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.

Higher Viewpoints: Part One, from arithmetic to algebra, the transition

This is a repost of a 1997 essay written for a seminar on Insight.

Higher Viewpoints: Part One

From Arithmetic to Algebra: the transition

by David Fleischacker
Draft Version 1

Copyright © 1997. All rights reserved

(This is a reflective commentary on one facet of sections 1 – 3 of chapter one in Insight.)

 

I. The Viewpoint of Arithmetic:

In Insight, Lonergan builds to the notion of a higher viewpoint after he has developed an understanding of clues, insight, concepts, questions, images, and definitions. A viewpoint is not merely a definition, but a set of systematically related definitions (and of the operations that underpin both the definitions and their systematic relations). It is not a single definition. Defining a circle, for example, is not a viewpoint, but it does arise out of a geometrical viewpoint, and contributes to it. The same is true of the distributive or commutative properties of algebra, or the power rule of calculus. They do not constitute an entire viewpoint, but they are components.

Lonergan illustrates lower and higher viewpoints with arithmetic and algebra. A mathematical viewpoint is constituted by rules, operations, and symbols (or numbers). The rules implicitly define the operations, and the operations implicitly define the symbols. What does he mean by this?

 

A. The Deductive Expansion of Arithmetic (the first horizontal development in mathematics):

Lonergan begins with arithmetic, more specifically with addition. One may count sheep or goats or troops in an army or persons inhabiting a town. The counting involves the operation of addition– one plus one plus one, and so forth. It is an activity relating quantities and defining them in terms of each other. The basic unit of this quantity can be symbolized, let us say with a “1” or “I”. Other symbols can be used to represent what one is doing when adding, such as “+” or “plus.”

Any number of symbols can be invented to represent operations (addition, subtraction, etc..) and numbers (some of which, Lonergan notes, are better conducive to the future development of mathematics than others because of their potential for leading to further insights). In order to simplify the ongoing definitions of numbers most cultures that developed mathematics introduced repeating schemes. Some introduced repetitions based on 30 or 60 (think of our clocks and watches). Our present system is based on repetitions of 10, so we developed a symbol for zero through nine, and then, once ten is reached, we add a place to the left indicating the number of “tens.” Then once the tens reaches beyond the ninth position, we add the hundreds, then thousands, and so on (Computers, you may have heard, are based on a binary, with ones and zeros).

From adding numbers we can develop, as Lonergan notes, a definition of the positive integers.

So,

1 + 1 = 2
2 + 1 = 3
3 + 1 = 4
etc., etc., etc..

Once the insight is gained, or in other words, when one understands what is meant by “etc., etc., etc.” then one can continue to indefinitely define any positive number. From this, one can create an entire deductive expansion of a viewpoint or horizon in arithmetic, and continue indefinitely to define the whole range of positive integers. One can also construct mathematical tables using 2s, 3s, 4s, etc..

2 + 2 = 4
4 + 2 = 6
6 + 2 = 8
Etc., etc., etc.. (“2” is added in a repeating fashion)

3 + 3 = 6
6 + 3 = 9
9 + 3 = 12
Etc., etc., etc.. (“3” is added in a repeating fashion)

4 + 4 = 8
8 + 4 = 12
12 + 4 = 16
Etc., etc., etc.. (“4” is added in a repeating fashion)

 

Then,

ETC., ETC., ETC. (For the entire process above)

Notice how all of the numbers are defined in terms of the operation of addition. It is the basic insight that grounds this operation which allows for the construction of an entire deductive expansion which creates a “world” or viewpoint, even if rather limited. It is a first, horizontal development of a horizon in mathematics.

 

B. The Homogeneous Expansion (the second horizontal development in mathematics)

One can become more creative, and begin to add a number to itself say three or four or five times.

So,

1 + 1 + 1 = 3
1 + 1 + 1 + 1 = 4
2 + 2 + 2 = 6
3 + 3 + 3 + 3 = 12
Etc., etc., etc..

 

And, instead of writing this with three 1’s or four 1’s or three 2’s or four 3’s, shorthand notation can be developed.

1 x 3 = 3
1 x 4 = 4
2 x 3 = 6
3 x 4 = 12

So, one introduces a different symbol, namely an “x” to indicate the number of times one is added to itself. Notice how this new symbol is still defined in terms of the old operation of addition. It means “adding” a number to itself so many times.

Creativity does not have to stop. If we can add one to another, then what happens if we take something away. We had three sheep, sold one, and now are left with two. This is the opposite of addition, and we can give it the symbolization of “-“(1) and call it subtraction. Again, one can develop charts of subtraction, just as with addition and multiplication. Likewise, just as one can reverse addition by removing something, so one can reverse multiplication by removing a number so many number of times. So, just as one can add 2 to itself four times to get 8, so one can remove 2 from 8 four times. This of course, is division.

Creatively constructing a viewpoint by introducing new symbols such as subtraction, multiplication and division, is what Lonergan calls a homogenous expansion. One has introduced new symbols relating numbers, but notice how everything is still thought of in terms of addition. Subtraction is the reverse of addition. Multiplication is the addition of a number to itself so many number of times. Division is the reverse of that. And if one wishes to add powers and roots, they likewise can be defined in terms of addition. A power is the multiplication of number by itself so many number of times.  Thus to define this in terms of addition, let the number that is powered equal y, and the power equal z.    Thus, the answer is defined as the number, y,  which has been added to itself y number of times, thus forming a group, a group which in turn is added to itself again y number of times, and repeating this formation of groups  z-1 number of times. This can be illustrated with 3 to the power of 4.  3 to the power of 4 is the same as 3 x 3 x 3 x 3.  The first group arises by converting 3 x 3 into 3 + 3 + 3. This group is then added to itself 3 times in order to get the equivalent of 3 x 3 x 3.  This results in a second group that can be written as [(3 + 3 + 3) + (3 + 3 + 3) + (3 + 3 + 3)].  Finally, take this second group and add it three times to itself.  The final answer comprises this third group, which can be written as

[(3+3+3) + (3+3+3) + (3+3+3)] + [(3+3+3) + (3+3+3) + (3+3+3)] + [(3+3+3) + (3+3+3) + (3+3+3)] = 81

A “root” is the reverse of this procedure. So, the 4th root of 81 requires breaking down the 81 into three groups, where the basic group, which when discussing powers was called the first group, is comprised of a number that has been added to itself its own number of times.  This number is the answer. So, even powers and roots can be thought of in terms of addition.

II. Algebra: The Higher Viewpoint ( a vertical expansion in mathematics)

The homogeneous expansion of arithmetic has not introduced any new rules. One can define each of the new operations in terms of addition (or the reverse of addition). New rules are only introduced when one starts “observing” patterns in arithmetic, and doing this initiates algebra (Lonergan notes that the image which leads to algebra is the doing of arithmetic). What does this mean? Lonergan notes that this “turn of question” that lead to the discovery of patterns in arithmetic occurred because of questions such as;

What happens when one subtracts more than one had?
Or what happens when division leads to fractions?
Or roots to surds?

Each of these refers to various problems that emerge in the homogenous expansion. Their answer lies in grasping patterns. Questions emerge which ask, what, in general, happens when one subtracts numbers, or adds numbers, or divides numbers, or adds powered numbers, etc.? Today, these initial patterns are given such names as commutative, distributive, and associative properties. Let us illustrate these laws.

A simple pattern is adding a number to zero.

1 + 0 = 1
2 + 0 = 2
3 + 0 = 3
4 + 0 = 4
5 + 0 = 5
6 + 0 = 6
etc., etc., etc..

The etc., etc., etc., again is introduced to gain the insight. A number added to zero results in an answer that is that number. This can be symbolized by creating a symbol that represents a number (or in other words, a variable).  Let that symbol be “A.” Below is the formulation of this pattern,

A + 0 = A

Another example is the multiplication of a number by 1.

