
What Even Is a Number? - ColinWright
https://notebook.drmaciver.com/posts/2019-02-18-08:58.html
======
cobbzilla
I had a fun mind-play with my kids; I asked them if numbers like "one" and
"two" really exist. They said yes, of course.

OK, I said, show me a plain old "one". Nope, that's one fork; nope that's one
ball; nope, you get the idea.

You cannot show just "one" unattached to anything else.

The concept of "one" is an idea. It only exists as a "real" concept in your
mind because it exists as the same concept in the minds of others, and it's a
very useful concept in describing the world around us.

This odd fact becomes ever more apparent when you find out about cultures that
have very different approaches to quantitative reasoning, like the Pirahã in
Brazil [1]. They don't have number words beyond "one", words roughly meaning
"some" and "many" are used for anything more than one.

[1]
[https://www.sciencedaily.com/releases/2008/07/080714111940.h...](https://www.sciencedaily.com/releases/2008/07/080714111940.htm)

~~~
carapace
Ah! I wrote the below before seeing your comment:

Numbers don't exist. Take the number two. You can have two apples but that's
not the number two. You can write the numeral "2", but that's not the number
two. It's one line. The word "two" has three letters, it's obviously not the
number two. In fact, the number two doesn't exist (or it's existence is not
contingent on an arrangement of matter/energy. No pattern of
matter/energy/space/time is actually THE number two.)

To me, it's wonderful to reflect on the epistemological status of numbers.
They don't seem to exist, but we talk about them as if they do. Certainly they
are useful. But they don't exist any more than, say, Sherlock Holmes does.

It's also much fun to reflect that, whatever the epistemological status of
number, that status is shared by computer languages and algorithms and much
else. E.g. the C language doesn't exist. The C standard(s) exist, many C
compilers exist, lots and lots of C code exist, etc., but the language itself
doesn't exist anywhere, any more than does the number two.

~~~
johnfn
Does an apple exist? You can show me an apple, but I would wag my finger in
exactly the same way you are to people trying to show you “one”. The apple
you’re showing me is not the same thing as the word “apple” which is an
abstract category that encompasses all types of apples - honeycrisp, Granny
Smith, etc. You could say a honeycrisp is like an instance of apple which is a
base class, so then it is indeed an apple. But then youd have to say an apple
is also a “fruit” and also a “food” even though those words also encompass
broad categories of things.

If you buy the above, you’d have to say that a single apple is also “one”.
After all, it’s just another broad category that a single apple fits in to. If
not, I think you’d have to reject the fact that a honeycrisp is an apple,
which seems untenable.

~~~
carapace
Reflecting on this stuff is my golf...

Even if I agree that THE number "one" is "just another broad category that a
single apple fits in to", I think the epistemological status of the Broad
Category is then itself just as problematical, existence-wise, as the original
concept of "number", eh? We can at least display a pair of apples and "count"
to "two" with them, etc. What hope do we have of enumerating (no pun intended)
the Broad Category of "one"? And what do we gain from introducing it?

------
cardamomo
I was once posed a related question by a three-year-old: "How do you make a
number?"

It was a simply worded but delightfully insightful wondering. I did my best in
the moment and answered, "One way we can make numbers is by counting." I've
continued to mull this question over since then. I'm drawn to this child's
conception of numbers as something we make. It makes sense to me as a teacher
of young children and a wonderer about math myself, yet it doesn't really
square with what we've been told about numbers in most educational settings.

~~~
kccqzy
One of the Peano axioms is that every natural number is either zero or the
successor of another. It's easy to explain to kids without jargons: a number
either counts nothing at all, or it counts things that have an extra item
compared to some other thing.

~~~
WilliamEdward
This kind of breaks down once you move past natural numbers, but for early
learners of mathematics i agree it's useful to think about.

