Just a quick note. I will be publishing a reflection every Friday at 3 pm. Most of these will be short pointers and thoughts about the writings of Bernard Lonergan.
by David Fleischacker
About two years ago, I started a new notebook on linking together the University and its life with that of the Holy Trinity. One of the areas that I wondered about was whether the Transcendental Notions (TN) could provide any type of analogy for understanding the three persons of the Holy Trinity. There are after all, three transcendental notions that Lonergan develops which are spiritual in nature, hence intrinsically independent of the empirical residue. These spiritual transcendental notions are Lonergan’s transposition of the agent intellect found in Aristotle and St. Thomas, and of the Light of Being (conscience, mind, etc) as found in the Platonists and St. Augustine (as a note, Augustine was clearly not a Platonist once you get into his head more thoroughly even if he learned much from them and borrowed some notions from them).
One of the immediate difficulties of course which one finds noted in Lonergan is that in finding an analogy for the Holy Trinity, we need to deal with acts or operations, not with anything in potency. The TN are a kind of potency, but much different than normal. These actually have the power or capacity to bring about self-transcendence. In St. Thomas (and Aristotle), these “lights” of the mind have the power to illumine, hence they act as agent causes. Most potencies do not have such capabilities. Hence the reason these lights are in a kind of actuality as well. Notice how some of the metaphysical terms and relations get stretched (but not violated! or confused). The TN are in a potency in relationship to the operations that arise, but in relationship to the potencies in the human subject to receive these operations they are in act. Many would say that this imprecision of the metaphysical terms and relations is why one needs to leave out the metaphysical, and turn to intentionality analysis. That is true in part, but if one does so, one as Lonergan notes in Insight, needs to run the full circuit, and return to metaphysics, both to refine the metaphysics, but also to articulate the intelligibilities discovered as belonging to being. To stay merely with a cognitive apprehension of conscious and intentional life leaves one ignorant of its “reality.” So the circuit does need to be run.
The reason I mention the circuit is because if one is to transpose the analogies for the Holy Trinity found in St. Thomas, then one needs to deal with some of the metaphysical points that he makes, such as God is pure act, and hence we need to find analogies in act that help us, and this is true of the Persons as well as of God. The Father is pure act, as is the Son, and the Holy Spirit.
Hence, are the TN in act enough for them to be used as analogies?
The TNs, though in a kind of potency, are also the “light” that makes possible the conscious and intentional operations. This means that in some manner, they are more in act than the operations. They underpin, penetrate, and transcend all operations. Still, there must be a reason that Lonergan did not turn toward these as analogies. He stuck with operations (eg. apprehension of the good, judgment of value of the good, love/decision of/for the good). I suppose one could argue that these operations are in part constituted by the TN, as the TN penetrate them. We could look at what that “penetration” means. It of course is not physical, but spiritual. Descriptively, it “illumines” the operation. It is what “receives” the operation. It is what “beholds” the operation. The TN is not only light, but also an intentional focus, hence can be described as the “eye” of the mind as well. I am tending to think that the TN is both light and eye (hence not distinct as these are physically in us — but I could be wrong). I suppose one could say the “eye” is the conscious subject as awakened in a TN and thus seeking an answer, hence waiting for an operation that mediates the answer. Then once the operation emerges, the subject as beholding the operation in the TN is an eye that beholds. The subject is however conscious through the TN, and thus the TN constitutes both the horizon and the subject as a gazing subject.
One of the areas that I explored a couple years ago in my notebook was whether there was a sufficient distinction and set of relations between the TN to result in some kind of analogy that sheds light upon the Father, Son, and Holy Spirit. Thus, does the TN of intelligibility have a kind of relationship to that of being/truth such that the former begets that latter. Of course, this does not happen without an operation. And it does not happen without the subject moving (raising the question for reflection). Likewise does the TN of goodness spirate from the TN of being? I cannot repeat all of the reflections here, but I can say that my reflections were not conclusive. I do intend however to start publishing these reflections in this particular sequence of blogs.
