## 3/5/14

### From Arithmetic to Nullary Operations

What are nullary operations? In Algebra and Mathematical Logic it is said that "nullary operations act as constants". How and why do nullary operations act as constants? The following conceptual path is taken to answer this question:

Arithmetic $$\rightarrow$$ Laws of Composition $$\rightarrow$$ Binary Operations $$\rightarrow$$ $$n$$-ary Operations $$\rightarrow$$ Nullary Operations

## From Arithmetic to Laws of Composition

Vaguely, Algebra is the study of calculating with algebraic operations, the most familiar of which are the elementary arithmetic operations of addition, subtraction, multiplication, and division. Each of these laws relates a pair of numbers to another number called their sum, difference, product, and quotient, respectively. Thus the sum, difference, product, and quotient of $$6$$ and $$2$$ is $$8$$, $$4$$, $$12$$, and $$3$$, in the order given.

Since the difference of $$3$$ and $$2$$ is $$1$$ but the difference of $$2$$ and $$3$$ is $$-1$$ the order in which the pairs of numbers are composed matters. Thus a law of composition must act on an ordered pair of numbers rather than just a pair of numbers (that is, ordered pairs account for this difference in differences). Again, to calculate the difference of $$5$$ and $$4$$ is to subtract $$4$$ from $$5$$ giving $$1$$ where as to subtract $$5$$ from $$4$$ gives $$-1$$.

A critical feature of these elementary laws of composition is that each ordered pair of numbers relates to a unique number e.g. the sum of $$2$$ and $$3$$ is $$5$$ and there is not a number distinct from $$5$$ which is also the sum of $$2$$ and $$3$$. The choice is made to fix this property when considering general laws of composition for objects other than natural numbers.

Definition. A (general) law of composition for a collection of objects $$A$$ associates with each ordered pair of objects from $$A$$ a unique object in $$A$$.
Notice that an additional constraint has been placed on this definition which has not been explicitly mentioned: it is required here that each ordered pair of objects from $$A$$ be associated with a unique object in $$A$$. A further generalization of a law of composition is a partial law of composition which associates some ordered pairs of objects from $$A$$ with a unique object in $$A$$.

For instance, the difference of an ordered pair of natural numbers (where the natural numbers are 0, 1, 2, 3, etc.) may be a negative number, which is not a natural number. Thus not every ordered pair of natural numbers has a difference which is itself a natural number. In fact, this only occurs when the number to be subtracted is smaller than the number from which the subtraction is performed. So, the law of subtraction is a partial law of composition for the collection of natural numbers.

But, if the collection of objects on which subtraction is performed is expanded to include not just the natural numbers, but also the negative numbers, this joint collection being called the Integers, then subtraction becomes a law of composition for the integers. Thus two components of the concept of a law of composition are relevant: the rule which dictates how an ordered pair of objects are to be combined, and the collection of objects to which the rule may be meaningfully applied.

## From Laws of Composition to Binary Operations

As is common in any mathematical investigation, once a new concept has been introduced it is prudent to compare it to common mathematical concepts. Often, after some time and critical reflection, certain mathematical concepts are seen as key to understanding others. Such a development is expedited with this question: how are laws of composition related to functions?

Functions may be seen as generalizations of elementary actions in arithmetic such as "add one" or "multiply by 2". Thus the act of adding one to $$2$$ gives $$3$$ and there is no other number which results from adding one to $$2$$.

Definition. A function from a collection of objects $$A$$ to a collection of objects $$B$$ is a rule which associates to each object in $$A$$ a unique object in $$B$$.
From the wording of this definition and the identification of the relevant features at the end of the previous section it is obvious that there is an interesting relationship between laws of composition and functions. Let $$A \times A$$ denote the set of all ordered pairs of objects from the collection $$A$$.
Proposition. A function from $$A \times A$$ to $$A$$ is a law of composition for $$A$$.
Functions which are themselves laws of composition are given a special name: binary operations. The reason for this is that a function from a set of ordered pairs is said to be a function of two variables which operates on the two variables by some rule to give the value of the function.

