Math on The VikiD Truth

Linear Algebra

Sat, 13 May 2023 21:34:56 +0400

Linear Algebra

Here is the one truth that ties every topic on this page together: linear algebra is the study of things that can be added and scaled, and of the machines that respect adding and scaling.

Straight lines, flat planes, arrows, sound waves, quantum states, the pixels in a photograph: the same handful of ideas runs through all of them.

The core idea, in plain words

The word “linear” means one thing: a machine $T$ is linear when

pythagoras

Fri, 28 Jul 2017 09:04:21 +0400

Pythagoras

Here’s the truth about the most famous theorem in the world: Pythagoras’ theorem isn’t really about triangles. It’s about how distance works when directions are independent of each other.

The core idea, in plain words

In a right-angled triangle with shorter sides $a$ and $b$ and longest side (the hypotenuse) $c$ ,

a^2 + b^2 = c^2.

Said plainly: walk $a$ steps east and $b$ steps north, and the straight-line distance back to where you started is $\sqrt{a^2 + b^2}$ . East and north don’t interfere with each other, they’re at right angles, and that’s exactly when their squares simply add.

Transitivity

Thu, 08 Dec 2016 12:04:27 +0400

Let S be a set and ~ a relation between any two objects in the set:
Reflexivity For every a in S, a ~ a
Symmetry For any a, b in S, if a ~ b -> b ~ a
Transitivity For any a, b, c in S, if a ~ b and b ~ c –> a ~ c

Transitivity: links chain together

The core idea, in plain words

A relation $\sim$ on a set $S$ is transitive when you can always take a shortcut:

Symmetry

Thu, 08 Dec 2016 12:04:14 +0400

Let S be a set and ~ a relation between any two objects in the set:
Reflexivity For every a in S, a ~ a
Symmetry For any a, b in S, if a ~ b -> b ~ a
Transitivity For any a, b, c in S, if a ~ b and b ~ c –> a ~ c

Symmetry: if it goes one way, it goes back

The core idea, in plain words

A relation $\sim$ on a set $S$ is symmetric when every connection runs both ways:

Reflexivity

Thu, 08 Dec 2016 12:03:01 +0400

Let S be a set and ~ a relation between any two objects in the set:
Reflexivity For every a in S, a ~ a
Symmetry For any a, b in S, if a ~ b -> b ~ a
Transitivity For any a, b, c in S, if a ~ b and b ~ c –> a ~ c

Reflexivity is the smallest of the three rules, and the easiest to wave past. Which is exactly why I want to slow down on it. A relation is reflexive when nothing is a stranger to itself.

Set Relations

Thu, 08 Dec 2016 12:00:32 +0400

Set Relations

Equivalence Relations

Definition

Let S be a set and ~ a relation between any two objects in the set:
Reflexivity For every a in S, a ~ a
Symmetry For any a, b in S, if a ~ b -> b ~ a
Transitivity For any a, b, c in S, if a ~ b and b ~ c –> a ~ c

The core idea, in plain words

A relation on a set $S$ is simply a set of ordered pairs $R \subseteq S \times S$ . We write $a \sim b$ when $(a,b) \in R$ . That’s it: any list of “who is linked to whom” is a relation. The interesting part is which properties the list has. Five matter most:

affinetransformation

Thu, 08 Dec 2016 10:53:33 +0400

An affine transformation is any transformation that preserves:

collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and
ratios of distances

That’s the textbook line. Here’s the truth behind it: an affine transformation is a linear transformation that’s allowed to move the origin. Stretch, rotate, shear, flip, and then slide the whole picture somewhere else.

The core idea, in plain words

A linear map must send the origin to the origin. That’s too strict for everyday geometry: simply sliding a drawing two centimetres to the right isn’t linear, because the origin moves. So we allow one extra term:

topology

Thu, 08 Dec 2016 10:46:32 +0400

A homeomorphism or continuous transformation, is an

equivalence relation and
one-to-one (injective) correspondence between points in two geometric figures or topological spaces that is continuous in both directions.
preserves distances is called an isometry.
Affine transformations

Those are the technical words. Here’s the truth behind them: topology is geometry with the ruler thrown away. It only cares what’s next to what, and whether you can get from here to there without tearing.

