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Mathematician’s Mind Redux, III

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:

  1. 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.
    1. One department follows upon another.
    1. Logically, they are discontinuous: each has its own definitions, postulates, and inferences.
    1. 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.
  2. 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.
  3. An actual element: The conjunction of the formal and material element. Heuristic structures of empirical methods operate in a scissors-like fashion:
    1. 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.
    1. 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.
    1. 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.

The Mathematician’s Mind Redux, II

Within Hadamard’s cycle, there is a movement of incubation to illumination in which ideas in a mathematician’s mind are surveyed in search of resolving a specified problem. In this process, ideas are evaluated based upon their relevance to the content of the problem and connections are sought between these ideas based upon their relevance to producing a solution to the problem. But what characterizes these ideas in the context of mathematics? As we have seen, an object of mathematics is, first, a possible existent and then, second, subject to quantities and their properties. Taken together, such an object is materially strictly intelligible. Furthermore, in being seen in a purely quantitative mode, such objects have their primary existence within an act of intelligence. As such, we shall identify their existence as a possible existence as first grasped in the intelligent consciousness through an act of understanding. It is this act that we now need to be more precise about in equating it with Hadamard’s notion of illumination.

Illumination is to be defined as the “supervening act of understanding”[1]. Being understanding in act, there is a process of grasping, formulating, seeing connections, all of which are being performed as part of the intelligent consciousness. Such could come from a process by which a series of ideas are seen as linked together, or by a sudden awareness of a collective of distinct ideas suddenly seen as a single interconnected network. In general, what we take to be formed is a whole idea for which the series or network of ideas are parts. Such a series or network is a result of grasping or formulating relations (as such occurs in the forming of meanings in which a relation is formed between sign and signified[2]) between the ideas. In viewing such acts as forming whole systems among ideas, there is seen a forming of unity, a unification, among ideas previously only seen as disparate parts[3]. 

Now, in terms of perceiving what is part of something material in its intelligibility, illumination operates in two ways: either by the formation of a similitude through abstraction, or by coming to understand how data, drawn from a similitude, may possibly be united into a whole. Furthermore, an enriching abstraction is an anticipation of an intelligibility to be added to sensible presentations: there is something to be known by illumination. What results is an erection of heuristic structures and the attainment of illumination, to reveal in the data what is variously named the significant, the relevant, the important, the essential, the idea, the form[JT1] .  

To connect these two modes of illumination to an account of knowledge that will ground the truth values of mathematical assertions, we bring in now an account of judgement and the way it fits into Hadamard’s cycle. As a body of knowledge, the subject-matter of mathematics is rendered, first, linguistically in terms of the norms of language along with appropriately chosen symbols that represents the terms, operations, and relations which then, in turn, will enable the expression of further equations and relations. A mathematical proposition is then a statement, declaration, assertion, etc. given in terms of this language and symbolic expressions that can be asserted to be true or false. Before specifying what is the proper structure of a mathematical proposition, we shall first review the method of deductive inference and its role in the production of proofs.

By a (deductive) inference we will mean a proposition having two further propositional parts: an antecedent and a consequent. Expressed linguistically in a sentence, typically the antecedent precedes the consequent in the sentence’s order. As a proposition, an inference is false when the consequent is false at once with the antecedent being true. One way to ensure that an inference be true is to link the consequent to the antecedent by a further true proposition, called the justification, which guarantees the consequent is true when the antecedent is true. In the language of syllogisms, the justification serves as the minor premise to the antecedent as the major premise. We will refer to an inference with a justification to be a verified inference. Finally, an inference is referred to as formal when it is verified, and the antecedent is assumed to be true.

Moving closer to the context of mathematics, we will focus on a special type of inference, called a theorem, for which the antecedent will be called the hypothesis, and the consequent is called the conclusion. A theorem is determined as verified when it is given a proof. This is an ordered sequence of propositions, called steps, in which (1) the first step is the hypothesis assumed as true, (2) each subsequent step serves as the consequent of a formal inference in which the antecedent is formed as a (possible) conjunction of previous steps and/or existing formal inferences (more on this anon), and (3) the final step is the conclusion. The dynamic act of forming a proof to a theorem is an extension of act of validating an inference. The formation of a justification to an inference is the act of verification. This follows upon an illumination that enables seeing the link between consequent and antecedent that necessitates the required verification. In fact, it is generally the scheme in the development of mathematics to identify properties of its proper objects by seeking conditions for their existence through formulating inferences which, in turn, require justification. This extends further to a process of formulating theorems and their verification through proofs.

