Footnotes

^[1] See (Tipler 1994). Tipler would add computing theory to nuclear power and rocketry as the sufficient conditions for permanent economic growth.

^[2] Of course, certain sorts of objects will be objectively more suited to perform the function of money.

^[3] The most recent strong argument for the non-specifiable nature of personal skills and even the ability to discover mathematical proofs is presented by Roger Penrose in Shadows of the Mind (Penrose 1994).

^[4] One might say that the division into personal versus objective knowledge obscures the fact that some intentional states (e.g., dispositional expectations) shade over into propositional intentional states, i.e., World 3 status. Peoples' "constitutive" attitude to money is a good example, for it may only be a shock like a hyperinflation that jolts people into making their unconscious expectations explicit and placed in linguistic form. But we do need the distinction between Worlds 2 and 3 to talk about the transition from one state to the other.

^[5] For example, from the law-like statement "At atmospheric pressure silver melts at 960 degrees centigrade." one cannot derive a statement predicting a melting of silver at some definite spatio-temporal coordinates, or indeed even a purely existential statement to the effect "that some silver at some place and time has melted at 960 degrees centigrade at atmospheric pressure". But one can derive negative predictions, such as the statement "One will not observe the melting of silver below 960 degrees centigrade at spatio-temporal coordinates w,x,y,z at atmospheric pressure".

^[6] The logical form of metaphysical doctrines may be of the all-Some variety. For example, determinism may be stated thus: for every event there is a cause. Or more informatively, for every event x there exist a y and a z such that y is a lawful relationship describable by some true universal law u, and z is an event (set of initial conditions) preceding x, and x is predictable (deducible) from z in the presence of y (or of u) (cf. Popper 1982a, p. 196). This is clearly untestable by confrontation with a basic statement, for suppose someone presents the determinist with a putatively uncaused event. The determinist always has two defensive options. No matter how far you have searched for the cause of some unexplained event and failed, the determinist can say either that you failed to look hard enough for the initial conditions or that you have insufficient imagination to formulate the correct lawful relationship connecting the two events (the initial conditions and the event to be explained). He can say this because you cannot logically exclude the possibility that the very next search will identify the cause.

^[7] Popper deals with the methodological problem of picking the best curve through some given graphical points in sections 32 & 38 of Logic of Scientific Discovery (Popper 1980). Popper argues that given that one wants the most informative, and therefore most falsifiable, theory one should opt for the (theory) curve that has the lower degree of dimensionality, i.e., the one whose statement requires the smaller number of parameters. Theories with higher dimensionality require a greater number of basic statements to falsify them. For example, to refute the theory that all planetary orbits are circles requires only 4 singular statements, whereas to refute the theory that all planetary orbits are ellipses would require six singular statements.

^[8] With the spread of psychological knowledge, this is a little unfair to "everyday" explanations. The knowledge of visual illusions, for example, is quite wide spread, and with this knowledge one can make quite precise predictions about another person's subjective experience and possible introspective reports.

^[9] I am talking about melodies that can be appreciated by humans. Thus I am talking about discernible note lengths and pitches, which will obviously be finite in number, and melodies of finite length.

^[10] To deny the influence of logical relationships would be to imply something very strange indeed: that the way things are never has the slightest influence on what we think or are prepared to maintain. Correctly identifying errors in reasoning is on this view a purely accidental affair. But if the set of hypotheses we maintain in science is controlled even slightly by the process of trial and error elimination, in which the false hypotheses are cast from the body of science because they contradict true observation reports, then the maintainance of some hypotheses after each period of elimination is partly explained by their being true and the rejection of false ones is partly explained by the fact that they are false.

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