Abstracts

The Birth of Philosophy of Mathematics: Out of the Spirit of Neo-Kantianism

Bernd Buldt

IPFW, Department of Philosophy

The general thesis underlying this talk is that a historically informed view can no longer subscribe to a number of assumptions we find embraced without much ado by a majority of those who work in the philosophy or mathematics.  In particular the assumption of a continuity of subject  matters doesn't survive closer scrutiny; neither the objects of mathematics nor the philosophical reflection on them display a continuity that would justify expressions like ``continuity from Aristotle to Atiyah'' or ``the philosophy of mathematics from Plato to Putnam.''

The more specific thesis I shall try to defend is that philosophy of mathematics started its career around 1800 as a short-lived creature of Kant's epistemology and sunk into oblivion thereafter before it was reanimated by Carnap and other logical empiricists in the early 20th century.  On both occasions, Kantians and logical empiricists (Carnap) had doctrinal reasons for assigning mathematics a distinguished epistemological status that would elevate it above the sciences. These assumptions, as I hope to show, are not compelling but have a number of repercussions for how we conceive of philosophy of mathematics today.

Thinking and Logic in Wittgenstein’s Tractatus

Leo K. C. Cheung,  Hong Kong Baptist University

The Tractatus relates thinking intimately to pictorial representation.  Thinking is making pictures of facts, or thoughts.  A thought is a projection of a state of affairs.  The method of projection is to think of the sense of a thought.  When a thought is expressed perceptibly, it is a (linguistic) proposition.  In this way, the picture theory ties thinking, thought and language together. 

The picture theory also accounts for the fact that thinking is subject to the laws of logic.  The reason for this is that logic is the logic of pictorial representation.  This is crystallised in the thesis that the general propositional form is the general form of a logical operation.  The thesis amounts to saying that the general rule of pictorial representation (language or thought) and the general rule of thinking are the same general rule of logical operation. 

            In this paper, I shall substantiate these views of the Tractatus, and explain Wittgenstein’s reasons for them.  I shall argue, first, that the Tractarian view of pictorial representation can be given a group-geometric interpretation, understood in the sense of Felix Klein’s Erlanger Programm.  Second, the set {N, NN}, where N is the sole fundamental operation introduced in TLP 5.5, equipped with the natural composition operation forms a (mathematical) group.  The invariants of the group are the propositional forms.  It is a matter of fact that, amongst N and NN, we have chosen NN as the general rule of pictorial representation.  This will explain why Wittgenstein thinks the general propositional form is the general form of a logical operation, as well as his view of the unity of logic, language and thinking.  

Indexical Thinking and Meta-representationality

Nevia Dolcini, University of Macerata

The debate on the Theory of Mind (ToM) is currently very alive both in philosophical psychology and developmental psychology. To have a ToM is to be able to share and understand another’s experiences or, in other words, to behave and communicate inter-subjectively. According to most of the theorists, children are in possession of a ToM at the age of 4, a period in which they can use verbal language and in particular they show a proper usage of indexical terms. This result suggests a strong connection between indexicals and the meta-representational capacities (“I think that you think that I think...”) that are necessary to one’s possession of a ToM. Is there any element in the communicative competence shown by children younger than 4 that could be taken as an evidence of a ToM? I shall argue that even in preverbal children it is possible to find a communicative behavior, the indexical gesture, that requires a meta-representational capacity. Thus, given the functional analogy between indexical terms and indexical gestures, we shall find that preverbal children can be tested for ToM by appealing to whether they succeed in comprehending and appropriately producing indexical gestures.

The Theory of Quantity in Principia Mathematica

Sébastien Gandon, PHIER, Université Blaise Pascal, Clermont (France)

The last published part of Principia Mathematica (1913) is devoted to quantity and measurement. This work has not been much          studied in the literature, and this is quite surprising, considering the great importance of the topic. Indeed, Russell and Whitehead developed, in this part, their definition of real numbers, and constructed a very general and very refined theory of the application of numbers to the world. This theory was supposed to provide with a transition with geometry. Moreover, the mathematician N. Wiener, who has been one of Russell’s students, was deeply impressed by this work, whose he contributed to extend in an important series of papers.

My talk aims at presenting the main lines of Principia’s theory of quantity. I will try to link this work both with some preoccupations in the contemporary mathematical sciences, and with Russell’s and Whitehead’s overall project. I would like to draw attention about the fact that the content of Principia, part VI, seems to compel us to revise some deep-rooted opinions about what Russell’s logicism was.