1 x 1 = 1
2 x 1 = 2
3 x 1 = 3
4 x 1 = 4
5 x 1 = 5
etc., etc., etc..

If one recognizes the pattern, then one notices that a number multiplied by one, gives the number. Hence, this insight can be symbolized.

A x 1 = A

The same is true with the various laws or properties (as they are actually called) mentioned earlier. The commutative property of addition states that A + B = B + A. The commutative property of multiplication states that A x B = B x A. The associative property states that (A + B) + C = A + (B + C). The parenthesis means add these numbers first. The associative property of multiplication states that (A x B) x C = A x (B x C). As an exercise right now, try expressing these patterns using actual numbers and the “etc., etc., etc.” as I did above.

You can practice this further by opening any algebraic text, and examining the numerous rules about addition, multiplication, division, powers, roots, addition of powers, multiplication of powers, multiplication of roots, and the inverses of each of these formulas (subtraction of powers and roots, division of powers and roots, etc..)

Notice how one is understanding the operations in a manner beyond that of addition. One begins to grasp, for example, that multiplying two negatives leads to a positive, that dividing a negative into a negative also leads to a positive and many other characteristics. These recognized patterns then begin to form new rules, which constitute the higher viewpoint called algebra. These rules guide one in solving problems, since they implicitly define how one is to carry out operations and define the new symbols of A’s, B’s, and C’s, which represent variable numbers (this will be discussed with more precision and examples in a later commentary). The rules constitute a vertical expansion of the mathematical horizon.

Like arithmetic, algebra also has a deductive and homogeneous expansion, or, at least something analogous. This is for a later section.

 

David Fleischacker

Copyright © 1997. All rights reserved

1. For a history of mathematics that discusses these symbols, see a book that is frequently recommended in Lonergan circles, Carl Boyer, A History of Mathematics (New York: John Wiley & Sons, Inc., 1991).

Part 7: Finality, Final Cause, and the Good in “Finality, Love, and Marriage”

by David Fleischacker

“For the final cause is the cuius gratia, and its specific or formal constituent is the good as cause.” (Finality, Love, Marriage, 19)

This quote” falls within the section on vertical finality in “Finality, Love, and Marriage.” In the paragraph before this quote, Lonergan was introducing the difficulties in apprehending finality. It is something that can be easily overlooked by the positivist because “quite coherently, any positivist will deny final causality.”  Instead, the positivist will only admit efficient causality.  Lonergan in short, defines finality in terms of final causality.  One wonders if he had yet broken from Aristotelian science sufficiently to provide an adequate account, even heuristically, of marriage and the marital acts.  I suspect many would say no.

To restate this difficulty, if vertical finality is defined in terms of final cause, it would seem that Lonergan has not adequately defined finality yet.  There is some truth in this as I stated in the last blog — since it seems that in light of Lonergan’s formulation of horizontal finality, he had not yet reached his more general formulation of finality that one finds in Insight, where it is isomorphic with the notion of being, and not related to essence alone (though it is related since essence or form is a component of being).  Just as one can expand on the notion of horizontal finality, so one can do the same with vertical. Let us push this a bit in this blog to see where it goes.

As the quote up top indicates, Lonergan is defining final cause in terms of the good.  In this section on vertical finality where he criticizes the positivist, Lonergan notes that the positivist can acknowledge  motives and terms, but only as efficient causes. The blindspot of the positivist is the denial of these motives as good and these terms as good.  In short, Lonergan is saying that in the potency, there is an orientation to the good that he calls finality.

One can find this orientation to the good throughout Lonergan’s later writings. In fact, it becomes more prominent, not less, as he formulates in a clear fashion the fourth level of consciousness and delineates the capacity for self-transcendence not in terms of one notion (being) as he does in Insight, but in terms of three transcendental notions — intelligibility, being, and the good (Method in Theology, 34 – 35 or 104- 105). As well, one can think of his formulation of the human good in chapter 2 of Method in Theology, especially the notion of the “terminal good” (Method in Theology, 51). In both cases, whether one thinks of the transcendental notion of the good/value or in terms of terminal value, these operate in the same manner as Lonergan’s formulation of a final cause in 1943.  In other words, the transcendental notion of value operates like the notion of being, and hence it is isomorphic with the good. And terminal value is the good as a term that is truly good.

Though their is a similar heuristic element to final cause in 1943 and Lonergan’s formulation of the notion of value later, there is an expansion.  Just as an expansion occurs in relating finality (whether horizontal or vertical) to the notion of being in Insight, so now one can isomorphically relate finality to the entire capacity for self-transcendence, which is constituted by the three transcendental notions–intelligibility, being, and the good.  To do this is not to say that what Lonergan defined as finality in 1943 is wrong, but rather it is to open up its meaning to the entire nature of the universe of intelligible and existing goodness.

Think about how Lonergan’s development of the capacity for self-transcendence actually points out a limitation in Insight.  Lonergan would have formulated the good in Insight in a manner similar to Aquinas, as convertible with being.  This point would be true later as well, but it receives some nuances.  The good  as a distinct transcendental notion in later writings, hence distinct in the human subject’s apprehension of the good, especially the hierarchy of the good/value, indicates the differential of something as existing (or some occurrence of a conjugate form as occurring) versus something as good.  In 1943, Lonergan introduced this goodness to being in terms of a final cause.  In other words, being and the good are more explanatorily developed in later writings but still operative in earlier writings. Final cause is not eliminated so much as explanatorily developed.  The manner that he used it in 1943 is still valid within its frame work.

Why was it and is it still valid? This validity is similar to how the Newtonian formulation of gravitation is valid within general relativity, but it is a more limited account. One can transpose the 1943 Lonergan into 1983 by formulating finality as the metaphysical and meta-ethical isomorphism with the capacity for self-transcendence.  This would further open the heuristic exploration of marriage and love that he formulated in 1943, and place his insights within a larger framework.  Already in the last blog, I have started to do this by uniting horizontal and vertical finality in terms of potency as one finds in Insight.   One can do more by relating finality to emergent probability as the emergent good.  The upwardly directed dynamism of finality for intelligibility, being, and the good/value (I have been using good partially because I do get tired of the relativistic overtones of the term “value” in modern culture).  Such a finality would apprehend the universe in its proportionate existence as an emerging good.  This recognizes the universe as an ultimate friendly universe in its very nature.  This also means that the entire intelligibility and being of marriage is not only real but good — and so getting that meaning right is crucial if the historical and traditional breakthrough into marriage is not to fade into the shadowland of scotosis or individual bias or group bias or the general bias (on the notion of scotosis and bias, see Insight, chapters 6 and 7).

Part 6: Horizontal and Vertical Finality in Marriage, Love, and Finality

By David Fleischacker

Four ideas about the generic relationship of horizontal and vertical finality stand out in Lonergan’s 1943 essay – “Finality, Love, and Marriage.” It is important to note that I have not passed much beyond exploring the first section of the 1943 essay, which makes general statements about finality.  In his later sections, he treats of love and the personalist elements of marriage within the framework of finality, and so these later elements will be crucial to comprehend what he contributed with this essay.  That will be for future blogs.

 

First Idea: Horizontal as Essential, Vertical as Excellent

The first deals with linking horizontal to what is essential, and then vertical to what is excellent.  This is mentioned a number of times (see for instance pages 18, 22-23, and 48).  Essence refers to what something is (Lonergan uses nearly the same formulations of horizontal, vertical, and absolute finality in his 1976 essay, Mission and Spirit, but instead of essence he just writes that the proportionate end “results from what a thing is” — A Third Collection, page 24).  Essence constitutes a kind of limit to the types of activities or operations that a thing can engage upon or into which it can develop.  The excellent refers to a higher level perfection that can emerge from the lower.  Lonergan’s use of essential and excellent is directly linked to the course he was teaching on marriage at the time,  and to Casti Cannubii, in which the essential and excellent ends of marriage were distinguished.  I find it interesting that he defines horizontal in terms of essence, but he does not define vertical in terms of excellence.  Rather, he defines vertical finality in terms of a dynamic emergence of properties that arise from a conjoined plurality.

