Sorin Bangu

I identify and characterize a type of noncausal explanation in physics. I first introduce a distinction, between the physical properties of a system, and the representational properties of the mathematical expressions of the system’s physical properties. Then I introduce a novel kind of property, which I shall call a dual property. This is a special kind of representational property, one for which there is an interpretation as a physical property. It is these dual properties that, I claim, are amenable to noncausal (mathematical, in fact) explanations. I discuss a typical example of such a dual property, and an example of an explanation as to why that dual property holds (the explanation of the quantization of the linear momentum).

In his Remarks on the Foundations of Mathematics, Wittgenstein claims, puzzlingly, that ‘the proof creates a new concept’ (RFM III-41). This paper aims to contribute to clarifying this idea, and to showing how it marks a major break with the traditional conception of proof. Moreover, since the most natural way to understand his claim is open to criticism, a secondary goal of what follows is to offer an interpretation of it that neutralizes the objection. The discussion proceeds by analysing a well-known geometrical proof.

The paper focuses on the lectures on the philosophy of mathematics delivered by Wittgenstein in Cambridge in 1939. Only a relatively small number of lectures are discussed, the emphasis falling on understanding Wittgenstein’s views on the most important element of the logicist legacy of Frege and Russell, the definition of number in terms of classes—and, more specifically, by employing the notion of one-to-one correspondence. Since it is clear that Wittgenstein was not satisfied with this definition, the aim of the essay is to propose a reading of the lectures able to clarify why that was the case. This reading shows that his better known views on language and mind expressed in Philosophical Investigations illuminate his conception of mathematics.

When considering mathematical realism, some scientific realists reject it, and express sympathy for the opposite view, mathematical nominalism; moreover, many justify this option by invoking the causal inertness of mathematical objects. The main aim of this note is to show that the scientific realists’ endorsement of this causal mathematical nominalism is in tension with another position some of them also accept, the doctrine of methodological naturalism. By highlighting this conflict, I intend to tip the balance in favor of a rival of mathematical nominalism, the mathematical realist position supported by the ‘Indispensability Argument’ – but I do this indirectly, by showing that the road toward it is not blocked by considerations from causation.

The paper discusses to what extent the conceptual issues involved in solving the simple harmonic oscillator model fit Wigner’s famous point that the applicability of mathematics borders on the miraculous. We argue that although there is ultimately nothing mysterious here, as is to be expected, a careful demonstration that this is so involves unexpected difficulties. Consequently, through the lens of this simple case we derive some insight into what is responsible for the appearance of mystery in more sophisticated examples of the Wigner problem.

Factivism is the view that understanding why a natural phenomenon takes place must rest exclusively on (approximate) truths. One of the arguments for nonfactivism—the opposite view, that falsehoods can play principal roles in producing understanding—relies on our inclination to say that past, false, now superseded but still important scientific theories (such as Newtonian mechanics) do provide understanding. In this paper, my aim is to articulate what I take to be an interesting point that has yet to be discussed: the natural way in which nonfactivism fits within the unificationist account of scientific explanation. I contend that unificationism gives non-factivists a better framework to uphold their position. After I show why this is so, toward the end of the paper I will express doubts with regard to the viability of de Regt’s (2015) kind of non-factivism, based on the idea that understanding should be captured in terms of (scientific) skill.

: Some of the great physicists' belief in the existence of a connection between the aesthetical features of a theory (such as beauty and simplicity) and its truth is still one of the most intriguing issues in the aesthetics of science. In this paper I explore the philosophical credibility of a version of this thesis, focusing on the connection between the mathematical beauty and simplicity of a theory and its truth. I discuss a heuristic interpretation of this thesis, attempting to clarify where the appeal of this Pythagorean view comes from and what are the arguments favoring its acceptance or rejection. Along the way, I sketch the historical context in which this heuristic interpretation gained credibility (the quantum crisis in physics in the 1920s), as well as the more general implications of this thesis for physicists' metaphysical outlook.

