The curious role of nothing in mathematics | Data Structures Math Foundations 187 @njwildberger

updated 8 years ago

The curious role of nothing in mathematics | Data Structures Math Foundations 187

Dispelling limit confusions and cheating | Sociology and Pure Mathematics | N J Wildberger

Insights into Mathematics 2024-06-19 | There are serious confusions about the role of "limits" in pure mathematics, and in this video we try to clarify the difficulties that are wide-spread in the subject, but purposely unacknowledged. We do this by explaining why there are really three, not two, kinds of "limiting" behaviour of "infinite series" in analysis.

The notion of a series satisfying the Cauchy condition plays an important role.

Video Contents:
00:00 Three kinds of limits for series
5:44 Converting a series to a sequence
8:02 Limit of a series/sequence
11:37 The Cauchy condition
15:36 False fact re convergence of Cauchy sequences
18:27 The big cheat: creating limits out of thin air

A talk at the 6th Mathematical Transgressions meeting on the role of Arithmetic | N J Wildberger

Insights into Mathematics 2024-06-04 | This is a short heads-up for a talk that I gave recently at the 6th Mathematical Transgressions meeting (June 3-4 2024) which was online and organized by Barbara Baranska from the Department of Mathematics, University of the National Education Commission, Krakow, Poland. The title of my talk was "Towards a logical rational arithmetic as a foundation for maths educations and research".

My talk is followed by an interesting talk by Lucja Farnik of the University of the National Education Commission, Poland entitled " Selected manifestations of Lakatos' quasi-empiricism in contemporary algebraic geometry". I hope you enjoy it as well.

The chair of this session was Prof. P. Blaszczyk whose work is also quite interesting in these foundational directions.

Here is the link to these talks:

youtube.com/live/yAmhpyAmsj4

Lets crack the Riemann Hypothesis! | Sociology and Pure Mathematics | N J Wildberger

Insights into Mathematics 2024-05-21 | Modern pure mathematics implicitly assumes that we are able to perform an unbounded, or infinite number of arithmetical operations in order to bring into being "real numbers" and "values of transcendental functions" such as "exp" and "log". We use these superhuman powers to reconsider J. Lagarias' equivalent reformulation of the famous Riemann Hypothesis concerned with the zeroes of the so-called "Riemann zeta function".

This allows us to resolve this famous conjecture with a computation, which is essentially similar in difficulty to the calculation of cos(7) or zeta(5) or exp(H_6) where H_6 is the 6th Harmonic number.

Fellow pure mathematicians: are we not able to think clearly about what we are actually able to do when it comes to mathematical computation? We don't need to pretend, just to obtain "pleasant results" that distort the real nature of the mathematical world. Let's rather replace philosophical justifications using obscure axiomatic frameworks with simple alignment with what our computers can do.

Video Contents:
00:00 The Zeta function
8:28 Analytic continuation
14:53 Equivalent formulation of R.H.
20:48 Lagarias' inequality and cracking the R.H.

Classical to Quantum | Complex numbers in Fourier Series and Quantum Mechanics | Wild Egg Maths

Insights into Mathematics 2024-05-10 | This is a video from the Playlist "Classical to Quantum" which is at our sister channel Wild Egg Maths. In this series we are planning on looking at a variety of topics in modern physics, particularly Relativity, Quantum Mechanics and the Standard Model, from a pure maths viewpoint.

Currently we are looking at Harmonic Analysis on Circles and Spheres, and showing how to rethink this theory using only a rational algebraic approach. This means no "real numbers", no "transcendental functions" and no "infinite processes"!

This will present a major new opportunity for both pure mathematicss and theoretical physics research, and will extend in a variety of directions.

In this video we introduce complex numbers as simplifying agents to understand just the elementary harmonic analysis on the unit circle in the plane. The Laplacian operator plays an important role, as does the notion of a harmonic polynomial (or polynumber as we prefer to call them). It will turn out that this approach allows us to completely rethink also integration theory on the circle, and in fact spheres and hyperbolas more generally.

Check out the entire Playlist for "Classical to Quantum", (Members of Wild Egg Maths channel):

youtube.com/watch?v=BL9GzEBtHrY&list=PLzdiPTrEWyz4VlOppC5CN0D0GjrjvBGKy

Video Contents:
00:00 Fourier Series and spheres
4:13 The 0 Dimensional Sphere
6:23 A containment hierarchy
10:22 Hyperboloids
13:43 Bi-polynumbers
18:05 Why complex numbers in QM?
20:23 Complex bi-polys
23:20 The complex basis
24:24 Proof of fundamental differential relation
30:34 Harmonic bi-polynumbers

Wildberger solves the twin prime conjecture!! | Sociology and Pure Maths | N J Wildberger

Insights into Mathematics 2024-04-29 | Does it really make sense to "go to infinity?", or to "take the limit of an infinite process?", or to "calculate an infinite sum?" Well in this lecture, we depart from our usually rigorous approach to pure mathematics, and accept the standard orthodoxy that we ARE allowed to do all these things, and then show that this leads to an essential collapse in number theory. Almost all the major unsolved problems can more or less straightforwardly be dealt with by simply applying this super power to make the crucial calculations to determine truth values of propositions involving an unbounded number of cases.

We initiate this exciting new chapter in number theory by solving the Twin Primes Conjecture: not by any sophisticated argument, but just by making a pedestrian computation: at least pedestrian by the current standards of modern analysis and number theory, which have no qualms in "computing cos(7) or zeta(5)" or a host of other "transcendental function values." In fact the heart of the matter is simply that the series 1/2+1/4+1/8+... ostensibly converges to 1.

Hopefully this should be a long overdue wake-up call, at least to students and young people entering the field. Are we not obliged to think clearly about what we are actually talking about in pure mathematics???

There is a much richer, more careful and beautiful mathematics out there, where everything actually makes 100% sense, and where there are multitudes of explicit examples to support our favorite notions. Our computers are soon going to be all over this terrain: but it's not too late to change gears and join the future. And in particular, start doing number theory in a more careful, interesting and ultimately rewarding fashion.

A big thanks to all my Patreon supporters!

Pure maths has painted itself into a corner | Sociology and Pure Maths | N J Wildberger

Insights into Mathematics 2024-04-23 | It is long past time that pure mathematicians as a community address the serious foundational weaknesses that beset almost all areas of the discipline outside of combinatorics and some adjacent areas. This is also hugely important for students of pure mathematics and those wishing to embark on a career either as a maths teacher or a researcher in mathematics.

Our AI machine friends/competitors will soon be breathing down our mathematical throats. Let's correct our errors and misunderstandings ourselves, before we are humiliated into doing so by our computational devices.

Ernst Machs approach to physics definitions | Sociology and Pure Physics

Insights into Mathematics 2024-04-01 | There is a curious parallel between definitional difficulties in physics and in mathematics. The Austrian physicist and philosopher Ernst Mach (1838 - 1916) advocated a particularly empirical approach to how fundamental concepts in physics ought to be introduced: by linking definitions to explicit measurements.

