Here is a nice question: "Is every bounded continuous function uniformly continuous?" If all you know of uniform continuity is the symbolic logic, you won't be able to solve this. If you have the intuition, then it becomes rather easy.
Regardless, I argue calculus/analysis should be taught using logic notation and formal proofs.
This is a very good point, one that the OP is probably not aware of. I estimate that I taught somewhere between 50 and 100 classes in calculus in my nearly 20 years of college teaching. The vast majority of the students in those classes were planning to go into engineering of some kind. I would also estimate that no more than 5% of my students would go on to pursue a degree in mathematics, and even that percentage could be an overestimate.Watch how fast the engineering departments pull their students out of there and start teaching calculus themselves.
I literally all of my math teachers would give me a set of rules to follow to get a correct answer (and indeed, sometimes I would intuit how I got to that correct answer through dutifully following the steps to get there) but they would never go into how those steps were pioneered--what bits of logic justified using those steps.
Speaking of "tracking", do you think an organization like Khan Academy could provide us with useful data on what people learn and how?Here is what I'd do if I could.
I think that the "New Math" got it backward. At least in the earlier years, I'd introduce math concepts in a quasi-empirical fashion. I'd also end the segregation of subjects, end spending 1 year on algebra, 1 year on geometry, 1 year on trigonometry, and the like. Each year, I'd teach a mixture of math topics, starting at simple ones and moving to more complicated ones. I'd start with simple versions of algebra and geometry and statistics, and move to more fancy ones in later years. I'd go slow on Euclidean-geometry constructions, but I'd get into coordinate systems and analytic geometry rather heavily, complete with making lots of graphs. As to reasoning, I'd teach the difference between deduction and induction, and how induction when treated as deduction is a fallacy: affirming the consequent.
I like the idea of education being tracked, so one can teach more advanced math in a "math and science" track. Stuff like infinite sets, formal logic, abstract algebra, and the like. One can introduce group theory with symmetries of physical objects like flowers, to have something easy to picture.
Tell you what Mr. Rainbird, I'll have a go at what you're proposing and I'll check back with you to let you know how I'm getting on. I'm a teacher, background in the Arts, speaker of 3 languages other than English and currently trying my hand at AP Chemistry and Physics, primarily with Khan Academy. I'm fascinated by the linguistic elements of learning, semantics etc and recently thought I'd buy a tee-shirt which said "I speak Physics". Currently, I'd say I'm functioning at 8th/9th grade level Math. So, where do you propose I jump in? I'm impressed by your passion and erudition, let's see where we can take this.
Thanks for the well-argued reply. From your first piece I went to formal logic, and read Chapter 1 of Peter Smith's book. Loved it, really loved it. I've ordered the book. So, even if never read anything else on your list, I feel that's going to be a real gem. I'll have a look for Royden and Fitpatrick. Let's see. So many books, so little time.Well, take a crack at it man. Most of the materials can be found on Scribd.com, which means you can download pdf's for a small monthly subscription (but you can search them all in one day, download and limit costs I suppose).
I'm finding formal logic, set theory and especially category theory to be essential in abstract algebra and measure theory. Category theory really shines a light on the "black box" that is known as a function or maps, and then allows one to get intimate with the notion of maps -- which are key in, as far as I can tell, nearly every branch of mathematics.
I might suggest Royden and Fitzpatrick (real analysis) and Hungerford/Lang (graduate algebra) as congruent (or instead of) reading to Rudin and Dummit and Foote. The great thing about this list is that the more abstract books take this logically airtight system of sets and categories, topologies, mappings etc, and then go on to build and build until applications are found. Then, when working with applications you have a framework of an airtight logical system which you understand how to use, and also know is the foundation of whatever it is you're working on. It's comforting to know that the rules of sets and mappings are always the justification for whatever it is you are doing, and that you already know all the logical operations and possibilities, and reasons/explanations for anything that is possible in any situation since you already know the set theory, the logic and the category theory. The only statements that are not logically airtight are the axioms, which will need to be accepted as is, without justification. The thing is that many of them basically seem reasonable as is. But some people actually don't accept some or others and right now a good example is the continuum hypothesis and Forcing Axioms vs. V = Ultimate L. The thing about these disagreements is that they are simply a matter of opinion, and since they're axioms of a system, it can be a matter of nothing else other than opinion. This is where metaphysics and epistemology, and philosophy in general, comes into play. These topics help you come to grips with accepting the axioms as they are without reasons other than the plausibility of their statements, or perhaps, by careful argumentation and reflection, an outside philosophical idea may help persuade you to accept the plausibility of an axiom, or what have you. Perhaps the opposite also could happen.
It's neat how to really understand the majority of lower division, and even high school, math, it requires nearly all of a University's graduate course catalogue. Which, again, I know the majority of people never even get the chance to take because graduate school is something people typically avoid, and admissions is very competitive. And this trend continues because as new research is published, new applications will be discovered, and will likely be taught to high schoolers one day in the future. The ironic thing is that this cyclic syphoning effect of PhD holders' discoveries being stripped of all the theoretical curriculum which led to these very discoveries are then recycled and re-taught to young kids with as little explanation as possible. I think I just fundamentally take issue with this cyclic progression in math education. But, ultimately these discussions are usually smothered by people who say that teaching PhD coursework to kids is worse than teaching the results of the world's best discoveries to kids without any census as to how these discoveries were discovered and why they are reasonable or correct. I'm not so sure that the latter is any better sounding than the former. Anyway, my opinions are out there.