## Archive for the ‘Mathematics’ Category

### Category Theory

June 10, 2019 I’ve been interested in Category Theory for a substantial number of years, but have never dedicated the time necessary to learn it properly. But it seems to me that these are great days (salad days?) for learning at least the rudiments of the subject. There are now wealths of copious materials available on-line for free for one’s self-study and enrichment, as well as new and classic treatises on Categories and their theory.

Category Theory has been called “abstract nonsense”, but while it is very abstract, it is hardly nonsense. Like set theory, it can be used to study the foundations of mathematics. Like algebra, it can be used to study generalized structure and relationships in math. Like the assortment of tools that is computer science, it can be used to study the essence of logic and computation. And like calculus, it can be used for all sorts of applied and scientific purposes.

Like all good math, Category Theory (CT) generalizes concepts that lie at the heart of many branches of mathematics. These concepts allow the mathematician to see similarities between these different branches, and carry them over into others as well. You might think “maths” is all one thing, and it is, roughly, but like science, it has evolved into a myriad of subjects and specialities. Ontologically (or perhaps even categorically), CT is listed as a topic under algebra, and it in turn has its own distinct branches.

And like all good math, CT benefits from a judicious choice of definitions and properties, that balance generality with precision to great expressive advantage. This balance between abstraction and concreteness gives it the power and utility that it has enjoyed for the better part of seventy-five years. This makes CT rather a new-comer to mathematics, but please don’t mistake youth for lack of expertise.

If I wanted to represent CT emblematically, I might suggest the diagram above. At root, a category merely consists of a collection of objects and the pairwise morphisms (or arrows) between them. But additionally, the arrows must also obey a small set of conditions, so the objects usually have a similar nature. This nature can be quite abstract though, and one of the most familiar “concrete” categories is that of sets and the functions between them.

If arrows are the mappings between objects, then “functors” are mappings between categories, that once again have to obey some rules to maintain structure. So you can think analogically that arrows are to objects as functors are to categories and you wouldn’t be too wrong. Next, you can imagine generalizing to a higher level that there are mappings between functors, and “natural transformations” are indeed defined to be so.

There are many types of entities and characters that inhabit the theory that serve to propel the plot threads along. Some turn out more important than others, but most are essential to the overarching tale. Mathematics as storytelling? What an interesting and novel concept!

https://www.math3ma.com/blog/what-is-category-theory-anyway

Why Category Theory Matters

https://en.wikipedia.org/wiki/Lists_of_mathematics_topics

https://en.wikipedia.org/wiki/Outline_of_category_theory

https://en.wikipedia.org/wiki/Category_(mathematics)

https://en.wikipedia.org/wiki/Category_theory

https://plato.stanford.edu/entries/category-theory/

https://ncatlab.org/nlab/show/category+theory

https://bartoszmilewski.com/

Rina Zazkis, Peter Liljedahl / Teaching Mathematics as Storytelling

[*6.180, *11.104]

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### The Lambda Cube

April 25, 2019

In mathematical logic and type theory, the λ-cube is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new way of making objects depend on other objects, namely

1. terms allowed to depend on types, corresponding to polymorphism.
2. types depending on terms, corresponding to dependent types.
3. types depending on types, corresponding to type operators.

The different ways to combine these three dimensions yield the 8 vertices of the cube, each corresponding to a different kind of typed system.

So in the diagram above, we have emblazoned the names of these type systems ordered from lower left to upper right:

• λ→: the simply typed lambda calculus, our base system
• λ2: add 1. above to λ→, giving what is also known as System F or the Girard–Reynolds polymorphic lambda calculus
• λP: add 2. above to λ→
• λ_ω_: add 3. above to λ→
• λP2: combine 1. and 2., λ2 and λP
• λω: combine 1. and 3., λ2 and λ_ω_
• λP_ω_: combine 2. and 3., λP and λ_ω_
• λC: combine 1., 2., and 3., giving the calculus of constructions

https://en.wikipedia.org/wiki/Lambda_cube

http://www.rbjones.com/rbjpub/logic/cl/tlc001.htm

https://en.wikipedia.org/wiki/Calculus_of_constructions

https://en.wikipedia.org/wiki/System_F

[* 11.86, *11.87]

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### Recipe for Mathematics

April 3, 2019 Guided only by their feelings for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefullness.

