Saturday, May 18, 2024

A Scientific Theory of Truth

(This is part 7 of a series about the scientific method.)

The over-arching theme of this series is that science can serve as a complete worldview, that it can answer deep philosophical and existential questions normally associated with philosophy or religion.  I gave a small example of that in the last installment where I showed how the scientific method can be deployed to answer a fun philosophical riddle.  Here I want to show how it can tackle a much deeper question: what is truth?

Note that what I mean by "truth" here in this chapter -- and only this chapter -- is not scientific truth, but philosophical truth.  Remember, I have already disclaimed the idea that science finds philosophical or metaphysical truth.  It doesn't, it finds good explanations that account for observations, which are valuable because one of the properties of a good explanation is that it has predictive power.  But a good explanation is not necessarily true in the philosophical sense.  Newton's theory of gravity, for example, turns out to be completely wrong from a philosophical point of view, but it is still useful because it makes accurate predictions nonetheless.

The word "truth" is sometimes used in science as a shorthand for "theory that is sufficiently well established and makes sufficiently accurate predictions under a sufficiently broad range of circumstances that we proceed as if it were the (philosophical) truth even though we know it's not."  When I want to emphasize that I am referring to philosophical or metaphysical truth I will capitalize it.  Science seeks (and finds!) lower-case-t truth, not upper-case-t Truth.

But this does not mean that (upper-case-t) Truth is beyond the realm of scientific inquiry!  Remember, the scientific method is to find the best explanation that accounts for all of your observations, and one of your observations (if you are a normal human) is a constant stream of overwhelming evidence that there is lower-case-t truth out there, and the obvious explanation for that is that there is upper-case-T Truth out there, and the former is somehow a faithful reflection of the latter.  Furthermore, what we know about lower-case-t truth can put constraints on upper-case-T truth.  Science might not be able to tell us what the Truth is, but it can tell us with very high confidence what it is not.

So how do we use the scientific method to tackle a philosophical problem?  We can't do it directly because "What is Truth?" is not a properly framed scientific question.  Scientific inquiry must begin with a Problem, something we observe that cannot be explained by current theories.  "What is Truth?" is not a valid Problem statement because Truth is not something we observe.  To start a scientific inquiry we have to somehow transform this into a question about something we can observe.

Fortunately, there is a general way of doing this for philosophical questions: we can observe that people wonder about what Truth is!  We can further observe that people have some intuitions about what Truth is (or at least what truth is), and that some of these intuitions are common across a wide swath of humanity, to the point where someone who does not share these intuitions can be considered mentally ill.  For example, here are some things that are widely regarded as true:

All triangles have three sides.

The sun rises in the east.

The sky is blue.

Humans are mortal.
And here are a few examples of statements that are widely regarded to be not true, a.k.a. false:
Some triangles have four sides.

The sun rises in the west.

The sky is green.

Humans are immortal.
We can now advance a some hypotheses to explain these observations:

  1. There is a metaphysical Truth out there, and that out intuitions are somehow in contact with this Truth.
  2. Our intuitions about truth are illusions.  There are no actual truths.  What we call "truth" is nothing more than a social construct into which we are indoctrinated.  (If you think this sounds ridiculous, believe me, I sympathize.  But this is actually a hypothesis that is taken seriously in some academic circles.  It's called post-modernism.)


Before we go on to discuss the relative merits of these hypotheses (though I guess I've already tipped my hand here), let's consider some more examples:

  1. Richard Nixon had eggs for breakfast on the morning of January 1, 1962.
  2. Coffee tastes good.
  3. The United States is a Christian nation.
  4. Gandalf was a wizard.
  5. Love is a many-splendored thing.
  6. Die Erde ist Rund.
  7. This sentence is false.
  8. Given a line and a point not on that line, there is exactly one line passing through the given point parallel to the given line.

Each of these examples is meant to illustrate a different subtlety with the notion of "truth".  The first one is either true of false, i.e. there is an actual fact of the matter regarding whether or not this statement is true, but that fact is almost certainly beyond our reach.  We can know that this statement is either true or it is false, but almost certainly we can't know which.

The second example is completely different.  It is an example of a subjective claim.  There is no "actual fact of the matter" with regards to the taste of coffee.  Some people like it, some don't.  Note that you can transform a subjective claim into an objective one by binding it to a particular person: "Coffee tastes good" is subjective, but "I think coffee tastes good" or "coffee tastes good to me" is objective.

The third example is harder to characterize.  At first glance is might appear to be a subjective claim, a matter of opinion analogous to the flavor of coffee.  But consider "Israel is a Jewish state", or "Saudi Arabia is a Muslim state."  Surely those are objectively true?  Surely there is more truth to them than "Israel is a Muslim state" or "Saudi Arabia is a Jewish state"?  I won't say anything more about this example right now, but keep it in the back of your mind because it will become important later in this series.

