Online research in philosophy

Although all mathematical truths are necessary, mathematicians take certain combinations of mathematical truths to be ‘coincidental’, ‘accidental’, or ‘fortuitous’. The notion of a ‘mathematical coincidence’ has so far failed to receive sufficient attention from philosophers. I argue that a mathematical coincidence is not merely an unforeseen or surprising mathematical result, and that being a misleading combination of mathematical facts is neither necessary nor sufficient for qualifying as a mathematical coincidence. I argue that although the components of a mathematical coincidence may possess a common explainer, they have no common explanation; that two mathematical facts have a unified explanation makes their truth non-coincidental. I suggest that any motivation we may have for thinking that there are mathematical coincidences should also motivate us to think that there are mathematical explanations, since the notion of a mathematical coincidence can be understood only in terms of the notion of a mathematical explanation. I also argue that the notion of a mathematical coincidence plays an important role in scientific explanation. When two phenomenological laws of nature are similar, despite concerning physically distinct processes, it may be that any correct scientific explanation of their similarity proceeds by revealing their similarity to be no mathematical coincidence.

A view of the sources of mathematical knowledge is sketched which emphasizes the close connections between mathematical and empirical knowledge. A platonistic interpretation of mathematical discourse is adopted throughout. Two skeptical views are discussed and rejected. One of these, due to Maturana, is supposed to be based on biological considerations. The other, due to Dummett, is derived from a Wittgensteinian position in the philosophy of language. The paper ends with an elaboration of Gödel's analogy between the mathematician and the physicist.

Many people think that mathematical models are built using well-known “mathematical things” such as numbers and geometry. But since the 19th century, mathematicians have investigated various non-numerical and non-geometrical structures: groups, fields, sets, graphs, algorithms, categories etc. What could be the most general distinguishing feature that would separate mathematical models from non-mathematical ones? I would describe this feature by using such terms as autonomous, isolated, stable, self-contained, and – as a summary – formal. Autonomous and isolated – because mathematical models can be investigated “on their own” in isolation from the modeled objects. And one can do this for many years without any external information flow. Stable – because any modification of a mathematical model is qualified explicitly as defining a new model. No implicit modifications are allowed. Self- contained – because all properties of a mathematical model must be formulated explicitly. The term “formal model” can be used to summarize all these features.

(I) MATHEMATICAL LECTURES. LECTURE I. Of the Name and general Division of the Mathematical Sciences. BEING about to treat upon the Mathematical Sciences, ...

Baker (2005) claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given without postulating mathematical objects and thus cannot be used by the mathematical realist. I also show that, despite this, mathematics keeps playing an important methodological role in the explanation and does not reduce to a merely computational or descriptive framework.

In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathematics has important consequences on our conception of mathematical knowledge, in particular mathematical growth.

Penelope Maddy’s original solution to the dilemma posed by Benacerraf in his (1973) ‘Mathematical Truth’ was to reconcile mathematical empiricism with mathematical realism by arguing that we can perceive realistically construed sets. Though her hypothesis has attracted considerable critical attention, much of it, in my view, misses the point. In this paper I vigorously defend Maddy’s (1990) account against published criticisms, not because I think it is true, but because these criticisms have functioned to obscure a more fundamental issue that is well worth addressing: in general – and not only in the mathematical domain – empiricism and realism simply cannot be reconciled by means of an account of perception anything like Maddy’s. But because Maddy’s account of perception is so plausible, this conclusion raises the specter of the broader incompatibility of realism and empiricism, which contemporary philosophers are frequently at pains to forget.

Moral theology is given force by punishment and reward, which is, in turn, comprehensible only in the presence of free will. Yet free will has been bedevilled with philosophical difficulties, not least among them the tension between omniscience and autonomy. The paper, building on a theory of temptation and sin published in /Mind/, gives a partial resolution to that tension using a mathematical argument.

This piece argues that neither character nor the lure/allure of the tempting object/subject may matter nearly so much as the structure--the mathematical structure--of the tempting situation. The implicature of this finding is that, perhaps, neither rehabilitation (which oftentimes just doesn't work even when--and after--it appears to work) nor retribution is the proper purpose of incarceration--rather simple incapacitation is. The effect of the finding is to dramatically limit the scope within which free will is actually operative in a range of situations. A further implicature of this finding is that so-called "sting operations" are completely immoral.