~~~
FakeComments
But other kinds of numbers can be made from that:

\- Differences of counting numbers (integers)

\- Ratios of numbers (rationals)

\- Limits to sequences of numbers (reals)

\- Solutions to polynomials made from numbers (complex)

You can think of each of these as adding a layer of behavior to the basic
“each number is nothing or one more than another number”.

~~~
btilly
I agree with your sequence until the last two.

The second to last, that should be Cauchy sequences.

And for the last, demonstrating that the algebraic closure of the reals is the
complex numbers from first principles is much harder than describing complex
numbers as pairs of reals with a multiplication rule that (a,b) * (c,d) = (ac
- bd, ad + bc) and then much later proving that it is algebraically closed
through complex analysis.

For those who don't know the proof, the idea is this. Liouville's theorem says
that if a function is differentiable everywhere, and it is bounded, then it
must be constant. Now suppose that p(z) is a polynomial. Consider the function
1/p(z). You can show that as z approaches infinity, it approaches zero. It is
not constant. Therefore it must not be differentiable or not bounded. It
doesn't take too much work from there to prove that it blows up somewhere, and
the spot that it blows up is a point where p(z) is 0.

Apply unique factorization for polynomials (see
[http://sites.millersville.edu/bikenaga/abstract-
algebra-2/po...](http://sites.millersville.edu/bikenaga/abstract-
algebra-2/polyufd/polyufd.html) for that proof) and you quickly get the fact
that the complex numbers are algebraically closed.

~~~
est31
A much easier to understand definition for the set of reals is probably to use
infinite decimal expansions that don't end in 9999 etc. Equivalence classes of
cauchy sequences is tougher to explain I'd say.

~~~
btilly
In that case, good luck coming up with a good definition around multiplication
where 3 * 0.33333... works out right. And then proving arithmetic properties
like the associative law. And then proving that when you do the reals in
decimal, you get the same system as the reals in binary.

It sounds harder, but is actually easier to go through the Cauchy sequence
definition and then point out that the decimal representation naturally gives
rise to a Cauchy sequence. So, for example, 3.1415926535... gives you (3,
31/10, 314/100, 3141/1000, ...). And as Cauchy sequences, of course, (1, 1, 1,
1,...) is easily proved to be the same as (9/10, 99/100, 999/1000, ...).

~~~
est31
Oh right, 3 * 0.33333 is an issue, yeah. Still, you need to define that
equivalence relation on the Cauchy sequences and then explain what a set
modulo a relation means.

~~~
btilly
This is all standard mathematics.

The equivalence relationship is that the sequence (x_1, x_2, x_3, ...) is
equivalent to (y_1, y_2, y_2, ...) if and only if the limit as n goes to
infinity of x_n - y_n = 0.

Formally, the real number represented by (x_1, x_2, x_3, ...) is the set of
all Cauchy sequences which are equivalent to that one. Since "equivalent to"
is transitive, any Cauchy sequence in that set will define the same set.

Addition and multiplication are defined elementwise. Proving that they are
well-defined is relatively straightforward. Their algebraic properties follow
for free. Any rational number q can be mapped to the Cauchy sequence (q, q, q,
...) which leads to a unique real number that we somewhat sloppily call q
again.

I've left some details out, but this construction is well-understood, and is
how we define the completion of a metric space.

------
dandare
> You can see some of that history of what we call some of the different sorts
> of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these
> represents a bitter argument about whether a new type of number should be
> allowed, all of which were eventually won by the people on the side of the
> new numbers...

Was there ever a proposed type of number that is today not considered a
number? Like i=squareroot(-1), was there ever an attempt to do math with
something like y=1/0 or another illegal construct?

~~~
msla
> was there ever an attempt to do math with something like y=1/0 or another
> illegal construct?

Wheels are a type of algebra where division by 0 is defined.

[https://en.wikipedia.org/wiki/Wheel_theory](https://en.wikipedia.org/wiki/Wheel_theory)

[https://math.stackexchange.com/questions/994508/wheel-
theory...](https://math.stackexchange.com/questions/994508/wheel-theory-
extended-reals-limits-and-nullity-can-dne-limits-be-made-to)

To call wheels "obscure" is to make them out to be better-known than they are,
however.

~~~
cobbzilla
I had not heard of Wheel theory, quite fascinating.

But eek! From the wikipedia article:

    
    
        0x ≠ 0 in the general case
        x − x ≠ 0 in the general case
    

Like, you get the ability to divide by zero, but at what cost? Multiplying by
zero is also not zero! Seems bonkers.

Are there any practical uses for this kind of algebra?