## From Binary Operations to $$n$$-ary Operations

By considering ordered triples or ordered quadruples of objects from $$A$$ a binary operation is generalized to an $$3$$-ary operation, being a function from $$A \times A \times A$$ to $$A$$, or a $$4$$-ary operation, being a function from $$A \times A \times A \times A$$ to $$A$$. Let $$A^n$$ denote the collection of all ordered $$n$$-tuples.
Definition. An $$n$$-ary operation on $$A$$ is a function from $$A^n$$ to $$A$$.
It is possible to return to the discussion of laws of composition and generalize them to $$n$$-ary laws of composition and then to show that an $$n$$-ary operation is an $$n$$-ary law of composition. An example of a $$3$$-ary law of composition is given by the rule which returns the sum of all three entries from an ordered triple of numbers e.g. under this rule $$1$$, $$2$$, and $$3$$ gives $$6$$.

## From $$n$$-ary Operations to Nullary Operations

A special case of the definition of $$n$$-ary operation given is when $$n$$ is taken to be $$0$$. A $$0$$-ary, or nullary, operation is a function from $$A^0$$ to $$A$$.

What is $$A^0$$? Since $$A^2$$ is the collection of all ordered $$2$$-tuples, or pairs, of objects from $$A$$, and $$A^1$$ is the collection of all ordered $$1$$-tuples from $$A$$, then $$A^0$$ should be considered as the collection of all $$0$$-tuples of objects from $$A$$. Note, an ordered pair, or ordered $$2$$-tuple, has two parts, a first entry and a second entry. Similarly an ordered $$1$$-tuple has only a first entry. Thus an ordered $$0$$-tuple has no entries and is consequently empty. It is common for $$\emptyset$$ to denote the ordered $$0$$-tuple. Furthermore, $$\emptyset$$ is also just called the empty set as any two collections of things which both contain nothing are empty and consequently equal to each other. Said another way, the empty set is that unique collection of things which has no objects within it (much like an empty bag).

Finally, since $$A^0$$ is, as has been said, the collection of all $$0$$-tuples, of which their is only one namely $$\emptyset$$, then $$A^0$$ is that collection which contains only $$\emptyset$$. Said another way, $$A^0$$ is a collection containing one object, and that object is the $$0$$-tuple, which is $$\emptyset$$. Thus, a nullary operation on $$A$$ is a function from a set containing one object, the empty set, to $$A$$. Thus a nullary operation is determined by specifying to which object it sends the empty set.

For example a nullary operation on the set of natural numbers is a function from the collection containing exactly the empty set to the collection of all natural numbers. So a nullary operation is only meaningfully applied to the empty set to give some natural number, such as $$2$$. That is, for each natural number $$n$$ there is exactly one nullary operation which when applied to $$\emptyset$$, the only meaningful thing to which it can be applied, it gives $$n$$. Furthermore, all nullary operations are obtained in this way. Thus, a nullary operation is equivalent to the value which it takes on. In the case of nullary operations on the natural numbers this means that each natural number may be seen as a unary operation on the set of natural numbers which takes on that natural number as its only value.

## But does it make sense?

No, in the context given it seems that to use a nullary operation is a pointless act of generalization mixed with a dose of nonsensical empty set arguments. This is the reaction had by most who follow this sequence of concepts in order to build up the idea of a unary operation. What this explanation demands is a suitable context where nullary operations are self evident and do not result from a random act of generalization. Such a context may be found in the development of set theory from category theory.

The standard development of set theory from category theory defines the theory of sets as an elementary topos with a natural number object where each epimorphism has a right inverse, and which is well pointed. The purpose of being well pointed is to assure that two functions may be distinguished if they do not agree for some argument, as a consequence each classical element of a set must be identified as a nullary operation. From this perspective, though it is currently seen as unfashionable, it is reasonable to desire that distinct functions be distinguished by a difference in their value at a single point, from which it follows that single points must be identified as nullary operations. It has yet to be seen if there is anything fundamentally significant about this equivalence, for now most treat it as nothing more than a matter of convenience.

