I hope that this Sampler will a) demonstrate the breadth and depth of the knowledge that Lonergan brought to Macroeconomic Dynamics, and b) inspire readers to compare their perspective to his in regard to science and macroeconomics. His thinking ranged over mathematics, natural science, method, history, philosophy, theology, and art. This sampler is arranged in groups of excerpts from particular books of Lonergan. Scroll down to see the arrangement.
CWL 3, Insight: A Study of Human Understanding
“The most significant book of the twentieth century,” (Philip McShane)
A distinction has been drawn between description and explanation. Description deals with things as related to us. Explanation deals with the same things as related among themselves. … description supplies, as it were, the tweezers by which we hold things while explanations are being discovered or verified, applied or revised. … [CWL 3, 291/316]
The concrete intelligibility of Space is that it grounds the possibility of those simultaneous multiplicities named situations. The concrete intelligibility of Time is that it grounds the possibility of successive realizations in accord with the probabilities. In other words, concrete extensions and concrete durations are the field or matter or potency in which emergent probability is the immanent form of intelligibility. (CWL 3, 172)
Now, at the root of classical method there are two heuristic principles. The first is that similars are understood similarly, that a difference of understanding presupposes a significant difference of data. The second is that similarities, relevant to explanation, lie not in the relations of things to our senses but in their relations to one another. Next, when these heuristic principles are applied, there result classifications by sensible similarity, then correlations, and finally the verification of correlations and of systems of correlations. But verified correlations necessarily involve the verification of terms implicitly defined by the correlations; and they do not involve more than such implicitly defined terms as related, for what is verified accurately is not this or that particular proposition but the general and abstract proposition on which ranges of ranges of particular propositions converge. … there is a fundamental heuristic structure that leads to the determination of conjugates, that is, of terms defined implicitly by their empirically and explanatory relations. Such terms as related are known by understanding, and so they are forms. Let us name them conjugate forms.[CWL 3, 435/460]
So we are brought to the profound disillusionment of modern man and to the focal point of his horror. He had hoped through knowledge to ensure a development that was always progress and never decline. He has discovered that the advance of human knowledge is ambivalent, that it places in man’s hands stupendous power without necessarily adding proportionate wisdom and virtue, that the fact of advance and the evidence of power are not guarantees of truth, that myth is the permanent alternative to mystery and mystery is what his hubris rejected. ¶ The real issue, then, is truth. Though it has concerned us all along, it will not be amiss to bring together at least the main points made on different occasions and in different chapters. Accordingly, we distinguish
- the criterion of truth,
- the definition of truth,
- the ontology of truth,
- truth in expression,
- the appropriation of truth, and
- the truth of interpretation. (CWL 3, 549/572-73)
The definition of truth was introduced implicitly in our account of the notion of being. For being was identified with what is to be known through intelligent grasp and reasonable affirmation; but the only reasonable affirmation is the true affirmation; and so being is what is known truly. Inversely, then, knowing is true by its relation to being, and truth is a relation of knowing to being. (CWL 3, 552)
Now the principles and laws of a geometry are abstract and generally valid propositions.