Now, the case of an inference to be false is a possibility and a justification may be proffered that invalidates that inference. In either case, it is the aim regarding any proposition to determine whether it be true or false, that is, to perform an act of judgement. In the specific circumstances of an inference, this act of judgement is one that either verifies or invalidates the inference, in either case we will refer to this process generally as a verification. 

In identifying this process as dynamic, we will refer to it as reflective. It is at this point that a movement is made to a further part of the mathematician’s consciousness beyond the empirical and the intelligent, specifically the rational. Following Lonergan, “It is the emergence and effective operation of a single law of utmost generality, the law of sufficient reason, where the sufficient reason is the [verification]. It emerges as a demand for the [verification] and a refusal to assent unreservedly on any lesser ground. It advances to grasp of the [verification]. It terminates in the rational compulsion by which grasp of the [verification] commands assent.”[4]

Now, the process of articulating and analyzing a proposition, enabling it to be spoken or written, requires the production of partial terms of meaning, such as symbols, words, and phrases, that coalesce into sentences. This is done by rendering acts of understanding into terms of meaning along with rules for combining those terms to represent more complex meanings. Structured in this way, a term of meaning is called formal if it may be affirmed or denied but is just assumed and a term of meaning is full if it is to be affirmed or denied. A proposition so rendered is to be called analytic. In the case of the rendering of inferences for verification, the antecedent and putative justification becomes rendered as the formal term while the term rendering the consequent is full.[5] Rendering allows for a further sensible representation of an analytic proposition through language, again either spoken or written.

One way to certify an analytic proposition as being true is when it can be properly made in reference to a fact or facts. Now a fact is to be understood first as concrete, i.e. as having contingent existence and occurrence. In being intelligible, it is independent of all doubtful theory, yet it may be made more precise through illumination and further formulation. Otherwise, a fact possesses a conditional necessity: it might not have been, it might have been otherwise; but as things stand, nothing can alter it now.[6] In making a judgement of a fact, data is observed through sense experience and then selected and subsequently formed into a unified whole as a similitude of that fact within the intelligent consciousness. Subsequent judgement is made by sense experience through powers of observation arising from a cumulative development of understanding exemplified by memory, experience, specialization, and expertise.[7]We will say that an analytic proposition is an analytic principle when the partial terms of meaning are existential, that is, they occur in their defined sense in judgements of fact. As facts have a contingent reality, affirming or denying an analytic principle will depend on the conditional necessity the facts bear in terms of current state-of-affairs. In this basic way, the proposition is referred as an outright analytic principle. Nevertheless, the terms and relations the principle asserts can ground further inferences that explore completely, generally, and ideally, the real range of fields to which the outright analytic principle access in a particular, fragmentary, or approximate manner. Viewed in this universal manner, the proposition is referred to as a serially analytic principle.[8] Furthermore, it is the case that the formation of a serially analytic principle from an outright analytic principle is by way of abstraction of the intelligible from the sensible as it pertains to the existential meaning of the partial terms within that outright analytic principle.

[1] Here we are identifying the definition of Lonergan’s insight (Insight p.ix) for Hadamard’s notion of illumination. Going forward, we will follow Lonergan’s account of insight in elaborating upon Hadamard’s illumination. 

[2] Ibid p. x

[3] “Understanding itself is an irreducible experience, like seeing colors or hearing sounds … It is the goal of the empirical, grounding the formation of concepts, definitions, hypothetical systems, pure implications. It is the grasp of unity (Aristotle’s intelligentia indivisibilium) in empirical multiplicity, and it expresses itself in systematic meaning. Strictly and primarily, the intelligible is the grasped unity; and it is only by their relations to that unity that other instances of the intelligible are intelligible.” B. Lonergan, “A Note on Geometric Possibility,” reprinted in Collection, Collected Works of Bernard Lonergan, volume 4, University of Toronto Press (1993) p. 102

[4] Insight, p. 322

[5] Ibid p. 305

[6] ibid, p. 331

[7] ibid pp. 280-283

[8] ibid p. 313

The Mathematician’s Mind Redux, I

After a long hiatus, I have decided to re-introduce and elaborate upon what I take to be an account of what it means to think beautifully about mathematics. In an effort to do so, I will be following a path that will engage, at least in its initial stages, the work of the mathematician Jacques Hadamard exploration of the psychological elements involved in a mathematician’s effort to invent, discover, and develop further her particular field of mathematics. After looking at a summary of his thesis and at how a particular historical vignette fits into it, I will quickly diverge by framing his thesis within an account of human understanding largely developed by Bernard Lonergan. 

