On the alleged simplicity of impure proof

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim

The "Artificial Mathematician" Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer

Role of immunohistochemistry for interobserver agreement of Peritoneal Regression Grading Score in peritoneal metastasis.

Pressurized intraperitoneal aerosol chemotherapy (PIPAC)-directed therapy is a new treatment option for peritoneal metastasis (PM). The 4-tiered Peritoneal Regression Grading Score (PRGS) has been proposed for assessment of histological treatment response. We aimed to evaluate the effect of immunohistochemistry (IHC) on interobserver agreement of the PRGS. Hematoxylin and eosin (H&E)-stained and IHC-stained slides (n = 662) from 331 peritoneal quadrant biopsies (QBs) taken prior to 99 PIPAC procedures performed on 33 patients were digitalized and uploaded to a web library. Eight raters (five consultants and three residents) assessed the PRGS, and Krippendorff's alpha coefficients (α) were calculated. Results (IHC-PRGS) were compared with data published in 2019, using H&E-stained slides only (H&E-PRGS). Overall, agreement for IHC-PRGS was substantial to almost perfect. Agreement (all raters) regarding single QBs after treatment was substantial for IHC-PRGS (α = 0.69, 95% confidence interval [CI] = 0.66-0.72) and moderate for H&E-PRGS (α = 0.60, 95% CI = 0.56-0.64). Agreement (all raters) regarding the mean PRGS per QB set after treatment was higher for IHC-PRGS (α = 0.78, 95% CI = 0.73-0.83) than for H&E-PRGS (α = 0.71, 95% CI = 0.64-0.78). Among residents, agreement was almost perfect for IHC-PRGS and substantial for H&E-PRGS. Agreement (all raters) regarding maximum PRGS per QB set after treatment was substantial for IHC-PRGS (α = 0.61, 95% CI = 0.54-0.68) and moderate for H&E-PRGS (α = 0.60, 95% CI = 0.53-0.66). Among residents, agreement was substantial for IHC-PRGS (α = 0.66, 95% CI = 0.57-0.75) and moderate for H&E-PRGS (α = 0.55, 95% CI = 0.45-0.64). Additional IHC seems to improve the interobserver agreement of PRGS, particularly between less experienced raters

Calibration to American options: numerical investigation of the de-Americanization method.

American options are the reference instruments for the model calibration of a large and important class of single stocks. For this task, a fast and accurate pricing algorithm is indispensable. The literature mainly discusses pricing methods for American options that are based on Monte Carlo, tree and partial differential equation methods. We present an alternative approach that has become popular under the name de-Americanization in the financial industry. The method is easy to implement and enjoys fast run-times (compared to a direct calibration to American options). Since it is based on ad hoc simplifications, however, theoretical results guaranteeing reliability are not available. To quantify the resulting methodological risk, we empirically test the performance of the de-Americanization method for calibration. We classify the scenarios in which de-Americanization performs very well. However, we also identify the cases where de-Americanization oversimplifies and can result in large errors

Recent progress in random metric theory and its applications to conditional risk measures

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measures. Section 2 gives the central framework in random metric theory, topological structures, important examples, the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals. Section 3 gives several important representation theorems for random conjugate spaces. Section 4 gives characterizations for a complete random normed module to be random reflexive. Section 5 gives hyperplane separation theorems currently available in random locally convex modules. Section 6 gives the theory of random duality with respect to the locally $L^{0}-$convex topology and in particular a characterization for a locally $L^{0}-$convex module to be $L^{0}-$pre$-$barreled. Section 7 gives some basic results on $L^{0}-$convex analysis together with some applications to conditional risk measures. Finally, Section 8 is devoted to extensions of conditional convex risk measures, which shows that every representable $L^{\infty}-$type of conditional convex risk measure and every continuous $L^{p}-$type of convex conditional risk measure ($1\leq p<+\infty$) can be extended to an $L^{\infty}_{\cal F}({\cal E})-$type of $\sigma_{\epsilon,\lambda}(L^{\infty}_{\cal F}({\cal E}), L^{1}_{\cal F}({\cal E}))-$lower semicontinuous conditional convex risk measure and an $L^{p}_{\cal F}({\cal E})-$type of ${\cal T}_{\epsilon,\lambda}-$continuous conditional convex risk measure ($1\leq p<+\infty$), respectively.Comment: 37 page

Representation of the penalty term of dynamic concave utilities

In this paper we will provide a representation of the penalty term of general dynamic concave utilities (hence of dynamic convex risk measures) by applying the theory of g-expectations.Comment: An updated version is published in Finance & Stochastics. The final publication is available at http://www.springerlink.co