Even if I discover that those reflections do provide an interesting analogy, there is still the further question about whether the analogy is an improvement upon that of the operations as such. I have a suspicion that they do not, but they might help to deepen my understanding of the operational based analogy (apprehension of the good, judgement of value of the good, decision for the good). Part of my reason for this suspicion is that God as pure act is the cause of the light that is in us, which we call the TNs. The TNs do allow us to grasp the unrestricted nature of the operations in God, but those are operations in God, not TNs. Just a few thoughts.
Why does Jesus need or want us to feed him? It would seem that the only appropriate relation to him is to allow him to feed us. Very true of course. At the same time, from the Cross, he cries out that he thirsts. He thirsts as St. Mother Theresa tells us. Jesus is in those whom we meet, especially the poor and the destitute. All of those who fall under the beatitudes. He thirsts in and through them for us to give him a bit of drink and food. It is part of the immense mystery of being a member of the body of our Lord. He knows us. He knows us in his divinity and he knows us in his humanity. As he hung on the Cross, he proclaimed the thirst of his entire body, as it exists in his mind and heart. This is the meaning of the unity of Christ and his body. In fact, it is a unity that each of us has with each other. When anyone thirsts, and it comes to dwell in us, it then comes to inform us as a constitutive act of meaning. Hence another’s thirst becomes our own. Likewise with Jesus Christ. We are his. And we are in him. He thirsts because we thirst. He thirsts because he became one of us. And as he fills that thirst, so we as part of him are to fill that thirst as well. This is the meaning of to abide and to mutually indwell.
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…
Message from David Alexander: Our next book selection is Beauty for Truth’s Sake: On the Re-enchantment of Stratford Caldecott. I want to suggest everyone procure their copy and we meet for the first discussion on 1/25/2017. The reading for the discussion is the Introduction and the first chapter, pgs. 11- 36.
The book is short, about 150 pages. We could either meet once a month and cover two chapters per session or meet every two to three weeks and cover one chapter of the six chapters per session, whichever you would prefer. Perhaps it will be easier to decide after our first reading, judging from the content.
As a qualification on how we are to understand judgment in Aristotle, please note that, in the kind of analysis which we find in Aristotle and also in the manner of his conceptualization and language, in our acts of judgment thus, a dual nature is distinguished or two natures are indicated in a way which seems to juxtapose one nature with another. Two natures exist instead of one nature. A synthetic, constructive element is alluded to, on the one hand, and, on the other hand, an affirmative, declarative element. Hence, questions exist (later questions were posed) which asked if Aristotle was successful in clearly distinguishing between the being of these two different aspects (existing as two distinct elements, each having its own distinct nature).1 Did he, in fact, clearly distinguish between acts of direct understanding and acts of reflective understanding which exist as acts of judgment since, in Aristotle, judgment engages in two different kinds of tasks. On the one hand, allegedly within our judgments, (1) a composition or a putting together of different concepts occurs or, on the other hand, a separation of concepts when we realize that some concepts should not be combined or joined with each other. If an act of direct understanding (which, as noted, Aristotle conceptualizes as an act of “simple apprehension”) moves through the instrumentality of an imagined fertile, apt image (existing as a phantasm) toward a single, distinct concept or a definition which expresses the fruit or the grasp of one’s prior act of understanding (in Aristotle’s understanding of the nature or the intelligibility of all our direct acts of understanding as we move from the being and the order of sense to the order and the being of understanding: ta men oun eidê to noêtikon en tois phantasmasi noei; the “intellect grasps forms in images”),2 a fortiori, if we should speak in this way about the being of a “simple apprehension,” then, to a greater degree, if we are to speak about how two or more concepts can be put together to reveal a greater unity or a link that exists between these concepts (leading to a larger, more general concept), then, in order to identify and to distinguish this species of intellectual act, we should or we must speak about the being of a “complex apprehension.” These exist allegedly as judgments. These judgments introduce an order which should exist among our ideas and concepts. However, if, for us, the intellectual object is not simply the apprehension of a conceptual complex unity but if, in fact, it is an understanding which wants to declare or know about the reality or the truth of one or more concepts (whether we should speak about simple concepts or about complex concepts), then, within this larger, greater, more demanding context, in Aristotle, a second understanding of judgment presents itself to us in terms of how it seeks to posit a relation or a synthesis which has been grasped by us in our prior acts of understanding. The object here is not essentially a synthesis, the apprehension or the grasp of a synthesis which points to a higher or a wider understanding of things but, instead, the taking of an already understood synthesis and further acts which would work toward an act of understanding which can conclude or move toward a declaration of its reality or a declaration of its truth (or which can deny the factuality of its reality or the factuality of its truth). This is so. This is not so. Either way, in affirmation or negation, a truth is known and it is grasped by us as known. In our awareness, a truth is known in terms of its reasonableness or cogency: hence, its being, its reality. The consciousness or experience that we have of evidence points to the being or the reality of a truth and, as an effect which would thus follow from this, with Aquinas, we would say about ourselves that “knowledge exists as one of the effects of truth” [cognitio est quidam veritatis effectus].3 The one comes from the other.
