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