Thanks a lot for the tutorial link, I was actually looking for a git tutorial because of the programming project from my university I mentioned in my submission!
No, it turns bilinear maps into linear one! If you have three R-modules (one can read K-vector spaces if unfamiliar with modules) N,M,P and a bilinear map N×M→P then there is a unique linear map N⊗M→P compatible with the map N×M→N⊗M which is part of the structure of a tensor product. (What's really going on here in fancy terms is the so called Hom-Tensor adjunction because the _⊗M functor is adjoint to the Hom(M,_) functor, but just thinking about bilinear and linear maps is much clearer)
If you ask a mathematician a tensor is an element of a tensor product, just like a vector is an element of a vector space. This moves the question to "what is a tensor product", which you can think about as a way to turn bilinear maps into linear maps (this is an informal statement of the universal property of the tensor product, you also need a proof of existence of such an object, but it's easy for vector spaces and alright for modules after seeing enough algebra)
In some (maybe most, but my experience is limited to a few) German university they have sinks, sponges and a thing to dry the blackboard after wiping it with a sponge in every class
A hypersurface is usually a codimension one object, while a manifold can have bigger codimension or even not be embedded in an higher dimensional space at all
You can think of a tensor as "an element of a tensor product" (just like a vector is "an element of a vector space") and look at the definition of a tensor product via its universal property, it basically turns bilinear maps into linear maps. This universal property is elegant and immediately gives the Hom-tensor adjunction, but to show that tensor products satisfying it actually exist you'll need an explicit construction, which can be done by quotienting a free module. This explicit construction might look unintuitive at first, but you're really quotienting by the minimal relations required for the "bilinearity" property of tensors to work out
A less dense book for commutative algebra is Miles Reid's Undergraduate Commutative Algebra, but it doesn't cover tensor product of modules which will definitely be needed for algebraic geometry. Also for everyone interested in Atiyah-Macdonald you MUST do the exercises, half of the book is in the exercises!
You'll need a good background in commutative algebra to learn algebraic geometry, if you have that and want to see the modern approach, schemes and everything else that Grothendieck did, I suggest Vakil's notes, they're freely available from his homepage. (Disclaimer: I only studied the first 15 chapters and the one on Kähler differentials which should be the 21st, but I suppose the second half is as good as the first, I'll find out for sure next term)
For another recent example Ryanair is being fined in Italy not because their practice of not including the hand luggage in the price of a ticket is illegal, but because it hurts comparability with other airlines
You said "acccording to classical mathematics", which is not constructive so I assumed you were working in ZFC or some similar set theory.
Also if we want to be precise I'd like to hear your definition of "finite definition", the only definition of "definability" I'm familiar with is relative to a model of a theory and if ZFC has models at all then it has countable pointwise definable models, but I guess that won't satisfy your idea of "finite definition"
So you're arguing that a set can exist even if its elements don't? Because we can surely give a unique description of the set of real numbers (the unique complete ordered field up to iso) and this description forces it to be uncountable
As far as I know CGT is mostly of interest to mathematicians, because there are some groups whose study require calculations that are absolutely impractical to carry out by hand, such as presentations of the sporadic groups. Or maybe you need to know how many nonabelian groups of order 2652 are there and throwing it to a computer is just the fastest way to find out or you need to compute conjugacy classes or stuff like that.
Computational algebra more broadly however has many applications to cryptography for example
Having just finished a geometric group theory course I was hoping to see something about the word problem for automatic groups, or at least an algorithm to compute Dehn presentations of f.g. hyperbolic groups, but it looks like a cool resource with plenty of interesting topics.
Another book the HN crowd might like is Shoup's "a computational introduction to number theory and algebra"[0], it is much more broad and with litte focus on implementations, but a very interesting book nonetheless
A similar "joke" in mathematics is that "a sheaf of groups is just a group of sheaves" (meaning that a sheaf of groups on a topological space X is a group object in the category of sheaves of sets on X)
There's actually a big debate between math educators on whether abstract algebra courses should follow a groups first or a rings first approach so it is totally possible to teach rings without appealing to groups. (I'd still argue that both approaches have their advantages and disadvantages and that there can be value in using groups to teach rings though, I just wanted to say it's not as obvious as you might think)
The different margin sizes on the left and right (swapped between odd and even numbered pages) are the standard for the book documentclass in LaTeX because that's the best for printing and binding!
Define (inside the system) the function S(1)=14 and S(n+1)=g(n) for all n\geq1 where g is any function which is not computable. I'm not convinced you can jump from "the function is uncomputable" to "the system cannot determine a single value of the function"
Plenty of math is done outside of ZFC, a lot of modern set theory for example, but that doesn't make it philosophy.
Even Grothendieck's EGA has a couple of arguments that cannot be carried out in ZFC in the generality he presents them in, but I wouldn't call EGA a philosophy treatise nor an algebraic geometer a philosopher
Most mathematicians don't care at all about CH and among those who do not all believe that ZFC+CH is more intuitive than ZFC+not CH (really it makes no sense to believe whether it is true or not unless you believe in some platonic universe of sets, but I do maths, not philosophy), for example Gödel himself thought that the continuum could be aleph_2, Woodin's Omega logic argued the same (but this is somewhat controversial) and the proper forcing axiom also implies that the continuum is aleph_2
I specified I'm working in standard first order logic. And I'm not sure what do you mean with the informal take.
Also intuitionist logic is not my area, but isn't it divided in inference rules for provability and (Heyting or Kripke) semantic for model theory and truth just like FOL?