Mathematics is of mind, human thought. Its domain of discourse, number, set, equation, proof, point, line, plane, sphere, infinity, indeed all mathematical concepts, are figments of human intuition. We sometimes see real instances of similar entities, lines, planes, etc., in our physical world. Yet such observations do not endow mathematical objects with existence independent of mind. Rather, we recognize that mathematics provides us a language, which by its nature is capable of abstractly describing aspects of our perceived reality.
All of mathematics is an abstraction, numbers, points, lines, planes, sets. They recall the forms of Plato, idealized objects. In our usage, these forms are given existence in so far as they relate to one another by definition within the abstraction that is mathematics. Unlike Plato, we do not assign mathematical objects a real existence independent of thought.
Unlike language definitions, which presume to capture the essence of the concept or object being defined, mathematical definitions usually only describe some behavior of an object, perhaps using hijacked English words assigned mathematical meanings. The actual meaning of an object (why someone would care about the object), rules for constructing an object, or even evidence that the object exists, are generally outside the purview of the mathematical definition.
Predication in natural language is by exemplifying, as with existent physical objects (e.g. Lilah is tall). But mathematics deals with abstract objects. A modified logic for abstractions is useful, where predication is by encoding (e.g. p is prime, in the sense that p possesses (encodes) the properties of a prime, or in other words, p satisfies the mathematical definition of primeness.
Aside: In mathematics education, the actual meaning of an object is too often left to be discovered by the reader/student, which makes student life much harder. A good teacher will endeavor to convey such meaning up front, by analogy and example using concrete instances from the real world or from the student’s prior mathematical experience. Such good teaching provides mental coordinates for maintaining one’s bearings and visualizing one’s context. Thus, meanings are essential to competent mathematical pedagogy, significantly enhancing the student’s ability to completely learn the material in one go-round.
Existence and Classification
Mathematicians seek answers to two essential types of mathematical question. The existence question asks if a mathematical object with assumed properties exists (in the domain of mathematical thought). The classification question asks how one should classify a mathematical object with given properties, placing it in a defined behavior class. Often, a goal of classification is to establish that a specific set of object properties identifies a unique object, perhaps by assuming there can be more than one with the given properties, and then showing that assumption leads to a logical contradiction. By exploring such questions with respect to patterns, mathematics becomes an essential tool for the sciences, wherever patterns in the natural world are detected.
Looking backward into the 19th century and before, existence questions increasingly involved questions of philosophy and an assumed dependence on, or derivation from, the objective (real) world. The resulting confusions were blown away when the domain of mathematics became understood to be entirely metaphysical, in the realm of human thought. While our intuitions are largely grounded in the objective world, mathematics is capable of exceeding the grasp of intuition.
Mathematicians approach questions by observing with the mind’s eye (as in Einstein’s thought experiments), aided perhaps by visual models, and perhaps by external marks on paper. But always the object of mathematical discovery is a thought construction, a mathematically defined entity connected logically to the existing mathematical framework.
To answer questions of existence, classification, and specific behavior, mathematicians employ two types of reasoning, inductive and deductive.
The inductive discovery technique observes the mathematical behavior of an object, intuits a general hypothesis explaining the behavior, and verifies that the hypothesis seems to work (can predict/explain logically consistent behavior). This technique has powered mathematics and all of man’s discovery since the dawn of our reign on earth. It is the truly fun part and the most creative part of mathematics.
Logic merely sanctions the conquests of the intuition.
My special pleasure in mathematics rested particularly on its purely speculative part.
If only I had the theorems! Then I should find the proofs easily enough.
Riemann (an odd statement from one who has a major conjecture named after himself that survives him to this day)
Plausible inference is the first step of inductive reasoning. It powers the creative discovery phase of many activities, creating a general hypothesis from observations and analogies interpreted by our intuition. Per Polya, “the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning.”
We can build our mathematics purely from such inductive reasoning (and a little luck). This was how mathematics was largely practiced prior to the 19th century. Although the Greek mathematicians had told us of the importance of demonstrable proof, slowing down to prove things could interrupt the creative flow (seriously interrupt, since some proofs wait centuries to be divined). Instead, we proceeded in a primitive manner, grading hypotheses by plausibility. Those most plausible (those with most people’s intuition in agreement, those with most evidence in support) would become popularly accepted fact. For two thousand years, the Unique Factorization hypothesis was accepted as fact without deductive proof, until Gauss turned it into a proven theorem.
There will always be doubt whether a plausible hypothesis states a fact. Much work can follow, referring to the hypothesis as accepted fact, only to have it be discovered that the plausible hypothesis was not factual after all. (Then it would be time to get out a really big eraser.) In modern mathematics, we call plausible hypotheses conjectures, and they linger in plausible limbo. They must be proved or disproved, no matter how plausible they seem.
Anticipating this issue, the classical Greek mathematicians added demonstrable proof to the toolkit, creating a formal mathematics whose truth was verifiable. The deductive reasoning of the proof complements the inductive method of concept discovery. Such formalized proof replaced hypothesis testing.