It follows that the mathematical expression of the principles and laws of a geometry will be invariant under the permissible transformations of that geometry. … Such is the general principle and it admits at least two applications. In the first application one specifies successive sets of transformation equations, determines the mathematical expressions invariant under those transformations, and concludes that the successive sets of invariants represent the principles and laws of successive geometries. In this fashion one may differentiate Euclidean, affine, projective and topological geometries. … A second, slightly different application of the general principle occurs in the theory of Riemannian manifolds. The one basic law governing all such manifolds is given by the equation for the infinitesimal interval, namely,
ds2= Σgijdxidxj [i, j = 1,2…n]
where dx1, dx2… are differentials of the coordinates, and where in general there are n2products under the summation. Since this equation defines the infinitesimal interval, it must be invariant under all permissible transformations. However, instead of working out successive sets of transformations, one considers any transformations to be permissible and effects the differentiation of different manifolds by imposing restrictions upon the coefficients. This is done by appealing to the tensor calculus. … Thus in the familiar Euclidean instance, gij is unity when i equals j; it is zero when i does not equal j; and there are three dimensions. In Minkowski space, the gij is unity or zero as before, but there are four dimensions, and x4 equals ict. In the General Theory of Relativity, the coefficients are symmetrical, so that gij equals gji; and in the Generalized Theory of Gravitation, the coefficients are anti-symmetrical. [CWL 3, 146 -147/170-71] [#41]
The point we are endeavoring to make, within the limits of our narrow premise, is that the notion of emergent probability is explanatory. (CWL 3, 124)
The abstract formulation of the intelligibility immanent in Space and in Time will be one of the possible sets of definitions, postulates, and inferences that systematically unify the relations of extensions and durations. All such possible sets of definitions, postulates and inferences are geometries. Therefore, the abstract formulation of the intelligibility immanent in Space and in Time will be a geometry. ¶ The expressions of the principles and laws of any geometry will be invariant. For principles and laws are independent of particular places and times, and so their proper experession cannot vary with variations of spatio-temporal standpoints. (CWL 3, 150)
It will be a basic position,
- if the real is the concrete universe of being and not a subdivision of the ‘already out there now’;
- if the subject becomes known when it affirms itself intelligently and reasonably and so is not known yet in any prior ‘existential’ state; and
- if objectivity is conceived as a consequence of intelligent inquiry and critical reflection, and not as a property of vital anticipation, extroversion and satisfaction
… On the other hand, it will be a basic counter-provision, if it contradicts one or more of the basic positions. … All counter-positions invite reversal. For any lack of coherence prompts the intelligent and reasonable inquirer to introduce coherence. But counter-positions, though coherent with one another, though the insertion of their symbolic equivalents into an electric computer would not lead to break-down, none the less are incoherent with the activities of grasping them intelligently and affirming them reasonably. For these activities contain the basic positions; and the basic positions are incoherent with any counter-position. … The only coherent way to maintain a counterposition is that of the animal; for animals not only do not speak but also do not make excuses for their silence. [CWL 3, 388/413] [#39]
Our analysis … acknowledged the existence of schemes of recurrence in which a happy combination of abstract laws and concrete circumstances makes typical, further determinations recurrent, and so brings them under the domination of intelligence. Moreover, it acknowledged that concrete patterns of diverging series of conditions are intelligible; granted both the requisite information and mastery of systematic laws, it is possible in principle to work from any physical event, Z, through as many prior stages of its diverging and scattering conditions as one pleases; and it is this intelligibility of concrete patterns that grounds the conviction of determinists, such as A. Einstein. … However, we agree with the indeterminists inasmuch as they deny in the general case the possibility of deduction and prediction. For while each concrete pattern of diverging conditions is intelligible, still its intelligibility lies not on the level of the abstract understanding that grasps systems of laws but on the level of the concrete understanding that deals with particular situations. Moreover such concrete patterns form an enormous manifold that cannot be handled by abstract systematizing intelligence for the excellent reason that their intelligibility in each case is concrete. There results the peculiar type of impossibility that arises from mutual conditioning. Granted complete information on a totality of events, one could work out from knowledge of all laws the concrete pattern in which the laws related the events in the totality. Again, granted knowledge of the concrete pattern, one could use it as a guide to obtain information on a totality of relevant events. But the proviso of the first statement is the conclusion of the second; the proviso of the second statement is the conclusion of the first; and so both conclusions are merely theoretical possibilities. For the concrete patterns form a non-systematic aggregate, and so it is only by appealing to the totality of relevant events that one can select the concrete pattern; on the other hand, the relevant totality of events is scattered, and so they can be selected for observation and measurement only if the relevant pattern is known already [CWL 3, 650/672-73]
