
What Would Happen If There Were No Number 6? - ryan_j_naughton
https://fivethirtyeight.com/features/what-would-happen-if-there-were-no-number-6/
======
OJFord
> If there’s no 6, said Caroline Turnage-Butterbaugh, a math professor at Duke
> University, then there can’t be a 7, 8 or 9 — or, really, any number greater
> than 5. Say you have a pile of seven items. You could take one away, ending
> up with the-number-formerly-known-as-6.

This seems like a weak argument: say you have a pile of five items; you could
add one, and then we have six after all.

I don't see why subtraction should be treated any differently?

I thought this was going to be more interesting than 'imagine the world in
base 6' \- what're the ramifications of a 'hole' in the middle? I recall
enough to know that fields _allow_ for it, but are the consequences at all
interesting?

~~~
opportune
If there were no number 6, there could be no infinite countable sets, because
all infinite countable sets have a bijection to Z, and 6 is in Z. Furthermore
there could be no finite sets with cardinality greater than 6 because those
have a bijection to Z_n for some n in Z. That would be true in both a
mathematical sense and a physical sense, since you could still physically
represent the quantity 6 with 6 particles. So any "world" with no 6 could have
at most 3 bits of information in it.

It's really a nonsensical question because 6 is both an abstract concept -
hence the reasoning about infinite and finite sets - and a real quantity you
can have of something. The normal arithmetic operators wouldn't even be well-
defined. The only way to make this universe consistent is if you made an
exception such as 5 + 1 = 7 and 7 - 1 = 5, but in that case you have simply
replaced the _symbol_ of the number k with the symbol of the number k-1 for
all k > 6.

It's just an absurdity, and in my opinion a poor question to write about for a
non-mathematical audience, because it will make people confused. It's like
asking the question what if there were no sets, or what if there weren't
addition.

Math is truth not limited by the constraints of our physical universe; it is
an absolute. There has to be a concept of 6 for there to be mathematics - or
anything - at all.

~~~
westoncb
I think your proof using the bijection amounts to: it wouldn't work because 6
wouldn't be in Z—but that is the essence of the proposition here: "what if 6
weren't in Z?" So you end up just dismissing the question.

Sure, it's a fairly arbitrary idea that isn't really going to go anywhere, but
maybe it can at list be a _bit_ more interesting.

For example, what if we treated operations resulting in what would be the 7th
member (counting from zero) of Z (i.e. '6') the same we treat division by
zero? If we are constructing the system axiomatically, we can follow the the
pattern with Fields, looking at their definition of multiplicative inverses
([https://en.wikipedia.org/wiki/Field_(mathematics)#Classic_de...](https://en.wikipedia.org/wiki/Field_\(mathematics\)#Classic_definition)),
which effectively excludes the possibility of division by zero. There's
probably a similar way of patching axioms related to addition and
multiplication so as to preclude the possibility of arriving at '6' through
application of those operations.

~~~
opportune
Ultimately the problem with reasoning like this is that it's so nonsensical
that things get confusing. For example, I'm sure we can agree that 6 is in Z.
If we remove 6 from Z, we are talking about another set: it's no longer Z but
Z', and because 6 is in Z but not Z' Z' != Z. Sure we can define new
operations and such on this set Z' but there is still only one Z.

That's different from asking about a world or a universe without 6 though. Z
exists independently of this universe or any universe. In a physical universe
with discrete elements, you'll probably want to count those discrete things,
and for that you have to use Z. Basically the argument I was making before was
that you're no longer counting things if you don't have 6, because you count
things with N, and 6 is in N. If you count things in N' which doesn't have
six, whatever number you use to count things after 5 _is_ 6 or you're not
counting. The main thing I'm saying is that you can make define a set that
doesn't contain 6 and for which operations are defined differently, but then
it's not Z, and there has to be a Z.

Also, note that we can derive/define the integers (and _essentially_ all of
what we call mathematics) from these axioms:
[https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory](https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory).
If theres no number 6, we need to pick a different set of axioms. At that
point you're arguably not even doing mathematics because these axioms are so
ubiquitously general across all of math.

~~~
westoncb
Hmm, I think makes sense regarding counting things: it's nonsensical to use Z
for counting things greater than five if you assert that it doesn't contain 6.

But at the same time, that's just one _use_ of Z. There are other uses of it
besides counting.

