![]() ![]() But how does one come up with a postulate? Are we free to assume whatever we want? What is to prevent me from assuming any kind of nonsense I want and then building a system of proofs from it - something like building lego buildings? Of course the lego structures will be 'internally consistent' in that it forms a complete world by itself, but for practical purposes, it would be totally useless. Not a great proof and written after it was proven that this could not be proved.Īs I understand it, the postulates/axioms are assumptions and they are used to construct theorems. On a side note, in 1890, Charles Dodgson (aka Lewis Carroll author of Alice in Wonderland) published a book with a "proof" of the parallel postulate using the first 4 postulates. We can see different versions of systems where the parallel postulate is false by assuming that either there are no parallel lines, or that for any line and point not on a line there are an infinite number of parallel lines. Later (1868) it was proved that the two systems were equally consistent and as consistent as the real number system. In the late 19th century (approximately 1823), three different mathematicians (Bolyai, Lobachevsky and Gauss) proved independently that there was a different system that could be used that assumed the 5th postulate was incorrect. Some really great proofs were created by mathematicians trying to prove the parallel postulate. Mathematicians kept trying to prove that the 5th postulate (commonly known as the parallel postulate) could be proved from the first four postulates and thus was unnecessary. I would say it's for people who are interested in applications (within mathematics) and also in a precise treatment of canonical theories.There was a big debate for hundreds of years about whether you really needed all 5 of Euclid's basic postulates. Some advanced math students may find it too slow, and some engineer oriented students may find it too rigorous - so it's not for everyone. I added this "strange" recommendation of a title because I felt nobody else would make it here, and this book has been valuable to me in studying on my own. Some canonical proofs were oddly left out, and are available as "internet supplements" from the authors' website (also in horrible translation) - a heroic attempt to save some paper perhaps.Īfter that you can go on to more advanced analysis books for which there are many recommendations on this website. But the treatment is rigorous, user friendly, and with many examples and solved exercises. I say "oddball" because the translation from Italian to English is so bad, it's comical. Hence my first recommendation is weird, and not often heard (it's also not old enough to be a classic): this odd-ball by Canuto-Tobacco, and its sequel. ![]() That is, to learn analysis rigorously in tandem with a healthy dose of specific examples (specific functions, say) and applications. I've had more success with the (usually non-American) way of approaching analysis by combining, from the outset, what Americans call "calculus" (more calculation oriented courses where you learn to integrate or differentiate various elementary functions, with a pinch of generality here and there) with the material of so-called "Analysis" courses. There are classics that everyone sees recommendations for I won't reiterate the names of famous apostles and babies, because they are good books but less user friendly than the modern ones. Next, there are many options to start learning calculus (ahem, analysis). Mathematics undergrads receive this intro-to-math material surreptitiously by taking a freshman course in discrete mathematics or elementary set-theory. And if you are past that, you might want a sort of general introduction to math, in order to get used to proofs, for example Liebeck's valuable book. ![]() Also, today you can even learn precalculus on Khan academy. You, as a reader, can abridge it yourself - usually it helps not to read math linearly. There are many books for this, for example Axler's, which is good but way too long for my taste. If, however, you wanna get serious about it, you should make sure you have what Americans call "precalculus" in place. Albeit almost as many useless photos and flashy design elements. They tend to be calculation-oriented, somewhat lax on rigor (proofs omitted or glossed over) with occasional real world applications, and (to their credit perhaps) many graphs of functions plotted out. If the former, engineers sometimes like the heavy glossy-paged books with photos of spaceships in them (like Stewart etc.). You'll have to decide if you want an engineering-oriented book or a "pure-mathematics" rigorous approach to calculus. I don't have experience with answering questions here, but I have experience with learning mathematics by myself. ![]()
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