Sunday, May 05, 2024

Languages are theories: debunking the new riddle of induction and flat-eartherism

This is the sixth installment in a series about the scientific method.  My central thesis is that science is not just for scientists, it can be used by anyone in just about any situation.

In part 2 of this series I gave a few examples of how the scientific method can be applied in everyday situations.  In this chapter I want to show how it can be used to tackle what is considered to be a philosophical problem, something called the New Riddle of Induction.  I already covered the "old" riddle of induction in an earlier chapter but I'm going to go back over it here in a bit more detail.

The "old" problem of induction is this: we are finite beings.  There are only a finite number of us humans.  Each of us only lives for a finite amount of time, during which we can only have a finite number of experiences and collect a finite amount of data.  How can we be sure that the theories we construct to explain that data don't have a counter-example that we just haven't come across yet?

The reason this is called the "problem of induction" is that the example most commonly used to motivate it is the (alleged) "fact" that all crows are black.  It turns out that this isn't true.  There are non-black crows, but they are rare.  If all the crows you have ever seen are black, then it seems not entirely unreasonable for you to draw the conclusion that all crows are black because you have never seen a counter-example.  But of course you would be wrong.

The "new" riddle of induction (NRI) was invented by Nelson Goodman in 1955.  It adds a layer to this puzzle by defining a new word, "grue', as follows:

An object is grue if and only if it is observed before midnight, December 31, 2199 and is green, or else is not so observed and is blue.

Goodman then goes on to observe that every time we see a green emerald before December 31, 2199, that is support for the hypothesis that all emeralds are green, but it is equally good support for the hypothesis that all emeralds are grue, and so we are equally justified in concluding that at the stroke of midnight on new years eve 2199, all of the world's emeralds will suddenly turn blue as we are in predicting that they will remain green.

Now, of course this is silly.  But why is it silly?  You can't just say that the definition of "grue" is obviously silly, because we can give an equally silly definition of the word "green".  First we define "bleen" as a parallel to "blue":

An object is bleen if and only if it is observed before midnight, December 31, 2199 and is blue, or else is not so observed and is green.

And now it is "green" and "blue" that end up with the silly-seeming definitions:

An object is green if and only if it is observed before midnight, December 31, 2199 and is grue, or else is not so observed and is bleen.

An object is blue if and only if it is observed before midnight, December 31, 2199 and is bleen, or else is not so observed and is grue.

The situations appear to be completely symmetric.  So on what principled grounds can be say that "grue" and "bleen" are silly, but "blue" and "green" are not?

You might want to take a moment to see if you can solve this riddle.  Despite the fact that philosophers have been arguing about it for decades, it's actually not that hard.

It is tempting to say that we can reject the grue hypothesis because it has this arbitrary time, midnight, December 31, 2199, baked into the definition of the words "grue" and "bleen", so we can reject it for the same reason we rejected last-thursdayism.  The grue hypothesis (one might argue) is not one hypothesis, it is just one of a vast family of hypotheses, one for every instant of time in the future.  In fact, if you look up the NRI you will find the definition of grue given not in terms of any particular time, but explicitly in terms of some arbitrary time called T.

This explanation is on the right track, but it's not quite right because, as I pointed out earlier, the green hypothesis can also be stated in terms of some arbitrary time T.  What is it about "green" that makes it more defensible as a non-silly descriptor than "grue"?

Again, see if you can answer this yourself before reading on.

The answer is that while it is possible to give a silly definition of "green" in terms of grue and bleen, it isn't necessary.  It is possible to give a non-silly definition of "green"; it is not possible to give a non-silly definition of grue.  It is possible to define "green" without referring to an arbitrary time; it is not possible to define grue without referring to an arbitrary time.

How can we know this?  Because the grue hypothesis makes a specific prediction that the green hypothesis does not, namely, that all of the emeralds discovered after time T will be blue, which is to say, they will be a different color than all of the emeralds discovered before T.

Goodman would probably reply: no, that's not true, all of the emeralds discovered before and after time T will be the same color, namely, grue.  But this is just word-play. If you take two emeralds, one discovered before T and one after, they will look different.  If you point a spectrometer at a before-T emerald and an after-T emerald, the readings will be different.  In other words, on the grue hypothesis you will be able to distinguish experimentally between emeralds discovered before T and after T.  The grue hypothesis is falsifiable, and it will almost certainly be falsified the first time someone discovers an emerald after time T.

The crucial thing here is that your choice of terminology is not neutral, it is a crucial component of the expression of your hypothesis.  To quote David Deutsch, in an aphorism that arguably sets the record for packing the greatest amount of wisdom into the fewest number of words: languages are theories.  An argument based on hiding questionable assumptions under a terminological rug can be rejected on that basis alone.

