Yes, it's not the apparent logic in the operations that counts but the choice of definitions (and these include the definitions of the operations). I wrote a paper for the "Mind-Body Conference" in 1975 that discussed this. The idea is that "reasonable things" are done within "stable neighborhoods of 'truth' " that can be thought of as regions whose boundaries are the definitions. Inside we pretend the definitions are true, while the larger view from above knows the neighborhood is arbitrary.This is an old idea (e.g. Euclid). The "modern" part of it is that the definitions are not assumed to be true outside the neighborhood. This is simple and powerful because the results are larger worlds that can be compared to others and to phenomena and experiment (and without setting up dogmas and religions).Because of the way human minds work, there will be tendencies to think the definitions are "actually true" (and so the logic inside the boundaries) if they and the conclusions are appealing. But the form of this knowledge helps keep us saner if we are diligent about drawing the maps and boundaries correctly.Much of science has this character, and the model helps to understand what it means to "know" something scientifically. Science is a negotiation between "what's out there?" and what we can represent inside our heads via phenomena on the one hand and the "boundaries and neighborhoods" on the other. Einstein's nice line about "math vs. reality" hits it perfectly ("As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.").Newton's Principia was a huge step along these (and many other lines). He completely separates out the math part in the first large parts of the work. And only then does he start looking to see how the math models map onto observed phenomena.To say it again, Science is partly about being very careful about how the definitions map to "out there".In terms of context, if you are aware that you are in contexts -- the first step! -- and aware and careful about the ones you are using -- the next steps! -- there is a chance that "reasonable thinking" might happen.

 > Science is partly about being very careful about how the definitions map to "out there".I keep tripping on this part. Definition as in understanding the boundary? As in trying to find the boundaries between better, perfect, and the impossible (ie sweet spot)?> Because of the way human minds work, there will be tendencies to think the definitions are "actually true" (and so the logic inside the boundaries) if they and the conclusions are appealing.Does this mean?: Humans are pre-disposed to a model of how the Universe works: Zeus's thunderbolts, witches, Saturn/Satan/Santa, Geo-centric, etc. Science is a set of guidelines to help prevent us from shoehorning the Universe into the inaccurate mental models we are pre-disposed to believing?> while the larger view from above knows the neighborhood is arbitraryCan the leap from Geocentric model to Kepler's work be an example of this? As in: Geocentric model becomes irrelevant with Kepler's discoveries and p.o.v.?
 1. The definitions are the boundary -- these are what are used to make the interior. The analogy is to definitions (used to be called axioms) in math. Simple for math (because it is only about itself). Difficult for science because we can't just make up the definitions, we have to try to find ones that have some mappings to "out there".A boundary for chemistry is the physics standard model. Within physics the model is pretty accurate but unsatisfying as far as knowledge. They would like to have a better boundary, and get the standard model (and the key constants) from it. But it makes a good boundary for chemistry to make excellent chemical models.Similarly (oversimplifying here) chemistry makes a good boundary of definitions for molecular biology.Note that in this scheme of thinking, the "knowledge and meanings" exist inside the boundary, but don't include the boundary.One of the issues addressed in this approach is how to make progress in "knowing" without infinite regresses. Philosophically, it is a kind of pragmatism.As I mentioned elsewhere, science is a negotiation between two different kinds of things not a set of truths. It has many things in common with mapping (and making good maps is a branch of science, and one of the real starts of real science).2. We are predisposed to believe things. Bacon's notion of why we needed to invent a "new science" is to create a set of processes and heuristics that would help us deal with and get around to some extent "what's wrong with our brains".As a young scientist, I got the warning that is given to most young scientists "Beware, you always find what you are looking for!"Some of the interesting examples of good science turning into belief revolve around Newton and both Maxwell's Equations and the orbit of Mercury (neither are "Newtonian"). And if you look at the history you'll see that Newton was a lot less Newtonian than many of his followers.And yes, we also seem to have some things that are easier to imagine than others -- gods, demons, witches, etc seem easy, but future floods etc seem hard.3. Sure. E.g. if you think things have to be circles, then this will be part of your implicit context for thinking about orbits. Geocentric used circular orbits and then epicycles to correct them and save the theory (and some great metaphors there for lots of human thinking). But it's important to realize that Copernicus also used circular orbits, and they also used epicycles to save that theory. Kepler worked with Brahe and admired him, so decided to trust his measurements. This led to a different model. The planets themselves didn't care about any of the models.The definitions are still not -true- and the neighborhood is still not the phenomena. It's just better. You only get Newton from Kepler, not Maxwell or the orbit of Mercury and then Einstein for both.
 A few years back, I wondered: Why is it I believe in things no else agrees with?I seemed to look at the different schools of thought, pick the best one that answers as many questions as possible, and believe in it until something better comes along (ie can it solve more problems than the previous school of thought). (But, I was at least somewhat aware it was not scientific or scholarly. A combination of logic, emotion, and preferences.)My peers in HS who went on to higher education (Stanford, Yale, Harvard, Worcester, MIT, etc.,) agree with the status quo, work within it, and have careers, children, so on. (Nothing wrong with that and there is no professional jealously on my part.)I suspect, however, they are shipbuilders, who are treated as explorers. Scientific researchers who find juicy hacks and turn them into over-priced drugs with dangerous side-effects.You, however, take on the far superior approach to accumulating and developing new knowledge. However, it seems to come more naturally to you than to your peers. (Even taking into account your mathematics/musician parents and good teachers you encountered.)Which cities and universities around the world have been most receptive to your lectures? (I would assume it would be some in Canada, Scandinavian countries, and China. With the least receptive being in the US.)How was poor Faraday able to contribute so much to science and technology despite the vast resources of the classical educated members of The Royal Institution and Royal Society? (Granted, there were many people who contributed before/during/after Faraday's time to allow him to make those discoveries. Then it took others like Maxwell to carry on even further.)
 In 2004 I wrote a tribute to the research community I grew up in "The Power of the Context" (http://www.vpri.org/pdf/m2004001_power.pdf) and this will possibly help with some of your questions. I was just one of many in this community, and as I said at a recent Stanford lecture "The goodness of the results depends primarily on the goodness of the funders". Every era has enough of the kind of people who like to "ask what would real progress be?" and then try to make it happen. The large differences in "real progress" have fluctuated as the good funding has fluctuated (right now and for quite a few years, there has been almost none).A very important first step in this is to put a lot of work into "learning to see the present" and then where it came from (this is a lot of work, and it's not what human minds generally want to do). This will free up most thinking, will open up many other parts of the useful past, and especially about much better futures.

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