> Z exists independently of this universe or any universe.

That's a pretty strong claim. Another possibility is that it's a purely human
construction, and even in _this_ universe it wouldn't exist if it weren't for
us. (To be clear though, I am specifically referring to the _set_ Z, which
doesn't exist without the human creation of Set Theory. Maybe other universes
would use something like Set Theory, but I think it would be hard to
demonstrate one way or the other.)

~~~
opportune
I talked about the use of counting to illustrate a point - not as "proof."
What I think you're not understanding is that the set of integers without 6 is
not Z. It is something else, and you can't just replace Z with this new set
and reason about it using our mathematics. That's what I was trying to say
when I said that Z exists independently of this universe or any universe: it
is a "unique" (up to bijection) set generated by axioms that we could just as
well define in a different universe or any universe, unless we are allowing
universes that don't follow our laws of logic, in which case anything goes.

~~~
westoncb
The original question was about the Z' you mentioned before, i.e. Z without 6.

> you can't just replace Z with this new set and reason about it using our
> mathematics

Sure you can. Many things change and will break of course, and I agree
counting doesn't work (which is the only reason I brought it up last time)—but
that's exactly what the question is about: which specific things change and
break. Some things would likely still continue to work though. For instance,
you could construct something like I was describing above, which we might call
Field-6, and do arithmetic and algebra with it. It wouldn't be as useful as a
normal Field over the real numbers (for example), but that doesn't mean it
wouldn't be a self-consistent thing describable within the framework of
contemporary pure mathematics. You could also prove any number of theorems
about it too—it's merely a question of why one would want to.

------
tynpeddler
Someone should have asked the kid what they meant by their question. I've
noticed that kids generate strange questions in the way you might generate a
grammatically correct, but nonsensical phrase using BNF. Asking them to
clarify their question either highlights that they actually have no clue what
they're looking for, or that their question is much much simpler than it first
appeared (for example, maybe the kid just meant what happens if the word "six"
didn't exist).

All that being said, the modulo world that the article mentions does't make
sense to me. If 6 doesn't exist, how could you have modulo 6?

~~~
CapacitorSet
>Asking them to clarify their question either highlights that they actually
have no clue what they're looking for

That wouldn't be very nice towards them, though. Many times children ask
questions not because they're interested in _their_ particular questions -
quite often they don't really have a clue of what they're asking, as you
mentioned - but they ask for the sake of learning something.

At least, that's the usual answer to "why children ask so many questions":
because they like to learn, more than because they have a specific question.

~~~
perl4ever
My own attitude when I was young was probably kind of akin to feeding random
input into a program to try to make it crash - how can I find a bug in reality
(or the grownups).

------
ars
I'm not so sure about not having 6 meaning 7 is impossible.

You could imagine a universe where if you have 7 objects it's literally
impossible to remove one. The universe will not let you. If you try you would
end up removing 2 objects no matter how you try.

It would need more. Say you have 2 piles of 3. If asked how many there are in
total, and you counted, you would get either 5 or 7. A kind of relativity,
where how many objects depends on who is checking.

This might break though - what if you labeled the objects to make them
distinguishable? What would be the distinguishing mark on the 7th object?

~~~
cbuq
You're missing something.

If 6 doesn't exists 7 can't exist. You can't just say 7 - 1 doesn't exist,
while 7 - 2 does.

If you remove 6 from the real numbers and say anything that _would_ result in
6 is impossible, doesn't 6 exist anyways?

The article isn't trying to make any claims of relativity, but suggests that 3
+ 3 = 0

~~~
kobeya
Sure it does. You can have a collection of 5 objects and a collection of 2
objects. Push them together and you have 7 objects. But for whatever reason
you just can't push any combinations together to form (a multiple of?) 6, or
break up any larger group in such a way that one of the resulting pieces would
have six elements. If you did it'd instantly break up into smaller clumps of
1-5 elements, or disappear, or whatever.

This isn't even that crazy. Sulfur with atomic weights 32, 33, 34, and 36 are
stable. S-35 is extremely unstable and decays rapidly enough to be dangerous.
You could say that 35 doesn't exist in the world of sulfur. You can start with
S-32 and throw a neutron at it and get S-33. Do it again and you get S-34. Do
it again and you get.. chloride. But hit S-34 with _two_ neutrons and you get
stable S-36.