In this video we discuss Mach's thinking, talk about the difficulties with several fundamental elementary concepts in physics, especially the problematic issue of "mass", which has seen considerable evolution over the centuries, up to modern quantum field theory.

And then we compare the situation with the corresponding difficulties in modern pure mathematics where definitions often float freely in a thought bubble ultimately pinned down by "prior understanding" and "intuition" rather than by explicit procedures for writing down expressions.

This discussion is particularly important when we try to understand Special Relativity.

Video Contents:
00:00 Ernst Mach
2:02 Key concepts of physics
5:10 Critical role of mass
12:16 Confusions in physics and in mathematics

The speed of light c is NOT a universal constant (I) | Sociology and Pure Physics | N J Wildberger

Insights into Mathematics 2024-03-14 | Einstein's Second Postulate for Special Relativity asserts that the "speed of light" c is the same in any inertial reference frame. Unfortunately, this is not a correct statement about the world.

To understand why, we will have to go back in time to the real beginning of Relativity, with the remarkable insight of Galileo Galilei in 1638 and its dramatic implications about the nature of space and time. We'll discuss the Michelson Morley experiment, and what it does and does not actually demonstrate. Various widely held beliefs or assertions in the physics community will be examined ... somewhat more critically than usual. And we'll see that there really is a difference between us and those Klingons when it comes to "measuring the speed of light".

Chords in Parity Staff Notation | Maths and Music | N J Wildberger

Insights into Mathematics 2024-03-01 | Parity Staff Notation (PSN) is an alternate, much simplified system for annotating music, which avoids sharps and flats, and steps away from the dependence of our current system on the architecture of the keyboard.

In this video we begin by getting a deeper understanding of chords and their interval sequences, comparing traditional and PSN notations, and focusing especially on inversions.

For those with an interest, I now have dozens of TikTok musical improvisations and other informal pieces, of various levels of quality no doubt, but hopefully with something of a different flavour or sound from what you might be used to. Check it out at tiktok.com/@UCXl0Zbk8_rvjyLwAR-Xh9pQ

This is part of the Maths and Music playlist, at:

youtube.com/watch?v=c7Bn81IXPZs&list=PLIljB45xT85DNcjH7DeaXE_ojA0tHCRI1

Letting go of Inertial Reference Frames | Sociology of Physics | N J Wildberger

Insights into Mathematics 2024-02-06 | Einstein's theory of Special Relativity has at its core the notion of an "inertial reference frame". Unfortunately this is an overblown concept which immediately distorts our understanding of our position in the world, and does not jive with the reality of our experience as galactic observers.

This is especially relevant when applied to cosmological issues involving spaceships travelling at uniform velocities with respect to each other and measurements that such inertial observers can make, and deductions they can infer about the relations between these measurements.

We need to think a bit more carefully, and be prepared to reconsider some fundamental assumptions here.

This video is part of the Sociology of Physics playlist. A big thanks to all my Patreon supporters!

Parity staff notation (PSN) for music | Mathematics and music | N J Wildberger

Insights into Mathematics 2024-01-31 | Parity staff notation is a simplified musical system which is not prejudiced towards the keyboard, which removes the need for sharps and flats, which treats the treple clef and bass clef in exactly the same way, and which has the possibility of dramatically enhance our understanding of music. Happily it can be put into practice just using standard music notation, or indeed actually just a lined notebook is sufficient. We will see that scales and chords become so much simpler to understand in this system.

This is part of the Maths and Music Playlist.

Video Contents:
00:00 Parity Staff Notation (PSN)
2:51 A 12 tone approach to black and white notes
7:39 Two Staves
17 :18 Two Uniform Scales
19:38 Major Scales
24:54 Example of Scales
27:24 Crossing Between Staves

Time Contraction and length dilation in SR | Sociology in Pure Physics | N J Wildberger

Insights into Mathematics 2024-01-23 | We present a simplified Euclidean version of the mathematics behind Special Relativity, in which we are able to appreciate some of the seemingly remarkable consequences of the Lorentz transformations such as time dilation and length contraction. As the title of the video suggests, in the Euclidean case there is an interesting twist.

With a bit of geometry and linear algebra, we see that the heart of the matter is just a change of coordinates. Nothing "physical" is happening here, except that we are just developing the consequences of having different points of view on the same "underlying story". In fact we move towards letting go of there being a well-defined "underlying story": there are only points of view and the relations between them.

This is an elementary talk that hopefully will be accessible to a broad audience, even with only some basic high school mathematics.

This is part of the Sociology and Pure Maths playlist available at youtube.com/watch?v=UZah3BqsU8w&list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua

Q Series via Box Arithmetic | Math Foundations 239 | N J Wildberger

Insights into Mathematics 2024-01-10 | We have a look at the interesting topic of q-series from algebra / analysis / combinatorics / number theory from the point of view of our new "box arithmetic" which prominently utilizes anti-boxes along with boxes. This is a chance to get some more familiarity with this curious new arithmetic in which the role of "nothing" is different from what we are used to.

To interpret Euler's pentagonal formula we introduce an unbounded extension of the current Box arithmetic.

This new way of thinking appears to have obvious computational advantages, in that the notation is capturing some of the arithmetic more efficiently. It also finesses the need for philosophical discussions about "variables".

Video Contents:
00:00 Euler's Pentagonal Number theorem
3:10 Box Arithmetic
7:18 More arithmetic with boxes
10:51 Unbounded extensions
15:10 An identity of Euler
21:44 Notation for multiplicities

A skeptical look at the Special Relativity narrative | Sociology and Pure Physics | N J Wildberger

Insights into Mathematics 2023-12-14 | The usual story of Special Relativity (SR) is built from two Postulates introduced by A. Einstein in his famous 1905 paper "On the Electrodynamics of Moving Bodies" in which he subsequently derives the Lorentz transformations and introduces the mind-bending notions of length contraction, time dilation and mass expansion as (relative) speeds approach that of light. But does this story really ring true??

Here we look critically at this standard narrative, and show that there is in fact another hero to the story (!): Galileo Galilei who really had the critical understanding that underpins the subject. In fact we claim that SR is a relatively simple logical consequence of taking Galileo's Invariance Principle to heart and working out carefully and consistently the logical and necessary implications of it.

We consider also the important insights of W. Ignatowsky, along with P. Frank / H. Rothe, from around 1910 about Einstein's second Postulate. And clarify the very important distinction between the notions of "Hypotheses" and "Postulates" in pure physics. We see that there is a serious confusion currently about even the basic nature of the statements being proposed here.

And let's also boldly introduce a very uncomfortable truth: that "Einstein's Postulate" about the constancy and universality of the speed of light "c" in all inertial reference frames has serious logical issues.

There is a considerable amount of sociology surrounding this topic, and so lots to think about in this direction.

