What am I talking about?
In mathematics, one makes statements with great certainty about
objects of whose nature one is ignorant. I say “There are infinitely
many prime numbers.”^{[1]}
and I know it to be true. I don't know how a number “exists”.
“There are infinitely many prime
numbers.”^{[2]} is a
sentence about the structure of the natural numbers (individual
numbers are meaningless without a structure). It states that however
large a natural number you pick, there are greater natural numbers
that are prime. In classical logic, if there happen not to be such
things as natural numbers, the statement is false.
Real Things?
For most of Western history, the majority of people who seriously
investigated the properties of natural numbers thought the natural
numbers were a real thing that existed. This might be because
Western matheamtics was in large part founded by a mystic named
Pythagoras who believed all the world was Number. The idea that
abstractions Really Exist somewhere became known as Platonism since
Plato originated and popularized the idea that you have Beauty,
Goodness, The Ideal Circle, the Natural Numbers… out there somewhere
more Really Real than the everyday world which might be thought of
as a mingling of their shadows. It's also known as Realism, because
it holds that mathematical abstractions are Real and Exist
independently of anything else.
Standards of Beauty change over time; there are competing claims of
Goodness, but everyone (Intuitionists and Constructivists accept
fewer theorems, but they don't come to conclusions that contradict
classical mathematics) has the same math. There's never been an
instance of nature acting against mathematics; on the contrary,
people invent newer, ever more abstract mathematical ideas and
someone finds a way to apply them to the natural world. Thus,
mathematical Realism stays while Beauty and Goodness are thrown into
the seas of cultural contingency.
Realism invites questions. Exactly how do these things exist?
Transcendently?^{[3]}
Outside the universe? Beyond space and time? That's how it's usually
taken. It avoids having to explain how something completely
immaterial could exist in space and how something eternal
could exist in time. It also matches the intuition that mathematical
statements would be true whether anything existed or not.
The most famous and compelling argument for Realism, the
Indispensability Argument of W. V. O. Quine, states that since our
best physical theories rely on mathematical abstractions, we ought
be as willing to accept the existence of those abstractions as we
are electrons. This is compelling, but the physical theories also
explain how electrons interact with each other and other charged
particles. The second derivative operator does not interact with a
moving rocket in the same way that the gravity of a planet does.
Furthermore, there are mathematical abstractions that no physical
theory depends on. A Realist position in which all statements about
the Fischer-Griess Monster Group are true only if some physical
theory is found that depends on them is not very Realist. This is
the main reason why Quine and others referred to his stance as
Empiricist.
There are more reasons to ask which abstractions are Real.
Leopold Kronecker famously said "God made the natural numbers;
all else is the work of man." Almost
every^{[4]} Realist
would say that the natural numbers exist, they
feel^{[5]}
too primitive and natural not to. While the natural numbers'
naturalness is almost certainly a property of humanity rather than a
property of the natural numbers, let us say they exist.
Let's throw in the integers and rational numbers, those are fairly
uncontroversial. What about the real numbers? Do they live up to
their name? The Finitists reject uncountable
sets^{[7]}; some go
further and, while accepting the existence of every natural number,
reject the infinite set of natural numbers. If they are correct and
real numbers do not exist, statements about them may be well-formed
and provable but false. We might allow other things to exist:
lambda calculi and set theories. Which set theories?
ZF?
ZFC?
Something else? There are constructive set theories. There are set
theories that have only a countably infinite number of finite sets.
This is my biggest problem with Realism. Unless you accept
ontological maximalism (everything that can exist does exist, where
'can exist' is usually some notion of 'follows from some consistent
set of axioms'), you don't know whether anything you say is true.
The things you're talking about might not exist and there's no way
to find out, unless you accept Quine's formulation and its
contingency of mathematical truth.
Realism is kind of weird. Let's make some conservative assumptions:
the natural numbers, lambda calculi, and some set theories, logics,
and other countable things exist. No uncountable anything. The
Natural Numbers are Real, truly, in themselves. They also exist,
Really, as multiple constructions in set theory, as multiple
constructions in logic, and as multiple constructions in lambda
calculi. They have to. If our Realist abstractions are to have any
meaning, the Number Five has to Really exist in the Natural Numbers
constructed from the Set Theory that Really exists just as much (if
not moreso!) as it exists when we pile up groups of five rocks and
see how many equal groups of rocks we can divide them into.
