First Principles Mathematics

This is a draft outlining original resources that could be made to describe the foundational constructions of mathematics with the goals of:

• Articulating the motivations behind different definitions and constructions, aiming to make definitions as obvious as possible, e.g. with a suitable definition of a jump, to further define that A rational function is continuous if it has no jumps.
• Developing a strong sense of not just the sufficient conditions for a proof, but what role each premise has in that proof, and how that informs intuition.
• Maintaining an early distinction between constructive algorithms and non-constructive proofs, and with clear justification of what an incomputable function is and why some topics deal with such fictional objects.
• Presenting statements in a way that strengthens one's intuition for what kinds of things are true and why they are true, and not allowing mystified confusion to be treated as profound.
• Building a mathematics that is based on real-world epistemic inferences like that of Euclid, rather than exploring the consequences of formal or informal rules of symbol manipulation.
• Building concrete mathematical systems before analysing abstract structures that those systems exemplify.
• Reasoning explicitly about the analogies and equivalences that often come up in mathematics, as basic concepts in the philosophy of mathematics, rather than leaving them as vague feelings until set theory and category theory and homotopy type theory make them concrete.
• Finding genuine improvements over the historical development of mathematical ideas, such as the use of linear algebra and vectors without a prior theory of complex numbers or quaternions.
• Making sure to develop explicit algorithms describing how repetitive manipulations (especially algebraic manipulations) can be generalised, when such manipulations are required for later theories.

Attempting to write a mathematics that achieves these goals will be valuable on its own for me the author, but hopefully these constructions will be interesting and insightful to those interested in the philosophy of mathematics, the foundations of mathematics, or the communication of mathematics.

I. Philosophy

Part I. will be the most esoteric, and will be quite dry to those not directly interested in the philosophy of maths; this part should be skipped and returned to as a reference if that does not sound appealing to you. In either case, however, it is worth noting that the philosophy that we lay out here is not a formal system of abstract symbols, the way that most math foundations such as ZFC are set up; instead we make claims about the material world itself, like a formal and thoughtful version of what is taught in elementary schools – for example, we will later come to define quantities as abstract properties and measurements of the objects we experience in the world, and numbers as names that represent these quantities, nothing more; to this end we cannot simply postulate a list of rules that describe which sets or types or categories do or do not exist, because in doing so we have already left the real world. Instead Part I. builds up a basic vocabulary that has more in common with epistemology and linguistics than with the axiomatic model theory that formalists tend to prefer.

Basic Arithmetic

This section is the beginning of the real mathematics, and in it we will build up the rational number system. In our desire to articulate and motivate our definitions explicitly we end up with a sequence of constructions that are much more involved than the definitions usually given (if at all) in mainstream mathematics. In their simplest state numbers are meant to represent how many of something we have; the goal of this chapter then, is to build up the natural, integral, and rational numbers while keeping track of what real life phenomena we are actually quantifying by using these different numbers.

In everyday life counting arises as a technique for coordinating physical objects, and the very beginnings of mathematics arise directly out of increasingly sophisticated methods of coordinating these physical objects. We can understand counting and whole-numbered quantities as a description of properties of physical collections, we can develop techniques of account for objects by manipulating symbols that signify these quantities, and eventually realize the constitutive power of these symbols as a basis for increasingly rich and expressive number systems. It is this last aspect of counting and common numbers that we aim to establish, and so we should describe:

• The visually discernible quantities one, two, three, e.g. *, ||, •••,
• Practical phenomenology of small quantities: connectedness, reflective symmetry, bijectivity,
• Incremental counting for reasoning about larger bijections (incrementation is a more practical, pre-numerical version of mathematical induction),
• Addition as an invariant of counting multiple collections,
• Multiplication as addition of equal sized collections,
• Arithmetic procedures, arithmetic expressions, arithmetic identities,
• Algebraic equations as symbols representing possible identities,
• Subtraction and divisors as possible identities,
• Basic number theory of Euclidean division, greatest common divisors, prime factors.

Integers and Rationals

While not all subtractions or divisions have a well defined result, the difference or ratio between pairs of numbers is still an important quantity of those pairs, and by taking those pairs as a new mathematical object, and identifying them based on their difference or ratio, we can construct new number systems. The integers are useful in accounting, and the rationals are useful in measuring physical quantities with arbitrary precision, rather than the fixed precision of any given unit of measurement. Once again we avoid both of these practical concerns and primarily construct and reason about these systems for their use as constituents of even richer objects such as polynomials, quadratic extensions, the rational number line, geometric manifolds, rational functions, and even Cauchy sequences and Dedekind cuts.

High level topics

With the full rational number line in place many aspects of mathematics become possible to explore:

• Rational functions, metric analysis, cut continuity, Cauchy continuity, uniform continuity,
• manifolds, topology, standard topological continuity
• limits, completion, (real numbers) uncountability
• basic differential geometry, tangent vectors
• countable/topological measure theory
• polynomials and simplifying algebraic expressions, field extensions
• linear algebra, vector spaces
• geometric algebra, rotors and spinors
• type systems, type checking, univalence
• formal construction of graphs, connected components, communicating classes