Topics seem more interesting when their motivations are made clear. This is especially true for abstract subjects such as mathematics.
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, ….
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
We often hear of two kinds of mathematics, applied and abstract. But that is a distinction without a difference. Rather, we should consider these the different viewpoints of the mathematician, who understands mathematics as the language that objectifies our reality, and who develops this language via a pursuit of pure reason, guided initially by intuition about the objects being described.
Shafarevich illuminates the real, objective utility of mathematical abstraction/coordinatization, through an example, an association of abstract mathematical objects with real phenomena. This example is useful in an exemplary way only. Its specific content, quantum mechanics, is neither understandable at this basic level, nor germane to the remainder of the discussion.
- state of a physical system
- a line φ in an ∞-dimensional complex Hilbert space
- scalar physical quantity
- self-adjoint operator
- simultaneously measurable quantities
- commuting operators
- quantity taking a precise value λ in state φ
- operator having eigenvector φ and eigenvalue λ
- set of values of quantities available by measurement
- spectrum of an operator
- probability of transition from state φ to state ψ
- |(φ, ψ)|, where |φ| = |ψ| = 1
Can such advanced algebra and functional analysis be taught without mentioning physical systems? Yes, and it most often is. But do we lose insight and understanding in the process? Undoubtedly.
The lesson is to keep any existent or potential application of the mathematics in our mind’s eye as we develop and teach it. Mathematics and physics go hand-in-hand and illuminate each other to our good advantage.
Beyond Intuition and Reality
Mathematics is particularly useful to us in comprehending our physical world, for often such use exceeds our intuition. Mathematics is further useful to us in comprehending our mathematical world, even when applicability to the physical world is non-apparent.
Human intuition, based in the physical world in which we evolved, serves us well in understanding this world, up to a point. But much of modern physics lies beyond the reach of our intuition, in dimensions that extend far beyond our experience. Then we need mathematical abstraction/coordinatization to show us the way, even though it taxes our understanding in those areas where intuition cannot follow. We must follow the mathematics blindly until recognition hopefully sets in and we are able to extend our intuition through reason.
There remains much of mathematics that currently has no known facility at all to explain our physical world. But mathematicians follow such logical trails to reach further understanding of our metaphysical mathematical world. And whenever the abstract mathematician constructs a new mathematical abstraction, it must still need to be coordinatized, so that it, in turn, can be measured, abstracted, and operated upon.
We continue to develop advanced mathematics beyond its 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 the continuum have no physical counterpart, yet the ediface of mathematics would be impossible without them. They are the bricks and mortar of an ediface that exists only in the mind.
Let’s define ‘mathematician’ most generally as a practitioner of advanced mathematics. Researchers in a field of study, who utilize mathematics to work with that field’s phenomena, are termed mathematicians. This includes the theoretical mathematician whose field of study is mathematics. The 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’.
Mathematicians spend long years picking away at hard problems with no guarantee of success. To understand this, we need to consider their motivation. 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.
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.
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 the 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…?
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 and behavior, 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).