With illumination, abstraction, the rendering of propositions elaborated, the elements and development of mathematics may now be described. First and foremost, a definition may be made by identifying with a term what is sensibly perceived and rendered according to its correct usage. For example, numbers, such as 1, 2, 3, etc., may be defined as what is assigned to discrete objects that are sequentially ordered, or, following Euclid, a straight line is a line lying evenly between its extremes. In Lonergan’s parlance, such definitions are nominal in form. In contrast, a circle may be nominally defined as a perfectly curved line, but Euclid defines it as a coplanar points equally distant from a fixed point. Such a definition has both sensible and intelligible features to it and, as such, is explanatory. In this form, further inferences may be made from knowing something to be a circle by this definition. In contrast, the definition of a straight line allows for no further non-trivial inferences. Instead, a postulate would be further needed to, say, there exists a line between two distinct points. The nominal definition of numbers, straight line, and circle along with the explanatory definition of a circle are outsight analytic principles while the postulate is a serially analytic principle.
For each of these definitions, there involves terms and relations. In forming them there occurs a basic illumination in which the terms and relations form a circle such that “the terms fix the relations, and the relations fix the terms, and the [illumination] fixes both.” From this, a refined definition of straight line may be formed, following Hilbert, in which a point and a straight line are such that two points determine exactly one line on which they both lie on. Here point and straight line are both fixed by the relation that two and only two distinct points determine a straight line. Such a definition is referred to as implicit and is serially an analytic principle.
Beyond the basic concepts in mathematics identified through definitions, the notion of quantity is intrinsic to its subject-matter. To be specify how we will understand this notion, let us make the notion of (natural) number implicit by identifying a term 1, a relation =, and an operation +. Further terms 2, 3, 4, etc. may be defined and related by 2 = 1+1, 3 = 2+1, 4 = 3+1, etc. Rules such as commutativity and associativity are further sensibly observed and verified as outright analytic principles that are subsequently understood serially through abstraction. A further operation of ´ may be defined and a rule distribution may be observed and abstracted as a serially analytic principle. By quantity, we will identify this way of defining the natural numbers as a general procedure: distinguish (1) rules, (2) operations, and (3) numbers for which (A) numbers are defined implicitly by operations, so that the so that the result of any operation will be a number and any number can be a result of an operation, (B) operations are implicitly defined by rules, so what is done in accord with rules is an operation, and (C) it is the trick to obtain the rules that fix the operations that fix the numbers.
To understand things as implicitly defined is to know them strictly as materially intelligible. Through symbolic representations[JT1] that sensible material may be granted to them to supply images for further manipulation and inferences. Such symbolism (following Lonergan) is apt “inasmuch as its immanent patterns as well as the dynamic patterns of its manipulations run parallel to the rules and operations that have been grasped by [illumination] and formulated in concepts.” In their sensible material form, manipulations may be performed in accord with the rules determining the operations by the acts of calculation and formation of equations. Resulting properties are arrived at either by given definitions and rules or by inferred deductions. This process provides clues, hints, and suggestions aiding to potential illuminations by experimentation which leads to either dead ends or promising pathways.
Furthermore, an apt symbolism must satisfy a principle of invariance in which the formation of analytic propositions rendered with them represent properties, relations, and equations that provide the means to fully express mathematical meaning. The mathematical meaning of an expression resides in the distinction of constants and variables, and in the signs and collocations that dictate modes of calculation, i.e. operations of combining, multiplying, summing, differentiating, integrating, and so forth. If the patterns formed in the rendering are subsequently manipulated so that the symbolic pattern of a mathematical pattern is unchanged, its mathematical meaning is unchanged. Moreover, if a symbolic pattern is unchanged by any substitutions of a determined group, then the mathematical meaning of the pattern is independent of the meaning of the substitutions. Finally, symbolism appropriate to any stage of mathematical development provides the similitude in which may be grasped by illumination the rules for the next stage.