The fourth example is actually controversial, at least among philosophers.  If you polled ordinary people on the street you would probably find pretty overwhelming agreement that this statement is true, especially when contrasted with, say, "Gandalf was an orc."  But some philosophers argue that any statement about Gandalf cannot be true because Gandalf doesn't actually exist, and non-existent things can't have properties.  So it cannot be true that Gandalf was a wizard, but neither is it true that he was not a wizard.  He simply wasn't anything.  Personally, I think this is ridiculous, and I would not even mention it but for the fact that this point of view was advanced in all seriousness by someone whose views I otherwise hold in the highest regard.

To me, the answer is pretty obvious: the sentence "Gandalf was a wizard" does not mean that Gandalf was an actual wizard in actual physical reality.  It means that within the context of the fictional world created by J.R.R. Tolkien, Gandalf was a wizard.  Or, if you really want to be strict about it, "J.R.R. Tolkien wrote that Gandalf was a wizard", which is clearly true.

The fifth example appears superficially to be a factual claim, but it isn't.  It's poetry.  I included it to show that there is something about factual claims that transcends mere syntax.

The sixth example is German for "the earth is round."  This example makes a similar point to the one before: the process of deciding whether or not a statement is true, or even of deciding whether or not it even makes sense to say that it is true or false, is not a simple one.  There is no straightforward procedure that you can apply to a string of letters to decide these things.

The seventh example is the famous Liar Paradox.  Superficially it appears that this sentence should be either true or false, but either possibility leads immediately to a contradiction.  Another interesting example, which is not considered nearly as often but which I think is equally interesting, is the opposite: "This sentence is true."  It can be true, or it can be false, and both possibilities are internally consistent.

This is also the case for the last example, which is the famous Euclid's fifth postulate.  Intuitively it seems like it should be true, and it also seems like it should be provable from some simpler assumptions, but humans searched in vain for such a proof for two thousand years before realizing that whether or not to consider this statement true or false is an arbitrary choice, and either choice leads to interesting and useful results.

The main point I'm trying to make here is that truth is complicated.  The road to truth winds through the vagaries of natural language and subjective experience, takes a few twists and turns through prejudice and personal opinion, before finally arriving...well, somewhere.  My personal experience is consistent, at least superficially, with the hypothesis that there is an objective reality "out there" which I share with other conscious beings.  Specifically, I am something called a "human being" living on the surface of something called a "planet" which I share with other human beings who do things like eat and sleep and build computers and write blog posts.  If you don't accept that, then I'd be interested to know how you account for what you are doing right this very moment as you read this.

For now, though, I am just going to assume that this is the case.  This is an example of engaging in Step 2 of the scientific method.  The first step is to identify a Problem.  In this case, the Problem is to account for the fact that humans profess to believe, sometimes vehemently, that certain things are true and other things are false.  It's possible that this is because all humans other than me are NPCs in a simulation, and I'm the only truly conscious being in the universe.  (This is called solipsism.)  It's hard (though not impossible) to refute solipsism, but here I'm not even going to try.  I'm just going to assume it's false, and that all my fellow humans really are what they appear to be.

So here is the actual hypothesis I am willing to defend, my proposed answer to the question of "What is truth?"

Ron's Theory of Truth: Truth is a property of propositions, which are ideas that stand in relation to some circumstance in objective reality (whose existence we have assumed for the sake of argument).  If the circumstance corresponding to that proposition actually pertains, then the proposition is true, otherwise it is false.
Notice that I have introduced two new words here: "proposition" and "idea".  I don't want to get into too much detail about these just now.  My purpose here is not to present an academically rigorous argument, merely to illustrate in broad brushstrokes how the scientific method can be used to attack problems that are often regarded as outside of the scope of scientific inquiry.  I will get much more precise about this later in this series, but for now just assume that "idea" means what it is commonly taken to mean: some vaguely identifiable thing that exists in someone's mind which can be somehow transferred into another person's mind.  This transferability is the defining characteristic of an idea; it is what distinguishes ideas from other things that might exist in someone's mind, like emotions or self-awareness.

Despite the fact that the idea of an idea (!) is pretty common, it is actually very hard to demonstrate.  I can't show you an actual idea.  All I can show you is a rendering or representation of an idea through some physical medium like writing or speech or dance or music.  So, for example, I can write:
The earth is round.
That looks like an idea, but appearances are deceiving.  What you are looking at is not something inside my brain (which is where ideas live) but patterns of light emitted from your computer screen, which, needless to say, is not part of my brain.  The marks you see on the screen get translated by your brain into an idea, but the marks and the idea are not the same thing.  The idea is the thing that ends up in your brain after seeing the marks, which in this case your brain interprets as letters and words.  Compare with:
Die Erde ist Rund.
or
地球は丸い

Those are completely different marks on the page, but they all denote the same idea, namely, that the earth is round.  That idea is a proposition because it maps onto things in objective reality, namely a planet called (in English) "earth" and a physical property called (again, in English) "round".  And that proposition is true because that thing actually has that property.

