Topics seem more accessible when their motivations are made clear. This is especially true for abstract subjects such as mathematics.
The vast majority of us, who have achieved basic numeracy and use our mathematical insight to interpret our world, will have to settle for the status of ‘mathematically inclined’. Our motivation is not to be wrong in life’s decisions, when mathematical insight enables the more correct choice.
Researchers, who utilize mathematics to work with their field’s phenomena, are termed mathematicians. This includes the theoretical mathematician whose field of study is mathematics itself. Mathematicians spend long years, first being educated in basic mathematical truths and logic, then in current applicable mathematics of their intended areas of expertise, then applying this knowledge to pick away at hard problems with no guarantee of success. To fully understand this, we need to identify their motivation.
The initial mathematical motivation is our curiosity, arising from questions posed to us by our intuition. Born of the objective (real) world, our intuition further suggests the necessity for our mathematics. Humans need a language that can precisely describe the objective world, express its organizing principles, and identify coordinates and operations via which we can measure, predict, manipulate, project, transform, ….
Curiosity motivates discovery. The curious child wants to know why. Along the way, the minds of certain observant curious children, our likely future mathematicians, will process many associations and from these associations they will sense patterns. When attempting to understand the why of the patterns, they will find mathematics a good friend.
In other instances I was not guided by [architectural] motives, being attracted only by curiosity, by the need to know the answer to an enigma, without reference to its importance in a general context.
Borel
With more mathematical experience, the potential and perceived utility of mathematics becomes a source of inspiration and appeal. Our modern world would be impoverished without the utility of mathematics. Theoretical mathematicians are further motivated by the generality, abstraction, and unifying power of mathematics.
Keeping It Real
Physical reality, as we perceive it, is subjective, of our senses. Mathematics allows us to render our experience objective, and thus to better understand, classify, and communicate what we observe. The power of mathematics to coordinatize a physical concept, to measure, abstract, and operate with it, enables our rapidly advancing understanding of natural law and phenomena.
The descriptive power of mathematics has evolved over many millennia. With regard to physical reality, we have used mathematics to inform in diverse ways:
- measure/estimate, beginning with simple enumeration
- compute (operate on) mathematical objects
- identify invariants
- discover dimensionality and shape
- identify pattern relationships
- explore macroscopic phenomena: dynamics, mass, energy, frequency
- express randomness and probability of occurrence
- explore atomic, relativistic, and quantum phenomena: wave functions
About Patterns
The essential spark that ignites our curiosity about reality may be recognition of patterns. Patterns are based in mathematical rules governing object behavior. Similar objects with the same constraints behave in a similar way. In this sense there is a mathematical basis for their pattern of behavior. Patterns form a principal connection between real world objects, their observed ‘behaviors’, and the corresponding mathematical objects and relations that we use to describe them.
The book Fearless Symmetry by Ash and Gross defines mathematics as the logical study of patterns. Since patterns are likely motivators of mathematical curiosity, let’s view the big pattern picture.
A pattern is a discernible regularity of an arrangement of entities, governed by some logical rule that facilitates prediction and contemplation. Within mathematics, patterns often arise in sets with structure, the objects of abstract Algebra. Notable among these are sets comprised of numbers (Number Theory, Arithmetic), points in space (Geometry), and solutions of equations (Algebraic Number Theory or ANT).
Applied or Abstract?
We often hear of two kinds of mathematics, applied and abstract. But that is a distinction without a difference. There is and will only ever be one ‘kind’ of mathematics. Rather, we should consider these the different motivations of the mathematician, who understands mathematics as the descriptive language that objectifies our reality. The ‘abstract’ motivation is a desire to more fully develop this language via a pursuit of pure reason, guided initially by intuition about the objects being described. The ‘applied’ motivation is a desire to solve problems presented by the real world, and thus the need to develop the mathematical tools to facilitate solvability. All mathematicians share both these motivations, to varying degrees.
Motivating Students of Mathematics
Problematically, mathematics is sometimes taught from reference books, written in a Bourbaki-like approach to mathematics as an application of pure reason. Such a practice was popular in the mid-1900s, and is hopefully on the wane. Denying the student insight into real world motivations for the subject can impede or dismiss entirely the student’s interest.
In mathematics, Shafarevich, for one, favors expository texts (filled with examples, but no proofs), introducing topics by talking about them conversationally. For example, his book Algebra 1, emphasizes accessibility over formalism and completeness. Such a text could throw in a proof or two to keep the reader in touch with the methodology of mathematics, but only if the proof illuminates some aspect of the surrounding mathematics. Reference books can then offer development insights of specific topics in a more complete, formal manner. For someone destined to teach mathematics at a secondary school level, study at the expository level is likely enough preparation.
