Judgment in Aristotle, two natures

As a qualification on how we are to understand judgment in Aristotle, please note that, in the kind of analysis which we find in Aristotle and also in the manner of his conceptualization and language, in our acts of judgment thus, a dual nature is distinguished or two natures are indicated in a way which seems to juxtapose one nature with another. Two natures exist instead of one nature. A synthetic, constructive element is alluded to, on the one hand, and, on the other hand, an affirmative, declarative element. Hence, questions exist (later questions were posed) which asked if Aristotle was successful in clearly distinguishing between the being of these two different aspects (existing as two distinct elements, each having its own distinct nature).1 Did he, in fact, clearly distinguish between acts of direct understanding and acts of reflective understanding which exist as acts of judgment since, in Aristotle, judgment engages in two different kinds of tasks. On the one hand, allegedly within our judgments, (1) a composition or a putting together of different concepts occurs or, on the other hand, a separation of concepts when we realize that some concepts should not be combined or joined with each other. If an act of direct understanding (which, as noted, Aristotle conceptualizes as an act of “simple apprehension”) moves through the instrumentality of an imagined fertile, apt image (existing as a phantasm) toward a single, distinct concept or a definition which expresses the fruit or the grasp of one’s prior act of understanding (in Aristotle’s understanding of the nature or the intelligibility of all our direct acts of understanding as we move from the being and the order of sense to the order and the being of understanding: ta men oun eidê to noêtikon en tois phantasmasi noei; the “intellect grasps forms in images”),2 a fortiori, if we should speak in this way about the being of a “simple apprehension,” then, to a greater degree, if we are to speak about how two or more concepts can be put together to reveal a greater unity or a link that exists between these concepts (leading to a larger, more general concept), then, in order to identify and to distinguish this species of intellectual act, we should or we must speak about the being of a “complex apprehension.” These exist allegedly as judgments. These judgments introduce an order which should exist among our ideas and concepts. However, if, for us, the intellectual object is not simply the apprehension of a conceptual complex unity but if, in fact, it is an understanding which wants to declare or know about the reality or the truth of one or more concepts (whether we should speak about simple concepts or about complex concepts), then, within this larger, greater, more demanding context, in Aristotle, a second understanding of judgment presents itself to us in terms of how it seeks to posit a relation or a synthesis which has been grasped by us in our prior acts of understanding. The object here is not essentially a synthesis, the apprehension or the grasp of a synthesis which points to a higher or a wider understanding of things but, instead, the taking of an already understood synthesis and further acts which would work toward an act of understanding which can conclude or move toward a declaration of its reality or a declaration of its truth (or which can deny the factuality of its reality or the factuality of its truth). This is so. This is not so. Either way, in affirmation or negation, a truth is known and it is grasped by us as known. In our awareness, a truth is known in terms of its reasonableness or cogency: hence, its being, its reality. The consciousness or experience that we have of evidence points to the being or the reality of a truth and, as an effect which would thus follow from this, with Aquinas, we would say about ourselves that “knowledge exists as one of the effects of truth” [cognitio est quidam veritatis effectus].3 The one comes from the other.

In Aristotle thus, depending on which passages or texts are being studied, a clear distinction does not exist between that which exists as understanding and that which exists as judgment (acts of direct understanding versus acts of reflective understanding) because judgment, in the language of “composition and division,” resembles acts of direct understanding in terms of the unities which are being grasped and understood by them (by our acts of understanding): unities which transcend pluralities and multiplicities as these exist initially among the givens of the data of our sense perception. However, in Aristotle, the being of judgments is such that they also seek to determine if a correspondence exists between that which exists as a form of mental synthesis within ourselves and that which exists as a species of real synthesis within the being of truly existing things (the being of truly existing objects). A real distinction accordingly exists between the type of answer that is given to this kind of question and the type of answer which is given to a question which asks about how concepts can be related to each other in ways that could lead to the understanding and eventually the expression of a new, more general concept.

On the basis then of this real distinction and as a species of new first principle, in the later work of Aquinas and also in the later work of Bernard Lonergan, clarifications were introduced into the thinking and the conceptuality of Aristotle’s analysis in a manner which attempted to introduce degrees of clarity that had not been too obvious to anyone or to most persons who had attempted earlier to read into the corpus of Aristotle’s philosophy in order to find, within it, a coherent understanding about how things exist within the reality of the world within which we all live (a reality which includes the kind of being which we have and which we are as human beings where our kind of being includes the kind of knowing which belongs to us as human beings and which does not belong to other kinds of living being). From an incoherent understanding about the nature of our human judgment (from an incoherent understanding about the nature of our human cognition), we can thus wonder if, for some in the subsequent history of reflection within philosophy, the result has been a defective, incoherent understanding about the nature of existing things where, in metaphysics, we turn to this science in order to move toward a comprehensive or a general understanding about the nature of all existing things qua the nature of being in general as it applies to all things which enjoy some form of real existence. What can be implied about the nature of our world if our point of departure is a particular belief or a particular understanding about the nature of our human knowing, an understanding which could be lacking in the degree of rationality which should belong to it?4

1Lonergan, Verbum, pp. 61-62.

2Aristotle, De Anima, 3, 7, 431b, as cited by Sala, Lonergan and Kant, p. 161, n. 72.

3Aquinas, De veritate, q. 1, a. 1, as cited by Sala, Lonergan and Kant, p. 147, n. 71.

4Randall, Aristotle, p. 6.

Aquinas, Summa Theologiae, 1a, q. 84, Notes

Notes on Aquinas, Summa Theologiae 1a, q. 84

ST 1.84: How does the human intellect know bodies (which are beneath it)

ST 1.84.1 Does the soul know bodies through the intellect

Objection 1: Bodies are understood by the senses, incorporeals by the intellect

Reply: This refers to the medium of knowledge, not the object

Objection 2: Sense:Intelligible :: Intellect:Sensible.  Since senses can’t understand intelligible things, the intellect can’t understand sensible things

Reply: Intellect is a higher power than sense.  While sense cannot know intelligibles, intellect can know sensibles.  Otherwise God wouldn’t have knowledge of sensibles.

Objection 3: Intellect understands eternal and unchangeable things, while bodies are always changing

Reply: Every movement presupposes something immovable.    Changing form requires unchanging matter.  Changing matter requires unchanging form.  Socrates may not always be sitting, but it is always true that when he does sit he stays in one place.

 

On the contrary: natural science, with its studies of motion and matter, would be impossible if the intellect had no knowledge of changeable things.

 

One possible answer: Certain Presocratics (Heraclitus) thought everything was in a state of flux, so certain knowledge of anything was impossible.

Refutation: Heraclitus was only led to this conclusion because of their reductive materialism.

 

Another possible answer: Plato’s theory of forms

Refutations:

 

  1. If forms are immaterial and immovable, how can we have knowledge of matter and motion?
  2. Plato begs the question: How can we know bodies, if we only know them through separate substances which differ essentially from them?
  3. Plato assumed that the form must exist in the knower in the same mode as in the thing known.  This is not true.

 

Whiteness is in bodies in different modes of intensity

The senses receive whiteness without receiving matter

The intellect receives whiteness under conditions of immateriality and immobility

 

ST 1.84.2 Does the soul know bodies through its essence?

It would seem so.  Knowledge is by assimilation, where the intellect “becomes” whatever essence it is to know.  Thus, the intellect knows all things by the essence it has assimilated, which essence has become the intellect’s own essence.

 

But Augustine says we know things through our senses.

 

Presocratic view: the material intellect receives forms materially

Refutations:

  1. Matter is only form potentially, and knowledge is of actualized form
  2. Why doesn’t matter outside the soul possess knowledge?

 

Platonist view: The immaterial soul receives forms immaterially

 

Only God knows things through his own essence; humans and angels don’t

 

ST 1.84.3 Does the soul know through innate species?

Our knowledge is different from angelic knowledge.  Our knowledge must be brought into act, while angelic knowledge is always in act.  Moreover, the angelic intellect is completed by the angelic form and is not in potentiality with respect to anything.  The human intellect is in potentiality with respect to things it does not know.  Prime matter is in potentiality with respect to its substantial form.

