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Objects and objectivity: Alternatives to mathematical realismPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2011 (English)Doctoral thesis, monograph (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Umeå: Institutionen för idé- och samhällsstudier , 2011. , xii, 177 p.
##### Series

Umeå studies in philosophy, ISSN 1650-1748 ; 10
##### Keyword [en]

Philosophy of mathematics, mathematical realism, ontological realism, semantic realism, platonism, the semantic argument, the indispensability argument, the non-uniqueness problem, Benacerraf's dilemma, the irrelevance challenge, Field, Carnap, Balaguer, Yablo, the internal/external distinction, fictionalism
##### National Category

Philosophy
##### Research subject

Theoretical Philosophy
##### Identifiers

URN: urn:nbn:se:umu:diva-43692ISBN: 978-91-7459-180-4OAI: oai:DiVA.org:umu-43692DiVA: diva2:415209
##### Public defence

2011-06-04, Humanisthuset, Hörsal F, Umeå universitet, Umeå, 10:15 (English)
##### Opponent

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Available from: 2011-05-13 Created: 2011-05-05 Last updated: 2011-05-09Bibliographically approved

This dissertation is centered around a set of apparently conflicting intuitions that we may have about mathematics. On the one hand, we are inclined to believe that the theorems of mathematics are *true*. Since many of these theorems are existence assertions, it seems that if we accept them as true, we also commit ourselves to the *existence* of mathematical objects. On the other hand, mathematical objects are usually thought of as *abstract objects* that are non-spatiotemporal and causally inert. This makes it difficult to understand how we can have knowledge of them and how they can have any relevance for our mathematical theories.

I begin by characterizing a realist position in the philosophy of mathematics and discussing two of the most influential arguments for that kind of view. Next, after highlighting some of the difficulties that realism faces, I look at a few alternative approaches that attempt to account for our mathematical practice without making the assumption that there exist abstract mathematical entities. More specifically, I examine the *fictionalist* views developed by Hartry Field, Mark Balaguer, and Stephen Yablo, respectively. A common feature of these views is that they accept that mathematics interpreted at face value is committed to the existence of abstract objects. In order to avoid this commitment, they claim that mathematics, when taken at face value, is *false*.

I argue that the fictionalist idea of mathematics as consisting of falsehoods is counter-intuitive and that we should aim for an account that can accommodate both the intuition that mathematics is true and the intuition that the causal inertness of abstract mathematical objects makes them *irrelevant* to mathematical practice and mathematical knowledge. The solution that I propose is based on Rudolf Carnap's distinction between an *internal* and an *external* perspective on existence. I argue that the most reasonable interpretation of the notions of mathematical truth and existence is that they are *internal* to mathematics and, hence, that mathematical truth cannot be used to draw the conclusion that mathematical objects exist in an external/ontological sense.

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