First I want to have a better understanding of what my mind does in the process of solving mathematical problems, proving theorems and general propositions in mathematics. Why do I want this? Because I plan to develop a deeper understanding of Objectivist epistemology compared to the one presented by Mortimer J. Adler’s “The Four Dimensions of Philosophy” and “Ten Philosophical Mistakes”. Adler’s “object of thought” vs Rand’s “mental entity.”
A second reason for deepening my knowledge of calculus and related topics in topology and abstract algebra is to be able to read Penrose’s “The Road to Reality.” Penrose suggests in his preface to the book that by working through the exercises he presents, an interested reader can gain better knowledge of the mathematics that powers quantum physics. His exercises worked fine for me up to about chapter 5 where he asks for proof of his expansion of exponentials in radial coordinates as an exercise for the interested reader. I recognized the process, but I don’t think I ever mastered in the first time around in my calculus, or physics, courses. Then he goes way deeper into Complex Analysis than I ever did. I disposed of my Complex Analysis book when we were paring down our possessions. Fortunately Penrose recommended a complex analysis textbook by one of his students. It is available for free on-line, and I am excited for the chance to study it.
Third, my high school assessment of psychology let to math as the “perfection” of human thinking, thus the pinnacle of psychological study. Inspired by Isaac Asimov’s “Second Foundation” novel.
My reading Feynman’s biography by James Gleick “Genius” clarified the distinction for me that Penrose had alluded to, understanding more than just manipulating symbols but grasping the meanings of the equations in as near a sensory manner as possible. Rand speaks of this in her essays about rationality, but never dives deep into it the way the physicists do.
Coincidentally, I am reading a history of economic thought in “Hayek’s Challenge” by Bruce Caldwell. The history cites epistemological errors in the use of abstract concepts by brilliant people in the early 1900’s. Sadly I still run across many of these same kinds of errors in current social, economic, news, and other reporting.
My best guess is that many of these people come from a literary background and have no hands on appreciation for the role of abstract concepts used as tools to understand reality. It as if I told them about a the concept of a circle, they would want to see empirical proof that it exists, else what good is it? How many studies have we heard about over the years that dig into the causes of poverty, and they always come up with the same answer. I can remember at least three that I have listened to, while commuting to work, over the decades. It is reassuring that they all get the same answer, but does the endless accumulation of data ever going to give something essentially new? It is like going out and gather data on all the examples of circles in the world in an attempt to prove the concept of circle. Each area of knowledge has its own modes of investigation, and proof. History is not the same as mathematics, chemistry is not the same as economics. The problem with economic research is trying to meet the demand for forecasting in order to support state policy.
Politics motivated the money spent in educating economists, hence economists doe the best they can to support providing the desires of politicians to “control” the economy for the benefit of the “common good,” e.g. more votes, more money for politicians. I believe this phenomena explains I. Kant’s theories of reason, and ethics. To support his Prussian King.
Example of learning, the definition of a function:
My initial steps were a cursory review of material that I “thought I knew.” Well maybe I missed some of the nuances, such as. the definition of supply set, that I glossed over because it seemed pretty trivial. The contents of the domain are drawn from the supply set.
The resistance to learning the formal definition of a function was that I had an intrinsic preference for curves instead of ordered pairs. Because I feel very bored with infinite series, number theory, set theory, fields, all the algebraic reasoning that goes into this stuff.
As I continue to study math, I make more and more connections between fields of math that I had not appreciated before. Algebra (abstract algebra), matrices, calculus, series, polynomials, the limits of what may be calculated.
As I observe the back and forth of imagination and logic I develop a much greater personal appreciation for imagination. Some how this study has also allowed me to recognize the allusions to this imagination-logic process by all sorts of people. Allusions that I would have not and did not notice in the past.
Even current debate in mathematical theory about teaching and understanding calculus, a geometrical, image based (imaginative) approach or a more formal abstract explanation based in proven number theory, delta-epsilon proofs. The debate is still current and driving other lines of mathematical investigation.
What would make this discussion more interesting would be the demonstration of the various findings and writings of these uses and discussions of imagination and logic. Imagination comes before logic in our thinking, both developmentally and functionally, but many people stop at imagination because it moves so quickly and feels so satisfying, but really it is a short cut to thinking by skipping verification of imagination by the use of logic and establishing consistency with prior knowledge. Even Kant mentions the importance of confirming imagination by logic unless it is his special intuition.
[TODO: Quote Kant on above comments about verification of imagination]
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