In Aristotle thus, depending on which passages or texts are being studied, a clear distinction does not exist between that which exists as understanding and that which exists as judgment (acts of direct understanding versus acts of reflective understanding) because judgment, in the language of “composition and division,” resembles acts of direct understanding in terms of the unities which are being grasped and understood by them (by our acts of understanding): unities which transcend pluralities and multiplicities as these exist initially among the givens of the data of our sense perception. However, in Aristotle, the being of judgments is such that they also seek to determine if a correspondence exists between that which exists as a form of mental synthesis within ourselves and that which exists as a species of real synthesis within the being of truly existing things (the being of truly existing objects). A real distinction accordingly exists between the type of answer that is given to this kind of question and the type of answer which is given to a question which asks about how concepts can be related to each other in ways that could lead to the understanding and eventually the expression of a new, more general concept.
On the basis then of this real distinction and as a species of new first principle, in the later work of Aquinas and also in the later work of Bernard Lonergan, clarifications were introduced into the thinking and the conceptuality of Aristotle’s analysis in a manner which attempted to introduce degrees of clarity that had not been too obvious to anyone or to most persons who had attempted earlier to read into the corpus of Aristotle’s philosophy in order to find, within it, a coherent understanding about how things exist within the reality of the world within which we all live (a reality which includes the kind of being which we have and which we are as human beings where our kind of being includes the kind of knowing which belongs to us as human beings and which does not belong to other kinds of living being). From an incoherent understanding about the nature of our human judgment (from an incoherent understanding about the nature of our human cognition), we can thus wonder if, for some in the subsequent history of reflection within philosophy, the result has been a defective, incoherent understanding about the nature of existing things where, in metaphysics, we turn to this science in order to move toward a comprehensive or a general understanding about the nature of all existing things qua the nature of being in general as it applies to all things which enjoy some form of real existence. What can be implied about the nature of our world if our point of departure is a particular belief or a particular understanding about the nature of our human knowing, an understanding which could be lacking in the degree of rationality which should belong to it?4
1Lonergan, Verbum, pp. 61-62.
2Aristotle, De Anima, 3, 7, 431b, as cited by Sala, Lonergan and Kant, p. 161, n. 72.
3Aquinas, De veritate, q. 1, a. 1, as cited by Sala, Lonergan and Kant, p. 147, n. 71.
4Randall, Aristotle, p. 6.
Notes on Aquinas, Summa Theologiae 1a, q. 84
ST 1.84.1 Does the soul know bodies through the intellect
Objection 1: Bodies are understood by the senses, incorporeals by the intellect
Reply: This refers to the medium of knowledge, not the object
Objection 2: Sense:Intelligible :: Intellect:Sensible. Since senses can’t understand intelligible things, the intellect can’t understand sensible things
Reply: Intellect is a higher power than sense. While sense cannot know intelligibles, intellect can know sensibles. Otherwise God wouldn’t have knowledge of sensibles.
Objection 3: Intellect understands eternal and unchangeable things, while bodies are always changing
Reply: Every movement presupposes something immovable. Changing form requires unchanging matter. Changing matter requires unchanging form. Socrates may not always be sitting, but it is always true that when he does sit he stays in one place.
On the contrary: natural science, with its studies of motion and matter, would be impossible if the intellect had no knowledge of changeable things.
One possible answer: Certain Presocratics (Heraclitus) thought everything was in a state of flux, so certain knowledge of anything was impossible.