Combinatorics: Combinatorics

Thu, 08 Dec 2016 06:23:01 +0400

Combinatorics: Combinatorics

General reference works (handbooks, dictionaries, bibliographies, etc.)
Instructional exposition (textbooks, tutorial papers, etc.)
Research exposition (monographs, survey articles)
Historical (must also be assigned at least one classification number from Section 01)
Explicit machine computation and programs (not the theory of computation or programming)
Proceedings, conferences, collections, etc.
Enumerative combinatorics
Combinatorial choice problems (subsets, representatives, permutations)
Factorials, binomial coefficients, combinatorial functions
Exact enumeration problems, generating functions
Asymptotic enumeration
Partitions of integers
Partitions of sets
Combinatorial identities
Combinatorial inequalities
$q$-calculus and related topics
Umbral calculus
None of the above, but in this section
Designs and configurations
Block designs
Triple systems
Difference sets (number-theoretic, group-theoretic, etc.)
Orthogonal arrays, Latin squares, Room squares
Matrices (incidence, Hadamard, etc.)
Finite geometries
Other designs, configurations
Matroids, geometric lattices
Packing and covering
Tessellation and tiling problems
Polyominoes
None of the above, but in this section
Graph theory
Trees
Degree sequences
Topological graph theory, imbedding
Distance in graphs
Coloring of graphs and hypergraphs
Perfect graphs
Directed graphs (digraphs), tournaments
Signed, gain and biased graphs
Graphs and groups
Enumeration of graphs and maps
Extremal problems
Paths and cycles
Connectivity
Eulerian and Hamiltonian graphs
Graphs and matrices
Generalized Ramsey theory
Isomorphism problems (reconstruction conjecture, etc.)
Graph representations (geometric and intersection representations, etc.)
Hypergraphs
Dominating sets, independent sets, cliques
Factorization, matching, covering and packing
Structural characterization of types of graphs
Graph labelling (graceful graphs, bandwidth, etc.)
Random graphs
Graph minors
Graph algorithms
Applications
None of the above, but in this section
Extremal combinatorics
Extremal set theory
Ramsey theory
Transversal (matching) theory
Probabilistic methods
None of the above, but in this section
Algebraic combinatorics
Symmetric functions
Tableaux, representations of the symmetric group
Combinatorial problems concerning the classical groups
Group actions on designs, geometries and codes
Group actions on posets and homology groups of posets
Association schemes, strongly regular graphs
Orthogonal polynomials
None of the above, but in this section