Written in 1945, Jacques Hadamard wrote the work The Mathematician’s Mind: The Psychology of Invention in the Mathematical Field in an effort to understand the process by which a mathematician comes to discover and invent her mathematics.[1] In the preface to the 1996 paperback edition, P.N. Johnson-Laird summarized the steps by which a mathematician goes from addressing to ultimately resolving a mathematical problem. Here they are as Laird-Johnson described them:[2]

  1. Preparation. You work on a problem, giving your conscience attention to it. 
  2. Incubation. Your conscious preparation sets going an unconscious mechanism that searches for the solution. Poincare wrote that ideas are like the hooked atoms of Epicurus: preparation sets them in motion and they continue their dance during incubation. The unconscious mechanism evaluates the resulting combinations on aesthetic criteria, but most of them are useless.
  3. Illumination. An idea that satisfies your unconscious criteria suddenly emerges into your consciousness.
  4. Verification. You carry out further conscious work in order to verify your illumination, to formulate it more precisely, and perhaps to follow up on consequences.

As Laird-Johnson notes, the articulation of these stages of the creative process can be traced back to Graham Wallace’s The Art of Thought (1926) and found in certain degrees in the writings of Helmholtz and Henri Poincare. Since we see this process as a continuous, periodic one, we will refer to it as the Hadamard cycle.  In the forgoing, we will give an account of mathematical knowledge, as centered on the mathematician qua human knower as the source of that knowledge. 

            In an effort to examine this cycle closely, let us take a look at how it occurs by repeating an account of invention and discovery given by Henri Poincare which Hadamard gives particularly close attention to.

“For fifteen days I strove to prove there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at the work table, stayed an hour or two, tried a great number of combinations and reached no result. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, making a stable combination. By next morning I had established the existence of a class of Fuchsian functions, those which come from hypergeometric series; I had only to write out the results, which took only a few hours.

Then I wanted to represent these represent these functions by the quotient of two series; this idea was perfectly conscious and deliberate, the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and I succeeded without difficulty in forming the series I have called theta-Fuchsian.

Just at this time I left Caen, where I was living, to go on a geological excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paced the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with my conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’s sake I verified the result at my leisure. 

Then I turned my attention to the study of some arithmetic questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside, and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness, and immediate certainty, that the arithmetic transformations of indeterminate ternary quadratic forms were identical with those of non-Euclidean geometry.”[3]

As Hadamard has done, we will want to dwell on how his cycle plays out regarding this concrete account of Poincare. This we will do by elaborating an account of human consciousness which fills in the details of Hadamard’s cycle in contradistinction to the analysis given by Hadamard[4].

            The course we will take will be to first identify the nature of human consciousness in light of Hadamard’s cycle and then relate that account to the role of invention and discovery specific to mathematics. In the process, we will the subject-matter of mathematics with the dynamic process of acquiring and developing mathematical knowledge. This we will do by unpacking what we mean first by the general act of knowing and of identifying knowledge and then specify what is meant by mathematical knowledge.

            To characterize and explicate knowledge properly, it will be important, up front, to begin removing any biases as to the status of the objects of mathematics as being subjective versus objective (in other words, invented versus discovered). This will involve resolving within the human knower the tension between how things that are external are known and how knowledge is acquired internally, i.e. the tension between what is received by the senses and what is discerned within the intellect. To that end, we identify both together as cognitive acts that the human subject is intimately aware of and which is what fundamentally defines the very nature of consciousness[5]. 

            To parse this nature more precisely, we first identify the act of consciousness that involves sensing, perceiving, and imagining as the perceptible. This is to be distinguished with an intelligent consciousness whose acts seek to obtain the intelligible through inquiry, illumination, and formulation. Here it is worthwhile to quote Lonergan directly regarding the precise nature of this intelligent part of consciousness:

“On this level cognitional process not merely strives for and reaches the intelligible, but in doing so it exhibits its intelligence; it operates intelligently. The awareness is present but it is the awareness of intelligence, of what strives to understand, of what is satisfied by understanding, of what formulates the understood, not as a schoolboy repeating by rote a definition, but as one that defines because he grasps why that definition hits things off.”[6]

It is important at this stage to dwell more deeply on two points regarding how consciousness operates: (1) the transition from the empirical to the intelligent and (2) the precise nature of understanding.

            In the movement from the empirical to the intelligent, there is first the recognition that the mind receives within its cognition images of external objects passively received through the senses. From these, an intentional directiveness may be consciously employed to select from those images, specific ones, which will be referred to as similitudes, from which intelligence seeks to grasp them individually or together. Let us consider these similitudes as data for intelligence. But what is the relationship between these similitudes and the external objects which provide the images through the senses? To answer this question, we now need to consider the general nature of abstraction. 

