Negatives numbers from anti msets

3,060 views Jan 30, 2023 Math Foundations

How do we set up negative numbers in this new arithmetic based purely on multisets or msets? The answer is surprising: we must adapt notions from 20th century particle physics: the idea of an antiparticle that shares many properties of a particle, but that mutually annihilates when in contact with the original particle. And the physical concept of "particle/anti particle creation from the vacuum" plays a useful role in this new arithmetic! This is an exploratory video where we look to discover by elementary means some of the implications in adopting this anti symmetry to the world of arithmetic formed from msets. It is certainly not logically tight: we still have to convince ourselves that this is a consistent world of arithmetic -- and of course simply "assuming" that things work out (like modern analysts do with their "arithmetic of real numbers" ) is not on the drawing board! Quite a lot of potential surprises here in this video. Hope you enjoy it! Very pertinent to this discussion is the earlier video "The curious role of "nothing" in mathematics | MF 187 | N J Wildberger is at • The curious role ... and the subsequent ones.

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foreign [Music] I'm Norman Wildberger thanks for joining me we are in the process of largely

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outlining a new approach to arithmetic and today very important and interesting lecture

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where we try to tackle the problem of how to create negative numbers into our system

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so it's quite a long lecture today there's quite a lot of stuff in it I'm going to also

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say a few philosophical or meta-mathematical things beforehand to sort of set the stage

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and also there's a tentative aspect or an aspect of this being a proposal okay it's not that we

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have a a current valid logically correct Theory established yet so we're working towards that

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so there will be an element of exploration today as we discover things perhaps for the first time

The antimultiset

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so we're going to be introducing negative numbers from a notion called an anti-multi-set and the

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basic idea is contained already in a prior lecture a couple of lectures in the math Foundation series

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notably Math Foundations 187 the Curious role of nothing in mathematics and the subsequent videos

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following that so that what I'm doing today is largely an extension or modification of what I

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did there okay where I already introduced the notion of an anti-objects and the important

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role of nothing in this multi-set framework but we're modifying things by not having the mark as

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the central fundamental object but as the empty mset or a zero as the fundamental thing okay but

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otherwise you'll see there's a lot of parallel to this and so if you're interested in this situation

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which I hope you are please go and and have another view of this video or have a look at

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it for the first time if you haven't already done so okay so currently we have a framework which kind of goes on but it starts with very simple level just the zero multi-set which is an empty

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multi-set that we have natural numbers which are just m sets of zeros one two and three and then

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we have poly numbers which are msets of natural numbers that one there called Alpha zero there's

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another one here's an mset of natural numbers which we can interpret it as the polynomial 2

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plus Alpha zero plus Alpha zero cubed and then multi-sets of poly numbers like this thing here

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this is an mset of poly numbers because that's a poly number that is that is that is and it has the

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interpretation of a polynomial in more than one variable so Alpha zero squared corresponding to

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that plus two alpha one alpha 2 corresponding to these two elements plus Alpha four corresponding

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to this one and we can go further we have a quite a an obvious sort of inductive or

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sequential process in in extending not just objects but also operations to further levels

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down and we also have a nice way of visualizing this kind of thing in an alternative way using

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trees in the language of computer science but notably the objects that are appearing here

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there's there's numbers appearing but they're all natural numbers so we today want to move to try to

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extend our number system to include negative numbers that's a very interesting challenge

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okay so broadly our aim is going to be to introduce a new number minus one

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which I'm also going to write as one bar so actually for a lot of purposes I prefer this

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bar notation for negative numbers and I'm going to largely be following that and I urge you to also

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consider using that okay so it's very useful because then this minus sign is not confused

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with the subtraction operation okay and it also minimizes the the necessary use of brackets in a

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lot of situations so in either case I'm going to use one bar typically what we want to establish

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about this number is that ordinary one plus one bar equals zero and also that one bar times one

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bar equals one so those are two crucial properties of this new number that we're trying to introduce

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and then after we have one bar then we're going to introduce more General negative numbers so that

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we get a kind of a integer number line with the natural numbers extending to the right like this

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and negative numbers one bar two bar three bar etc and we want the usual loss of arithmetic that are

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familiar to us in the natural number domain to extend to this larger domain of integers

