Judgment and the Recovery of Being

by Dr. David Fleischacker

Lonergan’s explanatory formulation of the interior structure of judgment dismantles one of the great culprits of the modern world that has left vast reaches of the Western world in a dark age. It is dark because it thwarts self-transcendence precisely in one of the great powers of the human mind.  Judgement makes possible a real presence of a person to that which is.  It mediates a true encounter with intelligible being. In other words, authentic judgment allows being to dwell within one.  This darkness is the real forgetfulness of being.  Heidegger was only partially right. He did recognize something that was true about the fallen state of us.  But he still left one with out the ability to enjoy and rejoice in the goodness of even the littlest beings in the world.  Those little, finite beings–trees, rocks, the human body, stars, planets–were merely ontic things.  For him Being– the Ontological–was all that mattered, and even that notion lacks in Heidegger the liberty that Lonergan comes to discover. It is after all a transcendental notion.

When one proclaims that all is mere perspective, or one announces that one can never be sure of what truly is, or one thinks of reality as out there but not in here (in my head), then one is proclaiming that being is fundamentally unknown.  It is as Kant said, in the noumena.  This is the darkness in which today we are chained and enslaved.  It is a self-inflicted cave of own’s own mind, and if one is completely honest, then Derrida is right, even that cave is a mere trace. It too resides in the darkness.  Even my own thoughts flow in the differance of lost presence.

For most, I think the world of entertainment and work keeps them from facing this haunting darkness which they have absorbed since their day of birth.  Many do escape into a world of common sense and do not bother with these questions.  But if pushed in a direction they do not like, then as an instinctual mechanism of self-defense, they pull out the darkness of the no-nothings.

I remember one day saying to a friend, “don’t you know that you can’t find happiness in hockey — he loved hockey to the neglect of nearly everything. He was able to deconstruct my simple quest with one stutter of his vocal cords and a brush of air sent my way in the wave of a hand.  I knew what he meant.  He meant you can’t really know the answer to what you are asking.  Don’t bother me with it.

Lonergan does not answer this deconstructive shallowness with the same brush of air and grunt.  No such simplicity can be found with his response.  Yet, amazingly, in one book he sends to the grave this particular darkness for any who want freedom from these chains that have been growing and entangling the Western world for 500 or more years.  I suppose one could argue that it has been longer and started with the nominalists, but the other day, someone I know — Dr. Chris Blum — pointed out rightly that without the founders of modernity (Descartes, Hume, Kant, etc.), these nominalists would have been forgotten.

Lonergan in one book opens the doors to the cave. That book is Insight. He let’s in some light. We can discover that the shadows and traces of being are not our genie lamp. With the great skill of a gifted surgeon, Lonergan, at the beginning of the book, asks the reader to examine in themselves the act of understanding. It begins a journey into a massive world of interiority and self-appropriation.  The attentive and careful reader who takes this journey is not asked to trust the writer in the end, though one must trust along the way.  He leads the reader from insight in math and science to that of common sense and things, all before he turns to the excavating work of exploring judgment.

It is a brilliant plan as anyone knows who has seriously read the text.  His first eight chapters remove the rocks that block the path to light and freedom, and then finally he removes the hinges of the locked doors of the cave.

Starting in chapter 9, he then begins to open the door.  In chapter 11, the reader gets asked to walk out of the cave unless he or she is too afraid to do so and simply refuses to see the beauty and the landscape of being.

In the next couple of chapters, through the notion of being and then of objectivity, Lonergan provides an explanatory account of why we can be present to being, and why being can dwell within us.  It gives the subject who has dwelt in the cave of the modern world a new wineskin and a new garment.  More technically, it is a new heuristic foundation to taste the beauty and glory of the real universe of being.

I could repeat Lonergan’s answer with regard to the conditions required for true judgments and the principle notion of objectivity, and why these happen in us all the time.  But for the full meaning of these explanatory formulations to burst forth and make sense, one really does need to travel down all of those earlier chapters of Insight first.

Hence, this blog you are reading is merely an invitation to those who have some inkling that perspectivalism and relativism are unhappy conclusions, and that traces of others are not so joyful as their real presence in filial and agapic bonds of love.