            Generally defined, abstraction is an operation of the intellect that in understanding what a thing is “distinguishes one thing from another by knowing what one is without knowing anything of the other, either that it is united to it or separated from it” but are in fact “are one in reality.”[7] Before addressing the possibility of abstraction, we identify two principles that undergird it as an operation of the intellect. The first is: the aspect which is considered separate must not depend for its intelligibility on the other aspects for which it is mentally separated (Principle of Independent Intelligibility); the second being: that which is considered separate cannot be asserted to exist apart from the thing and its other aspects with which it really does exist (Principle of Dependent Existence).[8]

From these principles, we may identify what constitutes a similitude of an image received from an existing thing by the senses. Initially, we understand such a thing as composed of matter, that is, as (possibly) existing and subject to change. Further, we may take it to be an individual that possesses a concrete particularity in time and place. In what ways it possesses such features can only be identified by the senses and, as such, is structured in terms of sensible matter. Separate from these features, but integral to it as existing, a thing exists in a material form and, as such, is subject to counting, measurements, ordered and so forth which we refer generally as being ordinable. Understanding a percept strictly under these conditions alone, it is by way of the intelligence alone that such features are grasped. It is from this perspective that matter in a thing understood in its existence and as subject to quantities (broadly understood) alone is said to be constituted of intelligible matter. It is then by abstraction that the image of a thing understood in its materially intelligibility alone is what constitutes the similitude of the thing in the intelligent consciousness. By identifying intelligible matter as what can or does exist materially subject to being ordered we are identifying two different ways for something to be composed from such by understanding it as either (1) by abstraction from what actually exists, or (2) by what can possibly exist. In both cases, knowledge of a thing in being materially intelligible is obtained quantitatively either by analysis through forms of measurement or by understanding quantities per se and their properties. This is how we will initially understand what the subject-matter of mathematics is. In order to make this more precise, we will first take a closer look at Hadamard’s notion of illumination in connection with the act of understanding and then examine what is meant by quantity in the broadest sense. 

[1] Jacques Hadamard, The Mathematician’s Mind: The Psychology of Invention in the Mathematical Field, Princeton Science Library, Princeton University Press, 1996

[2] Ibid p. x

[3] H. Poincare, “Mathematical Creation,” reprinted in The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method

[4] To that end, we will be largely channeling the work of Bernard Lonergan, specifically Insight: A Study of Human Understanding(Philosophical Library New York, Darton Longman and Todd, 1957) and Understanding and Being (Collected Works of Bernard Lonergan, vol. 5, The Robert Mollot Collection, University of Toronto Press, 1990)

[5] “Consciousness is meant an awareness immanent in cognitional acts,” Lonergan, Insight, pp. 320, 322

[6] Ibid, p. 322

[7] Thomas Aquinas, The Division and Methods of the Sciences, Questions V and VI of his Commentary on De Trinitate of Boethius, translated with introduction and notes by Armand Maurer, The Pontifical Institute of Mediaeval Studies, Toronto, Canada, 1963, p. 30

[8] Thomas C. Anderson, “Intelligible matter and the Objects of Mathematics in Aquinas,” New Scholasticism 

A Sense of Sweetness: In Search of a Reformed Theology of Beauty

I am hoping to be on sabbatical in 2022-23, and to that end have been developing a detailed project proposal for the book I want to write about a Reformed theology of beauty. Since I would value input on this plan, I thought it might be helpful to post one iteration of the proposal here.

My Question: I am a theologian working within the Reformed tradition. At the same time, my theological work is focused on the theology of beauty. Sometimes the connection between those two statements is strained. I often encounter the idea that the Reformed tradition is aesthetically barren, or even that it is opposed to all beauty. Even people within the tradition sometimes think this. But the Reformed tradition offers and has always offered an alternative aesthetic, not an anti-aesthetic.

(more…) “A Sense of Sweetness: In Search of a Reformed Theology of Beauty”

Creational Thinking – Part III: Abstraction and Order

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.

(more…) “Creational Thinking – Part III: Abstraction and Order”

Creational Thinking – Part II: Things, Abstractions, and Formations

 We begin our journey in developing a notion of creational thinking by giving an account of physical objects in hylomorphic terms and describe how by abstraction human understanding distinguishes such an object in terms of the sensible and the intelligible. In the course of laying out such an account, we will be expanding upon our previous descriptions, which was largely dependent upon the thought of Thomas Aquinas, by channeling the thought of Bernard Lonergan, through his magnum opus Insight[i], and Erich Przywara, through his magnum opus Analogia Entis[ii].