 

ST 1.84.4 Are intelligible species derived by the soul from separate forms?

The objections proceed by analogy from sense, also noting that the intellect requires something actually intelligible in order to be brought into act.

But if we know by separate forms, there is no reason why our souls should be united to bodies.  Both Plato and Avicenna’s views are considered and deemed insufficient for explaining the existence of the body.

 

In replying to the objections, St. Thomas says there is no analogy between sense and intellect.  He concedes that divine ideas are the ultimate source of our knowledge, but we attain to this knowledge by the actualization of phantasms by agent intellect.

 

ST 1.84.5 Does the intellect know material things in the eternal types?

St. Thomas states that Augustine was “imbued with the doctrines of the Platonists.”

 

St. Thomas makes a distinction:

  1. Knowing the eternal types as objects.  The souls of the blessed know things in the eternal types in this way.
  2. Having the eternal types as a principle of knowledge.  Just as our senses see by the light of the sun, our intellect sees by the intellectual light which is in us, even as pilgrims.

 

The human intellect requires both intelligible species and intellectual light in order to know.

 

St. Thomas shows that Augustine was not as Platonist as he originally seemed.

 

ST 1.84.6 Is intellectual knowledge derived from sensible things?

Augustine’s argument that intellectual knowledge does not come from the senses

  1. Sensible things are always changing, but what is never the same cannot be perceived
  2. We cannot tell through the senses whether we are seeing a real image or whether we are seeing an imaginary image (dreaming), but the intellect is capable of perceiving truth
  3. What is lesser cannot act on what is greater, so the senses cannot produce knowledge in the intellect
  4. We know some immaterial things

 

Democritus thought all knowledge was by a discharge of atoms

 

Plato thought the soul forms within itself the species, after the sensible organ receives the sensible object.  The soul is roused by the species of the thing.

Augustine said the body was the messenger to the soul.

The senses rouse the intellect to the act of understanding.

 

Aristotle said sense is an act not of the soul alone, but of the “composite.”  The sensible is received in the sense by a discharge or some other operation.  Then agent intellect derives the phantasm and ultimately the intelligible species from the sensory input.

 

Sensible things are only the material cause of knowledge.  Agent intellect is also required.

 

ST 1.84.7 Can we understand without turning to phantasm?

No, because:

  1. When imagination is hindered by frenzy or lethargy, all intellectual operation is also hindered.
  2. We all understand by forming phantasms to serve as examples
  3. It belongs to a nature to exist in an individual.  This cannot happen apart from corporeal matter.  Since we apprehend the individual through phantasm, we need phantasm in order to see the universal nature existing in the individual.

 

Interesting facts:

When knowledge is not actualized, intelligible species dwell in the passive intellect.

The intelligible species is NOT the likeness of the individual thing.

Our intellect’s proper and proportionate object is the nature of a sensible thing (?)

 

ST 1.85.1: Does our intellect understand corporeal and material things by abstaction from phantasm?

3 grades of cognitive powers:

  1. Sense organ: object is form existing in individual matter
  2. Human intellect: object is form existing in individual matter, but not AS existing in this particular matter
  3. Angelic intellect: object is form existing apart from matter

2 types of abstractions

  1. Composition or division, i. e. Judgement of truth or falsehood
  2. Simple or absolute consideration: consider something and prescind from its principles of individuation

3 types of matter:

  1. Individual (“this flesh and these bones”)
  2. Common/signate (“flesh and bones”)
  3. Common intelligible matter (quantity, number, dimension, figure)

(things like “being,” “unity,” “power,” “act” abstracted from all matter)

 

The phantasms are made more fit for the abstraction therefrom by intelligible intentions.”

 

ST 1.85.2: Are intelligible species the object of our intellection?

No, intelligible species are the species qua, not species quae intelligitur.

Otherwise, science would be about words, not things.

 

Also, we would have no way of judging the truth or falsity of our ideas.

 

Intelligible species is a likeness of the thing understood, as heat in the heater is a likeness of the thing heated

 

Abstract universal exists in the mind, but the universal which is understood exists in its instantiations.

 

2 operations of the sensitive part:

  1. Impression (totally passive)
  2. Formation (imagination, active)

2 operations of the intellect

  1. Impression (passive intellect receives intelligible species)
  2. Formation (active intellect forms definition or judgement)

ST 1.85.3 Whether knowledge of the most universal species is prior?

Sense knowledge is prior to intellectual knowledge.  Therefore in some manner we know the particular first.

However, in the intellect we know what animals are, albeit indistinctly, before we can distinguish rational animals from non-rational animals.

 

In the senses, we know something is a body before it gets close enough to be known as an animal.  A child calls all men father before he learns to distinguish his father.  Therefore, universal sense knowledge is prior.

 

Something’s genus is derived from its common matter; the species comes from the form.  Nature ultimately intends the species, not the genus.

 

ST 1.85.4 Whether we can understand multiple things at once?

No, just as an object cannot have multiple colors at once, an intellect cannot have multiple intelligible species at once

 

We can, however, understand multiple things in a single species

 

ST 1.85.5 Whether our intellect understands by composition and division?

Yes, because our intellects achieve perfection by degrees.  We must make a judgement (i. E. a composition or division) about the state of our knowledge in order to attain more perfect knowledge.

 

Intellectual operations are in time insofar as we have to turn to phantasms

 

2 types of composition in a material thing

  1. Form with individual matter: Man is an animal (form with common matter); Socrates is a man (individual matter with form)
  2. Substance with accident: Socrates is white (Socrates as a man possessing whiteness)

 

2 types of composition in the intellect

  1. The whole with the part (form/species with common matter/genus)
  2. Subject predicated of accident (identification of Socrates with whiteness)

 

ST 1.86.6 Can the intellect be false?

The intellectual light cannot err, because the proper object of intellect is the quiddity of the thing.  Provided we are abstracting quiddities from phantasm or deriving first principles from the operation of the intellect itself, the intellect cannot be false.

 

In composition or division, the intellect can err.

 

ST 1.86.7 Can one man understand better than another?

  1. No, in the sense that all men understand through the same intelligible species.  Either you have the intelligible species or you don’t.
  2. Yes, in the sense that one man may perceive the intelligible species with greater keenness than another, as a man with 20-20 vision will see visible objects better than another.
  3. Yes, in the sense that intellect’s operation depends on the sensitive and imaginative faculties.  Someone with a better memory, for instance, can understand things better.

 

ST 1.86.8 Is the indivisible known before the divisible?

  1. Yes, in the sense that we know a quiddity indistinctly as a unity before we know it distinctly by understanding its parts.
  2. Yes, in the sense that we know simple quiddities of things before we engage in complex composition or division.
  3. No, in the sense that abstractions like “point” or “unity” are known only by privation.  A point is “something which has no parts.”  We know things with parts before we can consider something without parts.  This is because these concepts are opposed to natural, corporeal reality.

Square root of two as an irrational number

Square root of two as an irrational number
by Br. Dunstan Robidoux OSB
edited by Mr. Michael Hernandez MA

When Lonergan discusses inverse insight in the first chapter of his Insight: A Study of Human Understanding, he presents a mathematical example to illustrate the nature of inverse insight as an act of understanding which realizes that an expected, desired intelligibility is not to be reasonably nor rationally expected. (1) In some situations, in some inquiries, to anticipate in the type of intelligibility sought is to perdure in “barking up the wrong tree” and to waste time by asking irrelevant questions. However, since Lonergan’s example pains readers who have never acquired any easy familiarity with mathematics and who have lost what familiarity they once had, this paper will parse out the discussion in ways which should help. Let us begin.

Lonergan’s argument consists of the following sequence of numbered propositions:

Proposition 1: The square root of 2 is some magnitude greater than unity and less than two

Proposition 2: One would expect it to be some improper fraction, say m/n, where m/n are positive integers and by the removal of all common factors m may always be made prime to n.

