link to: NotesonManentpapers
link to:Heilbroner_Chapter 2
link to:Heilbroner_Chapter 2
[This is a reprint of a 1997 posting]
Higher Viewpoints: Part Two
From Algebra to Calculus:The Emergence of the Power Rule
A Thought Experiment
DRAFT VERSION 2
by David Fleischacker
Copyright © 1997. All rights reserved
September 4, 1997 (Originally written in 1992)
(First presented as supplementary notes in a seminar on INSIGHT held September 13, 1993)
The following is an exercise in creating a dynamic image which leads to the insight underpinning the power rule in calculus. This image is a particular “play” with algebraic equations and geometric graphs and definitions. Furthermore, I have intentionally set up diagrams, or symbols, in particular ways so as to illustrate the importance of images in order to get the insight. This exercise does not explicitly distinguish between the rules of calculus and the rules of algebra, but all this is not a far step once the exercises have been performed.
The general outline of the paper begins with some definitions. An understanding of arithmetic and some other basic definitions in math are presupposed. Once some key definitions are established, then we proceed to the setting up of the dynamic image and the thought experiment which leads to the power rule.
Part I. Some Preliminary Definitions (skip to part II)
(1) The Definition of a Point:
The definition we are using for a “point” is that any “x” and “y” on a coordinate system will define a point. The coordinate system in this case is two dimensional. Here is a general diagram of it;
3|………. * (5,3)
0 1 2 3 4 5 6 7 x-axis
The y-axis is the vertical line and the x-axis is the horizontal line on the coordinate system. Technically the two lines are perpendicular and intersect at a point which we have label (0,0). Every point will be given the form (x,y) where x is the number on the x-coordinate and y is the number on the y-coordinate. Thus, the point identified by the “*” on the graph is 5 units on the x-coordinate and 3 units on the y-coordinate (5,3).
The slope of a line is found by taking a segment of the line and measuring its rise and dividing by the measure of its run. The rise is the distance on the y-coordinate axis in a given segment on the line itself. The run is the distance on the x-coordinate axis in the same given segment (d) of the line. So, to get the slope of a line, simply select two points on that line [such as (x1,y1) and (x2,y2) in the diagram. Examine how you can figure out the slope from knowing two points on a line.].(1)
One may wonder why such numbers are used. The reason depends upon the problem one is solving. For now, let us say that we are just putting the numbers into a type of pattern, and later the reason will become clear.
The definition (geometrical) of a tangent is a line which passes through a curve on one and only one point on that curve. Thus, to “tilt” the line one way or the other would necessarily result in the contact of a second point on the line with a second point on the curve.
Notice, when the tangent moves to points “higher” on the curve, the slope of the tangent increases. In other words, the ratio of rise/run increases. When the tangent is moved lower on the curve, then the slope decreases.(2) This raises the question about the existence of a relationship between the slope of these tangents to the curve.
(5) Definition of a function:
The next step is to introduce the notion of function. Instead of a curve, one can actually figure out an algebraic function for the curve. Here, we cannot enter into the tricks of how that is performed. But we can go in the reverse direction, namely start with a function and then draw a curve using it.
A function equates variables to one another through the familiar operations of addition, subtraction, multiplication, division, roots, and powers. So, in the equation Y = X2,(3) the function uses equality and the operation of “powers” in order to relate two variables, namely “x” and “y.” In this equation, “x” and “y” are fixed, such that if you know “x” or “y” you can calculate the other (I do not wish to discuss imaginary numbers or other problems which arise in this activity, for we are staying with real numbers). So, if x equals 2, then y equals the square of 2, or 4. If x equals 3, then y = 9. One can set this up in a graph (see “PLOTTING A FUNCTION.”)(4)
In the diagram to the right, the function is plotted as a curve. One simply plots a point where “x” and “y” meet on the coordinate system. In addition to the six numbers plugged into the function one could include many more. This curve then approximates to the function, and the more points one calculates and the more dots one marks, the closer the approximation (If one could plot the infinity of points on the curve, one would have a continuous curve which would entirely represent the function, but since the curve is material and imaginative, it only approximates, hence the imagination struggles to keep up with intelligence).
Part II: Image and Insight underpinning the Power Rule
(1) The Slope of a Tangent
Before we move to the actual image that leads from algebra to calculus, we need to discuss how one arrives at the slope of a tangent of a curve. You may ask why, and again you will have to wait and see. It is simply another way of organizing the data or numbers for the purpose of understanding the Power Rule. The following set of diagrams will reveal one way to approach the slope of a tangent.