For one misses the real point to efficient causality if one supposes that it consists simply in the necessity that conditioned being becomes virtually unconditioned only if its conditions are fulfilled. On that formulation, efficient causality would be satisfied by an infinite regress in which each conditioned has its conditions fulfilled by a prior conditioned or, perhaps more realistically, by a circle illustrated by the scheme of recurrence. However, the real requirement is that, if conditioned being is being, it has to be intelligible; it cannot be or exist or occur merely as a matter office, for which no explanation is to be asked or expected, for the non-intellwble is apart from being. … (CWL 3, 655-56
The real is being. (CWL 3, 673)
… every counter-position leads to its own reversal; for it is involved in incoherence as soon as the claim is made that it is grasped intelligently and affirmed reasonably; and an intelligent and reasonable subject cannot avoid making that claim. (CWL 3, 673)
Our subject has been the act of insight or understanding, the eternal rapture glimpsed in every Archimedean cry of Eureka. Understanding meets questions for intelligence and questions for reflection. The unrestricted act meets all at once; for it understands understanding and all the intelligibility based on it; and it understands its own understanding as unrestricted, invulnerable, true. What is known by true understanding is being, and the being known by understanding’s self-knowledge is primary being, self-explanatory, unconditioned, necessary without any lack or defect. … God is without defect, not because the act of understanding is complemented by further acts, but by a single act that at once is understanding and intelligible, truth and affirming, goodness and loving, being and omnipotence. (CWL 3, 684)
In man there are three levels of development, namely, the biological, the psychic, and the intellectual. So one may consider
- any level in itself
- any level in its relations to other levels
- the harmonious or conflicting process of development on all three levels in any individual
- the cumulative, historical process of development in a multiplicity and succession of individuals
Clearly, the only complete consideration is the fourth. … (CWL 3, 741/ )
the relentless modern drift to social engineering and totalitarian controls [CWL 3, 745/ ]
the operative principle in the breakdown and disintegration of civilizations [CWL 3, 747/ ]
CWL 21, For a New Political Economy
[7/22/19] … real analysis (is) identifying money with what money buys. … And that is the source of the problem in real analysis. If you want to treat money that doesn’t make a difference, you can have a beautiful liberal monetary theory. But it doesn’t say the way the thing works. [CWL 21, xxviii] [4]
Our aim is to prescind from human psychology that, in the first place, we may define the objective situationwith which man has to deal, and, in the second place, define the psychological attitude that has to be adopted if man is to deal successfully with economic problems. Thus something of a Copernican revolution is attempted: instead of taking man as he is or as he may be thought to be and from that deducing what economic phenomena are going to be, we take the exchange process in its greatest generality and attempt to deduce the human adaptations necessary for survival. [CWL 21,42- 43]
the real issue is the value of the dummy (money in divided exchange rather than barter)… the relative value is its usefulness….the scarcity of the dummy is attended to by the technicians of the technical rules governing its issuance. Whether it issues from the printing press or from the credit structure makes no difference. The economic value lies in the human effort against scarcity… the exchange value is the ratio or proportion in which are exchanged the different categories of objects for which men strive because they are useful and scarce….It is now necessary to state the necessary and sufficient condition of constancy or variation in the exchange value of the dummy. To this end we compare two flows of the circulation: the real flow of property, goods, and services, and the dummy flow being given and taken in exchange for the real flow….Accordingly, the necessary and sufficient condition of constant value in the dummy lies in its concomitant variation with the real flow….More briefly, if there is concomitance between the two flows, then the proportion in which dummies and goods exchange remains the same. If there is lack of concomitance, then this proportion changes. But exchange value is a proportion. Therefore, the concomitance of the two flows is the condition of constant exchange value. (CWL 21, 38-39)
The alternative to constant value in the dummy is the alternative of inflation and deflation. Of these famous twins, inflation swindles those with cash to enrich those with property or debts, while deflation swindles those with property or debts to enrich those with cash; in addition to the swindle each of these twins has his own way of torturing the dynamic flows; deflation gives producers a steady stream of losses; inflation yields a steady stream of gains to give production a drug-like stimulus. [CWL 21, 37-38]
Now if continuity is defined by equality of sales at the final markets in successive instants or turnovers, the necessary and sufficient condition of continuity is that
DA’ = DA’
and
DA” = DA”.
