Ontology and the Foundations of Mathematics

2022 ◽  
Author(s):  
Penelope Rush
Keyword(s):  
Foundations Of Mathematics ◽  
Mathematical Realism ◽  
Epistemological Problem ◽  
Access Problem ◽  
Independent Reality ◽  
Mathematical Reality

This Element looks at the problem of inter-translation between mathematical realism and anti-realism and argues that so far as realism is inter-translatable with anti-realism, there is a burden on the realist to show how her posited reality differs from that of the anti-realist. It also argues that an effective defence of just such a difference needs a commitment to the independence of mathematical reality, which in turn involves a commitment to the ontological access problem – the problem of how knowable mathematical truths are identifiable with a reality independent of us as knowers. Specifically, if the only access problem acknowledged is the epistemological problem – i.e. the problem of how we come to know mathematical truths – then nothing is gained by the realist notion of an independent reality and in effect, nothing distinguishes realism from anti-realism in mathematics.

Category Theory and Foundations

2018 ◽  
Author(s):  
Michael Ernst
Keyword(s):  
Category Theory ◽  
Mathematical Thinking ◽  
Mathematical Practice ◽  
Foundations Of Mathematics ◽  
Ongoing Debate ◽  
Technical Adequacy ◽  
Theoretical Foundations

In the foundations of mathematics there has been an ongoing debate about whether categorical foundations can replace set-theoretical foundations. The primary goal of this chapter is to provide a condensed summary of that debate. It addresses the two primary points of contention: technical adequacy and autonomy. Finally, it calls attention to a neglected feature of the debate, the claim that categorical foundations are more natural and readily useable, and how deeper investigation of that claim could prove fruitful for our understanding of mathematical thinking and mathematical practice.

Explaining Perceptual Knowledge

2018 ◽  
Author(s):  
Barry Stroud
Keyword(s):  
Perceptual Experience ◽  
External World ◽  
Satisfactory Explanation ◽  
Perceptual Knowledge ◽  
Epistemological Problem ◽  
The Way

This chapter offers a response to Quassim Cassam’s ‘Seeing and Knowing’, which challenges some of the conditions Cassam thinks the author has imposed on a satisfactory explanation of our knowledge of the external world. According to Cassam, the conditions he specifies can be fulfilled in ways that explain how the knowledge is possible. What is at stake in this argument between Cassam and the author is the conception of what is perceived to be so that is needed to account for the kind of perceptual knowledge we all know we have. That is what must be in question in any promising move away from the overly restrictive conception of perceptual experience that gives rise to the hopelessness of the traditional epistemological problem. The author suggests that we should explore the conditions of successful ‘propositional’ perception of the way things are and emphasizes the promise of such a strategy.

Are the Colours of Things Secondary Qualities?

2018 ◽  
Author(s):  
Barry Stroud
Keyword(s):  
General Notion ◽  
John Mcdowell ◽  
Secondary Qualities ◽  
Independent Reality ◽  
Metaphysical Status

This chapter challenges the notion that the colours we believe to belong to the objects we see are ‘secondary’ qualities of those objects. Such a notion is endorsed by John McDowell, who has explained why he thinks the author is wrong to resist it. McDowell recognizes that the author’s focus on the conditions of successfully unmasking the metaphysical status of the colours of things is a way of trying to make sense of whatever notion of reality is involved in it. However, the author argues that the notion of reality he is concerned with is ‘independent reality’, not simply the general notion of reality. He also contends that an exclusively dispositional conception of an object’s being a certain colour cannot account for the perceptions we have of the colours of things.

scholarly journals Social constructivism in mathematics? The promise and shortcomings of Julian Cole’s institutional account

Synthese ◽  
2021 ◽  
Author(s):  
Jenni Rytilä
Keyword(s):  
Social Constructivism ◽  
Mathematical Practice ◽  
Social Constructs ◽  
Socially Constructed ◽  
The Social ◽  
Characteristic Features ◽  
Core Idea ◽  
Social Entities ◽  
In Virtue Of ◽  
Mathematical Reality

AbstractThe core idea of social constructivism in mathematics is that mathematical entities are social constructs that exist in virtue of social practices, similar to more familiar social entities like institutions and money. Julian C. Cole has presented an institutional version of social constructivism about mathematics based on John Searle’s theory of the construction of the social reality. In this paper, I consider what merits social constructivism has and examine how well Cole’s institutional account meets the challenge of accounting for the characteristic features of mathematics, especially objectivity and applicability. I propose that in general social constructivism shows promise as an ontology of mathematics, because the view can agree with mathematical practice and it offers a way of understanding how mathematical entities can be real without conflicting with a scientific picture of reality. However, I argue that Cole’s specific theory does not provide an adequate social constructivist account of mathematics. His institutional account fails to sufficiently explain the objectivity and applicability of mathematics, because the explanations are weakened and limited by the three-level theoretical model underlying Cole’s account of the construction of mathematical reality and by the use of the Searlean institutional framework. The shortcomings of Cole’s theory give reason to suspect that the Searlean framework is not an optimal way to defend the view that mathematical reality is socially constructed.

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