If you glance through the liturgy of the hours and the divine office, you will notice the frequency that the word light is used. Of course, it is in the context of prayer, and it is an attribute of Jesus Christ and of all the persons of the Holy Trinity. It is not by accident. Light in the physical world is the closest analogy to the spiritual that is found. Aristotle recognized that sight is the most liberated of the senses and the one that is closest to the nature of the human mind, especially what he called the Agent Intellect and Plato called the light of Being that illumines our minds and allows us to search out and then gaze out upon the world of being that is behind and beyond the world of appearances. St. Augustine also picks up on on this light and expands it from the light of being to the light of conscience, the light of faith, and the light of glory. Light is at the very essence of the human soul. St. Theresa Benedicta (Edith Stein) picks up on this in her book The Science of the Cross, and recognizes that this inner light is the essence of the human soul, and it is the inner region of the soul where God dwells, and unfortunately, the place from which we as human beings tend to be exiled. Which means we are exiled from our selves. But Christ wants us home, to return to abide with Him forever. God is light. And so are we.
In terms of the transcendental notions then, these are the differentiate lights that Lonergan has articulated in an explanatory manner. As long as we exist, that can be actuated. But they can also remain merely as a capacity for self-transcendence. When the interior question for understanding arises with regard to an experience, then this light has awakened. It is the same quest in which an insight emerges, and this can then be described as an illumination or an ah-ha experience. Notice how this parallels physical light. When light shines off into space with nothing to illumine, it seems dark. But as soon as something comes into its realm, like a moon, then it illumines the object, and suddenly our eyes have something to see. The same with the quest for understand. And insight (act of understanding) allows the eye of the mind (as Augustine frequently uses the term), and eye intrinsic to the quest (a question is intrinsically conscious and intentional) to see (intend) something. Furthermore, once seen, then we can speak it, and this parallels the need to conceive of the insight.
Once the insight is conceived, then a new transcendental notion can be actuated in our capacity for self-transcendence. There are two sequences of actuated quests here. The first regards the correctness of the insight. Is my insight into a representative democracy correct? The second is whether we have a factual object understood by the insight. Is our country a representative democracy? The first has to be answer correctly before the second can be answered factually. Both arise in the quest for truth or being. To answer this quest for truth, first one needs to gather evidence and then reach a kind of insight into whether the evidence is sufficient. Lonergan calls this a reflective insight, which traditionally is named a grasp of sufficient evidence. Such an insight then allows one to answer the quest and pronounce a judgment. Yes, that is the meaning of democracy. Or yes, we live in a representative democracy. Or maybe I realize that I do not understand the nature of a democracy, or that we do not live in one. These too are judgments, answers the that quest “Is it so?”
As the mind rises into the world of truth and being through judgment, then the possibility of a new light or transcendental notion emerges. Judgments of fact attune us to the world that is. Judgments on the correctness of insight not only make possible a grasp of judgments of fact, but also these open up to us worlds of possibility and even probability. We can then wonder about doing something in light of the possibilities that could become factual. There is as Lonergan articulates a scale of feeling that intentionally awakens to objects of fact and objects of possibility. Naturally, these response to the greater possibility (the greater intelligibilities and factualities) with more vibrancy or more commonly, with more energy. This can become vastly disordered of course. But nonetheless, our entire souls awaken within this light of goodness, of value, to the possibilities of our creative free participation in the coming to be of this world, and even facets of our own existence. Appreciation, thankfulness, and love respond to this world of intelligible facts that are good. Free decision is the possible response to intelligible possibilities and probabilities that then brings about factual goods. This light is that of conscience. It is the full realization of the transcendental notion of value. As the landscape of intelligibility grows, the landscape of possibility, probability, and factuality grows. That is the growth into an entire scale and horizon of the good.
Ultimately it also is the expansion of the actuation of the entire capacity for self-transcendence that then awaken to the question about unrestricted answers to the potentially unrestricted quests. That is as Lonergan articulates in chapter 4 of Method in Theology, the emergence of the question of God. If one has been appropriating one’s own interiority, one realizes as well that this is also an awakening to the very inner essence of one’s own soul, to the capacity for self-transcendence. As that capacity becomes actuated, its full actuation, the full actuation of the transcendental notions, which is aptly described as an unspeakable illumination through God’s outpouring love who gives himself as a Triune self to the subject, is then a state of being in love without conditions.