Steiner defines naturalism in opposition to anthropocentrism, the doctrine that the human mind holds a privileged place in the universe. He assumes the anthropocentric nature of mathematics and argues that physicists' employment of mathematically guided strategies in the discovery of quantum mechanics challenges scientists' naturalism. In this paper I show that Steiner's assumption about the anthropocentric character of mathematics is questionable. I draw attention to mathematicians' rejection of what Maddy calls ‘definabilism’, a methodological maxim governing the development of mathematics. I contend that because definabilism is anthropocentric, its rejection casts doubts on Steiner's assumption.

This book is meant as a part of the larger contemporary philosophical project of naturalizing logico-mathematical knowledge, and addresses the key question that motivates most of the work in this field: What is philosophically relevant about the nature of logico-mathematical knowledge in recent research in psychology and cognitive science? The question about this distinctive kind of knowledge is rooted in Plato’s dialogues, and virtually all major philosophers have expressed interest in it. The essays in this collection tackle this important philosophical query from the perspective of the modern sciences of cognition, namely cognitive psychology and neuroscience. _Naturalizing Logico-Mathematical Knowledge _contributes to consolidating a new, emerging direction in the philosophy of mathematics, which, while keeping the traditional concerns of this sub-discipline in sight, aims to engage with them in a scientifically-informed manner. A subsequent aim is to signal the philosophers’ willingness to enter into a fruitful dialogue with the community of cognitive scientists and psychologists by examining their methods and interpretive strategies.

In this paper I criticize one of the most convincing recent attempts to resist the underdetermination thesis, Laudan’s argument from indirect confirmation. Laudan highlights and rejects a tacit assumption of the underdetermination theorist, namely that theories can be confirmed only by empirical evidence that follows from them. He shows that once we accept that theories can also be confirmed indirectly, by evidence not entailed by them, the skeptical conclusion does not follow. I agree that Laudan is right to reject this assumption, but I argue that his explanation of how the rejection of this assumption blocks the skeptical conclusion is flawed. I conclude that the argument from indirect confirmation is not effective against the underdetermination thesis.

The question as to whether there are mathematical explanations of physical phenomena has recently received a great deal of attention in the literature. The answer is potentially relevant for the ontology of mathematics; if affirmative, it would support a new version of the indispensability argument for mathematical realism. In this article, I first review critically a few examples of such explanations and advance a general analysis of the desiderata to be satisfied by them. Second, in an attempt to strengthen the realist position, I propose a new type of example, drawing on probabilistic considerations. 1 Introduction2 Mathematical Explanations2.1 ‘Simplicity’3 An Average Story: The Banana Game3.1 Some clarifications3.2 Hopes and troubles for the nominalist3.3 New hopes?3.4 New troubles4 Conclusion

Modern mathematical sciences are hard to imagine without appeal to efficient computational algorithms. We address several conceptual problems arising from this interaction by outlining rival but complementary perspectives on mathematical tractability. More specifically, we articulate three alternative characterizations of the complexity hierarchy of mathematical problems that are themselves based on different understandings of computational constraints. These distinctions resolve the tension between epistemic contexts in which exact solutions can be found and the ones in which they cannot; however, contrary to a persistent myth, we conclude that having an exact solution is not generally more epistemologically beneficial than lacking one.

The Principle of Indifference is a central element of the ‘classical’ conception of probability, but, for all its strong intuitive appeal, it is widely believed that it faces a devastating objection: the so-called (by Poincare´) ‘Bertrand paradoxes’ (in essence, cases in which the same probability question receives different answers). The puzzle has fascinated many since its discovery, and a series of clever solutions (followed promptly by equally clever rebuttals) have been proposed. However, despite the long-standing interest in this problem, an important assumption, necessary for its generation, has been overlooked. My aim in this paper is to identify this assumption. Since what it claims turns out to be prima facie problematic, I will urge that the burden of proof now shifts to the objectors to PI; they have to provide reasons why this assumption holds.