![]() The same response seems appropriate here: there's no reason to cast aspersions on people whose work rests on the axiom of choice but, if it bothers you, then see how much of their work you can do without the axiom of choice. Well, maybe some constructivists do that, but more often they offer the better response of showing how many of the same results you can recover without requiring that logical machinery. I think one wouldn't expect, for example, a constructivist mathematician to disparage classical mathematics because it relies on such inelegant logical machinery as the law of the excluded middle. To be sure, your results only apply within that structure, but that's the way of mathematics, that things are proven only within some structure. You can use infinitesimals perfectly well without having to get into the weeds of how they're constructed. This was just what I meant to say-the details of the construction shouldn't matter, only the axiomatics of the structure that's been constructed. > You really think that it makes sense to require the axiom of choice to prove that the derivative of x^2 is 2x as Robinson's ultrafilter construction does? If a student takes a class and unexpectedly finds they want (or need) to take further classes in the area, they don't want to be find out later that, "Oh, sorry, we didn't think you would pursue this subject, so we gave you the version of the class that didn't prepare you for the next one." I guess every math class could be very different if you knew it was the last math class that students would take, but you don't know that, and it's problematic to separate students according to which ones will study the material further before they're even exposed to it. It also foreshadows the kinds of proofs that students might do in real analysis if they decide to pursue an engineering or physical science major. It makes sense to me it shows that the techniques are based in math and not in magic, and it lays the groundwork to introduce Taylor series later. First pictures to develop some visual intuition, then a gentle foray into how to make the intuitions more rigorous using limits, sequences, and series, and then techniques for solving certain differentiation and integration problems. I guess that was pretty normal once upon a time? I remember that was how my calculus textbooks did it. > But the worst part of the experience was the method of starting with epsilon/delta and limits, to "explain" what was going on, and then throwing that away to get on with solving problems using differentials and integrals Perhaps that's the penalty we have to pay.) However, I acknowledge that including this extra material makes textbooks larger and thus more expensive. I think that many mathematicians don't realize that many of us ordinary mortals just don't think the same way they do, so including more visual descriptions and diagrams, typical applications and historical background do actually help one's understanding. (I recall the significance was lost on me when I first studied the subject for the same reasons. Including this diagram and having some discussion about the significance of the relevance of the i, e and ᴨ in this famous relationship would have put the subject into better context. In some ways the Wiki page does a better job in that it has a diagram of the 'Three-dimensional visualization of Euler's formula' (its application to circular polarization, etc.). ![]() For example, the discussion of Eula's Formula on page 879 is somewhat incomplete. This would save the student time in taking notes.Īnother minor criticism I have is that like many texts on mathematics it lacks both detail on the background to mathematical concepts and their applications. That said, it's a shame the PDFs are in image format instead of text as it makes it more difficult to copy notes etc. Jerome Keisler, every bit helps to promote a math subject that frightens off so many. Questions about nature of information and computation and their unified view are addressed along with application of info- computational approach to knowledge generation.This is a nice gesture by H. We apprehend the reality within a framework known as natural computationalism, the view that the whole universe can be understood as a computational system at many different levels ? from quantum mechanical world, to biological organisms including intelligent minds and their societies. Those two complementary ideas are used to build a conceptual net, which according to Novalis is a theoretical way of capturing reality. Information and computation are inextricably bound: There is no computation without informational structure, and there is no information without computational process. The book presents investigations into the world of info-computational nature, in which information constitutes the structure, while computational process amounts to its change. ![]()
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