Dear friends, we will have our annual Epiphany potluck luncheon on Sunday February 5 at 12:15 pm, immediately after midday office which begins at 12:05 pm. Please feel free to bring friends. The more, the merrier. We give thanks to God for all good things, most especially the gift of friendship. with love to you all…
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
c2 = 2
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
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.
To move toward an initial understanding about what could be meant by a “transposition of meaning,” one can look at what Matthew Lamb says in “Lonergan’s Transpositions of Augustine and Aquinas: Exploratory Suggestions,” The Importance of Insight: Essays in Honour of Michael Vertin, eds. John J. Liptay Jr nd David S. Liptay (Toronto: University of Toronto Press, 2007): 3-21. As Lamb cites Lonergan about what happens in transposition, it is one thing to understand or to come to grips with a theological understanding which had been enjoyed by someone like Augustine or Aquinas and which they had spoken about within an earlier context of meaning. But, after one has truly and properly understood what another has understood and said in the context of their own day, one must find a way to take this same understanding and meaning and bring it into a newer context of understanding and meaning – a new, broader context of understanding and meaning which is the result of later achievements in the history of science and philosophy. Cf. Lonergan, “Horizons and Transpositions,” Philosophical and Theological Papers 1965-1980, p. 410. By means of what is new, one takes hold of the old (the old refers to the valid insights of previous acts of understanding) and, by a second step, one raises these true insights to a greater degree of perfection (working with “new notions” present in “science, scholarship, [and] philosophy.” Cf. Lonergan, Philosophical and Theological Papers 1965-1980, p. 298. Amid shifts of horizon and changes in philosophical and cultural perspective, a legitimate line of development can be traced as one speaks about the meaning of certain truths in a way which adds to the meaning of the truth that is known in previously known truths. In this context, one can speak about new meanings being added to old meanings.
In a certain sense then, one might say that, in the language of Paul Ricoeur, through a “hermeneutics of recovery” which goes back and looks for what is “best in the speech, writing, and action” of persons and groups who had lived and worked at earlier times, one engages in a work of interpretation which is akin to what happens in a “hermeneutics of transposition.” In both cases, one brings meanings into a new context of meaning (into a larger context). Cf. Lance Grigg and Hugo Meynell, “Reflections on the Essence of Critical Thinking,” Divyadaan: Journal of Philosophy & Education 21/3 (2010): 372. However, while a “hermeneutics of recovery” can be used as an agent of cultural transmission (bringing meanings from an earlier cultural context and tradition into a new current cultural context and tradition through a kind of conversation which can create a fusion of horizons between two different points of view), the creation of a fusion across cultural and historical divides needs to be distinguished from a “hermeneutics of transposition” which seeks not only or merely to bring old meanings into a new cultural context in the hope that changes will fruitfully occur within a given cultural matrix.
Yes, initially, the bringing of old meanings into a new cultural context is what happens and, initially, this is all to the good. However, more importantly, at a higher level, what is needed is a representation (a new articulation of truths) which, in some way, are already known. They exist, for instance, within the reception of a given theological tradition or in the reception of a tradition of belief and faith. But, in their day, these truths had been known through a conceptuality that differs from the kind of conceptuality which is currently operative in a changed cultural order. The good then which needs to be achieved through transposition refers to a new, higher synthesis of meaning (a new, higher synthesis which urges the necessity of expanding or growing in the depth and width of one's understanding and judgment). As Pope Leo XIII had urged in his encyclical Aeterni Patris: vetera novis augere et perficere. One best proceeds in one's thinking and understanding of things if one can augment and perfect what is old by means of what is new. Cf. Lamb, “Lonergan’s Transpositions of Augustine and Aquinas: Exploratory Suggestions,” p. 4; Lonergan, Philosophical and Theological Papers 1965-1980, p. 298. To understand what is meant by a “fusion of horizons,” see Gadamer, Truth and Method, pp. 306-307, and other relevant texts in this same work.