No wonder people people say mathematical Realism sounds like weird
religious mysticism! Start thinking that way and you'll fall into
the Tree of Life with the Natural Numbers at the crown, flowing
through lesser abstractions into the world. Except the Tree of Life
has a top and a direction of flow. With Gödel
numbering^{[8]} we
can spin it around and put logics at the top. Their theorems and
rules of inference would stand supreme, reflected in the natural
numbers and flowing down into the world. We could throw out the tree
of life entirely and have a try at
Indra's^{[9]} net,
with abstractions reflected in other abstractions, each complete in
itself and constructible in others, shining upon the world. Georg
Cantor, Master of Infinity, believed in the Absolute Infinite, the
Infinite that contained all other infinities. Too infinite to be a
number, each of its properties reflected in the things it comprised.
He also thought the it was
God^{[10]}.
Cantor was a mathematical Realist and ontological
maximalist. He believed that everything consistent (lacking internal
contradictions) that followed from some axiomatic system was Real.
Squishy Organic Stuff?
Traditional mathematical Realism is dualist. There's matter, and
there's math. Dualism has all sorts of philosophical problems, like
how your two substances interact. Also, nobody takes it seriously.
It's socially condemned. So, people come up with alternatives. One
of the most recent, championed by George Lakoff (like most things
favored by George Lakoff, it isn't very good), is the Embodied Mind
theory of mathematics. This school of thought tries to explain
mathematics as a behavior born of evolution and instinct. To the
extent that this is true, it is trivial. Professor Lakoff tries to
get rid of the idea of general reasoning over logical abstractions
and reduce all of mathematics to a few basic metaphors related to
interacting with the physical world.
It fails, for one, because it assumes that children learning their
multiplication tables think and reason about the natural numbers in
the same way and with the same internal abstractions as number
theorists proving a theorem. Professor Lakoff's theory is written in
terms of representation rather than relation. This is the biggest
problem. As you can see above, there are many ways to construct one
abstraction in terms of another, and Professor Lakoff's way of
building mathematics from metaphors requires that each method of
construction lead to a different mental object. This fails utterly
at capturing how mathematicians actually think. It also violates
the most fundamental attribute of mathematics: that its subject is
structural and relational. Lakoff's account of the predictive power
of mathematics shows where his entire notion of the Embodied Mind
(even apart from mathematics) goes wrong.
He explains that, since humans evolved to survive in the physical
world, they should expect that their minds and metaphors would be
very well suited to modelling the physical world. This sounds like a
very reasonable, logical answer. It's false. You, as a human, are
very, very, very bad at probability. Astoundingly bad. You have
crude heuristics for running away from things that might be snakes
in the grass, but they're awful at making accurate predictions. This
should be enough to kill off Lakoff's explanation. Humans can
develop probability theory and build abstract mental machinery to
make up for their more ‘embodied’ aspect's failure. Mathematics also
works remarkably well at grasping quantum electrodynamics, which has
nothing to do with the ancestral environment. The biggest flaw in
the current crop of Embodied Mind theories is that they assume that
(to borrow Daniel Kahneman's term) our minds comprise System 1 and
nothing else. Embodied Mind theories may one day be quite valuable
in education or predicting systematic errors, but their authors will
need to do better than writing the word ‘metaphor’ repeatedly
sprinkled with an occasional PHRASE IN CAPITAL LETTERS.
Lies?
One of my favorite answers to the question of what mathematics is
about is ‘Nothing!’. Hartry Field declared that mathematical objects
do not exist and all statements about them are false. He called
it fictionalism: the belief that mathematics is a useful
fiction. I adore this theory, not because I believe it, but because
of the work Professor Field did to support it.
In Science without Numbers, he recreated Newtonian
mechanics and gravitation without numbers. Instead of numbers he
used regions of space-time and notions of congruence and
betweenness. It's a triumph and one of the most awesome things I've
ever read. It makes me awfully happy, but I'm not convinced. For one
thing, Field ends up with a very abstract, rigorous, and structured
system. It doesn't look like a demathematicized science to me, it
looks like a beautiful system of calculus invented by aliens. It is,
too. It maps very well onto Calculus. Field attempted a proof that
mathematics does not conflict with any purely physical theory. He
thought the de-mathematicization and lack of conflict together could
explain the unreasonable effectiveness of of mathematics. It doesn't work for
me. To me, Field's demathematicization is math (also assuming
the reality of space-time regions independent of anything else plus
all the heavy logical machinery he used racks up a lot of
metaphysical debt), while showing that mathematics is not
inconsistent with known physical theories seems insufficient
to explain why mathematics and the world should have anything to do
with each other.
Nonexistent Things?
Once upon a time there was a man named Meinong. He rejected the idea
that you couldn't make true statements about nonexistent things.