Mathematical logic and foundations

Thu, 08 Dec 2016 06:21:18 +0400

Mathematical logic and foundations

General reference works (handbooks, dictionaries, bibliographies, etc.)
Instructional exposition (textbooks, tutorial papers, etc.)
Research exposition (monographs, survey articles)
Historical (must also be assigned at least one classification number from Section 01)
Explicit machine computation and programs (not the theory of computation or programming)
Proceedings, conferences, collections, etc.
Philosophical and critical
General logic
Classical propositional logic
Classical first-order logic
Higher-order logic and type theory
Subsystems of classical logic (including intuitionistic logic)
Abstract deductive systems
Decidability of theories and sets of sentences
Foundations of classical theories (including reverse mathematics)
Mechanization of proofs and logical operations
Combinatory logic and lambda-calculus
Logic of knowledge and belief
Temporal logic
Modal logic
Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
Probability and inductive logic
Many-valued logic
Fuzzy logic; logic of vagueness
Logics admitting inconsistency (paraconsistent logics, discussive logics, etc.)
Intermediate logics
Other nonclassical logic
Logic of natural languages
Logic in computer science
Other applications of logic
None of the above, but in this section
Model theory
Equational classes, universal algebra
Basic properties of first-order languages and structures
Quantifier elimination, model completeness and related topics
Finite structures
Denumerable structures
Ultraproducts and related constructions
Model-theoretic forcing
Other model constructions
Categoricity and completeness of theories
Interpolation, preservation, definability
Classification theory, stability and related concepts
Models with special properties (saturated, rigid, etc.)
Properties of classes of models
Set-theoretic model theory
Effective and recursion-theoretic model theory
Model-theoretic algebra
Models of arithmetic and set theory
Model theory of ordered structures; o-minimality
Models of other mathematical theories
Other classical first-order model theory
Logic on admissible sets
Other infinitary logic
Logic with extra quantifiers and operators
Second- and higher-order model theory
Nonclassical models (Boolean-valued, sheaf, etc.)
Abstract model theory
Applications of model theory
None of the above, but in this section
Computability and recursion theory
Thue and Post systems, etc.
Automata and formal grammars in connection with logical questions
Turing machines and related notions
Complexity of computation
Recursive functions and relations, subrecursive hierarchies
Recursively (computably) enumerable sets and degrees
Other Turing degree structures
Other degrees and reducibilities
Undecidability and degrees of sets of sentences
Word problems, etc.
Theory of numerations, effectively presented structures
Recursive equivalence types of sets and structures, isols
Hierarchies
Computability and recursion theory on ordinals, admissible sets, etc.
Higher-type and set recursion theory
Inductive definability
Abstract and axiomatic computability and recursion theory
Applications of computability and recursion theory
None of the above, but in this section
Set theory
Partition relations
Ordered sets and their cofinalities; pcf theory
Other combinatorial set theory
Ordinal and cardinal numbers
Descriptive set theory
Cardinal characteristics of the continuum
Other classical set theory (including functions, relations, and set algebra)
Axiom of choice and related propositions
Axiomatics of classical set theory and its fragments
Consistency and independence results
Other aspects of forcing and Boolean-valued models
Inner models, including constructibility, ordinal definability, and core models
Other notions of set-theoretic definability
Continuum hypothesis and Martin’s axiom
Large cardinals
Determinacy principles
Other hypotheses and axioms
Nonclassical and second-order set theories
Fuzzy set theory
Applications of set theory
None of the above, but in this section
Proof theory and constructive mathematics
Proof theory, general
Cut-elimination and normal-form theorems
Structure of proofs
Functionals in proof theory
Recursive ordinals and ordinal notations
Complexity of proofs
Relative consistency and interpretations
First-order arithmetic and fragments
Second- and higher-order arithmetic and fragments
Gödel numberings in proof theory
Provability logics and related algebras (e.g., diagonalizable algebras)
Metamathematics of constructive systems
Linear logic and other substructural logics
Intuitionistic mathematics
Constructive and recursive analysis
Other constructive mathematics
None of the above, but in this section
Algebraic logic
Boolean algebras
Lattices and related structures
Quantum logic
Cylindric and polyadic algebras; relation algebras
Lukasiewicz and Post algebras
Other algebras related to logic
Categorical logic, topoi
None of the above, but in this section
Nonstandard models
Nonstandard models in mathematics
Other applications of nonstandard models (economics, physics, etc.)
Nonstandard models of arithmetic
None of the above, but in this section

History and biography

Thu, 08 Dec 2016 06:16:20 +0400

History and biography

General reference works (handbooks, dictionaries, bibliographies, etc.)
Instructional exposition (textbooks, tutorial papers, etc.)
Research exposition (monographs, survey articles)
Proceedings, conferences, collections, etc.
Computational methods
History of mathematics and mathematicians
General histories, source books
Ethnomathematics, general
Paleolithic, Neolithic
Indigenous cultures of the Americas
Other indigenous cultures (non-European)
Indigenous European cultures (pre-Greek, etc.)
Egyptian
Babylonian
Greek, Roman
China
Japan
Southeast Asia
Islam (Medieval)
India
Medieval
15th and 16th centuries, Renaissance
17th century
18th century
19th century
20th century
Twenty-first century
Contemporary
Future prospectives
Biographies, obituaries, personalia, bibliographies
Schools of mathematics
Universities
Other institutions and academies
Collected or selected works; reprintings or translations of classics
Sociology (and profession) of mathematics
Historiography
Bibliographic studies
Miscellaneous topics

