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Feb
2 |
awarded | Necromancer |
Oct
1 |
answered | Why have mathematicians used differential equations to model nature instead of difference equations |
Sep
24 |
awarded | Autobiographer |
Jan
10 |
comment |
Is there a simple topological proof for a topological theorem about $S^2$?
I'm not sure what you would count as a valid topological proof, but J. Butterfield and C. J. Isham have a very nice categorial formulation. Their series of papers "A topos perspective on the Kochen-Specker theorem" approaches the problem in the form: in dimension > 2, there are no global elements of the spectral presheaf $\mathcal{O}_{d}^{op} \rightarrow$Set. |
Dec
19 |
answered | Examples of toposes for analysts |
Dec
12 |
comment |
Is there an explicit expression for the imaginary part of some non-trivial zero of zeta,
@GregMartin: Clearly there needs some amount of rigorisation here. There are infinite series with rational values that sum to the zeroes. There are a number of other infinite processes that may be considered fairly well known. What is meant here by explicit expression? Which constants are well known? I remember a competition question that involved line intersections in a complicated curve where I had to ask the professors if we needed to prove the Jordan curve theorem or could assume it. We had not been taught it formally yet. It pays to require clarity, like GH from MO, here. |
Oct
31 |
comment |
Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic
@KeshavSrinivasan: Then you do no justice to your understanding by adding exclamation marks after the point that exponentiation is not total in these systems. As even he points out, it is not total on modern computers. And finite computation does indeed provide a basis for models of predicative systems like Q*. However, as noted in the literature, you need to express such models with functions that restrict intermediate proof stages on notions of "surveyability" and other finiteness constraints. |
Oct
31 |
comment |
Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic
@KeshavSrinivasan: I think your comments on "crazy" show you to have misunderstood quite a lot. The distinction on predicativity he makes are clear, and he gives clear reasons. Existence of numbers is found in a new construction of them, not in a selection from a pre-existing set. We can't build numbers by ranging over them all. This may have many different metasemantic origins, but is a general statement among most hard finitists and ultrafinitists. It's why predicativity is important. It's an attempt to make numbers meaningful, instead of assumed. See pages 1-2 and 173-180 of his book |
Sep
23 |
comment |
Probability that a stick randomly broken in five places can form a tetrahedron
There are results going back much farther than the 2009 paper you mention. The edge lengths are basically an N=4 Euclidean Distance Matrix. Schoenberg gave sufficient results here, and you can see a good summary of the additional "fifth Euclidean property" or "relative angle inequality" (expressed in matrix terms) in any good summary like (ccrma.stanford.edu/~dattorro/EDM.pdf ). The volume of the inequality tetrahedron over the entire space of reachable distances (integrated over the probability at each point) should start an answer. |
Sep
11 |
awarded | Critic |
Jul
12 |
comment |
Textbook recommendations for undergraduate proof-writing class
@FrankThorne: I completely agree with your statement on the two different skills and suggestion on combinatorics for proof. However, I would also push such an education to much earlier than college, and this is what I do with my own children. Structuring reasoning into a proof is an important life skill that comes up in basic reasoning all the time. Combinatorics is a great proving ground as there are simple problems with 2-4 steps of proof to build on that use only basic addition, multiplication, and logic. |
Jul
2 |
comment |
procedure-based (as opposed to definition-based) concepts
Extensional semantics in computer science are regularly investigated. These are typically fixed point theories and are typical of many denotational programmes. In these semantics, the topics being investigated are things like (partial) correctness. Intensional semantics programmes, on the other hand, look at program definition and can be used to reason about computational complexity and related features of the procedures. I think that is likely the best formal framework in which to view your question. |
Jun
28 |
comment |
Essential reads in the philosophy of mathematics and set theory
I'm not saying it isn't instructive to learn from Kant the direction of much of that philosophy, but I wouldn't look to his work as being seriously defensible today. And I don't think it's purpose was to benefit mathematics either, simply to use it in a different programme. But if one wants to focus on the mathematical content of his work, I would recommend his Prolegomena before CPR. It is more focused on the examples of his philosophy, including much arithmetic, logic, and geometry. (And the occasional claim that the inverse-square law of forces is required by the area of spheres). |
Jun
28 |
awarded | Commentator |
Jun
28 |
comment |
Essential reads in the philosophy of mathematics and set theory
I'm not sure why your answer was voted down, but I do think that Kant is a horrible place to look for mathematical philosophy. In attempting to demonstrate the synthetic a priori, he uses basic mathematical statements that are now easily seen to not be a priori. His examples in Euclidean geometry are challenged by the fact that our geometry is apparently non-Euclidean. The logic of statements about the world appears to be a nondistributive orthomodular logic. Number relies on distinguishability, a much more troubling issue than Kant believed... (cont.) |
Jun
27 |
answered | Essential reads in the philosophy of mathematics and set theory |
Jun
3 |
comment |
Is there a nice characterisation of topoi with nice meta-logical properties?
@TheUser: Although I think it is probably a fair characterization that this response is polemical, the points being made seem more general than just Henkin semantics (and it seems to me that Andrej is not making his claim only about them). The point stands that there are other ways to provide interpretations that avoid metalogical difficulties, beyond just that provided by Henkin semantics. For example, Artemov's LP appears to be another direction for providing a higher-order semantics with these desirable properties. |
May
31 |
comment |
Widely accepted mathematical results that were later shown wrong?
@Kjetil: (also unknown): This is the central example in the book. It starts right there in the beginning with "A problem and a conjecture" and continues for pages and pages. I suspect you are thinking of a different book if you cannot see it. |
May
24 |
answered | Nested Sequence of Integers |
May
23 |
comment |
Why don't more mathematicians improve Wikipedia articles?
@András: Notice, I did not say that they could not be improved. That was not my point at all. I said that the barrier to improvement increases with every improvement. Do you think a high-school student could improve the entry on K-Theory? I think there are a number of high-school students who could have contributed originally to getting the Trigonometry entries edited. In the beginning, there were no entries on Trigonometry. Do you see how this can cause contribution rates to decrease over time, unrelated to anything to be worried about? |