A multiset approach to arithmetic

We introduce a new framework for basic arithmetic and algebra, using the data structure of a multiset, or mset. This is an unordered collection of mathematical objects in which repetition is allowed. But what constitutes a "mathematical object"? One way of proceeding is to begin with the simplest possible mset, which is the empty mset, and build up from there. This is analogous to well -known treatments from set theory, but the consequences when we use msets instead are profound.

We are able to give simple and natural inductive definitions of natural numbers, polynumbers, and higher extensions which we call multinumbers. Crucially the operations are also defined in a very general fashion.

I'd like to thank my Patreon supporters for encouragement and helpful comments.

Video Content: (Thanks to phi Architect)

00:00 Introduction and history of multiset development

4:20 A multiset (mset) is an unordered collection allowing repetitions

7:20 A natural number (NAT) is an mset of zeroes

9:40 A polynumber is an mset of natural numbers

11:55 A multinumber is an mset of polynumbers

14:40 Addition of msets

19:15 NAT is closed under addition and commutative, associative

21:40 Multinumbers are also closed under addition

22:20 Multiplication of msets of msets

31:00 Each "type domain" is closed under addition and multiplication

32:30 The meaning of "poly"

36:15 Distinction of mset and list

38:30 Mathematics as a topic in computer science

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0:00

foreign [Music] Hello everone and today's a very exciting video for me because it's a culmination of quite

0:14

some years of interrupted thought so it's a point of view towards arithmetic and in fact algebra

0:22

which is based firmly on a data structure point of view but it's not sets it's rather multi-sets

0:32

so what I'm proposing to do here today is to create a new framework for arithmetic with natural

0:38

numbers first of all and then with polynomials and then we'll see that we'll be able to extend

0:44

this arithmetic Beyond to more complicated things and so we're going to end up with a a vision of

0:52

arithmetic and algebra which is perhaps somewhat unfamiliar to you and seems to extend in you know

0:59

vast new directions so it's a really interesting and if you're interested in exploring this is a

1:05

great topic for for thinking about and exploring it turns out that this point of view that I'm

1:11

giving you is actually quite parallel in many ways to what happened in the early days of set theory

1:18

when set theory was being used to try to create a structure for the foundations of mathematics and

1:24

uh the empty set played a very important role then as it does for us now but the critical

1:30

difference is that we are moving away from sets to a more general approach to data structures.

1:36

So I talk a lot about the LOMS system okay representing lists, ordered sets, multi-sets,

1:42

and sets so for me sets are just one of these four basic types of data structures,

1:48

and while a computer scientist is going to be leaning towards lists as fundamental objects and

1:53

that's fine for us we're going to be thinking about the multi-sets as an unappreciated uh

2:00

kind of framework for thinking about mathematics which has a lot of potential and a lot of power.

2:06

And in some ways it's sort of unfortunate that the history of mathematics circumvented

2:11

multi-sets and focused all the attention on sets okay so in case you haven't watched a

2:17

lot of my videos the idea that mathematics is based on set theory is in my view a complete

2:23

Distortion of what's really happening okay so we need to have a much more broader point of view and

2:30

along with that we need to be much more solid and we need to be much more explicit we need

2:36

to focus on what we can actually write down so forget about philosophical pronouncements about

2:42

what happens when you go to infinity or or do an infinite number of things which you can't actually do so we're going to be always looking to avoiding that kind of talking which is essentially a kind

2:55

of a vacuous talking not supported by explicit computation okay so real number arithmetic is a

3:01

fantasy as far as I'm concerned fundamental theorem of algebra is just wishful dreaming

3:06

you could evidence yourself that by factoring x to the 7 plus x minus three into quadratic and linear

3:14

factors: just do it or admit that you can't do it! and you'll see that the fundamental theorem of

3:20

algebra is not really true. So there's all kinds of really serious problems with the foundations of

3:26

mathematics as I of course talk about very often in this series, but today I want to make a start

3:34

at another kind of approach which is in some sense parallel to things that I've done already I've talked a lot about marks even from the early days of this series Marx and anti-marks and so on

3:45

so this is another point of view that's sort of parallel sort of competing has a lot of overlap

3:51

but is a little bit different I would like to say also first of all a big thank you to my patreon

3:57

supporters and in particular benefited by having some reactions to a previous video that I posted

4:05

in this direction uh and my patreon supporters especially Lucas lufaro gave some some interesting

4:11

suggestions and and encouragement so so big shout out to all of my patreon supporters okay so what