~~~
ajkjk
Likely no (the peculiar distributive laws seem unusable to me). But I have
come across similar rules that I think may have a real usage. It sometimes
seems to me that it might make more sense to keep 'factors' attached to 0s,
such that x-x = (1-1)x = 0x might be 'sound', and to keep track of 'powers' of
0s by having factors of 0 attached to 0s: (0)0 = 0^2 != 0, etc.

If you keep factors like this, then you could implement L'Hopital's rule
without the result only being true if considered under a limit: say, lim(x->0)
(5x^2/ 3x^2) = 5/3 could be computed as (5)0^2 / (3)0^2 = 5/3.

This is not like Wheels though; it requires that any power of 0 be a distinct
number. I of course have no idea if it is sound or meaningful, but I do find
myself thinking about it a lot.

~~~
hopler
Those "factor" zeros look like infinitesimal forms, which are well studied.

~~~
ajkjk
There is a similarity, but no one claims infinitesimals are _literally 0_, and
certainly no one would claim that 2-2 = 2 epsilon.

------
jd007
There is a video
([https://www.youtube.com/watch?v=S4zfmcTC5bM](https://www.youtube.com/watch?v=S4zfmcTC5bM))
from the PBS Infinite Series that covered this topic as well, for people that
what a visual version of the explanation. I'm still sad that the
series/channel was shutdown though, such a great series.

PS: this article/video essentially defines the naturals from the fundamental
set theory axioms. This other video
([https://www.youtube.com/watch?v=KTUVdXI2vng](https://www.youtube.com/watch?v=KTUVdXI2vng))
from the same PBS series shows how you can then use the naturals to construct
other types of common numbers, up to the reals.

------
tim333
As a follow on people may enjoy Feynman's lecture on Algebra
[http://www.feynmanlectures.caltech.edu/I_22.html](http://www.feynmanlectures.caltech.edu/I_22.html)
and some audio clips [http://www.feynman.com/the-animated-feynman-
lectures/](http://www.feynman.com/the-animated-feynman-lectures/)

>To discuss this subject we start in the middle. We suppose that we already
know what integers are, what zero is, and what it means to increase a number
by one unit. You may say, “That is not in the middle!” But it is the middle
from a mathematical standpoint, because we could go even further back and
describe the theory of sets in order to derive some of these properties of
integers. But we are not going in that direction, the direction of
mathematical philosophy and mathematical logic, but rather in the other
direction, from the assumption that we know what integers are and we know how
to count. ...

------
Someone
_”This means we eventually end up either with an empty sequence (the set is
empty) or with a sequence with only + operations in it”_

For valid starting points, that is true, but it doesn’t follow from the text.
The loop has an error exit ( _”If the sequence starts with − then something
has gone wrong”_ ), and the text doesn’t show that you won’t get there for
valid inputs, and showing that isn’t trivial.

For example, if the sequence is ++---+, the first iteration removes the second
and third item, leaving +--+, and the second iteration removes the first and
second item, yielding -+.

The ‘program’ crashes there because the input is invalid, but proving that it
never will crash for valid inputs without resorting to “that’s how integers
behave” isn’t trivial.

------
lmarinho
My usual answer is that math isn't too concerned with what things _are_ , but
with _how they work_ with each other. So the answer would be: anything that
works like the Peano axioms says it should. The "what it is" can be filled in
later, and that's what makes math so powerful.

------
xamuel
Article seems very verbose and only really addresses the natural numbers (it
mentions negatives and rationals in passing).

In case anyone just wants actual answers without reading pages and pages of
prose, here is one set of constructions (there are other competitors too).

The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural
2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to the reader:
define appropriate arithmetical functions on the naturals.

Define a relation ~ on pairs (a,b) of naturals by saying (a,b)~(c,d) if and
only if b+c=a+d. Exercise to the reader: ~ is an equivalence relation. Its
equivalence classes are called "integers". The equivalence class containing
(a,b) represents the integer b-a. For example, the integer -1 is the
equivalence class {(1,0),(2,1),(3,2),...}. Exercise: define appropriate
arithmetical functions on the integers.

Define a relation ~ on pairs (m,n) (n nonzero) of integers by saying
(m,n)~(p,q) if and only if mq=pn. Exercise: Show ~ is an equivalence relation.
Its equivalence classes are called "rationals". The equivalence class
containing (m,n) represents the rational m/n. For example, 1/2 is the
equivalence class {(1,2),(-1,-2),(2,4),(-2,-4),...}={(k,2k) for all nonzero
integers k}. Exercise: define arithmetic operations on the rationals.

