Buried Treasure

A week ago, I took the USMLE step 1, an 8-hour Goliath of a test. As daunting of a prospect as that is, the study process was much more extensive. I studied for twelve hours a day for six weeks.

I had intended to spend those six weeks memorizing a whole lot of facts. But I eventually, I found that I wasn’t just learning facts; I was learning structures. This took a lot of the drudgery away, since the latter are quite a bit more fun to study.

Now that boards are over, I find myself stepping away from medicine and looking towards other fields. Do other fields, like medicine, produce elaborate structures from the underlying facts and principles? Must they? Are some resultant structures better than others? In medicine, there often is a right answer (especially on boards). Is the same true of other fields? Where, if at all, does the rubber meet the road?

Medicine

Not long after I started studying, I learned that the USMLE sought to ask me questions which I had never been asked before. And, by some point, there simply weren’t enough facts to ask about, at least without venturing into the obscure. (The USMLE certainly wasn’t above this, but, luckily, highly obscure questions tended to be the exception rather than the rule). The only way to produce new questions, then, was to test the relationships between the facts, rather than the facts itself: while there are many facts, there are many, many more relationships between facts. It’s in these relationships that the USMLE, and the QBank writers, found enough fodder to produce thousands of questions, while still producing many which were novel.

Let’s consider an example. We have the following facts:

  • Fat soluble vitamins include
    • A: deficiency leads to night blindness
    • D: deficiency leads to osteomalacia
    • E: deficiency leads to neuroacanthocytosis
    • K: deficiency leads to coagulopathy
  • Fat malabsorption leads to deficiency in fat-soluble vitamins. Causes of fat malabsorption include:
    • Celiac disease
    • Cystic fibrosis
    • Cholestasis
    • Crohn’s disease

We have two lists of four. Given these 8 facts, though, we can produce 4 • 4 = 16 combinations of the facts, as follows.

  • A 52-year-old woman with pale stools and itchy palms (signs of primary biliary cirrhosis, a cause of cholestasis) presents with a broken wrist after a minor fall (a sign of osteomalacia, indicating Vitamin D deficiency).
  • A 14-year-old boy with a history of recurrent pneumonia and chronic diarrhea (cystic fibrosis) presents with confusion and papilledema (signs of an intracranial bleed, due to Vitamin K deficiency)

And so on. Add to that that there are probably at least four causes of cholestasis; four other diseases, besides Vitamin D deficiency, can cause osteomalacia; Vitamin K deficiency can present in at least four different ways. So, we see how, with just a few starting facts, we can produce an explosive number of questions which ask about the relationships between the facts. Thus the boards asks questions whose answers cannot be memorized. Their answers must be produced.

I picked up on their strategy, and so I tried to adapt. I became rehearsed in identifying which facts seemed like they would be more likely to produce, and focused more energy on memorizing those particular facts. The thicker the web of interconnections which housed those facts seemed to be, the more interested I was in memorizing them. This way, I felt that I would be more prepared to answer questions which tested the relationships between the facts.

And I considered this to be a win-win. After all, the boards want doctors that can produce a diagnosis and treatment on the fly. Meanwhile, I found it more enjoyable to produce than to regurgitate.

Medicine itself, I found, is pleasurable, because it gives one the opportunity to witness the birth of myriad manifestations from just a few underlying principles. How do other fields compare, in terms of breadth and depth? Is complexity a given, or must it be sought out and discovered?

My search for a complicated field turned fruitful early, when I thought back to an old fascination of mine, Conway’s Game of Life.

Conway’s Game of Life

Conway’s Game of Life is a cellular automaton described as a zero player game. Remarkably—much like in the case of medicine—we see the arisal of complex and myriad manifestations from just a few underlying rules. In the case of Conway’s Life, in fact, the rules are much simpler. The game starts with a grid space, where certain squares are either filled in (alive) or not filled in (dead). Then, the behavior of the squares in future generations is given by the four rules:

  1. Any live cell with fewer than two live neighbors dies, as if caused by under-population.
  2. Any live cell with two or three live neighbors lives on to the next generation.
  3. Any live cell with more than three live neighbors dies, as if by over-population.
  4. Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction.

And that’s it. Simple, right? But the manifestations are extensive. A wide variety of complicated structures can be produced from these simple rules.

Game_of_life_animated_glider

The glider, one of the simplest spaceships.

Gospers_glider_gun

The glider gun, which continually produces gliders.

Conways_game_of_life_breeder_animation

The puffer-type breeder, which moves along, leaving glider guns in its wake.

Consider also the acorn. Acorn is a methuselah, meaning that it’s one of a handful of patterns which take an uncharacteristically long time to stabilize. Most patterns—especially those, like Acorn, consisting of seven or fewer initial tiles—stabilize in just a few generations. They reduce to still lifes, oscillators, spaceships, or they annihilate completely. In other words, their behavior becomes predictable. Acorn, on the other hand, remains chaotic for 5206 generations, by which time it has produced 633 cells, including 13 escaped gliders. Acorn can be viewed here; it’s the default pattern shown on the grid.