4:19

we're going to do is we're going to use this data structure point of view to get a new framework for arithmetic and then ultimately algebra and everything's based on multi-sets and what is the

4:28

multi-set it's a way of collecting things in an unordered fashion where repetitions are allowed

4:36

and I like to think of this as like boxes this is really box Theory

4:42

so here is a multi-set okay it's a container okay I can open up the container let's see what it has

4:49

in it okay oh look there's a few things in it uh there's this plug uh there's a couple of markers

4:56

these two markers are pretty well the same these two red ones so duplicates are allowed okay and

5:02

then there's this other thing what is this um well actually this is another container okay here's

5:07

another little container it's a little plastic container that has a a SanDisk drive inside so

5:13

it's actually this is like a little box inside this bigger box and that's a key point right

5:19

that we can think about boxes or collections or multi-sets of multi-sets of multi-sets etc

5:27

okay and the containment relations between these boxes or multi-sets becomes really interesting

5:34

okay so we start off at a very simple level by considering not something complicated like this

5:42

but something rather simpler like this an empty box okay an empty box or an empty multi-set

5:52

has the potential to put lots of stuff in there but currently there's nothing in it this is going to be our fundamental mathematical object it's the starting point of our mathematics

6:06

okay so we're going to denote it by pair of brackets with nothing inside so that's the empty

6:13

set or for us better the empty M set because has the potential of putting lots of things

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inside it we'll give it a name we call it zero and sometimes in diagrams will represented by

6:30

a box or a square like this or a rectangle with the possibility of putting various stuff in there

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so this object just by itself is already a type we can say we have a type of object zero type

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there's only one object that has this type the empty set or the empty mset or a zero so

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this is going to be our starting point and in some sense we're just going to use this

6:59

thing over and over again and put boxes inside other boxes to create arithmetic and algebra

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and we'll be able to go quite far just in this one lecture as you will see so even if you're

7:13

very used to arithmetic in algebra you're going to be introduced to some exciting new things

7:19

in this video okay so the next step up natural numbers so a natural number is an mset of zeros

7:32

the type of natural number denoted Nat that's the type here's some examples here's an mset of zeros

7:42

does it have any zeros in it well it has no zeros in it but we still consider an empty

7:48

set of zeros to be an mset of zeros so that's an important point. Let me just talk a little

7:55

bit about that okay so here is our as an amp set okay currently it has it's an mset of markers okay

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it's an mset of markers if I take one of these markers out it's still an mset of markers if I

8:10

take another marker out well it's still an mset of markers it happens to be the empty mset of markers

8:20

so the empty mset is actually or zero is actually a very flexible object but in particular it

8:26

is also a natural number for us so this new type of object includes the former type zero

8:37

okay and then we give these things some names the obvious ones so the set or mset with just

8:43

a single zero in it will be called one the mset consisting of zero and zero will be called two

8:51

and here is three and so on and so forth and in diagrams if we wanted to represent this

8:58

another way maybe simpler for for primary school kids we could maybe use some other symbol maybe like small circles to represent zero so what we usually call 13 then could

9:09

be Illustrated to a primary school student by a box or in this case a rectangle with

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13 little circles so these little circles are sort of playing the same role as the

9:24

marks these sort of Elementary marks that I've talked about earlier but there is a distinction because now these basic marks are not sort of Elemental things they are actually empty M sets

9:39

okay so we have a definition of what a natural number is it's an mset of zeros so those of

9:47

you who have watched a lot of my videos will know that I like to approach polynomials and polynomial algebra in terms of what I call poly numbers which are sort of more data theoretic

9:57

and I use these extensively even in the algebraic calculus course these are sort of workhorses for for doing algebra so here we're going to introduce them in a slightly different way but it's really

10:08

the same thing as we've already been talking about but just from a slightly different point of view so what is a poly number in this context a poly number is an M set of natural numbers

10:20

so we're going One Step Up further starting with zero the natural numbers and now poly numbers

10:28

and the type poly okay so here is a poly number why well it's an M set and what does it have in

10:37

it one one two those are all numbers so this is a multi-set of natural numbers

10:45

it is therefore a poly number here is another problem number and okay admittedly there could

10:52

be a little bit of confusion depending on how much space you think there is there this is meant to be 13 okay uh these are sort of like the adjacent so this is a a multi-set with just a single number in

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it but in particular it's a multi-set of natural numbers so it's a poly number here's another

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poly number because it's an mset of natural numbers because remember zero is a natural number

