Introducing (finally!) Box Arithmetic
Introducing (finally!) Box Arithmetic | Math Foundations 237 | N J Wildberger
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5,100 views Sep 3, 2023 Box Arithmetic
Box Arithmetic is a new way of understanding what arithmetic, and much of associated mathematics, is actually truly about. The most important new understandings are that 1) the most powerful data structure for foundational work is not a set, and is not a list, but is rather a multiset, or mset: where elements are unordered and repetitions are allowed and 2) that the particle / anti particle duality famously discovered by Paul Dirac in 20th century physics has a deep and remarkable analog in the foundations of arithmetic. When we put these two together, we get Box Arithmetic.
Here we give an overview of how we propose to reconsider the developments arising from mset arithmetic in the light of this powerful box / anti box duality.
I would especially like to thank my Patreon supporters for their generous support over the years! Your help and encouragement has propelled me forwards in more ways than you think.
Introduction
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foreign
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[Music] I'm Norman Wildberger I've been introducing something of a new point of view
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towards arithmetic fundamental arithmetic using the mset approach where we consider
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data collections which are unordered and where repetitions are allowed as the central sort of
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building blocks of our arithmetic and our algebra and we've had some pretty good success and we've
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gotten a fair ways down but we can sort of see there's a lot further to go there's a lot of
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details and you know and careful structuring that's required but at this stage here we've
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introduced some really novel elements having to do with anti-particles or the mathematical
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equivalents of them and so it's time I think to to sit back and to look at what we've already
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done and to try to reframe it so when we're developing mathematics it's important to not
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always just be at the forefront, but also to occasionally go back to what we've done
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earlier you know and try to understand well what's really what have we done here and have
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we done it in an optimal way considering how the thing is unfolding as we're proceeding
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so oftentimes the initial definitions or notation or whatever that we create aren't really optimal
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but that only becomes apparent sort of further on down the road and we have to be prepared to
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have this sort of reevaluation possibility that's what we're going to do today so it's not a very
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long lecture there's only two slides so first I'm going to just review what sort of more or less
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we've been doing and then we're going to introduce box arithmetic Okay so we've introduced arithmetic
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via msets and starting with zero the the empty mset is just a box say with nothing in it now
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typically I've been writing things in a linear fashion but that's more or less sort of just
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in a chord with our usual way of writing things for today's talk it'll be a little bit better if
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we move to this more obviously two-dimensional approach where the Box aspect of things becomes
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more apparent so once we have zero then we Define a natural number to be simply an mset of zeros
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a collection of zeros so in other words empty boxes which are unordered and which of course
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repetitions are allowed so this for example is how we think about the usual number two
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then we go down one step we define Polynumbers. A polynumber is an mset of natural numbers so
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here is an mset the big box and inside it there's there's a natural number and
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this is also a natural number because it's a sort of a special case of of this one here
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so we end up with this family of nested objects in mathematics usually called a sort of filtered
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object as opposed to a graded object so at every stage the new things that we're invoking include
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already the previously defined things so at the next stage we define multi numbers which
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are msets of poly numbers so here's an example of a multi number this thing here you can see is the
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poly number that we've just defined and here are a few others so in the usual way that I'm sort of
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wanting to link things up with ordinary polynomial algebra this was corresponding to 1 plus Alpha
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squared this is corresponding to one here and this one here to Alpha squared and this thing here well
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a little bit of thought and familiarity with what we've done we'll show you that this is equal to 2
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plus Alpha 0 plus Alpha zero Alpha two now in going from here to here what I've done is I've
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sort of introduced again some new notation which isn't really necessary at this stage but when you
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go to this stage it does become necessary so we're going to rewrite this as 1 plus Alpha 0 squared
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so just in the world of poly numbers this is sort of extraneous and unnecessary but when we move to
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multi numbers where we get more variables with different indices than the ordinary Alpha gets
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replaced with Alpha zero okay then we can keep going this is sort of an obvious inductive or
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recursive approach to this thing and in terms of operations everything comes down to the unique
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operation of addition of msets which is a unique thing that in the world of data structures the
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msets have that the other data objects don't have the ability just to dump everything out into a box
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thereby adding not just two but arbitrarily number of msets to get another mset so for example here's
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an mset here's an mset, when we add them we just dump the contents into another box and we get that
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thing there then multiplication is defined in terms of addition so a times B if a and b are
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msets in this world defined as the mset of things in the form a plus b we look at all possible sums
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a from A, b from B and they form an mset and then that's sort of the pattern that carries on the
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next operation say caret is defined in terms of this way if C and D are msets then C caret D
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is the mset consisting of all products c times d where little c belongs to Capital C and Little d
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belongs to capital D and obviously we can sort of keep going so the whole thing is set up so
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that recursion or sort of inductive approach is really natural just it unfolds in front of us
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but admittedly our intuition for what happens sort of at these lower levels is at this stage
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pretty minimal but if we just sort of stick to multiplication or maybe a caret also then we
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uh and down say down to this level here then we recover quite a lot of ordinary arithmetic or
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algebra but crucially up to this stage only with natural number coefficients if we want to have
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negative number of coefficients well then that's what we did next we introduced negatives and The
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crucial thing there was that it was almost forced upon us to start thinking in terms of particles
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and anti-particles the way the physicists do so there's an electron maybe and here's a positron