Frege’s Private I

Tomis Kapitan, Northern Illinois University

At one point in his essay Gedanke, in the course of talking about the significance of the first-person, Frege wrote that “everyone is presented to himself in a particular and primitive way in which he is presented to no one else.”  Accordingly, thoughts constituted by such modes of presentation cannot be communicated.  This passage has been problematic for commentators, some of whom argue that Frege is wrong to think that there are private first-person senses (Perry 1977) and others who contend that it is mistaken to think that Frege is endorsing private first-person senses (May 2006). 

I will not speculate on Frege’s actual view of this matter, but I think that there is space within the overall Fregean framework for private reference-determining senses that correspond to indexicals, senses that, while irreducibly indexical, are fully descriptive (contrary to the suggestion of some, e.g., Evans 1985).  Indexicality involves identifying and referring to items in virtue of the spatio-temporal perspective of those items within one’s experience, a perspective that is inescapably private yet communicable.  To maintain this perspectival view, however, requires abandoning the conception that communication through indexicals requires the speaker and hearer to have access to identical thoughts.  Instead, indexical communication is fully explainable through systematic coordination among distinct thoughts.  It remains to be seen whether this view can be reconciled with Frege’s insistence upon the objectivity of thoughts.

Kevin Klement, University of Massachusetts, Amherst

Most advocates of the so-called "neo-logicist" movement in the philosophy of mathematics identify themselves as "Neo-Fregeans" (e.g. Hale and Wright): presenting an updated and revised version of Frege's form of logicism. Russell's form of logicism is scarcely discussed in this literature, and when it is, often dismissed as not really logicism at all (in lights of its assumption of axioms of infinity, reducibiity and so on). In this paper I have three aims: firstly, to identify more clearly the primary metaontological and methodological differences between Russell's logicism and the more recent forms; secondly, to argue that Russell's form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that Neo-Russellian forms of neo-logicism remain viable positions for current philosophers of mathematics.

Russell's Schema and Self-referential Paradoxes

Gregory Landini, The University of Iowa

Investigating a theorem Russell used in discussing of paradoxes of classes, Graham Priest distills a schema and then extends it to form an Inclosure Schema which he argues is common structure underlying both class-theoretical paradoxes (such as that of Russell, Cantor, Burali-Forti) and the paradoxes of “definability” (offered by Richard, König-Dixon and Berry). This paper shows that Russell’s theorem is not Priest’s schema and questions the application of Priest’s Inclosure schema to the paradoxes of “definability”

Mathematical Reasoning, Following Peirce and Frege

Danielle Macbeth, Haverford College

Mathematicians often use visual metaphors in talking about their mathematical practice. If Peirce is right, such talk is not merely metaphorical; the necessary reasoning of mathematicians is, he thinks, essentially diagrammatic. My aim is, first, to clarify the nature of diagrammatic reasoning in the paradigm case: Euclidean geometry. I then show how reasoning in Frege’s logical language Begriffsschrift can be seen to be diagrammatic in essentially the same sense. In the end, we will see, Frege and Peirce together enable us to understand how reasoning from concepts can be fruitful or ampliative despite being strictly deductive, through reason alone.

With Charity Toward None: A Fond Look at the Philosopher’s Principle of Charity

Charles McCarty, Indiana University (Bloomington)

 A series of short — and sweet, for that — arguments to the conclusion that the Principle of Charity known, or unknown, to recent analytical philosophers is false without much qualification. This, at least, sets the philosophers’ principle off from principles of the same name current in rhetoric and informal logic. It also allows us to touch critically upon a batch of related topics, clouded by verbal mist, among them radical interpretation and the indeterminacy of translation.

On Quine’s Two Concepts of Truth: Immanent and Transcendent

Chienkuo Mi,  Soochow University, Taiwan

The concept of truth is ambiguous in the philosophical jargon. Since Tarski showed how we can avail ourselves of the truth predicate by his theory of truth, the concept of truth has been understood in two different ways. On the one hand, the truth predicate involved in the T-sentences, say “‘snow is white’ is true if and only if snow is white”, can be interpreted as truth for (or relative to) a particular language. The concept of truth is indeed understood as “truth-in-L” for any particular language, and “truth-in-x” for the language x and “truth-in-y” for the language y are two different concepts. On the other hand, the truth predicate involved in the T-sentences can also be interpreted as truth for all particular languages. That is to say, the concept of truth is understood as a general concept that can be applied to every particular language x, y or z.

