Monday, May 27, 2024

Truth, Math, and Models

(Part 8 in a series on the scientific method)

In the last installment I advanced a hypothesis about what truth is, which is to say, I suggested a way to explain the broad consensus that appears to exist about truth.  That explanation was: there is an objective reality "out there", and true statements are those that in some sense correspond to the actual state of affairs in that objective reality.  This was problematic for statements like, "Gandalf was a wizard" because Gandalf doesn't actually exist in objective reality, but that was accounted for by observing that the actual meanings of sentences in natural languages often goes beyond the literal.

But there is one aspect of truth that is harder to account for, and which would appear at first glance to be a serious shortcoming of my theory: math.  Most people would consider, for example, "1+1=2" or "7 is prime" to be true despite the fact that it's hard to map those concepts onto anything in objective reality.  I can show you "one apple" or "one sheep", but showing you "one" is harder.  The whole point of numbers, and mathematics and logic in general, is to abstract away from the physical.  Numbers qua numbers do not exist in the real world.  They are pure abstraction, or at least they are supposed to be.  Mathematical truth is specifically intended not to be contingent on anything in the physical world, and so it would seem that my theory of truth fails to capture mathematical truth.

Some philosophers and religious apologists claim that it is therefore impossible to ground mathematical truth in objective reality, that the existence of mathematical truth requires something more, some ethereal realm of Platonic ideals or the Mind of God, to be the source of such truths.  It's a plausible argument, but it's wrong.  Mathematical truth can be understood purely in terms of objective reality.  Specifically, mathematics can be viewed as the study of possible models of objective reality.  In this installment I will explain what I mean by that.

There are a lot of different examples of (what is considered to be) "mathematical truth" but let me start with the most basic: elementary arithmetic.  These include mundane truths about numbers, things like "two plus three equals five" or "seven is prime."  It would seem at first glance that numbers used in this way don't refer to anything in objective reality.  I can show you two of something but I can't show you "two" in isolation.

There is an easy answer to this: numbers in common usage are not nouns, they are adjectives.  The reason I can't show you "two" without showing you two of something is the same reason I can't show you "green" unless I show you a green thing.  Adjectives have to be bound to nouns to be exhibited, but that doesn't mean that "green" does not exist in objective reality.  It does, it's just not a thing.  Green is a color, which is a property of things, but it is not itself a thing.  Likewise, "two" is not thing, it is a quantity, which is a property of (collections of) things.  And the reason that two plus three equals five is that if I have two things and I put them together with three other things the result is a quantity of things to which English speakers attach the label "five".  Likewise "seven is prime" can be understood to mean that if I have a quantity of things to which English speakers attach the label "seven" I cannot arrange those things in a complete, regular rectangular grid in any way other than the degenerate case of putting them all in a line.

But this explanation fails for straightforward extensions of the natural numbers, like negative numbers or irrational numbers or imaginary numbers.  I can show you two apples, and I can explain addition and subtraction in terms of putting groups of apples together and taking apples away, but only for positive numbers.  I cannot explain "three minus five equals negative two" in terms of starting with three apples and taking five away because that is just not physically possible.  Likewise I cannot show you a square with a negative area, and so I cannot explain the square roots of negative numbers in terms of anything physical (at least not easily).

There are two more cases where the numbers-are-adjectives theory fails.  The first is truths that involve generalizations on numbers like "There are an infinite number of primes."  That can't be explained in terms of properties of physical objects because we live in a finite universe.  There are not an infinite number of objects, so if numbers are meant to describe a quantity of a collection of actual physical objects, then there cannot be an infinite number of them either.

Finally, there are a lot of objects of mathematical study beyond numbers: manifolds, tensors, vectors, functions, groups, to name just a few.  Some of these areas of study produce mathematical "truths" that are deeply weird and unintuitive.  The best example I know of is the Banach-Tarski "paradox".  I put "paradox" in quotes because it's not really a paradox, just deeply weird and unintuitive: it is possible to decompose a sphere into a finite number of parts that can be reassembled to produce two spheres, each the same size as the original.  That "truth" cannot be explained in terms of anything that happens in objective reality.  Indeed, the reason this result seems deeply weird and unintuitive is that it appears to directly contradict what is possible in objective reality.  So the Banach-Tarski "paradox" would seem to be a counter-example to any possible theory of how mathematical truth can be grounded in objective reality.  And indeed it is a counter-example to the idea that mathematical truths are grounded in actual objective reality, but that is not news -- we already established that with the example of negative numbers and imaginary numbers.

