Explanation in mathematical conversations:An empirical investigation

Analysis of online mathematics forums can help reveal how explanation is used by mathematicians; we contend that this use of explanation may help to provide an informal conceptualization of simplicity. We extracted six conjectures from recent philosophical work on the occurrence and characteristics of explanation in mathematics. We then tested these conjectures against a corpus derived from online mathematical discussions. To this end, we employed two techniques, one based on indicator terms, the other on a random sample of comments lacking such indicators. Our findings suggest that explanation is widespread in mathematical practice and that it occurs not only in proofs but also in other mathematical contexts. Our work also provides further evidence for the utility of empirical methods in addressing philosophical problems

Variants of the Selberg sieve, and bounded intervals containing many primes

For any m≥1, let H m denote the quantity MathML. A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1≤70,000,000. This was then improved by us (the Polymath8 project) to H1≤4680, and then by Maynard to H1≤600, who also established for the first time a finiteness result for H m for m≥2, and specifically that H m ≪m3e4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H1≤12, improving upon the previous bound H1≤16 of Goldston, Pintz, and Yıldırım, as well as the bound H m ≪m3e2m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H1≤246 unconditionally and H1≤6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h1,h2,h3), there are infinitely many n for which at least two of n+h1,n+h2,n+h3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the ‘parity problem’ argument of Selberg to show that the H1≤6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound MathML or H m ≪m e2m under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for H m when m=2,3,4,5

Variants of the Selberg sieve, and bounded intervals containing many primes

For any m≥1, let H m denote the quantity MathML. A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1≤70,000,000. This was then improved by us (the Polymath8 project) to H1≤4680, and then by Maynard to H1≤600, who also established for the first time a finiteness result for H m for m≥2, and specifically that H m ≪m3e4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H1≤12, improving upon the previous bound H1≤16 of Goldston, Pintz, and Yıldırım, as well as the bound H m ≪m3e2m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H1≤246 unconditionally and H1≤6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h1,h2,h3), there are infinitely many n for which at least two of n+h1,n+h2,n+h3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the ‘parity problem’ argument of Selberg to show that the H1≤6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound MathML or H m ≪m e2m under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for H m when m=2,3,4,5

Variants of the Selberg sieve, and bounded intervals containing many primes

For any m≥1, let H m denote the quantity MathML. A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1≤70,000,000. This was then improved by us (the Polymath8 project) to H1≤4680, and then by Maynard to H1≤600, who also established for the first time a finiteness result for H m for m≥2, and specifically that H m ≪m3e4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H1≤12, improving upon the previous bound H1≤16 of Goldston, Pintz, and Yıldırım, as well as the bound H m ≪m3e2m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H1≤246 unconditionally and H1≤6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h1,h2,h3), there are infinitely many n for which at least two of n+h1,n+h2,n+h3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the ‘parity problem’ argument of Selberg to show that the H1≤6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound MathML or H m ≪m e2m under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for H m when m=2,3,4,5

Variants of the Selberg sieve, and bounded intervals containing many primes

For any m ě 1, let Hm denote the quantity lim infnÑ8ppn`m ´ pnq, where pn is the nth prime. A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1 ď 70000000. This was then improved by us (the Polymath8 project) to H1 ď 4680, and then by Maynard to H1 ď 600, who also established for the first time a finiteness result for Hm for m ě 2, and specifically that Hm ! m3e4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H1 ď 12, improving upon the previous bound H1 ď 16 of Goldston, Pintz, and Yıldırım, as well as the bound Hm ! m3e2m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further, and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H1 ď 246 unconditionally, and H1 ď 6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture we show the stronger statement that for any admissible triple ph1, h2, h3q, there are infinitely many n for which at least two of n ` h1, n ` h2, n ` h3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds, or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the “parity problem” argument of Selberg to show that the H1 ď 6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound Hm ! mep4´ 28 157 qm, or Hm ! me2m under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for Hm when m “ 2, 3, 4, 5

Homogeneous length functions on groups

A pseudolength function defined on an arbitrary group G = (G, . , e, ()(-1)) is a map l : G -> 0, +infinity) obeying l(e) = 0, the symmetry property l (x(-1)) = l(x), and the triangle inequality l(xy) <= l(x) + l(y) for all x, y is an element of G. We consider pseudolength functions which saturate the triangle inequality whenever x = y, or equivalently those that are homogeneous in the sense that l(x(n)) = nl(x) for all n is an element of N. We show that this implies that l(x, y]) = 0 for all x, y is an element of G. This leads to a classification of such pseudolength functions as pullbacks from embeddings into a Banach space. We also obtain a quantitative version of our main result which allows for defects in the triangle inequality or the homogeneity property

Open science: Many good resolutions, very few incentives, yet

In recent years, a movement has emerged, which assembles itself under the umbrella term “Open Science”. Its intent is to make academic research more transparent, collaborative, accessible, and efficient. In the present article, we examine the origins, various forms, and understandings of this movement. Furthermore, we put the aims of individual groups associated with Open Science and the academic realities of their concepts into context. We discuss that much of what is known as Open Science can be viewed through the prism of a social dilemma. From this perspective, we explain why the concept of Open Science finds a lot of support in theory, yet struggles in practice. We conclude the article with suggestions on how to foster more Open Science in practice and how to overcome the obstacles it is currently facing.