Throughout his intellectual career, Carnap had developed original views on the nature of mathematical knowledge, its relation to logic, and the application of mathematics in the natural sciences. A general line of continuity in his philosophical work is the conviction that both mathematics and logic are formal or non-factual in nature. Carnap’s formality thesis can be identified in different periods, connecting his early contributions to the foundations of geometry and general axiomatics from the 1920s with his later work on the general syntax of mathematical languages in Logical Syntax. Given the centrality of this idea, how precisely did Carnap understand the formality thesis concerning mathematical knowledge? How was the thesis characterized at different stages in his philosophical work? The aim in the chapter will be to retrace the development of Carnap’s thinking about the formality of logic and mathematics from the 1920s until the late 1930s. As we will see, in spite of his general adherence to the thesis, there were several significant shifts in his understanding, corresponding to changes in his conceptual framework. Specifically, one can identify a transition from a semantic account of formality related to his study of axiomatic theories to a syntactic formalism developed in Logical Syntax.

Both neuroscientists and literary translators aim to understand how invariance of meaning can arise across different forms. But literary translators must render experiences that are more subtle, more contextual, more entire, than a glimpse of a cat’s tail. Thus, for the neuroscientist who seeks to understand invariances, the goals of literary translation match the highest levels of scientific aspiration. From a neuroscientific perspective, it is fascinating that literary translation is possible at all.

Symmetry is introduced as a basic notion of physics and, in particular, for soil mechanics also. Isotropy and anisotropy are discussed. A special case of isotropy of space is the principle of material frame indifference which plays an eminent role in the development of constitutive equations. The geometric scaling is discussed together with the notion of a simple material, which is – often unconsciously – basic in geotechnical engineering. Invariance with respect to stress and time scales is discussed. Mechanical similarity and the associated Pi theorem is shown to be the basis for the evaluation of so-called physical simulations with model tests.

We are now halfway through our book. It may appear to be a coincidence that insight and creative thinking appear at the “center” of a book on problem solving, a coincidence that the gestalt psychologists would surely have liked. Contemplating how the young Gauss solved an arithmetic problem, and considering how a mutilated checkerboard problem is solved, while realizing that the solution of these problems are analogous to how snowflakes look, may come as a surprise. It should not because all three are based on symmetry and invariance. Before discussing how “ordinary” insight problems are solved, this chapter describes the insights that led Galileo, Archimedes, and Einstein to their scientific discoveries. Physicists have known for over a century that there would be no science based on the natural laws, and no natural laws in the first place, if there were no symmetry in nature. So what encouraged this to happen just over a 100 years ago? In 1918, Emmy Noether formulated and proved her mathematical theorems that revolutionized physics. Her theorems showed how the conservation laws can be derived from the symmetry of these laws by applying a least-action principle. The review of symmetry in scientific discovery presented in this chapter provides the stage for a new formalism of problem solving that may apply not only to the sophisticated areas of science, but also to “ordinary” brain teasers, as well as to the TSP and the 15-puzzle.

Chapter 5 presents methods for assessing structural reliability under incomplete probability information, i.e., when complete distributional information on the basic random variables is not available. First, second-moment methods are presented where the available information is limited to the means, variances, and covariances of the basic random variables. These include the mean-centered first-order second-moment (MCFOSM) method, the first-order second-moment (FOSM) method, and the generalized second-moment method. These methods lead to approximate computations of the reliability index as a measure of safety. Lack of invariance of the MCFOSM method relative to the formulation of the limit-state function is demonstrated. The FOSM method requires finding the “design point,” which is the point in a transformed standard outcome space that has minimum distance from the origin. An algorithm for finding this point is presented. Next, methods are presented that incorporate probabilistic information beyond the second moments, including knowledge of higher moments and marginal distributions. Last, a method is presented that employs the upper Chebyshev bound for any given state of probability information. The chapter ends with a discussion of the historical significance of the above methods as well as their shortcomings and argues that they should no longer be used in practice.

In Chapter 1, I start by analysing the role of generalisations in scientific practice. Law statements or generalisations are involved in one way or another in explanation, confirmation, manipulation or prediction. I argue that these practices require a particular reading of the generalisations involved, namely, as making claims about the behaviour of systems. These practices therefore presuppose the existence of systems or things. Next, I look at the modal surface structure associated with laws. I use the term ‘surface structure’ to indicate that this structure may or may not be reduced to non-modal facts – as the Humean has it. I will sideline the debate about whether Humeanism is a tenable philosophical position. The positive claim I advance is that the modal surface structure can be explicated in terms of invariance relations – where I take invariance to be a modal notion.

This chapter expands a little on the idea that gravity is geometry, and then describes how the geometry of space and time is a subject for experiment and theory in physics. In a gravitational field, all bodies with the same initial conditions will follow the same curve in space and time. Einstein’s idea was that this uniqueness of path could be explained in terms of the geometry of the four-dimensional union of space and time called spacetime. Specifically, he proposed that the presence of a mass such as Earth curves the geometry of spacetime nearby, and that, in the absence of any other forces, all bodies move on the straight paths in this curved spacetime. We explore how simple three-dimensional geometries can be thought of as curved surfaces in a hypothetical four-dimensional Euclidean space. The key to a general description of geometry is to use differential and integral calculus to reduce all geometry to a specification of the distance between each pair of nearby points.

This chapter covers the Special Theory of Relativity, introduced by Einstein in a pair of papers in 1905, the same year in which he postulated the quantization of radiation energy and showed how to use observations of diffusion to measure constants of microscopic physics. Special relativity revolutionized our ideas of space, time, and mass, and it gave the physicists of the twentieth century a paradigm for the incorporation of conditions of invariance into the fundamental principles of physics.

Suppose one has a set of data that arises from a specific distribution with unknown parameter vector. A natural question to ask is the following: what value of this vector is most likely to have generated these data? The answer to this question is provided by the maximum-likelihood estimator (MLE). Likelihood and related functions are the subject of this chapter. It will turn out that we have already seen some examples of MLEs in the previous chapters. Here, we define likelihood, the score vector, the Hessian matrix, the information-matrix equivalence, parameter identification, the Cramér–Rao lower bound and its extensions, profile (concentrated) likelihood and its adjustments, as well as the properties of MLEs (including conditions for existence, consistency, and asymptotic normality) and the score (including martingale representation and local sufficiency). Applications are given, including some for the normal linear model.

We concluded the previous chapter by introducing two methods of inference concerning the parameter vector. Since the Bayesian approach was one of them, we focus here on the competing frequentist or classical approach in its attempt to draw conclusions about the value of this vector. We introduce hypothesis testing, test statistics and their critical regions, size, and power. We then introduce desirable properties (lack of bias, uniformly most powerful test, consistency, invariance with respect to some class of transformations, similarity, admissibility) that help us find optimal tests. The Neyman–Pearson lemma and extensions are introduced. Likelihood ratio (LR), Wald (W), score and Lagrange multiplier (LM) tests are introduced for general hypotheses, including inequality hypotheses for the parameter vector. Monotone LR and the Karlin–Rubin theorem are studied, as is Neyman's structure and its role in finding optimal tests. The exponential family features prominently in the applications. Finally, distribution-free (nonparametric) tests are studied and linked to results in earlier chapters.