Philosophy and History of Science seminar 05-06

'Universe creation on a computer'
abstract

According to a straightforward analysis dating back to medieval philosophy of language, the dictum of a sentence, which is in English usually a that-sentence, is to be conceived as a singular term. Given natural constraints of referential semantics, this singular term denotes an object. If, however, sentences involving dicta are construed as in modal logic or in adverbialist theories, no such commitment is made, because that-sentences are no longer treated as singular terms. I shall defend the view that that-sentences should not be construed as singular term and present propose a version of this view that avoids certain well known problems of operator and adverbialist accounts.

Modal logic is a well-established field, and the possible worlds semantics of modal logics has proved invaluable to our understanding of the logical features of the modal concepts such as possibility and necessity. However, the significance of possible worlds models for a genuine theory of meaning--let alone for metaphysics--is less clear. In this paper I shall explain how and why the use of the concepts of necessity and possibility could arise (and why they have the logical behaviour charted out by standard modal logics) without either taking the notions of necessity or possibility as primitive, and without starting with possible worlds. Once we give an account of modal logic we can then go on to give an account of possible worlds, and explain why possible worlds semantics is a natural fit for modal logic without being the source of modal concepts.

In his classic 1975 paper "Outline of a Theory of Truth", Kripke commented:

" the necessity to ascend to a metalanguage may be one of the weaknesses of the present theory. The ghost of the Tarski hierarchy is still with us." (Kripke 1975, p. 74).

In this talk, I shall ramble on about several inter-related topics, associated with the semantic paradoxes: (1) the inconsistency of the naive disquotational theory of truth; (2) Tarski's indefinability theorem (and other related results, and generalizations); (3) the possibility or otherwise of semantic closure; (4) the hierarchy resolution(s) of the paradoxes; (5) the notion of "interpreted language L"; (6) the relation "interpreted language L1 is a meta-language for interpreted language L2"; (7) the "Revenge Problem".

The Tarski-hierarchy proposal is frequently misunderstood by those who seem to think that Tarski *imposed* a hierarchy on semantic notions, as an antecedent constraint. However, this is not true. Quite the opposite. Tarski *proved a theorem*---namely, the indefinability theorem---which shows that for a certain class of interpreted languages L, the set of truths in L is *not* definable in L, although it is clear that the set of truths is definable in an extension of L. More generally, the result shows that a sufficiently rich language L cannot be its own meta-language. Some relevant notions require further clarification. I think that the most important of these are the notions: "L is an interpreted language" and "L1 is a meta-language for L2".

The primary reason for insisting on the Tarski-hierarchy resolution is negative: all non-hierarchy attempts to resolve the paradoxes (with the possible exception of Graham Priest's dialetheism) seem to lead to a Revenge Problem. So, does the Tarskian hierarchy solution have a Revenge Problem; and does dialetheism have a Revenge Problem?

I have some thoughts on these matters, although I do not have a satisfying conclusion. I suspect that the semantic closure problem is related to Dummett's notion of "indefinite extensibility", in the sense that truth is always an indefinitely extensible concept (just as, plausibly, the notion of collection is indefinitely extensible). However, at the moment, I do not understand Dummett's notion very well.

I shall defend the remarkable thesis that I knew what I was doing when I agreed to be an expert witness for the defence in the recent Pennsylvania trial concerning ID. Along the way, I shall discuss the role of philosophers as experts on the nature of science, the status of intelligent design as science, and the problems of arguing these matters in a fraught political environment that is mediated by the idiosyncrasies of the US legal system. I shall also sketch what I regard as an intellectually interesting strategy for the future of ID, though I claim no authority over those who are best placed to execute it.

In this paper, we investigate what can be salvaged from the original project of "Logicism" and what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic used in the reduction, (2) allow the addition of analytic-sounding principles to logic so that the reduction is not to "logic alone" but to logic and truths knowable a priori, and (3) revise the conception of "reducible". We try to show how the current versions of neologicism fit into this classification scheme, and then focus on a kind of neologicism which we take to have most potential for achieving the epistemological goals of the original logicist project. We argue that that the "weaker" the system, the more likely it is to be a new form of logicism, and show how our preferred system, though extremely weak, can nevertheless "reduce" arbitrary mathematical theories to logic and analytic truths, if given a legitimate new sense of "reduction".