I want to bring this back to the liturgy of the hours and divine office now. This unfolding of this liturgy throughout the year cultivates the human subject, and through it, God tills the landscape of one’s horizon at all levels of conscious intentionality, from experience up through moral acts to purify the totality of one’s soul so that it is more and more permeated by the transcendental notions. Then one becomes more and more a light in this world.
The liturgy of the hours and divine office however as liturgical however revolve around an even more potent set of lights. Those lights correctly understood and entered are the sacraments. The sacraments are gifts from God that literally started in the motor sensory world that if rightly received permeate the entire human subject from the lowest to the highest levels of conscious intentionality, and these do so by actuating the capacity for self-transcendence in its state of being in love without restriction. This state as long as it lasts and as much as it is lived from then increases the realization of each transcendental notion as it awakens along the paths of emerging conscious intentionality, and it transforms the entire landscape of conscious intentionality. The Eucharist is the supreme realization of all of this. One literally is moving into the inner reaches of the Holy Trinity. This is why it is described as an eternal banquet. At a mass, one’s motor-sensory being literally participates in a physical encounter with that which has been transubstantiated into the ascended incarnate Son of God who opened the doors for humanity to enter the inner life of the Holy Trinity. We are literally able to move closer in space and time to this reality. This reality though emerges into all levels of conscious intentionality through acts of faith and gifts of grace. The Father sends His Son into us through their Holy Spirit, literally and spiritually. Is this necessary? No. It is a gift and promise. God makes it possible for us to receive him into our beings in a manner similar to how Mary when Gabriel came to her and she proclaimed her fiat. The Son was then literally conceived in her. When we sit in the physical presence of the Holy Eucharist, the body-soul-humanity-divinity of the Son, God promises to enter us if we are rightly ordered, receptive, and pure. It reaches its height when we say and Amen to Him, and we are feed physically and if rightly purified, spiritually with Him. As incarnate beings, this way of being given the Son then illumines the totality of our conscious intentionality in a manner like nothing else. We can then understand what someone like John Henry Newman meant when he said that our presence in and reception of the Holy Eucharist is the closest we get to heaven while on earth. It is heaven, the eternal banquet. Now that is a light.
Just some thoughts.
By David Fleischacker
I am aware of at least two theological teachings that make significant use of the notion of indwelling. The first deals with the indwelling of God in the soul, and most would think of the indwelling of the Holy Spirit. The second deals with the mutual indwelling of the Father, Son, and Holy Spirit. To indwell is a profound notion, and I think Lonergan can help us unpack it.
In volume 2 of the Triune God–the one on systematics–and in a number of other places, Lonergan writes about how the known is in the knower, and the beloved in the lover. It is this type of presence and consciousness that articulates what happens to us when we know and love God, and each other.
To grasp the full scope of this, one must fully break with the extroverted notion of knowing. In the extroverted notion, the object remains “outside” of the knower, and hence love of the object also is perceived as a love of that which is outside of one. But once one shifts to the interior nature of the act of understanding that has been affirmed true in judgment, then that which is understood indwells in the human subject. This indwelling takes place because understanding and knowing (Judgment) is isomorphic with the form and act of the reality understood and known. When the judgment is not merely a judgment on the correctness of understanding (eg. understanding the nature of democracy), but rather is a judgment of fact (eg. this is a democracy), then the reality thus known as fact indwells in the knowing of the knower. It is a presence of the reality that constitutes the realization of the subject. The “other” really is in one, and even more precisely, constitutive of one.
Then, with this cognitive indwelling, there arises the possibility that the reality can dwell within the very orientation of one’s capacity for self-transcendence. This is what it means for something known to dwell in one’s heart. This is the more complete realization of indwelling.
Existentially we have all experienced this indwelling at some point in our lives. When we have first fallen in love, witnessed the birth of one of our children, or said yes at one’s wedding, one has experienced an indwelling. The same experience happens when a loved one dies. We feel like we have died. The basis of these experiences is the nature of how realities come to dwell in each of us. This is a profound reality.
When we turn to faith, and to a Transcendent being, we then begin to realize the greatest meaning and character of indwelling. The mystics are some of the most articulate, but because few have glimpsed such a level of indwelling, few have any insight into what they mean. Individuals like Saint Theresa of Avila, Saint John of the Cross, or more recently, Saint Theresa Benedicta (Edith Stein) give us glimpses into the way that God lives at the center of the human soul (this by the way is explained in understanding the transcendental notions and how these notions are created participations in unrestricted intelligibility and intelligence, being and rationality, goodness and responsibility), and that our journey to God is simultaneously a journey into the authentic self. But to travel this route, much has to be purified and opened up, something which the mystics can teach us far beyond what one finds in Lonergan. But using Lonergan’s call for interiority analysis can help to further clarify this journey within an explanatory context. One can link the Christian mystics to Lonergan’s way of self-appropriating our cognition, our volition, and most profoundly our capacity for self-transcendence as it culminates in a state of being in love with God. This would allow one to develop an explanatory account of indwelling. Here, all that I have done is given a few clues.