Refutation: Heraclitus was only led to this conclusion because of their reductive materialism.
Another possible answer: Plato’s theory of forms
Whiteness is in bodies in different modes of intensity
The senses receive whiteness without receiving matter
The intellect receives whiteness under conditions of immateriality and immobility
ST 1.84.2 Does the soul know bodies through its essence?
It would seem so. Knowledge is by assimilation, where the intellect “becomes” whatever essence it is to know. Thus, the intellect knows all things by the essence it has assimilated, which essence has become the intellect’s own essence.
But Augustine says we know things through our senses.
Presocratic view: the material intellect receives forms materially
Platonist view: The immaterial soul receives forms immaterially
Only God knows things through his own essence; humans and angels don’t
ST 1.84.3 Does the soul know through innate species?
Our knowledge is different from angelic knowledge. Our knowledge must be brought into act, while angelic knowledge is always in act. Moreover, the angelic intellect is completed by the angelic form and is not in potentiality with respect to anything. The human intellect is in potentiality with respect to things it does not know. Prime matter is in potentiality with respect to its substantial form.
ST 1.84.4 Are intelligible species derived by the soul from separate forms?
The objections proceed by analogy from sense, also noting that the intellect requires something actually intelligible in order to be brought into act.
But if we know by separate forms, there is no reason why our souls should be united to bodies. Both Plato and Avicenna’s views are considered and deemed insufficient for explaining the existence of the body.
In replying to the objections, St. Thomas says there is no analogy between sense and intellect. He concedes that divine ideas are the ultimate source of our knowledge, but we attain to this knowledge by the actualization of phantasms by agent intellect.
ST 1.84.5 Does the intellect know material things in the eternal types?
St. Thomas states that Augustine was “imbued with the doctrines of the Platonists.”
St. Thomas makes a distinction:
The human intellect requires both intelligible species and intellectual light in order to know.
St. Thomas shows that Augustine was not as Platonist as he originally seemed.
ST 1.84.6 Is intellectual knowledge derived from sensible things?
Augustine’s argument that intellectual knowledge does not come from the senses
Democritus thought all knowledge was by a discharge of atoms
Plato thought the soul forms within itself the species, after the sensible organ receives the sensible object. The soul is roused by the species of the thing.
Augustine said the body was the messenger to the soul.
The senses rouse the intellect to the act of understanding.
Aristotle said sense is an act not of the soul alone, but of the “composite.” The sensible is received in the sense by a discharge or some other operation. Then agent intellect derives the phantasm and ultimately the intelligible species from the sensory input.
Sensible things are only the material cause of knowledge. Agent intellect is also required.
ST 1.84.7 Can we understand without turning to phantasm?
When knowledge is not actualized, intelligible species dwell in the passive intellect.
The intelligible species is NOT the likeness of the individual thing.
Our intellect’s proper and proportionate object is the nature of a sensible thing (?)
ST 1.85.1: Does our intellect understand corporeal and material things by abstaction from phantasm?
3 grades of cognitive powers:
2 types of abstractions
3 types of matter:
(things like “being,” “unity,” “power,” “act” abstracted from all matter)
“The phantasms are made more fit for the abstraction therefrom by intelligible intentions.”
ST 1.85.2: Are intelligible species the object of our intellection?
No, intelligible species are the species qua, not species quae intelligitur.
Otherwise, science would be about words, not things.
Also, we would have no way of judging the truth or falsity of our ideas.
Intelligible species is a likeness of the thing understood, as heat in the heater is a likeness of the thing heated
Abstract universal exists in the mind, but the universal which is understood exists in its instantiations.
2 operations of the sensitive part:
2 operations of the intellect
ST 1.85.3 Whether knowledge of the most universal species is prior?
Sense knowledge is prior to intellectual knowledge. Therefore in some manner we know the particular first.
However, in the intellect we know what animals are, albeit indistinctly, before we can distinguish rational animals from non-rational animals.
In the senses, we know something is a body before it gets close enough to be known as an animal. A child calls all men father before he learns to distinguish his father. Therefore, universal sense knowledge is prior.
Something’s genus is derived from its common matter; the species comes from the form. Nature ultimately intends the species, not the genus.