Proof employs deductive reasoning, using propositional logic, to establish the inferred general hypothesis as fact, by reasoning from prior known facts and current established conditions. Using this extended toolkit, the Greek mathematicians formally established both geometry and number theory as verified theories, still today, after 2500 years, vibrant as essential fields of mathematical exploration. Euclid’s Elements is the surviving text that tells us of their wonderful achievement. Euclid’s mathematics demonstrates use of deductive logic to establish new facts from existing facts.
This great chain of facts must be bootstrapped from some intuited facts beyond the reach of reason and its logic. Thus, mathematics is axiomatic at its core, based in a minimal set of a priori objects (mathematical definitions and predicates) and formulae (object behaviors). Such primitive concepts and assumptions are conceived by intuition, and are established as a logically consistent foundation through application of first order predicate logic. Axioms and definitions so derived provide the basis for deductive reasoning in mathematics. Deductive reasoning is discussed more fully below, under the mathematical logic heading.
The Mathematics Construction Project
Did the ancient scholars perhaps invent mathematics, or perhaps discover it? Discovery is a loaded word. It often suggests the object of interest is outside of us in our natural world, so ‘discover’ may create a confusing mixed message. We will sort these semantics later. Invention is closer to the nature of mathematical thought, but usually suggests choice by the inventor, and the chosen rules of logic permit no such freedom, except at the periphery where fundamental assumptions are themselves under investigation.
The best sense of mathematical creation is as an abstract construction within the mind. The entity being constructed is the memeplex we call mathematics, where mathematicians are the construction crew, the blueprint is the chosen logic (several are possible), and the foundation is a set of chosen assumptions and definitions (one has free will to assume, but it is best to choose wisely and not to assume too much). The practice of classical mathematics has bound these two variables, logic and assumption, as described in later sections which discuss logic and foundations.
Curiosity, inspiration, ingenuity, a sense of discovery, all empower mathematicians to suggest new memes for adding to our group construction. It is hard work because to successfully augment the construction, one needs working knowledge of a significant part of the entire structure.
In mathematics alone, each generation adds a new storey to the old structure.
I have been guided by a sense of the architecture of this edifice, to which we continue to add new wings and new floors while renovating the parts already constructed …
Once in a while, a great inspiration will lead to design of an entire new storey or a grand new hall. Then mathematicians will grab their tools and join the crew that builds out the new mathematical space. But more usually, a modest inspiration will add an extra shelf in a closet somewhere, or perhaps repair a hole in a wall. Each new addition begins with the inductive discovery process. Before the addition’s design can be accepted, the deductive validation process must ensure a perfect (logical) match between the new meme and the existing ediface (memeplex).
Different proof memes may be interchangeable. It is not unusual to find some theorems with five or more distinct proofs. Some are known to have hundreds of distinct proofs. Each proof by itself establishes factual certainty within the logic and foundations of the theory. More proofs do not make the theorem more certain and integral. But multiple proofs can make the surrounding mathematics richer.
For the indefinite future, there will be rewarding, challenging work for the mathematics construction crew. Since it is of mind, multiple minds working together may be expected to advance the construction project more rapidly. More complete understanding of mathematics requires our greatest human ingenuity, enhanced through teamwork. Thus, we may need to emend the romantic model of a lone genius-type, laboring years in solitude. The greatest leaps will still be expected from such extreme, solitary labor of a most brilliant mind, but most of future mathematical progress will result from participation in that great, continuing conversation between mathematical minds around the globe and across the centuries.
The memeplex that is our mathematical structure is described in books written by people who are skilled at visualizing mathematical memes and describing them in words and symbols.
We will use the abstract metaphor of a manuscript to refer to the description of the totality of our constructed mathematics. This manuscript is a metaphor for a great library of math books, some written millennia ago, some written this year. The manuscript is the documentation (User Manual) for our memeplex.
The Manuscript Can Be Edited
Mathematics is emended every day; those new truths are then written in our mathematical manuscript.
Fundamental change close to our core truths can also be imagined and implemented. Even the essential components of formal mathematics are not fixed. Various choices of logic and axioms have been proposed since Euclid’s time. Their choice determines the utility of the mathematics that result, the limits of the mathematics that result, and the ease with which that mathematics can be developed.
A specific set of choices, which we have refined for the last few centuries, has served us without change for the last 100 years or so. In that time, however, some cracks have been discovered in more esoteric areas that will ultimately require attention of very smart people.
We might intuit that exercising choice in logic and assumption will produce different mathematics variants. Universality will then require showing equivalence of any mathematics model variants.
Can mathematical sufficiency be validated, providing for a future Eureka moment – our millennia-long thought construction is completed? We now suspect this will not come easily nor from within the current construction, since no set of consistent axioms can validate itself.
The Manuscript Cannot Be Burned
If mathematics were in any way arbitrary, able to escape the bonds of our logic, its practice would become a mere game. Mathematics would lose appeal and mathematicians would loose motivation.
Fear not. Unlike other human thought projects (languages, religions, laws, …), if sentient beings were to reconstruct mathematics again from its beginnings somewhere else, it would look like our mathematics. For it is mathematics’ basis in pure reason that makes mathematics ‘the manuscript that cannot be burned’. If mathematics is ever lost, it will be retrievable again in equivalent form by another group of rational minds.