To get from the rationals to the reals, see
[https://en.wikipedia.org/wiki/Dedekind_cut](https://en.wikipedia.org/wiki/Dedekind_cut)

~~~
12elephant
Your castle is built on sand. What, then, is {}?

~~~
xamuel
One of the axioms of set theory is the axiom of the empty set, which states
that there exists at least one set which has no elements. Another axiom of set
theory is the axiom of extensionality, which states that two sets are equal if
they have the exact same elements: from which it follows that all sets without
elements are identical, i.e., there is only one set without elements. We call
that the emptyset.

Other axioms of set theory are used to formalize other steps in my post.
[https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms)
For example, the Axiom of Infinity is used to formalize the handwavy part
where I said "In general, the natural n+1 is {0,...,n}".

~~~
12elephant
An _axiom_ is simply _something made up_.

At the end of the day, all of our Mathematics rests on foundations that are
made up.

It's difficult to say _what_ the empty set is. Because it isn't really
anything at all.

~~~
throwawaymath
There are legitimate criticisms you can levy against set theory, but I'm
starting to lose you here. I'm not really following your point anymore - this
seems like arguing about whether or not mathematics is invented or discovered.

Are you trying to argue coming up with new definitions isn't worthwhile if the
axioms don't have a foundation in reality? If so, why? If not, what _are_ you
saying?

~~~
12elephant
The original post is called "What even is a number?". The comment I replied to
tried to answer that question with "some formalism".

What I'm saying is: questioning foundations leads you to new foundations. New
foundations that you can then question all over again. It's turtles all the
way down.

If you're actually looking to use math, this is a futile exercise. It works,
so just use it. Essentially, I am re-iterating von Neumann's statement:

> In mathematics you don't understand things. You just get used to them.

~~~
throwawaymath
In that case I'd agree with you. I'm not particularly keen on rehashing
foundations of mathematics either.

------
guest2143
I was surprised by no discussion of ordinality. Cardinality was something I
also specifically expected.

Counts of things are described here, but the greeks went to great lengths with
geometry to discover numbers (and even killed people who came up with
irrational answers)

~~~
kccqzy
Most of computer science deals with finite things. As such, it isn't necessary
to introduce cardinal numbers or ordinal numbers. Natural numbers suffice. If
you do, you quickly need to get into territory that requires lots of math
background like well orderings, cardinal comparability hypothesis, transfinite
recursion etc.

------
mabbo
I read this as "What is an even number" and spent the entire article stoked
for when he got to the part about a number being even.

Still, lovely article.

~~~
oh_sigh
Don't even numbers have a pretty easy definition? If prime factorizartion of
the number contains at least a 2

~~~
mabbo
Oh sure, but then the article started talking about _sheep_ and _rocks_ and
I'm like "Oh boy, I bet I totally don't understand what it means to be
_even_!"

~~~
aidenn0
Well you can reduce the +/\- notation further by allowing yourself to (after
removing all +/\- pairs) to remove pairs of sequential + tokens, and if it
reduces to an empty string then it's an even number, otherwise it's an odd
number.

------
_bxg1
It's extremely obvious that the author is a computer scientist

~~~
fredliu
Exactly! I didn't know the author, but couldn't help but check on his
background after reading the blogpost.

It's almost a dead giveaway for excerpts like 'The notion of a counter is not
that of a single concrete class of object, it is an abstract description of
the behaviour of a system providing certain operations satisfying certain
rules.' , and 'A number is not any one thing, it is any one of any number of
things that implement some operations (and there are different types of number
depending on what operations you want to implement).'

~~~
_bxg1
And the very first thing he did was define natural numbers as a grammar of
operations with rules for interpretation

------
ddebernardy
> What even is a number?

It's a category. 1 is the category of singletons, 2 is the category of pairs,
etc.

As to what _numbers_ are, they're an ordered collection of categories.