By the way, for those who are not able for various reasons to move into the explanatory account of the freedom and light of true judgment, do not worry.  Lonergan’s account reveals that good sound judgment gives you that liberty even when you are unable to explain why.  You really can love–in a mutual indwelling presence–your friend, your spouse, your child….and God, even if the how remains a mystery.

 

Isomorphic Existentialism

Existential Isomorphism

By Dr. David Fleischacker

I would like to make a simple statement. The finality of the human person is one of existential isomorphism.

Why Existential?

I am sure some will think that I have committed an error in tying the word existential to isomorphism.  Some would be disturbed if they knew what I meant.  Some of the dead might twitch a bit. Nietzsche I am sure would turn in his grave. Most of the 20th century existentialists might will themselves to rise from the dead and burn me at the stake and insist that God is still dead. They might call upon their leader — Friedrich, Friedrich, where art though — so that he could lead them in their inquisition with his sharpened words and golden pen.   So, let me be clear as to my fears of the power of these willful mongers.  Will to power and its maturation in the 20th century notion of self-realization are not what I mean by linking the two terms. Yet, there is a truth in the 20th century existentialists that I would like to return to the world of being and goodness and beauty.  As St. Augustine said about heresies, there is always a great truth in them which is why they can arrest people and capture their imaginations.  The same is true I would argue with Existentialists such as Sartre.  That nugget of truth is that human beings do have something to do with their coming to be in this world (or in their self-destruction).

In other words, I want to recover the rightful place of human freedom or decisions.  I want to place it back into a normative framework of a naturally ordered universe that has its nature in a finality that is oriented as Lonergan argues in Insight toward increasing intelligibility and being and goodness. These transcendentals are the norm of the normativity of all existence, especially when they become conscious and active in the human soul as an actuation of the capacity for self-transcendence.  It takes wisdom to figure this out.

So, what about isomorphism?

In Insight Lonergan argues that the structure of cognition is isomorphic with that of being.  Hence, intellectually patterned experience, insights into conjugate and central forms, and judgments affirming those insights as true are isomorphic to conjugate and central potency, form, and act of beings.

J (judgement)   –>    Conjugate and Central Act

U (understanding)–>    Conjugate and Central Form

E (experience)–>  Conjugate and Central Potency

It is not just any E, U, and J that matters to this isomorphism.  The relevant conscious and intentional operations are those that have moved into explanatory accounts of this world–hence insights that emerge in intellectually patterned experience, and then are verified in judgments about the truth of those explanatory insights.

What this means is that in true explanatory knowledge, the human soul has come to be a mirror (as St. Thomas notes) of that which it knows, and it knows that which it knows by becoming a mirror to that which it knows.

Adding the term “existential” goes beyond what Lonergan does in Insight. And as mentioned, I want to expel it of the licentious willfulness that one finds in 20th century existentialist philosophers. I want to recover an older meaning of existence found in St. Thomas and Aristotle, one that links together being and becoming into a harmonious unity.  The act of will is only an act of will when it is based on an intelligibility, and thus it is an authentic volitional act when rooted on form, not on nothingness (which actually is impossible because we cannot create from nothing).  It really combines some of Lonergan’s later developments in Insight with those of his later life, namely the link of metaphysics and its isomorphism with intellectually patterned consciousness to the moral order and the level of decision.  In short, when decisions are based upon the fullness of the cognitive isomorphism with being, then one’s decisions shift one to an explicit participant in the unfolding potency of being [as a note, even one who operates in the world of common sense is a participant in the unfolding potency of being, but only implicitly.  Common non-sense however is evil because it is a failure to participate in this finality of the universe.], and thus participate in a moral isomorphism with the emergent universe and its finality.

I would like to add one other piece that identifies a more complete existential isomorphism, namely when the entire neural and motor-sensory operations, along with their landscape of emotions and passions join in on the isomorphism. For this to take place, the neural and motor-sensory levels need to reach an integrity in which they are intelligibly ordered in the higher levels of the moral and cognitive isomorphism (see what Lonergan does in his last chapter in Insight “Special Transcendent Knowledge”).  In other words, all levels of development when united in a sublating or subsuming fashion into the highest reaches of conscious intentionality form an authentic existential isomorphism of the soul with an emergent universe.

Interestingly, the university when setup right has as its specific end this existential isomorphism in which the totality of the person (organic and neural, motor-sensory, intellectual, rational, volitional, religious) is mediated toward this unity with the finality of the universe.