After all, unicorns have one horn. Nemesis is a twin star to the sun
that caused the extinction of the dinosaurs. I can speak about the
Natural Numbers whether they exist or not. He made existence a
property something could have like redness or tallness. Some things
happen to exist and some things happen not to exist. Some things are
impossible (those either lacking properties that grant them mass and
weight and extent in space and time, or those having contradictory
properties).
The theory of nonexistent objects requires that all nonexistent
objects…nonexist— Square circles, prime numbers with fifty divisors.
This is what the phrase ‘metaphysically extravagant’ was made for.
(No, really, it was!) Meinong's theory was that for every set
of properties, there is an object. Some objects had the property of
existence. I like ontological maximalism as much as the next guy,
but, like Georg Cantor, Master of Infinity, I'm only interested in
objects that are consistent. Bertrand Russell destroyed Meinong's
theory, causing it to explode into a mess of paradoxes. There have
been attempts to rehabilitate it, but they lack the appeal of the
original.
Just playing around?
Mathematical Formalism is the belief that mathematically
true statements are statements about the evolution and manipulation
of formal systems. One variant, Term Formalism is
concerned with syntactic manipulations of large (possibly infinite)
vocabularies of primitive terms. It was best elaborated by by
Haskell Curry^{[11]}.
He defined mathematical statements as true if it would be possible
to derive the associated relations of primitive terms from other
true relations of primitive terms. This is elegant, but is weighed
down by so much metaphysical debt in the form of reified logical
machinery and primitive terms that it falls back into Realism.
It's more interesting as a primitive base from which other things
can be constructed than as a metaphysics of mathematics.
The other variant of Formalism, Game Formalism, defines
mathematics as the manipulations of strings in accord with rules. A
statement is viewed as true when an appropriate string manipulation
yields it. This is the most popular escape from Realism. In
retrospect, this is surprising. It doesn't explain why mathematics
should describe the world so well. It doesn't bear any relationship
to how mathematicians think. Mathematicians do not take an arbitrary
string and apply arbitrary allowable manipulations to it. They think
about sets and functions and shapes. Automata theorists think about
string manipulations, but they think about them being done by
abstract machines working under complexity bounds. Furthermore,
statements are neither true nor false until someone has performed
the appropriate string manipulation, and some theorems, like whether
very large numbers are prime (large enough that to answer will take
exponentially longer than the lifetime of the universe), will
forever be neither true nor false.
I think the popularity of Game Formalism comes from people not
thinking about it very much. They like the connection between proof
and truth and don't grasp that the ‘proof’ in Formalism and the
‘proof’ in their heads have little in common. It lets them not be
Realists with a minimum of effort. I also suspect that the
intuitions of most Game Formalists tend toward what is actually
Modal Structuralism but that they have never heard of Modal
Structuralism. I might be biased.
I used to be a Game Formalist.
An Idea Objects to the Company I Make Him Keep
One night, as I was sleeping, a figment of my imagination came to
me. He was a Realist^{[12]}
and he was not very happy with me. For, you see, I think that the
generalized continuum
hypothesis^{[13]}
is likely true. Kurt Gödel, Lord of Logic, proved that it could be
proved neither true nor false within the generally accepted axioms
of set theory. He believed it was false.
Georg Cantor, Master of Infinity, hypothesized the hypothesis. He
accepted Gödel's proof and thought his hypothesis was true. They
were both Realists; they are allowed to believe unprovable things
about mathematical abstractions.
The figment explained to me rather fiercely that I had no business
claiming to be a Formalist, since I certainly didn't believe it. If
I really believed it, I wouldn't have opinions on
proved-undecidable hypotheses. That's the thing that anyone
but a Formalist can do! By my stated beliefs it was
not merely unknowable, it must be and must forever remain neither
true nor false and I was a cad and a bounder who had just adopted
what seemed like an easy way out of a mentally challenging question
and I should be ashamed! (It was friendlier than you're probably
imagining.)
Having been informed of my error, I spent some time reading and
thinking about a way to believe in the truth of mathematical
statements that would not get me yelled at by the other things I
think about.
Possibly Things?
I settled on Modal Structuralism, the belief that a
statement about some mathematical object is a statement about how
any entity possessing the structural attributes defining that object
must behave in any possible world in which it exists, while
committing to the idea that at least one possible world has
something possessing those structural attributes. So, if I make a
statement about the real numbers, I am saying that in any possible
world where something has the properties of the real numbers, that
thing must behave the way I say it does, and that such a world is
possible. It might be this world if space really is continuous and
every straight line has the structure of the real line. If space is
pixellated then it's some other possible world.
