We continue to sketch an account of creational thinking as a way of understanding mathematics and its relationship to reality. In our narrative so far, we had begun to develop a perspective of objects of reality as creative by adopting a hylomorphic view of them as the sensible and intelligible united in matter. As objects of thought and understanding, they are unified wholes of data, what Lonergan refers to as things. In reference to the object itself, as received from by the senses, we will continue to refer as formations[i]. In reasoning about a formation through its thing in the mind, the key characteristic we identified as pertaining to its intelligible structure is that it is ordinable. To understand what that means is that it is first and foremost intelligible above and beyond the sensible. Furthermore, the ordinable is arrived at by being separated from the sensible through abstraction. What is ordinable in a formation is still discerned in a material way through quantities which constrain and are constrained by the formation’s essence. These are the measured quantities and correlations which are termed explanatory conjugates (a la Lonergan). We now seek to give an account of how such conjugates become framed within scientific theories and, ultimately, become further framed within mathematical theories. In Przywara-ian terms, this is to move from the morphological to the eidetic (from the morphe to the eidos). To that end, we will be adopting Lonergan’s account of enriching abstraction.
In a bit of foreshadowing, we will be elaborating abstraction as a bridge from the particular to the universal, from sensible reality to its intelligibility. As a bridge, the object of abstraction arising from the formation again serves to constrain what that formation is in its intelligibility by having its existence rooted in that formation. What we seek to delineate is the universal nature that that object of abstraction habituates, in its independent intelligibility, with its particular powers and operations. It is these latter features that we will be identifying by the process of enriching abstraction.
The first step that needs to be understood is what occurs by moving from the empirically observable realm of the sensible to the resulting experiential conjugates at the intelligible level by abstraction from the empirical residue as existing within the morphology of act. From this level, the explanatory conjugates of theories and laws can be formulated from the observed measures, patterns, and correlates. We may observe, as Lonergan does[ii], that the validity of such theories and laws as empirical principles can only be held in a provisional way in that such validity depends on the existential dependence of them being empirically verified wherever the proper conditioned and conditions hold within formations. What we are interested in is how such theories and laws can become portrayed within mathematical theories. One form that our answer will take is that the understanding of the act/object of the eidetic constrains the understanding of the act/object of the morphological. In order to unpack the content of this answer we will need to first understand what is occurring at each of these levels and then how the different levels relate.
To first clarify the level of morphology of act, in order to make an observed formation intelligible (beyond declaring mere existence) is to understand it first as ordinable in a particular way, by such acts as counting, measuring, diagramming, or through renderings in terms of established symbolic/iconic representations. As Lonergan describes this regarding the scientific:
“… there 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; and at the end of this movement there is the ideal goal that is to be attained when all aspects of the data except the empirical residue will have their intelligible counterpart in explanatory conjugates and ideal frequencies.”[iii]
The morphological in act is then seen as explanatory conjugates understood in the formation through the thing abstracted from it. It is from here that the eidetic in act comes to formulate theories or laws, intelligibly understood independent of its connection to the existential particulars that is being abstracted from. In terms of mathematics, though, what type of theories fall under the eidetic of act? How do they relate to that understood through the morphology of act? Lonergan portrays such a relationship as occurring in a ‘scissor-like’ fashion:
“… not only is there a lower blade that rises from data through measurements and curve fitting to formula, but there is also an upper blade that moves downward from differential and operator equations and from postulates of invariance and equivalence.”[iv]
Furthermore, he elaborates the possibility of the two blades as being united into a scissor in that:
“… there is the movement of mathematical thought that begins from the empirical residue and endeavors to explore the totality of manners in which enriching abstraction can confer intelligibility upon any material that resemble the empirical residue. Clearly, these two movements are complimentary. For the mathematician begins from the empirical residue which the empirical scientist would end; and if the mathematical exploration of intelligible systems is thorough, then it is bound to include the systems of explanatory conjugates that the empirical scientist will verify in their respective domains.”