— E. T. Bell

A smile fell on the grass.
Irretrievable!

And how will your night dances
Lose themselves. In mathematics?

— Sylvia Plath, from The Night Dances

https://en.wikipedia.org/wiki/Eric_Temple_Bell

https://hellopoetry.com/poem/710/the-night-dances/

https://poetrywithmathematics.blogspot.com/

[*11.14]

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### Degen’s Eight-square Identity

March 25, 2019 Here is another identity but this time corresponding to an eight-fold: the Eight-square identity of Ferdinand Degen found about 1818. You know the drill: it states that a product of two numbers that are each the sum of eight squares is itself the sum of eight squares!

(a12 + a22 + a32 + a42 + a52 + a62 + a72 + a82)(b12 + b22 + b32 + b42 + b52 + b62 + b72 + b82) =

…The sum of the expressions in the eight triangles written in the diagram above. (Please consult the Wikipedia entry below for the textual formulas, as it’s too hard to write in HTML.)

Note that the expressions above have an interesting symmetry, aside from the one on the upper left. Indeed, Euler’s Four-square Identity has a similar simpler symmetry. There is also a connection with Octonions if you are interested in digging for it. If you are anticipating that there is such a formula for sums of sixteen squares, there is, but not a bilinear one, and it is much more complicated!

https://en.wikipedia.org/wiki/Degen%27s_eight-square_identity

https://en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity

[*11.64]

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### Euler’s Four-square Identity

March 24, 2019 Here’s a nice little math identity that mathematician Leonard Euler wrote down in a letter dated 1748. It states that the product of two numbers that are sums of four squares is itself the sum of four squares.

(a12+a22+a32+a42)(b12+b22+b32+b42) =

(a1b1-a2b2-a3b3-a4b4)2 + (a1b2+a2b1+a3b4-a4b3)2 + (a1b3+a3b1+a4b2-a2b4)2 + (a1b4+a4b1+a2b3-a3b2)2

It can be proved with elementary algebra or even by quaternions!

https://en.wikipedia.org/wiki/Euler%27s_four-square_identity

[*11.62]

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### Four Forms Make a Universe, Part 2

March 17, 2019  This is a continuation of my last entry. Above is a different representation of the LICO alphabet, with the letters turned 45 degrees counter-clockwise, and rearranged into a symmetric pattern. The letters seem to arise more naturally in this orientation, but then Schmeikal rotates them into his normal schema.

And to the right is a diagram of the logical expressions that correspond to the letters above.

After making these new diagrams, I became inspired and made a few other figures to share with you.

These two versions, with triangles instead of line segments, and also with borders between adjacent triangles removed:  And these two versions, with quarter circles, and also with edges between adjacent quarter circles removed:  https://equivalentexchange.blog/2019/03/12/four-forms-make-a-universe/

[* 11.50, *11.58]

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### Four Forms Make a Universe

March 12, 2019  How could I not love a paper with this title? I’ve struggled with it for a bit, and I’ve only managed a couple of diagrams relating the author’s LICO (Linear Iconic) alphabet made up of 16 letters. However, I see that there are a few other papers by Schmeikal available on ResearchGate that look easier to understand. But also however, the first one says to read the “Four Forms” paper first!

At any rate, I present a sixteen-fold of the LICO alphabet, and another of the binary Boolean operators that are in a one-to-one mapping with LICO. There is much to understand from these papers, including much syncretism between various mathematical sixteen-folds, so please forgive me if I don’t explain it all with immediate ease. However, I believe it is well worth the effort to understand.

(Please note that the characters of the LICO alphabet are oriented so that the bottoms of the letters are downward, but the Boolean operators are oriented so that the bottoms of the equations are towards the right angles of the triangles.)

The title comes from the result that four elements of LICO can reproduce the other twelve via linear combinations. These four forms are 1) Boolean True (A or ~A), 2) A, 3) B, and 4) A=B. These are within the interior right-hand triangles in the LICO diagram. Of course, it is well known from Computer Science that the NAND operator (~A or ~B) can also generate all other fifteen operators, but this is by multiple nested operations instead of simple Boolean arithmetic. There are several other “universal” binary gates that can do this as well.

Two other representations that have four elements that can generate the other twelve via linear combinations come from CL(3,1), the Minkowski algebra. These representations are called “Idempotents” and  “Colorspace vectors”. Because of this algebra’s association with space and time in relativity, Schmeikal claims that LICO has ramifications in many far-ranging conceptualizations.