If you are interested to learn more, please have a look at my 2014 UNSW seminar entitled "Bats, echolocation and a Newtonian view of SR" along with my latest videos on Special Relativity in the "Classical to Quantum" series at the YouTube channel Wild Egg Maths. There I will be laying out in some detail an alternative, purely mathematical, approach to SR which circumvents the role of light, and clarifies that most of SR is actually an observational effect which necessarily follows from consistent application of Galileo's Principle.

A big thanks to my Patreon supporters, and to the Members of the Wild Egg Maths channel.

Go Lesson 14: A classic game between Guo Bailing and Wang Hannian (around 1600)

Insights into Mathematics 2023-12-07 | Here is my analysis of a classic early game of Go played in China around 1600 between Guo Bailing and Wang Hannian. I also give you some advice in how to improve your play as a beginner in Go. And tell you a story about a game I played with a precocious 6 - year old Japanese player.

It is important to watch good players play without even being able to understand their moves: this is a huge help in developing higher levels of skill in the game.

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

A new Staff Notation based on Parity | Maths and Music | N J Wildberger

Insights into Mathematics 2023-11-23 | Many people over the years have lamented that our current musical notation, based as it is on the rather idiosyncratic design of the piano's white and black key system, is a hindrance to musical learning and understanding. I fully concur with this view. It is long past time to consider rationally what better alternatives there are.

In this video I bring a mathematical orientation to this question, and propose a vastly simplified system, in which the parity distinction between even and odd is reflected in the visual distinction between black and white notes, in which sharps and flats are entirely removed, where the treple and bass clefs are read exactly in the same way, and where transposition from one key to another is so much simpler. This will be a major help in understanding also the structures of chords in musical theory.

In keeping with this I also propose a modification of our notation for the length of notes.

Box Arithmetic with Polynumbers | Math Foundations 238 | N J Wildberger

Insights into Mathematics 2023-10-15 | We introduce fundamental terminology and notation for Box Arithmetic, putting the duality between mathematical objects and their anti objects at the centre of the development. We also move to a more geometric, visual approach with actual boxes over inline representations.

Is this the arithmetic of the world?

Big thanks to my Patreon supporters and Members of the Wild Egg maths channel.

ChatGPT4.0 discusses real number arithmetic | Sociology and Pure Maths | N J Wildberger

Insights into Mathematics 2023-09-23 | Perhaps my fellow pure mathematicians are not really interested in engaging in a meaningful discussion about the foundations of mathematics, and especially the incredibly logically sloppy "arithmetic of real numbers" that they all like to believe in. But ChatGPT 4.0 has no such qualms!

Even though this remarkable machine is not highly trained in mathematics, it knows enough to more than hold up its end of an important discussion on what "real numbers" really are, and whether or not the current theory of arithmetic of such beasts makes sense. It goes for the Dedekind cut approach, so let's see how it meets the challenges I throw at it.

Pretty clearly we are on the cusp of an entirely new paradigm for learning about and investigating the world around us. Very exciting!

A very big thanks to all my Patreon supporters. And also to Members of my sister channel, Wild Egg Maths.

Here is a link to the chat with chatGPT: you can carry it on further if you like!

chat.openai.com/share/eeb65164-31c9-4a12-a546-6b0b21448346

Introducing (finally!) Box Arithmetic | Math Foundations 237 | N J Wildberger

Insights into Mathematics 2023-09-03 | Box Arithmetic is a new way of understanding what arithmetic, and much of associated mathematics, is actually truly about. The most important new understandings are that 1) the most powerful data structure for foundational work is not a set, and is not a list, but is rather a multiset, or mset: where elements are unordered and repetitions are allowed and 2) that the particle / anti particle duality famously discovered by Paul Dirac in 20th century physics has a deep and remarkable analog in the foundations of arithmetic. When we put these two together, we get Box Arithmetic.

Here we give an overview of how we propose to reconsider the developments arising from mset arithmetic in the light of this powerful box / anti box duality.

I would especially like to thank my Patreon supporters for their generous support over the years! Your help and encouragement has propelled me forwards in more ways than you think.

Standard Staff Notation Issues | Maths and Music | N J Wildberger

Insights into Mathematics 2023-08-20 | There are two major issues in representing music on a written page: how to systematically specify the pitch of a given note, and how to systematically specify the duration of a given note. We introduce some of the standard features of our system by looking at a famous Mazurka Op 7 No. 2 of F. Chopin.

In the Western world, our solution to the first of these issues is firmly rooted in the particular architecture of the piano, in which the white notes are seen as first class citizens, and the black notes as second class citizens. I want to stress the arbitrariness of this current system, as well as its intrinsic lop-sidedness. Unfortunately all musicians, whether or not they play keyboards, are forced to use this particular idiosyncratic system, and of course centuries of use and overfamiliarity renders it largely immune to critical examination.

The second issue of note duration is certainly somewhat more systematic, but still has some rather ad hoc elements, notably the use of the distinction between black noteheads and white noteheads to separate quarter notes from half notes (and whole notes).

However we will see that in fact interesting alternatives to both issues have been raised, and in our next video we will add our own contribution: a balanced "parity" system for music notation.

This is part of the Maths and Music Playlist: please check out the other videos in the series at youtube.com/playlist?list=PLIljB45xT85DNcjH7DeaXE_ojA0tHCRI1

You can check out my TikTok (music) channel: www.tiktok.com/@njwildberger The latest offering is my take on Frederic Chopin's Waltz Op 70 No 2 in A Major. Can amateur musicians offer up something of value? It's an interesting question which I hope to take up more in coming videos.

A big THANK YOU to all my Patreon supporters, and also to Members of this channel's sister YouTube channel: Wild Egg Maths.

Video Contents:
00:00 Introduction to staff notation
1:10 Mazurka F. Chopin
2:18 Current musical notation conventions
8:14 Main Issues: How to represent notes
13:20 Examples
14:04 How to represent length/extent of notes

Central polynumber algebra and a (baby) Weyl character formula | Math Founds 236 | N J Wildberger

Insights into Mathematics 2023-08-09 | The multiset approach to arithmetic that we are here developing can be applied to re-interpret and deepen our understanding of the 20th century's most important formula: the Weyl character formula for the (irreducible) representations of simple Lie groups and Lie algebras.

The special cases that we are going to describe connect with the two most basic ADE graphs, the graphs A1 and A2, corresponding to the representations of SU(2) and SU(3), which coincidently are arguably the two most important non-commutative Lie groups in modern particle physics and feature centrally in the Standard Model. The group SU(2) is also the three dimensional sphere which occurs in the four dimensional algebra of quaternions, which we quickly review.

We also briefly mention the great 20th century mathematician Hermann Weyl.

The Weyl character formula viewed correctly is a statement about polynumbers, not complex exponentials as found in almost all texts and papers on the subject. Irrationalities must be avoided if we are going to understand mathematics, and physics, properly!

