Which Mathematical Objects are Referred to by the Enhanced Indispensability Argument?

2017 ◽  
Vol 49 (1) ◽  
pp. 121-126 ◽  
Author(s):  
Vladimir Drekalović ◽  
Berislav Žarnić
Keyword(s):  
Indispensability Argument ◽  
Mathematical Objects ◽  
Enhanced Indispensability Argument

scholarly journals Mathematical explanation and indispensability

2018 ◽  
Vol 33 (2) ◽  
pp. 233 ◽  
Author(s):  
Susan Vineberg
Keyword(s):  
Causal Explanation ◽  
Mathematical Explanation ◽  
Indispensability Argument ◽  
Mathematical Realism ◽  
Explanatory Role ◽  
Mathematical Objects ◽  
Scientific Explanations ◽  
Enhanced Indispensability Argument

This paper discusses Baker’s Enhanced Indispensability Argument (EIA) for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such use still does not support realism about mathematical objects.

Naturalism in the Philosophy of Mathematics

Philosophy ◽  
2015 ◽  
Author(s):  
Mary Leng
Keyword(s):  
Natural Science ◽  
Philosophy Of Mathematics ◽  
Sole Source ◽  
Indispensability Argument ◽  
Mathematical Truth ◽  
Methodological Naturalism ◽  
Mathematical Objects ◽  
Ontological Naturalism ◽  
Epistemological Challenge ◽  
Genuine Problem

In the context of the philosophy of mathematics, the term “naturalism” has a number of uses, covering approaches that look to be fundamentally at odds with one another. In one use, the “natural” in naturalism is contrasted with non-natural, in the sense of supernatural; in this sense, naturalism in the philosophy of mathematics appears in opposition to Platonism (the view that mathematical truths are truths about a body of abstract mathematical objects). Naturalism thus construed takes seriously the epistemological challenge to Platonism presented by Paul Benacerraf in his paper “Mathematical Truth” (cited under Ontological Naturalism). Benacerraf points out that a view of mathematics as a body of truths about a realm of abstract objects appears to rule out any (non-mystical) account of how we, as physically located embodied beings, could come to know such truths. The naturalism that falls out of acceptance of Benacerraf’s challenge as presenting a genuine problem for our claims to be able to know truths about abstract mathematical objects is sometimes referred to as “ontological naturalism,” and suggests a physicalist ontology. In a second use, the “natural” in naturalism is a reference specifically to natural science and its methods. Naturalism here, sometimes called methodological naturalism, is the Quinean doctrine that philosophy is continuous with natural science. Quine and Putnam’s indispensability argument for the existence of mathematical objects places methodological naturalism in conflict with ontological naturalism, since it is argued that the success of our scientific theories confirms the existence of the abstract mathematical objects apparently referred to in formulating those theories, suggesting that methodological naturalism requires Platonism. A final use of “naturalism” in the philosophy of mathematics is distinctive to mathematics, and arises out of consideration of the proper extent of methodological naturalism. According to Quine’s naturalism, the natural sciences provide us with the proper methods of inquiry. But, as Penelope Maddy has noted, mathematics has its own internal methods and standards, which differ from the methods of the empirical sciences, and naturalistic respect for the methodologies of successful fields requires that we should accept those methods and standards. This places Maddy’s methodological naturalism in tension with the original Quinean version of the doctrine, because, Maddy argues, letting natural science be the sole source of confirmation for mathematical theories fails to respect the autonomy of mathematics.

scholarly journals The Eleatic and the Indispensabilist

2015 ◽  
Vol 30 (3) ◽  
pp. 415-429 ◽  
Author(s):  
Russell Marcus
Keyword(s):  
Scientific Theory ◽  
Indispensability Argument ◽  
Mathematical Objects

The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe.  Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory.  Eleatics argue that only objects with causal properties exist.  Mark Colyvan’s recent defenses of Quine’s indispensability argument against some contemporary eleatics attempt to provide reasons to favor the indispensabilist’s criterion.  I show that Colyvan’s argument is not decisive against the eleatic and sketch a way to capture the important intuitions behind both views.