Proposition 3: If this expectation correct, then the diagonal and the side of a square would be respectively m times and n times some common unit of length.

Proposition 4: So far from being correct, the expectation leads to a contradiction.

Proposition 5: If sqrt(2) = m/n, then 2 = m2/n2

Proposition 6: But, if m is prime to n, then m2 is prime to n2

Proposition 7: In that case, m2/n2 cannot be equal to two or, indeed, to any greater integer

Proposition 8: The argument is easily generalized, and so it appears that a surd is a surd because it is not the rational fraction that intelligence anticipates it to be

To understand the controversy about the square root of 2, let us look briefly at the historical origins of the problem.

First, with respect to numbers, the square root of 2 is some sort of number. Numbers fall into different types or species since the square root of a number is unlike the number whose square root is sought. Numbers rank as human inventions since they do not exist as purely natural entities apprehended by sense. They were invented as the human need for them arose. (2) Different needs, as they emerged, formed new types of numbers. Hence, the first type of numbers invented were the counting numbers, sometimes cited as natural numbers: 1, 2, 3, 4, 5…. (3) They arose as correlatives to designate quantities: how many of this or how many of that. For example, “3” identifies three sheep or three fish. The sequence of counting numbers is potentially infinite since the human mind can keep adding units of 1 to form an ever greater number. Subsets are similarly infinite in their sequences. The odd numbers, as in 1, 3, 5, 7…, are infinite as are the even numbers, 2, 4, 6, 8…. On a straight line, in one vector, each natural number can be represented by one point on a line ad infinitum. (4)

A second species of number emerges in whole numbers when counting proceeds in reverse: toward and beyond 1. Nought or zero emerges as a number to signify the absence of some item. The creation of this numerical designation signifies an “empty set” as in “the number of Eskimos living in our house is 0.” (5) The inclusion of 0 with the counting numbers thus creates a larger system of numbers than the old quantitative counting numbers. Enumeration now begins from 0 which can also be represented by a point on a line.

A third, more comprehensive set of numbers emerges when the reverse counting which had led to 0 continues backwards to include numbers that are now less than zero. The result is a potentially infinite set of negative whole numbers. When these numbers are then added to the numbers that have already been generated by counting from zero upwards (the positive whole numbers), the result is a set of numbers known as integers. An integer is defined as a positive or negative whole number as in 0, ±1, ±2, ±3, ±4 . . . (6) The negative and positive signs indicate direction: all these numbers are directed. On a number line, the negative numbers go to the left of 0 while the positive go to the right. Each number has a point.

Rational numbers deriving from a ratio or fraction of integers or whole numbers emerged when it became necessary to specify measurements which are parts of a number. How does one express a length which is between 4 and 3 meters or 4 and 3 cubits? Is a loaf of bread, equally divided among 5 persons, divided in a way where each piece has a numeric value of 1/5? Does the addition of 1 piece to another not result in a union with a numeric value of 2/5? A number designating parts thus consists of parts in its makeup. There are two halves: a numerator above a line and denominator beneath. (7) The denominator indicates how many intervals exist between two possible whole numbers while the numerator indicates how many of these intervals are pertinent in a given measurement. The denominator cannot be 0 since, otherwise, one would be indicating that no intervals or parts exist between two numbers. Why specify numerators for portions or parts that do not exist? A rational number is commensurate with given lengths that are being measured. A number which includes a fraction can be assigned a point on a line. The position is determinate.

In the 5th Century B.C., the Pythagoreans initially assumed that numbers measuring the sides of a triangle are rational where each number can be expressed as the ratio or quotient of two integers (or two whole numbers). (8) Divisors (or denominators) exactly divide into numerators as in ½, 1/10, and 1/100: a half (or .5), a tenth (or .10), and a hundredth (or .100). A ratio as the quotient of two numbers or quantities indicates relative sizes. (9) The ratio of one number to another is expressed in terms of a/b or a:b. It was assumed that a one-to-one correspondence joins straight-line segments of length with rational (whole) numbers. (10) In attempting to measure the diagonal of a square by taking a small part of one side as the measuring unit, one should be able to fit the measuring unit a fixed number of times within both the side and the diagonal. (11) All lengths are measurable and commensurate in terms of rational (whole) numbers. Two quantities are commensurable if their designating numbers are multiples: both numbers arise as products of common factors (a factor being a number that divides a given number exactly or completely (12)). For instance, 16 and 12 are commensurable since both exist essentially as multiples of 1, 2, or 4: each exactly divides into 16 and 12 and no other number exactly divides 16 and 12. By multiplying one or more of these numbers together, one arrives at numbers 16 and 12 (in conjunction with other possible numbers that are also commensurable). Similarly, 3 feet and 2 inches designate commensurable quantities since 3 feet contains 2 inches an exact or integral number of times. (13) Hence, according to Pythagorean assumptions and expectations, the length of a square’s diagonal whose side is represented by a rational number should be represented by another rational number.

On the basis of this belief in rational numbers and the corresponding commensurability of lengths, according to the Pythagoreans, “numbers are things” and “things are numbers.” All things are numerable in terms of whole numbers and their properties. (14) A cosmic harmony exists in the universe given the interrelation of things based on whole numbers where the relation between two related things can be expressed according to a numerical proportion or ratio. For example, in music, ratios of concord exist between musical sounds (pitch) and whole numbers since by halving the length of a string on a lyre, one can produce one note one octave higher. All harmonies can be represented by ratios of whole numbers and, by extending this principle to all things, through geometry one can explore the configurations of perfect solids in the belief that all lengths are measurable in terms of rational whole numbers.

A crisis emerged for the Pythagoreans when, possibly prior to 410 B.C., they realized that some numbers, though real (as existing), class as irrational because they cannot be written as whole numbers, as integers or as quotients of two integers. (15) No assignable point of a line can be given them. Some numbers do not exist thus as whole numbers as can be seen through a deduction from Pythagoras’ Theorem in geometry which describes the relation between the lengths of the sides of a right-angled triangle in the following terms:

In a right-angled triangle, the square on the hypotenuse [the side of right-angled triangle opposite the right angle] is equal to the sum of the squares on the other two sides. (16)

Thus, if the hypotenuse has a length c and the other two sides, lengths a and b, then c2 = a2 + b2. Now, if, in a square, the side length constitutes 1 unit, then

c2 = 1 + 1

Hence,

c2 = 2

Thence,

c = sqrt(2)

The diagonal is 2 units in length. (17) This number obviously designates some magnitude greater than 1 or unity but less than two where, initially, one naturally assumes that this number is an improper fraction expressing a whole number (an improper fraction being defined as a fraction whose numerator exceeds its denominator as in 4/3 versus 3/4, designating a proper fraction (18)). (19) However, if the square root of 2 cannot be expressed as a whole number, its irrationality in terms of whole number properties creates major problems given expectations which assume the adequacy of whole numbers. After all, conversely, if only rational numbers exist, the hypotenuse of every right-angled triangle will have a length that cannot be measured by any whole number. (20) It is incommensurable, non-measurable: in the relation between the diagonal d and an adjoining side s, d cannot be divided by any unit common to s an integral number of times. In trying to effect any measurements, the Greeks found that however small or large would be their measuring unit, it failed to fit within both the diagonal and the adjoining side a fixed number of times. (21) A measuring unit that would fit the adjoining side a fixed number of times would not fit the length of the diagonal. It was either too short or too long. Proofs demonstrating the irrationality of 2 came in a number of varieties.

Aristotle refers to a proof on the incommensurableness of a square’s diagonal with respect to a side that is based on the distinction between odd and even, an odd number being an integer that is not divisible by 2 while an even number is divisible by 2. (22) To understand how this argument works, a digression on prime numbers introduces the discussion.