In physics, Feynman, in his Lectures, asked himself the same question regarding pedagogy, answering that it is best to emphasize accessibility over correctness early-on. Armed with the ensuing insights, the study of the full-blown theory will be both more exciting and more productive. Of course, correctness will ultimately be ensured by a competent author. Feynman and Shafarevich are of the requisite caliber. For authors less well known, look for a sizable bibliography, over 100 references desirable.
It is true that the brightest minds can dispense with expository introductions and intuit instances of abstract objects, but even those minds would be more productive if provided a quick, highly accessible introduction. For the middle spectrum of intellect, there is a risk of alienation and defeatism resulting from a formalistic approach, and subsequent failure to learn the subject.
Where do we find the examplars for teaching about mathematical objects? Classical physics is run through with mathematics, enabling analysis of the natural world. The connections between “natural laws” and mathematics run deep and illuminate large swaths of mathematical thought. Understanding these connections is bound to make the subject more accessible. Also, other branches of mathematics, such as geometry and topology, can provide exemplars for conversing about algebra, for instance.
Basic to mathematics are its several number systems, each motivated by a need to coordinatize a newly defined mathematical object. It is there we find the numbers’ utility and reason for being. Discussion of number systems in a purely abstract sense makes no sense at all in an introductory text.
Mathematics at all levels finds physical exemplars. The extension of mathematical coordinatization to the ‘axioms’ of quantum mechanics (QM), the Dirac-von Neumann equations, concerns itself with the non-trivial (uncertain) probabilities associated with measurement outcomes of QM particles. This can motivate the study of the abstract topic of Hilbert spaces, well-suited to the realm of QM. The lesson is to keep any existent or potential application of the mathematics in our mind’s eye as we learn it.
In QM as a first example, we see mathematics subsuming the physics. This foundation places QM in the realm of metaphysics (mathematics), more so than of natural law. Feynman’s advice to physicists, complaining about the non-intuitive weirdness of QM: ‘Shut up and calculate’. Only through the specialized mathematical methods of QM, can we operate in a world beyond our physical insights.
Can such advanced algebra and functional analysis be taught without mentioning physical systems? Yes, and it still most often is. But do we lose insight and understanding in the process? Undoubtedly. Does the beauty of mathematics reach its full intensity? Not in my view. For instance, the beauty of QM mathematics derives from its providing a first glimmer of the rules of God’s dice game, a game whose rules are still far from our intuition.
Here we also encounter a good example of a philosophical divide between mathematical pragmatism and precision. Dirac was pragmatic when postulating his QM axioms, attempting to maximize understanding. Van Neumann objected and proposed a version with complete precision, but the precision potentially clouded one’s understanding. It seems the answer is to employ both motivations, pragmatism and precision. Let’s be pragmatic in initial approaches that can sharpen our underlying intuition, then explore the full analytical precision.
Further regarding mathematical pedagogy, motivation arises from perceiving connections between mathematical subject areas. Standard pedagogy is to first teach each subject area as a separable space, with its own unique concepts and methods, waiting until advanced studies to explore interdependencies. But the joy of mathematics is best stimulated by contemplation of a whole entity whose parts cooperate and support each other, and hence also support our deeper understanding. We need to keep such cooperation between mathematical areas in front of students from the beginning of their studies, and the exemplars will arise from our mathematical physics. Thus we can exploit the advantage of a less precise unity, prior to precisely focused detailed studies.
Beyond Intuition and Reality
We see the motivation to do mathematics beginning with our curiosity, exploring its utility for comprehending the natural world of our intuition. Human intuition, based in the physical world in which we evolved, serves us well in understanding this reality, up to a point.
But much of modern physics lies beyond the reach of our intuition, in dimensions that extend beyond our experience. We must follow the mathematics blindly until recognition eventually sets in and we are able to extend our human intuition through reason.
Conversely, we continue to develop advanced mathematics beyond any known application to physical understanding of our world, because it is there, latent, demanding of our conscious attention. For example, the basic mathematical concepts of the infinite and continuum have no understandable physical counterpart, yet the edifice of mathematics would be impossible without them. They are the limitless extrema of an edifice that we intuit to exist only in the human consciousness.
Other Prime Motivators
Beyond the limits of current understanding, mathematicians are motivated by an hypothetical endgame, an imagined universal mathematical framework, a memeplex of concepts and rules that addresses all logical eventualities, its every hypothesis decidable. We’ll discuss further below in the Foundations section why such a dream is likely unreachable. How close we can come is not yet determined.
At a visceral level, what keeps mathematicians going is the excitement of the chase, removing mysteries and lifting veils. And behind all these motivations is the aesthetic of mathematics, the knowledge that new and greater beauty is the true attraction and reward of success. It is this beauty that reminds us of Blake’s poetic query: What Immortal Hand or Eye…?
areas of interest 