Note that evaluating the truth of statements on my theory is a two-step process.  The first step is mapping a rendering or a representation of an idea, which here will almost always be marks on a computer screen, onto an actual idea, and the second step is mapping that idea onto reality.  The importance of the first step cannot be overstated.  It is capable of cutting huge swathes through the philosophical underbrush.  Many seemingly intractable philosophical problems fall before it.

Take the example of "Gandalf was a wizard."  That string can reasonably be interpreted in two different ways, one of which is true, and the other false.  It can be taken to mean, "Gandalf was a real person in the real world, and he was a wizard" or it can be taken to mean, "Gandalf, the fictional character in J.R.R. Tolkein's 'Lord of the Rings' was, within the context of that fictional world, a wizard."  Disagreements over the truth of "Gandalf was a wizard" are nothing more than quibbles over this particular ambiguity of the English language.

The example of "all triangles have three sides" can be resolved similarly.  The word "triangle" means "a shape with three sides" and so this statement really means "All shapes with three sides have three sides."  Which is pretty obviously true, and also not very interesting.

A more interesting example is "The United States is a Christian Nation."  This turns on what is meant by the ambiguous phrase, "Christian nation."  It might mean that the United States is a Christian theocracy, which it is not (at least not yet).  It might mean that the United States was intended to be a Christian theocracy, which it also was not.  Or it might mean that the majority of the people living in the United States self-identify as Christian, which is true.  Again, disagreements over this are disagreements over the meanings of words, not good-faith disagreements over actual facts.

Good-faith disagreements over actual facts are very rare in science.  This is one of the reasons that scientific disagreements are almost invariably settled without resorting to violence, which is in very stark contrast to other methods that humans have tried.

As an exercise, see how far you can get following this line of thought to attack the Liar Paradox, i.e. "This statement is false."  I'll give the answer to that in a future installment because this one is getting too long.  But as a hint, here are two incorrect answers.

Most people upon seeing this puzzle for the first time think that the resolution has something to do with the self-referential nature of the phrase "this statement".  That's not the case.  It's straightforward to construct a similar paradox without any self-reference.  Here is one way:

The following sentence is false.
The preceding sentence is true.
We can even stop playing fast-and-loose with the distinction between strings and propositions simply by giving names to strings:
S1: "The proposition denoted by string S2 is false."
S2: "The proposition denoted by string S1 is true."
And we can even do the same thing without labels by using a clever trick called Quining, which consists of filling in the details of a string that looks something like, "The string that you get when you follow this procedure ... is false", where the ellipses are filled in with instructions such that the string that you get when you follow those instructions are the exact string that you started with.  It's quite a neat trick, and it's the foundation of Godel's famous incompleteness theorem, wherein he constructs a string that essentially says, "This proposition denoted by this string cannot be proven by standard mathematics."

So self-reference is not the problem.

A second possibility is that the liar string does not denote a proposition.  Just because a sentence bears a superficial resemblance to a proposition doesn't mean it actually is one.  "Love is a many-splendored thing" bears a superficial resemblance to "love is an emotion", but the latter denotes a proposition while the former does not.  In order to be a proposition on my theory, an idea has to refer to objective reality somehow because truth is determined by correspondence to reality.  The words "love" and "emotion" refer to things in reality, but "many-splendored thing" does not; it's just a poetic rhetorical flourish.

No such problems are immediately evident in the Liar Paradox sentence.  It refers to an idea, and ideas are part of reality, and so we cannot reject it as a proposition on the grounds that it does not refer to reality.

I'll give you one final hint: the resolution of the liar paradox involves discharging a tacit assumption about propositions which turns out not to be true according to the definition of truth I've advanced here.  If you think you know the answer, put it in the comments.  (Note that I have comment moderation turned on.  This blog has been around for twenty years and it attracts a lot of spambots.)

In closing, I want to reiterate that the main point here is not to resolve the liar paradox (that's just a fun puzzle that happened to come up) nor any other hard philosophical problem, but merely to show how the scientific method can be applied to such problems.  Philosophers have been puzzling over what truth is for millennia; I can't provide an academically rigorous answer in 2500 words.  The best I can hope for is to show that these questions are not beyond the scope of scientific inquiry.

But stay tuned.  There's more to come.

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.