A prime number is a whole number with exactly two whole-number divisors, itself and 1. Some primes are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . . , 101, . . . , 1093

Prime numbers are the building blocks of other whole numbers. For example,

18 = 233 40 = 2225 105 = 357

This type of factorization is possible for all nonprime whole numbers greater than 1 and it illustrates the fundamental theorem in arithmetic known as the Unique Factorization Theorem (23) which says, as follows, about the prime decomposition of a whole number:

Any nonprime whole number (greater than one) can be written as the product of a unique set of prime numbers. (24)

Every prime integer shares the important property that if it divides a product of two integers, then it must divide at least one of the factors (prime numbers being only divisible either by themselves or by 1). This theorem is important in many parts of mathematics. In one simple consequence, when the square of any whole number is written as a product of primes, each prime occurs as a factor an even number of times. For example:

(18)2 = 1818 = 233233 = 223333

two 2’s four 3’s

(40)2 = 4040 = 22252225 = 22222255

six 2’s two 5’s

(105)2 = 105105 = 357357 = 335577

two 3’s two 5’s two 7’s

To prove that the square root of 2 is irrational, let us suppose that 2 is a rational number; that is, suppose that 2 = m/n, where m and n are whole numbers (necessarily greater than 1). Then:

2 = m2/n2

and so

2n2 = m2

Now, imagine that both n and m are written as products of primes where, for instance (using algebraic notation), n = xy while m = zpt. But, as previously noticed, both n2 and m2 must then have either an even number of 2’s or no 2’s. But, in the above equation, the prime 2 appears on the left an odd number of times either once (if n2 has no 2’s) or more than once (if n2 has an even number of 2’s) but, on the right, the prime 2 appears either an even number of times or no times. This is clearly impossible since, given the nature of primes, m2 equates with a number or produces a number that has either an even number of 2’s or no 2’s. A contradiction obtains despite the equals sign. Therefore, what can be wrong? The only thing that can be wrong is our supposition that 2 is a rational number. If this proof is applied to other primes in terms of square roots for 3, 5, 7, . . ., the same dilemma results. (25) Odd clashes with even to demonstrate the irrationality of these numbers. Hence, could all numbers be the kind of numbers that the Pythagoreans had postulated? Are they all rational?

In Boyer’s version of the mathematical proof demonstrating the incommensurableness of the square root of 2 through the contrast between even and odd, he argues as follows: (26)

1. Let d and s respectively signify the diagonal and side of a square and let us assume that they are commensurable: the ratio d/s is rational and equal to p/q, where p and q are integers with no common factors.

2. given the Pythagorean theorem d2 = s2 + s2 reconfigured as d2/s2 = 2 (since d2 = 2s2), if the ratio d/s = p/q (p and q being integers with no common factor), then (d/s)2 = p2/q2 = 2 or p2 = 2q2

3. therefore, p2 must be even since its equivalent 2q2 is divisible by 2 (which corresponds to the definition of an even number as a number divisible by 2).

4. hence, if p2 is even, p is even since p2 when decomposed into constituent prime numbers necessarily includes at least two instances of 2 as both a prime number and a factor, and the presence of 2 in p makes p an even number since it is divisible by 2 (which again corresponds to the definition of an even number).

5. as a result, q must be odd (not divisible by 2) since, according to conditions stated in aforementioned proposition 2, q is an integer with no factors common to p and so it cannot have 2 as a constituent prime factor.

However, letting p = 2r and substituting in the equation p2 = 2q2 with, hence, the result that 4r2 = 2q2, 4r2 = 2q2 as reconfigured becomes q2 = 2r2. Then q2 must be even; hence q must be even (according to the argumentation which had explained why formerly p2 and p must both be even). However, a contradiction follows if one argues that q is both odd and even. No integer can be both odd and even. As a consequence, it thus follows that the numerical relation between d and s is incommensurable. (27) The result is not a definitive whole number.

A third but second species of proof relying on a study and understanding of prime numbers demonstrates the absence of an anticipated whole number by adverting to the relation between d and s. If, indeed, d (a whole number) is decomposed into constituent prime numbers and s (a second whole number) is similarly decomposed, and if no factor is common between them, the improper fraction d/s can never be resolved into a whole number since, in every case, the denominator does not perfectly divide into the numerator to produce an anticipated, desired whole number. The result is always some sort of fraction which, by definition, is not an integer, a whole number.

A geometrical proof that evidences the existence of irrational numbers in general, and not 2 specifically, designates a third species of proof. (28) Its lesser abstractness suggests earlier origins predating the construction of later proofs using other types of arguments. When examining the sides and diagonals of a regular pentagon (defined as a five-sided polygon with all the sides possessing equal length) and the respective relations between s and d, if the diagonals of this pentagon are all drawn, they form a smaller regular pentagon whose diagonals can also be drawn to form a smaller regular pentagon ad infinitum. Hence, pictorially, the relation or ratio of a diagonal to a side in a regular pentagon is indeterminate because it is indefinite. It is irrational. Similarly, if a straight line is divided into two parts and one part is divided into two smaller parts, it will be possible to keep dividing lengths indefinitely. (29) No determinate end is reached. Our expectations meet with frustration as our inquiry encounters mysteries that occasion questions about the adequacy of our intelligible anticipations. What is to-be-known cannot be known too easily or simply.

1. Bernard Lonergan, Insight: A Study of Human Understanding, eds. Frederick E. Crowe and Robert M. Doran 5th ed. (Toronto: University of Toronto Press, 1988), pp. 45-6.

2. Leslie Foster, Rainbow Mathematics Encyclopedia (London: Grisewood & Dempsey Ltd., 1985), p. 43.

3. Foster, p. 43.

4. Foster, p. 43.

5. Foster, p. 43.

6. 6The Penguin Dictionary of Mathematics, 1989 ed. S.v. “integer.”

7. Foster, p. 44.

8. 8E. T. Bell, The Development of Mathematics (New York: Dover Publications, Inc., 1992), p. 61.

9. 9The Penguin Dictionary of Mathematics, 1989 ed. S.v. “ratio.”

10. 10Bell, p. 61.

11. Joseph Flanagan, Quest for Self-Knowledge (Toronto: University of Toronto Press, 1997), p. 33.

12. The Penguin Dictionary of Mathematics, 1989 ed. S.v. “factor.”

13. The Penguin Dictionary of Mathematics, 1989 ed. S.v. “commensurable.”

14. Carl B. Boyer, A History of Mathematics, 2nd ed. (New York: John Wiley & Sons, Inc., 1989), p. 72; Frederick Copleston, S.J., A History of Philosophy, volume 1: Greece & Rome part 1 (Garden City, New York: Image Books, 1962), pp. 49-50; A Concise Oxford Dictionary of Mathematics, 1991 ed., s.v. “Pythagoras,” by Christopher Clapham.

15. The Penguin Dictionary of Mathematics, 1989 ed., s.v. “irrational number.”

16. A Concise Oxford Dictionary of Mathematics, 1991 ed., s.v. “Pythagoras’ Theorem,” by Christopher Clapham.

17. Bell, p. 61.

18. 18Penguin Dictionary of Mathematics, 1989 ed., s.v. “improper fraction.”

19. 19Lonergan, Insight, p. 45.

20. 20Euclid quoted by Walter Fleming and Dale Varberg, College Algebra: A Problem-Solving Approach (Englewood Cliffs, New Jersey: Prentice Hall, n.d.), p. 16.

21. Flanagan, p. 33.

22. 22Boyer, p. 72; Penguin Dictionary of Mathematics, 1989 ed., s.v. “odd number,” and “even number.”

