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A point about using diagrams to make arguments: If you are attempting to convince a person that something is true, rather than just launching into your evidence and favorite arguments it is often most efficient to begin by asking a series of questions to determine precisely how the person disagrees with you. The questioning allows you to hone in on the most important sticking points that prevent the other party from coming to your conclusion. These points can then be attacked individually, preventing you from wasting time making arguments that the other party already agree with or refuting positions that he or she has never even considered. The reason that this relates to diagrams is that this method of argumentation can be viewed as a tree, with questions at each of the higher level branches, and arguments at the leaf nodes.
"In my experience, the most staunchly held views are based on ignorance or accepted dogma, not carefully considered accumulations of facts. The more you expose the intricacies and realities of the situation, the less clear-cut things become."
Mary Roach - from her book Spook
A side note: The only reason that prime numbers are defined in such a way as to exclude 1 and negative numbers is because mathematicians found this way of defining them a bit more useful than the alternative possibilities. Mathematicians generally desire for important theorems to be stated in a manner that is as simple as possible, and the theorems about primes are generally simpler if we exclude 1. There is a more detailed analysis of this question here:
Anyone who claims to be rational in all area of their lives is speaking with irrational self confidence. The human brain was not designed to make optimal predictions from data, or to carry out flawless deductions, or to properly update priors when new information becomes available. The human brain evolved because it helped our ancestors spread their genes in the world that existed millions of years ago, and when we encounter situations that are too different from those that we were built to survive in, our brains sometimes fail us. There are simple optical illusions, simple problems from probability, and simple logic puzzles that cause brain failings in nearly everyone.
Matters are even... (read more)
Some further suggestions for handling hard questions, gleaned from work done in mathematics:
Hard questions can often be decomposed into a number of smaller not quite as hard (or perhaps even easy) questions whose answers can be strung together to answer the original question. So often a good first step is trying to decompose the original question in various ways.
Try and find a connection between the hard question and ones that people already know how to answer. Then, see if you can figure out what it would take to bridge the gap between the hard question and what has been answered. For example, if the hard question you are trying to
I believe that the analysis of this problem can be made more mathematically rigorous than is done in this post. Not only will a formal analysis help us avoid problem's in our reasoning, but it will clearly illustrate what assumptions have been made (so we can question their legitimacy).
Let's assume (as is done implicitly in the post) that you know with 100% certainty that the only two possible payouts are $1 million and $0. Then:
expected earnings = p($1 million payout) $1 million + p($0 payout) $0 - (ticket price)
= p($1 million payout) * $1 million - (ticket price)
= p($1 million payout|correctly computed odds) p(correctly computed odds) * $1 million... (read 544 more words →)
I would like to add another reason why we might perceive high status individuals as being less intelligent (or talented) than they originally seemed. The effect under consideration is reversion to the mean. Often, a person gains high status (or, at least meaningfully begins the climb to having high status) as a result of one exceptional act or creation or work. If our average skill level is X, we may often produce works that require skill close to X, but occasionally produce works that require much greater or much less skill than X (due to natural variability or variance in our performance). We are much more likely to be recognized (gain high... (read more)
Perhaps it is true that our modest technology for altering brain states (simple wireheading, recreational drugs, magnetic stimulation, etc.) leads only to stimulation of the "wanting" centers of the brain and to simple (though at times intense) pleasurable sensations. On the other hand though, it seems almost inevitable that as the secrets of the brain are progressively unlocked, and as our ability to manipulate the brain grows, it will be possible to generate all sorts of brain states, including those "higher" ones associated with love, accomplishment, fulfillment, joy, religious experiences, insight, bliss, tranquility and so on. Hence, while your analysis appears to be quite relevant with regard to wireheading today, I am skeptical that it is likely to apply much to the brain technology that could exist 50 years from now.
Sure, many people treat technology like magic, but as it becomes an ever increasing part our our lives, it is hard to deny that the supply of jobs in science and engineering will increase, and subsequently that the number of scientists and engineers will grow to meet this demand. What is more, even if most people are not curious about the technology they grow up with, that does not preclude the possibility that increased technology correlates with increased interest in science. All it would take is one in 10 or even 1 in 20 people to be influenced by the technology they use.
Unfortunately, in practice, being as knowledgable about the details of a particular scenario as an expert does not imply that you will process the facts as correctly as the expert. For instance, an expert and I may both know all of the facts of a murder case, but (if expertise means anything) they are still more likely to make correct judgements about what actually happened due to their prior experience. If I actually had their prior experience, it's true that their authority would mean a lot less, but in that case I would be closer to an expert myself.
To give another example, a mathematically inclined high school student may see... (read more)