SOPhiA 2018

Salzburgiense Concilium Omnibus Philosophis Analyticis

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Programme - Talk

Bolzano and Contemporary Metaphysics
(Affiliated Workshop, English)

Analytic Philosophy's great grandfather Bernard Bolzano is well known for his contributions to logic and the philosophy of mathematics. It is less well known that Bolzano also made significant contributions to metaphysics, in particular to mereology and to the theory of objective non-causal dependence. Both fields have gained increasing attention in current analytic metaphysics, and it can be shown that Bolzano's ideas often contain important and original contributions to the contemporary debate. The workshop will bring together recent work on Bolzano's metaphysics from historical as well as from systematic perspectives. Speakers:

(1) Anna Bellomo
(2) Benjamin Schnieder
(3) Kevin Mulligan


16:00--16:05 Welcome
16:05--17:20 Anna Bellomo: Bolzano, geometry, and the part-whole principle
Short break
17:25--18:40 Benjamin Schnieder: tba
Short break
18:45--20:00 Kevin Mulligan: Grounding in Austro-German Philosophy (1890-1927)


Anna Bellomo: Bolzano, geometry, and the part-whole principle
Set theory, the current foundation of mainstream mathematics, originated in the mid 19th century. It is uncontroversial among historians of mathematics that some of Bolzano's mathematical ideas anticipated set theoretic concepts (see e.g. [1], p. 75). It is controversial, however, to which extent exactly Bolzano anticipated set theory. One of the points of controversy is whether or not Bolzano accepted the principle of equality of size which is at the heart of set theory. In set theory, two sets A and B are considered equally big just in case there is a one-to-one correspondence between the elements of A and the elements of B, that is to say, if for every element a in the set A, there is exactly one element b in the set B, and vice versa.
It has been argued in the literature that Bolzano accepted this set-theoretic principle of equality of size only in some parts of mathematics. More precisely, it is argued that Bolzano accepted this principle in what he calls ''pure mathematics'' (that is, arithmetic and analysis), but not in geometry. We will call this view ''Mancosu's hypothesis'', after [2]. Our aim in this talk will be to evaluate Mancosu's hypothesis.
If Mancosu's hypothesis is true, then Bolzano must have adopted in geometry a principle of equality of size which is different from the set-theoretical principle of one-to-one correspondence. The obvious candidate is what we will call the part-whole principle, which says that the whole is bigger than a proper part of it. The part-whole principle is a reformulation of one of Euclid's five common notions, and it is known that Bolzano's geometrical work was essentially of Euclidean inspiration.
Maintaining two separate criteria for comparisons of size -- one-to-one correspondence in pure mathematics and part-whole in geometry -- runs against Bolzano's overall preference for general and unifying treatments of mathematical concepts such as size. However, our hypothesis is that the necessity of adopting two different criteria for equality of size stems from Bolzano's views on pure mathematics versus geometry, and in particular on the objects that they are dealing with. To be more precise, our hypothesis is that the ''pure quantities'' (reine Größen) which are the subject matter of pure mathematics in Bolzano's view call for one-to-one correspondence, whereas the special quantities such as those in geometry (i.e. quantities in space) call for the part-whole principle.
The aim of this talk is thus to evaluate Mancosu's hypothesis by investigating Bolzano's views on pure quantities versus special -- in particular geometrical -- quantities. We will focus on Bolzano's later mathematico-philosophical production (from the 1830s onwards), that is, his mathematical views as spelled out in the Wissenschaftslehre, Paradoxien des Unendlichen and Mathematische Schriften. Our philosophical analysis will be aided by computational methods to collect and analyze textual evidence for or against Mancosu's hypothesis.

-- J. Ferreirós. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel: Birkhäuser, 1999.
-- P. Mancosu. ''Measuring the size of infinite collections of natural numbers: Was Cantor?s theory of infinite number inevitable?'' In: The Review of Symbolic Logic 2.4 (2009).

Kevin Mulligan: Grounding in Austro-German Philosophy (1890-1927)
This paper looks at some claims employong grounding and related notions such as foundation between 1890 and 1927: logical truths ground logical norms; facts ground truths & correctness of attitudes; essence grounds modes of being; causal order grounds temporal order; values ground oughts; non-normative facts ground in a sui generis way values.

Further details coming soon

Organisation: Stefan Roski (Hamburg).

Chair: Stefan Roski
Time: 16:00-20:00, 13 September 2018 (Thursday)
Location: SR 1.005

Anna Bellomo 
(University of Amsterdam, Netherlands)

Benjamin Schnieder 
(University of Hamburg, Germany)

Kevin Mulligan 
(University of Italian Switzerland, Lugano; University of Geneva, )

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