23. Clapham, p. 187.

24. Fleming and Varberg, p. 16.

25. Fleming and Varberg, p. 17.

26. Boyer, pp. 72-3.

27. Boyer, p. 73.

28. Boyer, p. 73.

29. Boyer, p. 51.

Commentary: Ft. 28 Ch. 1 Lonergan’s Verbum: Word and Idea in Aquinas

Footnote 28 in Lonergan’s Verbum: Word and Idea in Aquinas

The full text of this footnote in Bernard Lonergan’s Verbum: Word and Idea in Aquinas is given as follows: “See In I Sent. (Op. Ox.), d. 35, q. unica, n. 7 (ed. Vives, X, 544). Scotus argues that the divine ideas cannot be accounted for by adding notional relations to the divine essence; for the object precedes the knowing, and relations that precede knowing are not notional but real. The argument does not touch Aquinas’s real position, which is that the object as known is not prior and that the relations pertain only to the object as known.” My italics. In attempting an exegesis of this text, to try and explain its meaning, I shall focus on two elements that exist within it: two elements that conceptually point to the presence of two core elements. According to Lonergan’s judgment, in Scotus’s understanding of human cognition, “the object precedes the knowing” and in Aquinas, a contrary point of view can beregarded as a species of first principle in his understanding of human cognition where, for him, “the object as known is not prior.” If something else is prior, it should not be conceived as a real object. It should not be seen to exist as a known object. The object, as known, is constituted by our acts of cognition; principally, by our acts of reflective understanding (our acts of judgment).

To try and understand the difference in these two positions, I would like to argue that it would be wise to grasp what could be true in arguing or believing (with Scotus that “the object precedes our knowing” and what could be true in arguing or believing that the object, as known, is constituted by our acts of cognition, our acts of sensing leading to acts of direct understanding and then, from there, to acts of reflective understanding. We would want to move toward some kind of higher viewpoint that would allow us to notice where the truth lies or how the truth is distributed and how, at the same time, this difference or dispute points to two differing views about the nature of human cognition. Bluntly put, one point of view thinks in terms of intuition; the other, in terms of a discursive form of human reasoning. If, to intuition, a meaning or a definition is ascribed which says that it refers to a non-rational form of human cognition, then, from this point of view, we can begin to understand why a real distinction should be drawn between these two schools of thought. The truth of one position would seem to point to the errancy of the other position. One is true; the other, false.

We begin then with an initial take on why it is said by some that the “object precedes knowing” while others say that the “object is constituted by knowing.”

As an initial point of departure, please note thus that, when Lonergan speaks about our human experience of acts and different objects that are intended by our different acts and how we can move from these experiences to conclusions which would speak about the probable existence of that which exists for us as the human mind, the human intellect, the human will, and the human rational soul, he postulates the probability that this type of analysis was employed in the inquiries of both Aristotle and Aquinas. This approach is, of course, likely since we cannot assume or believe that Aristotle and Aquinas were lacking in the experiences of understanding which properly belonged to them as human subjects (experiences of understanding which would differ from the kind of experiencing which would exist in their various acts of human sensing). However, if, in some way, it can also be argued that both Aristotle and Aquinas were interested in engaging in a type of inquiry which wanted to ask about the conditions of possibility which need to be discovered and acknowledged by us if, at bottom, we are to understand why our acts of sensing and why our acts of understanding exist in the way that they do, then it cannot be denied that some available textual evidence points to a metaphysical kind of analysis which begins with a sense or an initial apprehension of being which exists for us as a species of a priori, relative to the asking of questions in a manner which is constitutive of our human cognition. Cf. Verbum, p. 58, n. 207, quoting Aquinas, Summa Contra Gentiles, 2, 83, 31. This experience or awareness of being (that which is or that which simply exists) does not follow or emerge in the wake of our later acts of reflective understanding. If being is intelligible, it follows that being is known (we can argue that it is best known) through our subsequent acts of understanding, our acts of direct understanding leading to later acts of reflective understanding which would exist as affirmations or negations of being through the mediation which exists whenever we refer to the kind of act which exists in our acts of judgment. However, as we advert to our experience of self, we cannot deny that our acts of sensing and our acts of inquiry presuppose something other which is simply given to us (a kind of “is” which refers to something which is quite other than ourselves, an other that we cannot deny without being involved in some kind of internal performative contradiction). We know, of course, that we know different instances of being through our judgments. But, if we begin with an initial a priori awareness of being which exists as a kind of pre-condition to our later acts of human cognition, is it not right to argue that, from this awareness of being as it exists without a specific determination of it (it is known by us in a vague manner), from this awareness, can we conclude that, within the context of our self-reflection, this awareness of being is to be associated with what we mean when we went to speak about “being” in a non-differentiated general sense? Cf. Roland Krismer, email, January 3, 2015.

Obviously, something other is initially given to us (it is the context of our individual being and existence) and if, in a metaphysical context, this being, as other, is taken up and then viewed as a species of first principle from which a number of conclusions and deductions can then be drawn, then, in addition to a cognitional form of analysis which moves us from our cognitive experience of self as a species of first principle from which a number of conclusions and deductions can also be drawn – an analysis which moves toward consequent apprehensions of being and which seeks to transpose or to reduce our apprehensions of being into elements and relations which immediately exist for us within our cognitive experience of self – we can also get into a form of analysis which is specifically metaphysical (and not cognitional). Instead of a situation where a context is determined by a principle which says that the subject knows an object (an object is constituted by a subject through the acts of a given subject), a situation is to be adverted to where another context is determined by a principle which says that an object influences or moves a subject. Some kind of object creates a given kind of subject. From objects, we move to acts. Indeed, in the metaphysics of Aristotle and Aquinas, both say that nothing in a condition of potency is able to realize itself. No potency can move from how it exists as a potency to how it would exist as within a condition of act. The potency of our human understanding in fact points to a kind of priority which is to be admitted when our analysis shifts toward an order of being (an order of real objects) which exists independently of whether or not it can be known by us through an order which would refer to the order of our human cognition (the order of our human knowing versus the kind of order which exists within the order of being). From the order of our human knowing, yes, we can move toward an order which exists within the proportionately known known which exists, for us, as the order of being or the order of reality (the order of metaphysics). However, the priority of being over knowing as we experience this priority, to some extent, explains and at the same time it also justifies the value of working with a form of analysis which begins with the order of being (the given of being) as a way that can move us toward the order which exists when we get into the structure of acts which is constitutive of our human cognition. The advantage of starting with this approach is the awareness which we would have that our human knowing exists within a larger horizon which already exists, a horizon which needs to be attended to and known for what it is if our human knowing is to benefit from an awareness which knows that this context conditions and informs the contours and the content of our individual human knowing, giving it a direction and an orientation that it might not have otherwise if our sole point of departure were to be an awareness that is only concerned with the givens of our inner consciousness of self as this consciousness refers to our various acts and data of consciousness. Cf. Roland Krismer, conversation, December 31, 2014.

As thinking, knowing beings (as thinking, knowing subjects), we exist within a world which has already been formed: the objects which are constitutive of a naturally existing human world that is joined to a naturally existing physical, chemical, biological world act upon ourselves to elicit inquires and investigations which can lead us toward a better understanding of this same world. Nothing which exists within our world (and nothing which exists within ourselves) is adequately explained if we cannot work with a form of analysis which moves from a species of metaphysical understanding toward a species of cognitional analysis and then, from there, back toward a species of metaphysical analysis. As Aquinas would have it, an ongoing form of interaction is constantly operative for us as we move between inner and outer conditions or as we find that, as human subjects, we are always caught within an interplay which pertains to the existence of inner and outer conditions. We never engage in any kind of inquiry without having a prior knowledge of some kind. We begin with acts of understanding which we already have (acts of understanding that we have not to work toward) and, from this understanding, we hope to add to the understanding and knowledge of things which we already have.