David Fleischacker, Ph.D.
[May 26, 2009]
If Newton’s physics and Dalton’s chemistry are related as a lower to a higher viewpoint, there must be some point of contact, just as numbers and operations were the points of contact between arithmetic and algebra. It seems that this point is mass. Newton and Dalton dealt with masses within the context of “relative weight.” Newton related objects in terms of masses, distances, accelerations, and forces, especially his well-known discovery of the law of gravitation. Dalton discovered patterns in the “relative weights” that lead him to some postulates about atoms and compounds. A significant difference arises though. Newton studied large objects, large meaning what can be seen such as marbles and planets. Dalton studied gases and mixtures of solids and liquids (especially gases), and then made postulates about objects that cannot be seen. The objects that they studied seem very different, so how can they be related as lower and higher viewpoints?
Before drawing some conclusions, a closer examination of Newton and Dalton is in order.
1. Isaac Newton: The Law of Gravitation
Newton studied the relation of objects in terms of mass, distances, accelerations, forces, and the gravitational constant. If we specifically examine his equation for universal gravitation, his focus will become clear. The equation requires little space to write,
F = (Gm1m2)/d2
Explanation of this formula requires far more than writing it out, and though a full explanation will not be given here (any physics text book will give an explanation and some examples, along with some problems to solve), some identification of each of the terms is in order. In brief, “F” stands for force. “G” for a gravitational constant that is relevant for any mass. “m1” stands for a mass. “m2” stands for a second mass. “d2” is the square of the distance between the masses. The equation relates only two masses. Relating more would be far more complicated. It says nothing about what kind of masses are used, whether they are planets or marbles. Furthermore, it is supposed to be true of any masses whatsoever, hence it received the title of the universal law of gravitation. But, in the concrete, rarely, if ever, are only two masses involved. This law presupposed something similar to the “vacuum” that is presumed in Galileo’s law of falling bodies In that law, without friction a feather and a marble would fall to the earth in the same amount of time. In Newton’s law, without any other masses, presumably, the equation would hold true. However, just as with object falling on earth are effect by friction, so planets are affected by a number of other masses in addition to the earth or sun. So, this law really does not fully explain the motions of any particular planet (In fact, Newton realized it did not explain the data better than Ptolemy’s circular theories, though it was a simpler explanation). Yet, it is an important first step, just as distinguishing acceleration from velocity was an important step toward the law of inertia, the notion of mass, and the law of gravitation.
2. John Dalton: The Atomic Theory and Relative Weights
Dalton developed a new atomic theory of mass from their weight relationships. He writes “In all chemical investigations, it has justly been considered an important object to ascertain the relative weights of the “simples” which constitute a compound.”(1) He goes on “Now it is one great object of this work, to show the importance and advantage of ascertaining the relative weights of the ultimate particles, both of simple and compound bodies, the number of simple elementary particles which constitute one compound particle, and the number of less compound particles which enter into the formation of one or more compound particle. Dalton, like Newton, speaks of “two bodies,” but unlike Newton, Dalton adds the concern with their combination, not their gravitational relation.
“If there are two bodies, A and B, which are disposed to combine, the following is the order in which the combinations may take place, beginning with the most simple:
1 atom of A + 1 atom of B = 1 atom of C, binary.
1 atom of A + 2 atoms of B = 1 atom of D, ternary.
2 atoms of A + 1 atom of B = 1 atom of E, ternary.
1 atom of A + 3 atoms of B = 1 atom of F, quaternary.
3 atoms of A + 1 atom of B = 1 atom of G, quaternary.” (Page 112)
Then he adds, “etc., etc.”
This is rather similar to what happens when one is discovering algebraic patterns within arithmetic.