In the theology of Antonio Rosmini (d. 1855), one finds an understanding about human cognition where human beings work from an initial, ideal, indeterminate notion or idea of being which underlies and penetrates every kind of human inquiry and which is necessarily presupposed by all our acts of human knowing. This notion of beign is implied in every judgment. Cf. Gerald A. McCool, Catholic Theology in the Nineteenth Century: The Quest for a Unitary Method (New York: Seabury Press, 1977), p. 120. Without it, nothing can happen in human knowing. This idea exists in an “essentially objective” manner as an intellectual object which immediately and perennially illuminates the mind from without without necessarily eliciting or effecting any effect in how one's mind is supposed to respond. Cf. Catholic Encyclopedia, 1913 ed., s.v. “ Rosmini and Rosminianism,” by D. Hickey. This ideal, initial, indeterminate notion of being is self-evidently and intuitively known by us through a form of mental seeing which can never err since, by this seeing, no judgments of any kind are being made (the seeing exists before or prior to any judgment) and errors only exist whenever we make judgments. What is seen in this notion of being is distinct from and is opposed to the mind that sees.
In the self-revelation of being which occurs, as noted, the mind makes no contribution of its own because what exists as “an unconditionally necessary object cannot derive its intelligibility from a contingent mind.” Cf. McCool, p. 120. However, in any later knowing of anything that can be known initially through an act of sense, the human mind works with this ideal notion of being to apply it to a datum of sense, converting this datum into an object of experience. In human knowing, a process of objectification creates divisions between subjects and objects. The existence of a real distinction here between an intellectual object and any act of mental seeing which occurs in intuition (a real distinction between knower and known) accordingly recalls the fact that, for us, a similar real distinction exists between light as it exists in a material, external way and any eye which sees or beholds the light which it externally sees. Only by an abstractive species of thinking can human beings come to realize that an initial, indeterminate notion of being exists innately within one's mind to guide it from within as a species of inner light. Without its already existing within one's mind as the “form of one's understanding” or as the “light of one's intelligence,” no kind of inquiry can occur about any given topic or issue. Cf. McCool, p. 122.
These things being said, however, it is not with point that Rosmini's notion of being closely resembles Heidegger's notion of being (as this has been adapted by him from the notion of being which one finds in the philosophy of Edmund Husserl). Cf. Michael Sharkey, Heidegger, Lonergan, and the Notion of Being, pp. 9-16, unpublished paper, presented at a meeting of the Lonergan Philososphical Society, Baltimore, Maryland, November 6, 2010. In Heidegger's own words, when the being of something is to be determined through inquiry, the being which is to be determined is “in a certain way already understood.” It exists as a “preunderstanding” that is given to one even if it exists as “unoriented and vague preunderstanding.” Cf. Martin Heidegger, History of the Concept of Time: Prolegomena, tr. Theodore Kisiel (Bloomington: Indiana University Press, 1985), pp. 143-144, as cited by Sharkey, pp. 9-11. While Lonergan speaks about an a priori notion of being that is purely heuristic and which is without any kind of conceptual or formal content (“notion of being” versus “concept of being”), in Heidegger's notion of being, various texts here and there refer to a prior understanding of being which is already given and operative in human inquiry and which is not to be confused with determinate anticipations of being which exist either as assumptions, or as prejudices, or as prior understandings in the context of a particular inquiry which is seeking to solve a problem or to move toward some degree of growth in the content of one's understanding. In any given inquiry which we conduct as human beings in the concrete world, an anticipated conceptual content commonly accompanies a genuine search for growth in understanding and truth which, per se, as a search for understanding and truth, is to be identified with Lonergan's heuristic notion of being and the operation of this notion within the dynamic of human inquiry. All human beings ask questions in a context that is partially guided and determined by presumptions and prejudices and by previous acts of understanding which legitimately exist as a prior partial knowledge of being. Heidegger speaks about the presence and the activity of “fore-understanding”: a fore or pre-understanding which refers to the “fore-structure of understanding.” Cf. Gadamer, Truth and Method, pp. 265-255, citing Heidegger, Sein und Zeit [Being and Time], pp. 312ff. And indeed, if we turn to Aristotle, Aquinas, and Lonergan, we find that the same point is made although in different words and within the context of a different conceptualization. No attempt to seek an understanding about anything occurs or proceeds from any prior total lack of understanding. Cf. Aquinas, Sententia super Physicam, 1, 1, 7; Sentencia Libri De anima, 3, 14, 8; De Malo, q. 6, a. 1; Summa Theologiae, 1a, q. 85, a. 3; 1a2ae, q. 97, a. 1; Frederick E. Crowe, “Law and Insight,” Developing the Lonergan Legacy: Historical, Theoretical, and Existential Themes, ed. Michael Vertin (Toronto: University of Toronto Press, 2004), p. 275 & n. 22. Certain things are already understood and known and, about certain things, no questions need to be asked. From a partial understanding of being, one only moves toward greater understanding or one tries to move toward a greater understanding.
