A Descriptive Metaphysics: An Analogy for the Science of Metaphysics

By David Fleischacker, PhD
First Draft

Metaphysics as a science seeks to build a comprehensive and generic viewpoint of the universe that is thus far known. How? It finds common features of all sciences and the common features of the objects of those sciences, and melds these features into an integral unity, a horizon of being.

I think it might be easiest to start getting a sense of metaphysics by setting up an every day analogy that will help to point to a more complete metaphysics.

Here are the points that I would like to invite reflection upon:

1. Events: including things, persons, and activities.
2. Frequencies of events: “How often does X take place?”
3. Webs of events and their frequencies.
4. Developmental stages and sequences of webs.

If we stay within our everyday lives, and come up with some examples of these points, I think we can go a long way toward developing a generic view of our worlds. This will be only a descriptive, everyday view, however the reason I chose these points is because these provide analogies for understanding Lonergan’s metaphsics.

An event is person or thing involved in some activity in our everyday worlds. In order to understand that event, we would need to begin raising some questions: What, why, how, where, when type questions. If I asked you “Who is so and so?” In answering the question, you might give me a name? Then I might be curious as to know why the person is here, or what this person does. All of these questions would be expansions of the same type of question, a question that helps me to understand the event. These types of questions can thus be called “questions for understanding.” What kind of questions do you think fit this type in your language?

Notice that in answering questions for understanding, we came to know this event (person, place, or action) through our senses. We can come to describe this event with our senses in a multitude of ways. If it is a thing, I might describe it by its color or shape or how it is to be used. If it is a person, I might describe what he or she looks like, or what the person does. If it is an action, I might describe the movements involved, or the sounds made. We can make use of any of our senses to answer this question for understanding.

Notice how much of the world is known to us in this manner. The inviting landscape that surrounds Seoul, the beautiful clothing, the sense of dignity in the people, the smell of the air, the design of the buildings, the layout of the campus, the smell of the Cherry blossoms, the excitement of students. All of this comes to us through our senses, and we can describe it in language, in art, in poetry, in business terms, and in many other ways. For all of us, this forms the greater part of how we understand the world around us.

Frequencies of events: A second type of question: How often?

I would like you to think about some of the phrases in your language that match the following ones:

“It happens every day”
“It rarely happens at all”
“It happens once in a while”
“It happens too much”
“She does this all the time”

Notice that each one of these statements is an answer to the basic question “How often?” It does not mean that we have quantitative numbers like someone in the fields of statistics would seek. But descriptively, we get a sense of the regularity of events and things in life. This is important for us to live. Think about your own life for a minute, and how you develop many daily expectations based on your past experiences that result in this sense of how often things take place. If you run a grocery store for example, and children are always dropping the jar of jam in one of the isles of the store, then you probably will make some changes in the placement of the jars, so as to reduce this type of an accident. Around your apartments or homes, many things get placed because of your sense of frequency. For example, you probably have a place to hang your coat near the entrance door rather than up in the attic or in the farthest room from the front door, simply because that is the regular (or frequent) location where you will need to put on your coat or take it off. How you setup a kitchen, or a school classroom, or a library, is closely tied to frequencies of use of different items or the frequencies of various activities. If we had no sense of “how often” things or events happen, we literally would be constantly building and setting up places, roads, farms, industries, museums in an impractical manners.

The relationship of “questions for understanding” and “questions for frequency” (“how often?”)

Notice that this second question, “How often?” really cannot be answered unless one has first answered the question for understanding. Only when we have some sense of the event (person, thing, action) can we then start paying attention to it and get a sense of its frequency or regularity. Some events may only occur once in the whole of history. They might be short lived or last for thousands of years. Others might recur over and over again each day, even each minute, like our breathing. Notice, that for me to get a sense of how often breathing takes place, I need to have formed some basic understanding of breathing first. Only subsequently, if I am paying attention to it, will I get a sense of its frequency. Let us look at another example, the frequency of people that drive on a road to the market. Notice, that the first step is that I have to understand “road” and “driving” and “people.” Then, if I am paying attention to this road and the drivers on it day in and day out, over a certain number of days and weeks, and I am also paying attention to the numbers of drivers on other roads day in and day out, then I will start to get a sense of whether this particular road is unusually busy compared to others or not. In the end, I will end up with simple conclusion, such as “This is a busy road.” It is a statement based upon many everyday observations of the roads over many days and week. I may not be counting this like a mathematician but I am forming memories.