ST 1.85.4 Whether we can understand multiple things at once?
No, just as an object cannot have multiple colors at once, an intellect cannot have multiple intelligible species at once
We can, however, understand multiple things in a single species
ST 1.85.5 Whether our intellect understands by composition and division?
Yes, because our intellects achieve perfection by degrees. We must make a judgement (i. E. a composition or division) about the state of our knowledge in order to attain more perfect knowledge.
Intellectual operations are in time insofar as we have to turn to phantasms
2 types of composition in a material thing
2 types of composition in the intellect
ST 1.86.6 Can the intellect be false?
The intellectual light cannot err, because the proper object of intellect is the quiddity of the thing. Provided we are abstracting quiddities from phantasm or deriving first principles from the operation of the intellect itself, the intellect cannot be false.
In composition or division, the intellect can err.
ST 1.86.7 Can one man understand better than another?
ST 1.86.8 Is the indivisible known before the divisible?
by David Fleischacker
Further, love is the act of a subject (principium quod), and as such it is the principle of union between different subjects. Such union is of two kinds, according as it emerges in love as process to an end or in love in the consummation of the end attained. The former may be illustrated by the love of friends pursuing in common a common goal. The latter has its simplest illustration in the ultimate end of the beatific vision, which at once is the term of process, of amor concupiscentiae , and the fulfilment of union with God, of amor amicitiae (“Finality, Love, Marriage,” 24)
Though there is more to say on finality, I am now turning attention to the meaning of love within the 1943 essay “Finality, Love, and Marriage.” On an initial review, and I think final as well, Lonergan was only beginning to move into a deeper explanatory account of love in 1943. His use of terms derived from faculty psychology and his notion of appetite illustrate this beginning. We must remember however that the use of faculty psychology does not make something false. What happens once one shifts into intentionality analysis is a transposition which sometimes results in a translation of a term into the intentional framework and, at others, an elimination of a term. For example, I would argue that the potential intellect gets translated into the capacity for self-transcendence, and hence expanded and united within the light of all the transcendental notions. Likewise, the agent intellect becomes translated into the transcendental notions, and thus more adequately expanded as well. Thus, Lonergan’s formulation of love in 1943, even if in faculty psychology, can be transposed, something which Lonergan had done by the time he wrote Method in Theology.
First, let’s look carefully at the 1943 text. This section is titled “The Concept of Love.” Notice Lonergan is using the term concept. However, in his opening line, he identifies love as utterly concrete.
The difficulty of conceiving love adequately arises from its essential concreteness and from the complexity of the concrete.(23)
Love is neither a concept or an abstraction, but of course in talking about it, one does have to conceive it.
In conceiving of love, Lonergan develops four aspects, the first two dealing with the nature and act of love itself, and the second dealing with the subject who loves. The first two clearly are formulated within faculty psychology. Love is an act of a faculty. A faculty is a kind of power that is constitutive of what a living thing is. It gives the living thing the ability to carry out certain type of operations. To get an insight into a faculty, one has to carefully analyze a whole landscape of operations and then in examining the operations, discover fundamental characteristics that unite those operations. So, seeing, hearing, tasting, touching, and smelling all have a material element to them, such that the very operation itself regards a spatial-temporal element. As well, these sensate operations allow one to be present and conscious of sense objects. And hence recognizing that all of these sense activities both have a conscious element and a material element would allow one to then formulate a common power or capacity that one has in these types of activities. This becomes the source of the insight into a particular faculty or power. Other operations transcend certain material limitations, and the principle examples of this are the activities of understanding and knowledge. One can posit a common power or faculty to these spiritual (non-material activities), such as the faculty if the intellect. Now on to each of the four aspects.
First Aspect: Love as an actuation of a faculty
Lonergan formulates love as a realization or actuation of faculty. Specifically, it is a faculty of appetite, and love is the central appetite – “it is the pure response of appetite to the good” (23) Other responses are derivative – desire, hope, joy, hatred, aversion, fear, and sadness. Hope is the expectation to become present to that which is love. Hatred is toward that which has harmed the good that is loved. Fear arises in response to the possible loss of the good that is loved. Sadness is the response to that good as lost. Joy is the enjoyment of the good as present. Love is key. It is central. There is nothing false in formulating love in this manner. Identifying it with a faculty, and a fundamental appetite is to recognize that it is a real power or capacity of the human person.