General Math

Thu, 08 Dec 2016 05:46:52 +0400

Map of Math

General Math

Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) General and miscellaneous specific topics General mathematics Mathematics for nonmathematicians (engineering, social sciences, etc.) Problem books Recreational mathematics Bibliographies External book reviews Dictionaries and other general reference works Formularies Philosophy of mathematics Methodology of mathematics, didactics General applied mathematics Theory of mathematical modeling General methods of simulation Dimensional analysis Physics (use more specific entries from Sections 70 through 86 when possible) Miscellaneous topics Conference proceedings and collections of papers Collections of abstracts of lectures Collections of articles of general interest Collections of articles of miscellaneous specific content Proceedings of conferences of general interest Proceedings of conferences of miscellaneous specific interest Festschriften Volumes of selected translations Miscellaneous volumes of translations Collections of reprinted articles

Dynamics

Wed, 07 Dec 2016 21:04:48 +0400

Math | Physics | Science

Dynamics

Dynamics is the study of how systems change over time. Dynamics Started as a formal system of study with the works of Leibniz and Newton in the 17th Century.

A Sketch on the History of Dynamics

17th Century
- 1666-1696 - Invention of Calculus - Newton & Leibniz
  - Major Developments in Calculus and Classical Mechanics
18th Century
- Analytical Works of Planetary Motion
19th Century
- 1890’s Poincare works on Geometric Approach
20th Century
- Birkhoff
  - Kolmogrov
  - Arnol’d
  - Moser
  - Lorenz
  - Ruelle
  - Takens
  - May
  - Feigenbaum
  - Winfree
  - Mandelbrot
- Ordinary Differential Equations the study of changes with respect to only one variable, mostly time
  - Partial Differential Equations the study of changes with respect to two or more variables, mostly time & space

Jordan B. Peterson

Wed, 07 Dec 2016 14:38:58 +0400

Maps of Meaning | Philosophy | Psychology

Jordan B. Peterson

Date 7th December 2016

Jordan Peterson is a Clinical Personality Psychologist. I am currently in Jordan Peterson mode. I get into these kinds of voracious moods whenever I find a new thinker that I admire or I am fascinated by. Some years ago it was Christopher Hitchens, Krishnamurti, etc… even before that it was Mandela, Ghandi, Swami Vivekananda, etc… More recently it was Daniel Dennet, Sam Harris, Stephan Molyneux, etc… Personality de jour is Jordan Peterson.

Category Theory

Wed, 07 Dec 2016 07:59:13 +0400

Category Theory

Category theory is abstract, about as abstract as mathematics gets.

It is the study of systems of related objects. The relations between the objects under study are called maps (also arrows or morphisms)

Maps bind Objects together in a relationship.

Often Objects are Represented by Letters and Maps by Arrow, such as:

 f g
 A ---> B --> C

where we have objects A, B and C, and arrows f & g.

Multivariable Polynomial Rings

Wed, 07 Dec 2016 07:57:41 +0400

Multivariable Polynomial Rings

Here is a truth that took me far too long to see: a polynomial in several variables is a description of a shape, and the ring of all such polynomials is the algebra of shapes.

Inner Product Spaces

Wed, 07 Dec 2016 07:55:13 +0400

Inner Product Spaces

Here’s the truth about inner products: a plain vector space has no idea how long anything is or what angle two vectors make. An inner product is the ruler and protractor you hand it.

Linear Operators and Matrices

Wed, 07 Dec 2016 07:54:36 +0400

Linear Operators and Matrices

Here is the truth that finally made matrices make sense to me: a matrix is not a box of numbers. It is a photograph of a machine, taken from one particular angle.

Vector Spaces

Wed, 07 Dec 2016 07:53:46 +0400

Vector Spaces

Here is the truth about vector spaces that no textbook says plainly enough: a vector is anything you can add to another of its kind and stretch by a number. That’s it. Arrows are just the most famous example.

Modules

Wed, 07 Dec 2016 07:53:03 +0400

Modules

Here’s a truth that tidied up half of algebra for me: a module is a vector space where the scalars are only allowed to come from a ring, so you can scale but not always unscale.