This distinction between horizontal and vertical finality seems to be lost by the time of Insight where the terms are not used at all. I would argue that this is a result of a broader, more general formulation of finality in which it is understood as proportionate or isomorphic with the notion of being (and hence the desire to know).  Finality is the “upwardly but indeterminately directed dynamism towards ever fuller realization of being” (Insight, chapter 15, section 6). As such, it is simply the potency of the universe for the emergence and maintenance of each and all intelligible being.  However, Lonergan does use the term “vertical” in a similar fashion to its use in 1943, though only once, when discussing the relations of a lower manifold to a higher order (Insight, chapter 15, section 7.3).  Hence he is not speaking of finality but of developmental relations. And instead of horizontal, he speaks of lateral developments.  In Insight, the only kind of distinction he makes regarding finality is in terms of minor and major flexibility (hence not horizontal and vertical), but explanatorily these are not different, especially in terms of the meaning of finality as a potency that is a directed dynamism (Insight, chapter 15, section 5). In Insight, finality as such is not merely within an individual, or species, or genus, but it is the potency for fuller being in each and all individuals, species, and genii.

There is one key notion that links the 1943 essay and Insight.  It is the notion of a concrete plurality which becomes forumlated into terms of both the non-systematic and statistic residues in Insight.  The potency of a plurality of acts ends up being central to understanding the open ended dynamism of each individual, species, and genus. Take for instance a carbon atom.  The carbon atom itself is a chemical conjugate, and if in act, it is an existing chemical atom.  Within an existing atom of carbon, the concrete plurality that is a potency for other chemical forms is the sub-atomic elements which are either quarks or compounds of quarks.  These quarks and compounds of quarks are the lower manifold pluralities that have the potentiality for being informed as other chemical elements or compounds.   In carbon, there are sufficient materials to form other elements or compounds through atomic or nuclear changes–as long as the total masses of these elements and compounds does not exceed that of carbon.  One could theoretically form hydrogen or helium, or any other elements up to carbon from a carbon atom. Hence this concrete plurality of sub-atomic elements is the location of the potency for dynamic change.

In 1943, Lonergan tended to limit this dynamic notion of finality to vertical finality, since such finality is based in the “fertility of a concrete plurality” (and this is equal to an indeterminate but directed dynamism to his use of vertical finality).  However, one finds this concrete plurality to some degree in his use of horizontal as well.  There are a few points in his 1943 essay where Lonergan identifies a statistical relationship between two events on the same horizontal level (namely the conjugal act and conception—see page 46, footnote 73), but he tends to identify horizontal as rooted in essence still, rather than the potency of a non-systematically related set of aggregates that can become “conjoined” into an order whether on the same plane of being or a higher plane.

As mentioned, later in his life, Lonergan does reintroduce vertical and horizontal finality in his third collection that reprints a 1976 paper titled “Mission and Spirit.” He more or less repeats the same definitions as given in 1943 but without the link of horizontal to essence/natural law and vertical to concrete plurality and statistical law. Hence has a similar meaning as in 1943, but he has the developments of Insight in the background, along with the question of evolution.  It is almost if he had re-read the 1943 piece, and decided to bring the notions of horizontal, vertical, and absolute finality to attention.  Just a few years earlier, in Method in Theology published in 1972, he introduces horizontal and vertical liberty (not finality) taken from Joseph de Finance (Method, pages 40, 122, 237-8, 269) but there is no clear indications of any connections to finality and the 1943 essay (one can make connections however).

Second Idea: Horizontal within the field of natural law and vertical within the field of statistical law

Horizontal finality  results  from  abstract  essence;  it holds  even when  the  object  is in isolation;  it is to a motive  or  term  that  is proportionate to essence.  But vertical  finality is in the concrete; in point  of fact it is not from  the isolated  instance but from  the conjoined plurality; and it is in the  field  not of natural but  of statistical  law, not  of the  abstract per se but  of  the  concrete  per accidens. (22)

Tied to linking horizontal to “abstract essence” is the idea that it is in the field of natural law (I am presuming this is what he is implying above, but I could be wrong) rather than in the field of statistical law. The notion of natural law as well as the location of horizontal finality are modified by the time of Insight.  “Nature” in Insight is formulated in terms of a heuristic notion that is like naming an unknown “X” that needs to be understood (Insight, chapter 2, section 2.2).  In this context, nature, and one could argue natural laws, and statistical fields are not distinct, but rather closely linked. Nature as transposed into correlations identify the conjugate forms, and statistics deals with ideal frequencies of the actuation of those forms.  Hence, in Insight, Lonergan differentiates nature (and natural laws) into correlations (or functional relationships) and their statistical ideal realizations.  I would argue as well that developmental operators also belong to the realm of “nature” for Lonergan.  Hence, in Insight, Lonergan will shift finality not only to potency, but to a potency that is an indeterminate but directed dynamism to fuller being. Lonergan was moving in this direction with vertical finality in 1943, but had not worked it out in terms of horizontal yet.

 

Third Idea: Horizontal is not dynamic, the Vertical is the source of dynamism.

The claim that horizontal finality is not dynamic on the one hand and that vertical is dynamic on the other is closely related to the above two ideas.   Because Lonergan was conceiving of horizontal finality in terms of essence and a type of static natural law, he had not thought through the dynamism for fuller being that actually takes place due to horizontal finality.  Since in Insight, he works out his notion of finality in terms of potency and then how this notion fits in with his general theory of development (Insight, chapter 15, sections 5 – 6), even what he is getting at with horizontal finality will turn out to be dynamic as well, because their can be fuller realization of being on the same genus of conjugates.  Think for example about the illustration of arithmetic development in chapter 1.  Arithmetic is presented the first of three levels in math, and there are both deductive and homogeneous expansions at this level.  These expansions are developmental in nature, and they arise in the potency of operating with numbers from the moment one “combines” or adds numbers.  Or look at Lonergan’s formulation of minor and major flexibilities of development. Both illustrate these same points since both “rest on an initial manifold” and thus are rooted in a kind of potentiality that Lonergan would have a called a plurality in 1943 (Insight, chapter 15, section 6).  Minor flexibility refers to something that can have some variation while it unfolds into its mature state, but it still reaches the same mature state.  In major flexibility, a thing can unfold in a new and surprising manner which results in a shift in its mature state (Insight, chapter 15, section 6).  This shift could be a different species but on the same genus, hence horizontal, or it could be to a new genus, hence vertical. An example would be if a grass became a shrub (I am presuming these are different species, and the complexity of the change from one into another may be extremely difficulty or unlikely—I do not want to enter into the explanatory challenges to this within the realm of biochemistry, genetics, and molecular biology, though I acknowledge the challenge) (If you would like to see a bit into my understanding of explanation within biology, take a look at my blog on Behe’s book Darwin’s Black Box).  In this case, the relationship of the initial potency to this new species is a horizontal finality.  However, the major shift could be a shift from a lower to a higher genus, in which case then a vertical development has taken place, and the relationship of the initial potency to this vertical realization is one of vertical finality. Such an example would be the shift from a vegetative form of life into something that is sensate.  This is perhaps best illustrated by the development of an animal (dog, cat, etc.) from a single zygote that starts with cellular operations (or what was traditional called the vegetative level–a lower genus) and then adds sensitive operations (a higher genus).

