Introduction and Invitation
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Welcome to Six --- an Elementary Course in Pure Mathematics meant for a general lay audience with a minimal amount of mathematical prerequisites! In this video we introduce the basic objects: the symbols 1,2,3,4,5 and 6 along with the basic tools to create more complex mathematical objects from simpler ones --- the data structures LOMS, standing for Lists, Osets, Msets, and just Sets. (Osets are short for Ordered sets, while Msets are short for Multisets). With this rather modest starting point, we get right to work in building up some fundamental initial concepts that will play a central role in the course. If you have an interest in mathematics and would like to get some real insight into how pure mathematics works, why not roll up your sleaves and join the course? Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated.
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[Music]
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hello
00:11
and welcome to six
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i'm noah barbara and this is my attempt
00:17
at introducing pure mathematical
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thinking
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to a very wide audience with minimal
00:26
prerequisites so this is going to be a
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course
00:29
in pure mathematics which is going to
00:33
investigate an interesting area
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that we can access in a very elementary
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way
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the audience is going to be
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motivated high school students and
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beyond
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any lay people who are interested
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in learning about pure mathematics
00:54
from the ground up the only
00:57
prerequisites
00:58
really at least to begin with are going
01:00
to be arithmetic with natural numbers
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roughly what you would have learned from
01:05
primary school education
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so very minimal prerequisites
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okay and the aim of the
01:14
course is to illustrate what pure
01:17
mathematics research is really like
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to give you a direct and tangible
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experience
01:23
of this and
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we're going to develop concentration
01:29
skills
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logical reasoning careful organization
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and hopefully a love of mathematics
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now usually in order to access pure
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mathematics which is a rather esoteric
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and
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usually thought of as difficult subject
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you require
01:49
many years of undergraduate education
01:51
followed by graduate school we're
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circumventing all of that
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and we're doing it in a way that
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doesn't really connect with any other
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things so we're just going to
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investigate
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some independent very elementary
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situation and we're going to explore
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this really
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interesting landscape around it and
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perhaps we're going to meet all kinds of
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interesting things
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so our aim is actually not to get to any
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particular point okay
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i'm going to just going to be explaining
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a
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branch of pure mathematics which is all
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about
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the number six so it's mathematical
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content
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which you're really not going to find
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anywhere else
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but ultimately we will be i can promise
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you touching base with
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really classical and lovely
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19th century mathematics that actually
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engaged the interest of a lot of really
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prominent mathematicians
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so there will be something at the end of
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it that's going to be
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really fascinating but that's not our
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aim our aim is just
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to explore so i hope you're going to
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join me for this
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i hope you'll take a notebook or maybe
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your laptop or
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ipad whatever and follow along and
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listen carefully to everything that i do
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and try to understand everything that i
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say
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and even if you're not used to enjoying
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mathematics i hope you'll
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get a sense of what the subject is
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really about
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careful definitions clear notation
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and an organized approach to analyzing
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patterns and
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making connections so we are going to
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start
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by taking six objects
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and we're going to consider these
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objects to be the numbers one
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two three four five and six so the
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entire course
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is going to be built up from these six
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initial objects
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we will generally write them in black so
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whenever you see
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numbers from one to six in black you
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should think of them as being our
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initial objects and we're going to be
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thinking about
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having containers for these six so we
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have a container of ones a container of
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twos and so on
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and we're able to take uh numbers
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uh from these containers and combine
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them to create
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more complicated objects okay
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so that's the the initial thrust is
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we're going to be
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creating objects from these six numbers
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and we are then going to look at the
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patterns that we can form
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from these now i want to stress that
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the actual arithmetic associated with
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these
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six numbers is going to be irrelevant so
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we're not going to be adding or
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multiplying or doing any kind of
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arithmetic with these objects
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you can think about these as just sort
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of short forms for the names
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of six things so i could have used pens
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or
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coins or any number of things but it's
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convenient
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to use these things because they're all
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very familiar and because there's a
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natural order to them
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which will be useful to us later on
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so what our aim is at first is to create
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expressions
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just involving grouping or arranging
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these kinds of objects
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in various ways and when we group
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these basic objects together we have
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some choices whether we're going to
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create
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patterns or groupings which are ordered
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or unordered and another
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choice that we can make is whether
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repetitions are going to be allowed
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or not allowed
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so this is a very important kind of
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distinction and this is going to
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provide sort of a theoretical framework
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for us moving forward
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we have to talk about how to organize
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data or information in
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various ways ordered or unordered
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repetitions allowed or not allowed
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so let's talk about the very important
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idea of organizing
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mathematical objects in mathematics
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there are generally
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four different ways of organizing data
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or other objects i like to call this the
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loms
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framework and there are four kinds of
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objects
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lists ordered sets multisets and
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sets and let's talk about what these
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things are
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and the notation for them
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notation is going to play a very
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important role in this course
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so we have to understand precisely
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what it is that we're writing down and
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what the meaning of the things that
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we're writing down
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so the first one a list a list is an