(more…) “Creational Thinking – Part II: Things, Abstractions, and Formations”

Creational Thinking – Part I: Introduction

In a previous series of posts (beginning here https://thinkingbeautifully.org/mathematical-understanding-as-seen-within-a-framework-of-beauty-part-1/), I described a particular perspective on thinking beautifully in mathematics. In that description, I aimed to channel the thinking of Medieval thinkers such as Thomas Aquinas and Bonaventure, but also by incorporating the thinking of such twentieth century philosophers as Bernard Lonergan and Michael Polanyi. Unfortunately, my grasp on the latter two thinkers was not sufficient to capture the depths of their ideas in developing the theory of mathematical knowledge that I am aiming for. I have since dug deeper in both of these authors’ works on human understanding, particularly Lonergan’s Insight, and I want to now expand upon my previous forays by articulating how mathematical knowledge fits into a larger mode of human understanding which seeks to see the human knower and knowable reality as united and understood in that unity as a divine creation. To that end, I will also be bringing in as an additional conversation partner, along with Lonergan, Erich Przywara through his magnum opus Analogia Entis. 

(more…) “Creational Thinking – Part I: Introduction”

Mathematical Understanding as Seen Within a Framework of Beauty (part 4)

Following upon my previous posts, where I began to explore the possibility of framing mathematical understanding within the Thomistic modes of beauty proportio and claritas, I now consider the third mode of beauty, that of integritas. This mode brings beauty in mathematical understanding to its fulfillment and serves to complete and unite the other two modes. For it is in the crafting, rendering, and communicating mathematical truths that integritas is to be sought and developed. In being a measure of beauty, integritas measures the extent to which a work, rendering, or communication expresses a subject-matter of mathematics in such a way as to uncover ever deeper proportios and orders, enable ever more profound claritas to be revealed, and so allow for ever greater works to be developed which, in turn, are themselves measured for their individual integritas. Furthermore, it is in the acts of verification through defining, axiomatizing, and producing proofs that we move from claritas to integritas. For, it is by the movement toward integritas that the proportios form order, as seen with claritas, and become formalized in order to be communicable to others. In the end, the aim is the advancing of mathematical knowledge to those within the community of mathematicians.

(Here is the link to part 3:

https://thinkingbeautifully.org/mathematical-understanding-as-seen-within-a-framework-of-beauty-part-3/)

(more…) “Mathematical Understanding as Seen Within a Framework of Beauty (part 4)”

Mathematical Understanding as Seen Within a Framework of Beauty (part 3)

In the first two parts of this series, I began outlining how mathematics and mathematical understanding can be framed within the Thomistic modes of beauty: proportio, claritas, and integritas. In particular, I defined mathematics as the science whose subject-matter is measurable orders: objects understood as parts united into whole, having a distinction of same or difference, related in proportio, and analyzed by measuring through quantities (broadly understood). In this part, we will begin a focus on how human understanding is an essential component to the development of mathematics by highlighting how the modes of claritas and integritas are integrally involved. The main point to be made is how understanding is arrived at in mathematics as it pertains to the disclosure of order and to the discernment of truths that pertain to it. The key here will be to articulate how being underscores both the unity of parts into a whole and our grasp of it.

(Here is the link to part 2:

https://thinkingbeautifully.org/mathematical-understanding-as-seen-within-a-framework-of-beauty-part-2/)

(more…) “Mathematical Understanding as Seen Within a Framework of Beauty (part 3)”

Mathematical Understanding as Seen Within a Framework of Beauty (part 2)

In part 1 of this series, I began to explore how mathematical reasoning can be understood within a Thomistic framework of beauty as expressed in the modes proportio, claritas, and integritas. In that post, I arrived at an initial description of the subject-matter of mathematics as pertaining to quantitative being understood as parts ordered into a unified whole and related to each other through a proportio. Objects of mathematics themselves are to be understood per se as ordered and potentially related to each other within a further ambient order. As objects of human reasoning, they arrive to our cognitive awareness as beings of our imagination, by means of abstraction from created beings, as received by our senses. Thus, they are understood as formations of intelligible matter. This type of matter is what underlies a created being’s size, shape, etc. and is therefore understood through quantities. Furthermore, changes to an object that preserves its form are understood through such quantities. In this post, my aim is to continue this exploration by defining what is meant by quantity, describe what underlies its nature, and examine how it relates to quantitative being in terms of its ordering.

(Here is the link to part 1:

https://thinkingbeautifully.org/mathematical-understanding-as-seen-within-a-framework-of-beauty-part-1/)

(more…) “Mathematical Understanding as Seen Within a Framework of Beauty (part 2)”