The exercise is to locate the point on the curve through which the tangent line passes, call it point A. Then choose any other point on that curve (5,25) and draw a line from point A to your chosen point. Since you have two points, you can figure out the slope (m2).
Then select a point closer to point A. Perhaps move to the other side (2,4). Although you cannot tell from the diagram, slope m1 is closer to the tangent slope (m3) than is slope m2. As one gets closer to point A, you will find a convergence upon some slope. From this convergence, you can actually approximate the slope of the tangent (m3).(5)
An example will reveal this convergence. We shall use the function “y = x2.” Let us say that we are interested in the slope of the tangent of this function at point (3,9). So, we need to approach the slope by drawing lines through points on the curve which are increasingly closer to (3,9). As the points approach (3,9) from both sides of the point, the lines drawn from (3,9) to those points will increasingly approach the slope of the tangent at (3,9).(6)
|Destination(3,9)(8)||Selected Point(x1,y1)(7)||Calculation(9)of Slope (m)rise/run = (9-y1)/(3-x1) = m|
|a. (3,9)b. (3,9)
|(9-1)/(3-1) = 8/2 = 4/1 —thus 4 is the slope(9-4)/(3-2) = 5/1
(9-16)/(3-4) = -7/-1=7/1
(9.00-6.25)/(3.0-2.5) = 2.75/.5 = 5.5/1
(9-12.25)/(3-3.5) = -3.25/-.5 = 6.5/1
(9-7.5625)/(3-2.75) = 1.4375/.25 = 5.75/1
(9-10.5625)/(3-3.25) = -1.5625/-.25 = 6.25/1
(9-8.7025)/(3-2.95) = .2975/.05 = 5.95/1
(9-9.3025)/3-3.05) = -.3025/-.05 = 6.05/1
Notice: As we moved from step “a” to step “i” you can see that the point (x1,y1) approaches the point (3,9) and the slope (m) approaches 6. So, perhaps the slope of the tangent at 3,9 on the function “y = x2 is 6. It at least approaches that number. If one continues to bring the points closer to (3,9), one will find that the number likewise continues to approach 6.(10)
The basic question is “what is the relationship between the slope of a tangent line and the curve itself?” A clue was given earlier, when we noticed a correlation between the location of the point on the curve and the slope of the tangent through that point. Obtaining an insight into this will be gained through a series of hypotheses about this relationship that serve as the playground for our inquiry.
(1) Hypothesis Number 1
In the next pieces of data, let us say that we have performed the above activity for the points (4,16), (5,25), (6,36), (7,49) on the same function and found the various approximations to slopes.
Slope (m) at (x,y)
Are there any patterns? Examine the numbers in both columns. There are many relations which could be examined, but to move toward our goal, notice the relationship between the “x” in the left column and the slopes in the right (each is boldfaced below).
What is the relationship? The relationship appears to be 2*x or 2x(11) (“*” means multiply, and in 2x, the multiplication symbol is implied).
x * 2 = m
3 * 2 = 6
4 * 2 = 8
5 * 2 = 10
6 * 2 = 12
7 * 2 = 14 (12)
Let the “2x” be named the “slope function” because it is the equation which relates the “x” to the slope of the tangent which passes through the point on a function at (x,y). Once again, we could ask whether this has significance. To ascertain this significance, return to the original equation of the function. It is “y = x2.” Do you see any pattern?
Both the square(13) in the function, and, on the other hand, the “slope function” have two’s in them. Perhaps the relationship between the slope of the tangent and the function involves the power which in this case is 2. To get the slope of any tangent on the function at any point (x,y), you simply multiply the power of the function by the “x”.
(2) Hypothesis Number 2
Let us turn to another function that is not complicated, such as “y = x^3”. If you perform all the suppositions and operations done on the earlier function, this is what you get
The pattern is not exactly the same. The relationship between “x” and the slope of any tangent is not 2x. In looking at the first point, (1,1), maybe it is 3x. But, in trying to multiply the x-coordinated in the second point (2,8) times 3, the number is six, not twelve which was the approximated slope of the tangent at this point. Let us draw up a quick list, placing 3x alongside the (x,y) and the slope (m)
Disappointed? The relationship between the function and the slope of its tangent is not simply multiplying the power by “x.” Look at the numbers again for a pattern. Try another function. Perhaps “y = x4” and add 4x alongside so that it will be consistent with the two earlier diagrams. This will keep things simple.
