Foundationalism is not the only game in town.

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THE GREAT INFLUENZA, John M. Barry, 2004

page 16: "This commitment to logic coupled with man’s ambition to see the entire world in a comprehensive and cohesive way actually imposed blinders on science in general and on medicine in particular. The chief enemy of progress, ironically, became pure reason. And for the bulk of two and a half millennia—twenty-five hundred years—the actual treatment of patients by physicians made almost n progress at all."

In what context was that remark made? It seems to apply to very particular things.

For example, I thinking of when I was trying to understand why a derivative works in calculus. I wish I would have taken the approach, okay, for now let me just learn the ridiculously easy procedure, and move on to the next section. That way I can stay ahead of the professor in the book, or at least even, and be alert and able to get some benefit from the next lecture. Yes, I want to understand, of course I do, and in Zen-like fashion I am remaining open to understanding, but no set schedule.

As far as I can tell, the best way to understand a lot of mathematics is to closely look at the concepts with what you already know and suss out at what point properties do and do not hold, like say when a function is or isn't continuous over all real numbers. Also, I find that if I have developed a solid understanding in the methods I am working with I am more able to recall them later; like how for an exam, I can remember a few concepts better than 100 formulas that I don't understand.

However, if you don't want to contribute to mathematics (like, say, if you want to be an engineer) then I can understand why foundational considerations wouldn't interest you.

Thinking in a broader, more historical and less personal context...

The efforts to make mathematics more rigorous, precise and formal in the 19th century led to greater communication between mathematicians, and thus more work and better work. For instance, until a rigorous foundation of the calculus was developed it was widely ignored or at least viewed with skepticism. Mathematics has progressed rapidly in the 20th century, make no mistake. Only in the last century or two has it become physically impossible to have a professional level of understanding in all fields of mathematics.

Even so, the foundations of mathematics themselves were not of great interest aside from a few people (most notably Frege) until paradoxes emerged, especially Russell's. These were distressing to the working mathematician. After all, besides whatever philosophical convictions they had, advanced mathematics was being widely-applied. It was in everyone's best interests that bridges built with apparently solid mathematical reasoning should not fall down (both figuratively and literally). Since then, solid foundations of mathematics have been developed that are free of paradoxes, so now pretty much all mathematicians feel secure that the way they reason is indeed valid (except for the small minority of constructive mathematicians). The bridges appear to be safe.

That said, in the investigations that followed the paradoxes it was found that, roughly speaking, a consistent system of arithmetic cannot prove certain true statements, in particular its own consistency. So there are limits to what can be determined, foundationally-speaking. This revelation led to an investigation of the limits of what can be reasoned mechanically. Computers at least may not have been as well-understood without an investigation into the Entscheidungsproblem yielding both the lambda calculus and Turing machines.

So overall, I'd say that efforts to make mathematics more precise have been both fruitful (for more than just mathematics itself) and necessary.

Hector, I think we might have a lot of disagreement. And that's okay, let's just try and be open and accepting about it.

Alright, I'm thinking of a physician who has made a standard diagnosis and is starting standard treatment (as if they are visualizing a table from a medical school textbook, these symptoms mean this disease), but they have a feeling something's not quite right, that something is not quite jelling. A good physician had learned to pay attention to these feelings. It's not just clunky left-brain. It's whole brain, right brain as well. So, this physician may well go forward with the standard treatment, and at the same time do a few more tests and see if there's more to the story.

The all time classic case is a police detective who has a gut feeling. That does not mean throw out all the evidence you've already gathered. But it might mean, look a little further, look a little broader, look in places you might not think to look.

You want to talk about bridges, the walkway that collapsed in the Hyatt Regency, Kansas City, 1981. During construction, workers noticed that the walkway was unstable under heavy wheelbarrow loads. This construction traffic was simply rerouted! And the workers may have correctly perceived the social climate. That if they had insisted on making an issue out of it, they would have been criticized, doghoused, etc. That it would have taken someone with exceptional skills. 'I'm not saying it is an issue. I'm saying it might be an issue. And yeah, I'm going to have to put it in writing. Doing my job . . .' And so it's not enough to have a perfect initial plan, even for something as seemingly straightforward as the inside of a building. You still have to watch how things go along.

We as Aspies have already had our overlogic phases, many organizations and people in professions have not. That potentially gives us a strength. We can better integrate theory and practice.

Okay, math, the all-time pure logic, pure reasoning. Except we humans are very good at seeing the whole narrative arc all at once. Why don't we play to strength? I think we should. And yet so much of school takes the hard approach. The computer skills are good (are valuable), the human skills are not. And that's not necessarily the case.