Simulations are useful for exploring the implications of a theory in a situation which we are unable to investigate directly. This may be because it refers to past or future events, or to situations which cannot be realised by real-world methods of investigation. Recent work in social simulation enables us to model the emergent behaviour of systems of actors whose complexity defeats both statistical methods and traditional linear mathematics. Suppose we had a simulation that could model scientific activity, such that philosophers, historians, sociologists, psychologists -- and even scientists – could agreed on its adequacy. Thus, for example, to understand how any rule of inference would work in practice, it would have to be implemented in a way that reflects the socially situated character of scientific thinking.

In this talk I describe a simulation which models features of science such as inferences based on changing evidence and communication between actors. Central to this approach is a model of experimentation conducted by agents who can interact with the world (via experiments) and with each other (by exchanging opinions). Simulation is iterative, allowing us to play out the consequences of our assumptions given certain properties of a situation and certain attributes of actors, e.g., that an actor is biased, or insensitive to the opinions of others, or has access only to particular actors or experiments. In this way we can evaluate the assumptions made in our models of science by evaluating their consequences. I will explore some implications for philosophy of science of this iterative approach to modelling science.

Formmalist accounts of mathematics, particularly pre-Hilbertian 'game formalism' of the type scathingly attacked by Frege, have fallen into disrepute. Most of Frege's criticisms are just and a fatal objection arises from the observation that the ontology of a formal calculus or game is itself abstract, essentially that of platonistic arithmetic.

I sketch a variant of game formalism immune, I claim, from these objections.The account of maths is a two-tiered one in which at the base level the sentences of the calculus lack truth-values, whilst at the higher contentual level, the level at which real mathematical assertions occur, the sentences are truth-valued. The truth-conditions are given by 'concrete' provability at the lower level, though these truth-conditions do not figure in the informational or literal content of the contentual mathematical assertions.

From 1880-1920, the question ‘What is Electricity?’ attracted a score of publications and much wider attention in popular science writing. This was not (merely) an outbreak of public metaphysical curiosity. For householders considering whether to install Edison and Swan’s electric light, the nature of electricity became a pressing concern. Painfully accustomed to both fraudulent water supply and lethal gas explosions, householders sought clarification on the commodity they would be paying for and the risks that arose in admitting this intangible agency to their homes. However, publications that purported to address the ontology of electricity typically evaded giving an answer, offering instead diversionary accounts of electromagnetic ether theory or of accomplishments (past or future) in electrical engineering. My paper explains that this was because contemporary electrical science simply could not give an unequivocal answer to the pressing question. Controversy lingered long among theorists as to whether electricity consisted of one fluid, two fluids, a mode of motion, a form of energy or a kind of entity hitherto unknown, and the question remained moot even after the advent of electron theory. I suggest that rather than resolving the nature of electricity, early twentieth century specialists sought to deal with such awkward questions from the public by limiting their disciplinary prerogatives to exclude what they construed as difficult ‘metaphysical’ matters – albeit considered rather differently by the public as matters of contractual specification pertaining to commercial consumption..

I am currently writing a short book for a popular audience that argues that the Middle East—more particularly southern Iraq—was the centre of three key developments in numeracy, all of which reverberated around the world and had a lasting impact on global society and intellectual culture.

First, the invention of recorded number enabled complex economies to be managed, and thus for cities to be developed in the late fourth millennium BC. Second, the development of the powerful and flexible sexagesimal place value system in about 2000 BC enabled abstract mathematics including algebra and geometry, and eventually opened the way to sophisticated mathematical astronomy in the Middle East and to the time-keeping system based on 60 that we know today. Third, Al-Khwarizmi’s clear explanation of the Indo-Arabic numeral system (which had previously been used only by Indian astronomers) brought about its adoption in the scientific communities of the Middle East in the ninth century and eventually into Europe and the rest of the world. Meanwhile, folk traditions of abacus and finger counting continued in the Middle East, as in the rest of the world, in commercial and practical contexts where recording was not important.

In the talk I shall touch on some of these key historical questions, and also discuss the challenges I am facing in making what has been considered a very abstruse branch of the history of science accessible and interesting to the general public.