The Web of Events and Their Frequencies

The third point that I would like to focus upon is webs of events and their frequencies. All of us have a sense of some of the webs of life. The interconnections of events (people, things, actions) in our homes, or neighborhoods, our work environments, our economic life, our political worlds are webs. A sense of a web begins to develop when we connect one event to another. In special cases, these events form into types of regular cycles, regular patterns of connections between events. When we grew up for example, in our neighborhoods, we started to get a sense of how neighbors and activities impacted each other. We began to get a sense of the interconnections of events and their frequencies. Perhaps when you rode your bicycle past the neighbor’s dog, he always barked at you, then kept on barking for another ten minutes, until one of the neighbors calms the dog down. The reason the dog barks at you is because he wants you to come and throw a stick which he loves to fetch, and you have thrown this stick for this dog many times over the years……

Think about some of these webs in your life. The point here is to get a sense of the generic meaning of “web” and cycles of events and their frequencies.

Families, economies, political orders, churches, volunteer organizes are all examples of vast webs of people and things involved in numerous activities. For example, if one increases the frequency of production in an industry, perhaps pollution increases, if pollution increases, then diseases of the lungs increase, if diseases of the lungs increase more people die. Or in another example, if people are driven by ultimate meaning and purpose, by a love that is unrestricted, and this impacts everything that is done, it increases the charitable acts that they perform day in and day out, it results in greater generosity in the simple and daily interactions with others. Such acts of charity awakens desires that fulfill and quiets desires that destroy. It results in changing the web and habits of one’s own life and in the world around.

Absence of the knowledge of a web makes us awkward in situations. We do not know what to expect from others, we do not know what they are doing or where they might be going, and what they might be expecting from us. This happens whenever we move to a new place, or start a new job, or a new family. It takes time to learn these webs, and notice how we cannot really live easily and well until we learn the web.

Developments of Webs

The last type of questions that I would like to explore in our everyday lives can be expressed as How does this web change over time? This can include when and why it came to be? Or when and why did it disappear? As with our everyday sense of family life, of business, of our neighborhoods, and our political orders, we also, especially as we grow older, gain more and more a sense of how these cycles or webs in life change. If we have paid close attention, we begin to grasp various stages of these changes as well. We might see how one stage in the growth of our grocery store or our in an auto industry led to the next. We might see how one stage in the growth of government led to the next. Though we might not have worked this out as a historian, one might have had enough experience, and paid attention to those experiences, remembering the events and their frequencies, and their formation into webs, and then changes in events and their frequencies over time, in order to get a sense of what was going forward from one web to the next, and why things changed. It leads one to an everyday sense of the stages of development of one web into another.

Web 1  Web 2  Web 3  Web 4 etc., etc., etc..

Notice how important this is as well. If we never get a sense of the stages of our lives, we probably cannot help our children through those stages. We remain bewildered by the changes, confused as to what to do next, just as if we had entered a new society for the first time, or a started a new family. Someone who is starting a business, and has no sense of how to start then grow such a “web” will most likely fail. If someone has no idea how children grow intellectually, then they will be at a loss as to how to setup a proper educational system, since such a system is a web designed to provide children with an environment to awaken the natural desire to know and learn, and exercise responsibility.

Creating a General Descriptive “View” of this Universe

Now I would like to focus us back upon the kind of general view of life that this creates. Notice that it is rather open ended. Events regard anything and anyone that can be known through our senses. The “how often” also regards anything that can be described through our senses. The web recognizes that every event and its frequency has many connections with others, and thus forms into vast web. Furthermore, then entire world that we know in these webs develops along stages. Thus, the more we know about such development, the more we can predict the most likely stages that will unfold in the future.

This view of events, frequencies of events, webs, and developments thus orients us in our worlds. If we want to learn about anything from family life to political life to religious life, we know how to start and build our views of a situation. Here are a few properties and consequences of this general view of life.