Group Actions

Wed, 07 Dec 2016 07:51:53 +0400

Group Actions

In Groups I said a group is the mathematics of “doing and undoing”. Here’s the follow-up truth: a group action is when those doings actually do something to something.

A group on its own is a bag of abstract moves. An action lets the moves loose on a set (corners, colourings, points, solutions of an equation) and asks what happens to it.

Fields

Wed, 07 Dec 2016 07:51:10 +0400

Fields

The truth about fields is the one I wish I had heard first: a field is a number system where you can add, subtract, multiply and divide (by anything except zero) and never get thrown out.

Exterior Algebras

Wed, 07 Dec 2016 07:50:48 +0400

Exterior Algebras

The truth hiding behind this intimidating name is a lovely one: exterior algebra is the algebra of oriented areas and volumes. It’s what the determinant was secretly trying to be all along.

The core idea, in plain words

Take a Vector Space $V$ . We invent a new product, the wedge $u \wedge v$ , which stands for the little parallelogram spanned by $u$ and $v$ , with a direction of travel round its edge. The rules:

Commutative Rings

Wed, 07 Dec 2016 07:50:21 +0400

Commutative Rings

The truth about commutative rings is a simple one: they are the rings where the order of multiplication doesn’t matter, and that one small courtesy makes almost all of school algebra possible.

Every time you expanded $(x + 1)^2$ into $x^2 + 2x + 1$ , you were quietly relying on it.

Rings

Wed, 07 Dec 2016 07:49:45 +0400

Rings

Here is the truth nobody told me at school: a ring is just a place where you can add, subtract and multiply, and the usual habits of arithmetic still work. Division is not invited.

Quotient Groups

Wed, 07 Dec 2016 07:49:25 +0400

Quotient groups were the place where abstract algebra first made me dizzy. The symbols $G/N$ look like division, and in a strange way they are. But here’s the truth: a quotient group is what you get when you deliberately go blurry.

Groups

Wed, 07 Dec 2016 06:46:59 +0400

Here is a truth I wish someone had revealed to me much earlier: a group is just the mathematics of “doing and undoing”.

That’s it. That’s the secret. Everything else is bookkeeping.

The core idea, in plain words

A group is a set of actions (or things) together with one way of combining them, such that four rules hold:

Number Theory

Wed, 07 Dec 2016 06:42:39 +0400

Math

Number Theory

I find Number theory one of the very hardest parts of Mathematics so this is probably the most contrived part of this wiki. You will find a struggle here but hopefully if you are having a hard time yourself, there maybe some value to you here below.

My fascination with Mathematics really started with numbers. No matter how hard I tried, I couldn’t really figure out what they were (Ontologically). Did they have a basis in reality and if not, what basis did they have and moreover, why did Reality and Math track so well if they didn’t have a basis in reality.

viKid Abstraction

Tue, 06 Dec 2016 08:10:32 +0400

Abstract Algebra | ZF Axioms | Math on Computers | Computing

abstract

viKid Abstraction

What is Abstraction?

Abstraction is at the heart of both Mathematics and Computer Science.

In fact I believe both Mathematics and Computer Science are the embodiment of forms of Abstraction.

I will save you from technical definitions, you can find that with a cursory search or at the links at the bottom of the page.

What I want to discuss here is more philosophical and I think we can begin to look at the subject from,

Aldous Huxley

Sat, 03 Dec 2016 00:03:54 +0400

Aldous Huxley

Aldous Huxley is (or was one) of my greatest hero’s when I was a young man. I first read Brave New World, Doors of Perception, Island & The Perennial Philosophy. I also read a few of his articles and other works. Each book had its own influence on me, carving out my thought processes, making me deeply think about life in all its dimensions. Huxley was really one of the principle reasons I ever went to thinking in the first place. He showed me just what can be achieved with the mind, a pen (and some psychodelics).