## 1/31/14

### Contents of Mac Lane and Birkhoff's Algebra

1. Sets, Functions, and Integers
1. Sets
2. Functions
3. Relations and Binary Operations
4. The Natural Numbers
6. Inequalities
7. The Integers
8. The Integers Modulo n
9. Equivalence Relations and Quotient Sets
10. Morphisms
11. Semigroups and Monoids
2. Groups
1. Groups and Symmetry
2. Rules of Calculation
3. Cyclic Groups
4. Subgroups
5. Defining Relations
6. Symmetric and Alternating Groups
7. Transformation Groups
8. Cosets
9. Kernel and Image
10. Quotient Groups
3. Rings
1. Axioms for Rings
2. Constructions for Rings
3. Quotient Rings
4. Integral Domains and Fields
5. The Field of Quotients
6. Polynomials
7. Polynomials as Functions
8. The Division Algorithm
9. Principal Ideal Domains
10. Unique Factorization
11. Prime Fields
12. The Euclidean Algorithm
13. Commutative Quotient Rings
4. Universal Constructions
1. Examples of Universals
2. Functors
3. Universal Elements
4. Polynomials in Several Variables
5. Categories
6. Posets and Lattices
7. Contravariance and Duality
8. The Category of Sets
9. The Category of Finite Sets
5. Modules
1. Sample Modules
2. Linear Transformations
3. Submodules
4. Quotient Modules
5. Free Modules
6. Biproducts
7. Dual Modules
6. Vector Spaces
1. Bases and Coordinates
2. Dimension
3. Constructions for Bases
4. Dually Paired Vector Spaces
5. Elementary Operations
6. Systems of Linear Equations
7. Matrices
8. Matrices and Free Modules
9. Matrices and Biproducts
10. The Matrix of a Map
11. The Matrix of a Composite
12. Ranks of Matrices
13. Invertible Matrices
14. Change of Bases
15. Eigenvectors and Eigenvalues
7. Special Fields
1. Ordered Domains
2. The Ordered Field $$\mathbb{Q}$$
3. Polynomial Equations
4. Convergence in Ordered Fields
5. The Real Field $$\mathbb{R}$$
6. Polynomials over $$\mathbb{R}$$
7. The complex Plane
8. The Quaternions
9. Extended Formal Power Series
10. Valuations and $$p$$-adic Numbers
8. Determinants and Tensor Products
1. Multilinear and Alternating Functions
2. Determinants of Matrices
3. Cofactors and Cramer’s Rule
4. Determinants of Maps
5. The Characteristic Polynomial
6. The Minimal Polynomial
7. Universal Bilinear Functions
8. Tensor Products
9. Exact Sequences
10. Identities on Tensor Products
11. Change of Rings
12. Algebras
1. Bilinear Forms
2. Symmetric Matrices
5. Inner Products
6. Orthonormal Bases
7. Orthogonal Matrices
8. The Principle Axis Theorem
9. Unitary Spaces
10. Normal Matrices
10. Similar Matrices and Finite Abelian Groups
1. Noetherian Modules
2. Cyclic Modules
3. Torsion Modules
4. The Rational Canonical Form for Matrices
5. Primary Modules
6. Free Modules
7. Equivalence of Matrices
8. The Calculation of Invariant Factors
11. Structure of Groups
1. Isomorphism Theorems
2. Group Extensions
3. Characteristic Subgroups
4. Conjugate Classes
5. The Sylow Theorems
6. Nilpotent Groups
7. Solvable Groups
8. The Jordan-Hölder Theorem
9. Simplicity of $$A_n$$
12. Galois Theory
2. Algebraic and Transcendental Elements
3. Degrees
4. Ruler and Compass
5. Splitting Fields
6. Galois Groups of Polynomials
7. Separable Polynomials
8. Finite Fields
9. Normal Extensions
10. The Fundamental Theorem
11. The Solution of Equations by Radicals
13. Lattices
1. Posets: Duality Principle
2. Lattice Identities
3. Sublattices and Products of Lattices
4. Modular Lattices
5. Jordan-Hölder-Dedekind Theorem
6. Distributive Lattices
7. Rings of Sets
8. Boolean Algebras
9. Free Boolean Algebras
1. Categories
2. Functors
3. Contravariant Functors
4. Natural Transformations
5. Representable Functors and Universal Elements
15. Multilinear Algebra
1. Iterated Tensor Products
2. Spaces of Tensors
6. The Exterior Algebra of a Module
7. Determinants by Exterior Algebra
8. Subspaces by Exterior Algebra
9. Duality in Exterior Algebra
10. Alternating Forms and Skew-Symmetric Tensors
Appendix
1. The Affine Line
2. Affine Spaces
3. The Affine Group
4. Affine Subspaces
6. Euclidean Spaces
8. Projective Spaces
10. Affine and Projective Spaces