Correction at 20:59 I mention the classsification of simple Lie algebras, and forget to include the other two families, of types B_n and C_n.

The tensor product of representations corresponds in some fashion to interactions between elementary particles, and for this the characters of the representations are very useful, as the questions reduce to computing ordinary products of these. We illustrate this in the SU(2) case of central polynumbers, showing how the Weyl character formula in this situation connections with tensor product multiplicities and can be computed using our integral polynumber algebra.

This lecture is also closely connected to some of the topics that we discuss in the Exceptional Structures in Mathematics and Physics via Dynamics on Graphs exploration series available to Members of the Wild Egg Maths channel, and also to Patreon supporters. In fact I will develop the contents of this lecture in more detail and in greater generality in that series.

Video Contents:
00:00 Central polynumbers and relations
4:02 A1 Weyl character formula
7:27 Examples of the A1 WCF
9:03 Quaternions
15:05 Hermann Weyl
20:44 The rational Weyl character formula
30:12 Corresponding characters

Scale Adjacency, Sharps and Flats | Maths and Music | N J Wildberger

Insights into Mathematics 2023-07-26 | The almost uniform property of the major scale that we discussed in our last video has an important consequence which is responsible for the current staff notation system of using sharps and flats to create key signatures for the 12 major keys.

Here we look at this from a mathematical point of view, using 12 tone chromatic scale notation. The circle of fifths or fourths, which is better described as the circle of 7 steps or 5 steps, naturally makes an appearance. And there is an interesting connection with the geometry of the clockface.

This video anticipates a wider ranging discussion about alternative ways of notating music.

This is indeed part of the Maths and Music Playlist: please check out the other videos in the series at youtube.com/playlist?list=PLIljB45xT85DNcjH7DeaXE_ojA0tHCRI1

Artwork created with the help of Midjourney, an amazing Bot that stands ready to put half of the planet's graphic designers out of business in the next few years. It's a new world coming, and us pure mathematicians are not immune, as we shall discuss in our Sociology and Pure Maths series!

You can check out my TikTok (music) channel: www.tiktok.com/@njwildberger

And a big Thank You to all my Patreon supporters at patreon.com/njwildberger

Video Contents:
00:00 Math: three sets from {1 2 3 4 5}
3:42 Circle of Adjacent Scales
12 :36 Major Scale
18:32 Using Traditional Notation
21:04 Sharps and Flats

Central polynumbers and SL(2) / SU(2) characters | Math Foundations 235 | N J Wildberger

Insights into Mathematics 2023-07-13 | Let's consider a novel approach to the representation theory of the Lie groups SL(2) and SU(2), which play a major role both in mathematics and physics. We give an elementary algebraic approach to this story which is, to my knowledge, not found in any of the many standard texts and articles which try to explain this subject. The general polynumbers C_n which appeared in the last lecture turn out to play a truly central role in this subject.

We are utilizing our mset approach to arithmetic, augmented by the particle / anti particle duality which we have taken from modern physics and placed centrally in our arithmetic with integral polynumbers.

This orientation is strongly motivated by our insistence that pure mathematics be done correctly! In other words, that "completed infinite processes" and the associated fantasy of "arithmetic with real numbers" have to be avoided; and so we want to frame everything in terms of rational numbers and their complex number extensions. Lots to think about here for students of both Lie theory and modern physics.

Alternating / symmetric polynumbers: a missing chapter of Algebra | Math Foundations 234 | N J W

Insights into Mathematics 2023-06-15 | We introduce subtraction into the world of arithmetic with integral polynumbers. This presupposes prior familiarity with negative numbers, which in earlier videos we introduced via the basic duality between msets and anti msets, or just amsets. The fundamental reflection symmetry denoted by sigma between natural numbers and their negatives allows us to define symmetric and anti symmetric or alternating polynumbers. This gives us an arena for very fundamental yet elementary investigations into Algebra which have largely been missed by our educational system focused, perhaps too much, on functions due to their role in analysis.

We introduce the basis B_n of symmetric polynumbers, as well as closely related basis A_n of alternating polynumbers. The algebraic relations between these become important when we investigate a third class C_n of central polynumbers which play a big role in many areas of mathematics, including q-series, representations of SU(2), hypergroups, quantum groups, and in physics also.

Video Contents:
00:00 The subtraction operation
04:46 Fundamental reflection symmetry
06:32 Symmetric Binomials B_n
09:17 Product formula for symmetric binomials
11:24 Alternating binomials A_n
14:14 Products of Alternating & Symmetric
19:40 Central polynumbers C_n
22:33 The Central Importance of central polynumbers

The major scale is almost uniform (and 42) | Maths and Music | N J Wildberger

Insights into Mathematics 2023-06-01 | Building from the discussion of uniform scales in our last video in this series, we show that the major scale, which is the central framework of modern western music, actually has a somewhat curious property of being "almost uniform". This feature of the major scale turns out to be intimately connected with our current staff notation involving sharps and flats to give "key signatures" to all 12 major, and minor, scales.

Again we lean heavily on our arithmetical, mod 12 approach to the chromatic 12 tone scale. There is an interesting mathematical question which arises from this discussion.

You can check out my TikTok (music) channel: www.tiktok.com @njwildberger

A big thank you to my Patreon supporters!

The question that I raise here about finding all the 7 note scales which have this almost uniform property has been answered by Federico Rocca, a frequent contributor to this channel and also the Wild Egg Maths channel. In his comment, pinned below, you can find the following list of interval sequences (in lexographic order) along with the list of 12 overlaps generated by sequential translations. He used a Python program to generate this: perhaps others might check it? [This is a "mathematical / musical classification": can you think of any others?]

Interval [1, 1, 1, 1, 1, 1, 6]: (7, 6, 5, 4, 3, 2, 2, 2, 3, 4, 5, 6)
Interval [1, 1, 1, 3, 1, 1, 4]: (7, 5, 3, 2, 3, 5, 6, 5, 3, 2, 3, 5)
Interval [1, 1, 1, 3, 1, 2, 3]: (7, 4, 3, 4, 3, 4, 6, 4, 3, 4, 3, 4)
Interval [1, 1, 1, 3, 2, 1, 3]: (7, 4, 3, 4, 3, 4, 6, 4, 3, 4, 3, 4)
Interval [1, 1, 1, 4, 1, 1, 3]: (7, 5, 3, 2, 3, 5, 6, 5, 3, 2, 3, 5)
Interval [1, 1, 2, 1, 3, 1, 3]: (7, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4)
Interval [1, 1, 2, 2, 1, 1, 4]: (7, 4, 4, 2, 4, 4, 6, 4, 4, 2, 4, 4)
Interval [1, 1, 2, 2, 1, 3, 2]: (7, 3, 4, 4, 4, 3, 6, 3, 4, 4, 4, 3)
Interval [1, 1, 2, 2, 2, 2, 2]: (7, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2)
Interval [1, 1, 2, 3, 1, 2, 2]: (7, 3, 4, 4, 4, 3, 6, 3, 4, 4, 4, 3)
Interval [1, 1, 3, 1, 3, 1, 2]: (7, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4)
Interval [1, 2, 1, 2, 1, 2, 3]: (7, 3, 3, 6, 3, 3, 6, 3, 3, 6, 3, 3)
Interval [1, 2, 1, 2, 1, 3, 2]: (7, 3, 3, 6, 3, 3, 6, 3, 3, 6, 3, 3)
Interval [1, 2, 2, 1, 2, 2, 2]: (7, 2, 5, 4, 3, 6, 2, 6, 3, 4, 5, 2) --- The major

There are some interesting features about this list, which I will discuss in a future video. Note for example that only four of the fourteen overlap sequences does not have a 6 centrally placed. Note also that the major scale is the only one on the list with interval sequence consisting of just 1's and 2's.