These things being thus said, if we acknowledge that, in the context of our concrete experience of life, we have both kinds of knowledge (a prior knowledge of things prior to inquiry and a later knowledge of things that is reached later through inquiry), then, from this context, we can speak about two notions of object. A first notion speaks about an object that exists apart from our human knowing and which determines our human knowing and a second notion of object speaks about an object as it exists within the context of our human knowing and which is constituted by the different acts that are constitutive of our different human acts of cognition. Bluntly put, and as a kind of introductory summary, for those who regard knowing as a species of confrontation between a subject and an object, “everything that is known, is known inasmuch as it is an object” while, on the other hand, for those who subscribe to an understanding of cognition which thinks in terms of an identity which exists within a knowing subject, “everything that is known, is known inasmuch as it is in act.” Cf. Fred Lawrence, “Jerusalem and Athens in Lonergan’s Integral Hermeneutics,” Divyadaan (2008): 95. In the later kind of knowing, knowing exists as a kind of perfection. To some extent, human knowers change (they are inwardly changed to some extent) when, within their experience of cognition, they participate in the greater or the other reality of the being of a thing which they have come to understand and know. In other words, if we are to understand how we can speak about two different notions of object and how they can be distinguished and related to each other in a way that can join these two notions into a higher unity that we can understand, we can attend to an understanding of human cognition which begins with an elementary understanding of our human cognition (an understanding which points to a notion of being which thinks of it or which would want to conceive of it to exist as a species of external, outer object) and then, from there, we can move to a notion of object which is informed by forms of cognitive experience which differ from the givens of sense whenever we refer to our acts of sense and the content or the data that come to us through our different acts of sense. We refer here to a form of cognitive experience which exists where an objective order of things can begin to participate in the subjectivity of a given knower through a change which occurs within the subjectivity of a knower in a way which joins object and subject into a union which, previously, had not been known. It had not existed in any kind of prior way.

To understand a notion of object which thinks of it as something which is quite other than ourselves and which exists independently of ourselves, we can begin thus with a strong, common temptation which suggests to us, as human beings, that human cognition should be viewed as essentially a simple, single act of intuition which is best understood if it is seen as fundamentally akin to the nature of what happens in ocular acts of perceiving and seeing. An elementary object acts from without as an agent object to elicit a response from the passivity of an elementary subject: a knower who exists only as a sensing subject or who tends to think or believe that his or her knowing exists essentially as an act of sensing. Cf. Lonergan, Insight, p. 207. Within this context, partially for reasons of simplicity, it is so easy for us to say that knowing is just taking a look: we see what is there to be seen and knowing happens by way of what comes back to us from our taking a look. Cf. Lonergan, Understanding and Being, p. 159. An object or objects are seen to be already constituted (they fully and entirely exist in their being and reality) before they are attended to by us and before they are grasped by us through our later acts of understanding and the subjective species of operations which exists when we refer to our various acts of understanding. In general, between knower and known, an unbridgeable gap exists. Between knower and known, a confrontation exists. Necessarily, it exists. Knowing does not happen within a subject but in a way that exists from outside a subject or by way of bracketing a subject’s existence (putting it to the side). Nothing happens within a subject in a way which could point to the necessity of a contribution that a subject would make or should make in any acts of cognition that are truly proper to us as human subjects. In this type of approach, we would not need to speak about a need for some kind of development that would or should occur within the life of a subject. Knowing does not exist within the life of a subject and it does not require any kind of change in the life of a given subject. The less a subject understands and knows, the better. A subject then would be more open to pure acts of reception as these would exist by way of agent objects that would be acting from without to effect a stimulus of some kind in the consciousness of a would be knower (whether we would wish to refer to a sensible consciousness or some kind of intellectual consciousness which is thought to resemble the kind of consciousness which belongs to our acts of sense).

As an aside thus (and also by way of illustration), from the priority of objects over subjects (or, in other words, from the apparent obviousness of this priority), we can then understand why, in Plato, it would naturally follow for him that Ideas, Forms, Natures, Concepts, or Essences must always subsist in themselves in a very distinct world which is constituted by these same Ideas, Forms, Natures, Concepts, or Essences. Hence, Ideas, “the Ideas are ta ontõs onta, what really is.” Cf. Lonergan, Early Works on Theological Method 1, p. 70. Within this context, if we should want to refer to a possible relation which could possibly exist between ideas and our human acts of understanding, ideas exist or they are seen to exist as instances of objective being (as esse obiectivum) and not as terms or as objects which are constituted by the acts of understanding that have been given to us. If the Ideas that we happen to know about subsist in themselves, not so or much less so is this the case when or if we should try to refer to how Ideas, Forms, Natures, Concepts, or Essences are all joined to each other in a manner which is constitutive of an already existing general scheme of things (an order or world-order as distinct from individually existing Ideas, Forms, Natures, Concepts, or Essences). Within this context (the priority of objects over subjects), we can only move toward a general order or a general scheme of things (understanding or knowing about a general order of things) if we first attend and work with separately existing Ideas, Forms, Concepts, or Natures that we somehow already initially know: Ideas, Forms, Natures, Concepts, or Essences which exist independently of our acts of understanding (be they human or divine). Ideas first exist on their own and, as persons, we can contemplate these same ideas or we can understand these same ideas. Cf. Ronald Vardiman, “Notes on Conceptualism,” n.d. The absence of acts of understanding which can explain or point out why this “x” could be related to this “y” or why the existence of “x” is conditioned by the existence of “y” and why too the existence of “y” is conditioned by the existence of “x” – these absences of understanding all serve to explain why Ideas, Forms, Natures, Concepts, or Essences cannot be understood from within a context which would want to know about the existence of real relations: how, in themselves, for instance, a relation or a connection between elements “x” and “y” is constitutive of the being or the meaning of an Idea, a Form, a Nature, a Concept, or an Essence. Cf. Brian Himes, “Lonergan’s Position on the Natural Desire to See God as Corroborated by Aquinas’s Doctrine of Creation by Participation and His Nominal Definition of God as Ipsum Esse,” Boston College, School of Theology and Ministry, June 30, 2011, pp. 6-7, http://www.lonerganresource.com/pdf/contributors/Himes%20-%20Lonergan’s%20Position%20o% 20the%20Natural%20Desire%20to%20See%20God.pdf (accessed November 4, 2013).

This order or priority of objects over subjects which we find in Plato accordingly explains why Plato speaks about a “first order of ideas” that refers to Ideas or “intelligibles” and a “second order” which refers to gods who contemplate this order of Ideas or “intelligibles.” Cf. Lonergan, Understanding and Being, p. 237; Ontological and Psychological Constitution of Christ, p. 255; Early Works on Theological Method 1, p. 70. If you will, we can say that the priority of objects over subjects in Plato is used by him to construct a theology which reflects this order in the subsequent order of relations which exists within his theology. Easily, for us, we go from an understanding that we have about the nature of our human cognition into a theology is grounded in the depth of understanding which we may have or not have as this understanding refers to the nature or the order of our human cognition.

In the subsequent history of philosophy and in developments after Plato, a conception of knowing which thinks in terms of confrontation and the priority of object over subject can accordingly be found in thinkers such as Clement of Alexandria, Origen, and Plotinus and then later on in how it seems that St. Augustine speaks about the dynamics of our human cognition. See, for instance, Peter Beer, Introduction to Bernard Lonergan, pp. 170-171, on how Clement of Alexandria and Origen both conceive of human knowing in terms which speak about a spiritual looking or gazing that occurs through an intellectual or spiritual kind of seeing as this occurs employing the “eyes” of our minds, the “eyes” of our understanding. In medieval philosophy, Duns Scotus was a prominent exponent of this same point of view about the nature of our human cognition (our human knowing occurs intuitively and by way of a species of confrontation through the presence of some kind of spiritual or intellectual form of seeing which beholds truths or realities that are somehow seen) and, in modern philosophy, this same point of view is perpetuated in the philosophy of Rene Descartes and Immanuel Kant. Prominent 20th Century exponents include Edmund Husserl and Martin Heidegger despite a common interest which each has in showing how our human intentionality (our cognitive human desire) functions as a constitutive element within our human cognition, eliciting or pointing to a development in the life of a human subject (a human being who would be existing as a human subject). Cf. Linus Kpalap, “The Knower and the Known,” unpublished paper given at Sogang University, Seoul, Korea, June 3, 2010, p. 7.