Conway’s Game of Life proves to be an extremely rich source of study: scholars of the game have put considerable effort into the search for patterns and principles. Many have described the complex patterns which arise from simple structures as a form of beauty.

And the computer science manifestations are vast.

The game of life can be used as a counter. If two gliders are shot at a block in the right way, the block moves closer to the source of the gliders. Meanwhile, if three gliders are shot at a block in just the right way, the block moves farther from the source of the gliders. In either case, the movement of the block can be used to measure the passage of time, or the amount of gliders which have hit it. Of course, a glider itself can be used to measure the passage of time. But it moves much faster than the blocks would. Note that a glider gun produces a glider only once every 14 or greater generations, depending on the glider gun. So the block would move once every 28 or 42 generations.

The game of life can also simulate boolean operators. AND, OR and NOT can be constructed using gliders.

The game of life can be used to build a finite state machine, which has computational power equivalent to that of a universal Turing machine. So, the game of life is said to be Turing complete.

The game of life is undecidable. The decidability question asks: given any initial position, can it be determined whether or not a second later position will occur? It turns out that the answer is no, and so the game of life is undecidable. This is a corollary of the halting problem.

I don’t pretend to understand much, or any, of the computer science surrounding Life. But it’s clear to me that the potential for study is vast. And it’s remarkable that such complexity is born from such simple rules.

Conway’s Life, then, might be used as a veritable case study for a system which is, as is medicine, complicated and interesting.

Conway’s Life, however, might supply a bit more intrigue. In medicine, the rules are fixed (we didn’t choose how the body works). The rules happen to give rise to complicated structures, but need this be the case? For medicine, it’s hard to say. In games of life, though, we get to choose the rules. What if a cell with three neighbors dies, as opposed to one with two? Or a cell with four neighbors lives on, but not one with three? In fact, people have produced games of life apart from Conway’s. However, one source writes that the “vast majority of them produce universes that are either too chaotic or too desolate to be of interest”.

This observation proves instructive to us. Life is no accident. Alternative rulesets exist, but they’re just not as good. Life, meanwhile, is interesting. Scholars of cellular automata spend a lot of effort, and take a great deal of pleasure, in studying Conway’s game. And they don’t spend this effort studying other worlds, which they describe as either desolate or chaotic.

In medicine, the subject of study is fixed. In other fields, though, there’s more room for choice. Given a choice of worlds, then, we may as well choose to study one which, like Conway’s, yields rare intrigue.

Philosophy of language

We can think of Conway’s Game of Life as residing in a “game space”, consisting of all possible rulesets which go on to produce cellular automata.

Likewise, philosophy of language consists of a space of all possible theories, principles, and approaches. Philosophy of language, then, provides an instructive example, because here, too, certain approaches might yield complexity, or fail to do so. In other words, these theories might be desolate, chaotic, or anywhere in between. Let’s consider a few of them.

In The Two Dogmas of Empiricism, Willard Van Orman Quine takes on the analytic/synthetic distinction. This is a storied concept: it was introduced by Kant; it had been discussed in various forms earlier by Hume (relations of ideas vs. matters of fact) and Leibniz (truths of reason vs. truths of fact); similar distinctions include a priori / a posteriori, necessary / contingent, and inductive / deductive, and logical / empirical. In spite of its history, though, Quine takes aim.

Kant said that an analytic statement is one which “attributes to its subject no more than is already contained within the subject”. In other words, the sentence is true by virtue of meaning alone and independently of fact. For example:

  1. No unmarried man is married.

This statement is true by virtue of its meaning, and in fact, it’s the meaning of the logical particles alone which makes this so. All we need to declare (1) true is knowledge of the function of “no” and “un-”. On the other hand:

  1. No bachelor is married.

rests not on the meaning of logical particles but on the meaning of the word bachelor. And it’s (2) that Quine takes issue with.

(2) is analytic so long as “bachelor” and “unmarried man” are synonymous. If they are, a substitution can be performed, and (2) reduces to (1). However, how can we be sure that these two words are indeed synonymous?

We could consult a dictionary. But dictionaries don’t represent some sort of divine truth; rather, they simply contain the work of some lexographer, who has determined, according to his prior knowledge and life experience, that bachelor and unmarried man are synonymous. But if this determination rests on experience, and not on fact, then bachelor and unmarried man cannot be substituted within the confines of logic alone. So (2) can only be said to be a synthetic truth; it cannot be said to be an analytic one.

Grice and Strawson fire back at Quine in In Defense of a Dogma.

They do take some time to poke holes (and some quite large ones) in Quine’s argumentation. Interestingly, though, the paper, at its spirit, seems aimed at questioning not Quine’s logical process, but the rather, the result achieved by that process. In other words, Quine’s logic may be sound, and his result may well be true. However, true might not be synonymous with instructive or useful.

Grice and Strawson employ a brilliant and highly compelling thought experiment, which proves that Quine’s theory is, if not wrong, then simply absurd in the context of natural language.