1. The world is filled with events and their frequencies, some of which form into regularly recurrent cycles. The sun regularly rises, then sets, goes around the earth only to rise again, then set again. People regularly need to eat, agricultural systems get setup, and thus on a regular basis, in a linkage of webs, food is distributed to stores, people buy the food, and prepare it.
2. The existence of one web can set the stage for the growth of more webs. Hence the creation of the steam engine set the stage for the creation of steam-boats and steam driven trains.
3. Thus, the universe as descriptively perceived is a series of webs that set stages for future webs, and it does so with greater or lesser regularities.
4. World process seems to go in various directions, though when developmental stages are present, then it goes with some degree of regularity in a certain direction of growth, though this is not guaranteed by any means. One can see a great business on the rise, and then something happens unexpectedly, and it collapses. A church can be serving many people one decade, then through failures decline in a day, and close its doors. A great family growing in character and virtue can be rendered asunder by many causes.
5. If enough time exists, then it seems likely that many possible events and their frequencies, and webs might come to be. It also seems possible that many webs will not necessarily lead to any new kinds of development either. Dead ends are possible. One might make a new kind of DVD machine, only to watch it end up in the pile of other technologies that never made it.
6. Webs that are further down the line in terms of stages will be expected to be fewer in number on the whole, because more needs to take place before these come to be. Hence, one would find more societies to possess more farms than to possess universities.
7. If something has more stages, then one will need more time or resources. Hence, for example, one will need many more years of life to develop a sense of metaphysics than a sense of arithmetic.

Finality, Love, Marriage: Tensions and Contradictions in Love, Part 9

by David Fleischacker, Ph.D.

I have decided to return to an earlier sequence of blogs that are commentaries on Lonergan’s 1943 essay, “Finality, Love, and Marriage” — an essay which can be found in the 4th volume of his collected works.  My objective is twofold; 1) to understand Lonergan’s intentions in the essay, 2) to examine what will happen to the essay in light of Lonergan’s later writings on Insight and Method in Theology.

This blog will examine a single paragraph in the essay — section 2.2 — titled “Tension and Contradictions.”  I will start with a quote,

“But besides this multiplicity of aspects, to be verified in any instance of love, there also is a multiplicity of appetites and of loves generating within a single subject tensions and even contradictions.”

Section two of this essay as a whole is focused on the “concept of love.”  In 2.1, Lonergan articulated how love is “the basic form” of all appetite, and how appetite is a principle that unites subjects both in seeking and enjoying a common end.  In 2.2, the section we are considering in this blog, Lonergan brings up the simple fact that appetites are never single in the human subject.  These multiple appetites have multiple ends that are loved and that unite subjects in both seeking and enjoying.  But because of this multiplicity there are “tensions and even contradictions.”

If love is the basic form that unites subjects, why then does it cause tensions and contradictions.  Let us begin the answer by starting with a more precise understanding of the multiplicity.  Notice Lonergan’s three examples of appetite.

  1. Hunger
  2. maternal instinct
  3. rational appetite

The proper object of the first is “my goal” — more specifically nourishment of my body. The proper object of the second is the good of the child.  The proper object of the third is the “reasonable good.”  The way to these objects is specific as well, hence food for the first, care of the child for the second, and the discovery of the reasonable good for the third.

One can imagine that the appetite for my good, the appetite for the good of another, and then an appetite for what is absolute will find themselves in conflict some day if not every day. However Lonergan notes that the third appetite  is the doorway to a liberation from this interior war.  The third “moves on an absolute level to descend in favor of self or others as reason dictates.” (as a note, Newman says this in a number of his writings).  Somehow, reason will discover and project a harmony in the multiplicity.

What happens in the later writings of Lonergan help to expand and fill out the way that reason transcends “on an absolute level” the tensions and contradictions, and can “descend” to set a harmonious order of human action in the multiplicity of appetites.

Insight and Appetite

In Insight, Lonergan seems to have the same notion of appetite operative even though he is heading into a transposition of faculty psychology into intentionality analysis.  One sees this transposition in

  1. Lonergan’s formulation of image and affects via Freud and depth psychology.
  2. His linking of image and affect to neural demand functions.
  3. The linking of image, affect, and neural demand functions into patterns of experience.
  4. The three levels of human development that are interlocking higher and lower levels of integration and operation (organic development, psychic development, and intellectual development) — (chapter 15)

And this list is not exhaustive. These developments provide a glimpse into Lonergan’s deepening understanding of the nature of the levels and the relationships of the levels of being within the framework of the development of the human person.  Lonergan does not yet use the language of horizon, but but he is using “view” and “viewpoint” which reminds one of Newman’s use of the terms (and a number of other figures over the previous century).

In Insight, the language about the mind itself is transposed into intentionality.  Data of sense is combined with data of consciousness to provide the starting point for self-understanding and self-knowledge.  With the clarification of the notion of being (chapter 12), which transposes the agent intellect–and the manifestation of that notion in questions for understanding and questions for reflection, concepts, and judgments–the nature of the mind itself is more directly reflected in the terms and relations.  This in turn provides a basis to answer the most fundamental problems of modernity and post-modernity in their challenge to epistemology and metaphysics.

