Colin McLarty
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Why did Bourbaki's Élements omit the theory of categories?
22 votes

These are good answers and I have nothing to add on the particular reasons. To sum it up I would say it was impossible to write a comprehensive Elements of Mathematics based on category theory, ...

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Reflection principle vs universes
14 votes

Answering this question depends strongly on exactly what you want from Higher Topos Theory, because expressing high logical strength is a different goal from expressing an aptly unified logical ...

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Authorship of Grothendieck universes
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14 votes

Pierre Cartier has told me everyone at the time (i.e. everyone in those circles) knew Pierre Samuel wrote the appendix. Incidentally this makes a third person breaking the general rule that all ...

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What is the source of this famous Grothendieck quote?
14 votes

I would be surprised if the purported Grothendieck quote is really his. He does not lean to the short and sweet. It sounds more like an adaptation of another thing Deligne says in "Quelques idées ...

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Idea of using etale site
13 votes

I can tell you how they are related. Before Riemann people would say, for example, the complex square root function (for $z\neq 0$) is two valued, but for any small region of (non-zero) complex ...

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Where does the notation $\pi_1(X,x)$ for the fundamental group first appear?
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11 votes

Notice that in Poincare's setting there is no base point $x$, since he regards the fundamental group as a group of permutations of fundamental regions -- though he also knew the path interpretation, ...

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When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?
11 votes

Proper classes come up when you exhaust the means of forming sets. You need a set when you need to know the means of set theory have not been exhausted -- for example when you want to go on and form ...

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Problems for developing mathematical visualization expertise
8 votes

As to an example from number theory, Mumford style drawings of arithmetic schemes really helped me to get the difference between split, ramified, and inert ideals in (rings of integers over) algebraic ...

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The conceptual difference in notations of Cat
7 votes

Here I take terminology fromnLab: As long as 2-category means strict 2-category this usage is exactly the same as using $M$ to name both a manifold and its set of points. If $Cat$ is meant to be ...

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Does the bundle of germs of functions $f:X\to \mathbb R$ have the same sheaf of sections as $X\times \mathbb R$?
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7 votes

These bundles have the same sections. But which is simpler depends on your point of view. $X\times \mathbb R\to X$ is simpler set theoretically but is `more complicated' topologically in the the ...

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product operation: name and notation
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7 votes

Probably $\langle f, g\rangle : C \to A \times B$ is most often just called the arrow to the product. You are right it should not be called a product arrow. People who want a specific name for the ...

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Equivalence between Lowenheim-Skolem Theorem and Godel Completeness
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6 votes

On Question (a), yes this is correct. On Question (b), almost yes. If you drop the assumption that ϕ is satisfiable, then Skolem has given a sound and complete procedure for refuting a formula in a ...

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What is known about the reverse mathematics of algebraic number fields?
6 votes

In short, algebraic number theory ``flies below the radar'' of current Reverse Mathematics. First, after working on this a while, I think it is fair to say most familiar theorems on algebraic ...

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Lingering foundational question about sheaves of abelian groups
6 votes

I take it you mean sheaves on topological spaces. I think it is valuable to grasp both of two approaches. One (which works essentially the same way for sheaves on sites) is to first see that ...

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(Finite) Models of two subtheories of Peano Arithmetic
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6 votes

Infinite models are partly classified by two theorems of $PA^{(-1,2)}$ the subtheory of $PA$ without axioms 1 or 2. Then I will describe the finite models completely. In $PA^{(-1,2)}$ if axiom 1 ...

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Rigid analytic spaces vs Berkovich spaces vs Formal schemes
6 votes

A discussion of Berkovich spaces and rigid analytic spaces, with references is at http://ncatlab.org/nlab/show/Berkovich+space. For all three, also with references, see http://en.wikipedia.org/wiki/...

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Who first realized that the shortest distance between two points is a straight line?
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5 votes

In order to be a question about mathematics this would have to ask not when the fact became common knowledge -- since it is already known to bees and dogs as mentioned -- but when it got expressed as ...

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Show that $A$ cartesian closed need not imply $A^J$ is cartesian closed.
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5 votes

I think it will not be easy to find an example with $J=2$. Since a cartesian closed category has finite products, the result cited in David White's answer shows the category $A$ for such an example ...

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Finite order arithmetic and ETCS
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3 votes

Ah, Thomas Forster's 1998 paper Forster T. (1994) Weak systems of set theory related to HOL. In: Melham T.F., Camilleri J. (eds) Higher Order Logic Theorem Proving and Its Applications. HUG 1994. ...

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Can Hasse-Minkowski be proved without class field theory?
3 votes

So far the answer to the question is no. Lam was speaking in very broad terms, and O'Meara's wish for a direct proof is unmet. When Lam spoke of using the Dirichlet theorem on primes in ...

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Are there any consequences of the initial object in Set being unique, while the isomorphism class of terminal objects is nontrivial?
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3 votes

The distinction is meaningful in many senses. Certainly it is on ZF foundations, and contrary to what Dylan Wilson might seem to suggest, the distinction can be stated formally in the Elementary ...

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Algebraic numbers and the complex projective line minus three points
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3 votes

The comments are all correct. Deligne is citing one half of what is called Belyi's theorem, though experts feel this half was actually clear much earlier due to Weil. Weil used a version of the ...

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Ideals of etale structure sheaves
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3 votes

I must apologize for posting a false answer. in writing up a proof i discovered a gap which grew to a counterexample. In fact not every sheaf of ideals of an etale structure sheaf is finitely ...

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When was Bounded Zermelo set theory first formulated?
2 votes

I wish there was more direct evidence. For now, it seems likely Bounded Zermelo originated as one unnamed member of a family of theories arising from equiconsistency results. The article Jensen ...

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The importance of generating series in Algebraic Geometry
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2 votes

Your example from Witten makes one point: generating functions can make differential operators summarize combinatoric/algebraic information -- basically by making differential operators express ...

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Elementary proof of bounds on factor polynomials
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1 votes

The Henselian strategy of Zassenhaus uses $p$-adic bounds rather than the integer exponential bounds of Gelfond's theorem. Weinberger and Rothschild develop this approach over algebraic number ...

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Interpreting the Galois theory of finite extensions of $\mathbb{Q}$ in PA
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1 votes

It looked straightforward to me, but proof theory can be tricky. Emil's comment convinces me I am not overlooking any pitfall here. PA and EFA interpret the Galois groups of all finite extensions ...

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The Hasse Minkowski theorem in Peano arithmetic
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1 votes

The community has spoken by silence. No one has worked on this.

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Rigidity, moduli space, and moduli field
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0 votes

While there are a lot of ideas in the comments, I think the upshot is that if you want to read Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” (http://www.math....

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