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ordered
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collection where repetitions are allowed
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so it's ordered means there's a first
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there's a second etc
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and repetitions are allowed so here for
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example is a list
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it's first of all enclosed in these
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square brackets
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the entries are four and then three and
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then three and then five
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and they are ordered by the presence of
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these commas
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that separate subsequent elements so
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first there's four
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then there's a comma then there's three
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then there's a comma then there's three
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then there's another comma there's five
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and we can see that repetitions are
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allowed because there are two
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threes notice that the objects that
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we're actually using are all from our
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fundamental
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six allowed objects that's going to be
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happening
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throughout the course the actual
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ingredients to everything are just one
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of these six
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starting numbers okay
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now the next data structure is an
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ordered set
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or short o set should probably underline
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that too because that's important
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okay and oh set so what is an origin
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it is a collection which is ordered
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but now no repetitions are allowed
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no duplication and the notation for that
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is
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these curve brackets sometimes called
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braces
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okay that kind of brace and
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there's commas to indicate that things
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are still
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ordered so this is the ordered set 6
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comma 1 comma 2.
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our next data structure is a multi-set
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or for short m set
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it is unordered but repetitions
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are allowed
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it's unordered so there are
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no commas that's the way we represent
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an unordered collection
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there are no commas that put the
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elements in their proper place that
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separate them and kind of lock them in
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you can think about the contents here
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like the contents of a box
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the elements are allowed to move around
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and the object itself is not
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changing the brackets are the square
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brackets the same kind of brackets
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that we have in a list and that tells us
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that repetitions
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are allowed okay so the the brackets
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tell us whether repetition
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is allowed or not and the commas or
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spaces
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tell us whether the thing is ordered or
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not
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so multi-sets are unordered and
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repetitions are allowed and the final
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one
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which is actually in modern mathematics
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by far
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the most familiar and widely used
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is the set which is unordered
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and repetitions are not allowed
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and our notation for a set is going to
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be consistent with what we've said up
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here
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the fact that it's unordered means that
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there's no commas
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separating the elements just
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spaces and the fact that no repetitions
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are allowed
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is the same as for an o set these are
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these
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brace brackets so we can
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look at these four kinds of data
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structures and we can tell
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just from the form whether we're talking
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about a list
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an ordered set an m set or a set
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now in classical mathematics sets
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are dominant but for us
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that's not going to be the case we are
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going to maintain
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uh flexibility and in fact a kind of a
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feeling of equality between these
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four different ways of organizing things
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that's going to be very important so
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it's crucial
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that you spend some time staring at this
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and
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get yourself used to the notation that
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we're going to be using
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now in my math foundation series i talk
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quite
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extensively about these data structures
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and in my
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algebraic calculus course we actually
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use them so you may have had some
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experience with that if you watch some
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of my other videos
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and if you want a little bit more
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information you can go to the math
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foundations
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series and find out more about these but
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this is really
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all you really need to know is what
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these four types of
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ways of organizing mathematical objects
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are and how the notation
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tells us which one is in front of us at
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any given time
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so just to reinforce that let me put all
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that information in a little array
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with four boxes and over here we have
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the ordered
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objects and the unordered objects in
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this first row and then in the second
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row
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so the list and the ordered set are
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ordered the multi-set or m
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set and the set are unordered
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and over here in the columns the columns
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represent whether
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repetitions are allowed here they're
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allowed
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and over here they're not allowed so
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there's a list there's an example of a
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list
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it's ordered and repetitions are allowed
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here's an o
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set there's an example of it
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the uh it's it's ordered
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because of the commas
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and its repetitions not being allowed in
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this case
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as you can see by the type of brackets
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over here is an m set or a multi-set
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it's
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unordered and repetitions are allowed
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unordered because the spaces
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uh between them replace the commas so
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spaces means that you can move things
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around it's unordered
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and these brackets tell us that
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repetitions are okay
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just like in the list and finally the
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set
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has these braces telling us that
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repetitions are not
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allowed but it's unordered as we can see
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by
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the spaces between elements instead of
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commas so for example with this set you
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could
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change the position of the six and the
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four
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and it would be the same set
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with the ordered set you're not allowed
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to change the
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the positions of say the two and the
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three that would be a different ordered
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set