Edit: To whoever downvoted this comment, kindly explain. Insofar as I'm aware
this is textbook maths, psychology, and philosophy.

~~~
throwawaymath
If you define the natural numbers this way you'll run into Russell's paradox.
For practical purposes that's fine and this is an elegant, modern restatement
of Frege.

But if you need to avoid Russell's paradox the sets defining each natural _n_
can't contain _n_ as an element. The easiest such construction was outlined
elsewhere in this thread by xamael. Under your category theoretic
construction, every natural number _n_ is defined as the category of sets
having cardinality _n_. But then every _n_ th category will necessarily
contain infinitely many sets with cardinality _n_ that also contain _n_. That
causes the paradox.

Unfortunately I don't think you can construct the naturals in a category
theoretic way while avoiding Russell's paradox since any category by
cardinality will fall into that trap. But if you don't need to mind that
problem, this is neat.

Responding to your edit: I didn't downvote you; in fact I upvoted this comment
because it's correct and it was gray at the time of my writing. My comment is
just a point of clarification.

~~~
edflsafoiewq
I don't understand how Russell's paradox comes in. The set of all eg. pairs
does not contain itself.

~~~
throwawaymath
If you've defined the natural number 2 to be an arbitrary set with cardinality
2, you're including sets which contain the number 2. That's the basic form of
Russell's paradox.

If you define the natural number 2 to be the category of pairs, your objects
are the sets with cardinality 2, and your relations between objects are
equivalence relations. As a consequence your category 2 will contain sets
which contain itself.

~~~
ball_of_lint
Why is that a problem?

------
toddwprice
I love articles like this. Thanks for sharing. Ultimately math is just a
language invented to describe the world. Which is what any language does. And
it's a really useful language for sharing certain kinds of intelligence. I got
into a discussion with my son last night about sine waves. Which don't really
exist, per se, but they are a useful language construct for thinking about
waves (especially sound waves as we were discussing). Shared intelligence like
this is, as some have maintained, a major differentiator between humans and
pre-humans. So the very fact we can argue about what is a number is part of
what it is to be human.

~~~
tim333
It may be more than a language to describe the world. Take Euler's identity,
e^(i*pi)=-1. One of the most striking equations in maths and it doesn't really
describe anything physical in particular and true regardless of the nature of
the world.

~~~
mzs
I can visualize it as triangles.

[https://www.math.toronto.edu/mathnet/questionCorner/epii.htm...](https://www.math.toronto.edu/mathnet/questionCorner/epii.html)

------
lqet
Possibly related: [https://www.smbc-
comics.com/comics/1485876000-20170131.png](https://www.smbc-
comics.com/comics/1485876000-20170131.png)

------
maxbendick
Here's a silly definition:

A number is a sequence of 1 or more digits, optionally beginning with a '-'
and optionally followed by a '.' and one or more digits, where digits are the
characters {'0' '1' '2' '3' '4' '5' '6' '7' '8' '9'}.

Examples: 1, 25, 3.2, -8

"1 + 2" is not a number, but its evaluation is

------
theWheez
This appears quite similar to the difficulty in explaining Monads (from the
programming perspective rather than the mathematical)

------
ngcc_hk
I think the counting is natural to human. The corresponding theory op used is
not natural. If one has to use that, one start to use writing to do
correspondence. Or use shell/good/coin.

That is for +integer.

Not sure about how natural is real number other than pi using geometry. And
irrational also use geometry.

Use algebra and group-ring-field is very late.

------
aidenn0
The article ends with "Almost certainly not, but if so why would we care?"
which I find (along with its inverse "Almost certainly, but if not why would
we care") to be my answer to many philosophical questions.

------
chasereed
This really just boils down to: Counting is an intuitive thing for us and it's
useful to mathematically define it.

~~~
aidenn0
Can you expand? I think the author chose counting because of its
intuitiveness, but they then go on to show that a very simple definition of
counting can lead to some non-intuitive concepts, _and_ that even restricting
yourself to that simple definition of counting can lead to classifying
something controversial (i.e. infinity) as a number.

------
galaxyLogic
Number is an abstraction. What is an abstraction? It is a definition for a set
of things by stating what is common to all things considered to be in that
set.

"1" is the set of all things which have exactly one element. It is an
abstraction for all such things.

What does it mean to "have one element"? I think that's a more difficult
question. Maybe it should be an axiom?

------
dnprock
I often hear people say mathematics is the language of science. Since learning
about cryptocurrency, I came to realize that more precisely, mathematics is
the language to describe energy transformation.

So a number can be seen as a metaphysical representation of energy.