Just a thought that has tremendous ramifications.

From David Fleischacker

Just a quick note.  I will be publishing a reflection every Friday at 3 pm. Most of these will be short pointers and thoughts about the writings of Bernard Lonergan.

Kindly,

David Fleischacker

Trinitarian Reflections: The Transcentdental Notions and God, blog 1

by David Fleischacker

About two years ago, I started a new notebook on linking together the University and its life with that of the Holy Trinity.  One of the areas that I wondered about was whether the Transcendental Notions (TN) could provide any type of analogy for understanding the three persons of the Holy Trinity.  There are after all, three transcendental notions that Lonergan develops which are spiritual in nature, hence intrinsically independent of the empirical residue.  These spiritual transcendental notions are Lonergan’s transposition of the agent intellect found in Aristotle and St. Thomas, and of the Light of Being (conscience, mind, etc) as found in the Platonists and St. Augustine (as a note, Augustine was clearly not a Platonist once you get into his head more thoroughly even if he learned much from them and borrowed some notions from them).

One of the immediate difficulties of course which one finds noted in Lonergan is that in finding an analogy for the Holy Trinity, we need to deal with acts or operations, not with anything in potency.  The TN are a kind of potency, but much different than normal.  These actually have the power or capacity to bring about self-transcendence.  In St. Thomas (and Aristotle), these “lights” of the mind have the power to illumine, hence they act as agent causes.  Most potencies do not have such capabilities.  Hence the reason these lights are in a kind of actuality as well.  Notice how some of the metaphysical terms and relations get stretched (but not violated! or confused).  The TN are in a potency in relationship to the operations that arise, but in relationship to the potencies in the human subject to receive these operations they are in act.  Many would say that this imprecision of the metaphysical terms and relations is why one needs to leave out the metaphysical, and turn to intentionality analysis. That is true in part, but if one does so, one as Lonergan notes in Insight, needs to run the full circuit, and return to metaphysics, both to refine the metaphysics, but also to articulate the intelligibilities discovered as belonging to being.  To stay merely with a cognitive apprehension of conscious and intentional life leaves one ignorant of its “reality.”  So the circuit does need to be run.

The reason I mention the circuit is because if one is to transpose the analogies for the Holy Trinity found in St. Thomas, then one needs to deal with some of the metaphysical points that he makes, such as God is pure act, and hence we need to find analogies in act that help us, and this is true of the Persons as well as of God.  The Father is pure act, as is the Son, and the Holy Spirit.

Hence, are the TN in act enough for them to be used as analogies?

The TNs, though in a kind of potency, are also the “light” that makes possible the conscious and intentional operations.  This means that in some manner, they are more in act than the operations.  They underpin, penetrate, and transcend all operations.  Still, there must be a reason that Lonergan did not turn toward these as analogies. He stuck with operations (eg. apprehension of the good, judgment of value of the good, love/decision of/for the good). I suppose one could argue that these operations are in part constituted by the TN, as the TN penetrate them.  We could look at what that “penetration” means.  It of course is not physical, but spiritual.  Descriptively, it “illumines” the operation.  It is what “receives” the operation.  It is what “beholds” the operation.  The TN is not only light, but also an intentional focus, hence can be described as the “eye” of the mind as well.  I am tending to think that the TN is both light and eye (hence not distinct as these are physically in us — but I could be wrong).  I suppose one could say the “eye” is the conscious subject as awakened in a TN and thus seeking an answer, hence waiting for an operation that mediates the answer.  Then once the operation emerges, the subject as beholding the operation in the TN is an eye that beholds.  The subject is however conscious through the TN, and thus the TN constitutes both the horizon and the subject as a gazing subject.

One of the areas that I explored a couple years ago in my notebook was whether there was a sufficient distinction and set of relations between the TN to result in some kind of analogy that sheds light upon the Father, Son, and Holy Spirit.  Thus, does the TN of intelligibility have a kind of relationship to that of being/truth such that the former begets that latter.  Of course, this does not happen without an operation. And it does not happen without the subject moving (raising the question for reflection).  Likewise does the TN of goodness spirate from the TN of being?  I cannot repeat all of the reflections here, but I can say that my reflections were not conclusive.  I do intend however to start publishing these reflections in this particular sequence of blogs.

