Insight Chapter 2, a version
Archival Number: A400
Author: Lonergan, B.
pt 33, first paragraph (ts 24): the first paragraph of the chapter as it appeared in the earlier typescript reads:
1. Mathematical and Scientific Insights.
So far our illustrations of insight have been drawn from the field of mathematics. There have been examined the definition of the circle, the transition from arithmetic to algebra, the distinction between different kinds of infinite sets. However, our basic illustration initial example was not mathematical but mechanical. It was Archimedes' discovery of the principles of displacement and of specific gravity. sets. It is true that we began from the story of Archimedes' discovery of principles of displacement and specific gravity. But then we were content merely to indicate the more obvious features of insight and made no attempt to analyse the precise nature of the origin and development of scientific knowledge. Such an analysis must now be tackled.
pt 33, line -2 (ts 24): ts has `... the desired measurements. From the measurements to the formula Then, he discovered ...'
pt 33, line -1 (ts 24): ts has colon rather then semicolon after rule
pt 34, line 4 (ts 24): ts has `Strangely, the same thing something similar happens ...' Change made by hand.
pt 34, line 9 (ts 24): ts has crossed out at this point: In the fourth place, however, there is a profound difference between the definition of the circle and the law of falling bodies. The insight that grounds the geometrical definition is a grasp of necessity and impossibility: if the radii are equal, the curve must be round; if the radii are unequal, the curve cannot be round. But the insight that grounds the formulation of the law involves no grasp of necessity or impossibility. However well the law is understood
pt 34, line 15 (ts 25): ts has no comma after bodies
pt 35, lines 3ff. (ts 25): ts has `... investigator will tend to give his images endow his images the closest possible tend to endow his images with the closest possible approximation to the laws he conceives. But while his imagination will do its best, the sensible data will make no effort at best, while ...'
pt 35, line 6 (ts 25): ts has no comma after none the less
pt 35, line 15 (ts 25): ts has no comma after method
pt 35, line -13 (ts 25): ts has colon after immanent
pt 35, line -10 (ts 25): ts has colon after things
pt 37, line 1 (ts 26): ts has no comma after inquiry
pt 37, line -17 (ts 27): ts has: But similarities are of two kinds. No indication of change. (So too in manuscript B).
pt 38, line 4 (ts 27): ts has no comma after precisely
pt 38, line -7 (ts 28): ts has `... says let x ... can say let some ...'
pt 39, line 1 (ts 28): ts has no comma after Physics
pt 39, line 3 (ts 28): ts has no comma after then
pt 39, line -14 (ts 28): At this point pages from ms B are substituted for ms A. The replaced pages are in A 385 above.
pt 39, line -6 (ts 39): ts has `fifteen minute'
pt 41, line -2 (ts 86.3): ts does not have `incidentally'
pt 43, line -2 (ts 86.5): ts has no comma after momentum
pt 44, line 10 (ts 86.6): ts has prescientific
pt 44, l. - 1 (ts 76): ts has `transformation' for `transformations'
pt 49, line -1 (ts 90.5): ts has no comma after both
pt 51, line 1 (ts 90.6): ts has: For "random" may be defined as "any whatever provided specified conditions of intelligibility are not fulfilled."
pt 52, line -1 (ts 91): ts has: ... and the word, "random," has ...'
pt 53, -8 (ts 92): ts has, without any indication of a change: ... but successful diagnoses are not studied in fixing death rates.
pt 60, l. 17 (ts 97): ts: `just as the limit we considered under consideration lies beyond more terms than can be conceived ...'
pt 62, lines 15 and 16 (ts 98): ts has no commas before but and and
pt 63, par. Secondly (ts 100): prescientific is not hyphenated in ts
pt 64, l. 10 (ts 101): ts has `... of rates or of relative actual frequencies.' No change is indicated. This seems correct; pt should be changed.
pt 64, l. -1 (ts 102.1): ts has no comma before and
pt 65, l. -7 (ts 102.2): ts has no comma after Still
pt 66, pars. `I shall be asked' and `However, if': ms has here the one paragraph that appeared in the first ed., in place of these two paragraphs, which were introduced in the second ed.
pt 67, l. -17 (ts 105): ts has `do' for `does'
pt 68, l. 1 (ts 105): ts has `Indeed, that fallacy would wreck our whole analysis: for we have granted that single events may be deducible and, in that sense, certain yet the same events as a group may form a coincidental aggregate and so, when investigated with the generality made possible by statistical would wreck our analysis. Not only ...'
pt 68, ll. 5, 7 (ts 106): ts has no comma after verified
pt 69, l. -4 (ts 107): For the last sentence, ts has `Chapters XI to XVII endeavor to grasp within a single view the totality of views on knowledge, objectivity, and reality, for all proceed from the empirical, intellectual, and rational consciousness of the concrete subject.'
pt 69, end (ts added to chapter): Typescript has here an appendix to chapter 2. Handwritten at top (by BL, I think): I do not know whether this remains or not. The appendix is somewhat different from that which appeared in the discarded pages (see above, A 385), and is as follows:
Appendix to Chapter II.
On the Use of the Terms "Classical" and "Statistical."
