> 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?
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.
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...), 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.
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. 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.
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.)
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.
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.
There are multiple interpretations to the question, clearly.
Removing a tick mark from an infinite number line would just cause all the other tick marks to slide closer together like billiard balls, no?
Or is the question really about what happens if we render six as a forbidden, unaddressable element? Maybe equations normally resulting in an integral six, or involving an integral six return an error stating: undefined? Not unlike a divide by zero error?
Consider rationalizing division by zero. One could interpret the outcome of such an operation as always resulting in zero, instead of an error, but what would that mean when modeling real world events? If you have zero, and you want to divide it into N number of parts, well, you’d still have zero things, and so what? But what if you had somethings, and you wanted to divide them all into zero parts? Does that mean the operation represents the total annihilation of those things?
We can imagine a situation where total annihiliation by division is possible, but is it worth building that interpretation into most models, when an operation like that isn’t often applied to circumstances, and such an outcome is rarely requested?
My understanding is that the math professor interpreted the question to mean "what if our universe, rather than acting in so-far-seeming correspondence with ZFC set theory, was one that was best explained by a different axiomatic set theory—one that didn't entail the existence of "6" as a mathematical object? What universe would we be in, and what would it be like?"
> 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?
Why? There's nothing interesting to say. If you have no 6, either you renamed it ( http://www.smbc-comics.com/index.php?id=3913 ) or you've stopped working in a useful structure.
In particular, ℤ without 6 is not an algebraic field, because fields are closed under addition and multiplication.
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?
> Someone should have asked the kid what they meant by their question.
I honestly think that this answer values the children's point of view more than the article's urge to not ignore them by launching into theses on the deep impossibility of what a missing integer might mean to the universe.
All the adults in the article interpreted the question to mean 'what if six things couldn't exist?' and proceeded to formulate complex mathematical implications.
None of the adults in the story asked, "what do you mean"? And none of the adults said "maybe 3 plus 3 would be 7", or maybe "7 could be pronounced 'six'", or what if the kid was asking about one-handed people after hearing about ancient kings who cut hands off of rule breakers, and was really just wondering in kid style whether it'd be better if we have a base 5 number system instead of base 10.
>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.
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).
I think teaching children to interrogate their own reasoning is the most useful thing you can teach them. I've met a fair number of adults who ask the same kind of nonsensical questions as children do because they don't know how to analyze their own thinking.
I think the idea is that 6 modulo 6 would just be 0, so in that system the only integers used are 0 through 5. Six is used as a concept in structuring the system, but it would not be present in the usage of the system.
Take it from the other direction. Assume a world with an integer "sqorkle", between the integers 3 and 4. It's >3, <4, and not equal to any other integer.
In our world that doesn't have sqorkle, what is 3 modulo sqorkle? Since is less than the modulus, it should just be 3, right? 4 modulo sqorkle: 1, I guess, being 1 above sqorkle? OK, so what about 4 mod 3? It's defined as 1 in our world, but in a world where 3+2=4, that can't still be true, right?
Mathematical constructs are based on patterns that we see in the world. Change the nature of the patterns, and you invalidate the mathematical constructs that we've built up. I'd think that since our constructs would be invalid in sqorkle-land, trying to apply them there will just end up with contradiction. Ditto, trying to apply our math to 6-less-land.
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?
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."
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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 :\
"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."
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.
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?
yeah that was my first thought too. Why not go to base 6. Don't know why you'd have to fire up a program to do base 6 arithmetic though? it's just like binary or decimal or octal or hexadecimal(1+1=10, 9+1=10, 7+1=10, F+1=10)
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?
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.
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 x0=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 x0=1 is a false statement for any value of x.
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.
Because we don't need it: we have limits! The constant i and its associated field, ℂ, is needed so we can solve the half of the quadratic equations which have no real roots. Dividing by zero has no use case (that I know of) that isn't covered by taking the limit as x->0 of 1/x.
We have... in a sense. There are number systems wherein 0 is treated rather like dx in terms of being a formal entity upon which algebraic manipulations can be performed; in such systems we can say for example that sin(0)/0 = 1 while sin(0^2)/0 = 0.
Do you have a reference for this? I would believe you if you said "an infinitesimally small value" rather than 0. If it's true then I'm very surprised. For one, 0 = -0, so if sin(0)/0 = 1, then -1 = -sin(0)/0 = sin(0)/-0 = sin(0)/0 = 1
The non-handwavy answer is that mathematics is defined by a set of axioms through which all mathematics follow via logic and defining new structures / relations. If there were no 6, given the same set of axioms we could simply define something representing the same concept of 6 with a different symbol, because the concept of 6 is consistent with our axioms. Dividing by 0 is inconsistent with our axioms, so even if we define x = 1/0, we would be able to arrive at conclusions which contradict our axioms.
We can if we want. The problem is that there is no useful way to define division by zero, and have it work well with all the other rules of arithmetic that we want to continue working.
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.
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.
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.
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.
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.)
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.
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 {}!
Ah yeah I was working on the assumption that we could keep the real numbers (article makes no mention of them), since for classical problems they're all you really need (maybe we can keep 5.9(repeating)?)
Why do you assume that without the number 6, all dimensions would be a lattice? There's nothing that forbids them from being dense (at least until Planck scale).
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.
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?
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.
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!
One of the very first things in maths I taught my kids -- after they could count to 20 -- was binary ("robot numbers!") and from there alternate number bases [stuck well with the first, not so well with second, not at that point with any others]. Think that came from the same place as "what if there's no 6".
I took the approach of how aliens would count if they didn't have "enough" fingers.
Someone told me that once; perhaps the reason we naturally count in base 10 is because we have 10 fingers. If there was no six, I also think that we'd be counting in base 5
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.
"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
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.
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?