In terms thus of a history of subsequent ramifications which have occurred within the later development of Catholic theology, see also Georges Van Riet’s study of epistemology as this has existed amongst Thomists since early in the 19th Century. Van Riet’s L’epistemologie thomiste: Recherches sur le probleme de la connaissance dans l’ecole thomiste contemporaine (Louvain: Editions de l’Institut Superieur de Philosophie, 1946) divides the cognitional theory of Thomist philosophers into two basic divisions: “those whose epistemology is fundamentally a matter of confrontation” (“seeing and confronting”) [primacy of object over subject] and “those who hold an epistemology that is based upon understanding, comprehension, intelligence” [an object as known exists as the term of an act of reflective understanding]. Cf. Lonergan, Understanding and Being, p. 103; p. 177. Amongst Catholic philosophers, if, following Scotus, knowing is seen essentially as “taking a look,” acts of judging are said to be the “taking [of] a look.” Being or reality is that which is looked at. It is that which can be perceived. And so, for Etienne Gilson, understanding (or judging) is but a form of seeing. Judging exists as a species of intuition and it can be referred to in terms which speak about judgmental intuitions: “By our senses we perceive the sensible, but intellect can see being in the sensible.” Lonergan’s italics. Cf. Lonergan, Early Works on Theological Method 1, p. 119, quoting from E. Gilson, Réalisme thomiste et critique de la connaissance, p. 225. The judgmental intuition which we find in the cognitional philosophy of Etienne Gilson and Joseph Owens clashes, however, with the judgmental finality which we can find within the cognitional philosophy of Bernard Lonergan and Karl Rahner. Cf. Michael Vertin, “Rahner, Lonergan, Knowing, and Teaching,” paper presented to the Catholic Theological Society of America, June 2004, pp. 3-4. In both Gilson and Owens (and also in William of Ockham), if we should want to speak about judgment and exercises of judgment, judgments of existence would say about reality that something is present or that it is given: something is directly seen or perceived. Instead of a judgment which would say that “this is” or that “this reality exists,” it would be said instead that “a reality is present there,” or “there is a particular reality there,” or a reality is “situated there,” or “an existent reality is present there.” Acts and data of sense are not transcended by any acts of the human mind which would move towards “that which is” or “that which exists” since these same acts of the mind (intellect, understanding, judgment) only co-operate with human acts of sense simply or only to affirm that which is already grasped by us in our human acts of sense. Cf. Early Works on Theological Method 1, pp. 119-120; Marie-Dominique Philippe, Retracing Reality A Philosophical Itinerary, trans. Brothers of St. John (London: T & T Clark, 1999), p. 154; p. 156. The kind of passivity which exists with respect to our acts of sense is extended and we say that it is applied to all our acts of understanding and judgment in a way which alludes to a like passivity (a radical form of passivity) which seems to exist with respect to all our differing acts of cognition and so, by this means or by this omission, no notice is taken about how, in our human inquiry, active potencies can be properly alluded to: potencies, movements which point to operative forms of efficient causality which exists within our human cognition: activities which would immediately point to a work of constitution which occurs within our human cognition as we engage in acts, actions or activities which set us apart as knowers from every other kind of cognition (whether the knowing of animals, the knowing of angels, or the knowing which only belongs to God).

The partial absence of passive potency within our human cognition (according to relative determinations of it) serves thus to explain why it can be said that the objects which we know through our cognition are known through apprehensions of intelligibility and truth that are constitutive to some degree (to a degree which should be recognized). Through inquiries that lead to direct acts of understanding, an intelligible universal component is detached from sensible material components and this kind of separation supposes an interest and efforts on our part to achieve this kind of desired separation (even as we admit that our acts of understanding exist as receptions and not as activities which we can simply do through our willing of them). In another but similar way too, through inquiries that lead us to acts of reflective understanding, through the kind of self-reflection which occurs in our prospective acts of judgment, we advert to conditions of possibility which need to be given or fulfilled if we are to move from apprehensions of meaning to apprehensions which know that an understood meaning is to be viewed as a true meaning (a real meaning). In our self-reflection, we look for evidence which could exist for us as a species of first principle, triggering an act of understanding which could exist as an affirmative conclusion or affirmative judgment. One kind of first principle refers to basic laws of reason (for instance, with respect to the law of contradiction, does a contradiction exist within my reasoning as I move from premisses to conclusions?) and a second kind of first principle refers to grounding acts and date of sense (am I presently experiencing a datum of sense or a datum of consciousness which points to the truth of a given conclusion?). In the mere givenness that exists within our acts of sense, through the kind of immediacy which exists within this givenness and also through the mere givenness or the immediacy which exists when we refer to the basic laws of our human reasoning, perhaps here, in a limited way, we can speak about how our acts of cognition can be seen to exist in a way which points to intuitions. Where immediacy exists within our human cognition, we are tempted to speak about our acts of cognition as if they exist as intuitions although, on the other hand, where immediacy does not exist within our human cognition, within this context, we are tempted to speak about the discursiveness of our human cognition: its gradualness and its condition of incompletion as we move from something that we know to something that we can come to know by an addition that possibly we can make to the sum which we already know. Every advance in our understanding and knowledge points to the possibility of other possible advances and how or why our cognition exists as a combination of active and passive potencies that, in some way, work together to reveal how, through our subjectivity, it is possible to move toward experiences of objectivity that are given to us within a range of data which refers to the data of our consciousness, the inner consciousness which we have of ourselves as human beings (experiencing, understanding, and knowing). Outer, external objects or other, external things come to exist within our consciousness of self through the experience of intelligibility that exists within ourselves through the consciousness of self which exists within ourselves.

By way then of a kind of conclusion, we can notice that experiences of immediacy in cognition suggest or point to an understanding of cognition which thinks in terms of intuition. The kind of immediacy which exists in sense resembles the immediacy which we can also experience when we allude to the fundamental laws that are constitutive of our human acts of reason: the laws of identity, excluded middle, and contradiction. Sometimes too, if immediacy is regarded as a legitimate species of first principle, some acts of understanding can be regarded as intuitions. Upon self-reflection, we know that many of our acts of understanding have not to be gained or acquired in any way through any of our human efforts. They already exist and sometimes our knowledge of them is revealed to us in other contexts when we advert to a question that we are asking and when we realize that our question or our questioning is supposing something else that we do not question: a something else that we already understand and know. Our experience of immediacy in our human cognition accordingly suggests that intuition is not entirely alien to the performance of our human cognition. A place exists for it.

However too, it is to be admitted that the asking of questions, the onset of human inquiry, is to be regarded as another kind of first principle which points to the incompleteness or the discursiveness of our human cognition. In a manner which differs from animal kinds of knowing, our human cognition does not exist as something which is already given to us as some kind of inborn pattern or instinct. In contexts where instinct determines the behaviour of any kind of living thing, it is not necessary to think about alternative courses of action and to think about whether or not a given course of action should be put into effect rather than some other possible course of action. In instinct, we find a species of determination. Similarly, by way of another contrast, it is not said about angels or of God that their knowing is characterized by any form of potentiality. It is said that, although a creature, an angel does not need to ask any questions. Everything has already been given to an angel from another kind of intellectual being who, also, is without any need or compulsion to ask questions and engage in any inquiries. Everything is fully understood. Everything is fully known. Hence, for us, as human beings, the lack of completeness in our knowing when this is coupled to the possibility of growth which belongs to us as human knowers – together, these two variables point to a discursiveness that is peculiar to us in the kind of cognitive life which properly belongs to us as contingent human beings. The discursiveness of our cognition sits as the center of things within our cognitive life and intuitions exist more or less as boundaries or as a kind of frame that surrounds the discursiveness of our cognitive operations. Our knowing always begins with an understanding of things and a seeing of “things” (a perception of bodies) that we already have (an understanding that we have no need to ask questions about) and the working out or the development of our cognition through acts that lead from our acts of sensing to our acts of direct understanding and then from there to our acts of reflective understanding – these acts all lead us toward acts of cognition that possess an obviousness or an unquestionableness which, in its own way, points to an immediacy or the mere givenness of content and data that seems to come across as an intuition (as a species of intuitive grasp) even as we admit that these experiences of immediacy are conditioned by prior acts of inquiry which point to the discursiveness of our human cognition. Whenever we have anything which appears to us as something which is self-evident, we associate the experience of self-evidence with the deliverances of intuition although, as we have already indicated, inquiries are often needed before a given meaning or a given truth can be regarded at some point as something which should be self-evident to us in our experience of understanding. Paradoxically, through the inquiry and the discursiveness which belongs to our various human acts of reasoning, we can come to know about the existence of immediacy within human cognition and also about the absence of immediacy which exists within our human cognition and so, by this means, we can come to know about the nature of our human intuition and how it relates to a more fundamental second nature which refers to the discursiveness of our human reasoning and how a strategy of acts exists within the structure or the form of our human acts of inquiry. Conceptually or abstractly, an exclusivie disjunction exists between an understanding of human cognition which speaks about intuition and another understanding of human cognition which speaks about its discursiveness. However, when we attend to the data or the experience of our human performance as this is given to us in our experience of self, we find where these two understandings of cognition have a grounding which points to an order which joins these two kinds of cognition with each other although, as noted, the greater context is an understanding which points to a discursiveness which is properly distinctive of the kind of cognition which belongs to us as human beings whenever or as we exist in our subjectivity as living human subjects.