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then what we want to do or as part of this process is we want to extend these numbers to

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poly and multi and Beyond so that poly numbers and multi numbers and further things may also

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use these negative numbers as constituents to build up these more complicated structures and

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so we end up with the an extension of the usual laws of algebra to incorporate negative numbers

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and hopefully we'll get some new insights in this way so let me say a few words about

Native numbers

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Native numbers it's important to realize before we start okay that negative numbers have been a

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source of insecurity of questioning of um of skepticism for a long time

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most ancient civilizations even if they were reasonably proficient with arithmetic and algebra

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for example even going back to the Babylonians but also the Greeks and the Arabs and Chinese

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and so on most of these civilizations had had ways of avoiding the explicit usage of negative numbers

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so for example if we want to solve the equation x squared plus x minus 1 equals zero that quadratic

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equation they would have preferred to rewrite it as x squared plus x equals one so there's only

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natural numbers appearing this unsureness about what negative numbers are and what their role is

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extends even into the modern era Descartes Rene Descartes famously the father of modern geometry

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and a key figure in in 17th century mathematics was dubious about the role of negative numbers and

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explicitly tried to avoid them in his geometry so the usual Cartesian coordinate axes that we

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teach undergraduates as being you know a result of Descartes thinking actually with its usage of a

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number line with both positive and negative values is a little bit foreign to Descartes thinking he

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preferred to think in terms of having two axes labeled by positive numbers positive numbers okay

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so he restricted himself to what we might call the sort of upper quadrant so we have to have

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some respect for the difficulty in establishing a sound theory of of integers and you might like to

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think personally okay just yourself how you go about establishing a proper theory of integers

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I know you've all learned about integers somewhere in primary school and you've sort of been taught these rules you know and perhaps you can remember back in grade five or whatever when

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you were told that minus two times minus three is plus six and you maybe you're asking well

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you know if -2 represents a debt like instead of having two apples I have minus two apples means

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that I owe somebody else two apples um what does it mean to multiply minus two with minus three and

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get plus six and how do we really establish the laws of arithmetic in this integer domain have

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you actually gone through that yourself even if you're a professional mathematician chances are you probably haven't actually okay so this is something that we sort of assume a sort

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of background yeah we all have the uh faith in this but do we actually have a coherent Theory so

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that's that's a starting point here okay we have to recognize that there's something potentially

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problematic or difficult or challenging about negative numbers and today I'm going to be

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challenging you further by introducing them to you in a very novel way okay that surprisingly

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really ultimately relies on 20th century physics for its for its intellectual support so let's

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begin our exploration and try to deduce what likely we have to have if we want to have a

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consistent theory of negative numbers extending this multi-set arithmetic that we already have

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okay so the primary thing that we're aiming for is first of all a minus one or a one bar so let

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me remind you that in our system the number one is the mset consisting of zero and zero

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of course is the empty mset so this new thing that where interested in this minus one or one

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bar it's going to also be an mset because all of the things in our world so far are msets

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so we composite that this is a new amp set with something in it but we don't know what what that

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thing is that's inside here okay so let's put it represented with a question mark okay so what do

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we need to have happen we want that one plus one bar is zero so one that's mset zero one bar is

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this m set with a question mark in it okay and we have to add these two things how do you add msats

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well you dump them both out into another M set so we're creating a new amp set and the zero goes in

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that M set and this question mark whatever it is also goes in there and what we want to have

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happen here is that the result of that is zero which is the empty M set so this question mark

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has to have the property that if you put it in a box with zero the result is an empty box

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that's necessarily what's required in our framework there's no getting around it so this is a very unfamiliar situation to you know mathematics

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in the last uh 10 000 years there's nothing in mathematics that's really like this that gives

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us any kind of Prior intuition about such a thing we don't have situations where we have one thing

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being completely canceled with something else to create nothing not zero but nothing okay

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this zero and this question mark are creating nothing okay but 20th century physics completely

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remarkably has discovered such a phenomena and crucially it's one of the most important

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developments in fact of 20th century physics that was the discovery essentially by Paul Dirac

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of anti-particles starting with the anti-particle for the electron which is called the positron

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which is a particle which is very similar to the electron in most aspects has the same mass same

