James Ladyman

According to logical positivism, so the story goes, metaphysical questions are meaningless, since they do not admit of empirical confirmation or refutation. However, the logical positivists did not in fact reject as meaningless all questions about for example, the structure of space and time. Rather, key figures such as Reichenbach and Schlick believed that scientific theories often presupposed a conceptual framework that was not itself empirically testable, but which was required for the theory as a whole to be empirically testable. For example, the theory of Special Relativity relies upon the simultaneity convention introduced by Einstein that assumes that the one-way speed of light is the same in all directions of space. Hence, the logical positivists accepted an a priori component to physical theories. However, they denied that this a priori component is necessarily true. Whereas for Kant, metaphysics is the a priori science of the necessary structure of rational thought about reality, the logical positivists were forced by the history of science to accept that the a priori structure of theories could change. Hence, they defended a notion of what Michael Friedman calls the ‘relativised’ or the ‘constitutive’ a priori. Carnap and Reichenbach held that such an a priori framework was conventional, whereas Schlick seems to have been more of a realist and held that the overall relative simplicity of different theories could count as evidence for their truth, notwithstanding the fact that some parts of them are not directly testable. All this is part of the story of how the verification principle came to be abandoned, and how logical positivism transmuted into logical empiricism.

Realism about computation is the view that whether or not a particular physical system is performing a particular computation is at least sometimes a mindindependent feature of reality. The caveat ’at least sometimes’ is necessary here because a realist about computation need not believe that all instances of computation should be realistically construed. The computational theory of mind presupposes realism about computation. If whether or not the human nervous system implements particular computations is not a natural fact about the world that is independent of whether we represent it as doing so, then the computational theory of mind fails to naturalise the mind. Realism about computation is also presupposed by attempts to use computational principles such as Landauer’s Principle to dispel Maxwell’s Demon. Realism about computation has been challenged by Hilary Putnam and John Searle among others. Various arguments have been put forward purporting to show that any physical system of sufficient complexity trivially implements all computations. Ladyman et al. offer a precisification and general proof Landauer’s Principle. In order to do this they present an analysis of what it is for a physical process to implement a logical transformation. In this paper, their analysis is explained and its implications for realism about computation and the use of Landauer’s Principle in foundational debates is assessed.

There has recently been a good deal of controversy about Landauer's Principle, which is often stated as follows: The erasure of one bit of information in a computational device is necessarily accompanied by a generation of kTln2 heat. This is often generalised to the claim that any logically irreversible operation cannot be implemented in a thermodynamically reversible way. John Norton and Owen Maroney both argue that Landauer's Principle has not been shown to hold in general, and Maroney offers a method that he claims instantiates the operation Reset in a thermodynamically reversible way. In this paper we defend the qualitative form of Landauer's Principle, and clarify its quantitative consequences. We analyse in detail what it means for a physical system to implement a logical transformation L, and we make this precise by defining the notion of an L-machine. Then we show that logical irreversibility of L implies thermodynamic irreversibility of every corresponding L-machine. We do this in two ways. First, by assuming the phenomenological validity of the Kelvin statement of the second law, and second, by using information-theoretic reasoning. We illustrate our results with the example of the logical transformation 'Reset', and thereby recover the quantitative form of Landauer's Principle.

Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics might be expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the ‘HoTT Book’. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory (but is compatible with the homotopy interpretation), and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. 1 Introduction2 What Is a Foundation for Mathematics? 2.1 A characterization of a foundation for mathematics 2.2 Autonomy3 The Basic Features of Homotopy Type Theory 3.1 The rules 3.2 The basic ways to construct types 3.3 Types as propositions and propositions as types 3.4 Identity 3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types 5.1 Tokens as mathematical objects? 5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy 7.1 Framework 7.2 Semantics 7.3 Metaphysics 7.4 Epistemology 7.5 Methodology8 Possible Objections to this Account 8.1 A constructive foundation for mathematics? 8.2 What are concepts? 8.3 Isn’t this just Brouwerian intuitionism? 8.4 Duplicated objects 8.5 Intensionality and substitution salva veritate9 Conclusion 9.1 Advantages of this foundation.