Some Reflections on Kleene's Introduction to Metamathematics and Bourbaki's Theory of Sets

Recently I have been reading Kleene's Introduction to Metamathematics and have found his descriptions of the various foundations for mathematics exciting. I am excited not only to find such well written descriptions of the methods of metamathematics, but also to find another mathematician whose beliefs appear to be quite close to my own, at least in his description of metamathematics. I do not claim to know to what extent his beliefs about non-mathematical things are in line with my own as I have found through experience that a mathematicians beliefs about mathematics, though they should logically have a strong influence on their beliefs about non mathematical things, tend to have a sporadic impact if any on their non-mathematical beliefs.

His text has been inspiring both because of its clear descriptions of formal mathematics in a way which is much more prescient than the others that I have read, and noticeably accessible in its accommodations of a wider scope of investigation into intuitionistic logic, and because of its support of my belief that there must be greater effort put into describing the mechanics of mathematics, that is the details of the actions which are themselves mathematical. Specifically, the recognition that to the formalist the objects of mathematics are certain texts which represent a formal description of some area of mathematics without appealing to the meaning that is usually attached to the objects described by the formalism. As much as mathematics is meant to discover things which are timelessly relevant it would seem the formalists have made a key observation which will persist, in some form, for at least the next century if not the next millennium.

Unsurprisingly, while I've been reading Kleene's IM I have been comparing its observations to Bourbaki's Theory of Sets. Much of the criticism of the work of Bourbaki is focused on its Hilbertian foundations. There are also those who dislike that Bourbaki only mildly mentions the work of Gödel, Kleene, Bernays, Cohen, Church, Schoenfield, Tarski, Brower, Gentzen, Skolem, and all the other mathematicians, logicians, and mathematical logicians who have contributed to our modern understanding of the foundations of mathematics.

The claim that Bourbaki's Theory of Sets is Hilbertain is strict in that it uses a form of Hilbert's Epsilon Calculus as its logical foundation. Instead of using Hilbert's \(\epsilon\) notation, Bourbaki uses a distinct \(\tau\) and \(\square\) notation with linkages. Informally, if \(R(x)\) is a relation and if there is an object which satisfies the relation \(R(x)\) when substituted for \(x\) then \(\epsilon x R(x)\) represents such an object, otherwise \(\epsilon xR(x)\) is a thing about which nothing meaningful can be said. Again, this vague description is not meant to be anything other than an informal description. I do not yet know enough about the development of Hilbert's epsilon calculus to make a formal comparison between its use in his program, and its use in Bourbaki's theory of sets.

Though I am not deeply familiar with Hilbert's epsilon calculus in general, I am familiar with the logic of Bourbaki, and find it interesting that the logical symbols \(\tau\) and \(\square\) are used, together with an informal notion of substitution, to define the existential and universal quantifiers. Additionally, the modern role of the axiom of choice is reduced entirely to certain statements involving the \(\tau\) notation. This is yet another point of contention with most modern mathematical logicians, as some believe this method "hides" the use of the axiom of choice in the formal language rather than as an additional axiom on a footing equal to the pairing axiom or power set axiom.

As far as Bourbaki is concerned, the purpose of their description of formal mathematics is to avoid circumlocutions leading up to their development of their theory of sets. Bourbaki does not set out to show how their theory of sets sits among the differing foundations of mathematics, they simply wish to get as quickly as possible to a description of sets so that they can proceed with their treatment of the rest of modern mathematics. Said another way, it is not the purpose of Bourbaki to formalize mathematics, but rather to present it as being describable using the basic notions of set theory. The belief that mathematics may be almost entirely based on set theory is often given the hyperbolic name "Cantor's Paradise" so as to give credit to Cantor's independent discovery and creation of the precursors to modern set theory.

It is admitted by Bourbaki that their informal goal is to use set theory as a way of classifying and organizing modern mathematics. Specifically, the final chapter of Theory of Sets is entitled Structures and the second section is entitle Species of Structures. Though the definition of structure is given formally, it includes the formal language of set theory developed earlier in the book. The basic idea is that a structure is an element of some set which is built from the fundamental constructions permissible in Bourbaki's set theory. For example, within the classical theory of sets a topology \(\mathcal{T}\) on a set \(X\) is a collection of subsets of \(X\) which satisfy two relations:

  1. if \(\mathcal{U}\) is a subset of \(\mathcal{T}\) then \(\bigcup \mathcal{U} \in \mathcal{T}\); and
  2. if \(\mathcal{U}\) is a finite subset of \(\mathcal{T}\) then \(\bigcap \mathcal{U} \in \mathcal{T}\).
Thus, a topology on a set \(X\) belongs to \(P(P(X))\), the power set of the power set of \(X\). Furthermore, the collection of all topologies on a set \(X\) is a subset of \(P(P(X))\) whose members satisfy these two topological relations. Consequently, the set of all topologies on a set \(X\) is a unique member of \(P(P(P(X)))\), so that what it means to be a topology on \(X\) is governed in part by a specific set of axioms and membership to a set constructed from \(X\) by iterating the power set operation.

A somewhat simpler example is the structure of relations on a set, which are completely described by being a member of the cartesian product of the set under consideration. That is, a relation on a set \(X\) is a member of \(X \times X\) and the collection of all relations on a set \(X\) is \(P(X \times X)\). So that \(R\) is a relation on a set \(X\) if and only if \(R \in P(X \times X)\). Here it is clearer that what it means to be a relation, or what the structure of a relation is, on a set \(X\) is characterized by the construction of the power set of the cartesian product of \(X\) with itself.