In 1943, Lonergan tended to see horizontal finality in a static manner because of his formulation of it in terms of a kind of an essence (interestingly, he defines it similarly in his 1976 essay, so one wonders a) whether I am right in thinking that he had a more static notion of essence in 1943, or b) whether Lonergan had thought through the notion in light of Insight when he reintroduced it in 1976).  However, his illustrations show that there are dynamic elements to this as well.  I already mentioned the statistical link between the conjugal act and conception.  Another example is the link between fecundity, the conjugal act, and the adult offspring (41).  Adult offspring require development of course.  And Lonergan is thinking of adult offspring in terms of the matured and differentiated organic operations, not in terms of how these are then sublated into educated adult offspring, or religiously educated adult offspring.  And so, when he uses the phrase “adult offspring” alone, he identifies it as having a horizontal relationship to fecundity and the conjugal act (which is the actualization of fecundity in the union of two semi-fecundities).  Yet, there is clearly a developmental relationship, similar to what he later calls a homogeneous expansion or development.  Perhaps a more significant example is when he talks about the two levels above the organic, namely the life of reason and the life of grace.  With the life of reason, he talks about how the potency for a life of reason in both the man and the woman at this civil level is horizontally related to the historically unfolding good life (42).  This clearly recognizes a developmental unfolding which is not fully determinate, yet  dynamically directed by the desire to be intelligent and reasonable.

Fourth Idea: The vertical emerges all the more strongly as the horizontal is realized the more fully

If, then,  reason  incorporates sex as sex  is in itself,  It will incorporate it as subordinate to its horizontal end , and  so marriage will be an incorporation of the  horizontal finality of sex  much more  than of sex itself;  nor is this to forget vertical finality, for vertical and  horizontal finalities  are  not alternatives, but  the vertical emerges all the more strongly as the horizontal is realized  the more fully. (46)

Notice here that there is a kind of dynamic element implied in the horizontal.  Again, if we grasp that horizontal finality is a potency for dynamic realization of being (development), and it is distinct from mere flexible ranges of operations that are already in place, then such development is from an initial potency, and it is the dynamism in that potency that gives rise to it.  And this dynamism can be more fully or less fully realized.  If one turns to Lonergan’s arithmetic illustration in Insight, one can see this point right away.  Arithmetic provides the “image” for algebra.  The more arithmetic one does, the more one will apprehend general intelligible patterns in arithmetic. These patterns are algebraic. Adding A to B results in the same answer as adding B to A.  This results in the rule that states A + B = B + A, an algebraic rule.  And this is just one rule. If one has engaged in a number of operations over and over again not only with addition, but with subtraction, multiplication, division, powers, and roots, then one will begin to grasp all kinds of other patterns.  One will see that multiplying A to B results in the same answer as B to A.  In contrast, one will notice that one cannot do the same for subtraction or division (A-B does not equal B-A, nor does A divide by B equal B divided by A — unless A and B are the same).  Algebra emerges all the more strongly as arithmetic is realized the more fully.  The same is true in chemistry and biology.  The more that one carries out rightly designed experiments on matter, the more fully insights arise into patterns of elements and compounds.  And the more these are unfolded within living organisms, the more one understands higher order organic properties.  Through DNA and biochemical processes, one grasps more fully the organic operations of the cell. And as one unpacks these cellular operations within multicellular organisms, the more one grasps the operations of those multicellular organisms (eg. respiration, immunity, digestion, etc.). These cognitive expansions horizontally and vertically have ontological parallels within all developmental entities.

It is important to note that the vertical cannot emerge and be sustained without the proper operations of the horizontal.  If you eliminate the realization of the finality of stem cells which maintain and perfect cellular systems, then the systems will cease to function (respiration, immunity, etc).  And if these organic events and schemes cease, then sensitive operations and schemes will cease, along with the potential for deductive and homogeneous expansions of sensitive operations.  And if sensitive operations and their development fail, then there will be no insight into images, or judgment based on insights and evidence, or decisions based on judgments of fact and value, or the development of viewpoints.  The lower has to flourish for every higher order to flourish that is dependent upon the lower.

In terms of finality, this means that the more vibrantly that the horizontal finality is realized, the more fully the vertical can be realized.   Both the horizontal and vertical are rooted in potency, and the fact that the same potency is for both, and that the realization of higher is dependent on the lower, means that the two finalities are always necessarily dependent upon each other.  The ability to see is rooted in the neural networks that are tied to sight.  This potency to see is realized horizontally when the eyes are opened, the optical neurons are activated by light waves, and the associate and sensory cortices are integrating the neural activities initiated by the light waves.  It is realized vertically when these lower activities arise into a conscious sensory percept.  A similar relationship accrues to the development of sensory operations in relationship to the development of neural patterns, and these developments are actualizations of the finality, or potentiality, of each of these levels.

And interestingly, the higher can come to assist in the greater realization of the lower.  Problems, needs, and wants that arise with regard to higher level operations require a type of expansion–sometimes a shift, and even potentially a conversion–of the lower orders such that these can then bear fruit for the higher.  Lonergan in his later life sometimes called this the top down element, or the gift element, that allowed for the flourishing the lower.  This does not mean that the higher is free or independent of the lower.  Rather the fulfillment or realization of the higher still depends upon the lower, even if the higher is self-assembling, and acts as a mediator of the perfection of that lower. So to continue the illustration of an animal, associations of the percepts can take place through a kind of willful use of the sensory organs (cortices, etc.).  An animal can “pay attention” to this or that, it can shift its body or head to see or hear or taste or touch or smell something.  This higher order guidance and activation of the lower level neural manifolds allows for the further unfolding of those lower manifolds so that they can contribute to the construction of associative memories, imaginative constructs, and even feelings.

This point about the dependence of the vertical upon the horizontal–the dependence of the dynamic unfolding of the higher upon the lower–is expressed in a mulitude of ways in Lonergan’s later writings. In Insight, examine chapter 8 on things, or chapter 15 on explanatory genera and species.  In Method in Theology, see his formulation of the levels of consciousness and the levels of the functional specialities.  In 1943, the fecundity that is actualized through a union of two semi-fecundities has a horizontal relationship to adult offspring.  The more fully this fecundity is realized at the horizontal level, the more fully it provides for a realization at the higher vertical levels and ends (good life and eternal life).  One could differentiate these ends and levels in light of Lonergan’s later writings both within the parents (the four/five levels of consciousness) and the levels of the child.  One could as well, place these within the unfolding of all levels of being from quarks to the actuation of the capacity for self-transcendence in a state of being in love with God.

I intend on saying more about these higher levels in later blogs. At this point, I wanted to just comment on a basic metaphysical principle regarding the relationship of the lower to a higher level of being, whether that being is conscious or not.  If one eliminates the finality at a lower level, one destroys the possibility of the emergence of the higher.  And the more that the lower flowers, the more that the higher can flourish.

Grasping this finality within marriage sets up a heuristic that allows one to explore marriage in a differentiated and integrated way.  The differentiation is over the different generic levels of reality as sublated within human historical process.  As integrated, these provide the mode of inquiry into the relations of higher and lower orders of intelligibility as well as the potency for new types of operations and new levels of conscious life.   In this 1943 essay, Lonergan  introduced this heuristic first so that he could then suggest specific ways to explore the meaning of marriage.

Book Review: Michael Behe’s Darwin’s Black Box

by David Fleischacker

I have been enjoying Michael Behe’s book Darwin’s Black Box (first published in 1996, with an update in 2006).  It brings out a significant challenge in thinking through evolution and so it is worth reading. However, I do not agree with his ultimate conclusion or even his explicit criterion for validating his argument.  The central point of his argument is that once one turns to the molecular and biochemical understanding of organisms, one finds systems with an irreducible complexity that could not result from the gradual steps of evolutionary development. In addition to his presentations of specific biochemical systems that are irreducibly complex, Behe supports his position with the lack of any serious biological arguments that explain such gradual steps that construct these complex biochemical systems.