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the type of natural number included the former type of zero so zero is a natural number

11:26

and in fact this thing here is also a poly number because it's an mset of natural numbers how many

11:35

natural numbers are in here well there aren't any but it's still an empty mset of natural numbers so in particular this thing which is of course zero is also a poly number okay so we

11:47

have zero the type zero contained in the type not contained in the type of poly and now something

11:57

one level further which is a little bit fancier okay and uh it will take some getting used to

12:04

you know what this actually means but it's sort of a natural thing to do we sort of have this natural hierarchy now we can just sort of keep on going so a multi number is by definition an M set

12:16

of poly numbers an M set of poly numbers and there's the type multi okay so look at this thing

12:26

here there's an M set what's this comma doing here oh no so here here's the answer okay here's the M

12:33

set yes brackets here and what's inside this m set well it has this object what is this thing

12:40

this is an M set with numbers in it one one so M set of natural numbers so this is a

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poly number here's another poly numbers these two happen to be equal so this whole thing is an

12:56

M set of poly numbers so this is a multi number here's another multi number it's an M set which

13:05

happens to have one poly number in it the poly number consisting of the multi-set 3 5 and 11.

13:14

how about this thing is this an M set of poly numbers well is zero a poly number well yes we

13:21

just said it once see this thing is zero and this we've said is a poly number so this is a poly number so this is an M set of poly numbers in particular it's a multi number

13:31

what about the empty set well it's also an M set of poly numbers the empty msat of poly numbers

13:39

so it also includes so again multi includes poly which includes Nat which includes zero

13:46

so we have this inclusion inclusion of types zero Nat poly multi each included in the other in the

13:55

following sense in the sense that if you have an object of this type then it is automatically an object also of this type therefore also an object of this type therefore also an object of this type

14:07

and I hope that it's sort of obvious that the way we're going from one step here to another

14:13

uh is directly generalizable okay we could easily imagine just keeping on going once we've defined

14:20

multi numbers we could Define some higher objects which are M sets of multi numbers okay and later

14:29

on we're gonna we're gonna be thinking about that kind of thing but already here we have a very rich

14:36

framework for arithmetic and algebra all right but if we want to have arithmetic we

14:42

need operations as well as numbers or objects so there is a fundamental operation in this

14:52

world of multi-sets which is very very general and it's addition okay so this

14:58

is the beauty of the multi-set framework which distinguishes it from the set theory world in the

15:04

set theory there's no such thing as addition of sets but in the multi-set world there is

15:10

okay it's crucial to understand and to appreciate the importance of this operation this very general

15:17

operation so how do you add msets you just combine their contents into a single M set

15:26

let me show you physically okay so here are two M sets they're boxes this is a little box that has a

15:36

SanDisk in it okay and here this one here is what does it have in it oh it has two two markers in

15:44

it okay shall we add these two M sets add this m set and this mset I'm going to show you how to do

15:51

that okay we need another box okay hey here's here's a box okay so what am I going to do I'm

15:58

going to take this box and this box here these two boxes and I'm going to combine their contents

16:08

one at a time because I only have two hands okay and there's the there's SanDisk that's also going

16:13

in there okay there are the original boxes I don't need them because I've put everything foreign

16:24

everything in this new box and so there I hope you can see here there's our new box it's got

16:30

a SanDisk in it there it is there and it has our two markers okay that's how you add msets

16:42

let's have a look at some examples with the kinds of objects we've been talking about okay so here we have some some objects okay and notice that in this

16:54

case I actually have three things okay and what are these things well these are um these things are are what well this is a this is a poly number because it's a multi-set

17:07

of natural numbers here's another problem so this is here this thing here is a multi number here's another multi number here's another multi-number but in particular these are msets

17:18

so we don't require that we just have two episodes we could do this with more than two msets we could

17:24

have several msets and add them at the same time just by dumping all their contents into a box

17:30

happily things are unordered and repetitions are allowed so there's no need to fuss around uh and

17:37

and remove duplicates as you would if you were working with set theory it's just absolutely simple minded you just dump them out in the box and there you go but in particular it's

17:47

not intrinsically a binary operation so how do you how do you add these things well you just get

17:52

a bigger box represented by those brackets there and you put the set three in the M set four the

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mset 112 the M set four the zero okay that's also a that's a a natural number okay so in particular

18:09

it's a poly number so there's the zero and there four is there and the one two one is there so

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we're just dumping the contents of these three insets into a single msat that's how you add

18:20

okay the operations name will be the usual Edition so if we have you know if this is a