## 1/28/14

### Contents of Mac Lane's Mathematics Form and Function

CHAPTER I
Origins of Formal Structure
1. The Natural Numbers
2. Infinite Sets
3. Permutations
4. Time and Order
5. Space and Motion
6. Symmetry
7. Transformation Groups
8. Groups
9. Boolean Algebra
10. Calculus, Continuity, and Topology
11. Human Activity and Ideas
12. Mathematical Ideas
13. Axiomatic Structure
CHAPTER II
From Whole Numbers to Rational Numbers
1. Properties of Natural Numbers
2. The Peano Postulates
3. Natural Numbers Described by Recursion
4. Number Theory
5. Integers
6. Rational Numbers
7. Congruence
8. Cardinal Numbers
9. Ordinal Numbers
10. What Are Numbers?
CHAPTER III
Geometry
1. Spatial Activities
2. Proofs without Figures
3. The Parallel Axiom
4. Hyperbolic Geometry
5. Elliptic Geometry
6. Geometric Magnitude
7. Geometry by Motion
8. Orientation
9. Groups in Geometry
10. Geometry by Groups
11. Solid Geometry
12. Is Geometry a Science?
CHAPTER IV
Real Numbers
1. Measures of Magnitude
2. Magnitude as a Geometric Measure
3. Manipulations of Magnitudes
4. Comparison of Magnitudes
5. Axioms for the Reals
6. Arithmetic Construction of the Reals
7. Vector Geometry
8. Analytic Geometry
9. Trigonometry
10. Complex Numbers
11. Stereographic Projection and Infinity
12. Are Imaginary Numbers Real?
13. Abstract Algebra Revealed
14. The Quaternions-and Beyond
15. Summary
CHAPTER V
Functions, Transformations, and Groups
1. Types of Functions
2. Maps
3. What Is a Function?
4. Functions as Sets of Pairs
5. Transformation Groups
6. Groups
7. Galois Theory
8. Constructions of Groups
9. Simple Groups
10. Summary: Ideas of Image and Composition
CHAPTER VI
Concepts of Calculus
1. Origins
2. Integration
3. Derivatives
4. The Fundamental Theorem of the Integral Calculus
5. Kepler's Laws and Newton's Laws
6. Differential Equations
7. Foundations of Calculus
8. Approximations and Taylor's Series
9. Partial Derivatives
10. Differential Forms
11. Calculus Becomes Analysis
12. Interconnections of the Concepts
CHAPTER VII
Linear Algebra
1. Sources of Linearity
2. Transformations versus Matrices
3. Eigenvalues
4. Dual Spaces
5. Inner Product Spaces
6. Orthogonal Matrices
8. The Principal Axis Theorem
9. Bilnearity and Tensor Products
10. Collapse by Quotients
11. Exterior Algebra and Differential Forms
12. Similarity and Sums
13. Summary
CHAPTER VIII
Forms of Space
1. Curvature
2. Gaussian Curvature for Surfaces
3. Arc Length and Intrinsic Geometry
4. Many-Valued Functions and Riemann Surfaces
5. Examples of Manifolds
6. Intrinsic Surfaces and Topological Spaces
7. Manifolds
8. Smooth Manifolds
9. Paths and Quantities
10. Riemann Metrics
11. Sheaves
12. What is Geometry?
CHAPTER IX
Mechanics
1. Kepler's Laws
2. Momentum, Work, and Energy
3. Lagrange's Equations
4. Velocities and Tangent Bundles
5. Mechanics in Mathematics
6. Hamilton's Principle
7. Tricks versus Ideas
8. The Principal Function
9. The Hamilton-Jacobi Equation
10. The Spinning Top
11. The Form of Mechanics
12. Quantum Mechanics
CHAPTER X
Complex Analysis and Topology
1. Functions of a Complex Variable
2. Pathological Functions
3. Complex Derivatives
4. Complex Integration
5. Paths in the Plane
6. The Cauchy Theorem
7. Uniform Convergence
8. Power Series
9. The Cauchy Integral Formula
10. Singularities
11. Riemann Surfaces
12. Germs and Sheaves
13. Analysis, Geometry, and Topology
CHAPTER XI
Sets, Logic, and Categories
1. The Hierarchy of Sets
2. Axiomatic Set Theory
3. The Propositional Calculus
4. First Order Language
5. The Predicate Calculus
6. Precision and Understanding
7. Gödel Incompleteness Theorems
8. Independence Results
9. Categories and Functions
10. Natural Transformations
11. Universals
12. Axioms on Functions
13. Intuitionistic Logic
14. Independence by Means of Sheaves
15. Foundation or Organization?
CHAPTER XII
The Mathematical Network
1. The Formal
2. Ideas
3. The Network
4. Subjects, Specialties, and Subdivisions
5. Problems
6. Understanding Mathematics
7. Generalization and Abstraction
8. Novelty
9. Is Mathematics True?
10. Platonism
11. Preferred Directions for Research
12. Summary