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spin that kind of thing but crucially it has opposite charge and moreover has this property

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that if electron and a positron get close together then boom they mutually annihilate each other

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in a flash of light with some some photons being emitted so they have rest energy that energy has

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to go somewhere and the energy is carried off by by two photons going in opposite direction so that

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linear momentum is conserved so this is a totally remarkable thing and especially when it became

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clear that this is not just a phenomena involving electrons and positrons the positron was found a

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few years after Dirac postulated its existence but moreover and sort of spectacularly it's now

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the case that that it appears that almost every in fact every particle that is known all these

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Elementary particles have corresponding antiparticles which essentially have the

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same kind of relation that they more or less share most of the properties with the original particles except that when you bring two of them together the particle and the

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anti-particle boom they annihilate each other in in with the releasing some kind of energy

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now this is a remarkable thing I'm going to have to say more about that but that's really our motivating idea that this is exactly what we're needing here minus the

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energy uh transmission okay zero combined with something rather gives us nothing

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so what we're going to do is we're going to introduce such a new object into our mathematics

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okay we have to create an anti-particle for our fundamental object which is zero the empty

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mset and I'm going to give it the name zero with an a above it a of course standing for

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anti-particle we're going to call it anti-zero so the anti-0 is this expression it's a new

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symbol okay with a property that if you have an M set containing zero and zero a

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then they mutually cancel each other out and we just get an empty M set

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so the question well what actually is this thing we're not going to answer that question

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any more to say well it is only a symbol we're dealing with a language of symbols okay and

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just like this is a sort of a symbol this is a new symbol that we're introducing in order that we're able to ultimately come up with a minus one

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and it's pretty well necessary if we're going to do arithmetic in this multi-set kind of framework

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so curiously if this arithmetic had been discovered centuries earlier then probably the

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physicists would have been much more inclined to have discovered the anti-particle kind of

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symmetry that exists in nature which is such a profound and remarkable symmetry perhaps earlier

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along the lines of the physics I I do have to remark one more additional thing however that this

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this symmetry that we have is is a broken symmetry in that the actual world that we live in is

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predominantly made out of matter particles and not antimatter so antimatter is something that we only

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can cook up in under rather extreme circumstances or perhaps in cosmic rays or something but it's

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not typical okay at least as far as we can see everywhere we look we see matter not a lot of

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antimatter and uh so this imbalance between the matter and the anti-matter is a as a huge problem

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for for physicists and for cosmologists it's one of the central problems in 21st century physics is

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to explain this apparent asymmetry between these two kinds of things which otherwise seem to be

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quite um parallel so now I want to explore with you some consequences of this fundamental idea of

Consequences

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having this somewhat symmetrical object anti-zero added to our arithmetic so we're to develop some

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consequences but I want to emphasize that this is in an exploratory fashion that is we do not

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know actually if this proposed arithmetic that I'm laying out here for you actually is a consistent

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arithmetic okay we do not know that we would like that to be true okay and and maybe there's some

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indications that it might be but this is still a very important question and we we can't get around

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it and in particular what we are definitely not allowed to do is what the modern analysts do with

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respect to real numbers which is basically to say well we don't actually have a proper foundation

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for real numbers so let's just stomp our feet and sort of axiomatically suppose that we have

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a set of real numbers that obeys the following Pleasant properties that we would like to have happen okay we are not allowed to do that we have to establish everything correctly in a bona fide

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way right so there's there's a lot of work to be done here in first of all checking whether

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or not what I'm proposing actually makes sense and maybe it doesn't it may be one of you will next week discover some contradiction okay it's possible I'll wear that okay that'll be fun then

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we can revamp things or look in an alternate direction or how we have to modify things

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okay but we're not just going to assume that what we want to be true is going

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to be true just because we want it to be sorry analysts but we want to do things correctly here

Possible Implications

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all right so let's proceed with some possible implications of the existence of this anti-zero

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let's look at this equation 1 times 0 equals zero which is a valid equation in our system

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so far we want that still to be valid so this is a basic idea that all of the prior arithmetical

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rules that we currently have for the objects that we currently have we want them still to

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be valid even when we're extending the arithmetic to include these new kinds of objects okay what