Since equations (8) DE’ + DE” = DI’ + DI” and (9) G’DI’ = G”DI” are independent, it follows that continuity in its simplest form has a twofold condition. First, from equation (8), total primary and secondary income must be spent. Second, from equation (9), primary income moving to the secondary final market must equal secondary income moving to the primary final market. (CWL 21, 47-48)
An initial and provisional theorem of continuity was enounced in a preceding chapter (§24). Now it may be indicated in its full generality. ¶ The analysis has revealed that the economic system is a pattern of aggregate dynamic relationships arranged in different kinds of velocity and accelerator rhythms. In the real order there are the primary and secondary rhythms, with the former accelerated by the latter. In the monetary order there are the rhythms of excess release from the redistributional area to the primary and secondary rhythms; and again, the former accelerate the latter. ¶ Now the general theorem of continuity is that this complex machine has a nature that must be respected. Absolutely, there is no necessarily right value for the monetary accelerators DT’, DT”, DC’, DC”; again, absolutely, there is no necessarily right values for the six multipliers C’, C”, T’, T”, G’, G”. But what is true is this: as soon as a few of these are determined, the rest become determined within narrower limits, for all form part of an organic whole; to violate this organic interconnection is simply to smash the organism, to create the paradoxical situation of starvation in the midst of plenty, of workers eager for work and capable of finding none, of investors looking for opportunities to invest and being given no outlet, and of everyone’s inability to do what he wishes to do being the cause of everyone’s inability to remedy the situation. Such is disorganization. Continuity, on the other hand, is the maintenance of organization, the stability of the sets and patterns of dynamic relationships that constitute economic well-being in a society. ¶ While the provisional theorem of continuity (§24) did regard the static phase, it is important to observe that the general theorem regards any phase. There is a general historical movement of ideas, opportunities, and decisions integrating into that major rhythm in which transformations are followed by exploitations only to bring forth new and deeper transformations. Within this broad historical scheme of things, the role of any age, and still more of any country, is but a small thing: the past was settled by our forebears, and the future will be in the hands of posterity; only the present is ours, and it is only within the limitsthat we make of the present what we wish. Our starting pint is already determinate: we have to face things as they are; we may never lose sight of them or attempt to reckon without them. But not only is there ever a broad and unalterable datum of things as they are; there are also the limitations which this datum imposes on things as we are going to make them. ¶ The theorem of continuity is the abstract and formal aspect of such limitations in the economic order. At the moment the exchange process is static or expanding or contracting. We may like it so or we may wish it different. But in any case there is some determinate range of values of the multipliers and of the monetary accelerators – of C’, C”, T’, T”, G’, G”, of DC’, DC”, DT’, DT” – that corresponds with such a decision. Moreover there has to be an internal coherence between these values, and to violate this coherence is to rout economic organization. Just as the movements of the controls of an airplane must be coordinated and all coordinations are not possible at all instants, so also he economic machine as its controls, which can be moved only in concert and only in a limited number of ways at any given time. ¶ Such is the general theorem of continuity. In the abstractand in a general way, it affirms that the economic process can proceed only within the limits of equilibrium of the various phases. To step outside them is to bring about a general breakdown. (CWL 21 73-5)
CWL 15, Macroeconomic Dynamics: An Essay in Circulation Analysis
Now, when the surplus stage of the process is effecting a long-term acceleration of surplus activity but as yet not affecting basic activity, one may expect successive values of Q” to increase in geometrical progression. This gives an initial period, in which the graph of dQ”/Q” is approximately a level straight line. Next, as the surplus expansion develops and devotes more and more of its activity to the long-term acceleration of the basic stage, one may expect no more than a uniform acceleration of the surplus stage. This gives a second period in which dQ”/Q” is curving downwards with successive values in decreasing geometrical progression. Thirdly, as the expansion approaches its maximum in the surplus stage, dQ” reverts to zero and Q” becomes constant. In this third period dQ”/Q” is again a level straight line but now coincidental with the x-axis; h” is zero, but h’Q” may be great for a notable period to effect a long-term acceleration of the basic stage which, however, gradually declines as replacement requirements begin to mount. [CWL 15, 122-24]