However, if acts of prior understanding or if acts of prior misunderstanding are distinguished from an unqualified a priori which simply refers to a prior understanding of being (a prior act of understanding which is to be equated with an a priori understanding of being), then one is dealing with a different kind of hypothesis (i.e., a different kind of claim). Hence, to the degree that Heidegger adheres to a point of view which holds to an a priori notion of being which exists as an a priori understanding of being (an understanding which somehow already exists and which has been intuited in some way and whose meaning is but gradually unpacked and specified through subsequent inquiries that one might be making), it follows that Heidegger's notion of being presupposes or points to a notion or understanding of cognition which thinks in terms of dualism, confrontation, and intuition. Human knowing is grounded in a mysterious, prior confrontation of some kind (a confrontation that exists between a subject and an object). An inquiry into the Husserlian roots of Heidegger's notion of being best indicates how or why Heidegger is able to speak about a prior indeterminate notion or sense of being that is somehow later drawn out or explicated when, in one's later acts of understanding, one moves from an initial experience or understanding of being as a totality into articulations or explications of this totality which distinguish parts or elements within the totality of being and which also indicate how these parts or elements are related to each other in certain ways. Categories are invoked as means that can be used to distinguish parts or elements from each other although in a manner which can indicate how these same parts or elements are, in fact, related to each other. With respect to the possible transitions which can occur as a potential human knower moves from a sense of the whole to a sense of parts or aspects that can related to each other in a certain way, see Sharkey, p. 11, who distinguishes, in Husserl's understanding of human cognition, three different kinds of intuition which appear to exist together (“there from the beginning” in sensible experience) but which yet allow one to move from one kind of intuition to another as one's attentiveness shifts back and forth (from whole to part and then back to whole). “Sensuous intuitions” are distinguished from “synthetic categorial intuitions” and these, in turn, are distinguished from “ideational categorial intuitions.”
Hence, later on, when we move from philosophy to theology, if we can correctly argue that Heidegger's Husserlian notion of being exerts a determining influence within the Trinitarian theology of Karl Rahner (however partial is this influence), we can perhaps conclude that, in Heidegger's notion of being, we can find clues or suggestions which can help explain why Rahner tends to refrain from working with psychological analogies in his proffered systematic theology of the Trinity. If, in some way, knowing tends to be conceived in terms of some kind of intuition or, better still and perhaps more accurately, if knowing is being regarded in a way which has not fully separated itself from an intuitional understanding of cognition (an understanding which reduces all human acts of knowing to a simple single act that is akin to an act of sense), then this sense or understanding about the nature of human cognition cannot be used too easily as a fit analogy for thinking about how one might want to think and speak about processions within God (a plurality of processions which points to three persons while yet also pointing to the truth of God's essential oneness). For a fit analogy, for a better analogy, one must more fully and explicitly enter into the details of a discursive understanding of human cognition (an understanding which knows that human knowing consists of a plurality or assembly of different acts which are all ordered to each other in a way which evidences an internally constitutive, dynamic, inner unity). Cf. Conversations with Michael Sharkey, November 6, 2010; Conversations with Roland Krismer, November 29, 2010.
