A widely held metaphysical myth is that there is the way the world is, and that the way that it is can be characterized or represented in purely nominalistic terms. The indispensability of mathematics to the languages of the sciences shows definitively that this is false. There are important implications of these facts for the practice of metaphysics. I try to deflect the possibility that the indispensable use of mathematics in the sciences screens us off from what the world itself is like; I try to deflect the Kantian possibility that scientific descriptions of the world can only provide appearances (descriptions of the world that are false to things in the world themselves because they must be characterized in the language of applied abstracta). That is, I mean to deflect the possibility that the way the world really is is a thing-in-itself that’s out of reach of our abstract-saturated scientific descriptions of things in the world. I try to show that instead we can characterize aspects of the world that are really out there, and that we can do this despite our scientific languages enabling these characterizations only in terms of applied mathematical formalisms.
It is a widespread view that inferences can be either deductive, that is, necessarily truth preserving, or ampliative, that is, not necessarily truth preserving. This view, however, is inadequate because there are inferences, such as abductive inferences, which are neither ampliative nor truth preserving. In this paper an alternative classification of inferences is proposed, as well as a justification of both deductive, non-deductive and abductive inferences which takes into account their role in knowledge, distinguishing their justification from their usefulness. It is argued that the justification of deductive, non-deductive and abductive inferences raises similar problems and is to be approached much in the same way.
This paper focuses on Galen’s (2nd century CE) distinctive views about the epistemic status of medicine. I will argue that, while taking the Aristotelian views as his starting point (see An.Pst. I; Metaph., I.1-2), Galen develops a more extended and flexible conception of scientific knowledge, adapted to the distinctive features of medicine. He regards medicine as a demonstrative science that also involves empirical and conjectural features when its general theorems come to be applied to individual patients. The use of logical methods, however, makes the good doctor able to minimise the possibility of erring in his practice. Interestingly, then, Galen seeks to integrate the treatment of empirical and contingent matters within the domain of demonstrative knowledge. Hence, e.g., Galen's interest in a (still rudimentary and non mathematical) idea of probability, according to which the stochastic and accidental character of medicine can be seen as approximating to truth and certain knowledge. This is closely connected with the further anti-aristotelian idea that science should extend to sensible particulars (in fact, Galen’s comes close to the view that each man is determined by a quasi-leibnizean individual form corresponding to the distinctive ratio of its elementary components). Finally, Galen argues that logical demonstrative methods have an intrinsic heuristic value and he seeks to transpose the ‘analytical’ geometrical method of resolution of problems into the domain of medicine.
Herbert Simon’s approach to the study of rational choice was originally based on a criticism of classical game theory as it was usually applied to economics and organization science. The model of rational choice he pointed out was that of “bounded rationality”, which profoundly influenced both Artificial Intelligence and cognitive science. Once it was applied to Darwinian evolution, game theory raised much controversies, and Simon’s claims on bounded rationality and related concepts (e.g., docility) seem to converge with some criticisms to the application of classical game-theoretical models to evolutionary theory. Simon’s claims might thus be considered as more than a starting point for evaluating an integrated approach to the study of behavior.
In Knowledge and its Limits Timothy Williamson maintains that «knowing is the most general factive stative attitude, that which one has to a proposition if one has any factive stative attitude to it at all». In a language the characteristic expression of a factive stative attitude is a factive mental state operator (FMSO). Williamson’s proposal is that «if F is any FMSO, then ‘S Fs that A’ entails ‘S knows that A’». However, Williamson does not prove that every FMSO conforms to his principle that factive-stative attitudes entail knowledge. I shall consider a possible counterexample. If it is a genuine counterexample, knowledge is not the most general factive stative attitude.
According to many philosophers and scientists, we are on the verge of solving some of the most venerable philosophical issues, thanks to the astonishing progress of the neurosciences. I will in particular discuss some proposals that concern free will and the nature of morality. My general conclusion will be that, if we have many reasons for believing that neurobiology can enrich our understanding of the features of the human mind, there is no sound reason for thinking it will ever explain them all.
In this paper I consider various conceptions of rigor, what the benefits of rigor so conceived are, and have been supposed to be, what role(s) logical reasoning has been taken to play in the attainment of rigor and whether and/or under what conditions it may indeed serve in such a role(s). Special attention will be given to the relationship between rigor and the formalization of logical reasoning.
The empiricist view of logic is the view that logic is not justified a priori, but by its empirical success in applications. It is claimed that this view was first introduced by Quine in his famous 1951 paper: 'Two Dogmas of Empiricism'. Quine used the example of a new logic for quantum mechanics which had earlier been suggested by Birkhoff and von Neumann. This example was later taken up and developed by Putnam. In this paper, however, it is suggested that quantum logic does not give a decisive argument in favour of the empiricist view of logic, but that such an argument is provided by the successful application of non-classical logics in artificial intelligence (AI).
