21–30 of 89 posts
Earlier quoted context omitted.
I haven't read the paper, but if you define the reals as constructive Cauchy sequences you'd be forced into a coinnductive definition. Is that equivalent?
I'm not familiar with constructive Cauchy sequences. The usual way of using Cauchy sequences is to quotient them by the ideal of Cauchy sequences that converge to 0.
The hardest ( ie , best) math prof I ever had (I was in EE, he was in the math department) used to say that "engineers can teach calculus, but their students cannot go on to teach calculus". I'm impressed by someone who knows a subject deeply enough to take that long of a view. Personally, I just like the engineering view of dx and dy as simply being new variables (with caveats that we immediately forget). Which is w…
Theorem 9.1. Let f be a continuous real-valued function. Let y be a real number. Let b ≤ c be real numbers with f(b) ≤ y ≤ f(c). Then f(x) = y for some x in [b, c].
Why is this well-formed? Once you say "Let y be a real number", I'm free to pick any real number, which means that there might not be a b and c such that f(b) ≤ y ≤ f(c). Now, I obviously understand what is said, but shouldn't this be formulated more like:
Let f be a continuous real-valued function. Let b ≤ c be real numbers from the domain of f. Let y be a real number in the closed interval bounded by f(b) and f(c) ([f(b), f(c)] or [f(c), f(b)], depending on whether f(b) ≤ f(c) or not). Then there exists an x in [b, c] such, that f(x) = y.
The way this and other theorems, definitions, etc. are formulated in the article bugs me, because I must go back and re-qualify variables based on information deduced from things introduced, after the variable in question was introduced.
One thing that bugs me about this, is that a lot of theorems seem to be formulated backwards. To give an example: Theorem 9.1. Let f be a continuous real-valued function. Let y be a real number. Let b ≤ c be real numbers with f(b) ≤ y ≤ f(c). Then f(x) = y for some x in [b, c]. Why is this well-formed? Once you say "Let y be a real number", I'm free to pick any real number, which means that there might not be a b and…
This focus on calculating derivatives as opposed to actually understanding the concept and why the calculations work that way is, I think, why so many struggle with it.
Hm, I always thought Spivak was calculus for mathematicians.
http://ocw.mit.edu/courses/mathematics/18-014-calculus-with-...
One thing that bugs me about this, is that a lot of theorems seem to be formulated backwards. To give an example: Theorem 9.1. Let f be a continuous real-valued function. Let y be a real number. Let b ≤ c be real numbers with f(b) ≤ y ≤ f(c). Then f(x) = y for some x in [b, c]. Why is this well-formed? Once you say "Let y be a real number", I'm free to pick any real number, which means that there might not be a b and…
One thing that bugs me about this, is that a lot of theorems seem to be formulated backwards. To give an example: Theorem 9.1. Let f be a continuous real-valued function. Let y be a real number. Let b ≤ c be real numbers with f(b) ≤ y ≤ f(c). Then f(x) = y for some x in [b, c]. Why is this well-formed? Once you say "Let y be a real number", I'm free to pick any real number, which means that there might not be a b and…
The "let" should really be read as "for all". If you do so, your objection disappears since quantifying over the empty always imply the consequent.
The hardest ( ie , best) math prof I ever had (I was in EE, he was in the math department) used to say that "engineers can teach calculus, but their students cannot go on to teach calculus". I'm impressed by someone who knows a subject deeply enough to take that long of a view. Personally, I just like the engineering view of dx and dy as simply being new variables (with caveats that we immediately forget). Which is w…
Why do you claim this? You don't have to have seen calculus to appreciate how trig functions are defined, how to manipulate them, or how to use them in applications.
As a math professor, I personally like the fact that we teach trig and exponential/logarithmic functions before calculus. They are (as you well know) exceedingly rich examples which illustrate why calculus is interesting and useful, and knowing them already enables the student to study calculus without excessive digressions.
Earlier quoted context omitted.
The "let" should really be read as "for all". If you do so, your objection disappears since quantifying over the empty always imply the consequent.
You also need to translate "with" as a forall (pi binder) as well. Then, you'll merely note that the formula is unsatisfiable without demonstrating a proof that y is within the proper bounds.