Sets and Structures

Thu, 01 Dec 2016 11:24:20 +0400

Math | Abstract Algebra | ZF Axioms | Set Theory | Number Theory} Groups | Quotient Groups | Rings | Commutative Rings | Fields | Group Actions | Modules | Vector Spaces | Linear Operators and Matrices | Inner Product Spaces | Tensors | Exterior Algebras | Theory | Polynomial Rings | Category Theory |

Sets and Structures

Set Theory is the Machine Language of mathematics. It’s a very low level language, close to the machine, so to speak. In set theory with define really primitive operations on even more primitive types data. Set theory in a way isn’t as illuminating as high level mathematics but in another way, it is the light that makes all of math possible.

Logic - Modus Tollens

Wed, 30 Nov 2016 23:43:22 +0400

Logic - Modus Tollens

Modus tollens is modus ponens’s sharper sibling, and the logic behind every good experiment. The truth: if a prediction fails, the idea that made it is in trouble.

The core idea, in plain words

If you know “if $p$ then $q$ ” and you know $q$ is false, you may conclude $p$ is false.

Logic - Modus Ponens

Wed, 30 Nov 2016 23:42:52 +0400

Logic - Modus Ponens

Modus Ponens is latin for “mode that affirms”

 p→q
 p
 _____
 ∴ q

Modus ponens is the rule of reasoning we all use a hundred times a day without naming it. The truth: it is the engine that turns an “if” into a “therefore”.

The core idea, in plain words

If you know a conditional “if $p$ then $q$ ” is true, and you know $p$ is true, you may conclude $q$ .

VikiD Logic

Wed, 30 Nov 2016 23:36:06 +0400

VikiD Logic

Logic is the study of abstract, hypothetical connections, involving necessity.

These connections are between ‘Statements’, ‘Propositions’, or ‘Assertions’.

Classified into two groups, valid vs invalid.

Modus Ponens is the very most basic type of connection. It says,

if A => B
 A
 ---
 ∴ B

that is, if A implies B, and we “assert” A is “valid”, then we can conclude B is also “valid”.

Jeremy Bentham

Wed, 30 Nov 2016 22:31:25 +0400

Jeremy Bentham

(1748 - 1832)

 He is best known as the founder of Utilitarianism, which
 he saw as the underlying moral principle on which his legal
 and social reforms should be based. Although his influence
 during his life was perhaps minor, his impact was greater
 in later years as his ideas were carried on by followers
 such as John Stuart Mill, Robert Owen and John Austin.

@ philosophybasics

Epicurus (341–270 BCE) s

Wed, 30 Nov 2016 22:20:24 +0400

Epicurus (341–270 BCE) s

 Hellenistic school founded by Epicurus (341–270 BCE) that emphasized
 achieving happiness by minimizing pain and pursuing pleasure.
 1. Atomism
 a. Slight swerve
 b. Free Will and the slight swerve
 2. Image-flakes and perception
 3. Happiness: decrease pain, increase pleasure
 4. Decrease pain by eliminating fears
 a. Death
 b. Gods
 5. Three kinds of desires
 6. Pleasure through moderation
 7. Widsom and the virtues

@ utm

Thales of Miletos

Wed, 30 Nov 2016 10:38:36 +0400

Thales of Miletos

This story should be something that is instantly recognizable:

 His involvement in local politics is also rather anecdotal
 in nature, but Thales apparently impressed both sides of the
 ongoing conflict between the Lydians, Medes and Persians over
 the fate of the region of Ionia,

 when he predicted an eclipse of the sun

 which brought fighting to a standstill. He was also
 reportedly involved in the negotiations which followed the
 hostilities, and managed to obtain favourable terms for Miletus.

http://www.philosophybasics.com/philosophers_thales.html

Ludwig Wittgenstein A Memoir by Norman Malcom

Tue, 29 Nov 2016 01:22:44 +0400

Ludwig Wittgenstein A Memoir by Norman Malcom

I found this book for GBP 3.00 in a small second hand bookshop in Hampstead, London and I simply had to have it. It is a funny thing but I first realised that I had fallen in love with philosophy reading a book about Wittgenstein. It was a surreal moment, it is as if the book read back to me, the contents of my own mind. I was shocked;

Ludwig Wittgenstein

Tue, 29 Nov 2016 01:15:41 +0400

Philosophy | Economics | Politics | Ethics | Aesthetics

Ludwig Wittgenstein

 Wittgensteins's Tractus may be called a synthesis of the theory
 of truth-functions and the idea that language is a picture of
 reality.

