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• Lonergan's Insight

4/28/2007: Chapter 2: Differential Equations

Section 2.4

  1. Recall: Concerned with the clues.

  2. Lonergan sets up three steps in generalizing the example from algebra (re: the clues).
    1. Name the unknown: “The nature of ….” (2.2)
    2. Similars are similarily understood (abstraction to universals)
      1. Anticipates that similar same data will have the same insight. True whether
        1. data in relation to senses (known before one has turned to a search for, or an anticipation of “nature of…” something)
          • hence the nature of X (where X is understood in the same way no matter place or time)
        2. or data in relation to each other (become the proximate materials for understanding the “nature of…”) Hence, one turns to organizing this data only if one is on the search for the “nature of….”, however seeking insights into this nature in terms of coorelations/functions.
    3. "Indeterminate correlation to be determined/indeterminate function to be determined”
      1. Anticipates similar data will be understood in the same way, and that understanding is formulated in a universal definition which is a determinate function or correlation.
  3. Notice how we heuristically anticipate something at each step:
    1. Anticipate the nature
    2. Anticipate the universality of the insight
    3. Anticipate the correlation/function.

     

  4. One general summary of the emergence of modern scientific insights and definitions.
    1. Merely Descriptive
    2. Descriptive insights leading to the identification of the “nature of…”
    3. The “nature of…” leading to more refined descriptive categorizations.
    4. The descriptive categorizations leading to possibilities of relating the data
    5. The emergence of the "indeterminate correlation to be determined/the indeterminate function to be determined". (In calculus, differentiation equations are ways of finding functions, hence it leads to an extremely powerful way for anticipating the answer).
      1. Analogy: just as the discover of patterns in arithmetic is algebra, and such patterns are formulated in algebraic rules/laws (eg. a + b = b + a), and then these algebraic laws can then be used to solve problems; so the discovery of patterns in algebra is calculus, and such patterns are formulated in the laws/rules of calculus (eg. power rule), and then these laws/rules can be used to solve problems. In the case of Algebra, the solution is an arithmetic number. In the case of calculus, the solution is an algebraic function.
    6. This anticipation then leads to a series of more precise collections of data in their relations to each other, which in turn provides the “image” into which this highly potent anticipation of a function can become determined.
    7. Once determined, then the definition in a functional equation emerges.


Click here for a printable outline.


 

 

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