Dalton then proceeds to discuss the actual relative weights of different substances that were known. Hydrogen was given a base weight of 1, and to this all the other “simples” or “ultimate particles” can be determined. Carbon is five times the weight of hydrogen, hence it has a relative mass weight of 5. Oxygen is seven times hydrogen, so it has a relative weight of 7. Water is a binary combination of hydrogen and oxygen, so it has a relative mass weight of 8. From this, he then unites the rules for combining bodies with their discovered relative weights to formulate another law which presupposes the law of the conservation of mass. The weights of binary, ternary, and quaternary compounds should be equal to the combined weights of the “simples” that constitute the compounds. Still, analyzing and synthesizing these “simples” and compounds is not an easy matter, and Dalton develops some rules of thumb.(2)
After developing these rules of thumb, Dalton then proceeds to explain which actual weights are combinations of simples, binaries, ternary, etc., and what those simples, binaries, ternaries, etc., might be. For example, he then discussed how one might reason that water is a binary of hydrogen and oxygen.
3. The Higher Viewpoint
So, what is the link between Dalton and Newton? The link can be grasped by paying closer attention to the experiments and theories each relied upon and developed. Newton’s law of gravitation applied not only to planets but to any mass object. The gases, solids, and liquids of the chemist are some of those objects. Gases, liquids, and solids have weight, and weight is a combination of a mass and gravitation. Newton was concerned with relationships between any masses, relationships which were defined in terms of their respective distances, and the changes in their velocities (or lack of such changes). So, he described force as a product of mass times acceleration, or force as a product of a gravitational constant multiplied by the two masses, then divided by the distance between them. Dalton does not use Newton’s law of universal gravitation as the lower viewpoint in which he discovers patterns and laws of a higher viewpoint. He only uses the notion of weight, but because he refines it in terms of relative weights, the real difference is due to a difference of mass. When developing “relative weights” what really distinguishes the objects is the mass, because the “gravitational component” is equal.(3) So, what distinguishes Newton’s concern from Dalton’s is that Dalton wanted to discover patterns of different mass relations, Newton wanted an explanation of weight itself. It would be many centuries before the actual formulas of physics could be utilized in the lower viewpoint as a phantasm or image for the higher viewpoint of chemistry.(4) At this point, problems in the combining of weight was the starting point for chemistry just as negative numbers, fractions, and other arithmetic problems were the starting points for algebraic rules.
Dalton’s concerns or horizon form a higher viewpoint because he is developing new principles and laws regarding weights and the combining of weights into compounds.(5) He is not developing a fully elaborate higher viewpoint of all aspects of Newton’s theories and formula’s, but it is a higher viewpoint with regard to one dimension, and that is weight, and implicit in weight, mass. (I will continue to articulate this point in further revisions of these notes because the point of “physics” at which Dalton’s viewpoint arises is much like the initial development of the higher viewpoint of algebra from the problems of negative numbers or of calculus from the power rule, and ignoring all the other areas of arithmetic from which algebra can formulate its new rules, or the other areas of algebra, from which calculus can build its rules).
A further inquiry would bring us to grasp the relationship of Dalton and Mendeleev. Is Mendeleev’s periodic table a higher viewpoint to Dalton’s atomic theory, or is it a homogeneous expansion? That is a further question, which would be worthwhile to investigate.
- John Dalton, “A New System of Chemical Philosophy,” in Breakthroughs in Chemistry, ed. Peter Wolff (New York: A Signet Science Library Book, 1967), 111.
- Dalton lists seven rules. “1st. When only one combination of two bodies can be obtained, it must be presumed to be a binary one, unless some cause appears to the contrary. 2nd. When two combinations are observed, they must be presumed to be a binary and a ternary. 3rd. When three combinations are obtained, we should expect one binary and the other two ternary. 4th. When four combinations are observed, we should expect one binary, two ternary, and one quaternary, etc. 5th. A binary compound should always be specifically heavier than the mere mixture of its two ingredients. 6th. A ternary compound should be specifically heavier than the mixture of a binary and a simple, which would, if combined, constitute it; etc. 7th. The above rules and observations equally apply, when two bodies, such as C and D, D and E, etc. are combined” (115). As a note, Dalton was also one of the first to develop symbols of these “simples” and compounds (recall the need for phantasm to obtain insight).
- If the masses of the objects were greater, then they would affect the overall gravitational force, but like most of the objects that Galileo studied, there mass is insignificant (which is why “light” and “heavy” object fall to the earth with the same acceleration, baring any significant friction). These relative masses would hold even if the gases, liquids, and solids were on a different planet, or on the moon, hence the real term that distinguishes is the difference of the masses between the gases, liquids, and solids.