scholarly journals The indispensability argument and the nature of mathematical objects

2018 ◽  
Vol 33 (2) ◽  
pp. 249 ◽  
Author(s):  
Matteo Plebani
Keyword(s):  
The Other ◽  
Indispensability Argument ◽  
Heavy Duty ◽  
Mathematical Objects ◽  
Other Hand ◽  
Unstable Position

I will contrast two conceptions of the nature of mathematical objects: the conception of mathematical objects as preconceived objects (Yablo 2010), and heavy duty platonism (Knowles 2015). I will argue that friends of the indispensability argument are committed to some metaphysical theses and that one promising way to motivate such theses is to adopt heavy duty platonism. On the other hand, combining the indispensability argument with the conception of mathematical objects as preconceived objects yields an unstable position. The conclusion is that the metaphysical commitments of the indispensability argument should be carefully scrutinized.

FACING INCONSISTENCY: THEORIES AND OUR RELATIONS TO THEM

Episteme ◽  
2013 ◽  
Vol 10 (4) ◽  
pp. 351-367 ◽  
Author(s):  
Michaelis Michael
Keyword(s):  
Classical Logic ◽  
Paraconsistent Logic ◽  
Indispensability Argument ◽  
Mathematical Objects ◽  
The Face ◽  
Inconsistent Objects ◽  
Inconsistent Theories

AbstractClassical logic is explosive in the face of contradiction, yet we find ourselves using inconsistent theories. Mark Colyvan, one of the prominent advocates of the indispensability argument for realism about mathematical objects, suggests that such use can be garnered to develop an argument for commitment to inconsistent objects and, because of that, a paraconsistent underlying logic. I argue to the contrary that it is open to a classical logician to make distinctions, also needed by the paraconsistent logician, which allow a more nuanced ranking of theories in which inconsistent theories can have different degrees of usefulness and productivity. Facing inconsistency does not force us to adopt an underlying paraconsistent logic. Moreover we will see that the argument to best explanation deployed by Colyvan in this context is unsuccessful. I suggest that Quinean approach which Colyvan champions will not lead to the revolutionary doctrines Colyvan endorses.

scholarly journals Putnam’s indispensability argument revisited, reassessed, revived

2018 ◽  
Vol 33 (2) ◽  
pp. 201 ◽  
Author(s):  
Otávio Bueno
Keyword(s):  
Philosophy Of Mathematics ◽  
Indispensability Argument ◽  
Mathematical Objects

Crucial to Hilary Putnam’s realism in the philosophy of mathematics is to maintain the objectivity of mathematics without the commitment to the existence of mathematical objects. Putnam’s indispensability argument was devised as part of this conception. In this paper, I reconstruct and reassess Putnam’s argument for the indispensability of mathematics, and distinguish it from the more familiar, Quinean version of the argument. Although I argue that Putnam’s approach ultimately fails, I develop an alternative way of implementing his form of realism about mathematics that, by using different resources than those Putnam invokes, avoids the difficulties faced by his view.

scholarly journals Unification and mathematical explanation

2021 ◽  
Author(s):  
Robert Knowles
Keyword(s):  
Divide And Conquer ◽  
Mathematical Explanation ◽  
Indispensability Argument ◽  
Mathematical Explanations ◽  
Enhanced Indispensability Argument ◽  
One Step ◽  
The Right

AbstractThis paper provides a sorely-needed evaluation of the view that mathematical explanations in science explain by unifying. Illustrating with some novel examples, I argue that the view is misguided. For believers in mathematical explanations in science, my discussion rules out one way of spelling out how they work, bringing us one step closer to the right way. For non-believers, it contributes to a divide-and-conquer strategy for showing that there are no such explanations in science. My discussion also undermines the appeal to unifying power in support of the enhanced indispensability argument.

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