 Out of this synthesis arises a third main ingredient of the book,
 its doctrine of that which cannot be said, only shown.

 Ludwig Wittgenstein A Memoir by Norman Malcom

Date: 30/11/2016

He had a habit of writing thing down in note books. Like him, I too have a scattering of note books but unlike him, mine are not so organised or dated. There is a beauty to note taking. Something in it that sharpens the mind. It makes you think with greater precision and clarity and acts as a reference at future dates often bringing back more than what is immediately written down.

Tensors

Tue, 29 Nov 2016 00:06:14 +0400

Abstract Algebra | ZF Axioms | Math

Tensors

Einstein’s Summation Convention

Get the pdf here: summation (pdf)

Understanding Gravitation and all kinds of modern physics requires a working knowledge of Einstein’s summation convention.

And here is the truth behind the word itself: a tensor is a machine that eats several vectors and returns a number, and is linear in each one it eats.

Scalars, vectors and matrices are all tensors. They’re the first three rungs of one ladder.

Abstract Algebra

Mon, 28 Nov 2016 23:50:04 +0400

Abstract Algebra

To me, Algebra is a branch of Mathematics that deals with symbols and the rules for manipulating those symbols. It really is the foundation of all Mathematics because it sets up and sets out the written language of math. Without algebra we would have no powerful method of communicating our Mathematical ideas (to humans or computers) and progress would essentially ground to a halt.

The Foundations of Mathematics

Mon, 21 Nov 2016 22:49:09 +0400

The Foundations of Mathematics

By Ian Stewart & David Tall

Back to Math

This page is my companion to Ian Stewart and David Tall’s The Foundations of Mathematics (Oxford University Press, first edition 1977, second edition 2015). It’s the book I’d hand anyone who has done school maths and wants to know what’s underneath it. Here’s the truth it reveals: mathematics is not built on numbers. It’s built on logic and sets, and the numbers come later, constructed by hand.

Richard Rorty

Sun, 20 Nov 2016 22:05:50 +0400

Richard Rorty

The American philosopher Richard Rorty, thought that there might in fact still be something behind the world as it appears to us. However, he thought that this could play no role for us as human beings, so he instead suggested that we increase solidarity among human beings rather than look for ultimate Truth (with a capital T) or ultimate Reality (with a capital R).

Rorty concluded that there is no position outside historically situated language games from which to distinguish mind from world.

Philosophers to Study

Sun, 20 Nov 2016 22:03:41 +0400

Philosophers to Study

Pre-Socratics

The Primary Source here is, philosphybasics by Luke Mastin. The Links below are really trying to pick the most memorable parts of each philosophers work to bring it to life in our imaginations & our memories.

Thales of Miletos (c. 624 - 546 B.C.)
Anaximander (c. 610 - 546 B.C.) Greek
Anaximenes (c. 585 - 525 B.C.) Greek
Pythagoras (c. 570 - 490 B.C.) Greek
Heraclitus (c. 535 - 475 B.C.) Greek
Parmenides of Elea (c. 515 - 450 B.C.) Greek
Anaxagoras (c. 500 - 428 B.C.) Greek
Empedocles (c. 490 - 430 B.C.) Greek
Zeno of Elea (c. 490 - 430 B.C.) Greek
Protagoras (c. 490 - 420 B.C.) Greek
Gorgias (c. 487 - 376 B.C.) Greek
Democritus (c. 460 - 370 B.C.) Greek

Classical Greek

Hellenistic

Epicurus (341–270 BCE)

Eastern

Medieval

Renaissance

Elightenment

Modern & Contemporary

Jeremy Bentham (1749 - 1832) Ludwig Wittgenstein
Friedrich Nietzsche (1844-1900)
Aldous Huxley (1894 – 1963)
Jordan B. Peterson 1962 (age 53–54)

Postmodernists - Constructionists

Richard Rorty

Quantum Mechanics

Mon, 01 Aug 2016 23:49:10 +0400

Quantum Mechanics

Here’s the truth that the mystique around quantum mechanics hides: underneath the weirdness, quantum mechanics is linear algebra on complex vectors, plus one strange rule about what happens when you look.