- Gases became important because they, as a matter of fact, were able to be produced from mixing substances, and these gases tended to be divided into what we now call elements. Dalton was one of the first to postulate that these were elements, or as he named them, “simples.”
- Also, notice the similarities to arithmetic and algebra. Arithmetic wanted to get numbers through the operations of addition, subtraction, multiplication, division, powers, and roots. Algebra discovered patterns in adding, subtracting, multiplying, dividing, powering, rooting. Similarly, Newton wanted to related masses through distances, accelerations, gravitational constants, and forces. Dalton discovered some patterns in a particular range of these related weights (that range being limited to the weights of gases, solids, and liquids on earth that can “combine”).
By David Fleischacker, PhD
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.
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.
- maternal instinct
- 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
- Lonergan’s formulation of image and affects via Freud and depth psychology.
- His linking of image and affect to neural demand functions.
- The linking of image, affect, and neural demand functions into patterns of experience.
- 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.
Dear friends, we will have our annual Epiphany potluck luncheon on Sunday February 5 at 12:15 pm, immediately after midday office which begins at 12:05 pm. Please feel free to bring friends. The more, the merrier. We give thanks to God for all good things, most especially the gift of friendship. with love to you all…
Square root of two as an irrational number
by Br. Dunstan Robidoux OSB
edited by Mr. Michael Hernandez MA
When Lonergan discusses inverse insight in the first chapter of his Insight: A Study of Human Understanding, he presents a mathematical example to illustrate the nature of inverse insight as an act of understanding which realizes that an expected, desired intelligibility is not to be reasonably nor rationally expected. (1) In some situations, in some inquiries, to anticipate in the type of intelligibility sought is to perdure in “barking up the wrong tree” and to waste time by asking irrelevant questions. However, since Lonergan’s example pains readers who have never acquired any easy familiarity with mathematics and who have lost what familiarity they once had, this paper will parse out the discussion in ways which should help. Let us begin.
Lonergan’s argument consists of the following sequence of numbered propositions:
Proposition 1: The square root of 2 is some magnitude greater than unity and less than two
Proposition 2: One would expect it to be some improper fraction, say m/n, where m/n are positive integers and by the removal of all common factors m may always be made prime to n.
Proposition 3: If this expectation correct, then the diagonal and the side of a square would be respectively m times and n times some common unit of length.
Proposition 4: So far from being correct, the expectation leads to a contradiction.
Proposition 5: If sqrt(2) = m/n, then 2 = m2/n2
Proposition 6: But, if m is prime to n, then m2 is prime to n2
Proposition 7: In that case, m2/n2 cannot be equal to two or, indeed, to any greater integer
Proposition 8: The argument is easily generalized, and so it appears that a surd is a surd because it is not the rational fraction that intelligence anticipates it to be
To understand the controversy about the square root of 2, let us look briefly at the historical origins of the problem.
First, with respect to numbers, the square root of 2 is some sort of number. Numbers fall into different types or species since the square root of a number is unlike the number whose square root is sought. Numbers rank as human inventions since they do not exist as purely natural entities apprehended by sense. They were invented as the human need for them arose. (2) Different needs, as they emerged, formed new types of numbers. Hence, the first type of numbers invented were the counting numbers, sometimes cited as natural numbers: 1, 2, 3, 4, 5…. (3) They arose as correlatives to designate quantities: how many of this or how many of that. For example, “3” identifies three sheep or three fish. The sequence of counting numbers is potentially infinite since the human mind can keep adding units of 1 to form an ever greater number. Subsets are similarly infinite in their sequences. The odd numbers, as in 1, 3, 5, 7…, are infinite as are the even numbers, 2, 4, 6, 8…. On a straight line, in one vector, each natural number can be represented by one point on a line ad infinitum. (4)
A second species of number emerges in whole numbers when counting proceeds in reverse: toward and beyond 1. Nought or zero emerges as a number to signify the absence of some item. The creation of this numerical designation signifies an “empty set” as in “the number of Eskimos living in our house is 0.” (5) The inclusion of 0 with the counting numbers thus creates a larger system of numbers than the old quantitative counting numbers. Enumeration now begins from 0 which can also be represented by a point on a line.
A third, more comprehensive set of numbers emerges when the reverse counting which had led to 0 continues backwards to include numbers that are now less than zero. The result is a potentially infinite set of negative whole numbers. When these numbers are then added to the numbers that have already been generated by counting from zero upwards (the positive whole numbers), the result is a set of numbers known as integers. An integer is defined as a positive or negative whole number as in 0, ±1, ±2, ±3, ±4 . . . (6) The negative and positive signs indicate direction: all these numbers are directed. On a number line, the negative numbers go to the left of 0 while the positive go to the right. Each number has a point.