Second Aspect: Love of a beloved as first principle
The second aspect is that it is the principle – “the first in an ordered series” – that initiates a process to its end, which is that which is loved. One can think of simple vital desires for example. The desire for food is not only the “form” of the end process by which one goes out to find, hunt, or grow food, but it is the first principle of that entire process, and it has as its object the end, the food itself. In the case of love it is the beloved. The beloved becomes the first principle that moves the person in love to the beloved.
Third Aspect: Unification of subjects toward an end
The third aspect highlights that the act of love, the act of this fundamental appetite, this first principle of movement to the beloved as term, bonds the subjects who are in love based upon their common pursuit of an end. Those who have not yet reached the end, and rather are still in pursuit of it, become bound when pursing that end collaboratively. Lonergan draws this out further through Aristotle’s notion of friendship in a later section of his essay. Notice that here, Lonergan does not specify the end that is pursued, because any good ends pursued can unite individuals to each other. This pursuit also perfects the human subjects as such, and thus bonds them to each other for each other, but that is the point of the next aspect.
Fourth Aspect: Love of Beloved as United, as Consummated
The fourth aspect highlights that love as realized unites subjects as mutual persons who enjoy the good that each is, a mutual unity that is based upon the good that each person is and has become. The ultimate example of this aspect that Lonergan identifies is the beatific vision, “which at once is the term of process… and the fulfillment of union with God” (24).
It is important to note that Lonergan says these are simultaneous aspects (23). The differences between each is a different focus upon what is “utterly concrete.” By simultaneous he means that one does not happen without the other, even if the individuals involved may be focusing in upon one of the aspects and not the others.
Contrast to love in Method in Theology
There is not only a clear difference of words between 1943 and 1972, but a clear difference in scope. Lonergan by 1972, was able to formulate love in terms of insights that he had into the structure of consciousness, specifically in terms of the capacity for self-transcendence, and the different states of being of that capacity. One not only has the notion of potency in a capacity, but it is a potency that has a directly relationship to states (which is derived from statistical notions – the difference between actual frequencies from ideal frequencies gives one an understanding of the state of something), and it includes a clear differentiation of the notions that constitute the capacity as a whole – the transcendental notions. Lonergan thus could formulate love not as merely an actualization of a faculty, but one might say the actualization of the faculty of all faculties, the base of all bases. Love is basic because it orientes all levels of consciousness. All the questions that one pursues are guided by that which one loves. In other words, the state of being orients all the operators of human development at all levels of conscious intentionality. Love is the actuation of the capacity for self-transcendence, and the more profound it is, the more it underpins, penetrates, and transforms all of one’s horizon.
This does not negate the insights Lonergan had in 1943, but it does formulate these insights more clearly, and it expands upon what he understood of love. It is still utterly concrete, and so concrete that nothing that human beings do escape it, because even getting up in the morning means there is some basic actuation of the capacity, some basic state of one’s being. It is an actuation of a kind of faculty, but not just among others. Rather, it regards the capacity for any human intentional operator and operation. It is a central appetite, but it is also a the central finality of all human activities. The transposition of faculty psychology into intentionality analysis reinforces what Lonergan says about love in 1943 and expands it. Furthermore, the last two aspects can be understood more deeply. When one understands that love is a realization of the capacity for self-transcendence, and that all other operators and operations thus emanate from this realization, then one comes to understand the more comprehensive scope upon which subjects can be bound to each other both as they self-transcend, and as they reach the fulfillment of their self-transcendence. This is especially true when one transposes the beatific vision into a perfection of the human capacity for self-transcendence by the gift that is the Transcendent, the ultimate meaning and ultimate value because the Transcendent is the only true realization of the capacity. Lonergan’s reflections upon Christology and Trinitarian theology draw this out even more (and one might add his work on grace).
Just a few things to think about as we start this exploration on Lonergan’s notion of love in “Finality, Love, and Marriage.”
Square root of two as an irrational number
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.