The merit of his book is that it does raise the validity of evolutionary theory in light of the developments taking place within biology due to studies in biochemistry.  Behe does recognize the provisionality of his argument at times though at other times he presents his conclusions with complete certitude. As well, he recognizes that his insights regard only some biochemical systems, and not all. He does acknowledge that some systems at the molecular level can be explained by evolution so he does not consider his focus to be a comprehensive theory of organic life, but rather a focused inquiry that has ramifications for both intelligent design and for the explanatory scope of evolutionary theory (which he thinks is more limited than the field of biology recognizes)

At the same time, I would argue that the argument suffers a bit. Behe argues that a number of complex biochemical schemes are irreducibly complex.  Irreducibly complex means that key parts of a system contribute to the whole system and that removing any of the parts results in the loss of the function of the system as a whole. This functional whole seems to refer to two possibilities. In one case it is like a scheme of recurrence in which the main focus is upon one of the events in the scheme that is crucial for organic function, such as is the case of ATP in Kreb’s cycle within mitochondria.  ATP is one of the molecules constructed in the scheme of recurrence that we call Kreb’s cycle, and because it is a central energy molecule, it is crucial for many processes in the cell, and thus is sometimes referred to as the functional reason for the whole of the cycle.  Another meaning to functional whole however is what Lonergan would call the higher conjugate form that is built upon a lower matrix of conjugate forms (see explanatory genus and species in Insight – chapters 8 and 15).  So, something like an immune response is a conjugate form that exists within an aggregate of genetic/biochemical events.  In both cases take away one of the “parts” — which could be an event within a scheme or an event within an aggregate that constitutes a lower level matrix for a higher conjugate form–and the “functional whole” is lost.

Irreducible complexity alone is not an argument against evolution.  Part of the suffering is that it lacks an adequate account of the heuristic structures operative in biology and the isomorphic metaphysics that is implicit in those structures.  Key that is missing is the shift from descriptive definitions to implicit definitions, and then how this shift moves into an explanatory horizon that then begins to move into an explanatory account of development, whether of single organisms or organisms within ecological relationships.

Why is the shift from description to explanation such a key piece that is missing in his arguments? Because evolutionary theory was initiated from a series of descriptive traits that had been observed by Darwin and Wallace (and earlier folks who developed different explanations).  Descriptive knowledge identifies characteristics and activities of things through conjugates that are derived from the senses.  So, the blue bird has certain colors and shapes that are used to describe its bodily features, colors and shapes that derive their meaning from a relationship between the bird and our senses.  Our senses are attuned to a rather large range of objects in our spatial-temporal world.  We can see shifts in light patterns with our eyes (and of course the associative cortices and the sensory cortex involved in constructing the input that comes through the eyes), which we can articulate as colors or shapes or sizes.  We can hear shifts in auditory sound waves with our ears. We can smell patterns and changes of the chemicals that are found within our atmosphere through our noses and the chemical make-up of solids, liquids, and even gases that touch the sensory neurons of our taste buds.  Through touch we can feel textures and contours and temperatures.  All material objects that have a certain mass size can be detected by our senses.  Many physical objects in our world fall within our sensory capabilities.   So, when we talk about descriptive knowledge, one of the elements that Lonergan recovers against modernity is the degradation of such knowledge.  It is not imaginary.  It has a validity to it and a crucial place in our lives.

Behe does seem to suffer a bit from this modern mistake.  Descriptive knowledge is not false or misleading.  It is incomplete of course, but it is true as far as it goes.  This happened in physics. Copernicus argued that the sun no longer goes around the earth. But, now one can admit that because of relativity Copernicus was not right, at least entirely.  The sun does rise and set when one sets the frame of reference (the X, Y, and Z of a three dimensional manifold — and one must also include t as a fourth dimension) as one’s own sensory framework, or even the earth.  It is not that Copernicus was entirely wrong. One can set the sun as the center of that frame of reference.  Or one can set the center of the Milky Way galaxy (presumably a black hole).  Behe seems to want to say that in this more primitive descriptive world, one might come up with the idea of evolution, however, when you move to the “black box” of molecular and biochemical explanation, then one moves to reality and away from myth.  I say “seem” simply because of his phrase “black box” though he does not explicitly say that the data of Darwin was unreal.  One correction here is to suggest that discoveries made through descriptive knowledge do have their relevance when put within the right frame of reference. Behe does not adequately deal with the kind of descriptive knowledge that is involved in validating evolution (or the modern synthesis that integrated Mendel and Darwin, since it falls within the realm of explanation constituted by explanatory rather than implicit definitions — see chapter 1 of Insight).

Furthermore, scientific description is not merely another frame of reference based on the relations of things to us, but it is the way that science both collects its data and verifies its theories.  Concrete inferences of laws–whether classical or statistical or developmental–all require implementing and verifying those laws through descriptively articulated data. Ultimately, scientific description and scientific explanation are complementary to each other.  What is discovered and proposed in one cannot ultimately conflict with the other if they are both true (or converging provisionally upon what is true).

With regard to biology, I would argue that many do not adequately understand the stage of development in which it currently resides.  This is largely due to the complexity of the field.  Organisms as Lonergan outlines in Insight, chapter 15, require that one shift into a grasp of operators and development.  And these only arise after one has introduced correlations and statistics.  Furthermore, there are preliminary stages within the descriptive world.  There are descriptive conjugates that are preparatory for explanatory conjugates.  Describing what happens when liquids are mixed or when objects are projected or when the planets move in their orbits prepares the way for the world of correlations. Describing whether something happens for the most part, or infrequently, or all the time prepares the way for ideal frequencies whether these are based on descriptive, explanatory, or implicitly defined conjugates.  And with regard to development, one gets a sense that things grow and change in their descriptive conjugates before one discovers the operators that transform one system of conjugates into another.

At this stage in the development of biology, at least over the last century, I would say it largely resided within the world of explanatory definitions.  In an explanatory definition, one of the terms is descriptive, the other is explanatory.   One good example of this is Mendel. Notice that his theory of genetics incorporated one descriptive term, and one explanatory term.  Phenotype is descriptive. Genotype is explanatory (he use the term hereditary unit).  However, later in the century, genetics moved to implicit definition through Watson and Crick, who related genes to proteins (three nucleotide sequences are paired to an amino acid — the building blocks of proteins). Since their introduction of this implicit definition, there has been a vast expansion taking place.

Evolutionary theory when it was first introduced was like the explanatory definition.  All of the traits mentioned by Darwin are descriptively understood.   As such, they had not reached the level of explanatory definitions, let alone the level of implicit definitions. So what makes it like explanatory? Well, he identified a number of descriptive conjugates that seemed to be related across species through some kind of parental origin.  Notice, this springs from a recognition that organisms do come from other organisms (progeny come from parents), and that progeny are never exactly the same as the parental organism (s).  It is important to note that there is not a clear sense that evolution results or can result in development. Rather, it is the emergence of an adaptive, and advantageous, change.

In the early 20th century, Evolution moved into another explanatory level with the modern synthesis. Once Mendel was discovered, Evolutionary theorists went to work to integrate Mendel’s breakthroughs.  That synthesis was largely generic and heuristic because now biological explanation had to incorporate genetics.  This synthesis took another leap once one introduces molecular and biochemical analysis into the science. I would argue that this latest synthesis shifted the images in which biology operates, and this shift has allowed for recognizing new patterns that constitute organic life, and these patterns are defined implicitly.  Watson and Crick provide only one example.