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is this here is one it's the empty set contained in the set and zero of course is the empty set

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and now crucially this empty mset can be replaced with the mset consisting of zero and anti-zero

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so like what the physicists call particle anti-particle creation okay this is an important

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idea that to the physicist nothing like a vacuum is is not exactly you know completely empty

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because bubbling out of the vacuum you can create these particle anti-particle pairs just like we're

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doing here this is very parallel to roughly the kinds of things that they're doing okay so this

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has to still be uh you know a reasonable rule so what happens when we apply the usual laws for

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multiplying this thing by this well the usual law is that we should take the amp set consisting of

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sums of elements here and elements here so that would be zero plus zero and zero plus anti-zero

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well zero plus zero is zero and zero plus anti-zero we don't know what that is yet but we

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do know that the result of this thing had better be the right hand side which is the empty set

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so this object zero plus zero a or zero plus anti zero better be anti-zero because we wanted

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to cancel with this zero to get nothing so we get a rule we get the rule that zero plus anti-0 is

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anti-zero and if we did it the other way around symmetrically anti-0 plus zero is anti-0 so there

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we have a first rule of arithmetic combining this new object anti-zero with a prior object zero

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and let me remind you that this is sort of consistent with the way we think zero works that when we take zero and we add it to something we get that other something okay and here's another

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rule that we want to have satisfied zero times zero is zero mset times empty and set should equal

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empty mset but this empty mset can be thought of as the mset 0 anti-0 similarly this one is zero

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anti-zero and when we think about it this way to multiply those two we have to form all possible

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Pairs and add them so we have to have zero plus zero as one entry zero plus anti-zero is another

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entry anti-zero plus zero is another entry and anti-zero plus anti-zero is another entry

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now zero plus zero is zero zero plus anti-zero we've just just established that's at the zero

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and this one here is also anti-zero but anti zero plus anti-zero we don't know yet what that is

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nevertheless we can do some cancellations because this zero and this anti-zero are

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together in this m set so they are going to mutually annihilate each other so they're gone

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what we have left is a single anti-zero there and this combination anti-0 plus add to zero

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and hopefully we're going to get the right hand side which is the empty mset

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so again we must have some mutual cancellation we have an anti-zero here so this thing here

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had better be zero because that's the thing that annihilates with anti-zero to get nothing

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so we can conclude that anti-zero plus anti-zero equals zero okay so for our next derivation let's

Another derivation

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do something similar to what we just did but a little bit more generally let's consider an mset consisting of some object M okay where m is some mset or object in this arithmetic and we're again

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multiplying it by zero typically we want that to be zero okay that's what we sort of expect

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so in terms of msets there's mset M 0 is of course just the empty M set

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and that again can be replaced with the mset consisting of 0 and anti-0

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so what we now do this multiplication we get something similar as before but a little bit more generally we get M plus zero as one of the entries at M plus anti zero is the other

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entry M plus zero is M and here we have M plus anti-zero and the right hand side ought to be

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the empty set so we see that for any M set or object M in this arithmetic if this is going

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to be valid then there has to be an object which cancels m in the same way that anti-0 cancels zero

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so this is a a huge extension we are necessarily forced to recognize that this particle

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anti-particle symmetry extends far beyond the zero anti-zero pairing just like Drax electron positron

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kind of symmetry actually ended up extending much further to in fact General particles in fact these

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days it's recognized that that's valid even for you know at the atomic level and in principle

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even at the molecular level and even at the level of human beings it's possible that there could be an anti-norman you know exactly more or less the same as me but made entirely of anti-particles

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okay so the same kind of thing is happening here we have this big enlargement of this this symmetry

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okay on with this divide between matter and antimatter and the physical level between msets

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and these anti-m sets in this arithmetical framework okay so what should we call this

Antim

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thing well obviously we should call it M with a superscript a okay we'll call it anti-m okay so

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we want this anti-m to have the property that if you have M and anti-m in an mset like they're just

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objects in an mset that they find each other and they cancel they're gone and we just get

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nothing inside the empty this empty mset and what we've established up here okay is that

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that this particular combination that capital M plus anti-zero is going to be this anti-m in other

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words this anti-m is actually M plus anti-zero and we could do it another way around anti-0 plus m

