It would be really helpful if Cambridge University publishes video lectures of Math Tripos, at least some of them.(I believe David Tong’s lectures on QFT are the only CU lecture videos available on the web).

The problem with the MOOC’s is that they expect us to finish a course in a period of time that many might be unfortunately uncomfortable with.Also,they are not always available and I believe these are the two major drawbacks of MOOC’s.

My suggestion for CU is that they should not go for MOOC’s rather publish the video lectures in the way they want it and must be available to anyone deeply interested in Mathematics.Also,the lecture notes that are available in the Archimedeans are really great and I believe they will supplement the video lectures,if published,very fine.

Thank You.

]]>Please refer to pages 23-28 of Apostol Calculus-1 which will clarify your doubt.

]]>1. Show that this ordering extends in the obvious way to an ordering on the field of rational functions (i.e. fractions with polynomials as numerator and denominator), giving an ordered field.

2. Show that this ordered field contains an isomorphic copy of the reals as an (ordered) subfield.

3. Show that this copy of R is bounded in the field of rational functions.

Thus, when we prove that R has the Archimedean property (for any real x there exists an integer n with n>x), we have to use the least upper bound property of R (in spite of the fact that Q has the same property).

]]>A good example to illustrate Batata’s point is group theory. We can study something like the symmetry group of the cube or the group of all invertible real matrices in a concrete way, or we can look at the group axioms and their consequences. The latter is a perfectly valid abstract approach even though many objects satisfy the axioms: indeed, that is one of the main virtues of the abstract approach to group theory.

But maybe your emphasis is on the word “completely”. Perhaps what you are saying is that we cannot deduce everything about a concrete object from certain properties unless those properties characterize the object. The natural response to that is that we often don’t want to deduce *everything*: we have certain specific statements in mind that we want to prove, and these may follow from properties that don’t characterize the object in question, in which case we will have proved a generalization of what we set out to prove.

Cambridge’s course Analysis I ]]>

That is not quite right, that’s absolutely not necessary to an “abstract” approach. Maybe this confusion is easy to make in real analysis because most of the time we think only of decimals as a concrete example of a complete ordered field and that is (maybe) good enough. Take for example ring theory, you have to have in mind a couple of rings to “get the feel” for the theorems we study.

Last but not least, everything is unique up to some isomorphism (equivalence relation) in mathematics, that is the purpose of isomorphisms (morphisms, in general), to enable us to regard different objects as “the same” when studying specific properties.

Well, since it is Cambridge, I first thought that the course is actually called “Nunbers and Sets” (old English + tradition?) until I saw “Numbers and Sets” a few paragraphs below.

]]>*Thanks — corrected now. (Is it any defence that I actually wrote ‘Nunbers’? No, perhaps not.)*

The idea of making mathematical videos has occurred to me, but I have nothing that could be called a plan.

]]>*Many thanks. I had corrected another of those but didn’t spot that one. I’ve changed it now.*

1 is the least upper bound for . Any number greater than 1, such as 2, is *an* upper bound for .

I can’t understand above sentence. Isn’t 1 the upper bound for A?

]]>*Many thanks — corrected now.*

Thanks. I’ll go and correct that.

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