Video Contents:
00:00 Moving towards ley signatures and staff notation
00:10 Understanding and analysing musical notes
00:34 The key of D and a Prelude of J S Bach
02:09 Diminished and uniform scales
06:09 Major scale is almost uniform! (and 42)
10:10 Translates of minor scale I(s)=[2,1,2,2,1,3,1]
12:24 Translates of another scale I(s)=[2,1,2,1,2,2,2]

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF

The curious world of integral polynumbers | Math Foundations 233 | N J Wildberger

Insights into Mathematics 2023-05-18 | Arithmetic with multisets, or msets, incorporates negative numbers and integral arithmetic by employing the particle / anti-particle duality that has played such a big role in 20th century physics, pioneered by the great British physicist Paul Dirac.

Once we have integers, we can expand our arithmetic of polynumbers or polynomials in two different ways: one is to introduce integers as possible coefficients, and the other is to introduce them as possible powers, or exponents. This gives us a very different, perhaps even surprising, approach to this larger integral polynomial arithmetical world. We want to think about these objects here not in a calculus view, where a polynomial is considered as a special kind of "function", but rather in a more purely algebraic way involving again ideas from physics in the form of discrete mass distributions or measures.

In this way multiplication corresponds to traditional convolution of measures. And the entire story supports an important new kind of symmetry, that will play a major role in the further development.

NOTE: There's a typo with the last number of the first poly (a 4 instead of a 3) in the first slide: it should be 2+x+3x^4=
__
| 2
| 1
| 0
| 0
| 3 not 4 (thanks Federico Rocca!)

Video Contents:
00:00 Introduction
01:33 Integral Polynumbers and Ipoly
04:08 Integers as coefficients
07:37 BIG difference between Integral Polynumbers and functions
14:31 In algebra: avoid the "function " concept!
19:11 IPoly as measures/ densities
25:14 RPoly : Rational Polynumbers
29:40 The reflection symmetry
34:21 Symmetric /alternating integral polynumbers

Uniform scales and group theory (mod 12) | Maths and Music | N J Wildberger

Insights into Mathematics 2023-05-07 | We apply some elementary group theory to study the 12 tone chromatic scale and its subgroups, which correspond to uniform scales. The fact that the number 12 is so highly divisible strongly influences the musical possibilities here. Besides the notion of a subgroup, the related concept of a coset of a subgroup also plays an important role.

This is part of a Playlist where we look at the intimate relations between mathematics and music. This video is a very clear example of the power of some abstract mathematics to clarify what is going on with structures in music.

Video Contents:
00:00 Scales with a particularly uniform property
03:13 Corresponding subgroups and scales
03:40 The generating subgroup (chromatic scale)
04:14 The two-step uniform scale
05:12 The three-step uniform scale (diminished)
06:00 The four-step uniform scale (augmented)
06:46 The 6-step uniform scale
07:20 Uniform scale
08:00 Cosets of a group and translations

Chords and the Mathematical Fretboard | Maths and Music | N J Wildberger

Insights into Mathematics 2023-04-10 | We look at how the 12 tone chromatic scale system provides a powerful tool for learning to navigate around a guitar and to identify and locate chords, with particular emphasis on triads and tetrads (four note chords). This is a step away from the current piano-focused system of musical nomenclature and terminology which is fundamentally at odds with the more translation invariant symmetry encompassed by the modern guitar.

It will also be an important step as we move to consider more general and canonical ways of applying mathematical notions to simplify and rationalize our understanding of music and its theory.

Video Contents:
00:00 Introduction
01:51 Basic structure of a guitar and the fretboard
09:19 From one string to the next
21:26 Definition of a chord
22:14 Chords with three notes: Triads
26:03 Chords with four notes: Tetrads
29:13 Types of triads
33:42 Finding triads on the fretboard

New Directions for Mathematics Education and Research | Channel Trailer 2023 | N J Wildberger

Insights into Mathematics 2023-03-26 | Welcome to "Insights into Mathematics" -- a YouTube channel devoted to careful explanation of a wide range of mathematical topics, in detail and with lots of examples. From N J Wildberger, Honorary Professor at the University of New South Wales (UNSW) in Sydney Australia, and an unorthodox voice advocating a much stronger logical structure and re-evaluation of many standard topics.

In this video, you get an overview of the Channel, along with a summary of my concerns about modern pure mathematics research, and some specific examples of why the current orthodoxy doesn't really work, despite huge amounts of wishful thinking on the part of the professoriate. A key point is that our connection with computational reality is ever-widening, and getting to the point of being stretched way to far. Ultimately our conceptual theories, no matter how abstract, have to be translatable into computational form.

There IS a new, more logical, and coherent mathematics out there waiting to be discovered, developed, and applied to make a better framework for mathematics education; and this YouTube Channel, and our sister channel Wild Egg mathematics courses, is at the heart of this exciting restructuring.

A big thank you to all my Patreon supporters, and also to Members of the Wild Egg Maths channel at youtube.com/c/WildEggmathematicscourses.

Video Contents:
00:00 Introduction to Insights into Mathematics
05:55 Pure maths (2023) does not work logically
14:33 "Real number arithmetic " is a dreaming

Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

More arithmetic with negative msets | Math Foundations 232 | N J Wildberger

Insights into Mathematics 2023-03-14 | Let's illustrate this new box arithmetic using anti msets to define negative numbers, with some more explicit examples. This involves using integers not just for coefficients of polynomumbers / polynumbers, but also to crucially extend the possibilities for exponents to negative numbers. This introduces a vital symmetry into the world of algebra with polynumbers that is almost entirely missing from usual algebra courses: where essentially Laurent polynomials appear not in a calculus / analysis context, but as pure objects of analysis.

This will have major consequences for our understanding of algebra going forward.

A big thank you to all my Patreons supporters and Members of the Wild Egg mathematics courses channel.