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and in particular if you have um M being anti-zero you play this same game with anti-zero which is an object in our in our world

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then the ante of the anti-zero should be a zero because if we if we replace M here with anti zero

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then over here we have anti-0 plus anti-zero which we've just established is zero okay so

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that establishes that if we take the anti of zero and then we take the anti of that in the following

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in that sense that I've just described then we're going to get back zero and now I'm going to play a very similar game as to the previous ones except that I'm going to move up or down in the operation

Carrot

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ladder so instead of thinking about multiplication I'm going to think about the carrot operation so

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let's look at the equation one carat zero equals zero which is a valid equation in our system

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so if it's a valid equation then it should still be valid after we replace the empty

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set here on the left hand side with the set 0 anti-zero so when we do this carrot operation

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let me remind you that what we're doing is we take all possible Pairs and we have to multiply

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them so we have to take zero times zero and zero times anti-zero the zero times zero is

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zero so this thing here again since we want this to be zero okay maybe I should write that

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we want this to be zero so it follows that this combination zero times anti-zero head

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better be anti-zero and other way around anti-zero times zero equals anti-zero

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and I've put triple exclamation mark here okay so please make note of this this is

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um an unfamiliar aspect of this arithmetic okay so perhaps some of you are getting a little bit

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nervous and this is this is a cause for pause here okay because we are declaring that we have

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some object so that when we multiply it by zero we do not get zero anymore you know usually that's

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always what happens when you multiply by zero you multiply by zero you're going to get zero but here in this world it appears that zero times anti-0 is not zero it's rather anti-zero

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okay so then we can do the same kind of thing replacing the one with zero zero carat zero

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equals zero that gives us this equation do the multiplications zero times zero zero times zero

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a a zero times zero a 0 a 0 or 0 to the a zero to a and this is zero zero times

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anti-zero we've just established that's anti-zero this one's also anti-zero we've just established

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that uh these two here so this unknown one here can be determined because now again we have these

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cancel okay we just get anti-zero and this anti-0 times anti-zero and since we want that to be

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the empty mset we must have that this combination is actually the thing that cancels anti-zero which is zero so we get anti-zero times anti-zero equals zero

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so we have two things multiplying together basically squaring getting zero but the

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actual object itself is not zero it's another sort of on unpredictable kind of uh thing

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it's counter to our usual intuition great so now we can write down addition and multiplication tables for zero the zero level of our arithmetic which now contains not just zero

Addition and Multiplication

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but also anti-zero so that that zeroth level there are now two objects so there's an addition table

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zero plus zero is zero zero plus anti-zero is anti-zero anti-zero plus zero is anti-zero

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and anti-zero plus anti-zero is zero and there is a multiplication table which is

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zero times zero is zero zero times anti-zero is anti-zero anti-zero times zero is anti-zero and

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anti-zero times anti-zero is zero note notably that these tables are pretty well the same

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except for the operation sign there they're exactly the same so at the level of zero the

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operations of addition and multiplication cannot be distinguished okay they cannot be distinguished

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so here's another exercise if you like this kind of thing um show that for any mset or object M

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which is not equal to anti-zero we have M times anti-zero same as anti-0 times m is

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anti-zero and this is quite interesting because of this requirement here that we

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have to restrict ourselves it's not true this is not true if m is equal to anti-zero itself

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because if m is equal to anti-0 anti-0 times anti-zero that's equal to zero not anti-zero

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okay so that that's already suggesting that that's a quite an interesting sort of thing

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that we have to be mindful of this sort of exemption to this general rule okay there's

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something interesting happening here and it makes us uh be motivated to really investigate further

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the internal consistency of this arithmetic is this arithmetic really consistent or not

Negative Numbers

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okay so having spent quite a lot of time on these anti-zeros and the various properties now let's

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move to negative numbers okay that's a different kettle of fish because we are still using the anti

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property but we're sort of putting it one level down in an mset so specifically what we're going

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to do now is introduce minus one or preferably one bar Define that to be the mset consisting of

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anti-zero let me remind you that one is the amset consisting of zero so now minus one is the amp set

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consisting of anti-zero and if we add the two of them well then obviously we get an mset consisting

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of an anti-0 and a zero they annihilate we just get the empty set so yes indeed one bar plus one