What Behe catches upon is that the theories that had largely been developed from descriptive understandings of traits seem rather shaky in light of the shifts to molecular/biochemical accounts of the organism. He is right in a certain manner. The complexity of biochemical pathways involved in a number of organic activities are mind boggling.  It is hard to fathom how these could have developed. Behe argues that these are impossible to account for in some kind of evolutionary development.

Where I think he has been mistaken is thinking that one should be able to develop an evolutionary theory of a biochemical process at this stage in the history of the field of biology.  In the transition from explanatory to implicit definitions, it is natural that one first has to development a viewpoint that is constituted by implicit definitions which is adequate before one could then begin grasping the operators that unfold a deductive or homogeneous expansion, or a vertical expansion.

Another element that is missing from Behe, and nearly all other biologists or chemists, is the shift from lower levels of organic life to higher genera of sensate and rational life. This shift is far more difficult than from simple to complex organic processes (a horizontal shift). These are shifts from a lower to a higher level genus. These vertical developments add a new meaning to the complexity of evolutionary development.  If Behe understood Lonegan’s articulation of higher and lower genus and species, he could strengthen his argument more. But then he might also have the breakthrough into generalized emergent probability as well, which would, at minimum, severely modify his view of biochemistry and the basis of his entire argument. I say at minimum because the argument of evolution is still an argument based on evidence, hence one of fact, even if it never rises beyond a provisional analytical principle.

So, what would the shift to biochemical images and implicitly defined organic conjugates do to the validity of evolution?  I think it does weaken it a bit in terms of the degree of certainty that many hold evolution today.  Largely, it still is at the stage of an explanatory definition.  Its terms and its evidence are descriptive conjugates.  We are a long way from reaching the periodic table of organic life.  The traditional set of organic charts (kingdom, phylum, class, order, family, genus, species) are based on descriptive traits, though these are being modified daily as a result of biochemistry, molecular biology, cellular biology, and genetics into “evolutionary trees.”  We are even further from developing an adequate set of developmental operators of individuals, species, and genii. In reality, biologists really do not have an explanatory or an implicit definition of species (notice that saying something is reproductively compatible is not saying “what” it is, hence they have not articulated the key explanatory conjugates the form a species within the genera of organic/vegetative/cellular life).

However, just because the development of the discipline of biology is not yet beyond the explanatory definitions in the field of evolution (it has begun to move to implicit definitions in genetics and some other conjugates of biology), does not mean that the explanatory definitions are wrong.  They are based on evidence.   And hence, just because there seems to be some irreducible difficulties when one begins to examine biochemistry and molecular biology, that alone is not sufficient to reject evolution as a theory.  One still has to explain heredity, the differences of progeny from parents, and the ramifications of these differences over time. Evolution is one way to do that.

As Pat Byrne in his essay on “Lonergan, Evolutionary Science, and Intelligent Design,” argues, one of the things that Behe is missing is a grasp of emergent probability [Patrick Byrne, Revista Portuguesa de Filosofia T. 63, Fasc. 4, Os Domínios da Inteligência: Bernard Lonergan e a Filosofia. / The Realms of Insight: Bernard Lonergan and Philosophy (Oct. – Dec., 2007), pp. 893-918]. Emergent Probability, as Byrne notes is ultimately derived from Lonergan’s cognitional theory, not from empirical theories in the sciences and hence it is relatively independent from the development of specific scientific theories. Relatively independent because it is developed from the actual methodological operations involved in the sciences.  Cognitively and metaphysically, the classical and statistical heuristic structures point to the potential of a dynamically oriented developmental universe.  Still, as Lonergan notes, one has to argue what in fact the universe is about.  It does not need to be developmental or evolutionary.  Darwin’s theory has gained much weight through evidence that supports a matrix of descriptive conjugates that relate parents to progeny through generations in an environment where adaptation is possible and probable.  Molecular biology, genetics, biochemistry, and cellular biology are moving biology into the realm of a fully explanatory discipline that is built upon provisionally verified implicit definitions.  This shift is and will continue to bring with it challenges to the old explanatory definitions that had emerged in the field, including that of Darwin and those who advanced his theory in the modern synthesis (note that this synthesis relied on Mendel, not Watson and Crick).  A yet even newer synthesis is arising and Behe’s book highlights the challenges to evolution of this shift even if, in the end, Behe turns out to be wrong and that evolution will rise even stronger once the new explanatory biology matures. But then again, he could be right. Maybe evolution is not right.  I tend to think the evidence supporting the explanatory definitions of evolution theory have a significant weight that is left untouched by Behe’s arguments, and thus provide a valid way of proceeding in biology as it unfolds its new image (symbolic constructs from biochemistry) that is underlying its shift to implicit definitions. I suspect evolution will rise more nuanced in the end, and more like emergent probability, and hence stronger in the end. There many reasons for this beyond biology as well. There is evidence for emergent probability, quite a bit.  There are analogous types of developments in almost all other realms of being (human history, social development, dogmatic development, etc.)

As a note, I have not addressed Behe’s views on intelligent design. That is another discussion and has some serious defects I would argue.  I just wanted to discuss his views of evolution at this point.

Part 5a: Horizontal Finality, a note

… marriage is more  an incorporation of the finality of sex than  of sex itself. (Finality, Love, Marriage, 45)

On vacation, so this will be short.  Horizontal finality refers to the relationship of a initial potency to an unfolding of that potency both deductively and homogeneously on the same genus of being.  Hence, the relationship between the spermatazoa and the oocyte to the multi-cellular organism is one of horizontal finality. This same finality Lonergan argues is what is sublated vertically both into the man and into the woman personally at the levels of understanding, judgment, deliberation, and the actuation of the capacity for self-transcendence.  However, that which is sublated in the man and in the woman is not the zygote, but rather the generative procreative act of the union of the man and the woman. It is that act which is the “conjoined plurality” and most important, it is the finality of that conjoined plurality, that is sublated in a man as a husband, but only when he has committed himself to the woman “until death do us part” and likewise into the woman as a wife, but only when she has committed herself to the man “until death do us part.”  Take note. This finality can only arise in virtue of the conjoining of two semi-fecundities that are ordered toward such a union.  And they can only be sublated within a context of understanding, rationality, and good will that grasps the higher orders of the good life and eternal life which contextualize the conjoined union in a conjugal act.  In other words, a union based on the potency of the conjoined plurality of male and female is much different than any other types of human unions.  It is rooted in the very nature of the semi-fecundities as complementary, and this complementarity is what constitutes the finality.

I cannot depart without mentioning the sad case of the supreme court decision today.  It has been long in the making of course, arguably centuries.  It means that nothing of this natural finality is recognized as constitutive of marriage anymore.  The good news is that the reality of the finality exists whether a state or a people recognize it or not.  It is a bit like the worth of life.  No state constitutes such worth.  A state can only recognize it and it should develop policies that support it. The same is true with regards to the finality constituted by a concrete plurality called male and female.  The state does not constitute it because it is a given within the fabric of the organic world as sublated within psychic, intellectual, rational, volitional, and the finality of the entire universe.  It is sad because such a finality does have a unique and crucial role within the unfolding of the emergent universe, and to not recognize it results in a serious blind spot and deformation within human existence.  It is the strange reality of sin that it can violate the finality of emergent probability.  But never for long without devastating consequences.  History has proven this point.

Just some quick thoughts.