## 1/25/14

### Organizing Mathematics and Saunders Mac Lane

In my post Reading, Writing, and Reworking I mentioned that I was reading MacLane's Mathematics Form and Function. It is a delightful book which brings the central parts of modern mathematics together into a single text without requiring the reader to slog through the technicalities of proof. That is not to say that his development is entirely informal. He has taken great care in the organization of his mathematical narrative. MacLane states "organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics." This book is his attempt to sketch mathematics from human activity to rigorous formalism. MacLane shows that such an organization is both economical and understandable.

It is not a primary focus of his text to reveal how the passage from human activity to formal functionalism (as he calls it) is an instance of analysis and synthesis. The human activities mentioned by Mac Lane include simple things such as grouping, selecting, observing, and other equally vague notions. It is the inherently vague character of his descriptions of human activity which provide the perfect starting point for his analysis throughout the remainder of the book. At each chapter one is compelled to return to his list of human activities and to see if each corresponds in a "comfortable" way to the formal description being given. For example, the act of counting leads to the idea of next and finally the formal concept of succession and order. He does not intend for these correspondences to constitute a defensible argument, but rather they represent the intuition which has gone into his organization of the chapters and concepts that follow.

Since his goal is not to produce an instance of the analytic and synthetic method he leaves it to the reader to evaluate the fitness of his descriptions to address the vague human activities identified in the introductory chapter. Once he arrives at the mathematical topics which are currently principle to any modern study of mathematics he has achieved his goal and only rarely looks back at the robustness of the path taken. As if to openly acknowledge this fact he places some emphasis at the end of the text on future avenues of inquiry and isolates the furthering of mathematical understanding as being equivalent in value to mathematics as a whole as the usual practice of proof and conjecture.

Mac Lane concludes, quite reasonably so, that there does not seem to be a single origin of mathematics, or a single foundation for that matter, but rather a myriad of starting points each of which reveals some different part of significantly similar ideas born from yet another starting point. This conclusion is still today contrary to the beliefs of many working mathematicians who are under the impression that the mathematics which they study is THE mathematics, and that for their mathematics there is a single foundation from which it springs.