Modal Structuralism has a lot going for it. When I think about sets
or the real line or functions, I'm thinking about sets, the real
line, or functions because the structure is what matters,
not the construction. It addresses the predictive power of
mathematics. Theorems about a mathematical object predict the
behavior of some aspect of the world when that aspect of the world
models the structure of that mathematical object. It requires one
reinterpret every mathematical\n statement to be about structures
modeling something in a possible world, but I don't mind that. More
concerning: what the heck is a possible world?
Possible worlds evolved as a tool in logic to evaluate statements
that involve the world being other than it is. The statement “If
there were a present king of France he could be named Louis.” is
true if, in at least one possible world that is pretty similar to
ours but in which France has a king, that king is named Louis.
Possible worlds aer usually defined as complete descriptions of a
world with consistent propositions and histories. Modal
Structuralism also has one of the problems of Realism: what do I
admit as possible? We're back to the same arguments over whether to
admit everything consistent, like Georg Cantor, Master of Infinity,
or to reject anything infinite like Leopold Kronecker. I,
personally, side with Cantor. (By now, you've probably guessed that
I like Georg Cantor, Master of Infinity, a lot.)
Why is there all this stuff here?
I once read a book called Why Does the World Exist?. It
was a wonderful whirlwind tour of all sorts of mad ideas trying to
explain why there's anything at all. We start with the idea that
Nothing is such a strong force for
annihilation that it eventually annihilates bits of itself and
creates something. Others argue that nothing is impossible. The one
that influenced me most was the argument that the world exists
because of a primordial need for
goodness^{[14]}.
This gentleman claimed that what made any world at all exist was of
much less interest than why this world in particular exists. Thus,
he said, all worlds containing beauty and goodness were the ones
that came into being. I don't buy this idea because holding beauty and
goodness as objective values is absurd. It made me think, though. We
know that the world exists, so having more worlds exist isn't an
extravagant leap. Some physical theories already suggest multiple,
non-interacting universes. (Each such theory would be its own
‘world’.) Positing something that sifts the possible worlds and
actualizes some of them is much more extravagant than multiplying
the number of worlds. Thus, the best answer, with respect to Occam's
Razor, to “Why does this world exist and not some other?” or “Why
does this subset of worlds exist and not some other subset?” is
“What makes you think that? All possible worlds exist.” I later
discovered this belief is called Modal Realism.
So, Real Things?
It took me longer than it should have to realize it, but believing
both an ontologically maximalist form of Modal Structuralism and
Modal Realism compelled me to believe in the Reality of all
mathematical objects. This came as a shock, considering how much
effort I'd put into avoiding mathematical Realism. Now, my
confidence in Modal Realism is fairly low for a belief I claim to
have. It's like I'm tidying up my mental house and want everything
in its place, and Modal Realism seems the neatest and tidiest for
now.
Without realizing it, I had run head-first into Max Tegmark's
Ultimate Ensemble. Tegmark claims that not only do all mathematical
objects exist, but nothing but mathematics exists. Having arrived by
the scenic route, this seems more plausible than when I first heard
of it. It was a natural step. I'd committed to all these possible
worlds containing all these things that model mathematical objects
derived from axiomatic systems, and there was only one way to make
it simpler.
Electrons, photons, quarks, and gluons, Ws, Zs, taus, and muons,
neutrinos, gravitons, and the
Higgs^{[15]} all
have no internal life. Each exists only in its interactions. We have
internal lives, they're visible in our behavior. They're, to some
degree, measurable through examinations of our brains. We ourselves
are made of electrons, quarks, photons, and gluons with the odd
W or Z popping into being and a neutrino zipping away. Everything
that happens in the universe is a matter of structure and relation.
We need the structure. We don't need the, well, stuff. Things have
no essence, only relation, and the solution to the Dualism inherent
in mathematical Realism is to throw out everything but mathematics.
The answer to what puts the fire in the equations is that all
equations have fire pre-installed. Burning. Somewhere.
You win this round, figment.
Footnotes
[1] Prime numbers are natural numbers
divisible only by one and themselves. Euclid, an ancient Greek
explorer who wrote the definitive text on the geography of Flatland,
proved that however many primes you have discovered, there must be
at least one more.
It's simple and elegant and goes like this. Take all your primes and
multiply them together. We'll call that The Product. Add one to The
Product and you'll get The Sum. It might be the case that The Sum is
prime. If it is, you're done, because it couldn't have been on the
original list of primes.
If The Sum isn't prime, then it must have a Prime Factor. If that
Prime Factor were on the list it would have to divide The Product,
but to divide both The Product and The Sum, the Prime Factor would
have to divide one. And it can't. So it isn't. Therefore, how many
primes you may have, there are always more.