Part 5: Horizontal and Vertical Finality, and further note on Conjoined Plurality in Finality, Love, and Marriage

by David Fleischacker

 

Quick note on horizontal and vertical finality

I am not going to say much on this today, simply because I am still trying to formulate my findings in a more precise manner.   In Insight, Lonergan is able to develop a formulation of horizontal and vertical in terms of the lower and higher viewpoints and levels of being, and these levels of being are identified as genera.  Hence the developments on a single level of being, a single genus, are horizontal, and the finality of the potency for those developments on that level is horizontal finality. Note that in Insight, the notion of development includes but is not limited to the notion of finality–this is a distinction that Lonergan does not seem to make in 1943 in this essay.  In 1943, Lonergan roots horizontal finality in the essence of the thing.  Thus horizontal finality is cast in terms of a potency within an essence for a set of operations or ends that are proportionate to the essence.  This is getting at the same thing as is found in Insight, but the language is more compact, and he did not introduce the notion of explanatory genera and species to clarify the meaning of horizontal and vertical.  Yet he seems to have something close to a genera in mind when he differentiates the levels of being. I am still working out the precise meaning of this differentiation and how it compares to his formulation in Insight.

 

Further note on Conjoined Plurality.

One other thing that I do want to add is a further note on “conjoined plurality.”  When one introduces a conjoined plurality (coming together of conditions to inaugurate a conditioned), this constitutes a realized horizontal finality, and this becomes a realized horizontal finality only if it is a sublation into a higher order. In the human being, this realized horizontal finality did not take place because of a circular scheme of recurrence at the level of the conjoining (level of spontaneous nature to use Lonergan’s 1943 language). So what does this mean for the man and woman? The conjoined union of a man and woman does not take place because of organic schemes of recurrence. Rather, one has to move to higher level operations to account for the union. The man and woman have motor sensory operations that bring them together, and these in turn, because they are human, are sublated within yet higher levels of conscious operations.  These motor-sensory operations themselves would not be completed without intellectual, rational, and volitional operations (even if these are minimized to hedonistic utilitarian or narcissistic pursuits).  So, the higher level operations complete the lower.  Let’s put this another way. In contrast, the schemes of procreation in plants are completed by vital, physical, organic schemes. But in human beings one has to introduce motor-sensory operations and intellectual/rational/volitional operations in order to account for the conjoining of the conditions at the level of nature (vital, physical, organic), a conjoining which then inaugurates the finality to adult offspring on the one hand, and a sublating relationship into the life and relationship of the man and the woman on the other.  In short, the conjoined union of a man and woman does not have its origins in organic schemes of recurrence. There are no such schemes. Rather, at the level of organic nature, these remain a plurality, a kind of aggregate, until these conditions are brought together through operations at the level of experience, understanding, judgment, and decision, all of which are rooted in a state and actualization of the capacity for self-transcendence.  And notice that each higher level completes something in the lower.  Understanding grasps a link between conditions and a conditioned, and judgment affirms the link as true.  If these conditions have not been completed, then a decision is an act that fulfills a condition, and if sufficient, a conditioned comes to be, and the decision then transforms “being.”  More can be said on this, but in general without an actuation of the life of reason (of understanding and of judgment) and without the actualization of the moral level, there would not be a human conjoining of these conditions that inaugurate the horizontal and vertical finality of fecundity.

Part 4a: “Conjoined Union” in Finality, Love, and Marriage

by David Fleischacker

 

As I was thinking about the meaning of a “conjoined union,” the key kind of potency in such a union is not merely finality, but a kind of realized finality. Realized because it is then sublated within the higher levels of being and/or consciousness.  The “conjoined” refers to a set of conditions that converge, and out of which convergence some type of event emerges.  All conditions arise from previous conditions except when something is created from nothing, which is accomplished solely by a divine act.  The conditions can be completely unique, hence fully non-systematic, or they can arise with various kinds of regularities.  In Insight, Lonergan called these regularities “schemes”.  I would like to note three types of regularities.

1. Source regularity.  When a regular set of events arises from a resource that has a significant supply of the conditions needed for the occurrence of the event, then we have source regularity.  Examples include the energy emitted from the sun and gasoline — though there is a circular scheme with this later examine but we are developing schemes upon it which are beyond the circular rate of renewal.  You can think of this as a one way trip.  It is a kind of entropy, but along the way things happen.   We are not talking of circular schemes here, in which the sets of conditions end up replenishing the originating point.  Rather, this is a sufficient pool that then provides a regularity of subsequent events that can be calculated statistically, although usually there is a slow decline of the statistical regularity until the source is depleted.  These are non-renewable resources, and depending on the overall structure of the universe, it may be on this kind of a course.

2. Circular schemes of recurrence.  This is what most reader’s of Insight think about in terms of regularity.  When we think of event A fulfilling conditions for the occurrence of B, and B of C, and C of A, or any more complicated sequence such that the conditions form a cycle or circuit.  These would be renewable resources. However, Lonergan would say that these schemes of recurrence fall within a larger non-systematic environment.  Hence, as long as the conditions remain the same, this circular or even flexible range of circular schemes will remain intakes.  The “remain the same” means both that the positive conditions remain in place, and that no interfering conditions come online.

3. Schemes of development.  Some things become “regular” because of regular schemes of development.  Examples include the regular formation of adult offspring, or the regular sequence of ecosystem development (for example when a fire burns a number of acres), and the ecosystem undergoes a kind of rebirth.  One of the more interesting and recent discoveries is that of stems cells. These have a development relationship to matured functional cells within multi-cellar organisms. These system maintains the cell/organ systems in multi-cellular organisms.

Whenever conditions come together and an event or new thing arises, they form a conjoined union of a plurality.  One can calculate the statistical probabilities of such conjoining of conditions that leads to an event or a thing–though sometimes this proves to be impossible on a practical level.  And, in every case, one can discover a finality in the potency of the pluralities.  Whether the plurality is one of hydrogen and oxygen molecules in an kind of aggregate that contains some electrical energy being exchanged, or one is speaking of the male and female semi-fecundities, there is a finality in all of these conditions that includes both horizontal–if these stay within the same genus of correlations and functional relations (and this would include both deductive and homogenus types of expansions — see chapter 1 of Insight)–or vertical if one examines the relationship from a lower to a higher genus.

With these convergence of conditions whether through source regularity, circular schemes of recurrence, or schemes of development, we can further clarify Lonergan’s introduction of conjoined plurality in his 1943 essay.  He notes that a specific finality arises in the conjugal act (conjoined plurality of two semi-fecundities) that is to adult offspring on the one hand and to higher orders of reason and charity within and between the man and the woman on the other.  The conjugal act follows the pattern of a source regularity that then follows up with a scheme of development. Why this specific act?  Because this specific act is the coming together of two correlative pluralities (male and female semi-fecundities), that is then a realization of the finality of these pluralities. Why a realization? Because this conjoining in the conjugal act is what can be sublated into the higher orders both of reason and charity that constitute the relationship of the man and woman properly as husband and wife (there are other ways of conjoining man and woman as well, but these have other horizontal and vertical finalities), and that is then a realization of a key step in the horizontal finality to adult offspring, and a vertical finality with the adult offspring to an educated adult offspring and a Christianly educated adult offspring.

I will treat more about this relationship of horizontal and vertical finality in the next blog.

Part 4: Statistics and Finality in Finality, Love, and Marriage

By David Fleischacker

 

In part two of this series, I mentioned that the “repetitive” element of the physical, vital, and sensitive spontaneity is differentiated into schemes of recurrence based on classical laws and statistical probabilities, and then schemes of development with one stage being not only an integrator but also an operator, hence possessing a finality, for later stages.

Concrete plurality and statistics

One element that does seem to stay the same between 1943 and 1957, though is explored more fully in Insight, is the relationship between the concrete plurality and its statistical possibilities that constitutes the potentiality that is horizontal and vertical finality.

As to the difficulty that frequently procreation is objectively impossible and may be known to be so, distinguish motives and ends; as to motives, the difficulty is solved only by multiple motive and ends; as to ends, there is no difficulty, for the ordination of inter­course to conception is not a natural law, like ‘fire burns,’ but a statistic laws which suffices for an objective ordination.[1]

It is important to note that even though the relationship of the conjugal act to conception is statistical, it has an objective ordination to the end of adult offspring. If one backs up in the article a bit, this statistical element is linked to a concrete plurality.