Perhaps MacLane's greatest accomplishment in this book is to present a narrative which is not in any way mystical or fantastical yet lays bare some of the essential points of mathematics which must be addressed by any meaningful theory of knowledge or, in particular, philosophy of mathematics. It was believed by Russell that the logical machinery used in his and Whiteheads Principia Mathematica constituted a single system for building and evaluating the various equivalent starting points which lead to our modern mathematical ideas. Sadly he was only able to latch onto the reductions of his time, that of set theory as a foundation for mathematics. We have since discovered that set theory clearly comes in many flavors, and that it might not be as fit for a foundation of mathematics as previously believed, though Bourbaki's Elements of Mathematics shows that the discovery of new mathematics as well as the organization of old mathematics can benefit from a decidedly set theoretic description. Gödel and others have also revealed additional doubts as to the fitness of certain logical reductions in any foundation of mathematics, which themselves have yet to be put in a remedial context.

The following questions came to my mind while reading his book.

1. What methods are there for evaluating the fitness of his analytic and synthetic descriptions?
2. Is there a well written book with a title akin to Mathematics as Based on the Theory of Categories? Specifically one which is not needlessly "heavy handed" in its insistence on a strictly category theoretic description of math. I'm looking for something like Bourbaki with a less restrictive dogma.
3. What role does Russell's method of positive doctrine play in the presentation of alternate origins of each mathematical concept?
4. Is there a well written evaluation of the progress made since the publication of Mac Lane's text?
5. To what extent are MacLane's descriptions analytic or synthetic? The purpose being to use MacLane's work as an outpost into the vague notions of analysis and synthesis.

## 1/21/14

### Review of Posts Past

I like blogs that review their posts and progress. Here I provide context, commentary, and criticism on my past posts to summarize, revise, and evaluate their effectiveness and relevance.

In the past three years I've written 65 posts (not including this one) on 23 topics. The math posts written before 2/24/12 follow the standard structure of mathematical exposition: definition, proposition, proof. Those written after do not contain a single mathematical expression. I attribute this sharp divide to my project and my departure from graduate school.

Since leaving graduate school I have focused on collecting, organizing, and analyzing math and physics. This past year I realized that private progress may not be relevant to my larger goals; amassing knowledge without sharing it seems unwise. Thus education has become a new and vital interest of mine. Rather than stockpiling mathematical facts I wish to facilitate their dispersal by building narratives which fit the facts and avoid distracting obstacles.

To satisfy my educational desires (and to make money) I tutor students in math and physics. As a tutor I work hard to demystify calculus, statics, dynamics, algebra, and geometry by explaining how abstract concepts tend to be related directly to basic human activities. For example, I might explain modular arithmetic by winding ropes around barrels, or the composition of functions as the successive movements of a block on a table. By connecting something abstract to its concrete origins I help students see that the abstract is not removed from reality: math may be mystifying but it need never be mystical.

I have found that knowledge of mathematics and physics helps me to connect the abstract to the concrete. In addition to understanding the logical origin of a concept one must know the way in which the logical development corresponds to human activity. This not only establishes relevance in the eye of the student but also helps build intuition. Thus I have been working hard to pin down the logical origins of the principle parts of math and physics in a way which lends itself to such concrete analogies.

Those narratives which seem to help students the most follow a simple yet general problem solving process. My template for problem solving is Polya's How to Solve It. Polya's system uses questions to shift attention and focus the mind on what is most important to finding a meaningful solution to a given problem. I say "meaningful solution" because it is often the case that students find solutions to problems without knowing whether their results are relevant to the problem. This pattern will only become more prominent as Google improves its search engine.

Over the past month I have become more familiar with the principles and standards used in America to guide the way in which math is taught at the elementary, middle, and high school levels. Though I have yet to compile a detailed list of the differing opinions on what constitutes a reasonable mathematics curriculum at these levels, I have noticed that much of the available information tends to focus on telling students about math rather than showing them mathematical reasoning in action. I do not think that my comments are anything more than vague conjecture, but I intend to tease out the relation between mathematical fact and mathematical education.