The core idea, in plain words: the state of a system is a vector. Change over time is a linear operator. Measurement turns amplitudes into probabilities. Almost everything below is one of those three sentences in action, and the pages on Vector Spaces, Linear Operators and Matrices, Inner Product Spaces and Tensors are the toolkit.

The Zermelo-Frankel Axioms

Fri, 29 Jul 2016 11:44:02 +0400

The Zermelo-Frankel Axioms

The ZF-Axioms are kind of a bed rock in Mathematics. A place where everything can begin. Now this is not the only way to formalise mathematics. There many theories around. I haven’t got there yet. Category theory and Homeotopic type theory come to mind here.

Now, let the caveat be stated here clearly.

I am not a mathematician, I am not formally trained in the topic and I do understand there may be errors here. If you are so disposed and would like to free the Internet of error. Please let me know! I will be happy to be schooled in this manner. What appears below is how I have understood the subject and it may be helpful to you.

Relativity

Tue, 28 Jun 2016 01:27:13 +0400

Relativity

Special, General, and Cosmological

Here’s the truth that Einstein revealed, and that still sounds wrong the first time you hear it: space and time are not separate stages on which things happen. They are one four-dimensional fabric, and different observers slice it differently.

The core idea, in plain words

Special relativity (1905) rests on two sentences:

The laws of physics are the same for every observer moving at constant velocity.
The speed of light $c$ is the same for all of them.

The second sounds harmless until you take it seriously. If a train passenger and a platform watcher both measure the same speed for the same flash of light, then they must disagree about distances and durations. Moving clocks run slow; moving rulers shrink along their motion. What everyone agrees on is not time, not distance, but the spacetime interval:

Mathematics for the viKid

Tue, 10 May 2016 23:27:07 +0400

Mathematics for the viKid

I didn’t start to appreciate mathematics until after University. It started with book by Professor Ian Stewart called The Foundations of Mathematics. It was a short white book that looked very complicated. Yet I was determined to figure out just what math was.

How did math work?

Why was it so effective at describing natural phenomenon?

Math on The VikiD Truth

Linear Algebra

Linear Algebra

The core idea, in plain words

pythagoras

Pythagoras

The core idea, in plain words

Transitivity

Transitivity: links chain together

The core idea, in plain words

Symmetry

Symmetry: if it goes one way, it goes back

The core idea, in plain words

Reflexivity

Reflexivity: everything is related to itself

Set Relations

Set Relations

Equivalence Relations

The core idea, in plain words

affinetransformation

The core idea, in plain words

topology

Combinatorics: Combinatorics

Combinatorics: Combinatorics

Mathematical logic and foundations

Mathematical logic and foundations

History and biography

History and biography

General Math

General Math

Dynamics

Dynamics

A Sketch on the History of Dynamics

Jordan B. Peterson

Jordan B. Peterson

Category Theory

Category Theory

Multivariable Polynomial Rings

Multivariable Polynomial Rings

Inner Product Spaces

Inner Product Spaces

Linear Operators and Matrices

Linear Operators and Matrices

Vector Spaces

Vector Spaces

Modules

Modules

Group Actions

Group Actions

Fields

Fields

Exterior Algebras

Exterior Algebras

The core idea, in plain words

Commutative Rings

Commutative Rings

Rings

Rings

Quotient Groups

Groups

The core idea, in plain words

Number Theory

Number Theory

viKid Abstraction

viKid Abstraction

Aldous Huxley

Aldous Huxley

Sets and Structures

Sets and Structures

Logic - Modus Tollens

Logic - Modus Tollens

The core idea, in plain words

Logic - Modus Ponens

Logic - Modus Ponens

The core idea, in plain words

VikiD Logic

VikiD Logic

Jeremy Bentham

Jeremy Bentham

(1748 - 1832)