Rational numbers deriving from a ratio or fraction of integers or whole numbers emerged when it became necessary to specify measurements which are parts of a number. How does one express a length which is between 4 and 3 meters or 4 and 3 cubits? Is a loaf of bread, equally divided among 5 persons, divided in a way where each piece has a numeric value of 1/5? Does the addition of 1 piece to another not result in a union with a numeric value of 2/5? A number designating parts thus consists of parts in its makeup. There are two halves: a numerator above a line and denominator beneath. (7) The denominator indicates how many intervals exist between two possible whole numbers while the numerator indicates how many of these intervals are pertinent in a given measurement. The denominator cannot be 0 since, otherwise, one would be indicating that no intervals or parts exist between two numbers. Why specify numerators for portions or parts that do not exist? A rational number is commensurate with given lengths that are being measured. A number which includes a fraction can be assigned a point on a line. The position is determinate.
In the 5th Century B.C., the Pythagoreans initially assumed that numbers measuring the sides of a triangle are rational where each number can be expressed as the ratio or quotient of two integers (or two whole numbers). (8) Divisors (or denominators) exactly divide into numerators as in ½, 1/10, and 1/100: a half (or .5), a tenth (or .10), and a hundredth (or .100). A ratio as the quotient of two numbers or quantities indicates relative sizes. (9) The ratio of one number to another is expressed in terms of a/b or a:b. It was assumed that a one-to-one correspondence joins straight-line segments of length with rational (whole) numbers. (10) In attempting to measure the diagonal of a square by taking a small part of one side as the measuring unit, one should be able to fit the measuring unit a fixed number of times within both the side and the diagonal. (11) All lengths are measurable and commensurate in terms of rational (whole) numbers. Two quantities are commensurable if their designating numbers are multiples: both numbers arise as products of common factors (a factor being a number that divides a given number exactly or completely (12)). For instance, 16 and 12 are commensurable since both exist essentially as multiples of 1, 2, or 4: each exactly divides into 16 and 12 and no other number exactly divides 16 and 12. By multiplying one or more of these numbers together, one arrives at numbers 16 and 12 (in conjunction with other possible numbers that are also commensurable). Similarly, 3 feet and 2 inches designate commensurable quantities since 3 feet contains 2 inches an exact or integral number of times. (13) Hence, according to Pythagorean assumptions and expectations, the length of a square’s diagonal whose side is represented by a rational number should be represented by another rational number.
On the basis of this belief in rational numbers and the corresponding commensurability of lengths, according to the Pythagoreans, “numbers are things” and “things are numbers.” All things are numerable in terms of whole numbers and their properties. (14) A cosmic harmony exists in the universe given the interrelation of things based on whole numbers where the relation between two related things can be expressed according to a numerical proportion or ratio. For example, in music, ratios of concord exist between musical sounds (pitch) and whole numbers since by halving the length of a string on a lyre, one can produce one note one octave higher. All harmonies can be represented by ratios of whole numbers and, by extending this principle to all things, through geometry one can explore the configurations of perfect solids in the belief that all lengths are measurable in terms of rational whole numbers.
A crisis emerged for the Pythagoreans when, possibly prior to 410 B.C., they realized that some numbers, though real (as existing), class as irrational because they cannot be written as whole numbers, as integers or as quotients of two integers. (15) No assignable point of a line can be given them. Some numbers do not exist thus as whole numbers as can be seen through a deduction from Pythagoras’ Theorem in geometry which describes the relation between the lengths of the sides of a right-angled triangle in the following terms:
In a right-angled triangle, the square on the hypotenuse [the side of right-angled triangle opposite the right angle] is equal to the sum of the squares on the other two sides. (16)
Thus, if the hypotenuse has a length c and the other two sides, lengths a and b, then c2 = a2 + b2. Now, if, in a square, the side length constitutes 1 unit, then
c2 = 1 + 1
c2 = 2
c = sqrt(2)
The diagonal is 2 units in length. (17) This number obviously designates some magnitude greater than 1 or unity but less than two where, initially, one naturally assumes that this number is an improper fraction expressing a whole number (an improper fraction being defined as a fraction whose numerator exceeds its denominator as in 4/3 versus 3/4, designating a proper fraction (18)). (19) However, if the square root of 2 cannot be expressed as a whole number, its irrationality in terms of whole number properties creates major problems given expectations which assume the adequacy of whole numbers. After all, conversely, if only rational numbers exist, the hypotenuse of every right-angled triangle will have a length that cannot be measured by any whole number. (20) It is incommensurable, non-measurable: in the relation between the diagonal d and an adjoining side s, d cannot be divided by any unit common to s an integral number of times. In trying to effect any measurements, the Greeks found that however small or large would be their measuring unit, it failed to fit within both the diagonal and the adjoining side a fixed number of times. (21) A measuring unit that would fit the adjoining side a fixed number of times would not fit the length of the diagonal. It was either too short or too long. Proofs demonstrating the irrationality of 2 came in a number of varieties.