Even if I discover that those reflections do provide an interesting analogy, there is still the further question about whether the analogy is an improvement upon that of the operations as such.  I have a suspicion that they do not, but they might help to deepen my understanding of the operational based analogy (apprehension of the good, judgement of value of the good, decision for the good).  Part of my reason for this suspicion is that God as pure act is the cause of the light that is in us, which we call the TNs.  The TNs do allow us to grasp the unrestricted nature of the operations in God, but those are operations in God, not TNs.   Just a few thoughts.

More later.

Feeding the thirst of Jesus Christ

Why does Jesus need or want us to feed him? It would seem that the only appropriate relation to him is to allow him to feed us. Very true of course. At the same time, from the Cross, he cries out that he thirsts.  He thirsts as St. Mother Theresa tells us.  Jesus is in those whom we meet, especially the poor and the destitute.  All of those who fall under the beatitudes.  He thirsts in and through them for us to give him a bit of drink and food.  It is part of the immense mystery of being a member of the body of our Lord.  He knows us.  He knows us in his divinity and he knows us in his humanity.  As he hung on the Cross, he proclaimed the thirst of his entire body, as it exists in his mind and heart.  This is the meaning of the unity of Christ and his body.  In fact, it is a unity that each of us has with each other.  When anyone thirsts, and it comes to dwell in us, it then comes to inform us as a constitutive act of meaning.  Hence another’s thirst becomes our own.  Likewise with Jesus Christ. We are his.  And we are in him.  He thirsts because we thirst.  He thirsts because he became one of us.  And as he fills that thirst, so we as part of him are to fill that thirst as well. This is the meaning of to abide and to mutually indwell.

Higher Viewpoints: Part Two From Algebra to Calculus: The Emergence of the Power Rule

[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;

y-axis
5|
4|
3|………. * (5,3)
2|
1|________________
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).

(2) Definition of Slope:definition_slope

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.

 

 

(3) Definition of a Tangentdefinition_tangent

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.

 

 

 

 

 

(4) Tangent and the Curve: An important cluepatterns_tangents

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. plotting_functionSo, 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)

c. (3,9)

d. (3,9)

e. (3,9)

f. (3,9)

g. (3,9)

h. (3,9)

i. (3,9)

(1,1)(2,4)

(4,16)

(2.5,6.25)

(3.5,12.25)

(2.75,7.5625)

(3.25,10.5625)

(2.95,8.7025)

(3.05,9.3025)

(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)

(x,y)(3,9)

(4,16)

(5,25)

(6,36)

(7,49)

rise/run (m)6/1

8/1

10/1

12/1

14/1

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).

(x,y)(3,9)

(4,16)

(5,25)

(6,36)

(7,49)

rise/run (m)6/1

8/1

10/1

12/1

14/1

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

(x,y)(1,1)

(2,8)

(3,27)

(4,64)

(5,125)

rise/run (m)3/1

12/1

27/1

48/1

75/1

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)

(x,y)(1,1)

(2,8)

(3,27)

(4,64)

(5, 125)

rise/run (m)3/1

12/1

27/1

48/1

75/1

3x3

6

9

12

15

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.

(x,y)(1,1)

(2,16)

(3,81)

(4,256)

(5,625)

rise/run (m)4/1

32/1

108/1

256/1

500/1

4×4

8

12

16

20

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.

(x,y)(1,1)

(2,8)

(3,27)

(4,64)

(5, 125)

rise/run (m)3/1

12/1

27/1

48/1

75/1

3x3

6

9

12

15

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.

(x,y)(1,1)

(2,16)

(3,81)

(4,256)

(5,625)

rise/run (m)4/1

32/1

108/1

256/1

500/1

4×4

8

12

16

20

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.

Reponse?

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.

  1. 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.
  2. We are staying with a simple function. This is all we need for a basic grasp of the power rule.
  3. 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.
  4. 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.
  5. Chosen point.
  6. The is the point at which the tangent contacts the curve.
  7. This is calculated from the algebraic definition of a slope on page 1.
  8. 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.
  9. 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.
  10. 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.
  11. This means the power of 2.
  12. 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.
  13. 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.”

Part 8:  Love in Finality, Love, and Marriage

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.”

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).