This shift to interiority analysis greatly expands upon Lonergan’s articulation of “reason.”  Insight and Judgment constitute the acts which when rightly exercised are isomorphic with being.  His explorations of insight in math, science, common sense, and philosophy provide a deeper understanding both of the harmony and unity of intelligence and reason.  When he shifts to an explanatory account of epistemology and the general character of metaphysics in the second half of the book, he completes the circuit which allows one to discover not only the harmony of the mind, but the harmony of being within a framework of generalized emergent probability.   This significantly expands what can be said about unity and plurality of appetites, and the contraditions they might generate, and why reason can objectively discover a unified order in the multiplicity of appetites.

Let me suggest the heuristic solution to discovering this harmony.  In Insight,  occurrences and events fall within the framework of systematic and non-systematic process, which is developed more precisely into schemes of recurrence, conditioned series of schemes, and at the height of generic intelligibility, emergent probability itself.

If one shifts from a mere multiplicity of occurrence (hence a statistical apprehension defined by classical correlates) to emergent process, then one begins to grasp a horizon-scape (land, water, sky) in which these appetites are naturally ordered.  Hunger for example is within the nutritional cycles of the cells and cellular systems of the body.  Eating is part of these systems in its sublation into the cultural mores of a people.  The same is true of all appetites.  They have their natural rhythms and cycles. These really do form a harmony within the context of generalized emergent probability, which incorporates even dead ends and catastrophes (see chapter 4 of Insight).  But even dead ends and catastrophes do not involve a real conflict of appetites, even if a conflict of appetites can cause catastrophes. The gravest catastrophes result in death, and no one subject to death has an appetite for such an end.

Yet, to the casual observer, there does seem to be real conflicts. Are these objective? If so, how?  The answer is yes.  Ultimately for such a conflict of appetites to occur, there needs to be a free subject who can violate the intelligible order of emergent probability.  Intentionality analysis reveals the culprit.  It is the fallen spirit.  The one who rejects the light of intelligibility, being, and the good.  In other words, the subject who fails the dictates of seeking and finding understanding, truth, and value.  Such failures of spirit (spirit = that which is intrinsically independent of the empirical residue) will result in a failure to descend to unite the multiplicity of appetites into their order within the emergent world.  In turn, this failure will descend into the fabric of the polis and even the cosmos.    That is a larger scope of tension and conflict however than we find in section 2.2.  Yet, this larger expanse of being and the world that can be known by reason sheds some light on the source of the tensions and conflicts.  Motor-sensory and intersubjective appetites will conflict when a wise order is privated. That natural and wise order is one of emergent probability.

Method in Theology and Appetites

In terms of Method in Theology, the tensions and contradictions of the appetites can be easily transposed into intentionality analysis. Every cognitional and moral operator and operation is both conscious and intentional. As conscious, these intellectual, rational, and volitional levels of operators and operations intrinsically allow the subject to be self-present, but not necessarily understood and known.  As intentional, one can transpose the relationship of a specific appetite to a specific object.  Hunger can be transposed into the intention of food, motherly care into the intention of the well being of the child, rational appetite into the intention of intelligibility and being.  No conceptual revolution here.

Some areas of appetite are expanded in Method.  Borrowing from Dietrich von Hildebrand, Lonergan constructs a simple generic map of affective appetites — non-intentional states and trends are organized under one column, intentional feelings under another (and these sort into those at the level of sensate experience–pleasure–and those at the level of decision–value).   This generic pattern is further expanded when Lonergan focuses upon the intentional feelings that respond to value. In chapter two, he formulates the scale of values — vital, social, cultural, personal, and religious/transcendental.  One can see the beginnings of this scale in 2.2, with the examples mentioned above.  Hunger belongs to vital values, motherhood to intersubjective and social values (one could argue personal emerges as well), and rational appetite constitutes a type of cultural value (descriptive as well as explanatory).  Both in Insight, and more so in Method, the scale is expanded in scope.

Arguably the most significant transpositions and expansions found in Insight and Method are the Transcendental Notions (only that of Being is explicitly formulated in Insight, and in Insight it is identified as a notion, which falls within the framework of heuristic notions which are components of heuristic structures).  These notions are integrators of systems and operators of development.  This shift to the transcendental notions allows Lonergan to recast a more developed understanding of the dialectic of human development.  In Insight, that dialectic sprung from a tension of levels (eg. sensate and intellectual).  In Method, is becomes more precisely articulated as a dialectic of authenticity and inauthenticity. To return to 2.2, the tension considered is on the same level, that of motor-sensory appetite and conflicts between these appetites.  However, as the essay proceeds, Lonergan will move to a dialectic closer to that found in Insight, but not quite as precise.