Set this up in the same manner as the first two sets because keeping a consistency in the setups improves the chances of recognizing patterns. Sit back again, and look at the numbers.
Look at the “y = x3” data again.
Notice that if you multiply the “3x” by the “x” again, you get the slope.
3x * x = m
3 * 1 = 3
6 * 2 = 12
9 * 3 = 27
12 * 4 = 48
15 * 5 = 75
Then turn to the “y = x4” data.
Notice that the pattern does not follow when you multiply 4x times x.
4x * x does not equal m, except when “x” is 1.
4 * 1 = 4 does follow the pattern
8 * 2 = 16 does not equal 32, which is the slope
12 * 3 = 36 does not equal 108
16 * 4 = 64 does not equal 256
20 * 5 = 100 does not equal 500
Look at the numbers again. Notice that if you multiply the outcome of what you just did (4, 16, 36, 64, and 100) with “x”, you get the slope.
4 * 1 = 4
16 * 2 = 32
36 * 3 = 108
64 X 4 = 256
100 X 5 = 500
Now let’s see. To get the slope of the tangent when the function was “y = x2” then the “x” was only multiplied once, by the power. When the function was “y = x3” then the “x” was multiplied twice, once by the power and then by itself. When it was “y = x4” then the “x” was multiplied three times, once by the power and twice by itself. If you carry out the same activities with the function “y = x5“, you will find a similar pattern.This time the “x” was multiplied four times, once by the power and three times by itself.
Notice the pattern? Not only do you have to multiply the x more times when the powers of the function increase, but the times you multiply happen to be exactly one less than the power. You compile the pattern as follows;
if y = x2, then the slope of a tangent on that function is 2 times x or 2x.
if y = x3, then 3 times x times x or 3x2.
if y = x4, then 4 times x times x times x or 4x3.
if y = x5, then 5 times x times x times x times x or 5x4.
What this pattern solves is the slope of a tangent on a function by finding what was called the “slope function.” If you think about it more, a simple rule can be devised from the original function. Let the power = n. Then if the curve is defined by the function y = xn, then to get the slope of the tangent along this function simply multiply “x” by “n” and give the “x” the power of “n-1.”
xn ———–> nx(n-1)
Examine more functions and try out the rule. It should work in every applicable case. Basically, it gives you a new way to figure out the slope of the tangent on a curve at any point you would like to examine. Simply carry out this rule, and then plug in the “x” of the point on the curve which you would like to investigate. It makes this task much easier. Instead of performing the rather involved task in finding the slope which we did earlier, now we just follow this simple rule. Not only that, but the rule is not an approximation like the slope found on page 7 (although it is still a “serial analytic principle”–see ch. 9 of INSIGHT).(14) One thing that should be noted in the applicability of this rule is that it only works for simply functions like x2, x3, x4, x5, etc.. Functions like “x2 + 2x + 3″ do not work with this rule. Finding tangents on those more complicated functions will require more work.(15)
What has been named the “slope function” in this example is, for those who have studied calculus, the derivative. The rule developed in which xn ——-> nx(n-1) is the familiar power rule. The process of applying the rule to a particular problem is called derivation.
This rule is only a first step in developing the mathematical viewpoint of calculus, and it, like arithmetic and algebra, has an analogous deductive and homogeous expansion.
1.The rise of a slope is equal to the distance on the y-axis, which, regarding segment “d,” is y2-y1. Likewise, the run of a slope is equal to the distance on the x-axis, which, regarding segment “d,” is x2-x1. Hence the algebraic definition of slope.
2.In practice, you would probably examine many curves and tangents to see if there is a pattern, not just one curve like we are doing. Using terms like “up” and “down” are really only relevant to the curve and tangents we are using. Furthermore, we are only drawing and staying in one quadrant of the coordinate system. The larger coordinate system extends into the negative y- and x-axis. These extension are not need though, for our concerns.
- Notice the “apt” symbolism. If “x” times “x” were used, it would not have the same probabilities of leading to insight into the power rule. Pay attention to the next few sections to verify this claim.
- We are staying with a simple function. This is all we need for a basic grasp of the power rule.
- It is easier to see the convergence of slopes of the lines drawn from point A to the points 1, 2, and 3 upon the slope of the tangent to point A.
- This exercise uses certain rules. One can compare these to the higher rule which eventually emerges from this kind of “play,” namely the power rule.