Bernd Schmeikal / Four Forms Make a Universe, in Advances in Applied Clifford Algebras (2015), Springer Basel (DOI 10.1007/s00006-015-0551-z)

Bernd Schmeikal / Free Linear Iconic Calculus – AlgLog Part 1: Adjunction, Disconfirmation and Multiplication Tables

Bernd Schmeikal / LICO a Reflexive Domain in Space-time (AlgLog Part 3)

https://en.wikipedia.org/wiki/Spacetime_algebra

[*9.145, *11.50]

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### Octonions

February 25, 2019 Octonions are the fourth, the last, and the greatest of the so-called normed division algebras based on the Real numbers. When I first learned about the Octonions way back when, I didn’t like them because they weren’t associative like the Quaternions, the Complex numbers, and the Reals. But now I’m fine with that, and they may be important for new theories of physics!

Octonions have the general form:

a0e0 + a1e1 + a2e2 + a3e3 + a4e4 + a5e5 + a6e6 + a7e7

Where the coefficients ai are Real and the bases ei have (something like) the following relations:

e0 = 1 (and -1 = e12 = e22 = …)

e1 = I = e2e3 = e7e6 = e4e5

e2 = J = e5e7 = e3e1 = e4e6

e3 = IJ = e1e2 = e6e5 = e4e7

e4 = K = e5e1 = e6e2 = e7e3

e5 = IK = e7e2 = e1e4 = e3e6

e6 = JK =e5e3 = e1e7 = e2e4

e7 = IJK = e6e1 = e3e4 = e2e5

In addition, if any of the products like e2e3 = e1 are reversed you get the negative, so e3e2 = -e1.

Non-associativity is demonstrated by going through the list of triples:

(e1e2)e3 = e32 = -1

e1(e2e3) = e12 = -1

(e1e2)e4 = e3e4 = e7

e1(e2e4) = e1e6 = -e7

(e1e2)e5 = e3e5 = -e6

e1(e2e5) = e1e7 = e6

so it is hit or miss I guess. Also note that e7 = (IJ)K = -I(JK). And for all (eiej)ek and ei(ejek), if they are not equal, is one equal to the negative of the other? And do I have to multiply them all out to find out?

https://en.wikipedia.org/wiki/Octonion

http://math.ucr.edu/home/baez/octonions/

https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

http://mathworld.wolfram.com/Octonion.html

http://homepages.wmich.edu/~drichter/octonions.htm

https://ncatlab.org/nlab/show/normed+division+algebra

[*11.33]

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### Quaternions

February 22, 2019 The Quaternions are a number system that enlarges the Complex numbers, just as the Complex numbers enlarge the Real numbers. In fact, Quaternions can be thought as special pairs of Complex numbers, just as Complex numbers can be thought as special pairs of Real numbers.

Quaternions can be used for all sorts of wonderful things, such as rotations in 3D space, instead of using matrices. Above is a pitiful diagram (although better than my last one) of the Quaternion units 1, i, j, and k used in the typical representation a + b i + c j + d k. Please read about them in the links below and be amazed!

https://en.wikipedia.org/wiki/Quaternion

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

http://theanalyticpoem.net/quaternions/

http://mathworld.wolfram.com/Quaternion.html

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/

https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/

[*10.160]

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### The Free Energy Principle

November 27, 2018 Below is a link to a fascinating article about Karl Friston, whose research on the “free energy principle” (also known as active inference) tries to explain how biological (or even artificial) systems maintain or even increase their organization. Without much explanation, here are some details. The model is a system with four main variables:

• Sense (s)
• Action (a)
• Internal States (r or μ)
• Hidden States (ψ)

Sense and action divide the internal states of the system from the hidden states external to it. This division is called a “Markov Blanket”. I’ve tried to show the equations between the variables correctly but I’m not too sure if I have them right. They seem to change from paper to paper. F is an expression of the free energy, but I’m not sure what f is at the moment, except for being some sort of “flow”.

https://www.wired.com/story/karl-friston-free-energy-principle-artificial-intelligence/

https://en.wikipedia.org/wiki/Free_energy_principle

https://en.wikipedia.org/wiki/Markov_blanket

http://rsif.royalsocietypublishing.org/content/15/138/20170792

[*11.6]

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