Johannes Lenhard

This paper examines a looming reproducibility crisis in the core of the hard sciences. Namely, it concentrates on molecular modeling and simulation (MMS), a family of methods that predict properties of substances through computing interactions on a molecular level and that is widely popular in physics, chemistry, materials science, and engineering. The paper argues that in order to make quantitative predictions, sophisticated models are needed which have to be evaluated with complex simulation procedures that amalgamate theoretical, technological, and social factors – leading to problems with reproducibility. Thus, for methodological reasons, the predictive success causes a reproducibility problem.

The main thesis of the paper is that in the case of modern statistics, the differences between the various concepts of models were the key to its formative controversies. The mathematical theory of statistical inference was mainly developed by Ronald A. Fisher, Jerzy Neyman, and Egon S. Pearson. Fisher on the one side and Neyman–Pearson on the other were involved often in a polemic controversy. The common view is that Neyman and Pearson made Fisher's account more stringent mathematically. It is argued, however, that there is a profound theoretical basis for the controversy: both sides held conflicting views about the role of mathematical modelling. At the end, the influential programme of Exploratory Data Analysis is considered to be advocating another, more instrumental conception of models. 1. Introduction2. Models in statistics—‘of what population is this a random sample?’3. The fundamental lemma4. Controversy about models5. Exploratory data analysis as a model-critical approach.

This contribution defends two claims. The first is about why thought experiments are so relevant and powerful in mathematics. Heuristics and proof are not strictly and, therefore, the relevance of thought experiments is not contained to heuristics. The main argument is based on a semiotic analysis of how mathematics works with signs. Seen in this way, formal symbols do not eliminate thought experiments (replacing them by something rigorous), but rather provide a new stage for them. The formal world resembles the empirical world in that it calls for exploration and offers surprises. This presents a major reason why thought experiments occur both in empirical sciences and in mathematics. The second claim is about a looming aporia that signals the limitation of thought experiments. This aporia arises when mathematical arguments cease to be fully accessible, thus violating a precondition for experimenting in thought. The contribution focuses on the work of Vladimir Voevodsky (1966–2017, Fields medalist in 2002) who argued that even very pure branches of mathematics cannot avoid inaccessibility of proof. Furthermore, he suggested that computer verification is a feasible path forward, but only if proof is not modeled in terms of formal logic.

In this paper, we explore the extent to which issues of simulation model validation take on novel characteristics when the models in question become particularly complex. Our central claim is that complex simulation models in general, and global models of climate in particular, face a form of confirmation holism. This holism, moreover, makes analytic understanding of complex models of climate either extremely difficult or even impossible. We argue that this supports a position we call convergence skepticism: the belief that the existence of a plurality of different models making a plurality of different forecasts of future climate is likely to be a persistent feature of global climate science.

In this paper, we show that reproducibility is a severe problem that concerns simulation models. The reproducibility problem challenges the concept of numerical solution and hence the conception of what a simulation actually does. We provide an expanded picture of simulation that makes visible those steps of simulation modeling that are numerically relevant, but often escape notice in accounts of simulation. Examining these steps and analyzing a number of pertinent examples, we argue that numerical solutions are importantly different from usual mathematical solutions. They are do not merely approximate the latter, but introduce new problems, including issues of artificiality, stability, and well-posedness. Consequently, simulation modelling can attain reproducibility only to a certain degree because it is working with numerical solutions.

Modularity is a key concept in building and evaluating complex simulation models. My main claim is that in simulation modeling modularity degenerates for systematic methodological reasons. Consequently, it is hard, if not impossible, to accessing how representational structure and dynamical properties of a model are related. The resulting problem for validating models is one of holism. The argument will proceed by analyzing the techniques of parameterization, tuning, and kludging. They are – to a certain extent – inevitable when building complex simulation models, but corrode modularity. As a result, the common account of validating simulations faces a major problem and testing the dynamical behavior of simulation models becomes all the more important. Finally, I will ask in what circumstances this might be sufficient for model validation.

The English language has adopted the word Tardis for something that looks simple from the outside but is much more complicated when inspected from the inside. The word comes from a BBC science fiction series, in which the Tardis is a machine for traveling in time and space, that looks like a phone booth from the outside. This paper claims that simulation models are a Tardis in a way that calls into question their transferability. The argument is developed taking Molecular Modeling and Simulation as an example. There, simulation models are force fields that describe the molecular interactions and that look like simple and highly modular mathematical expressions. To make them work, they contain parameters that are adjusted to match certain data. The role of these parameters and the way they are obtained is seriously under-appreciated. It is constitutive for the model and central for its applicability and performance. Hence, the model is more than it seems so that working with adjustable parameters deeply affects the ontology of simulation models. This is particularly crucial for the transferability of the models: the information on how a model was trained is like luggage the model must carry on its voyage.