The universe just doesn't allow 35 it seems, for some sulfur-centric
definition of "allow."

\----

I thought the article was going to be more interesting than it actually was,
to be honest. I thought the experts would talk about benzene and its six
carbon atoms, and how a vital molecule needed for life would not be possible
and the ramifications of that. Instead benzine got just an off-hand remark.

Or the kind of mathematical system you'd get if you struck out multiples of
six, and what implications that might have for physics and chemistry. Or
whether a game of life is possible on a six-state (0..5) grid and if life
could plausibly evolve, etc.

Instead we got a boring counting argument that was trivially defeated :\

------
vinchuco
A good answer: [http://blog.darkbuzz.com/2014/03/counterfactuals-
math.html](http://blog.darkbuzz.com/2014/03/counterfactuals-math.html)

"Mathematician Terry Tao explains his version of counterfactual reasoning in
three posts on The “no self-defeating object” argument. As he explains, many
elementary theorems are very confusing to students because they are based on
constructing some impossible object. There can be no impossible object, so the
proof seems like a paradox. The contradiction is avoided by precise
mathematical definitions and analysis."

~~~
nathancahill
All of Terry Tao's writing is really good and approachable. Highly
recommended.

------
timb07
What if we approach the question from a psychological perspective instead, and
assume that human brains can't process the number 6?

You can have five apples, but nobody would ever suggest giving you another
apple, because that would be absurd.

You can see snow, but if you look at a snowflake under a microscope, you
wouldn't be able to see anything.

There's a street-gang of homeless children who are no longer 5 years old and
not yet 7 years old, because their parents forgot they existed the day after
their 5 and 364/365th birthday.

Isaac R., age 5 1/2, had better start preparing.

------
BillBohan
Modular arithmetic is mentioned but base 6 is not well covered.

I started gforth and entered:

3 3 + base ! ( avoiding 6)

I was in a world where there was no 6. 5 1 + . gives 10 as a result. 6 1 -
complains that 6 is an undefined word.

Decimal returns things to normal.

~~~
CapacitorSet
Bases are about the _representation_ of the number, not the number itself.
That's why a baby can identify a group of six items despite not knowing about
base 10 or digits in general.

For a more practical example, do you think the number twelve does not exist
because it takes two digits to represent it?

------
allemagne
We can show logically that "6" as a concept should exist as a consequence of
very fundamental ideas about numbers. If we knew that there was no "6" then
could we say that math and maybe logic itself as we know it has been been
disproved by contradiction?

------
marlokk
It would reveal that despite popular belief, 786.

------
danharaj
Non-classical foundations, violently non-set-theoretic, perhaps could cope
with this while allowing numbers greater than six.

Pretend there is a non-commutative set theory where observations of membership
do not commute akin to quantum mechanical systems. Perhaps it would be
possible for a model of such a theory to have all finite sets as valid states,
but sets of cardinality exactly 6 are forbidden.

Just extremely sketchy hand waving but it might make sense. Certainly
classical mathematics is out the window.

------
d--b
If it didn't exist then we'd invent it. Pretty much like we invented
sqrt(-1)...

~~~
soperj
why haven't we invented divide by 0 then?

~~~
tekromancr
Because dividing by 0 is a logical paradox, akin to "This statement is False".
Sure x=1/0 looks like a valid syntactic construction, it doesn't comply with
the logical system it exists within. Try looking at it using elementary
algebra;

x=1/0 must be equivalent to x _0=1. And since one of the axioms on which the
system is built is that any value taken zero times must be zero, we know that
x_ 0=1 is a false statement for any value of x.

~~~
soperj
Now try looking at square root of -1 using elementary algebra...

~~~
tprice7
The two situations are not analogous, there is a crucial difference.

Like tekromancr was getting at, you get a logical contradiction if you assume
the existence of 1/0, if you also assume the ring axioms. To elaborate, if 0x
= 1, then 0 = 1. It is also an axiom that 0 != 1, and even if you abandon this
axiom you only get one extra ring that only has one element. Boring!

You also get a logical contradiction if you add a square root of -1, if you
are assuming, say, the ordered ring axioms. But you can lose the order
structure and then there is no more contradiction.

If you want to add a new number x with 0x = 1, then you also need to propose
how one might alter the notion of a ring so that you don't get a logical
contradiction, and you are still left with something interesting. Just like
how we altered the notion of an ordered ring, by removing the ordering, so
that you don't get a contradiction when adding a square root of -1.