This leads to understanding how mathematics can be developed through higher viewpoints. Consider, for example, the movement from natural numbers to rational numbers. Here the subject-matter consists of ratios a/b of two natural numbers a and b. The objective is to establish how =, +, ´ work in this context. From the viewpoint of natural numbers, two such numbers a and b are viewed rationally as a/1 and b/1. In this manner, a/1 = b/1 if a = a´1 = b´1 = b, a/1 + b/1 = (a+b)/1, and (a/1) ´(b/1) = (a´b)/1. To extend to general ratios, illumination reveals and establishes that a/b = c/d provided a´d = c´b (experience will further verify). This further leads to, first, a/d + b/d = (a+b)/d then to a/b + c/d = (a´d)/(b´d) + (b´c)/(b´d) =
(a´d + b´c)/(b´d). Finally, illumination shows, and experience verifies that (a/b) ´(c/d) = (a´c)/(b´d). That the rules for =, +, ´ for rational numbers are verified as the same operations when restricted to the natural numbers verifies that the rational numbers are mathematically a higher viewpoint to the natural numbers. Now, this scheme may be extended. For given rules, operations, and numbers determined by an established procedure in which the rules fix the operations which, in turn, fix the numbers, a higher viewpoint consists in an illumination from which a new procedure arises from the operations performed according to the old rules and is expressed in the formulation of the new rules.
Continuing to follow Lonergan, we now view mathematics as comprising of series in which each term of a series is a department which consists of (1) rules governing and so defining operations in which operations proceed from some terms to others and so relating and defining them, and (2) a formalization developed as an elaborate technique in which definitions are worked out, postulates are added, and, from them, further conclusions are reached within the departed by the rigorous procedure of deductive inference. Each series, then, has a further formalization comprised of:
- A formal element: From abstraction, illumination goes beyond similitudes and data by intelligible unities which contain a reference to similitudes or data but add a component to knowledge that does not exist actually on the level of sense or imagination. Furthermore, there is the act of preparation, or “learning of mathematics”, is a dynamic process in which illuminations are gradually acquired necessary to understand mathematical problems, to follow mathematical arguments, and to work out mathematical solutions. Such an acquisition occurs in a succession of higher viewpoints.
- One department follows upon another.
- Logically, they are discontinuous: each has its own definitions, postulates, and inferences.
- Intellectually, they are continuous: symbolic representations of operations in the lower fields provides the similitudes in which intelligence grasps the idea of the new rules that govern the operations in the higher field.
- A material element: By the senses, data is first perceived in material that is individual, in particular time and place, from which understanding abstracts. The movement from experience to intelligence is not free but linked existentially: from lower to higher viewpoints, from the particular to the general, from the approximate to the ideal. From concrete instances of one, two, three, the mathematician explores the totality of natural numbers, of real numbers, of complex numbers, of ordered sets. In every field in which mathematics may be applied, the mathematician sets out to explore the whole of each region in which the fields occur. Besides its preference for the general, the complete, and the ideal, the development of mathematical thought also is restricted by its material element. Material as first received by the senses supplies mathematics with samples of the type of stuff in which mathematical ideas confer intelligibility and order. Such is the intelligible matter which goes beyond the determinate kinds of data that other sciences deal with, rather embodies all possible data.
- An actual element: The conjunction of the formal and material element. Heuristic structures of empirical methods operate in a scissors-like fashion:
- There is a lower blade that rises from data through measurements to formulae. It is a movement of empirical science from description to explanation, from proper domains of data to systems of laws that implicitly define the terms they relate, until the ideal goal is reached when all aspects of the data not strictly part of the sensible matter have their intelligible counterpart in explanatory systems.
- There is an upper blade that moves downward from equations and postulates of invariance and equivalence that begins from specific sensible material, the percept, and endeavors to explore the totality of manners in which abstraction can confer intelligibility upon any other material that resemble the determine data within that specified matter.
- The two movements are complimentary: The mathematician begins from the specific sensible material, the percept, where the empirical scientist would end. So, if the mathematical exploration of intelligible systems is thorough, then it is bound to include explanatory systems that the empirical sciences will verify in their respective domains. In fact, the process of developing a system is one which arises from measurements and patterns in data providing intelligible unity and correlations to particular instances.
Having given an account of illumination, verification, and preparation in Hadamard’s cycle, it remains to discuss incubation, which we next move to do.