I've already tipped my hand and told you that (my hypothesis is that) mathematics is the study of possible models of objective reality.  To defend this hypothesis I need to explain what a "model" is, and what I mean by "possible" in this context.

A model is any physical system whose behavior correlates in some way with another physical system.  An orrery, for example, is a model of the solar system.  An orrery is a mechanical model, generally made of gears.  It is the actual physical motion of the gears that corresponds in some way to the actual physical motion of the planets.

Mathematics is obviously not made of gears, but remember that mathematics is not the model, it is the study of (possible) models (of objective reality).  So the study of mechanical models like orreries falls under the purview of mathematics.  Mathematics obviously transcends the study of mechanical models in some way, but you may be surprised at how closely math and mechanism are linked historically.  Math began when humans made marks on sticks (or bones) or put pebbles in pots to keep track of how many sheep they had in their flocks or how much grain they had harvested.  (These ancient roots of math live on today in the word "calculate" which derives from the latin word "calculus" which means "pebble".)  And mathematics was closely linked to the design and manufacture of mechanical calculating devices, generally made using gears just like orreries, right up to the middle of the 20th century.

There is another kind of model besides a mechanical one: a symbolic model.  Mathematics has its roots in arithmetic which has its roots in mechanical models of quantities where there was a one-to-one-correspondence between marks-on-a-stick or pebbles-in-a-pot and the things being counted.  But this gets cumbersome very quickly as the numbers get big, and so humans came up with what is quite possibly the single biggest technical innovation of all time: the symbol.  A symbol is a physical thing -- usually a mark on a piece of paper or a clay tablet, but also possibly a sound, or nowadays a pattern of electrical impulses in a silicon chip -- that is taken to stand for something to which that mark bears no physical resemblance at all.  The familiar numerals 0 1 2 3 ... 9 are all symbols.  There is nothing about the shape of the numeral "9" that has anything to do with the number it denotes.  It's just an arbitrary convention that 9 means this many things:

@ @ @ @ @ @ @ @ @


and 3 means this many things:

@ @ @


and so on.

Not all symbols have straightforward mappings onto meanings.  Letters, for example, are symbols but in general they don't mean anything "out of the box".  You have to assemble letters into words before they take on any meaning at all, and then arrange those words into sentences (at the very least) in order to communicate coherent ideas.  This, too, is just a convention.  It is not necessary to use letters, and not all languages do.  Chinese, for example, uses logograms, which are symbols that convey meaning on their own without being composed with other symbols.  And symbols don't have to be abstract either.  Pictograms are symbols that communicate meaning by their physical resemblance to the ideas they stand for.

Mathematical symbols work more like logograms than letters.  A mathematical symbol like "3" or "9" or "+" generally conveys some kind of meaning by itself, but you have to compose multiple symbols to get a complete idea like "3+2=5".  Not all compositions of symbols result have coherent meanings, just as not all compositions of letters or words have coherent meanings.  There are rules governing how to compose mathematical symbols just as for natural language.  "3+2=5" is a coherent idea (under the usual set of rules) but "325=+" is not.

There is a further set of rules for how to manipulate mathematical symbols to produce "correct" ideas.  An example of this is the rules of arithmetic you were taught in elementary school.  The result of manipulating numerals according to these rules is a symbolic model of quantities.  There is a correspondence between strings of symbols like "967+381=1348" and the behavior of quantities in objective reality.  Moreover, manipulating symbols according to these rules might seem like a chore, but it is a lot easier to figure out what 967+381 is by applying the rules of arithmetic than by counting out groups of pebbles.

It turns out that manipulating symbols according to the right rules yields almost unfathomable power.  With the right rules you can produce symbolic models of ... well, just about anything, including, but not limited to, every aspect of objective reality that mankind has studied to date (with the possible exception of human brains -- we will get to that later in the series).

Mathematics is the study of these rules, figuring out which sets of rules produce interesting and useful behavior and which do not.  One of the things that makes sets of rules for manipulating symbols interesting and useful is being able to separate string of symbols into categories like "meaningful" and "meaningless" or "true" and "false".  Sometimes, for sets of rules that produce models of objective reality, "true" and "false" map onto things in objective reality, and sometimes they are just arbitrary labels.