Consider the following two conversations:

  1. A: My neighbor’s three-year-old child understands Russell’s Theory of Types.
    B: You mean the child is a particularly bright lad.
    A: No, I mean what I say—he really does understand it.
    B: I don’t believe you—the thing’s impossible.
  2. X: My neighbor’s three-year-old child is an adult.
    Y: You mean he’s uncommonly sensible or very advanced for his age?
    X: No, I mean what I say.
    Y: Perhaps you mean that he won’t grow anymore, or that he’s a sort of a freak, that he’s already fully developed.
    X: No, he’s not a freak, he’s just an adult.

B’s experience is one characterized by a failure to believe. But at least B knows what to prepare for. B knows that if the child is in fact produced, and can fully explain and criticize Russell’s Theory of Type, he will be forced to change his mind. Y, on the other hand, experiences a failure to understand. As he approaches the neighbor’s house, Y has no idea what to expect and what to prepare for. Grice and Strawson write: “And whatever kind of creature is ultimately produced for our inspection, it will not lead [Y] to say that what [X] said was literally true, but at most to say that [Y] now sees what [X] meant.”

Their piece does seem to render Quine’s approach absurd. Who’s right? Well, both probably are. Quine, along with Grice and Strawson, both separate the pool of all statements into those which are analytic and those which are synthetic. They simply draw the boundary in a different place, such that (2) falls towards synthetic under Quine and towards analytic under Grice and Strawson. We should then ask not which choice of boundary is more right, but rather which is more informative.

My belief is that Quine’s system is less informative. It’s probably no less true than Grice and Strawson’s. But once we accept it, we arrive at a dead end. On the other hand, under Grice and Strawson, further questions can be asked. What is at play here? How does this divide compare and contrast with a priori / a posteriori, and necessary / contingent? Are mathematical findings analytic? Is there a clean divide, or a gray area? Does it depend on who you ask? “All whales are mammals” might be synthetic to a sailor, but analytic to a marine biologist (proposed by Donellan). These are all fair questions, and will likely produce further intrigue and discussion. But they can only be asked once Grice’s and Strawson’s rules are accepted, and Quine’s are discarded.

My grasp of philosophy of language is certainly limited, and I’m not married to the idea that Quine’s universe is desolate. Rather, I just aim to argue that some worlds, within philosophy, might be more complicated–more Conway’s-like–than others. And we should strive to discover and study those worlds.

Why should we study them? Well, for one, it’s fun. I pointed out earlier that scholars of cellular automata choose to study Conway’s game, because it’s the most interesting to them. For some reason, balanced worlds are more interesting than chaotic or desolate ones. We need not know why this is true to say that it is.

We can guess, though. Conway’s game is probably interesting because it at least appears to illuminate some real or true phenomena. This is the case precisely because methuselahs like Acorn are hard to find. When something is both complex and rare, it seems to be valuable, almost because it must be.

We can imagine that there is some truth within the “game-space” of cellular automata. The challenge simply lies in finding it. In a desolate world, we’ll never find it. In a chaotic world, we might see it, but we wouldn’t be able to tell it apart from anything else. Only in a balanced world are we successful both in seeing the signal and ignoring the noise.

It’s like a metal detector at the beach. If it’s not sensitive enough, it never beeps. If it’s too sensitive, it beeps all the time. But if it’s balanced, it beeps only when it hovers over metal.

Worlds like Conway’s are interesting because beeps (or, rather, Acorns) are rare. And it’s exciting when the metal detector sounds, because it shows that we’ve at least found something. Indeed, it could be buried treasure, or it could be an old hubcap. It seems that we’ve stumbled on at least some sort of phenomenology, whether or not it’s valuable.

Is Acorn true, in some absolute sense, or does it just appear to be true? Is there something fundamentally real about chess, or even math? These are complex questions, and I’m not sure if there’s a good answer, or a good way to go about finding an answer.

We don’t need an answer, though, to acknowledge the fact that balanced worlds at least feel like they yield truth, and thus are more interesting to study. And the same is true in philosophy. If given a choice, we may as well choose Grice and Strawson’s world, over Quine’s. We stand a chance at finding buried treasure, but we have to make sure we’re digging in the right places.

Resources

  1. Conway’s Game of Life
  2. Kant: A Critique of Pure Reason
  3. Quine: Two Dogmas of Empiricism
  4. Grice and Strawson: In Defense of a Dogma
  5. Donnellan: Necessity and Criteria
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One comment on “Buried Treasure

  1. Till says:

    Hey Josh, I hope you´re fine now after spending so much time and effort in studying for the USMLE exam? Currently I am looking for med students who are interested in reviewing our new site which provides users with e.g. a huge USMLE QBank (there is already Step 1 for free and Step 2 about to come). Your feedback would help us a lot and maybe your readership is interested in more tips as well? Well, hope to hear from you soon. Just write an email to: t.vennemann@lecturio.com. Big thanks, Till

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