By way of a conclusion, however, which can possibly indicate how a bridge can be conceived to exist between Lonergan's understanding of being and the understanding of being that is commonly found in the transcendental philosophy of Martin Heidegger, Emerich Coreth, and Karl Rahner, one can take Lonergan's notion of being and ask about its conditions of possibility. Why does it exist or what is its ground? In his cognitive theory, Lonergan identifies a heuristic notion of being which can be found to exist as an operative principle within the dynamics of human cognition. But, if one asks about the grounds of this notion (where this notion comes from or why it exists), one is compelled to give an answer which refers to a metaphysics and the existence of a certain kind or type of being. A being or ontology of things explains why certain things act in the way that they do (why they are the subjects of certain acts and why also they are the recipients of other kinds of acts). Human beings exist in a certain way. They have come to exist in a certain way. From a more thorough understanding about the nature of human cognition, one naturally moves into some kind of ontology or metaphysics. And so, as one engages in this line of inquiry, on this basis, one can speak about Being or the existence of things as a fundamental presupposition. The order of Being enjoys a certain priority (it exists as a legitimate point of departure) although, as we have already noted, a simple prior understanding or knowledge of being (prior to the existence of any kind of inquiry) is to be sharply distinguished from a partial prior understanding and knowledge of being that is always operative to some extent in human acts of inquiry and understanding. Lonergan cannot argue and he does not argue that his notion of being exists as an absolute. It exists rather as a relative. It exists as a conditioned since it is explained by an order of being that is already given and which is always present. In the context of Lonergan's own thought, in the context of his analysis, from a fuller understanding of one's self as a human knower, one properly moves into a metaphysics. One can begin to understand the priority of metaphysics as this relates to the kind of priority which one discovers when one adverts to the existence of a pure desire to know that is found to be operative within the structure of our human cognition.
With respect thus to the existence of two priorities, as one begins to discover why one should speak about a priority which exists with respect to metaphysics, by asking questions about the grounds and the conditions of possibility for the existence of metaphysics, one is soon led to a set of answers which now refer to acts of understanding. If the being of things is intrinsically intelligible, it exists on the basis of some kind of rational ground. But, rational grounds presuppose an existence of reasons and considerations which can only exist within minds (within acts of understanding). In other words, as one understands the priority of metaphysics, one understands the priority of acts of understanding (the priority of cognition vis-a-vis the priority of metaphysics). And so, as we try to think together the tradition of thought that is found in Lonergan (and which is exmplified in his heuristic notion of being) and the tradition of thought that is found in the insights of Heidegger, Coreth, and Rahner (and which refer to other notions of being), it seems that the best solution is an approach which thinks in terms of a reciprocal or mutual priority. The mutual priority or mutual causality which one finds in how Aquinas understands the relation which exists between intellect and will (understanding and willing) serves as a similarly useful device for understanding why, in one sense, one can properly speak about a contrasting heuristic notion of being as this is found in the context of Lonergan's thought and why, in another sense, one cannot speak about a contrast if it is conceived to exist as an absolute. Within the tradition of German transcendental thought, differences exist among different thinkers and sometimes one wonders if these differences are explained more by the use of different starting points than by deficient understandings that are had about the nature of human understanding. If Lonergan's heuristic notion of being is more adequately understood as a relative, if the conditions of its existence can be more adequately understood, a context can be created that could better mediate the insights of Lonergan's thought into the corpus of transcendental thought as this exists within the German speaking world. From a transformation that can occur from within the context of traditional transcendental philosophy, more good can be effected. More good can be expected.
Br. Dunstan Robidoux, OSB
In speaking about human cognitive acts and especially about human acts of understanding, instead of speaking about laws of nature and the intelligibility that these laws have, Lonergan prefers to speak about another kind of meaning or, in other words, about another kind of law. A real distinction should be drawn between laws of nature which specify what a given thing can do and what is cannot do and what Lonergan refers to as “transcendental laws”: laws which account for a species of freedom which belongs to human acts of understanding but which does not belong to other created human acts. Cf. Lonergan, The Triune God: Systematics, p. 175. Human reasoning and understanding is not only intelligible. It is also intelligent. It functions as a source of intelligibility not only by discovering laws which already exist but also by functioning, to some extent, as a source or as an originator of law. It brings laws into being through operations that are eminently rational.
This transcendental desire functions as an inner first principle (an inner law) which governs all subsequent cognitive operations. Initially, this transcendental desire emerges in a completely spontaneous way in the lives of human beings. However, as this same desire moves to help create conditions which can move a potential knower toward later possible receptions of understanding, the initial experience of spontaneity which is given in this transcendental desire is supplanted by a second kind of experience which refers to experiences of rationality that are not properly understood if they are simply viewed as spontaneous.