Now, in terms of grounding the eidetic of act in the morphology of act in the construction of mathematical theories, Lonergan delineates one way they arise from by abstraction from sensible data by
“… a bias from the particular to the general, from the part to the totality, from the approximate to the ideal. If there exist concrete instances of one, two, and three the mathematician explores the totality of positive integers, of real numbers, of complex numbers, of ordered sets. If there exist edges and surfaces, the mathematician works out not one geometry, but the total series of possible geometries. If there are various fields in which it seems mathematics can be applied, the mathematician sets out to explore the whole of each region in which the fields occur.”[v]
It is first worth noting overarching theme of this portion of the text in which objects and theories of mathematics are constructed hierarchically starting from empirically verifiable instances that become generalized in abstraction. There are four important points to clarify about the development of this hierarchy:
- In the case of actually counting particular formations as 1, 2, 3, … this can be developed generally in an arithmetic direction by the addition of 1: 1, 2 = 1+1, 3 = 2+1, etc. In terms of counting formations, these are finite exercises in iteration, but as Henri Poincare observed the mathematician is not under such a constraint from the side of intelligence and may develop a mathematical object, such as the system of positive integers, through indefinite iteration.[vi] Thus an integer may be generically identified simply by n with the property that n+1 also is a positive integer. In order to obtain this system as an object of mathematics, concepts of same/difference need to be made explicit through =, the basic term of 1 is identified, and the operation + incorporated. All other terms of the system are grounded by the particulars 1, =, + together with first principles that have existential validity (for example through Euclid’s Common Notions[vii]) and verifiable in reality.
- Once such a mathematical system has been obtained by such operation of abstraction, that object in its intelligibility is understood as a movement from the morphology of act to the eidetic of act. From this perspective, we have what we term an ordinable system[viii] and is understood to consist o
- rules governing and so defining operations, and
- operations proceeding from some terms to others and so relating and defining them.
- To move to a next level of the hierarchy is to extend an ordinable system to one in which a new operation not definable on a terms generally of the system becomes defined generally on the next level. For example, subtraction (being the inverse of addition), not being generally defined on the system of positive integers, is defined on the system of integers. Similarly, multiplication is defined on the (positive) integers, but the inverse operation of forming ratios (i.e. division) is not generally defined on the system of integers. In the next level of the rational number system, ratios of integers are defined with the exception of dividing by 0. (This last point highlights a particular feature possessed by any ordinable system: the Principle of Non-Contradiction must be upheld.) This way of developing ordinable systems in a hierarchy is what Lonergan terms higher viewpoints.[ix]
- This way of developing mathematical objects as ordinal systems fits Lonergan’s notion of abstraction as enriching in which “insight goes beyond images and data by adding intelligible unities and correlations and frequencies which, indeed, contain a reference to images or data but nonetheless add a component to knowledge that does not exist actually on the level of sense or imagination.”[x] Furthermore,
“… there is the movement of mathematical thought that begins from the empirical residue and endeavors to explore the totality of manners in which enriching abstraction can confer intelligibility upon any material that resemble the empirical residue … the mathematician begins from the empirical residue which the empirical scientist would end; and if the mathematical exploration of intelligible systems is thorough, then it is bound to include the systems of explanatory conjugates that the empirical scientist will verify in their respective domains.”[xi]
Returning to formations, viewing such a one as having essence-in-existence, by abstraction we may understand the resulting thing morphologically in act as having an essence beyond existence. As we have described above, seen from this vantage point, those explanatory conjugates that can be found to pertain to the formation in turn may be seen in its intelligibility in a universal way as an eidetic in act. As such, those terms and operations that express those conjugates may become expressed as the terms and operations of an ordinable system having the rules of those operations generally defined. From this perspective, we will say that the formation is understood by participation in that ordinable system, or that the formation participates the ordinable system. Reciprocally, we will say that the formation is ordered by the inherence of the ordinable system in it.
In the next installment, we will show how the objects of applied mathematics arising from the move from the morphology of act to the eidetic of act can be extended to account for pure mathematical objects by an account for how act moves to object. Thus we will adapt Lonergan’s general account of mathematical understanding to Przywara’s morphology and eidetic in object.
[i] As borrowed from David Bentley Hart’s “Emergence and Formation: The Limits of Mechanism”
[ii] Insight pp. 326-334
[iii] ibid p. 337
[v] Ibid pp. 336 ff
[vi] See, for example, pp. 93-96 of Janet Folina, Poincare and the Philosophy of Mathematics, Palgrave, 1992
[vii] Euclid, Elements, Book I
[viii] What Lonergan calls departments of mathematics. See Insight p. 335.
[ix] Ibid pp. 37-43
[x] ibid p. 36. See also pp. 111-114
[xi] ibid pp. 337-338