Here is another example: consider, "The sun rises in the east."  Most people would consider that to be true.  But if you think about it critically, this sentence is laden with hidden assumptions, not least of which is (at least apparently) that the sun rises.  It doesn't.  The sun just sits there, and the earth orbits around it while rotating on an axis.  That makes it appear, to an observer attached to the surface of the earth, that the sun rises and sets even though it actually doesn't.  But that doesn't make "the sun rises in the east" false, it is just a deliberate misinterpretation of what those words actually mean in practice.  "The sun rises in the east" does not mean that the sun literally rises, it means that the sun appears to rise (obviously), and it does so in the same general direction every morning.  There is also an implicit assumption that we are making these observations from non-extreme latitudes.  At extreme latitudes, the sun does not even appear to rise in the east.  In fact, at the poles, the concepts of "east" and "west" don't even make sense -- at the poles, east and west literally do not exist!  (By the way, this is not just a trivial detail.  This exact same thing will come up again when we start talking about space-time, cosmology, and the origins of the universe.)

Note that "the sun rises in the east" is not an inductive conclusion, nor is it a hypothesis.  It is a prediction, one of many, made by the theory that the earth is a sphere rotating about an axis.  Furthermore, the fact that the sun rises and sets, together with the fact that this happens at different times in different places, definitively debunks the competing hypothesis that the earth is flat.  On a flat earth, if the sun is above the horizon, it must be above the horizon for all observers.  If the sun is below the horizon, it must appear below the horizon for all observers, and likewise if the sun is at the horizon.  This is in conflict with the observation that sunrise and sunset happen at different times in different locations.

Similarly, "all crows are black" is neither an inductive conclusion nor a hypothesis, but a prediction made by a very complex set of theories having to do with how DNA is transcribed into proteins, some of which absorb light of all visible wavelengths and so appear to be black.  "All emeralds are green" works the same way, but with one important distinction worth noting: in the case of crows, the hypothesis admits the possibility of occasional genetic mutations that result in non-black crows, which is in fact exactly what we sometimes observe.  (It also predicts that these will be rare, which is also what we observe.)

Emeralds are different.  Emeralds are green not because they contain proteins produced by DNA, but because they consist of particular kinds of atoms arranged in a particular crystalline structure with some particular impurities that make them look green.  It is possible to have other impurities that produce other colors, but in that case the result is not called an emerald but aquamarine or morganite.  All emeralds are green, without exception, because that is consequence of the definition of the word "emerald" plus the known laws of physics.  If a non-green emerald were ever discovered, that is, a mineral with the same chemical composition and crystal structure as an emerald but which was not green, that would be Big News.

Notice how easy all this was.  We didn't have to do any math.  We didn't have to get deep into the weeds of either scientific or philosophical terminology.  The hairiest technical terms I had to use to explain all this were "chemical composition" and "crystalline structure".

Notice too that we didn't have to debunk any of the specific arguments advanced by flat-earthers.  All we had to do is think about what "the sun rises in the east" actually means, and combine that with the fact that time zones are a thing, to generate an observation that the flat-earth hypothesis cannot explain.  Unless and until flat-earthers refute that (and they won't because they can't) we can confidently reject all of their arguments even if we have not examined them in detail, just as we can confidently reject claims of perpetual motion even if we have not examined those claims in detail.

In fact, we can reject flat-eartherism even more confidently than we can reject perpetual motion, and that is really saying something.  There are possible worlds where the second law of thermodynamics doesn't apply, the world we live in just happens not to be one of them.  It is a logical impossibility for the sun to rise and set at different times on a flat earth.  Simultaneous sunset and sunrise for all observers is a mathematical consequence of what it means to be flat.

The take-away message here is that the choice of terminology, the concepts you choose to bundle up as the definitions of words, is an integral part of the statement of a hypothesis.  Often the entire substance of a hypothesis is contained not in the statement of the hypothesis, but in the definitions of the words used to make the statement.

There are all kinds of problems that philosophers have argued about for decades that are easily resolved (and also bad science pedagogy that is easily recognized) once one comes to this realization.  It is a hugely empowering insight.  If someone tries to explain something science-y to you and it doesn't make any sense, it very well might just be that they haven't explained what they mean by the words they are using.  Science is chock-full of specialized terminology, and a lot of it sounds intimidating because, for historical reasons, scientists have adopted words with Greek and Latin roots (and sometimes German too).  These can sound weird, but the important thing to remember is that even weird-sounding words are just words, and they mean things just like more familiar words, and the things that they mean are often not nearly as intimidating as the words themselves.  Don't let weird words scare you.

The same can be said for math.  A lot of people are put off from science because it tends to have a lot of math, which they find to be off-putting.  But here is the empowering secret about math: math is just another language!  It is a very a very weird and specialized language, but a language nonetheless.  It uses a lot of unfamiliar symbols and notational conventions (the relative placement of symbols on a page matters a lot more than in other languages) but at the end of the day it's just marks on a page that mean something, and it's the meaning that matters, not the marks.  Keep that in mind any time things start to feel like they're getting too complicated.

[UPDATE] There are actually a lot of adjectives in English that act like "grue" and change their underlying meanings depending on time or other circumstances: normal, average, extraordinary, fashionable, affordable, polite, misspelled, technologically-advanced...  There are also some shape-shifting nouns, with the most prominent examples being "here" and "now".  What is it that makes these less silly than "grue" and "bleen"?  It is very simple: the changes captured by the definitions of these words reflect actual changes that happen in the world while the changes captured by the definitions of "grue" and "bleen" do not.

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