Inner Words as Objects of Thought (without footnotes)

Some Notes on the Inner Word as an Object of Thought

by Br. Dunstan Robidoux OSB

In a review of arguments that Lonergan marshals, Lonergan contrasts the necessity of an inner word in human understanding with the absence of such a necessity in divine understanding (where God exists as an unrestricted act of understanding). Key is the discursiveness of human reason. The absence of any discursiveness in divine understanding and knowing (or, in other words, the simplicity of God’s existence as an act of understanding) explains why we cannot conclude that the proceeding of an inner word necessarily exists in God.1 If an inner Word necessarily exists within God, its basis or ground is an actuality that exists without the necessity of there having to be any reasons or considerations which exist as prior conditions that need to be attended to and fulfilled. The unconditionedness of divine things can be spoken about or conceived in thoughts and words by us in the thinking and speaking that we do although this unconditionedness or this simplicity cannot be understood by us through any acts of understanding which exist in us because these acts (our own acts of understanding) always exist as conditioned acts and no conditioned act of understanding is able to understand an unconditioned act of understanding. Conditioned acts of understanding have have been brought into being through conditions which have been fulfilled or which must be fulfilled if anything is to happen in terms of the possible reception of an act of understanding. If A, then B. But A, therefore B.

Before attempting to understand more deeply why it is not possible for us to move from our contingent acts of self-understanding toward an understanding of God which demonstratively proves the existence of a Divine Word, we can simply note that, for us, inner words need to proceed for us if our understanding is to move toward satisfactory completions of one kind or another. First, without the proceeding of an inner word from a prior, direct act of understanding, we cannot have an idea or a meaning which we can detach from its point of origin (a point of origin which refers to its originating act of understanding). An idea or meaning cannot exist as an other that we can think about, ponder, and question in order then to think about what kind of reality perhaps exists in a given thing or in a given event. No further thinking can occur unless an initial idea or meaning has been transposed into an externalized form through an act of conceptualization that follows in the wake of a prior act of understanding. By this means, a form or inner meaning, through a conceptualizing act of understanding, can now exist as a negotiable point of reference, as a communicable concept, and as a datum of consciousness which differs from a datum of consciousness that is simply present to us in the experience of that which exists as an idea or meaning (the term of a direct act of understanding).2 Through the proceeding of an inner word, we can begin to transcend the initial identity which always exists in our every act of understanding (an identity which exists between an act of understanding and that which is being understood by us in an act of direct understanding). Through an inner word, an idea or meaning can be taken and worked on to see if it can be viewed later as a fact, as a true idea or as a true meaning. Through the forming of an inner word, an idea or meaning can also be re-combined with a material component as this exists in a generalized specification of matter which refers to common matter, and so, by this means, we can apprehend a meaning which defines and points to a thing which we can later judge to see if it truly and really exists.3

Second, specifically with respect to our acts of reflective understanding as this exists in judgment, without the proceeding also of an inner word from a prior act of reflective understanding, we cannot move from an apprehension about a sufficiency in evidence toward an affirmation that can speak about the truth of an idea or meaning.4

Third, without the proceeding of inner words that can accumulate into communicable bodies of knowledge, human learning cannot develop in a way which can lead to growth in the development of a tradition of thought which characterizes how progress can be made in the development of the various natural and human sciences. Without inner words, a universalized human order of meaning cannot begin to arise in a manner which can lead to the creation of a higher order of control for us as human beings as we try to find a way to live with each other in a context that already exists for us within a world that is constituted by physical, chemical, and biological nature. Inner words encourage the asking of new, additional questions and the later, possible receptions of newer acts of understanding which can reveal new ways of thinking about our world and about new possible actions that we can take to change the world that we are currently living within. From an order of meaning that tends to be immune from radical changes in the meaning of things, apprehensions of value can also come and these can also act as a stabilizing influence in the conduct of our lives (despite what trials and difficulties might come our way).

Fourth and lastly, without the mediation of inner words, we would not have a point of departure for moving into analogical forms of inquiry which encourage the reception of analogical acts of understanding. As inner words reveal an inner life which exists within our limited human understanding of things and a dialogue within our understanding which reveals an orientation which is directed toward unrestricted experiences of meaning and truth, inner words begin to function in a different manner. A denotative order of significance yields to a form of attribution which takes a properly understood predicate and which then properly applies it to two or more analogates as when we should say, for instance, that in human beings understanding exists and that in God understanding also exists.5 Understanding applies differently; it has a different meaning when we think about human understanding and then when we think about divine understanding although, as regards both God and ourselves as human beings, understanding exists as a real thing. It is a real property. And so, through this form of signification, inner words can be employed to speak about some meanings which are real although we do not adequately understand what these meanings mean when we work from meanings which are only adequately understood within a context of conditions as these exist within our concrete, contingent human life. Throughout, in the context of Lonergan’s discussion, because the proper object of our human cognition is not to be equated with the final or formal object of human cognition,6 inner words are to be regarded as provisionally necessary for our human cognition if our human knowing is to move from point A to point B in the reach or the progress of human understanding and knowing (if human knowing is to move from limited acts of understanding toward more comprehensive acts of understanding).

In conclusion thus, at this point, it can be said that the completeness of divine understanding (the absence of any discursiveness in the manner of its operation) precludes identifying any reason or consideration which would suggest that, in divine understanding, some kind of inner word must be postulated to exist if divine understanding is to exist and function in its own proper way. In divine understanding, no gap needs to be bridged, no movement or shift needs to occur in moving from a proper object of understanding toward any final object of understanding. Everything already exists within God since, within God, an absolute identity exists between God’s intentional, cognitive being and God’s natural, ontological being.7 The absence of any discursiveness in divine knowing reflects a perfect identity in divine understanding which, in turn, reflects a perfect identity in the nature of God’s being, God’s existence. The simplicity of God’s understanding points to the simplicity of God’s being (God’s being who exists as an unrestricted act of understanding). From God’s unrestricted self-understanding, everything else flows. Everything is understood. God’s being as an unrestricted act of understanding primarily exists as divine self-understanding since, in the life and being of God and unlike how human beings think and understand, nothing exists outside of God which God must first understand before he can then attend to the possibility of having any self-understanding. In the self-understanding which God has of himself and because God exists as the first principle or point of origin of every other thing that exists or can exist, everything which can possibly emerge from God is already fully understood. A knowledge of the latter (the secondary objects of God’s understanding) is given or subsumed within God’s knowledge of the former (God’s understanding of himself).8 In an explanation which can be gleaned from Aristotle’s reflections: in understanding something which is most intelligible, easily and immediately, we understand everything that possesses lesser degrees of intelligibility.9 If, on the one hand, our human self-understanding typically begins with an understanding of external things (external objects) and then, from there, we move more inwardly toward our self-understanding, a converse movement typifies the kind of understanding which refers to God.

Potency: Aristotle, Aquinas, Lonergan

Potencynotes

This paper exists as an excerpt belonging to a much larger work which attends to how acts of sensing, understanding, and judging are to be correlated with the metaphysical principles consisting of potency, form, and act. These reflections are constantly subject to revision as understanding hopefully advances and as better forms of expression come to mind.