Now, when the acceleration of Q” is uniform, the long-term potential of the surplus stage is increasing, and so the surplus stage is devoting more and more of its efforts to the long-term acceleration of the basic stage; then Q’ will be increasing at an increasing rate, and time series of its values may stand in a geometrical progression to make dQ’/Q’ a level straight line. When, however, Q” becomes constant, the acceleration of Q’ becomes uniform, and then dQ’/Q’ will curve downwards; and as replacement requirements begin to mount, this downward curve is accentuated until dQ’ reverts to zero. Thus, both dQ”/Q’ and dQ’/dQ’ are described as initially straight level lines that eventually curve downwards till the acceleration ratios become zero. One may well ask for an account of the movement of the acceleration ratios from their initial zeros to the level straight lines. ¶ There are two factors in such a movement: short-term acceleration and the period of generalization of a long-term acceleration. … [CWL 15, 126]
Lonergan’s meaning of ‘exploitation’ in this sentence moves in just the opposite direction from Marx’s usual meaning, … ¶Exploitation for Marx generally refers to the systemic relations of production that regularly cause the expropriation of the surplus part of the labor value produced by the workers so that it becomes the profits of capitalists, while the workers’ standard of living collectively approaches the mere subsistence level. That Lonergan was quite aware of Marx’s position is shown from a supplement handed out in his class in 1979: ‘We, on the other hand, have to distinguish basic and surplus [prices and quantities], P’ and P”, Q’ and Q”. for unless surplus is conceived as a distinct circuit with its own final market, the Marxists object that the basic final market is demanding payment not only for basic goods and services but also for surplus as well; hence the accusation [that] profit is robbing workers of part of the labor value of their ‘contribution.’ ¶ In the present context of Lonergan’s presentation of the surplus stage of the pure cycle of production, the point is that the new basis for overall production of the standard of living of the total economy brought about by the significant addition of more and/or new plant and equipment is to be exploited in the widening and deepening of basic production that raises the standard of living of everyone, but especially of the workers (or, as he used to say in class, ‘of widows and orphans’). The whole point of Lonergan’s analysis, then, is to emphasize what he also makes clear in the 1979 page mentioned above: ‘And now with the circuits distinguished, the crossover makes it manifest that it supplements the wages paid in the basic circuit, so that profits are not robbery and there is no need for the gifts of bank credit to supplement workers’ basic wages.’ (for a fuller quotation, see note 87 below) (CWL 15, 33-34, ftnt. 34)
Need the moral be repeated? There exist two circuits, each with its own final market. The equilibrium of the economic process is conditioned by the balance of the two circuits: each must be allowed the possibility of continuity … what cannot be tolerated, much less sustained, is for one circuit to be drained by the other. That is the essence of dynamic disequilibrium. [CWL 15, 175]
CWL 5, Understanding and Being
Newton set down laws of motion and proceeded to demonstrate that if a body moves in a field of central force, its trajectory is a conic section. He set out with a minimal cluster of insights, definitions, postulates, axioms and proceeded to account for the laws that had previously been empirically established, bringing them into a single explanatory unity. ¶A single insight yields a conception, a definition, an object of thought; but from a cluster of insights, you build up a system of definitions, axioms, postulates, and deductions. We have to note that a system is quite an achievement; systems are not numerous. [CWL 5, 52]
There’s the good of order. The good of order is very well represented negatively by an economic depression. In an economic depression there is no lack of material, there is no lack of capital, there is no lack of people willing to work, and there is no lack of people willing to buy, but things don’t run, they don’t work. You can prime the pump, and you get a single burst of water going round, but it doesn’t keep going round as when the economic system is functioning properly. You can keep priming and repriming and build up an enormous national debt, and things still aren’t clicking. … What is lacking is a good of order. And what is this good of order? It is the dovetailing of one thing with another that you have when the economic system is functioning properly. A makes shoes, B wants shoes and makes something else, and the whole thing clicks together and it works. The difficulties, of course, why the thing periodically does not work, the reasons for that are more complex. But there is a good of order. … Again, you have the good of order in a family, a family as an institution. … There is a good of order of the polity, the state. These objective schemes of recurrence – you have a good breakfast this morning, you have a good breakfast the next morning, and you keep on having a good breakfast every morning; you have a job to do that interests you today, and you’ll have it tomorrow and so on. … and the good of order is when the whole thing clicks together. It is good in the intelligible sense. It is the object of an insight. (CWL 5, 378-379; Please read these and following pages in CWL 5 in full.)