- Chosen point.
- The is the point at which the tangent contacts the curve.
- This is calculated from the algebraic definition of a slope on page 1.
- There is actually an algebraic equation which can be used to solve this problem definitively, but it is rather complex, and it is not needed for our purposes.
- In standard notion, although a multiplication sign is not used here, in “2x” what this means is 2 times “x.” In other places in this paper, multiplication may be signified by the capital X.
- Since any number divided by “1” is equal to the number, the slopes listed in this chart do not have the form “m/1.” So, that does not mean that we have eliminate the “run.” Instead, you should just assume the “1” is there.
- This means the power of 2.
- Proving this requires utilizing the algebraic equation used to solve the slope of a tangent on a curve. Not only is this equation more difficult to learn than the approximations we performed above, but it has many limits to its use. There is much guesswork which has to be waded through in order to solve problems using this method, whereas with calculus, the rules are very systematic.
- In calculus, the next step is usually the chain rule. In the same way that this present “thought” experiment was set up, so one could be performed with this second rule. It would be more difficult though.
***In step “a.”, draw a line from point (3,9) to (1,1). In calculating the slope of this line (under the third column), carry out the operations within parentheses first. So, in the above equation, first carry out (9-1) and (3-1), which will result in two numbers, 8 (the rise) and 2 (the run). Then divide the first number obtained with the second, resulting in a rise/run ratio of 4/1
^ The “^” means “to the power of.”
by David Fleischacker
Further, love is the act of a subject (principium quod), and as such it is the principle of union between different subjects. Such union is of two kinds, according as it emerges in love as process to an end or in love in the consummation of the end attained. The former may be illustrated by the love of friends pursuing in common a common goal. The latter has its simplest illustration in the ultimate end of the beatific vision, which at once is the term of process, of amor concupiscentiae , and the fulfilment of union with God, of amor amicitiae (“Finality, Love, Marriage,” 24)
Though there is more to say on finality, I am now turning attention to the meaning of love within the 1943 essay “Finality, Love, and Marriage.” On an initial review, and I think final as well, Lonergan was only beginning to move into a deeper explanatory account of love in 1943. His use of terms derived from faculty psychology and his notion of appetite illustrate this beginning. We must remember however that the use of faculty psychology does not make something false. What happens once one shifts into intentionality analysis is a transposition which sometimes results in a translation of a term into the intentional framework and, at others, an elimination of a term. For example, I would argue that the potential intellect gets translated into the capacity for self-transcendence, and hence expanded and united within the light of all the transcendental notions. Likewise, the agent intellect becomes translated into the transcendental notions, and thus more adequately expanded as well. Thus, Lonergan’s formulation of love in 1943, even if in faculty psychology, can be transposed, something which Lonergan had done by the time he wrote Method in Theology.
First, let’s look carefully at the 1943 text. This section is titled “The Concept of Love.” Notice Lonergan is using the term concept. However, in his opening line, he identifies love as utterly concrete.
The difficulty of conceiving love adequately arises from its essential concreteness and from the complexity of the concrete.(23)
Love is neither a concept or an abstraction, but of course in talking about it, one does have to conceive it.
In conceiving of love, Lonergan develops four aspects, the first two dealing with the nature and act of love itself, and the second dealing with the subject who loves. The first two clearly are formulated within faculty psychology. Love is an act of a faculty. A faculty is a kind of power that is constitutive of what a living thing is. It gives the living thing the ability to carry out certain type of operations. To get an insight into a faculty, one has to carefully analyze a whole landscape of operations and then in examining the operations, discover fundamental characteristics that unite those operations. So, seeing, hearing, tasting, touching, and smelling all have a material element to them, such that the very operation itself regards a spatial-temporal element. As well, these sensate operations allow one to be present and conscious of sense objects. And hence recognizing that all of these sense activities both have a conscious element and a material element would allow one to then formulate a common power or capacity that one has in these types of activities. This becomes the source of the insight into a particular faculty or power. Other operations transcend certain material limitations, and the principle examples of this are the activities of understanding and knowledge. One can posit a common power or faculty to these spiritual (non-material activities), such as the faculty if the intellect. Now on to each of the four aspects.