In ordinary usage, "classical and "statistical" are not opposed. The opposite to "classical" is "quantum," and the opposite to "statistical" is "mechanical." This usage may be illustrated by the fourfold classification of 1) classical mechanics (Newton), 2) classical statistics (Boltzmann), 3) quantum mechanics (Schrödinger, Heisenberg), and 4) quantum statistics (Bose-Einstein, Fermi-Dirac).
The trouble is that this fourfold classification seems incomplete. For relativity mechanics is opposed to classical mechanics and, while special relativity enters into combination with quantum mechanics (Dirac), general relativity seems as opposed to it as Einstein himself. Further, if these complications are not to be neglected, it is necessary to go behind the terminology to a systematic conception of the conceptions entertained by interpreters of physical theory. As is obvious, however, the purpose of this appendix is not to expound and to justify a systematic view but simply to clarify our linguistic usage view but simply to clarify the linguistic usage that we have found convenient by contrasting its assumptions with the assumptions that seem to underlie more common modes of speech.
From our viewpoint, then, the fundamental disjunction regards the interpretation of laws of the Newtonian and Einsteinian type. Such laws will be said to be interpreted concretely if they are taken to relate imaginable terms. The same laws will be said to be interpreted abstractly if they are taken to relate terms that are defined implicitly by the laws themselves.
On the first alternative of concrete interpretation, the law is completely determinate in principle. It is true enough that the law is expressed by a mathematical formula of wide generality and that further determinations will have to be added before any application to concrete instances can occur. It also is true that the further determinations cannot be deduced from the law itself. as a mathematical or as a physical formula. But on concrete interpretation the law is not simply a physical formula; it relates imaginable terms; and because terms are imaginable inasmuch as their various dimensions are assignable, it follows that for concrete interpretation the law is fully determinate in principle.
However, those that accept the first alternative split into two groups. The first group not only affirms concrete interpretation but also affirms that concretely interpreted laws of the Newtonian type exist. The second group agrees with the first in admitting concrete interpretation but differs from it by affirming that, if any such laws seem to be verified, the verification is mere macroscopic appearance. The agreement and difference of this first and this second group seem to me to correspond to the agreement and the difference that that unites and the difference that separates classical statistics and quantum mechanics. that unites and the difference that separates ordinary conceptions of classical mechanics and statistics and quantum mechanics.
On the second alternative of abstract interpretation, laws of the Newtonian and Einsteinian type are determinate but not fully determinate. They are determinate in their own abstract order. They are not fully determinate in two respects. In the first place, they are applied to concrete instances only by assigning precise numerical values to be substituted for their variables; and from abstract laws alone such precise numerical values cannot be deduced. In the second place, when one deduces some precise numerical values from other known numerical values, the deduction rests not simply on the truth of the abstract laws but also on the truth that such and such a conjunction of abstract laws is alone relevant to such and such a concrete process in such and such a concrete situation.
Now it is this second indeterminacy of abstract laws that is significant. For it is an indeterminacy that resides primarily not in the knowing subject but in the object to be known. Thus, to affirm the existence of a planetary system removes an indeterminacy in knowing, for it posits such and such a conjunction of laws as alone relevant to a given concrete process in a given concrete situation. Still, this removal of an indeterminacy in knowing rests, not on abstract laws alone, but on a set of concrete matters of fact; it rests on matters of fact that might be otherwise; and if the matters of fact were otherwise, the extraordinarily accurate predictions of astronomers would vanish and in their place there would be merely the unsolved 3-body problem. there would arise an insoluble problem in which the selection of relevant laws would depend on a diverging series of observations and the selection of the relevant observations would depend on knowledge of the relevant laws. to be known. For situations are of two kinds. In some concrete situations the relevant laws are concretely applicable in a closed or almost closed circle of mutual conditioning; and such is the case in the planetary system that has provided the most striking examples of accurate deduction and prediction. But there are other concrete situations in which the relevant laws are applicable only in a diverging and scattering series of ever more numerous and more remote conditions; and such would be the case if one attempted to deduce and predict the emergence or destruction of planetary systems.
Hence, from the viewpoint of abstract interpretation, one distinguishes between 1) abstract laws applied to schematic situations and 2) abstract laws applied to non-schematic situations. In both cases the abstract laws exist and govern every event that occurs; in neither case are the abstract laws mere macroscopic appearance. But in the first case accurate deduction and prediction are possible. In the second case there is no objective possibility of accurate deduction and prediction because there is no objective mediation between scheme that removes the indeterminacy of the abstract laws. deduction and prediction are possible objectively because the situation is schematic. And in the second case there is no possibility of removing the native indeterminacy of the abstract laws and proceeding to accurate deductions and predictions because the objective condition of a schematic situation is not fulfilled.The arguments for this abstract interpretation will appear in the course of Chapter III. It differs from the interpretative assumptions of classical mechanics and statistics inasmuch as 1) it restricts predictable events to schematic situations and 2) it denies schematic situations to be universally the sole situations that exist. It differs from the interpretative assumptions associated with quantum mechanics and statistics inasmuch as it rejects as invalid the inference that, because there are non-schematic situations in which predictions are not objectively possible, therefore laws of the Newtonian and Einsteinian type are mere macroscopic appearance. Finally, it is in the light of this abstract interpretation that we refer to laws of the Newtonian and Einsteinian type as classical interpretation, we feel justified in continuing to refer to laws of the Newtonian and Einsteinian type as classical and in opposing to them solely the statistical laws that have to be invoked in dealing with non-schematic situations.