The transition from this essay to Insight and Method will not in the end contradict or undermine the general arguments of the essay, but rather will strengthen the arguments as will be seen when we move to further sections.  Lonergan’s later writings provide a more penetrating heuristic for understanding the general character and specific features of the nature and life of marriage.

Next commentary on this essay will be upon friendship — 2.3.

 

Annual Lonergan Epiphany potluck luncheon, February 5

Dear friends, we will have our annual Epiphany potluck luncheon on Sunday February 5 at 12:15 pm, immediately after midday office which begins at 12:05 pm.  Please feel free to bring friends.  The more, the merrier.  We give thanks to God for all good things, most especially the gift of friendship.  with love to you all…

Square root of two as an irrational number

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.

Hermeneutics of Transposition vs Hermeneutics of Recovery

 

 

To move toward an initial understanding about what could be meant by a “transposition of meaning,” one can look at what Matthew Lamb says in “Lonergan’s Transpositions of Augustine and Aquinas: Exploratory Suggestions,” The Importance of Insight: Essays in Honour of Michael Vertin, eds. John J. Liptay Jr nd David S. Liptay (Toronto: University of Toronto Press, 2007): 3-21.  As Lamb cites Lonergan about what happens in transposition, it is one thing to understand or to come to grips with a theological understanding which had been enjoyed by someone like Augustine or Aquinas and which they had spoken about within an earlier context of meaning.  But, after one has truly and properly understood what another has understood and said in the context of their own day, one must find a way to take this same understanding and meaning and bring it into a newer context of understanding and meaning – a new, broader context of understanding and meaning which is the result of later achievements in the history of science and philosophy.  Cf. Lonergan, “Horizons and Transpositions,” Philosophical and Theological Papers 1965-1980, p. 410.  By means of what is new, one takes hold of the old (the old refers to the valid insights of previous acts of understanding) and, by a second step, one raises these true insights to a greater degree of perfection (working with “new notions” present in “science, scholarship, [and] philosophy.”  Cf. Lonergan, Philosophical and Theological Papers 1965-1980, p. 298.  Amid shifts of horizon and changes in philosophical and cultural perspective, a legitimate line of development can be traced as one speaks about the meaning of certain truths in a way which adds to the meaning of the truth that is known in previously known truths.  In this context, one can speak about new meanings being added to old meanings.

 

In a certain sense then, one might say that, in the language of Paul Ricoeur, through a “hermeneutics of recovery” which goes back and looks for what is “best in the speech, writing, and action” of persons and groups who had lived and worked at earlier times, one engages in a work of interpretation which is akin to what happens in a “hermeneutics of transposition.”  In both cases, one brings meanings into a new context of meaning (into a larger context).  Cf. Lance Grigg and Hugo Meynell, “Reflections on the Essence of Critical Thinking,” Divyadaan: Journal of Philosophy & Education 21/3 (2010): 372.  However, while a “hermeneutics of recovery” can be used as an agent of cultural transmission (bringing meanings from an earlier cultural context and tradition into a new current cultural context and tradition through a kind of conversation which can create a fusion of horizons between two different points of view), the creation of a fusion across cultural and historical divides needs to be distinguished from a “hermeneutics of transposition” which seeks not only or merely to bring old meanings into a new cultural context in the hope that changes will fruitfully occur within a given cultural matrix.

 

Yes, initially, the bringing of old meanings into a new cultural context is what happens and, initially, this is all to the good.  However, more importantly, at a higher level, what is needed is a representation (a new articulation of truths) which, in some way, are already known.  They exist, for instance, within the reception of a given theological tradition or in the reception of a tradition of belief and faith.  But, in their day, these truths had been known through a conceptuality that differs from the kind of conceptuality which is currently operative in a changed cultural order.  The good then which needs to be achieved through transposition refers to a new, higher synthesis of meaning (a new, higher synthesis which urges the necessity of expanding or growing in the depth and width of one's understanding and judgment).  As Pope Leo XIII had urged in his encyclical Aeterni Patrisvetera novis augere et perficere.  One best proceeds in one's thinking and understanding of things if one can augment and perfect what is old by means of what is new.  Cf. Lamb, “Lonergan’s Transpositions of Augustine and Aquinas: Exploratory Suggestions,” p. 4; Lonergan, Philosophical and Theological Papers 1965-1980, p. 298.  To understand what is meant by a “fusion of horizons,” see Gadamer, Truth and Method, pp. 306-307, and other relevant texts in this same work.