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over here you're allowed to change the
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one and the two you could move that one
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there and move the two over there
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that would be the same m set
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so one more important aspect of these
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things is
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a measure of how many elements there are
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in them
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okay so this uh
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this thing here has five elements one
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two three four five we're counting
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this one has four elements this one has
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three elements this one has four
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elements
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we can make that part of the
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the name okay by adding the number of
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elements
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in front of the data structure so for
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example
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here's another example of a list the
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list two
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four four this is a three list
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it's a three list because it has three
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elements
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here's an ordered set or an o set it's a
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2-0 set we write it like this 2-0 set
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because it has two elements
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here is a 5-m set it's an m-set
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with five elements and here is a four
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set it's a set with four elements
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so let me emphasize that this notation
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this setup is very logical
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okay but it's different from what often
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is done because
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classically the emphasis is on sets and
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usually we don't write sets this way we
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write them in classical mathematics with
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commas
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so we are separating our usage here from
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the classical one
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you have to remember that okay so when
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we write something like this you have to
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recognize this as a set
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when we introduce commas that means
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we're introducing an
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ordering does that make sense i hope
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it's sort of logical
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it is logical and it's sort of
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consistent
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but it's something that you have to
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remember
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all right so let's use these data
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structures to create
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some new objects out of our six
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initial objects first
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let's give our initial objects a name
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okay we are going to call them these
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primary objects one two three four five
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six we're gonna call them
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nodes have a look nodes
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right so this is a node that's a node so
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this is just a terminology that we're
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introducing we're gonna agree
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that node where it might have some other
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prior meaning in ordinary life
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for us node is going to refer to one of
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these
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six objects and to emphasize the fact
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that
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this is very specific usage of this
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terminology i'm going to introduce
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what i call german notation
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i'm going to capitalize the first
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letter of this word that's going to
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immediately remind us that this is a
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fundamental kind of object
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now what is this six that's floating
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beyond here
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this six is in blue it's not a black
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six so it's not the same as this 6 here
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this 6
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is telling us how many nodes there
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are we have a very simple world here
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so one of the things that we're going to
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be doing at least is
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counting things when we create them
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okay so there are six nulls
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okay now it's time to introduce some new
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objects
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and the first thing that we're going to
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introduce are the edges the
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new objects are going to be called edges
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and
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what they are are two sets
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of nodes two sets of notes
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what does that mean it means it's a set
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with two elements
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so it looks like this has this brace
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kind of uh you know
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brackets and there's no commas
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separating elements and there's going to
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be two elements
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and these elements have to be themselves
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nodes
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in other words they have to be amongst
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our basic nodes
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and we're going to introduce some
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abbreviated notation because we're in
17:43
this happy situation where
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we're always only going to be using
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these six numbers
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and that means we can save ourselves
17:51
quite a little bit of work by being
17:53
judicious
17:53
about our notation it's going to be very
17:57
important for us
17:58
okay so instead of writing set a
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b i'm just going to write a followed by
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b so in this world
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the world of this course a b does not
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mean the product of a
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and b it means the two set
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whose elements are a and b
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but with a further proviso that we're
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going to arrange things
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so that a is less than b
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right a and b are both elements in here
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they're amongst these notes
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and while the set
18:37
is unchanged if we permute
18:41
the elements we are going to write it
18:44
so that the smaller
18:47
entry of the smaller node comes first
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okay so what effectively do these edges
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look like
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well the simplest edge is one two
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okay one two that's our short form
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for the two set whose elements are one
19:05
and two
19:06
and then there's one three that's the
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next
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kind of logical one and then there's one
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four
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and then there's one five and then one
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six and then
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two three two four two five two six
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three four three five three six four
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five four six
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five six notice that i've listed them
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in a somewhat logical order starting
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with the
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simpler the smaller first
19:37
element in this representation
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so i'm using the natural ordering on on
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the six elements
19:46
to list these 15 edges
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in a reasonably sort of consistent
19:52
pattern
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and there are 15 of them because it's 1
19:55
2 3 4
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5 6 7 8 9 10 11
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12 13 14 15 such
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edges
20:07
so this is um i hope you see what we're
20:10
kind of doing is we're taking planet
20:12
earth
20:12
okay with all the you know complexity
20:16
and and human interactions and all kinds
20:18
of you know things
20:19
and we're we're taking a rocket ship and
20:21
we're leaving planet earth far behind
20:23
we're not aiming towards applications
20:26
okay we're now on this distant planet
20:29
and there's nothing there except for
20:31
these sort of six nodes
20:33
actually we have these containers of six
20:36
nodes so we have lots of good nodes
20:38
and we're just going to create stuff
20:40
okay
20:41
and that is going to be our our pure
20:44
mathematical exercise
20:47
very abstract very unconnected with
20:51
applications okay we're just wanting to
20:54
explore
20:55
what we can actually say about six
20:57
things
20:58
and this is a good place to start with
21:01
our six
21:01
nodes and our 15 edges
21:05
okay so we're going to now want to
21:07
introduce one more
21:09
type of new objects and then we'll be in
21:11
a nice position to finish this first
21:13
lecture
21:14
and i'll give you some homework okay so
21:16
the
21:17
15 edges that we've just
21:20
written down if you stare at them you
21:23
might
21:23
realize that any two of them
21:27
have a possible connection so we're
21:30
going to introduce a definition we'll
21:32
say that two edges are
21:33
incident precisely when they
21:36
share an element after all an edge is
21:40
actually a two set
21:42
so if we have two edges we have two two
21:44
sets and we can ask
21:46
do these two two sets actually contain a
21:49
common element or a common node
21:53
so for example the edge one two and the
21:56
edge 2
21:57
5 are incident because
22:00
they share the node 2.
