Frege says, at the end of a discussion of formalism in the Foundations of Arithmetic, that his own foundational program “could be called formal” but is “completely different” from the view he has just criticized. This essay examines Frege’s relationship to Hermann Hankel, his main formalist interlocutor in the Foundations, in order to make sense of these claims. The investigation reveals a surprising result: Frege’s foundational program actually has quite a lot in common with Hankel’s. This undercuts Frege’s claim that (...) his own view is completely different from Hankel’s formalism, and motivates a closer examination of where the differences lie. On the interpretation offered here, Frege shares important parts of the formalist perspective, but differs in recognizing a kind of content for arithmetical terms which can only be made available via proof from prior postulates. (shrink)

Frege famously held that numbers play the role of objects in our language and thought, and that this role is on display when we use sentences like "The number of Jupiter's moons is four". I argue that this role is an example of a general pattern that also encompasses persons, times, locations, reasons, causes, and ways of appearing or acting. These things are 'objects' simply in the sense that they are answers to questions: they are the sort of thing we (...) search for and specify during investigation or inquiry. I support this epistemological conception of objects by studying specificational sentences, a class of sentences which includes Frege's original example. I defend an analysis of such sentences as question-answer pairs, and show how to formally represent this analysis using game-theoretical semantics. (shrink)

Mathematical formalism is the the view that numbers are "signs" and that arithmetic is like a game played with such signs. Frege's colleague Thomae defended formalism using an analogy with chess, and Frege's critique of this analogy has had a major influence on discussions in analytic philosophy about signs, rules, meaning, and mathematics. Here I offer a new interpretation of formalism as defended by Thomae and his predecessors, paying close attention to the mathematical details and historical context. I argue that (...) for Thomae, the formal standpoint is an _algebraic perspective_ on a domain of objects, and a "sign" is not a linguistic expression or mark, but a representation of an object within that perspective. Thomae exploits a shift into this perspective to give a purely algebraic construction of the real numbers from the rational numbers. I suggest that Thomae's chess analogy is intended to provide a model for such shifts in perspective. (shrink)

In this paper we look at the manual analysis of arguments and how this compares to the current state of automatic argument analysis. These considerations are used to develop a new approach combining a machine learning algorithm to extract propositions from text, with a topic model to determine argument structure. The results of this method are compared to a manual analysis.

BĪRŪNĪ, ABŪ RAYḤĀN MOḤAMMAD b. Aḥmad (362/973- after 442/1050), scholar and polymath of the period of the late Samanids and early Ghaznavids and one of the two greatest intellectual figures of his time in the eastern lands of the Muslim world, the other being Ebn Sīnā.

We show how faceted search using a combination of traditional classification systems and mixed-membership topic models can go beyond keyword search to inform resource discovery, hypothesis formulation, and argument extraction for interdisciplinary research. Our test domain is the history and philosophy of scientific work on animal mind and cognition. The methods can be generalized to other research areas and ultimately support a system for semi-automatic identification of argument structures. We provide a case study for the application of the methods to (...) the problem of identifying and extracting arguments about anthropomorphism during a critical period in the development of comparative psychology. We show how a combination of classification systems and mixed-membership models trained over large digital libraries can inform resource discovery in this domain. Through a novel approach of “drill-down” topic modeling—simultaneously reducing both the size of the corpus and the unit of analysis—we are able to reduce a large collection of fulltext volumes to a much smaller set of pages within six focal volumes containing arguments of interest to historians and philosophers of comparative psychology. The volumes identified in this way did not appear among the first ten results of the keyword search in the HathiTrust digital library and the pages bear the kind of “close reading” needed to generate original interpretations that is the heart of scholarly work in the humanities. Zooming back out, we provide a way to place the books onto a map of science originally constructed from very different data and for different purposes. The multilevel approach advances understanding of the intellectual and societal contexts in which writings are interpreted. (shrink)

What does it mean to ‘give’ the value of a variable in an algebraic context, and how does giving the value of a variable differ from merely describing it? I argue that to answer this question, we need to examine the role that giving the value of a variable plays in problem-solving practice. I argue that four different features are required for a statement to count as giving the value of a variable in the context of solving an elementary algebra (...) problem: the variable must be in the scope opened by the problem statement; the values given must be in the range of the variable, which is determined by the problem; the statement giving the values must represent a complete solution; and it must be in a canonical form. This account helps us better understand elementary algebra itself, as well as the use of algebraic tools to analyze phenomena in natural language. (shrink)

This thesis modifies the wording of William Clifford’s 1877 evidence principle (that ‘it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence’) to propose an explicitly moral principle, restricted to descriptive beliefs (about what is or is not the case) and excluding prescriptive beliefs (about what ought or ought not to be the case). It considers potential counter-examples, particularly William James’s 1896 defence of religious belief; and concludes that the modified principle survives unscathed. It then searches (...) for suitable criteria for ‘sufficient’ and ‘insufficient’ evidence, first within a Bayesian framework enriched by pragmatic considerations, and finally by returning to Clifford’s conception of our shared responsibilities to our shared epistemic asset. Combined with elements of Edward Craig’s (1990, 1999) ‘state-of-nature’ theory of knowledge this gets us to a minimum threshold to avoid insufficient evidence and an aspirational criterion of sufficient evidence. (shrink)

By extending Husserl’s own historico-critical study to include the conceptual mathematics of more contemporary times – specifically category theory and its emphatic development since the second half of the 20th century – this paper claims that the delineation between mathematics and philosophy must be completely revisited. It will be contended that Husserl’s phenomenological work was very much influenced by the discoveries and limitations of the formal mathematics being developed at Göttingen during his tenure there and that, subsequently, the rôle he (...) envisaged for his material a priori science is heavily dependent upon his conception of the definite manifold. Motivating these contentions is the idea of a mathematics which would go beyond the constraints of formal ontology and subsequently achieve coherence with the full sense of transcendental phenomenology. While this final point will be by no means proven within the confines of this paper it is hoped that the very fact of opening up for the possibility of such an idea will act as a supporting argument to the overriding thesis that the relationship between mathematics and phenomenology must be problematised. (shrink)

Hossack’s project in this book is to provide a new foundation for the philosophy of number inspired by the traditional idea that numbers are magnitudes.

In §57 of the Foundations of Arithmetic, Frege famously turns to natural language to support his claim that numbers are ‘self-subsistent objects’:I have already drawn attention above to the fact th...