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anti anti-particles okay so this notion from physics ends up being intimately involved
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when we start constructing negative numbers and investigating their role and what I want
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to do today is I want to bring this particle anti-particle or mset anti-m-set Duality to
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the four I don't want us to just introduce this at some later stage I want to now go back and
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incorporate this right from the beginning okay so I think this is a key difference between this
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system and all prior systems of arithmetic and I think we need to sort of centrally position things
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so that we can see this Duality and you know and try to understand this Duality very prominently
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so that's what we're going to do now we're going to introduce box arithmetic okay so how are we
Introducing Box Arithmetic
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going to do this how are we going to introduce box arithmetic well what we want to do is we want to
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go back and we want to introduce anti-amsets and negatives as an essential part of the whole story
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from the start and I propose that we're going to do that by thinking about boxes and anti-boxes
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so we're going to move away a little bit from this idea that everything is written on a line with
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lots of brackets nested brackets all kinds of them and think more about representing the arithmetical
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structures in a two-dimensional fashion and we'll have a box approach but there will also
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be a tree approach that's completely equivalent so it's sort of a matter of taste or a matter of
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visual orientation or maybe you know if you're a computer scientist you're naturally going to
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be led to want to think about things in terms of trees and that's fine okay so boxes and anti-boxes
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we're going to proceed in a parallel way to what I've just described but now starting with this
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Duality first and foremost so to begin with we're going to want to consider just empty boxes and The
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crucial thing is that our empty boxes are now of two kinds there's black ones and there's red ones
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the black ones you should think of as being like the ordinary ones that we've already
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been talking about and the red ones are some new ones which are considered to be the anti-objects
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of the black ones and then what we do is we put these boxes or anti-boxes inside other boxes at
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anti-boxes exactly the way we've been doing previously but we're allowed to at any stage
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use either red boxes or black boxes so we're mixing these two things and the fundamental
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Duality between them as represented by these two examples so have a look here we have
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a black box and inside it we have this black box with a red box inside it and
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two red boxes separately and over here we have a red box with exactly the same stuff
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inside it okay so these are the same these are the same so these two boxes
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anti-boxes are exactly the same except that the bounding the outside boxes are different colors
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so we're gonna say that they are they are dual in this sense or they are antis
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of each other so this thing is X then this is going to be x super a or anti X
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now another way of representing that with a tree approach is that we're now going to
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use trees whose vertices are either black or red corresponding here right so this outside
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box here is represented by that black dot there and then inside it there is a red box
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represented by that red node and another red box represented by that red node there's also a black
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box represented by that black node and inside that one there's a red box represented by that red node
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and over here well it's the same story they're really exactly the same except for the top the
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top node is now here red instead of black and I've also allowed myself this Freedom that we are we're
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giving ourselves at the the trees are unordered so you know we can move these various ingredients
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around because of the unordered aspect of the msets that we're considering so we can also write
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this thing here for example if we wanted to this is really the same as we could make it like this
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okay that makes the quality or the The Duality between this one and this one more obvious but
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it's not necessary so you know but this thing here and this thing here are just different because of
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the top the entry the top node being a different color okay so now the crucial property between
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um with these black and red things and then their anti aspect is that they disappear they mutually
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annihilate each other if they're found in the same m box so the so the crucial fact here is that if
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we have a box or an anti-box with uh with the X in it okay x and x a in it all together then they
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cancel and we just get an empty box and that's the same if we have an anti-box with an X and
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an X anti in it we also just end up getting cancellation and we get we get a another red
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anti-box okay so that's the crucial aspect that we have this this Duality but these things cancel out
Interpreting Box Arithmetic
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and let's have a look at this thing here so how do we interpret this well the two red boxes is
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like a minus two because if we rub this thing off and just consider a black box with two red boxes
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inside it then that if we added it to a black box with two black boxes inside it then we would get
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a total with two black boxes and two red boxes inside and they would all cancel we would just
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get the empty empty box which is like zero so this is like minus two just with those two red
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boxes inside and what if we just had this black box with a red box inside it well that would be
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Alpha our basic you know poly number variable Alpha to the minus one because if we multiply
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that by a similar thing with with a black thing inside there in other words a black box with
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another black box with another Black Box inside it that's Alpha we multiply that with Alpha then the
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the the inner boxes have to be added okay the inner boxes have to be added and that
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means that when we we add them the red and the black ones are going to cancel
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we're just going to get an empty box so the result of multiplying those two things would
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be a black box with a single black box inside it which is the number one okay so that's why this
What is an unknown object
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thing here just by itself would be Alpha to the minus one so this whole thing here
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uh in in our world that would be minus two plus Alpha to the minus one
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now what about this thing here what is its meaning well as
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as an object in the mathematical Universe currently this is an unknown object we have not