Video Contents:
00:00 More Arithmetic with negative msets
04:26 Anti-objects!
07:14 Negative msets
09:09 Examples of integer arithmetic
13:06 Examples of integer polynumber arithmetic
19:42 Multiplicative arithmetic with integer polynumbers
28:55 Integral (Laurent) polynumbers
32:57 Arithmetic with integral polynumbers

Pitch, Saxes, and Transpositions | Maths and Music | N J Wildberger

Insights into Mathematics 2023-03-03 | Tones, notes and pitches are subtly different concepts. In music theory, the crucial role of transposition motivates us to a flexible approach to the musical duo decimal system of note naming. One way that this arises is through the various different keys that band and orchestral instruments come in. In particular, the saxophone family comes in 5 different versions, from soprano which is a B flat instrument to alto saxophone, which is an E flat instrument, all the way to a bass saxophone, which is again a B flat instrument.

This discussion is leading us inevitably to a major re-evaluation of our staff notational system for writing music on the page.

Please check out my TikTok (music) channel: www.tiktok.com/@njwildberger where I muse on alternate, improvised directions for (piano) music.

A big shout-out to my Patreon supporters and Members of the Wild Egg mathematics courses channel! Thanks so much.

Video Contents:
00:00 Introduction
00:34 Absolute pitches
01:24 Middle C tuning and powers of 2
03:35 Saxophone tuning and relative pitch
06:32 Transpositions
09:00 Notational system and transpositions

***********************
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 researchgate.net/profile/Norman_Wildberger

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 openlearning.com/courses/algebraic-calculus-one 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 patreon.com/njwildberger Your support would be much appreciated! :)

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are the Wild Egg Maths Playlists:

Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м

Tones, notes, duodecimals: and a Eureka moment for arithmetic | Maths and Music | N J Wildberger

Insights into Mathematics 2023-02-26 | Can music teach us something about basic arithmetic? In this video we will begin to appreciate that the answer might be: YES! We start by making the distinction between tones (of which there are twelve possibilities in our Western system) and notes, which are associated rather to frequencies and are in principle unbounded as we go up and down the keyboard (or fretboard etc).

We've seen that tones are best named in a duodecimal fashion, and then the question arises: how do we best name notes? This is an important issue that has not been given enough thorough attention in musical circles. However by clarifying it, we are let to a "eureka" moment where we suddenly realize that our own familiar arithmetic has a crucial bias in it towards negative numbers, and that there is actually an interesting "musical" alternative for negative numbers that ought to be given careful consideration. There are bound to be lots of potential repercussions from this simple insight!

To check out my TikTok (music) channel: www.tiktok.com/@njwildberger

A big shout-out to my Patreon supporters and Members of the Wild Egg mathematics courses channel! Thanks so much.

***********************
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 researchgate.net/profile/Norman_Wildberger

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 openlearning.com/courses/algebraic-calculus-one 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 patreon.com/njwildberger Your support would be much appreciated! :)

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are the Wild Egg Maths Playlists:

Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м

Modes of Minor and other Scales | Maths and Music | N J Wildberger

Insights into Mathematics 2023-02-06 | We use the mathematical 12 tone chromatic scale as a framework for an expanded view of modes in music, focusing first on minor scales, and then branching out to other scales. This frees us from the somewhat limiting pre-occupation with the usual Dorian, Lydian, Mixolydian etc modes for the major scale.

We use the notation introduced in the last videos for describing general scales and the associated interval sequences. Some basic mathematical concepts relating to transformations, typically rotations and powers are found to be useful here in this purely musical context.

And we start to see some non-trivial mathematics emerging from this purely musical set up also. Some interesting classification questions are just around the corner from this thinking. So Maths and Music do really interact with each other!

This is indeed part of the Maths and Music Playlist: please check out the other videos in the series at youtube.com/playlist?list=PLIljB45xT85DNcjH7DeaXE_ojA0tHCRI1

Artwork created with the help of Midjourney, an amazing Bot that stands ready to put half of the planet's graphic designers out of business in the next few years. It's a new world coming, and us pure mathematicians are not immune, as we shall discuss in our Sociology and Pure Maths series!

You can check out my TikTok (music) channel: www.tiktok.com/@njwildberger

Video Contents:
00:00 The importance of scales
00:50 The 12 tone clock - Major scale
02:14 Rotating an interval sequence (rho)
04:09 Quick review of modes of a major scale
06:40 Modes of a minor (harmonic) scale
10:15 Names of modes of a scale type
13:50 Examples of obtaining modes of a scale

***********************
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 researchgate.net/profile/Norman_Wildberger

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 openlearning.com/courses/algebraic-calculus-one 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 patreon.com/njwildberger Your support would be much appreciated! :)

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are the Wild Egg Maths Playlists:

Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м

Negatives numbers from anti msets | Math Foundations 231 | N J Wildberger

Insights into Mathematics 2023-01-30 | 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 youtube.com/watch?v=EbSJwDphAb8 and the subsequent ones.

The big mathematics divide: between exact and approximate | Sociology and Pure Maths | NJW

Insights into Mathematics 2023-01-26 | Modern pure mathematics suffers from a major schism that largely goes unacknowledged: that many aspects of the subject are parading as "exact theories" when in fact they are really only "approximate theories". In this sense they can be viewed either as belonging more properly to applied mathematics, or as being essentially provisional; awaiting a more precise and logically viable treatment.

This crucial distinction actually cuts across many areas of modern pure mathematics. It starts of course with arithmetic, and the difference between counting and measurement, that is between intrinsically exact and approximate evaluations, but appears also in modern notions of algebra, topology, function theory, number theory and many other disciplines.

In this video we give an introduction to this important distinction, culminating in some unsettling thoughts about the logical validity of the "Riemann zeta function" and that most revered unsolved problem in pure mathematics: the Riemann Hypothesis.

This video is part of the Sociology and Pure Maths playlist found at youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua

The Playlist on Solving Polynomial Equations (yes we really do it!) is at Wild Egg mathematics courses YouTube channel: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6

If you are interested in this topic, you might like to have a look at Curt Jaimungal's interesting interview with Prof Richard Borcherds, and my comment on that video re his response to a question I raised about the inexactness of "real number arithmetic". This video is at Curt's TOE channel: youtube.com/watch?v=U3pQWkE2KqM&t=1413s

Video Content
00:00 Exact versus approximate in mathematics
1:39 Associating applied maths to approximate values
4:58 Solving equations and ''real numbers''
12:05 Topological spaces
21:08 Functions
26:59 Number theory sigma and zeta functions
34:23 Riemann hypothesis issues

The mathematics of Scales and Modes | Maths and Music | N J Wildberger

Insights into Mathematics 2023-01-19 | We introduce a mathematical framework for understanding general scales and their modes. This is based on the 12 - tone chromatic scale, and is not restricted to major or minor scales. There is a natural connection with clock arithmetic and indeed also with the discrete calculus, as the difference operator on sequences plays a natural role.