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or one plus one bar in the other way around is indeed zero so this fundamental property that we

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require for a minus one or one bar is satisfied let's look at one bar times one there's one bar

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there's one we're multiplying so we have to take all possible sums of elements so we have an mset

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with anti-zero plus zero we know that that sum is anti zero and so we see that minus one times one

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is minus one and similarly one times minus one is one and In The Same Spirit minus one times

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minus one there's minus one there's minus one we multiply them we get anti zero plus anti-zero

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inside we know that particular combination is zero okay we've already established that so we get that

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one bar times one bar equals one okay now that we have minus one or one bar we can't

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introduce integers or negative numbers first so definition if n is a natural

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number then n bar otherwise known as minus n is going to be defined by one bar times n

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so for example 2 bar by definition that's one bar times two there's one bar the amp side consisting

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of anti-0 here's two the nset consisting of zero zero we multiply them so we have to form

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anti zero plus zero anti-zero plus zero each of those is anti-zero so we get anti-zero anti-zero

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so that's what two bar is as an M set it's the mset consisting of two anti-zeros in parallel

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way to the fact that two is the mset consisting of two zeros so let's do some arithmetic with

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integers say two plus three bar so there's two there's three bar or negative three adding the

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M sets we just dump them all together so we get a zero a zero anti-zero anti-zero anti-zero we get

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some cancellation because they're in the same mset so they can interact with each other okay

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this zero that anti-zero perhaps they annihilate this zero that anti-zero perhaps they annihilate

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we end up with a single anti-zero in the mset so that's one bar or minus one two times three bar

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well there's two times three bar we're taking the product so we have to form all possible Pairs and

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form their sums so zero plus anti-zero well that's anti-zero all of these are anti-zero and anti-zero

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anti-zero anti-zero anti-zero anti-zero we get six anti-zeros which is six bar or minus six

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so you can kind of see that the usual arithmetical laws for integers are going to be satisfied here

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in particular y minus 2 times minus 3 is 6 is going to be a self-evident simply because

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anti-0 plus anti-zero is zero okay in our final slide I want to show you that this

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notion of negation actually now extends much further so usually in algebra when

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we introduce negative things we have to sort of do it separately at various levels when we introduce polynomials we have to Define how we're going to Define negative uh multiplications

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etc etc here essentially the same point of view that we already have allows us to extend this

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negativity all the way down and again we we have to draw heavily upon this prior understanding of

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anti-m sets Okay so suppose m is an arbitrary pure mset so it's an mset just consisting of

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things that we already are working with no other kinds of mathematical objects

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so we're going to Define m bar or minus m if you like just to be one bar times m

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and now we can deduce that if m is a pure mset then what this really is doing okay it's it's

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taking the ante of each of the elements of M so M is a is a box okay which has a lot of little

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boxes inside these are the elements those boxes themselves may very well have all kinds of stuff

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inside each one of them that's fine but the negative of the entire big box is according

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to this what you get when you consider the Box consisting of the antes of each of the elements

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the ante of this little element the anti of this element the anti of this little element and so on

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okay that's what the negative of the entire mcset is so why is that well uh M Bar according to the

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definition is one bar times m one bar there it is there the M side consisting of anti-0

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M is the m set consisting of let's say little m's where m is an m so we have a whole bunch of these

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elements of M how do we multiply things well we have to take all possible pairs between these

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guys and these guys and we have to add them well there's only one of them here so we're just taking all pairs of the form O A and M and we're adding them so we're getting O A Plus M as we run Little

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M through capital M but O A Plus M if m is any M set is the anti of M remember that was one of the

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rules that we established that you add anti-zero to anything you're getting the anti of that thing

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so each of the elements of M is being anted by the effect of multiplying by minus one

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that's another way of thinking what negation is in this world

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okay it's unfamiliar it's something that we need to get our heads around uh in my

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next video I'm going to try to give you some more illustrations of how this works there's much more to be said here this has all kinds of very interesting consequences

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and still exploration to be made in terms of whether this actually you know Rings true and

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is a consistent Theory which we do not yet know all right so this is a a prospective arithmetic

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there is much to investigate and to think about here I'm Norma Wildberger thanks for listening!

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