Method in Theology
A theology mediates between a cultural matrix and the significance and role of a religion in that matrix. The classicist notion of culture was normative: at least de jure there was but one culture that was both universal and permanent … Besides the classicist, there also is the empirical notion of culture. It is the set of meanings and values that informs a way of life. It may remain unchanged for ages. It may be in process of slow development or rapid dissolution … When the classicist notion of culture prevails, theology is conceived as a permanent achievement, and then one discourses on its nature. When culture is conceived empirically, theology is known to be an ongoing process, and then one writes on its method…. [CWL 14,1971, xi]
The question of God, then, lies within man’s horizon. Man’s transcendental subjectivity is mutilated or abolished, unless he is stretching forth towards the intelligible, the unconditioned, the good of value. The reach, not of his attainment, but of his intending is unrestricted. There lies within his horizon a region for the divine, a shrine for ultimate holiness. It cannot be ignored. The atheist may pronounce it empty. The agnostic may urge that he finds his investigation has been unconclusive. The contemporary humanist will refuse to allow the question to arise. But their negations presuppose the spark in our clod, our native orientation to the divine. [CWL 14,1971, 103]
… we propose to describe briefly eight functional specialties in theology, namely, 1) research, 2) interpretation, 3) history, 4)dialectic, 5) foundations, 6) doctrines, 7) systematics, and 8) communications. …. [CWL 14,1971, 127]
CWL 11, 1970; The Triune God; Doctrines
… just as (John) declared life, truth, and love to be from God and represented them as being in the realm of light, so he declared hate, mendacity, and murder to be from the devil, attributed them to the world, and represented them as being in the realm of darkness. Hence, we must recognize in John a ‘world view’ that assigns human affairs to two opposite orders. Although these orders are represented symbolically by light and darkness and neatly attributed either to god or to the devil and the world, … (CWL 11, 655)
No less than scientific language, symbolic language intends a truth yet can be wrong. Just as there are truths known scientifically, so also are there truths communicated symbolically; and just as there are scientific opinions that have long been totally abandoned, so also are there myths … things have only one way of being, but humans do not have only one way of knowing. One who knows scientifically knows universally; but this universality does not belong to things (as if universals subsisted) but to the scientific way of knowing. One who apprehends and speaks symbolically is using a way of knowing that is full of vivid imagery and feeling, but nevertheless by reflecting one can distinguish between what are to be attributed to the things themselves and what are to be ascribed to this way of knowing. (CWL 11, 381)
Truths that are linked to one another can be set forth in two orders. We can begin from those truths that are prior with respect to us in order to arrive at those that are prior in themselves; and conversely we can begin from what are prior in themselves to arrive at what are prior with respect to us. The former order is called the order of discovery, of resolution into causes, or analytic; the latter is called the order of teaching, of composition from causes, or synthetic. CWL 11, 415
CWL 12, 1970, The Triune God; Systematics
Questions cannot be put in any order whatsoever. Some questions simply cannot be answered until others have been resolved. And sometimes the answers to one question immediately provide the answers to others. [CWL 12, 23]
… the questions are put in such an order that, once the first is solved, the solutions to the others follow with almost no difficulty. Therefore, because the later solutions are connected to the first as conclusions are connected to some principle, all solutions after the first seem to be the proper province of knowledge. [CWL 12, 25]
… if solving the first problem virtually solves all the others, the concepts and terms in which the first problem and the first solution are defined and expressed cannot be significantly changed if they are to serve to define and express the later problems and solutions. Clearly, then, it is not the arbitrary malice of professors but the interconnected questions and solutions themselves that demand both systematically formed concepts and a technical terminology that corresponds not to any concepts whatsoever but to systematic concepts. [CWL 12, 25]