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seen this kind of thing before it's like you know if I get the positron okay 1930s or maybe 1940 the
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positron was observed but before that you know the there weren't any positrons as far as people could
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tell so this thing here I mean what we could say is that what it is is the entity of this so we can
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say it's minus 2 plus Alpha inverse okay anti but what the meaning of that is is not entirely clear
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so what ends up happening is that when you put these red boxes inside a black box so
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in other words if you consider a box with possibly Red's boxes inside it or anti-boxes
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inside it perhaps under you know deep down in different levels then that kind of thing can
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still end up being represented in principle by a somewhat of an elementary or familiar object
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at least you know down to the level of multi numbers so beyond that we haven't much experience
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so the the red boxes or the the anti-boxes are useful in the sense that they are important
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constituents in the world of box arithmetic but just by themselves they have a kind of
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a strange ephemeral nature maybe like like quarks you know that they make sense when
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they're constrained in in in a nucleus or something but um just isolated quarks we
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don't really see them in nature they don't seem to play a role so while there is this
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symmetry box anti-box symmetry there's also a separation in our minds so we we also want to
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perhaps ultimately be more interested in the arithmetic on this side and think of the
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the anti-boxes as sort of objects which contribute to
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the richness of the arithmetic over here but by themselves maybe have an ephemeral nature
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so I realize this is all rather sort of you know strange maybe it's like Voodoo but I
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hope you've been convinced by my previous videos that we're on to something here that
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we really are touching base with ordinary mathematical reality but we're just looking
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at it in a very different way and we're sort of enshrining that very different way here we're not
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putting that under the carpet we want to really put it Center Stage and look at the consequences
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okay so that's what we're going to do we're going to be looking at thinking about box arithmetic
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being developed in a sort of consistent way so that this Duality is sort of
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almost always omnipresent and that this sort of what we might call a
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box this thing here this phenomenon here we might call this box anti-box creation
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slash annihilation
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so for example what that means is that when we we look at something like this thing here okay
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we could we could change this seemingly change it but actually not do anything real to it so
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if we put in in this say one of these red boxes if we put inside there a red one and a black one
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then the effect is to not change it we have not changed it
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so that means and that's that's true even over on this side you know whenever we put in uh two
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objects boxes or anti-boxes inside a single constituent but they're identical except that
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their top nodes are different colors then those two things will annihilate and so they are able
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to be presented this way so the kind of shimmering reality that the quantum physicists like to talk
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about where we have these fields and and these fields have all these emergent pairs of little
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particles you know that are being quickly created and then annihilated and they appear and then they
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disappear you know just for a split second you know and so this is some kind of really strange
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thing that that's how they think what we're seeing is something parallel to that here in this world
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that we're able to create these box anti-box Pairs and they and then they're annihilated a second
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later so if this is really strange to think about it the following way which which makes it maybe
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um more familiar so when you're doing ordinary arithmetic with fractions say three over five
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okay you you all know that you can replace that by multiplying top and bottom by whatever we want
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say like by two so three over five is the same as six over ten so what have we done there well we've
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introduced a new Factor on the top and the bottom the two factors happen to be the same but they're
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in different places so they're playing different roles sort of like this red dot and this black dot
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they're sort of the same except that they're they're different because one's red and one's
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black and correspondingly you know one of the twos is in the numerator one of the twos is in
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the denominator and so you can take three fifths and if you like you can quickly change it to six
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over ten or you can multiply it by three and get 9 over 15. right you're doing it something very
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similar you're you're you're creating something more complicated by adding some more factors but
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in these two different ways which are sort of balance so they're sort of kind of a duality
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so it's actually not that unfamiliar from some point of view
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okay so now I think uh with this sort of General orientation I think we can now go back to the
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development that we've already had in terms of zeros and natural numbers and poly numbers
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and multi numbers and sort of redo all those things but just starting with the Box anti-Box
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point of view so we just carry on down and it's I think an interesting exercise and we're going
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to have to look very carefully at that because we've seen that these particle anti-particle
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things have some Curious maybe even disturbing properties when we are doing basic arithmetic
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just with say the zero box and the anti-zero box so it might be that when we restrict ourselves to
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just boxed arithmetic that those curious aspects are are hidden from sight in fact that's sort of
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what I'm thinking will happen that those sort of anomalous kind of things that we're seeing are
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are not really visible parts of the the mathematical or arithmetical world that
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we're going to create we're ultimately going to be interested in box arithmetic but we want
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to have this symmetry because these guys are going to be crucial ingredients in the boxes
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so that's a lot of fun maybe some of you would like to you know try that out yourself and see
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how far you get in terms of building up a consistent Theory ultimately we want
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very careful definitions and and theorems and proofs Etc but you know exactly how we're going
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to do that that maybe still some ways away I'm Norman Wildberger thanks for listening [Music]