The canonical example of this is Euclid's fifth postulate: given a line and a point not on that line, there is exactly one line through the given point parallel to the given line.  For over 2000 years humans believed that to be true and were vexed when they couldn't find a way to prove it.  It turns out that it is neither true nor false but a completely arbitrary choice; you can simply choose whether the number of lines through a point parallel to a given line is one or zero or infinite.  Any of those three choices leads to useful and interesting results.  As a bonus, some of them turn out to be good models of some aspects of objective reality too.

Another way of looking at it is that mathematics looks at what happens when you remove the constraints of physical reality from a set of rules that model that reality.  More often than not it turns out that when you do this, what you get is a system that is useful for modeling some other aspect of reality.  Sometimes that aspect of reality is something that you would not have even suspected to exist had not the math pointed you in that direction.

An example: arithmetic began as a set of rules for counting physical objects.  You cannot have fewer than zero physical objects.  But you can change the rules of arithmetic to behave as if you could have fewer than zero objects by introducing "the number that is one less than zero" a.k.a. negative one.  Even though that concept is patently absurd from the point of view of counting apples or sheep, it turns out to be indispensable when counting electrical charges or keeping track of financial obligations.  So is it "true" that (say) three minus five equals negative two?  It depends on what you're counting.  Is it "true" that there are an infinite number of primes?  It depends on your willingness to suspend disbelief and imagine an infinite number of numbers even though most of those could not possibly designate any meaningful quantity of physical objects in our finite universe.  It the Banach-Tarski paradox "true"?  It depends on whether or not you want to accept the Axiom of Choice.  (And if you think the AoC seems "obviously true" then you should read this.)

There are many examples of alternatives to the usual rules of numbers that turn out to be useful.  The most common example is modular arithmetic, which produces useful models of things like time on a clock, days of the week, and adding angles.  Another example is p-adic numbers, which are like modular arithmetic on steroids.  It is worth noting that in modular arithmetic, some arithmetic truths that are often taken as gospel turn out not to be true.  For example, in base 7, the square root of two is a rational number (not just rational but an integer!).

Philosophers and religious apologists often cite mathematical "truths" as somehow more "pure" than empirical truths and our ability to perceive them to be evidence of the existence of God or some other ethereal realm.  Nothing could be further from the (empirical!) truth!  In fact, all mathematical "truths" are contingent, dependant on a set of (mostly tacit) assumptions.  Even the very concept of truth itself is an assumption!

With that in mind, let us revisit the liar paradox, to which I promised you an answer last time.  I'll use the two-sentence version since that avoids technical issues with self-reference:
1.  Sentence 2 below is false.

2.  Sentence 1 above is true.
The puzzle is how to assign truth values to those two sentences.  The reason it's a puzzle is that there are two tacit assumptions that people bring to bear.  The first is the Law of the Excluded Middle: propositions are either true or false.  They cannot be both, and they cannot be neither.  A simple way to resolve the paradox is simply to discharge this assumption and say that propositions can be half-true, and that being half-true is the same as being half-false.

The second tacit assumption that makes the Liar paradox paradoxical is the assumption that the truth values of propositions must be constant, that they cannot change with circumstances.  This is particularly odd because everyday life is chock-full of counterexamples.  In fact, the vast majority of propositions that show up in everyday life depend on circumstances.  "It is raining."  "I am hungry."  "It is Tuesday."  The truth values of those all change with circumstances.  Obviously, "It is Tuesday" is only true on Tuesdays.  Why cannot the truth values of the Liar paradox do the same thing?  We can re-cast it as:
1.  At the moment you contemplate the meaning of sentence 2 below, it will be false.

2.  At the moment you contemplate the meaning of sentence 1 above, it will be true.
The truth values then flip back and forth between true and false as you shift the focus of your contemplation from one to the other.  Note that both of these solutions can also be applied to the "this sentence is true" version, where all three of "true", "false" and "half-true/half-false" produce consistent results (though of course not at the same time).

Finally, note that we can also attack the Liar paradox experimentally by building a physical model of it.  There are many ways to do this, but any physical mechanism that emulates digital logic will do.  You could build it out transistors or relays or Legos.  All you need to do is build an inverter, a device whose output is the opposite of its input.  Then you connect the output to the input and see what happens.

