non-relativistic speeds, we wouldn’t be able to see this until the car had passed us, since the light would be physically blocked by the car— at such speeds, we can think of the speed of light as effectively infinite. Thus we would capture our usual side view in our photo.   But when the speed gets close to that of light, the time it takes for the light from each part to travel to the observer is significant compared to the speed of the car.  This means that when the car is a bit to your left, the contracted car will have moved just enough out of the way to actually let the light from the left rear tail light reach you.   This will arrive at the same time as light more recently emitted from the right rear tail light, and light from other parts of the back of the car that are in between.   In other words, due to the light coming from different parts of the car having started traveling at different times, you will be able to see an angled view of the entire rear of the car when you take your photo, and the car will appear to have rotated overall.   This is the Terrell Rotation.

I won’t go into the actual equations in this podcast, since they can be a bit hard to follow verbally, but there is a nice derivation & some illustrations linked in the show notes.   But I think the most fun fact about the Terrell Rotation is that physicists totally missed the concept for decades.   For half a century after Einstein published his theory, papers and texts claimed that if you photographed a cube passing by at relativistic speeds, you would simply see a contracted cube.    Nobody had bothered carefully thinking it through, and each author just repeated the examples they were used to.    This included some of the most brilliant physicists in our planet’s history!   There were some lesser-known physicists such as Anton Lampa who had figured it out, but they did not widely publicize their results.   It was not until 1959 that physicists James Terrell and Roger Penrose independently made the detailed calculation, and published widely-read papers on this rotation effect.    This is one of many examples showing the dangers of blindly repeating results from authoritative figures, rather than carefully thinking them through yourself.


And this has been your math mutation for today.


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