Divine Insight wrote:I'm quite serious. Of course I don't mean to imply that none of our mathematics is any good at all. Obviously much of it does work quite well. But often times as mere approximations.
Wrong. You see Divine Insight, your claims and ideas concerning mathematics are just dead wrong. Here you claim that most of mathematics boils down to 'mere approximations.'
Sorry, but you're wrong. Did you know that there's actually an entire field in mathematics devoted to approximating solutions? It's called numerical analysis, and even high school calculus students are taught basic numerical methods, ie: Newton's method for approximating roots, along with Euler's method (the tangent line method), and the Trapezoidal rule and Simpson's approximation.
Are you familiar with any of these elementary numerical methods? Also, did you know that numerical methods are usually only employed when an explicit analytical solution is impossible to obtain? Also, did you know that numerically approximated solutions are acceptable if and only if the error of approximation is relatively small?
Divine Insight wrote:And yes we stand on the shoulder's of giants in many instances, but we also stand on the shoulders of men who have made grave mistakes and have turned mathematics in very bad directions.
Name three.
Divine Insight wrote:In fact, that's precisely what needs to be examined. Exactly who's shoulders are we standing on, and why do we trust their shoulders to be the correct place to stand?
We stand on the shoulders of people like Gauss, Euler, Jordan, Galois, Fourier, Laplace, Lagrange, Riemann, Fermat, Cauchy, Dirichlet, Cayley, Kolmogorov, Banach, Poyla, and (in my humble opinion) von Neumann.
Please feel free to explain in great detail how these great mathematicians got it wrong.
Divine Insight wrote:Our current mathematical formalism is not anywhere near as sound as many people think.
Your opinion is noted. Your opinion is also meaningless and appears to have no facts to back it up.
Divine Insight wrote:In fact, engineers actually don't use pure mathematical formalism anyway. They actually ignore the pure mathematical formalism and do what actually works, which requires ignoring the actual formalism.
This is a gross oversimplification. Yes, engineers only use basic calculus, linear algebra and differential equations. The mathematical theory behind calculus, linear algebra and differential equations is generally only studied by mathematicians because engineers have no use for it. Engineers take the results and the theorems apply them accordingly, while mathematicians prove the results and formalize the theory behind the results.
Engineers study calculus, mathematicians study analysis.
Divine Insight wrote:Richard Feynman put it very nicely once when he said something along the lines that the universe doesn't "do" Calculus.
Obviously. Calculus is a language used to describe natural processes, namely change.
Divine Insight wrote:Calculus is merely our way of approximating what the universe is actually doing as best we can.
Wrong, already refuted above.
Divine Insight wrote:That's not to say that Calculus in particular is "wrong". But it's certainly not a description of "how" the universe behaves. It's merely our way of getting numerical approximations that match up really well with the quantitative behavior of the universe.
Wrong, already refuted above.