In the case of a relay, there is enough mechanical delay that the result will be flipping back and forth.  It will happen fast enough that the result will sound like a buzzer, and indeed back in the days before cheap transistors this is often how actual buzzers were made.  If you build this circuit out of transistors then the outcome will depend a lot on the details, and you will end up with either an oscillator or a voltage that is half-way between 1 and 0.

If you put two inverters in series and connect the final output to the initial input you will have built a latch, which will stay at whatever condition it starts out in.  This is how certain kinds of computer memory are made.

The modeling train runs in both directions.  This will become important later when we talk about information.  But that will have to wait until next time.

2 comments:

Publius said...

3 is prime

@Ron:
>Mathematical truth can be understood purely in terms of objective reality.

Then later you write:

>So the Banach-Tarski "paradox" would seem to be a counter-example to any possible theory of how mathematical truth can be grounded in objective reality. And indeed it is a counter-example to the idea that mathematical truths are grounded in actual objective reality, but that is not news -- we already established that with the example of negative numbers and imaginary numbers.

So ... mathematics cannot be understood purely in terms of objective reality?

>Specifically, mathematics can be viewed as the study of possible models of objective reality.

"Possible models of objective reality" are abstract objects. Mathematics is abstract.

Just where do stand on abstract objects? Do abstract objects exist in and/or before particular things (realism), or are they only concepts of the mind (conceptualism), or they don't exist and are just language words (nominalists)?

>It turns out that manipulating symbols according to the right rules yields almost unfathomable power. With the right rules you can produce symbolic models of ... well, just about anything, including, but not limited to, every aspect of objective reality that mankind has studied to date (with the possible exception of human brains -- we will get to that later in the series).

Are you sure? Every aspect of objective reality?

>But you can change the rules of arithmetic to behave as if you could have fewer than zero objects by introducing "the number that is one less than zero" a.k.a. negative one. Even though that concept is patently absurd from the point of view of counting apples or sheep, it turns out to be indispensable when counting electrical charges or keeping track of financial obligations.

Don't need negative numbers for either of those things.

>Mathematics is the study of these rules, figuring out which sets of rules produce interesting and useful behavior and which do not. One of the things that makes sets of rules for manipulating symbols interesting and useful is being able to separate string of symbols into categories like "meaningful" and "meaningless" or "true" and "false". Sometimes, for sets of rules that produce models of objective reality, "true" and "false" map onto things in objective reality, and sometimes they are just arbitrary labels.

Is this you viewing mathematics as formalism, the philosophy of mathematics takes it as holding that mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than playing ludo or chess are normally thought to have?

It's hard to tell, in the end, what you think mathematics is, and what mathematical truth is.

>Is it "true" that there are an infinite number of primes? It depends on your willingness to suspend disbelief and imagine an infinite number of numbers even though most of those could not possibly designate any meaningful quantity of physical objects in our finite universe. It the Banach-Tarski paradox "true"? It depends on whether or not you want to accept the Axiom of Choice.

I was expecting this to be an essay of what you believe, not a survey of options.

Ron said...

> mathematics cannot be understood purely in terms of objective reality?

Thanks for pointing that out, that is definitely something I need to clarify.

The post above is the fourth of fifth draft I wrote on this topic. It has proven to be the most difficult installment yet. It started out being about Turing machines and the Church-Turing thesis, and ended up what you see above. The "Mathematical truth can be understood purely in terms of objective reality" claim is a leftover from an earlier draft. I should probably just take it out and put it in the next chapter.

> Just where do stand on abstract objects?

If you put a gun to my head and force me to choose one of your labels I'll take conceptualism. But I would say that mathematical objects exist in the ontological category of ideas, just like wizards. The difference between math and fiction is not as great as most people think. The only difference is the strictness of the rules that govern their construction.

> Are you sure? Every aspect of objective reality?

Pretty sure, yeah. But I will grant that at this point this is an unsubstantiated claim, and I need to make that clearer too. So thanks for keeping me honest here.

> Don't need negative numbers for either of those things.

True. I guess I should change "indispensable" to "handy".

> Is this you viewing mathematics as formalism

Yes. Again, not as clear as it should be. Math is a game, and some sets of rules turn out to produce models of reality.

> I was expecting this to be an essay of what you believe, not a survey of options.

Why would you expect that? The *whole point* of this series is that what I believe is irrelevant (except insofar as someone has decided that I'm a trustworthy source).

But this essay definitely needs another rewrite. Thanks for the feedback.