The main thesis of the paper is that in the case of modern statistics, the differences between the various concepts of models were the key to its formative controversies. The mathematical theory of statistical inference was mainly developed by Ronald A. Fisher, Jerzy Neyman, and Egon S. Pearson. Fisher on the one side and Neyman–Pearson on the other were involved often in a polemic controversy. The common view is that Neyman and Pearson made Fisher's account more stringent mathematically. It is argued, however, that there is a profound theoretical basis for the controversy: both sides held conflicting views about the role of mathematical modelling. At the end, the influential programme of Exploratory Data Analysis is considered to be advocating another, more instrumental conception of models. Introduction Models in statistics—‘of what population is this a random sample?’ The fundamental lemma Controversy about models Exploratory data analysis as a model-critical approach.

Both thought experiments and simulation experiments apparently belong to the family of experiments, though they are somewhat special members because they work without intervention into the natural world. Instead they explore hypothetical worlds. For this reason many have wondered whether referring to them as “experiments” is justified at all. While most authors are concerned with only one type of “imagined” experiment – either simulation or thought experiment – the present chapter hopes to gain new insight by considering what the two types of experiment share, and what they do not. A close look reveals at least one fundamental methodological difference between thought and simulation experiments: while thought experiments are a cognitive process that employs intuition, simulation experiments rest on automated iterations of formal algorithms. It will be argued that this difference has important epistemological ramifications.

This paper discerns two types of mathematization, a foundational and an explorative one. The foundational perspective is well-established, but we argue that the explorative type is essential when approaching the problem of applicability and how it influences our conception of mathematics. The first part of the paper argues that a philosophical transformation made explorative mathematization possible. This transformation took place in early modernity when sense acquired partial independence from reference. The second part of the paper discusses a series of examples from the history of mathematics that highlight the complementary nature of the foundational and exploratory types of mathematization.

In his Pensées, Pascal introduced the very influential distinction between the subtle intelligence and the geometrical intelligence. In the first part of the present paper Pascal’s distinction is considered by looking at his famous wager argument where Pascal acts as a skeptical philosopher and at the same time as an applied mathematician. This argument employs the esprit de finesse in a way that is of fundamental significance for the epistemology of mathematics. This claim will be backed up in the second part of the paper that explores Charles Sanders Peirce’s conception of diagrammatic reasoning. Peirce’s semiotically inspired epistemology of mathematics brings to the fore the significance of the “old”position of Pascal – one has to face the fundamental problems of application.

It is a commonplace that “the map is not the territory.” The phrase goes back to Alfred Korzybski, while the insight—formulated by insisting on an apparent triviality—is surely older and has received poetical expressions from Lewis Carroll and Jorge Luis Borges among others. The difference between map and territory is a difference of categories and is at the base of the representation relation. Hence maps play a prominent role in the discourse about representation which takes place in a number of disciplines and from different perspectives. Three of them are of special interest for the present paper.One influential view has been put forward by historian of art Ernst Gombrich in his classical lecture on..

Einleitung Die Kantische Philosophie der Mathematik ist nach einer weitverbreiteten Meinung in ihren Grundzügen überholt. Die moderne Mathematik gilt, ganz unkantisch, als analytisches Denken. Im folgenden soll für eine partielle Verteidigung von Kants Philosophie der Mathematik argumentiert werden. Sie hat nämlich den Gegenstandsbezug der Mathematik und deren Anwendungsrelation zu ihrem zentralen Problem gemacht. Für Kant war es die Anschauung, die den gegenständlichen Bezug ermöglichen sollte und in dieser Funktion ist sie, wie von einem anwendungsorientierten Standpunkt aus argumentiert wird, keineswegs überholt. Die Verteidigung ist aber nur partiell, da Kant von einem apriorisch fixierten Anschauungsbegriff ausging und es diesen durch einen „naturalisierten“ Begriff zu ersetzen gilt

Two conflicting interpretations of modern axiomatics will be considered. The logico-analytical interpretation goes back to Pasch, while the model-theoretical approach stems from Hilbert. This perspective takes up the distinction between logic as calculus ratiocinator versus lingua characterica that Heijenoort and Hintikka placed emphasis on. It is argued that the Heijenoort-Hintikka distinction can be carried over from logic to mathematical axiomatics. In particular, the model-theoretical viewpoint is deeply connected to a philosophy of mathematics that is not committed to a foundational perspective, but oriented more at applications and at mathematical practice.