So the crucial difference is, as far as I know, nobody has succeeded in doing
this when it comes to division by zero.

------
cam_l
It was once believed that no Australian Aboriginal languages had numbers above
5. Certainly I was taught this in school, though it seems to be not entirely
true.

Interestingly, for groups that which we don't have a record of specific words
for abstract numbers above say 5 or 6 or 7, they had words for specific items
in a sequence above these numbers. (The article gives the example of birth
order.) So, in answer to the kids question, the concept of six may survive the
lack of an abstract number.

[0] [http://blogs.slq.qld.gov.au/ilq/2014/09/09/indigenous-
number...](http://blogs.slq.qld.gov.au/ilq/2014/09/09/indigenous-number-
systems/)

[1] [https://theconversation.com/countering-the-claims-about-
aust...](https://theconversation.com/countering-the-claims-about-australias-
aboriginal-number-systems-65042)

------
merraksh
As I read this, the post has 6 points and 6 comments. Gotta fix it.

~~~
cweagans
You're doing god's work. :P

------
_nalply
I am sure there are many more or less difficult ways to imagine a world
without the number 6.

One way would be that operations giving 6 would be just undefined exactly like
division by zero. The domain of the inverse function omits the number zero.
Likewise, the x+1 function omits the number five in its codomain. And so on.

However what happens if you look at decimal representations of numbers?
Numbers like 16 aren't possible anymore, or are they? A save could be that we
give it an own symbol like ∄. So we can write 16 as 1∄ instead. Because after
all this exact point didn't disappear like zero didn't disappear just because
it is undefined for the inverse function. But this feels like a very crappy
cop-out.

Let me stop here. I am sure this gets arbitrary the more one tries to move
around in a world without the number 6.

------
nategri
From a physics perspective, I think you could actually get most of classical
physics to hang together without the number 6. I suspect you could re-
formulate or re-normalize yourself out of many potentially sticky situations.

Anything to do with quantum mechanics and particles falls utterly and
immediately apart though.

~~~
danbruc
I don't think so. If you follow the logic of the article to eliminate 6, you
essentially have to turn everything too big into integers mod 6. I guess you
could also no longer have rational or real numbers or only really limited
versions of them without access to all the integers. Losing the real numbers
kills classical physics.

It seems space could be at most a 6 x 6 x 6 lattice, maybe forming a hyper-
torus or something like that. And time, only 6 possible points in time, too.
Hopefully nobody starts walking around in that tiny universe and discovers
that there are more than 5 distinct places to visit. To really get rid of 6,
you probably have to limit the entire universe to 6 states, but even then I am
not really confident that 6 remains out of reach.

Note that there is also a tension between ordinals and cardinals, you can
number points in each dimension of the lattice universe with 0 to 5 but that
are of course 6 numbers. But I decided to ignore that, it just makes the idea
even harder to realize.

~~~
schoen
A further weird problem with this, even given the one-dimensional universe
with only six locations, is to form the power set of points in the universe.
Now it's true that there's nowhere in the universe to write down the elements
of the power set because no representation of them fits anywhere in the
universe at a single time. But one might still feel that this power set is
well defined. (You could try to describe it as "How many groups of places in
the universe are there?", although indeed no useful representation of that
question can fit into the universe either.)

It contains 64 elements. ... oops!

~~~
opportune
In a sense it's not a weird problem so much as the entire concept of
forbidding a 6 is inconsistent with the axioms of mathematics. Reasoning about
a universe without 6 will inherently include contradictions because you can
always redefine something equivalent to 6 given our axioms.

~~~
schoen
I think that's a very reasonable way to put it.

For example,

[https://en.wikipedia.org/wiki/Peano_axioms#Set-
theoretic_mod...](https://en.wikipedia.org/wiki/Peano_axioms#Set-
theoretic_models)

does not depend on having any pre-existing set of 6 objects that you can point
out by ostension. Instead, you can construct one, given the concepts of ∅, ∪,
and {}!

------
arcbyte
No talk about Peano’s Axioms? From Math professors?!