18:26

b and c then this sum is denoted a plus b plus c and note again that it's not intrinsically a

18:33

binary operation it applies to all msets it's not just that we can add numbers right we can

18:39

have M sets up anything in principle and we're still able to perform this addition operation

18:47

it's really important that you digest the the fundamental aspect of this operation okay this

18:55

is something that is so Elementary but maybe has slipped by your Consciousness because you're not

19:01

used to thinking about msets as being fundamental but it's a crucial arithmetical structure that's

19:08

actually working in the entire world of msets okay so it's uh now easy observation that

19:18

the natural numbers are closed under addition in fact one step below that the zero

19:27

the zero type is also closed under addition let me show you that okay so here is now an empty M

19:34

set I've dumped everything previously B OMB out and here's another m the m set okay so suppose

19:41

I wanted to add these two M sets well I'm going to get myself another box okay and I'm going to

19:50

dump the contents of this one here and also dump the contents of this one here okay there

19:57

I've done it I've dumped all the contents of the two boxes and you can see it's still empty okay so empty set plus empty set equals empty set so zero plus zero equals zero so somewhat less

20:12

obvious but still pretty obvious is that if you have two natural numbers so three plus four say

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okay there's a picture of three it's an mset which has three zeros in it zero zero zero represented

20:27

by these little circles because it's a picture for primary school students and here's a box representing four you dump them out into another box well obviously you're gonna get there's the

20:37

first three and there's the next four and you're gonna get what we call seven in terms of the more

20:43

familiar notation that we've been talking about three is actually officially this thing here

20:48

's the M set zero zero zero four is the msat zero zero zero zero we put them all in an amp set so

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these guys go here and these guys say go here and then we can check that that's actually seven

21:02

note that poly is also closed under addition okay here's a poly number because it's an mset of

21:11

natural numbers here's another poly number here's another poly number here's another poly number if we add all four of them dump the contents dump the contents dump the contents dump the contents

21:21

get three three four zero seven and this empty set doesn't contribute and this is another multi-set

21:34

of natural numbers so it's another poly number so if we go up one level to multi numbers we

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see that also it's closed under addition for example here's a multi number it's an mset

21:47

of poly numbers that's a poly number poly number here's another mset of poly numbers that's a poly

21:53

number that's a polynomial that's a poly number we add them well obviously we get that one that one

21:59

that one that one that one we get another mset of poly numbers in other words another multi number

22:08

okay so we have addition now we want to go up a step okay and this really important step we're

22:16

going to go from addition to multiplication okay and it's really important to watch carefully what

22:23

I do here it's like magic okay a very simple kind of magic okay we're going to create multiplication

22:29

out of addition so how are we going to do that so the multiplication of mets of msets so strictly

22:35

speaking in the mset world you need msets of msets to make this this work okay as it turns

22:43

out all the things that we've been talking about are in fact msets of msets so strictly speaking

22:49

this is not really necessary I just point this out that if you were working with msets with other kinds of objects you you really want to restrict yourself to M sets of msets but this

22:59

is not really a restriction um in in the world that we set up so far okay so to multiply uh

23:05

am says what you do is you form the mset of sums of elements one from each mset so we're going

23:14

to use addition to Define multiplication okay so look carefully here is an mset and we're going to

23:24

assume that the contents A and B are themselves msets let me remind you that that that is

23:32

um a condition that's automatically satisfied for all the objects that we've been considering

23:38

here's another mset all of its elements are also msets so we're assuming that okay how are we going

23:47

to multiply this m set with this nset we're going to take pairs in this case because there's two of

23:53

them that were multiplying pairs the elements from this one with elements from this one I'm going to

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take this pair and we're going to add them in this case a plus X we're going to form a plus X this is

24:07

an M set that's an M set so a plus X makes sense that's addition of M sets this is another M set

24:14

then we're going to take a and Y we're going to add them A Plus y Take A and Z add them get a

24:20

plus Z take B and X add them get B plus X take B and Y add them you get B plus y take B and z

24:29

add them you get B plus Z so we're getting another M set which has new M sets in it these

24:38

new M sets are created from the original two M sets by taking sums of elements one from each

24:48

that's how you multiply msets all right let me illustrate this multiplication at each of the levels so at the zero level

24:58

zero times zero what do we get well we have to form an M set which consists of what which

25:05

consists of the sum of all entries of this thing with entries of this thing well there are no

25:11

entries of this thing there are no entries of this thing so there are no possible sums that we can make so the result is empty fine that's telling us that 0 times 0 equals zero now let's go to