So the species of structures which are examined in mathematics are, as is assumed by Bourbaki, of this form. Each species of structure has some set or collection of sets as its base, like \(X\) in the previous two examples, and may be specified by writing down to which set constructed from \(X\) the structure belongs as well as the basic axioms that the structure must satisfy.

Much of the detail has been removed from this informal description, and modern mathematicians have continued to develop these ideas, though it still seems that the notion of structure is meaningless outside of a formal description of mathematics. The modern notion of structure has been taken back from the theory of sets by the model theorists (and universal algebraists) so as to not only permit varying structures of set theory, but also provide a more comprehensive framework for structures describable using a formal language which might not contain a theory of sets. In this context, specifically the model theoretic, a structure is more commonly referred to as an interpretation.

This push towards the logical foundations of mathematics brings me back to my interest in Kleene's IM. Though Shoenfield's Mathematical Logic is considered the modern introduction to mathematical logic, Kleene's IM, as well as his own Mathematical Logic which may be bought as a Dover book, is a much slower and digestible than Shoenfield's terse and dense book. Both are beautiful and for those interested in modern mathematical logic the two texts, Kleene's IM and Shoenfield's ML, are required reading. One important component of Kleene's IM which is sadly missing from Shoenfield's ML is Intuitionistic Logic. Kleene gives a description of the three most well known approaches to the problem of finding a foundation for mathematics: the logicists of which Russell and Whitehead are the principle members, the formalists of which Hilbert is most well known, and the Intuitionists of which Brouwer is the most well known.

Though each school of thought is based on a differing set of beliefs each shares some characteristics with the others. In particular, the use of formal languages may be identified across each school. Though, it is interesting to note that Gödel's monumental paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems is rooted in his recognition that the logical mechanisms employed by Russell where not formally described, and that once they were certain questions of consistency and completeness naturally arose. Russell's description of mathematical logic was of a decidedly philosophic nature, occasionally relying on informal descriptions rather than formal syntactic rules, as he believed that his foundation of mathematics was a description not of a philosophic ideal, from Plato, but rather a tool for describing that section of philosophic logic which results in our modern study of mathematics. His thesis, that mathematics is reducible to logic, is often misunderstood as referring to the modern use of formal texts to describe mathematics. Rather, Russell would clearly identify formal texts as being part of mathematics, and the type of thing which must ultimately be justified from a philosophically defendable position, one which necessarily requires the acceptance of some metaphysic and logic which itself need not be a formal mathematics.

The distinction between the formal and informal components of mathematics is expertly revealed in Kleene's IM. It is the principle reason his exposition is so familiar to modern mathematical logicians. The study of metamathematics is the formal study of formal mathematics, which, apparently, requires the use of informal mathematics. The use of certain fundamentally informal mathematical notions forms the center piece of Brouwer's intuitionism, wherein he believes that the only mathematical arguments which are admissible are those which are supported by a sort of informal acquaintance with certain features of the natural numbers. One of these informal notions is that one can not realize a completed set of natural numbers. Thus a proof of a statement "for all natural numbers \(n\) it is that \(P(n)\)" where \(P(n)\) is some statement about natural numbers, must proceed without the assumption that there is a collection of all those natural numbers \(n\) for which \(P(n)\) is true and a collection of all those natural numbers for which \(P(n)\) is false, otherwise these joint collections would entail the realization of the whole collection of natural numbers. This excludes the use of the classical proof by contradiction which presupposes the existence of such sets. That is, such a proof requires you to accept that \(P(n)\) is "already known to be true or false" of each natural number, something which Brouwer rejects. Obviously these descriptions are informal, hence the value of Kleene's IM.

Kleene's metamathematics gives a framework for dealing not only with classical logics, but also those of the intuitionistic variety. This process is exactly analogous to that employed in the discovery and description of non-Euclidean geometries.

In the future I hope to provide clear and exact descriptions of these informal reflections. The tools for such an exposition appear to be in Kleene's IM and, if not, will likely be found in Shoenfield's ML.


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 \(A\). So 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.


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
    5. Addition and Multiplication
    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
  9. Bilinear and Quadratic Forms
    1. Bilinear Forms
    2. Symmetric Matrices
    3. Quadratic Forms
    4. Real Quadratic Forms
    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
    1. Quadratic and Cubic Equations
    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
  14. Categories and Adjoint Functors
    1. Categories
    2. Functors
    3. Contravariant Functors
    4. Natural Transformations
    5. Representable Functors and Universal Elements
    6. Adjoint Functors
  15. Multilinear Algebra
    1. Iterated Tensor Products
    2. Spaces of Tensors
    3. Graded Modules
    4. Graded Algebras
    5. The Graded Tensor Algebra
    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
  1. The Affine Line
  2. Affine Spaces
  3. The Affine Group
  4. Affine Subspaces
  5. Biaffine and Quadratic Functionals
  6. Euclidean Spaces
  7. Euclidean Quadrics
  8. Projective Spaces
  9. Projective Quadrics
  10. Affine and Projective Spaces


Contents of Mac Lane's Mathematics Form and Function

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
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?
  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?
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
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
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
Linear Algebra
  1. Sources of Linearity
  2. Transformations versus Matrices
  3. Eigenvalues
  4. Dual Spaces
  5. Inner Product Spaces
  6. Orthogonal Matrices
  7. Adjoints
  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
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?
  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
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
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?
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


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.