~~~
CapacitorSet
It seems to me that Turnage-Butterbaugh's argument implicitly referenced
Peano's axioms: if there's no number 6, then its successor doesn't exist, nor
does its successor's successor [an existence which would be otherwise
guaranteed by, ironically, Axiom 6]. Then it claims that "all the other
integers are out", which may be a reference to the reverse: if 6 is the
successor of 5, and 6 doesn't exist, then neither does 5, and therefore
neither do 4 or 3.

I think the professors were basing themselves on Peano, but it was omitted by
either the editor or the professors themselves for the sake of clarity.

~~~
phaedrus
Actually I would think that it divides the domain/codomain of the successor
function into two disconnected regions. I just "6 is missing" as meaning
succ(5) is undefined and that no x exists such that succ(x) = 7. You could
probably still prove a lot of conventional math in such a system, except with
extra annotations on equations like "given x != 5" much like how we have to
add the annotation "given x != 1" on an equation that uses fraction like "y /
(x-1)", to avoid invalid derivations from "y / 0".

How do you know we're not already in a universe which is missing a number,
which is why we have to add annotations constraining denominators to not be
zero?

------
tscs37
This reminds me of SCP 033 [ [http://www.scp-wiki.net/scp-033](http://www.scp-
wiki.net/scp-033) ] which explores this concept in a more fictional manner.

Simply put, the Missing digit corrupts the mathematical system it's used in.

I believe there is another entry for the result of division by zero which can
have rather apocalyptic consequences...

------
EnigmaticLion
While reading the article i thought if there's no number 6, then the universe
can't have more than 5 of anything. So perhaps 5 photons and that's it. There
cannot be anything else because then there would be 6 objects in the universe.

~~~
kobeya
Well there are six quarks. By a variant of the one-electron universe theory,
we might argue that all quarks are shared instances of the same prototypical
quark of that class. If we begin counting with 0, that's "5" quarks. So the
answer is maybe: this universe is what you get!

(Please don't take this seriously.)

------
et2o
Love that this question was asked by Isaac, who is 5 1/2 years old.

------
jxramos
there'd be no single digit (base 10) perfect number of course ;)

~~~
gicadin
This is a very interesting way to get around the problem. We could just work
in base5, ( or lower) and mathematics would still make sense!

~~~
Xophmeister
6 still "exists" in base 5, it's just that it equals...ahem...is congruent to
10.

~~~
vinchuco
So take it out, you end up with a set of previously-numbers objects you can't
do much with. The question is answered.

------
koliber
I think if there were no number six, the universe would continue to exists as
it does right now. We wouldn't notice anything different.

We live in a universe that is absent the integer called pleebix. We're not
sure where it would fit on the number line, but one of the candidate locations
is between seven and eight. We're OK. Our universe adjusted perfectly fine to
the fact that this integer is missing. We don't even notice the hole because
everything is ideally consistent around the fact that pleebix does not exist.

If six was missing, the universe would be similarly consistent. No one would
notice anything out of whack.

------
rdiddly
If there were no 6, Little Man would stay 5 1/2 forever and keep asking
childish questions!

------
Sangermaine
This is just Philosophy of Math 101. Not even that, more like Baby's First
Philosophy of Math.

~~~
opportune
Could serve as a useful introduction to the topic of consistency and
definitions, or what happens if you have inconsistent axioms.

------
tprice7
"Renate Scheidler, a math professor at the University of Calgary, said there’d
be no six-string guitars, so music would sound different"

Good thing they asked exclusively math professors for a response, so you know
you are getting an authoritative answer from someone with many years of
training and deep knowledge about the topic. LOL

~~~
beder
This reminds me of a time there was some unusual lottery result in my area,
like 123456, and a TV station asked my father, a statistics professor, to
comment that the odds of that happening were 1 in a million.

~~~
Nition
Should've made a bet with them that there'd be another one-in-a-million result
the following week.

------
CurtMonash
Then perhaps there wouldn't be a different Number Two every week.

/ThePrisoner

------
MaysonL
Won't that simply be arithmetic modulo 6? 5 + 1 = 0 = 3 * 2

------
joesb
It's more like "If logic doesn't exist".

------
dahart
If there was a 5 but no 6, wouldn't 6 be called 10?

------
Cryptid
That would be telling.

~~~
Animats
That's what I was thinking.

------
Vanit
Enjoy wrestling in the mud.

------
Upvoter33
that's weird, because we all know 7 ate 9 and everything is still ok.