22:04
while the edges 3 6
22:07
and 1 4 are not incident because they
22:10
don't share
22:11
any common notes and now we're going to
22:14
introduce
22:15
new objects a little bit bigger and
22:18
better than edges
22:20
which are called meats okay i want some
22:22
meat
22:23
well a meat is a two set of
22:27
non-incident edges
22:32
so you may have to slow down and really
22:34
get your head around this
22:36
but it's possible to understand it
22:37
completely i want you to understand
22:39
completely what we're doing so this is a
22:41
two set
22:42
okay so it's a set with two things
22:45
inside and those two things are
22:48
each edges that we've just defined
22:52
but the two of them have to have this
22:53
property that they are non-incident
22:56
in other words the nodes that appear are
22:59
not overlapping
23:01
so for example here is an example of
23:04
a meat it's a two set
23:08
with well an edge there
23:11
and an edge there and these two edges
23:14
we've just seen
23:14
are not incident so this is a viable
23:18
meat now we're going to save ourselves a
23:22
little bit of time
23:24
by introducing more streamlined notation
23:27
we're going to exploit again the fact
23:29
that we can write
23:30
our edges rather compactly so we can
23:33
write this
23:34
to set compactly by
23:37
putting the edges together with a little
23:40
period or dot between them
23:42
but we're going to do it in such a way
23:44
that the
23:46
first element of the first edge is
23:49
less than the first element of the
23:51
second edge
23:53
so maybe i might say that okay here
23:56
where
23:58
by convention we're going to agree that
24:01
the the one here is actually less than 3
24:05
in the order of the nodes we're going to
24:08
do that so that we have a canonical way
24:10
of describing this object
24:12
because we could rewrite this 2 set by
24:15
interchanging these two elements
24:18
okay but by agreeing that
24:21
the first element of the first edge here
24:25
is going to be less than in the ordering
24:27
of the first element in the second edge
24:29
we guarantee that there's always a
24:31
canonical way of writing
24:33
a given mate and now i claim that there
24:36
are exactly
24:38
45 mates and
24:41
now we come to your first homework so
24:43
homework number one
24:44
is to list all 45 mates
24:48
same kind of way that i listed the 15
24:51
edges
24:52
i want you to list all the 45 meats and
24:54
hopefully
24:55
in an organized way in a systematic
25:00
way in a way that makes it easy for
25:02
somebody
25:03
to check that you've got all of them and
25:05
in which there's a pattern hopefully
25:07
that
25:07
it's pretty clear sort of how to keep
25:09
going
25:11
so this homework is too onerous
25:15
then probably this course is not for you
25:18
but if you can say ah yes this is not so
25:21
bad i'm going to organize
25:23
my thoughts i'm going to understand what
25:25
these words mean and i'm going to write
25:27
out all 45 of these meats and i'm going
25:29
to do it and i'm going to check that
25:31
it's correct
25:32
then you have a an orientation towards
25:36
thinking carefully thinking logically
25:39
thinking in an organized way
25:40
and those aspects are sort of at the
25:42
heart of doing
25:44
pure mathematics so to summarize what
25:48
we've done today
25:49
we've introduced using a prior
25:52
discussion
25:53
of data structures crucially
25:56
introducing lists ordered sets
25:59
multi-sets
26:00
and sets we've defined first of all our
26:03
fundamental nodes
26:04
and then 15 edges and then
26:07
45 meters so we already have now
26:10
the start of a a small landscape
26:14
all built up from our six
26:17
notes
26:21
so there's going to be lots of
26:23
interesting things happening
26:25
i hope you're going to be joining me for
26:26
the rest of this course
26:28
i'm dylan walberger thanks for listening
26:41
you