11 ms·
What Would Happen If There Were No Number 6?
- jxramos 9y agothere'd be no single digit (base 10) perfect number of course ;)
- gicadin 9y agoThis is a very interesting way to get around the problem. We could just work in base5, ( or lower) and mathematics would still make sense!
- pbhjpbhj 9y agoOne 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.
- gfredtech 9y agoSomeone 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
- Xophmeister 9y ago6 still "exists" in base 5, it's just that it equals...ahem...is congruent to 10.
- nategri 9y agoFrom 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 9y agoI 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.
- nategri 9y agoAh 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)?)
- danbruc 9y agoNope. No 9 for you. Also what do you mean with repeating, you aren't suggesting strings with more than 5 symbols, are you?
- schoen 9y agoA 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!
- Cryptid 9y agoThat would be telling.
- Animats 9y agoThat's what I was thinking.
- merraksh 9y agoAs I read this, the post has 6 points and 6 comments. Gotta fix it.
- cweagans 9y agoYou're doing god's work. :P
- ars 9y agoI'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 9y agoYou'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 9y agoSure 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 :\
- allemagne 9y agoWe 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?
- OJFord 9y ago> 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?
- dogecoinbase 9y agoThis seems like a weak argument: say you have a pile of five items; you could add one, and then we have six after all. You presume the existence of a sixth item, so of course you can construct it.
- pavement 9y agoThere 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?
- 9y ago
- BillBohan 9y agoModular 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.
- soperj 9y agoyeah 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)
- CapacitorSet 9y agoBases 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?
- tynpeddler 9y agoSomeone 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?
- westoncb 9y agoI 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.
- MrQuincle 9y agoIt indeed makes sense to talk about modulo 10 in base 10 and in modulo 10 in base 6. Or would base 6 also not exist?
- khedoros1 9y agoTake 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.
- rflrob 9y ago
- d--b 9y agoIf it didn't exist then we'd invent it. Pretty much like we invented sqrt(-1)...
- westoncb 9y agoWe already invented it in that same sense—it was just a more obviously useful thing to invent ;)
- soperj 9y agowhy haven't we invented divide by 0 then?
- d--b 9y agoWe probably would have if it proved to be useful to anything.
- tim333 9y agoJavascript has it. 1/0==Infinity
- tekromancr 9y agoBecause 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.
- soperj 9y agoNow try looking at square root of -1 using elementary algebra...
- tprice7 9y agoThe 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.
- marlokk 9y agoIt would reveal that despite popular belief, 786.
- arcbyte 9y agoNo talk about Peano’s Axioms? From Math professors?!
- CapacitorSet 9y agoIt 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 9y agoActually 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?
- et2o 9y agoLove that this question was asked by Isaac, who is 5 1/2 years old.
- Sangermaine 9y agoThis is just Philosophy of Math 101. Not even that, more like Baby's First Philosophy of Math.
- opportune 9y agoCould serve as a useful introduction to the topic of consistency and definitions, or what happens if you have inconsistent axioms.
- timb07 9y agoWhat 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.
- Upvoter33 9y agothat's weird, because we all know 7 ate 9 and everything is still ok.
- tprice7 9y ago"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 9y agoThis 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 9y agoShould've made a bet with them that there'd be another one-in-a-million result the following week.
- danharaj 9y agoNon-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.
- vinchuco 9y agoA 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 9y agoAll of Terry Tao's writing is really good and approachable. Highly recommended.
- joesb 9y agoIt's more like "If logic doesn't exist".
- dahart 9y agoIf there was a 5 but no 6, wouldn't 6 be called 10?
- EnigmaticLion 9y agoWhile 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 9y agoWell 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.)
- deleted 9y ago[deleted]
- CurtMonash 9y agoThen perhaps there wouldn't be a different Number Two every week. /ThePrisoner
- rdiddly 9y agoIf there were no 6, Little Man would stay 5 1/2 forever and keep asking childish questions!
- MaysonL 9y agoWon't that simply be arithmetic modulo 6? 5 + 1 = 0 = 3 * 2
- tscs37 9y agoThis 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...
- cam_l 9y agoIt 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-systems/ http://blogs.slq.qld.gov.au/ilq/2014/09/09/indigenous-number... [1] https://theconversation.com/countering-the-claims-about-australias-aboriginal-number-systems-65042 https://theconversation.com/countering-the-claims-about-aust...
- koliber 9y agoI 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.
- _nalply 9y agoI 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.
- Vanit 9y agoEnjoy wrestling in the mud.