Aristotle refers to a proof on the incommensurableness of a square’s diagonal with respect to a side that is based on the distinction between odd and even, an odd number being an integer that is not divisible by 2 while an even number is divisible by 2. (22) To understand how this argument works, a digression on prime numbers introduces the discussion.
A prime number is a whole number with exactly two whole-number divisors, itself and 1. Some primes are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . . , 101, . . . , 1093
Prime numbers are the building blocks of other whole numbers. For example,
18 = 233 40 = 2225 105 = 357
This type of factorization is possible for all nonprime whole numbers greater than 1 and it illustrates the fundamental theorem in arithmetic known as the Unique Factorization Theorem (23) which says, as follows, about the prime decomposition of a whole number:
Any nonprime whole number (greater than one) can be written as the product of a unique set of prime numbers. (24)
Every prime integer shares the important property that if it divides a product of two integers, then it must divide at least one of the factors (prime numbers being only divisible either by themselves or by 1). This theorem is important in many parts of mathematics. In one simple consequence, when the square of any whole number is written as a product of primes, each prime occurs as a factor an even number of times. For example:
(18)2 = 1818 = 233233 = 223333
two 2’s four 3’s
(40)2 = 4040 = 22252225 = 22222255
six 2’s two 5’s
(105)2 = 105105 = 357357 = 335577
two 3’s two 5’s two 7’s
To prove that the square root of 2 is irrational, let us suppose that 2 is a rational number; that is, suppose that 2 = m/n, where m and n are whole numbers (necessarily greater than 1). Then:
2 = m2/n2
2n2 = m2
Now, imagine that both n and m are written as products of primes where, for instance (using algebraic notation), n = xy while m = zpt. But, as previously noticed, both n2 and m2 must then have either an even number of 2’s or no 2’s. But, in the above equation, the prime 2 appears on the left an odd number of times either once (if n2 has no 2’s) or more than once (if n2 has an even number of 2’s) but, on the right, the prime 2 appears either an even number of times or no times. This is clearly impossible since, given the nature of primes, m2 equates with a number or produces a number that has either an even number of 2’s or no 2’s. A contradiction obtains despite the equals sign. Therefore, what can be wrong? The only thing that can be wrong is our supposition that 2 is a rational number. If this proof is applied to other primes in terms of square roots for 3, 5, 7, . . ., the same dilemma results. (25) Odd clashes with even to demonstrate the irrationality of these numbers. Hence, could all numbers be the kind of numbers that the Pythagoreans had postulated? Are they all rational?
In Boyer’s version of the mathematical proof demonstrating the incommensurableness of the square root of 2 through the contrast between even and odd, he argues as follows: (26)
1. Let d and s respectively signify the diagonal and side of a square and let us assume that they are commensurable: the ratio d/s is rational and equal to p/q, where p and q are integers with no common factors.
2. given the Pythagorean theorem d2 = s2 + s2 reconfigured as d2/s2 = 2 (since d2 = 2s2), if the ratio d/s = p/q (p and q being integers with no common factor), then (d/s)2 = p2/q2 = 2 or p2 = 2q2
3. therefore, p2 must be even since its equivalent 2q2 is divisible by 2 (which corresponds to the definition of an even number as a number divisible by 2).
4. hence, if p2 is even, p is even since p2 when decomposed into constituent prime numbers necessarily includes at least two instances of 2 as both a prime number and a factor, and the presence of 2 in p makes p an even number since it is divisible by 2 (which again corresponds to the definition of an even number).
5. as a result, q must be odd (not divisible by 2) since, according to conditions stated in aforementioned proposition 2, q is an integer with no factors common to p and so it cannot have 2 as a constituent prime factor.