This we term vertical finality. It has four manifestations: instrumental, dispositive, material, obediential. First, a concrete plurality of lower activities may be instrumental to a higher end in another subject: the many movements of the chisel give the beauty of the statue. Second, a concrete plurality of lower activities may be dispositive to a higher end in the same subject: the many sensitive experiences of research lead to the act of understanding that is scientific discovery. Third, a concrete plurality of lower entities may be the material cause from which a higher form is educed or into which a subsistent form is infused: examples are familiar. Fourth, a concrete plurality of rational beings have the obediential potency to receive the communication of God himself.[2]

Notice the use of “concrete plurality.” From my reading, it has the same meaning as coincidental manifold in Insight. When a coincidental aggregate is understood in its finality, both horizontal and vertical, then that aggregate is a coincidental manifold. In each case, an aggregate of activities or materials have the potency to be formed into some higher order. In the types mentioned in the quote above, the second, third, and fourth are of particular interest in this essay. The parental contributions to the generation of an adult offspring provide a material that causes the vegetative and even motor-sensory levels in their child. But the motor-sensory level provides but a dispositive cause for the emergence and activation of intellectual, rational, and moral consciousness. This is because intellectual, rational, and moral consciousness is “headed toward the systematization, not of the particular animal that I am, but of the whole universe of being.”[3] These higher levels of consciousness cannot be caused by the lower sensitive manifold because these are intrinsically independent of the empirical residue. In other words, these are spiritual and thus the lower sensitive level is incapable of being a “material cause.” But the sensitive level is still a manifold, and needs to be for the higher levels of consciousness to operate (insight is into phantasm, for example, and cannot take place without phantasm). In other words, the higher orders of spiritual consciousness are extrinsically dependent upon the empirical residue, and thus the lower has a dispositive causal relationship to the higher. [4] Then, finally, in the reception of divine revelation, a concrete plurality of human beings as a community form the recipient of that gift, hence the relationship of that concretely plurality to the gift is a vertical finality of obediential potency.

 

Conjoined plurality and emergence

In every case, the concrete plurality must form a set of conditions for the emergence of a conditioned, whether on the same horizontal order or of a higher vertical order. So, there is a need for some kind of unification of the concrete plurality in order for the conditioned to emerge. A bit later in the essay, Lonergan will call this unity a conjoined plurality.

But vertical finality is in the concrete; in point of fact it is not from the isolated instance but from the conjoined plurality; and it is in the field not of natural but of statistical law, not of the abstract per se but of the concrete per accidens.[5]

This quote was discussed in the last blog with regard to “statistical law.” But now I want to draw attention to the conjoined plurality. Notice how the isolated instance is not the point of potentiality for vertical finality, but rather it is the conjoined plurality that forms that locus. This is ABSULUTELY key. There needs to be a coming together of the right conditions for vertical finality to become a real potentiality. These conditions and their convergence each have a frequency, and thus as well, an ideal frequency rooted in the ranges of possibilities. As organisms become more complex, this range increases just as there is an increasing flexible circle of ranges of schemes of recurrence, and one might add, of development. [6]

The conjoined plurality arises in a statistical manner, with actual frequencies converging on an ideal. And it is true whether one speaks of instrumental, dispositive, material, or obediential potency. All involve frequencies of conditions and the conditioned. If one does not have the right distribution of molecules within a tree, then carving it into a canoe will result in failure. There has to be an ideal distribution of the molecules that allow for what descriptively we would call a “straight tree with its grains running evenly. Or there has to be the right distribution of molecules in a bio-soup if there is to be the likelihood of the emergence of a self-replicating molecule.[7] Or sensitive images need to be in the right disposition if there is to emerge an insight. Or the individual receptive of divine revelation need to have the right disposition and sets of relationships to receive a public, communal divine revelation.

The statistics is a necessary element in finality. In Insight, Lonergan works this out metaphysically.

Finally, the foregoing account of potency, form, and act will cover any possible scientific explanation. For a scientific explanation is a theory verified in instances; as verified, it refers to act; as theory, it refers to form; as in instances, it refers to potency. Again, as a theory of the classical type, it refers to forms as forms; as a theory of the statistical type, it refers to forms as setting ideal frequencies from which acts do not diverge systematically; as a theory of the genetic type, it refers to the conditions of the emergence of form from potency.[8]

Notice here that he is saying in an extremely succinct manner how correlations that define conjugate forms, along with statistical ideals frequencies and finality (as well as development) are linked in terms of the basic metaphysical elements (potency, form, and act). This could be further unpacked into his theory of generalized emergent probability. Concrete plurality is naming a situation in which frequencies that converge on an ideal frequency provide the potentiality for the emergence of forms from potencies, hence new acts, with their frequencies. This is all articulated in general metaphysical terms and relations which reveals with precision a close unity between statistics and finality. That close relationship, as the quotes above indicate, already existed in Finality, Love, and Marriage, and Insight. Obviously, Insight has unpacked and expanded upon all the elements involved in this relationship, but fundamentally, the link seems the same. A statistically distributed plurality provides a probability for emergence, and the potency of this plurality for emergence is finality.

Fecundity, statistics, and finality

Now let us turn to fecundity and its realization.

….the actuation of sex involves the organistic union of a concrete plurality, and as such it has a vertical finality.[9]

Fecundity that is differentiated into two sexual genders is actuated through the “organistic union” of these genders.  In other words, it is in this union that vertical finality of fecundity emerges.  In a later blog, I will discuss the range of this vertical finality, because it includes both an intrinsic self-transcendence within the subjects who are sublating this finality into higher levels and ends of the human subject (notice how easy this will be to translate into the higher orders of conscious intentionality), as well as a  vertical finality within their “adult offspring.”  At the moment however, I want to highlight that the statistical features of this organistic union require that these be a union of two semi-fecundities.  It is the actualization of fecundity that is under consideration, and for that to take place within a plurality of semi-fecundities means that a unification has to take place for the actualization to be initiated.[10]

In short, the fact of a statistical, conjoined plurality or coincidental manifold is neither an elimination of the finality to an adult offspring nor to the finality to higher orders within the man and the woman and the child, but rather, it is the central locus of that finality.  It is the potentiality that is that finality.[11] It is that conjoined plurality (the conjugal act itself) that is integrated into the higher levels and ends of marriage.  Understanding this locus that is elevated is what would lead one to say as Lonergan did in Finality, Love, and Marriage that the “statistical law” that is found in the relationship of the conjoined plurality to concenption

…suffices for an objective ordination.[12]

 

[1] Finality, Love, Marriage, 46 footnote 73.

[2] Finality, Love, Marriage, 20.

[3] Insight, 515.

[4] Insight, 516.

[5] Finality, Love, Marriage, 22.

[6] Insight, 459.

[7] This is just one theory of the emergence of life, life being anything that can “reproduce” itself.

[8] Insight, 432 – 433.

[9] Finality, Love, Marriage, 43.

[10] There are other ways of course, given modern technologies, to actuate the adult offspring, but these usually involve by-passing and hence failing to actuate one or the other, or both of the semi-fecundities as such.  More on that later – once I finish exploring the meaning of this essay, I will then turn to some of its ramification in lights of current questions and debates. And of course, there are ways to eliminate the finality to an adult offspring by through hindering the actuation of one or the other or both of the semi-fecundities. Both by-passing and hindering involve a loss of the conjoined plurality within the man and the woman as subjects.

[11] Lonergan links potency and finality in Insight, 444-451.

[12] Finality, Love, Marriage, 46 foot 73.