First Aspect: Love as an actuation of a faculty
Lonergan formulates love as a realization or actuation of faculty. Specifically, it is a faculty of appetite, and love is the central appetite – “it is the pure response of appetite to the good” (23) Other responses are derivative – desire, hope, joy, hatred, aversion, fear, and sadness. Hope is the expectation to become present to that which is love. Hatred is toward that which has harmed the good that is loved. Fear arises in response to the possible loss of the good that is loved. Sadness is the response to that good as lost. Joy is the enjoyment of the good as present. Love is key. It is central. There is nothing false in formulating love in this manner. Identifying it with a faculty, and a fundamental appetite is to recognize that it is a real power or capacity of the human person.
Second Aspect: Love of a beloved as first principle
The second aspect is that it is the principle – “the first in an ordered series” – that initiates a process to its end, which is that which is loved. One can think of simple vital desires for example. The desire for food is not only the “form” of the end process by which one goes out to find, hunt, or grow food, but it is the first principle of that entire process, and it has as its object the end, the food itself. In the case of love it is the beloved. The beloved becomes the first principle that moves the person in love to the beloved.
Third Aspect: Unification of subjects toward an end
The third aspect highlights that the act of love, the act of this fundamental appetite, this first principle of movement to the beloved as term, bonds the subjects who are in love based upon their common pursuit of an end. Those who have not yet reached the end, and rather are still in pursuit of it, become bound when pursing that end collaboratively. Lonergan draws this out further through Aristotle’s notion of friendship in a later section of his essay. Notice that here, Lonergan does not specify the end that is pursued, because any good ends pursued can unite individuals to each other. This pursuit also perfects the human subjects as such, and thus bonds them to each other for each other, but that is the point of the next aspect.
Fourth Aspect: Love of Beloved as United, as Consummated
The fourth aspect highlights that love as realized unites subjects as mutual persons who enjoy the good that each is, a mutual unity that is based upon the good that each person is and has become. The ultimate example of this aspect that Lonergan identifies is the beatific vision, “which at once is the term of process… and the fulfillment of union with God” (24).
It is important to note that Lonergan says these are simultaneous aspects (23). The differences between each is a different focus upon what is “utterly concrete.” By simultaneous he means that one does not happen without the other, even if the individuals involved may be focusing in upon one of the aspects and not the others.
Contrast to love in Method in Theology
There is not only a clear difference of words between 1943 and 1972, but a clear difference in scope. Lonergan by 1972, was able to formulate love in terms of insights that he had into the structure of consciousness, specifically in terms of the capacity for self-transcendence, and the different states of being of that capacity. One not only has the notion of potency in a capacity, but it is a potency that has a directly relationship to states (which is derived from statistical notions – the difference between actual frequencies from ideal frequencies gives one an understanding of the state of something), and it includes a clear differentiation of the notions that constitute the capacity as a whole – the transcendental notions. Lonergan thus could formulate love not as merely an actualization of a faculty, but one might say the actualization of the faculty of all faculties, the base of all bases. Love is basic because it orientes all levels of consciousness. All the questions that one pursues are guided by that which one loves. In other words, the state of being orients all the operators of human development at all levels of conscious intentionality. Love is the actuation of the capacity for self-transcendence, and the more profound it is, the more it underpins, penetrates, and transforms all of one’s horizon.
This does not negate the insights Lonergan had in 1943, but it does formulate these insights more clearly, and it expands upon what he understood of love. It is still utterly concrete, and so concrete that nothing that human beings do escape it, because even getting up in the morning means there is some basic actuation of the capacity, some basic state of one’s being. It is an actuation of a kind of faculty, but not just among others. Rather, it regards the capacity for any human intentional operator and operation. It is a central appetite, but it is also a the central finality of all human activities. The transposition of faculty psychology into intentionality analysis reinforces what Lonergan says about love in 1943 and expands it. Furthermore, the last two aspects can be understood more deeply. When one understands that love is a realization of the capacity for self-transcendence, and that all other operators and operations thus emanate from this realization, then one comes to understand the more comprehensive scope upon which subjects can be bound to each other both as they self-transcend, and as they reach the fulfillment of their self-transcendence. This is especially true when one transposes the beatific vision into a perfection of the human capacity for self-transcendence by the gift that is the Transcendent, the ultimate meaning and ultimate value because the Transcendent is the only true realization of the capacity. Lonergan’s reflections upon Christology and Trinitarian theology draw this out even more (and one might add his work on grace).
Just a few things to think about as we start this exploration on Lonergan’s notion of love in “Finality, Love, and Marriage.”
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.
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)
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.
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.
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).
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 formulated 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 there 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.
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.