In this context thus, in the context of rationality or, more properly, within experiences of intellectual or rational consciousness, a different kind of freedom is thus experienced by us as human beings. Instead of a freedom which refers to acts which exist because a given thing possesses a certain kind of nature or inner principle of intelligibility (a freedom which is limited because it is determined by a nature or inner principle of intelligibility which a given thing has), a second kind of freedom is experienced which refers to an order of self-constitution which exists within rational human activity as one kind of intellectual act leads to or emerges from another species of intellectual act. A sovereign kind of freedom makes its presence felt when, as human beings, we engage in a creativity which exists within ourselves and which is endemic in our own acts of understanding: a creativity that transcends all other known categories and laws to construct new categories and determine new laws. But, how is this order of self-constitution to be explained? How is its reasonableness to be understood? The transcendental freedom which exists within human understanding is not to be understood as a source of chaos or as a begetter of disorder.
Turning now to what Aquinas has to say in the Summa Theologiae, 1a2ae, q. 71, a. 6, ad 4, Aquinas refers to natural law as a species of law which derives from a higher law which is to be identified as God’s eternal law but which also exists, in a secondary way, in laws which are not eternal but which have somehow been created by us in our acts of understanding and judgment. These laws exist in the “natural judgment of human reason” (in naturali iudicatorio rationis humanae). Natural laws exist implicitly within the structure of our human reasoning and so they can be found there if one tries to understand the form or the intelligibility of our human reason. One understands the normativity or the lawfulness of human understanding in its operations and effects if one accordingly attends to the inclinations and ordinations that are most proper to human living and which distinguish human life from the existence and life of other beings. Cf. Summa Theologiae, q. 91, a. 6. Laws exist within created or subordinate things to the degree that these same things are naturally or normally inclined to abide by a higher set of laws which account for the existence and the life of these lesser things. The proper inclinations or ordinations of things reveal not only higher laws to which these things are subject but, most importantly for us, they also reveal these same laws as they also exist within these things as constitutive principles. The participation of a thing in a higher reality which functions as a source or point of origin for law turns the participant (particeps) into an analogous or secondary source of law. And so, as one attends to those special inclinations which distinguish human beings from any other kind of things, one discovers laws which exist within those inclinations: laws that are proper to the created human condition. One discovers laws which are constitutive of human cognition and, at the same time, within these same laws, one discovers laws which are constitutive of divine understanding and knowing.
In knowing about these laws (as partial as one’s understanding may be), one begins to understand a bit more about the difference which should exist between intelligibility, on the one hand, and intelligence, on the other hand. As a given, as something which had been received by something else, intelligibility presents itself as a passive determination. The natures of things exist as intelligibilities. But, with respect to intelligence, as we think about it, we find that intelligence is essentially active. It is determinative. It functions as a determinating principle. It is a cause of intelligibility in other things in a way which also means that it is also a cause of intelligibility in itself through the operations which it performs. As intelligent operations construct or form a pattern of acts among themselves, an inherent intelligibility is revealed within these acts. An intelligibility is revealed which we can possibly also come to know. And so, for these reasons, it can be properly said about intelligence that intelligence forms or constitutes itself according to laws that are best referred to as transcendental laws which are to be distinguished from the intelligibility of a nature which functions as a principle of limitation. Intelligence in act determines what it does at any given time through decisions which are made because rationally it functions or operates on the basis of an inner principle which governs all of its subsequent operations: an inner principle which Lonergan refers to as a “transcendental desire”: a transcendental desire which can be used as a point of reference for the existence of transcendental laws that are to be identified with the generative laws of human cognition and understanding. Cf. Lonergan, The Triune God: Systematics, p. 175. A cognitional factor helps to explain why human understanding is characterized by a form of transcendence which is best referred to in terms which speak about self-transcendence. But, for a fuller explanation which best shifts into a metaphysical understanding of things that allows one to speak about a form of indwelling which exists within created human acts of understanding. Through the principle of rationality or reasonableness, one can say that higher laws exist within created acts of human understanding: higher laws which serve as a point of departure for moving toward new achievements of human meaning (achievements that are not restricted to current human achievements as these may exist in human life, society, and culture).
Recall a point that Aquinas makes in the Summa Theologiae, 1a2ae, q. 66, a. 5, ad 4. The wise man judges between which first principles should be proved and what should not be proved in any given discipline or science and the wise man also judges which principles should be used as a basis from which to construct an ordering of variables in any given science or discipline.