Logic must treat its terms (S,P,M) or propositions (p, q, r) as if they were homogeneous, to exhibit forms of valid deductive inference. But the kinds of representations that make successful reference possible and those that make successful analysis possible in mathematics and the sciences are often not the same, so that significant scientific and mathematical claims (or demonstrations) often juxtapose heterogeneous terms (or propositions).This disparity calls for a philosophical critique of logic.
The "philosophy of mathematical practice" is a segment or aspect of the philosophy of human inquiry. Many great mathematicians, from Cayley to Atiyah, reported what they did and experienced, even though few philosophers seemed to listen. On the other hand, while John Dewey and his followers carefully describe the process of inquiry in general, they hardly mention how that theory includes mathematical inquiry. The validity of heuristic and computational reasoning in mathematics – both pure and applied – is a glaring problem and paradox, crying out for explication.
The relationship between data and hypotheses is the core of the process ofampliation of knowledge. I argue that understanding such a relationship and such a process requires revising the related notions of inference, logic and knowledge from an informationalpoint of view, examining how information is generated, extracted, processed and transferred. In order to do that, a concept of information different from the classic one is needed. I will analyze the relationship between data and hypotheses from an informational point of view using some examples from sciences.
85 years ago David Hilbert delivered his famous lecture “On the Infinite”. He invoked Kant’s philosophy of mathematics to endorse the need for intuitive finitary reasoning in mathematics. And he invoked Kant’s notion of an “unconditioned” idea of reason as a model for his treatment of completed mathematical infinity. His point: The set theoretic paradoxes arise from treating the ideal as real, and that is what Kant taught us not to do. I am going to follow Hilbert in both of those moves: I will discuss Kant's analysis of the notion of a finite grasp, and I will look at his view about the problems that arise when we mix certain ideal ("unconditioned") ideas of reason into the realm of the empirically real. But rather than apply these moves to the mathematical infinite, I will apply them to problems that arise in finite discourse. I expect to discuss Moore's Paradox ("The cat is on the mat, but I don't know it") and the hangman paradox. I may also look at issues surrounding the "Master Argument" of Diodorus Chronos.
There is no place for Aristotle's distinction between perfect and imperfect syllogisms in the main stream of modern logic. Yet, a distinction of that kind is needed, if we are to say what a proof is, or if we are to explain how the use inferences can generate knowledge. While it is generally agreed that the question whether a certain logical relation obtains between some given sentences depends on the meaning of the sentences, an account of the concept of perfect syllogism seems to depend on the more controversial idea that the meaning of a sentence is somehow based on what is required to justify an assertion of the sentence. Some possibilities of giving such a systematic account will be discussed.
A comparison is made between the tree-like representations for the grounding of knowledge and of truths that are offered by Frege (Grundlagen §§ 3,4 and 17) and Bolzano (Wissenschaftslehre §220), while drawing upon historical precedents in Aristotle and Leibniz. The constructive truth-maker analysis via proof-objects is used to explicate two versions of grounding: one decidable and one infinite, but containing canonical grounds only. An analogy of the latter to Brouwer’s Bewiesführungen in his demonstartion of the Bar Theorem is noted.
In the 2008 paper 'Extreme Science: Mathematics as the Science of relations as such', I assimilated mathematics to science. In this paper I discuss the operation of assimilation as a ubiquitous phenomenon of ordinary language that is almost missing from current informal mathematical language. I point to assimilations in history that would not happen today, abandoned assimilations that were failures to make important distinctions, and current assimilations that are controversial. These examples lead me to suggest that the lack of assimilations in mathematical practice is an important reason for the dependability of arbitrarily long chains of reasoning uniquely in mathematics, a feature that is striking and sometimes considered mysterious.
A text is written as a linear succession of words that make up sentences, but modern theories of sentence structure represent sentences in a tree form. The way sentences compose into a deductive argument in logic and mathematics has likewise been written linearly, but modern theories of proof represent such arguments in tree form. It is shown mainly through historical examples, from Aristotle, Frege, Hilbert and Bernays, Hertz, Gentzen, and Jaskowski, that the tree form has decisive advantages over a linear arrangement.
The lecture will consider Michael Dummett’s contention, in The Logical Basis of Metaphysics, that semantic theories should be formulated in such a way that the logic of the object-language is maximally insensitive to the logic of the meta-language. The controversy over the status of the Barcan formula and its converse in quantified modal logic will be taken as a case study. It will be argued that the insensitivity in question is a less desirable feature of a semantic theory than Dummett suggests.
- Part 1 - Part 2 - Part 3 - Part 4 - Social dinner