Essay Subjects I May Choose To Write About In the Future (...and I just wrote down brief summaries in this post instead)

 

Essay subjects:


1 - I Don't Like Conditional Probability

2 - "Exercises Differentiation"--What Makes a Math or Other STEM Researcher Unique?



You know what, let me just write a couple of brief paragraphs on these ideas right now.


1.  Suppose you have an event A.  Then the probability of A, P(A), might be equal to 63%.  Then you have P(A|B).  Now the odds are, let's say, 2%.  Then consider the odds of P(A|(B and C and D)).  Now the odds are 44%.  How do you know when to stop?  This is where Turing's halting problem proof comes in:  Some events in the physical universe are *undecidable*.  Thus, we might have that E, F, and G are all necessary events to predict the true value of the probability of A.  If we say, "P(A)", we are saying, the probability of A *given the notion that we don't know about B, C, or D*.  We could in theory write "P(A|(P(B) = 60%))", but when we write P(A), we are assuming that all other events are unknown and not factored in.  Ultimately, we might even have that E, F, and G are undecidable.  So we have P(A|(B and C and D and E and F and G), but even though that's the most accurate version, we literally can't calculate it!


So what should we use instead of conditional probability?  I argue:  independent variables, which we can control or at least measure very well, and dependent variables, which we seek to learn more about.  Using linear regression, including both simple linear regression and multiple linear regression, we can find *CORRELATIONS* between independent and dependent variables...including with statistical models such as the model used in the "13 Keys to the White House" system.  When you don't understand a variable X, try to find a correlation between it and one or more other variables that you do understand well, and can obtain/manipulate values for relatively easily.


2.  Albert Einstein once said, "a mathematical equation stands forever."  I think this is true.  At the same time, in business there's this idea of "product differentiation," and so to an individual (aspiring or established) researcher, there's this sort of similar notion of "exercises differentiation."  Also regarding Einstein, there's this idea of thought experiments.  An exercise could be, a worked homework or exam problem, an attempt to solve an open problem, a lecture prepared explaining a particular mathematical concept to an audience, a thought experiment, a conceptual game designed to help you generate new ideas, or an effort to publish one's dissertation in graduate school.  An exercise is a "metaphorical rocket launch," and once you've done it, it stays in your brain forever.  An exercise stands forever, in your brain...even if you forget it.  Your brain has learned how to solve a particular kind of problem, and gained new understandings that were necessary for the completion of the problem...that's how exercises work on your brain, they force you to understand new things, unless they're very easy.


My argument is, if you consider two different mathematicians, A and B, we'll have that if the exercises that A has worked is a proper subset of the exercises B has worked, then, neglecting IQ differences for now, in general, A will most likely have a "set of mathematical capabilities" that is a subset of B's such set.  In general, the way to compete effectively with the competition is:  Do exercises--problem-solving tasks-- that your competitors in your research field do not know about and could not easily figure out how to find on their own.  That is the key to be a success as a researcher.


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