However, letting p = 2r and substituting in the equation p2 = 2q2 with, hence, the result that 4r2 = 2q2, 4r2 = 2q2 as reconfigured becomes q2 = 2r2. Then q2 must be even; hence q must be even (according to the argumentation which had explained why formerly p2 and p must both be even). However, a contradiction follows if one argues that q is both odd and even. No integer can be both odd and even. As a consequence, it thus follows that the numerical relation between d and s is incommensurable. (27) The result is not a definitive whole number.
A third but second species of proof relying on a study and understanding of prime numbers demonstrates the absence of an anticipated whole number by adverting to the relation between d and s. If, indeed, d (a whole number) is decomposed into constituent prime numbers and s (a second whole number) is similarly decomposed, and if no factor is common between them, the improper fraction d/s can never be resolved into a whole number since, in every case, the denominator does not perfectly divide into the numerator to produce an anticipated, desired whole number. The result is always some sort of fraction which, by definition, is not an integer, a whole number.
A geometrical proof that evidences the existence of irrational numbers in general, and not 2 specifically, designates a third species of proof. (28) Its lesser abstractness suggests earlier origins predating the construction of later proofs using other types of arguments. When examining the sides and diagonals of a regular pentagon (defined as a five-sided polygon with all the sides possessing equal length) and the respective relations between s and d, if the diagonals of this pentagon are all drawn, they form a smaller regular pentagon whose diagonals can also be drawn to form a smaller regular pentagon ad infinitum. Hence, pictorially, the relation or ratio of a diagonal to a side in a regular pentagon is indeterminate because it is indefinite. It is irrational. Similarly, if a straight line is divided into two parts and one part is divided into two smaller parts, it will be possible to keep dividing lengths indefinitely. (29) No determinate end is reached. Our expectations meet with frustration as our inquiry encounters mysteries that occasion questions about the adequacy of our intelligible anticipations. What is to-be-known cannot be known too easily or simply.
1. Bernard Lonergan, Insight: A Study of Human Understanding, eds. Frederick E. Crowe and Robert M. Doran 5th ed. (Toronto: University of Toronto Press, 1988), pp. 45-6.
2. Leslie Foster, Rainbow Mathematics Encyclopedia (London: Grisewood & Dempsey Ltd., 1985), p. 43.
3. Foster, p. 43.
4. Foster, p. 43.
5. Foster, p. 43.
6. 6The Penguin Dictionary of Mathematics, 1989 ed. S.v. “integer.”
7. Foster, p. 44.
8. 8E. T. Bell, The Development of Mathematics (New York: Dover Publications, Inc., 1992), p. 61.
9. 9The Penguin Dictionary of Mathematics, 1989 ed. S.v. “ratio.”
10. 10Bell, p. 61.
11. Joseph Flanagan, Quest for Self-Knowledge (Toronto: University of Toronto Press, 1997), p. 33.
12. The Penguin Dictionary of Mathematics, 1989 ed. S.v. “factor.”
13. The Penguin Dictionary of Mathematics, 1989 ed. S.v. “commensurable.”
14. Carl B. Boyer, A History of Mathematics, 2nd ed. (New York: John Wiley & Sons, Inc., 1989), p. 72; Frederick Copleston, S.J., A History of Philosophy, volume 1: Greece & Rome part 1 (Garden City, New York: Image Books, 1962), pp. 49-50; A Concise Oxford Dictionary of Mathematics, 1991 ed., s.v. “Pythagoras,” by Christopher Clapham.
15. The Penguin Dictionary of Mathematics, 1989 ed., s.v. “irrational number.”
16. A Concise Oxford Dictionary of Mathematics, 1991 ed., s.v. “Pythagoras’ Theorem,” by Christopher Clapham.
17. Bell, p. 61.
18. 18Penguin Dictionary of Mathematics, 1989 ed., s.v. “improper fraction.”
19. 19Lonergan, Insight, p. 45.
20. 20Euclid quoted by Walter Fleming and Dale Varberg, College Algebra: A Problem-Solving Approach (Englewood Cliffs, New Jersey: Prentice Hall, n.d.), p. 16.
21. Flanagan, p. 33.
22. 22Boyer, p. 72; Penguin Dictionary of Mathematics, 1989 ed., s.v. “odd number,” and “even number.”
23. Clapham, p. 187.
24. Fleming and Varberg, p. 16.
25. Fleming and Varberg, p. 17.
26. Boyer, pp. 72-3.
27. Boyer, p. 73.
28. Boyer, p. 73.
29. Boyer, p. 51.