25:24

natural numbers okay which are msets of zero so there's an mset of zeros there's an mset of zeros

25:33

so how are we going to multiply these two things well we're going to take all possible combinations entry here entry here or element here element here okay and we're going to add them

25:45

we could add them because the guys inside here are msets and so they can be added using mset addition

25:53

okay so let's do that so this 0 plus this zero there it is there this zero plus this zero there

25:59

it is there this zero plus this zero there it is there this zero plus this zero there it is there this zero plus this zero there it is there this zero plus this zero there it is there have I done

26:09

all the pairs yes I've done all the pairs now we have to evaluate these things what is zero plus

26:14

zero well it's zero so each of these is zero so we get zero zero zero zero zero zero and in ordinary

26:23

Hindu Arabic notation we've computed that two times three is uh six okay let's go to Paulie

26:36

let's multiply poly numbers so here's a poly number here's another poly number these are poly

26:44

numbers because they're msets of numbers that's a number that's a number natural numbers here's another mset of natural numbers so to multiply them we have to take pairs combinations and add

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them so how do you add the number two and the number one first pair well you get three and

27:05

then two plus one is three two plus zero is two and then three plus one is four three plus one is

27:10

four three plus zero is three so we get the mset three three two four four three it's another mset

27:18

of natural numbers so that's another poly number poly number times poly number equals poly number

27:25

and let me show you that this is not restricted to having two we can multiply in principle a number

27:34

of different msets so now I'm going to multiply three msets together sort of at the same time

27:40

so the mset03 the mset 1 4 the mset 2 7. these are all poly numbers because they're msets of natural

27:47

numbers so what do we need to do well we need to take combinations where we're taking one element

27:54

of each one of these so let's start with the zero one and two what do I have to do with those things

27:59

I have to add them they're M sets so the addition of M sets of these three is well defined I don't

28:07

have to specify which order we're doing it in there's three things we're just dumping them out

28:12

so zero plus one plus two is three zero plus one plus seven is eight zero plus four plus

28:21

two is six zero plus four plus seven is eleven okay I have to keep track I'm going to pick L

28:28

possible combinations how about three plus one plus two oh that's six three plus one plus seven

28:34

that's eleven three plus four plus two is nine and three plus four plus seven is a fourteen

28:42

so we've multiplied these three poly numbers to get this poly number

28:49

okay now one more step up to the multi numbers so let's do this okay so here's a multi-number why is

28:57

this a multi-number well because this thing inside it is a poly number and here's another poly number

29:05

it's a multi-set of natural numbers therefore a poly number here's another mset of natural numbers therefore our poly number so this is an M set of poly numbers here's another mset of poly numbers

29:15

note that 2 is indeed a poly number because we've seen that natural numbers are included

29:23

in the type poly numbers so what do we have to do well we have to take all possible pairs

29:30

elements here elements here and we have to add them okay so let's take this element here and

29:35

this element here these are msets there's an mset there's an absent how do you add them

29:42

um well we just dump the contents into a box so we're going to get 0 0 2 1 1. there

29:48

that's the result of adding this mset and this mset how about this mset with two okay well we

29:54

get zero zero two uh what is this thing as an mset well two is really the mset 0 0. if

30:02

we want to write it that way so we could put another two zeros there now we've added two

30:08

um how about this thing and and this thing well then we're going to get um I better write smaller zero zero two nine okay and now this one plus this one that's three eight one one and we also

30:23

need down here okay three eight we're adding two so we have to add two zeros and finally

30:30

this one here three eight nine okay and that's the end so there's the big bracket at the end

30:44

so we're getting one two three four five six just to check okay six different constituents

30:50

or elements that makes sense because this one has two elements this one has three elements so we should end up getting six elements so we can do multiplication in each one of these types okay

31:01

so a little bit of thought and just doing some calculations along the lines I've showed you

31:06

reveals that each one of these from type domains zero inside Nat inside

31:12

poly inside multi is closed under both addition and multiplication

31:19

and moreover the usual arithmetic laws that we're familiar with hold;

31:24

namely we have commutative law A plus B equals B plus A, A times B equals B times A

31:38

(A plus B) plus C equals A plus (B plus C) that's an associate of law for addition and here's an associate of law for multiplication and distributive laws (A times B) plus C equals

31:52

A times B plus A times C, where we use the usual convention that the multiplication takes

32:00

precedence over the addition that means you have to do this product first and you do this product