You can check out my TikTok (music) channel: www.tiktok.com/@njwildberger

Video Content:
00:00 The 12 tone clock scale
2:51 Mathematical analysis of scales modes
5:53 What is a scale?
9:41 Interval sequence of a scale
15:04 When two scales are of the same type
19:41 Simple observation of a scale
23:33 Modes of a scale

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Multiset arithmetic via trees | Math Foundations 230 | N J Wildberger

Insights into Mathematics 2023-01-08 | We introduce an alternative visual representation of pure msets as rooted (or roofed) trees as in combinatorics or computer science. This allows us to see the counting functions in a simple geometric way, and gives us a different view of the interesting new arithmetic supported by multisets.

We illustrate this arithmetic with some concrete examples, and also introduce some useful shortcut notations that help us navigate in this new domain.

Video Content:
00:00 Introduction
5:01 Notation for rooted/roofed trees
7:03 Multiplicity convention (left subindices)
10:39 Counting with trees
15:22 Closed boxes/nodes
20:25 Arithmetic with trees

***********************
Here are the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

The operation of caret / exponentiation (new!) via multisets | Math Foundations 229 | N J Wildberger

Insights into Mathematics 2022-12-31 | We want to extend our ladder of operations: addition, then multiplication then to a new higher operation which we will call "exponentiation" or better yet following the suggestion of viewer Cristopher Game simply “caret” in accord with the wedge or caret symbol.

This is in the context of a somewhat new arithmetic with multisets that we laid out in the two previous videos (MF 227 and MF 228), and that does not suppose any familiarity with prior arithmetical systems.

Along the way in this video we introduce an inductive sequence of "counting functions" that correspond to the hierarchy of Zero to Nat to Poly to Multi that we have established so far. The inductive / recursive aspect of this approach means that it is pretty easy to see how to extend what we are doing to further higher "levels".

We try to illustrate this new way of thinking about arithmetic in a concrete way with lots of examples.

PLEASE NOTE: in subsequent lectures I will shift to the use of the "caret" terminology: that means that what we are calling "the exponential of A and B" in this lecture will be replaced with "the caret of A and B". So we will use caret also as a verb: "to caret multisets", in the same spirit as "to add multisets" or "to multiply multisets". We will not use carrots however.

Happy New Year 2023!

Video Content: (thanks to phi Architect)
0:00 Introduction - historical context of sets
2:20 Recursive structures in msets
4:30 Review of types
7:00 Elements of an mset
7:45 General mset notation
8:50 Examples of msets and operations
10:45 Counting functions on pure msets
11:30 Def (Z)
11:45 Def (N)
13:40 Def (P)
14:50 Def (M)
19:15 Operations on msets - addition, multiplication
20:45 Caret operation - exponentiation
26:30 A slightly more complicated example
29:40 Properties of the caret operation
34:30 It's all trees

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

The big step from polynumbers to multinumbers!! | Math Foundations 228 | N J Wildberger

Insights into Mathematics 2022-12-25 | We continue to develop a new kind of arithmetic, based on the data structure called a multiset, or mset for short. After a quick review of the last lecture, we investigate how to move from polynumbers, which are the embodiment of polynomials in one variable in this context, to multinumbers.

This turns out to be a convenient framework for algebra over many variables, with an accompanying more flexible attitude towards notation. Our reward is a large scale enlargement of what "arithmetic" actually means. Computer Science, here we come!

A big thanks to all my Patreon supporters, and to Members of the Wild Egg mathematics courses channel.

Video Contents: (thanks to phi Architect)

00:00 introduction
2:10 Pure msets are formed with only [ ]
3:50 Basic principle: pure msets can be described completely, unambiguously
7:00 Operations on pure msets
7:20 Addition: dump the contents of added msets into a new mset
7:50 Multiplication: add distributed combinations of the contents of msets
10:15 Modifying polynumber terminology / notation
13:00 α0 ≡ [ 1 ]
14:00 α1 ≡ [ α0 ]
14:40 α1 is the first multinumber that is not a polynumber
15:00 Basic arithmetic with polynumbers
20:00 But there are more multinumbers!
21:05 More arithmetic examples
30:00 Algebra in variables - α0, α1, α2, ... - extend Poly to Bi Poly
33:20 creating a tight framework for Algebra
34:18 Next: on a strange vessel on uncharted waters?

***********************
Here are the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

A multiset approach to arithmetic | Math Foundations 227 | N J Wildberger

Insights into Mathematics 2022-12-17 | 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

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Dynamics on Graphs 20 | Rethinking Standard Model (physics) numerology | N J Wildberger

Insights into Mathematics 2022-12-13 | This is an introduction to the eight-fold way of particle physics, which led to the theory of quarks by Gell-mann and Zweig around 1964 and is at the heart of the current Standard Model.
Our aim is to eventually show how the numerology of this theory, meaning the positioning of particles with respect to quantum numbers connecting with the representation theory of the compact Lie group SU(3), has ultimately a much simpler description in terms of the Mutation and Numbers games on the graph A2, along with some associated elementary geometry and algebra.

As an introduction, in this video we give a quick overview of the Standard Model as it arose in the 1960's, including a discussion of quantum numbers and terminology covering Fermions, Bosons, Hadrons, Baryons, Leptons and Mesons. Then we present three important families of particles: the Baryon Multiplet or Octet, the Meson Nonet and the Baryon Decuplet. Each of these will be exhibited in a novel way that should be of interest to physicists.

The ongoing series on Dynamics on Graphs and Exceptional Structures in Maths and Physics can be found at our sister channel Wild Egg mathematics courses, and is available to Members of that channel at youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM

These videos are also accessible to Patreon supporters. Please consider becoming a supporter at patreon.com/njwildberger

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Introducing Modes of the major scale | Maths and Music | N J Wildberger

Insights into Mathematics 2022-11-29 | Let's introduce the classical modes of the major scale: Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian. But we want to do this in two different ways!

In our next video, we will be wanting to generalize this construction considerably, and also to free it from this current "piano centric" point of view. But this is at least a good start.

Careful viewers / listeners will note that I messed up some of my "rock and roll examples" for some of these modes. I admit to not having a lot of practice in these alternate musical universes -- but that's perhaps what makes them interesting. Can you find out where my mistakes were?

Video Contents:
00:00 Introducing modes
00:42 Scales, major scales
02:34 Modes using notes from a major scale
03:40 Rock and roll chords in different modes
09:02 Mixolydian ,Dorian, Lydian modes
10:36 Phrygian mode

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

A more rational / integral Scale Notation | Mathematics and Music | N J Wildberger

Insights into Mathematics 2022-11-07 | Let's use our logical mathematical notation for the 12 tones of the chromatic scale to discuss various important scales that are found in a variety of cultures and idioms, including the major, minor, blues, pentatonic and Indian or uniform scales. But we have to let go of the unfortunate "whole tone" and "half-tone" or "semitone" terminology. But this has important advantages in clarifying the underlying mathematical patterns behind a lot of music.