So, in learning something new in math, you can see the entire procedure, from beginning to end, the entire arc, what are we trying to accomplish. And see a couple of examples of this. Then, go back and fill in the details.

Going down a rabbit hole and trying to do something creative. You can do that the beginning of a summer, you'll have an open expanse of time and you'll come out before the next school year. You might be able to do this the beginning of winter break, plenty of time, you'll come out before the next semester. Trying it on a Thursday afternoon, no Friday classes, some Thursday, all day Friday, trying something unique and independent, Saturday, and you're come out of it Sunday, I don't know, that almost feels like too tight a schedule.

A school semester's kind of an athletic competition and you kind of need to skim.

Maybe the bridges were a bad example, because while your story about bridges is a good story it does not amount to an attack on the foundations of mathematics. It could be an attack on naive planning and perhaps bad design to begin with. Basically a worry of many mathematicians in the early 20th century was that paradoxes in mathematics may lead perfectly valid statements to absurd results, which is why there was such a big truth to at least have foundations of set theory that avoided the paradoxes.

I also hope you don't mean to assert that mathematicians and logicians don't trust their gut feelings, or keep creativity at bay. Sure you have to have a strong background in an area for the gut feeling to be reliable, like the experienced doctor or detective, and your proof as always needs to be rigorous, but that does not preclude the use of intuition or creativity. In fact, creativity is necessary in mathematics. If creativity was not needed, machines could be used to determine whether (say) the Riemann Hypothesis or Goldbach's Conjecture hold. And machines can't do this. Moreover, if you assume the Church-Turing Thesis (which remains rock solid with more than 70 years of evidence), the fact that machines can't do this was proved in 1936. In any logical system, including the foundations of mathematics, you need a certain level of ingenuity beyond that of a machine to decide whether a statement is logically-valid.

Mathematicians can take some pride in that. When they prove theorems, it appears that there is often some sort of leap of mental ingenuity there which cannot be mechanically simulated. I think that's a stronger claim (and it's been more persuasively-argued) than your assertions about gut feelings. If it was not the case, then mathematicians would be no better than calculators, and perhaps all mathematicians would have been replaced with machines by now.

I actually agree with this, to the letter. Mathematics doesn't have to be taught solely through impeccable definitions without motivation. Motivation is important too. There are still well-written textbooks that combine solid technical details with a suitable level of motivation and looking ahead. Perhaps there are not enough. I attribute this more to bad writing - and perhaps bad teaching as well - than an issue with the field itself.

The details are indeed crucial, though. You cannot be a strong mathematician without a firm grounding in the technical details, just like you cannot be a doctor without slogging your way through medical school. This applies even more today, with the amount that is already known in mathematics worldwide, than it did a century ago or more.

Not just bad design on the Hyatt walkways, although there was some of that. There was also lack of attention to the application. If you're interested see Henry Petroski, "To Engineer Is Human: The Role of Failure in Successful Design," St Martin's, 1985. Petroski waffles, and that's okay, as if he's struggling with the issue right in front of us. At one point saying the margin of safety was essentially 1, which I take to mean no margin of safety. And at another point, saying if there had not been the unusual stress of dancing, the walkways would probably still be standing and no one would know that the design was inferior, too little margin of safety, and that the construction changes further eroded margin of safety.

Another example, some years ago medical science did not know the mechanism of action for aspirin. And maybe it still doesn't, not precisely. And as long as we know what it does in medium steps in the patient's body, we can follow it and also make informed judgments of which patients it's appropriate for (as simple as seeing if a fever is going down and/or the ratio of good cholesterol to bad cholesterol is improving).

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I remember lying in bed in fourth grade mentally reviewing a probability problem in baseball. A guy's at second. You have one out. The next two batters bat .333. What's the chance of bringing him home? I kind of knew the answer. I mentally reviewed it in a tight fashion and became sure of the answer. Then when I did it on paper, I was absolutely sure of it. That's one way to understand.

I also remember talking an algorithms class in the computer science department at age 37. We were studying heuristic solutions to the stagecoach problem (yes, even though it was an algorithm class, we were doing some heuristics). I did not understand one in a tight fashion. So the morning of the test, I got another book from the library that had some actual examples, went over them, got the feel of doing them. And did that on the test. I did well on the test. PLUS, I gained a kind of understanding so that if I later had the time and inclination, I would be closer to having the understanding in a tight fashion.

Both are valuable tools, plus a whole lot more.