Accordingly, in this active intellectual consciousness we can distinguish a general fundamental light and further determinations of that same light. The fundamental and utterly general light is our created participation in uncreated light, the source in us that gives rise to all our wonder, all our inquiry, all our reflection. Again, we attribute to this (utterly general) light those most general principles that contain no determination drawn from experience; for example, the principles of identity, non-contradiction, and sufficient reason, or the precept that good must be done and evil must be avoided. Still, what is consciously and intellectually operative in us not only consists in this general light, but is further determined by our own conscious acts. Sensible data determine us after the manner of matter; acts of understanding determine us after the manner of form; grasping evidence, judging, and deliberating further determine us after the manner of second act as intellectually, rationally, and morally consciously active and functioning. (CWL12, 139
CWL 10, 1993, Topics in Education
… an entirely new type of definition was introduced by Hilbert in his formulation of geometry.[1] He called it implicit definition. An implicit definition drops the common matter to express only a relational form… The significance of implicit definition is that it does not pin down the meaning of the words ’point’ and ‘line’ to anything. Point, in Hilbert’s expression of geometry, can be a Euclidean position without magnitude, and a line can be a length without breadth or thickness lying evenly between its extremes. But a point can also be an ordered pair of numbers, where (a,b) is not the Cartesian notation for a Euclidean point, but just that ordered pair. And a straight line can be a first-degree equation: y = mx + c is determined by two ordered pairs, and two ordered pairs will determine a first-degree equation. Hilbert can mean by point and line the imaginable Euclidean point or line, the Cartesian algebraic expression for point and line, or anything else that will satisfy the relation “two of one determines one of the other,” no matter what they are. The definitions are in terms of relational form, with no attention to any common matter. The relational form selects any common matter that will be thought relevant. Implicit definition is a more abstract type of thinking that omits even the common matter. (CWL 10, 126)
On the other hand, quantum theory deals with what it knows to be processes that cannot be imagined. It is a higher level of abstraction, and getting away from anything that can be imagined is connected with the fact that quantum theory is statistical – it is not the only factor. ¶ So you can see how even the ideas of definition and abstraction have become much more fluid. Scientific thinking is much more versatile, much more attentive to all the possibilities of the fundamental act that is insight into sensible data. I have given a series of illustrations of this. (CWL 10, 126-27)
… , as to the notion of cause, Newton conceived of his forces as efficient causes, and the modern mechanics drops the notion of force; it gets along perfectly well without it. It thinks in terms of a field theory, the set of relationships between n objects. The field theory is a set of intelligible relations linking what is implicitly defined by the relations themselves; it is a set of relational forms. The form of any element is known through its relations to all other elements. What is a mass? A mass is anything that satisfies the fundamental equations that regard masses. Consequently, when you add a new fundamental equation about mass, as Einstein did when he equated mass with energy, you get a new idea of mass. Field theory is a matter of the immanent intelligibility of the object. [CWL 10, 154]
the conceptual system must be rich in implications. In other words, its basic terms have to be properly defined and its range of implications clearly determined. Then you have an empirical science and apply a canon of selection which picks out of the conceptual system the elements that can be verified. (CWL 10, 142)