This is part of the Mathematics and Music Playlist at this channel. I would like to thank all my Patreon supporters for their continued help and encouragement --- your support is really appreciated ! :)

Here is a link to that Tik Tok improv video where I use the uniform 2 step scale:
vt.tiktok.com/ZSRvoX3Ha

CORRECTION: The minor scale that I discuss here is the harmonic, not the melodic, minor scale.

Video Content:
00:00 Moving beyond the chromatic scale
03:18 Major and minor scales
05:27 Pentatonic scale
08:32 Uniform 2- Step scale

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Equality in Computer Science: with Michael Reynolds | Sociology and Pure Maths | N J Wildberger

Insights into Mathematics 2022-11-03 | This is a conversation with Michael Reynolds on some fundamental issues for programming and mathematics related to equality, assignment and conversion. Computer science is a powerful alternative framework for setting up solid foundations for pure mathematics, with many advantages over the current set-up, which is largely built up from philosophy.

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

The many faces of equality (in mathematics) | Sociology and Pure Maths | N J Wildberger

Insights into Mathematics 2022-10-28 | We have a look at some of the many subtly different uses of the "equal sign" in mathematics. This is particularly pertinent to the foundational issue of the difference between "description definitions" and "specification definitions" of mathematical objects, which occurs when we start to consider "infinite sets" and the mathematics supposedly built up from them.

It also connects indirectly to long standing questions in philosophy having to do with identity and being.

Many thanks to all my Patreon supporters! Your support is much appreciated.

Video Content:
00:00 Introduction
2:02 Usages of '=' in maths
4:14 Assigning a value to a ''variable''
6:36 Definition of quantity
8:26 Definition (of math object)
11:31 Specification (of math object)
13:16 Other variants

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Clock Mathematics and the 12 tone scale (chromatic) | Maths and Music | N J Wildberger

Insights into Mathematics 2022-10-21 | In this video we elaborate on the central importance of the 12 tone chromatic scale -- transcending the usual preoccupation in western music with "major" and "minor" scales -- and also the natural naming system for tones that we introduced last time.

We want to connect with the natural and familiar geometry of the clock and associated mod 12 arithmetic, and tabulate the various common intervals and our new names for them.

While this is all somewhat elementary, this does have the potential for a serious re-evaluation of not just standard music theory but also the relationship between mathematics and music. In particular we see possibilities to engage many students with mathematicss through their prior interest in music.

We also look briefly at the history of mathematicians contributing to the theory of music. Quite a few famous mathematicians, including Descartes, d'Alembert, Euler and others engaged with the theoretical aspects of arithmetic. So there is a lot of precedence for this attempt to rethink some of the basic structure of music.

The basics of "mod 12 arithmetic" are also reviewed for an audience without necessarily a mathematics background. So some of my regular viewers will find some of the discussion here rather simple and obvious. However we are setting the stage here for some really major new insights into mathematical notation as well which we will meet more extensively in future lectures.

Finally: a big thank you to all my Patreon supporters -- your help is so much appreciated!

Video Content:
00:00 Introduction
3:23 The twelve note chromatic scale
8:28 Ancient Greek mathematics
12:06 Intervals
15:06 Interval notation
22:31 Mod 12 arithmetic
28:17 Usefulness of numerical approach to musical notation

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

The Scientific Method and the question of Infinite Sets | Sociology and Pure Maths| N J Wildberger

Insights into Mathematics 2022-10-13 | Let's get some kind of serious discussion going about the differences in methodology and philosophy between the sciences and mathematics, and how these differences manifest themselves in the attitude towards the logical foundations of mathematics.

In particular we look at a bulwark notion of 20th century logical axiomatics: that the concept of "infinite sets" is meaningful, that there are such things, and that they necessarily form the foundation for much of modern pure mathematics. Is it possible to have a rational, scientific discussion about these questions?

This is one of a series of lectures on the Sociology of Pure Mathematics, inviting social scientists to have a go at unravelling some of the arcane goings on in this hallowed discipline, and which reflect very strongly on the education of tens of millions of young people around the world.

A big thanks to all my Patreon supporters --- your encouragement is much appreciated!

Video Contents:
00:00 Introduction
01:56 The Scientific Method
02:30 Hypothesis
03:10 Scientific theories as predictive
04:13 Science broadens as new facts come to light
04:44 A framework for mathematics?
07:00 Criteria for establishing truth in science

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Rethinking musical note naming from a mathematical point of view | Maths and Music | N J Wildberger

Insights into Mathematics 2022-10-08 | Let's step back a bit from our piano focused notation for naming musical notes, and adopt a more mathematical, logical terminology. This will have major advantages to our understanding of the structure of intervals, scales, modes and chords going forward. We use this notation to discuss the intervals that compose the most familiar triads, the major, minor, diminished and augmented triangles.

Video Contents:
00:00 Structure of music from a mathematical point of view
00:58 Ratios of frequencies
02:25 Current naming convention
08:13 Perfect fifth /intervals
10:00 Chords and triads
12:09 Minor triads
14:03 Augmented triads

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Advice | Formal power series and the miraculous relation between E (exp) and L (log) | Wild Egg

Insights into Mathematics 2022-10-05 | We advocate moving from "functions" as the central objects of analysis to "formal power series" or "polyseries". In some sense this goes back to Euler and even Newton, and provides a powerful combinatorial alternative requiring no waffling about "evaluations" at "real numbers" that cannot in fact be explicitly computed. The painful reality is that the current over-reliance on "transcendental functions" is largely wishful thinking, obviously unsupported by the computational capacity of our machines.

A central object in this new story, involving formal power series/polyseries, are the exp and log polyseries, and here we introduce grounded variants called E and L, which are miraculously inverses with respect to composition. Demonstrating this involves the formal Faulhaber Derivative D that plays such a big role in the Algebraic Calculus One course.

In fact this material is very much in the direction of the Algebraic Calculus Two course, which is under preparation.

This lecture is part of the Exploring Research Level Maths series of videos, available to Members of the Wild Egg Channel, which includes Playlists on Dynamics on Graphs, The Hexagrammum Mysticum, Algebraic Calculus and Curves and general Advice. In the Advice Playlist, we are currently looking at Orthogonal Polynomials in a novel two dimensional number theoretical way using maxel theory. This video is part of that development.

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Go Lesson 13: Honinbo Sansa VS Rigen (~1600) | Playing Go | N J Wildberger

Insights into Mathematics 2022-09-27 | A classic game from Imperial Japan -- the Shogun era, between two prominent players, Honinbo Sansa versus Rigen. The game was played sometime in the 1600's. It features quite fierce fighting between rival groups trying to surround and kill each other. Despite both players being seemingly on the edge of collapse at various points, the balance is maintained and much beautiful play emerges.

I give a commentary on this game, concentrating on issues that are understandable for a beginner.

This is part of the Playing Go series of lectures on this channel.

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-