[2] As you can see from the above, this
statement could be written more formally as "At least one natural
number is prime and for any subset of the natural numbers all of
whose members are prime there exists a natural number which is prime
and not contained in that subset." The first half of the
conjunction is important. If there were no prime numbers at all any
statement we might make about all prime numbers or all sets of prime
numbers would be true.
[3] I wonder if anyone has considered an
Imminent Realism where mathematics pervades all of space and time in
some immaterial sense. It's unclear what that would mean, but it's
unclear what it means for them to be beyond space and time.
Intuitively, Transcendent Realism feels a better match for the idea
that all mathematical abstractions exist, aloof, outside reality.
Imminent Realism would mesh better with the idea that only those
demonstrated in physical law exist, something of a match for
the Indispensability Argument.
[4] There are people called Ultrafinitists
who reject the existence not only of the set natural numbers, but
who reject the existence of very large natural numbers. They write
very interesting philosophy papers but not much interesting
mathematics.
[5] While I reject George
Lakoff's^{[6]}
‘embodied mind’ analysis of mathematics as not actually being very
good at describing mathematical reasoning, it is quite true
that what one is used to, including embodiment and the environment,
strongly influences ones notions of naturalness. I could imagine
minds living in a gaseous or liquid world (maybe a particularly
runny gel) whose most ‘natural’ number system is the real numbers.
Their ancient civilizations might invent wonders of analysis without
any idea what a prime number is. It might take millennia for anyone
to discover that a counterintuitive, unnatural subset of the
‘natural’ numbers exists and is of interest. The basics of number
theory could, for them, be post-doctoral level material.
[6] To be fair I reject most things written
by George Lakoff.
[7] A long time ago there was a man named
Georg Cantor, Master of infinity. He was the first to rigorously
define infinity. His most famous insight was that some infinities
are more infinite than others. The natural numbers are the least
infinite and were called countably infinite. He proved that the real
numbers were uncountably infinite by a very clever trick which I
will let Vi Hart
explain.
[8] Gödel numbering, invented as part of
the machinery for Gödel's famous incompleteness theorem, allows one
to turn theorems, or anything else that can be represented as a
finite string of finitely many symbols, into a natural number. One
assigns a number to every possible symbol. To encode a string, raise
the number for the first symbol in the string to the power of the
first prime, the number for the second symbol in the string to the
power of the second prime, and so forth, then multiply them all
together.
Once you have a Gödel numbering set up, properties about theorems in
a logical system then become number theoretic properties and rules
of inference become functions. Gödel numbering, by providing a
mapping between various things and the natural numbers, also provide
a convenient way to prove that various things are countably
infinite. Alan Turing, Dreamer of Machines, used it to prove the
countability of the computable numbers, for example.
[9] One of the few lightning-affine deities
who isn't an embarrassment.
[10] Note that Cantor, Master of Infinity,
believed the Absolute Infinite was an inconsistent idea.
Something that, by his definition of mathematical freedom, was
beyond mathematics. He also believed that his work on transfinite
sets was communicated to him from Heaven and that he had been chosen
to reveal it to the world. He was Catholic and did not, as some
claimed, try to ‘reduce God to a number’, saying instead that
transfinite numbers were ‘at the disposal of the Creator’ just like
everything else.
[11] After whom the programming language
Haskell is named (his wife once mentioned that he didn't like his
first name, to the chagrin of Haskell's developers). He also gave
his name to currying, turning a function taking multiple
arguments into a function taking one argument and
returning a function which takes the second argument, and so on
aside from the last which returns the value. The logician
Schönfinkel invented the concept before Curry did, so some people
(but not very many, for obvious reasons) use the term
‘schönfinkelization’ instead.
[12]
Wouldn't you be if you were a figment of someone's imagination?
[13] The generalized continuum hypothesis
states that infinite cardinalities come in neat succession one after
the other, that the first infinite cardinality is that of the
natural numbers and the second infinite cardinality is that of the
power set of the naturals and is also the cardinality of the reals,
and that the third infinite cardinality is that of the power set of
the reals and on and on with nothing in between. Its negation allows
cardinalities between those of the natural numbers and the
cardinality of their power set; specifically that the cardinality of
the reals may be less than the cardinality of the power set of the
natural numbers.
[14] Cynicism is, in the modern day,
probably the single most common sign of moral and intellectual
failure. Also, the Surgeon General would like to warn you that it
greatly increases your risk of developing soul cancer.
[15] I once saw an interview with Peter
Higgs in which he referred to it as ‘the particle that happens to
bear my name’ with an annoyed air. He is even more unhappy about it
being called ‘The God Particle’.