Quine has made use of Tarski’s theory of truth and developed his own disquotational view regarding the concept of truth. According to Quine’s view, the truth predicate is a device of disquotation. The truth predicate has the cancellatory force in Tarski’s paradigm: “‘Snow is white’ is true if and only if snow is white”, which is to say that ‘snow is white’ is true is just to say that snow is white. Ascription of truth just cancels the quotation marks. However, Quine recognizes that the disquotational feature of truth has to be immanent: to call a sentence true is just to include it in our language, in our own theory of world, or in our science. We understand what it is for the sentence ‘Snow is white’ to be true as clearly as we understand what it is for snow to be white in our language or with respect to our theory of world. Truth in its immanent sense is transparent. But neither our language nor our science can fix truth. Quine knows pretty well that truth should hinge on reality but not language, and that our theory of world can be proved wrong. It seems to be this concern that leads Quine to puzzle over a transcendent sense of truth, and allows the kind of truth to be something that scientists are always in quest of, or something that “looms as a heaven that we keep steering for and correcting to”.

It will be shown that Quine’s immanent concept of truth should be understood as “truth-in-L” in Tarski’s theory (or at least one of the two senses involved in the T-paradigm), and the transcendent concept of truth can be identified as the general concept of truth which is supposed to be applied to any particular language. But the question is: are the two concepts of truth different in kind or just different in degree? If the immanent concept of truth and the transcendent concept of truth are two different kinds of truth, then, we may wonder, what will be the relationship between the two? Does the understanding of the one help to explain the other? Or are they totally different and unrelated? Worse still, we may feel, will there be a tension between two of them—one is transparent and the other is relatively mysterious in nature? If the two concepts of truth are only different in degree, then with respect to what aspect or based on what criterion are the two concepts closely connected and then measured or evaluated? I will argue that both of Quine’s immanent and transcendent concepts should be accommodated at the linguistic level or in the semantic project, rather than approaching them from a metaphysical point of view.  

Francesco Orilia, The University of Macerata 

 The phenomenal entities that we directly experience in perceptions, dreams and hallucinations tend to be viewed as essentially private and ephemeral (fleeting), i.e., necessarily incapable of being directly experienced by more than one subject and incapable of re-occurring more than once. Among phenomenal entities are the “object-shaped” gestalts studied in Gestalt psychology. The traditional ontological dichotomy of universals and particulars is appealed to, in order to make a distinction between (phenomenal) gestalt-types and gestalt-tokens. It is then proposed that the former are not essentially private and ephemeral. As regards the latter, it is argued that they are indeed essentially private, but ephemeral at most only in a contingent sense. The The relevance of these claims for our perceptual judgments about external objects is briefly investigated at the end of the paper.

Chris Pincock, Purdue University

George IV and the Law of Identity

Graham Stevens, University of Manchester 

Russell’s theory of descriptions is intended by Russell to explain how George IV can sensibly ask whether Scott is the author of Waverley without thereby expressing a desire to know if Scott is Scott (as Russell says, without merely expressing an interest in the law of identity). Saul Kripke and Scott Soames have both pointed out that Russell’s argument is flawed: on the de re reading, we get ‘The x such that x authored Waverley is such that George IV wanted to know if x = Scott’, but here George has a de re attitude towards the value of the variable x, and that value is Scott. So George is interested in the truth-value of the proposition that Scott = Scott, and hence appears stuck with an interest in the law of identity after all. In this paper I consider and reject Nathan Salmon’s proposed solution to the problem, which involves distinguishing the proposition that a = a from the proposition that a has the relational property of self-identicality. I defend the view that the de re reading is best understood as attributing to George a demonstrative thought, and that the cognitive significance of this thought attaches to its character, not its content. This view is also problematic for reasons raised in Salmon’s book Frege’s Puzzle but I argue that (1) this particular case is unproblematic, and (2) consideration of this case reveals that the source of the general problem lies in Russell’s insistence on the principle of acquaintance. I will suggest that this principle must be abandoned if Russell’s theory is to avoid these kinds of problems.

  Scott Soames, University of Southern California