32:06

also before you add them together and over here A plus B times C is a times C plus B times C

32:14

okay so it's a great exercise to check these and sort of convince yourself that these are are valid

32:22

and you see that it's a very general kind of arithmetic it's not just arithmetic with numbers

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that we're doing here we're doing arithmetic in a broader context all right so I want to finish

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the lecture by clarifying the meaning of poly and since that's a term that I've already used

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quite extensively I want to connect with the former usage and convince you that these are pretty well the same things although there is a slight difference so we can do that by

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concentrating on the sort of simplest object when we're going from natural numbers to poly numbers

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which is not itself a natural number so if we take the set consisting of zero that's a poly

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number but it's also a natural number that's one okay so the simplest poly number which is

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not a natural number is this one it's the mset consisting of a single one let's call it Alpha

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okay let's do a bit of arithmetic with this Alpha suppose we look at Alpha plus Alpha okay which we

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could call two alpha what is that well you have to take this thing and you have to add it with itself

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so you have to get a box and you dump one one in and then you dump another one in well you're gonna have two ones in the box so it's gonna look like this

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okay what about three Alphas well number one in dump another one in dump another one in you're

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gonna get this thing here and so on so multiples of outfits clear sort of what they correspond to

33:54

how about multiplication suppose we take Alpha times Alpha so there's Alpha there's Alpha how

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do you multiply them well you have to take the sums of corresponding elements one from each one

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of these well there's only one element here and one element here so you have to take one plus one and one plus one Edition natural numbers that's two so you're going to get the mset

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consisting of a single two and if you take Alpha cubed that means you're going to multiply this

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thing by another one of these then you have to compute one plus two which is three so you can get

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the answer consisting of three so powers of alpha are sort of naturally corresponding

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to those sort of singleton msets that just contain a single natural number

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and so that allows us to interpret some more complicated poly like this okay so here's a more

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complicated poly number it's an mset of natural numbers zero zero one one two three three three

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so to make sense of this to organize it a little bit we can write it as a sum so this is the sum

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of the msat 0 0 and the mset one one one and the mset 2 and the mset three three three three okay

35:13

what is this thing this is the number two what is this thing well this is what we've said is three

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Alpha what is this thing that's what we've said is Alpha squared and what is this thing

35:25

well Alpha cubed was three so two alpha cubed will be this plus itself will be the msat three three

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and having four of them that'll be four Alpha cubed so this is a representation of what an ordinary algebra we would write as the polynomial 2 plus

35:45

3 Alpha plus Alpha squared plus four Alpha cubed and in my preferred poly number language that I've

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talked a lot about in in other videos I would represent this with a a list of vertical list

35:59

like this okay with the coefficients two three one four so we're seeing that this sort of form

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or definition of a poly agrees with this sort of new definition of a poly they're really the same

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but what's the difference the difference is that this is a list okay this is a list

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and this thing here is of course an mset and there's a crucial uh distinction between these

36:31

two things that's a really um useful to to appreciate so remember when we were looking at

36:36

polynomials or poly numbers like this there was a certain amount of ambiguity because we could add a

36:42

zero below here and we agreed that was going to be the same thing that did not change the poly number

36:49

we could add in fact any number of zeros below and that's not going to change the poly number

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even though officially as a list it's kind of changing there's a little bit of ambiguity there

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it's a general rule that when you have that kind of ambiguity it's because it's because the list that you're considering is really more fundamentally to be thought of as a as

37:12

a multiplicity list for an mset okay so this is an mset of zeros ones twos and threes and since the

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the natural numbers are naturally ordered zero one two three we can represent such an absent or maybe

37:28

make it more concise by just counting how many each we have we have two zeros we have three ones

37:34

we have one two and we have four threes so this is a list that is the multiplicity list of this m set

37:44

but I would like to suggest to you that actually

37:49

formally officially the mset is actually more fundamental it's at the heart of things and

37:58

now we see that this this list object is really just a secondary object that sort of makes it

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easier for us to represent this nset but what a poly really is is an mset of natural numbers

38:17

great so we've accomplished a lot um of course a question that will be on your minds is okay

38:22

but what about these multi numbers what's their significance um so I'm going to tell you about that in the next video and we're also moving towards um an alternate view of this thing which

38:34

is even more in spirit for computer scientists so this topic is going to move really really closely

38:42

to alignment with modern computer science as we'll see Maybe Mathematics will end up being

38:53

a topic in Computer Science!! I'm Norman Wildberger thanks for listening. :)

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