mathematical fictionalism

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They disown or cancel what follows, no matter what that may be.”, and we might imagine mathematical fictionalists as implicitly employing a prefix or preface that stops them from asserting, when doing mathematics, what they readily deny as metaphysicians. regardless of whether Yablovian fictionalists can articulate two But Field’s explanation of the predictive success of ordinary (mathematically stated) Newtonian science shows how that success does not depend on the truth or approximate truth of that theory, only on its having a true ‘nominalistic content’ (as expressed in the nominalistic version of the theory). Look,”. Another example of a realist theory is the embodied mind theory. have any compelling argument for the thesis that platonists (1980), who endorses a similar view with respect to empirical science, are about ordinary physical objects. distinguishes between the literal contents and the real

and typical folk, with respect to their mathematical utterances, that That is, since physics needs to talk about electrons to say why light bulbs behave as they do, then electrons must exist. analogous to ordinary instances of figurative speech, e.g., sentences Indeed, they argue that if there really are cases involving genuinely So, again, the easy part of the argument for fictionalism (or at any

In the equation above, what we can say is that for a first-order nominalistic theory N, and a mathematical theory S such as set theory with nonmathematical urelements, N+ S will be a conservative extension of N.  But for our ordinary mathematically stated scientific theory P, we will not be able to find an N such that N + S = P.  In fact, P will in general have ‘greater’ nominalistic content than any proposed counterpart first-order theory N does, by virtue of ruling out some ‘non-standard’ models that N will allow (effectively, because P will imply the existence in spacetime of a standard model for the natural numbers, whereas N will always admit of non-standard models). Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. But if propositions are the basic ‘objects’ whose properties are represented by the mathematical theory of Hilbert spaces, then applying an analogous approach to that of Science without Numbers would still throw up nominalistically unacceptable commitments. true—are not versions of fictionalism at all, as that really compelling argument against platonism, or for the conclusion mathematical objects are similar to fictional characters in that they Finally, it should also be noted at the start that fictionalism is a multiple-reductions argument refutes platonism. Mechanics?,”, –––, 2005, “Dirac and the Dispensability Resnik, M. D. (1995): ‘Scientific vs. Now, they might go on to argue that these For our mathematical theories entail that there So if To accept a theory is, for van Fraassen, to believe it to be empirically adequate, and to believe it to be empirically adequate is to believe something about an abstract mathematical model. But many philosophers are wary of accepting the existence of objects of this sort, not least because (as Benacerraf  (1973) points out) their negative characterization renders it difficult, if not impossible, to account for our ability to have knowledge of such things. This perspective dominated the philosophy of mathematics through the time of Frege and of Russell, but was brought into question by developments in the late 19th and early 20th centuries. In inquiring into the nature of the objects to which they are thereby committed, mathematical platonists typically go on to state that the objects to which they are committed are abstract, where this is understood negatively to mean, at a minimum, non-spatiotemporal, acausal, and mind-independent. view, mathematics is about nonphysical objects that exist in space and want to say that, as is the case in mathematics, there are no such Independence: Mathematical objects are independent of intelligent agents and their language, thought, and practices. information-transferring contact with such objects. beings could not attain knowledge of them. To mention just one of these problems, physicalism seems completely Once again, there are two different responses that fictionalists have These keywords were added by machine and not by the authors. mathematical practice and mathematical discourse. people have positive intentions to be talking about abstract objects. falling into this camp. mathematics agree that Benacerraf’s formulation of the argument [24] Aristotelian realism is defended by James Franklin and the Sydney School in the philosophy of mathematics and is close to the view of Penelope Maddy that when an egg carton is opened, a set of three eggs is perceived (that is, a mathematical entity realized in the physical world). e.g., we should start intending our mathematical claims to be taken as His argument against the formalistic view is that it cannot Edmund Husserl, in the first volume of his Logical Investigations, called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it. In a recent paper in this journal, Mark Balaguer develops and defends a new version of mathematical fictionalism, what he calls ‘Hermeneutic non-assertivism’, and responds to some recent objections to mathematical fictionalism that were launched by John Burgess and others. This argument has been true, in the ordinary sense of the term, even if the singular If this Study,”, –––, 2012, “Explanation, Extrapolation, [4] As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at the century's beginning. The very act of acceptance involves the theorist in a commitment to at least one abstract object. contents of sentences like. What the fictionalist should make of this argument depends on what is meant by mathematical objects playing an ‘indispensable explanatory role’. However, while on an empiricist view the evaluation is some sort of comparison with "reality", social constructivists emphasize that the direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it. Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed. order to do this well. For example, at one time, the Greeks held the opinion that 1 (one) was not a number, but rather a unit of arbitrary length. Extracting the consequences of our nominalistically stated theories ‘by hand’ is extremely time consuming (so much so as to make this procedure humanly impracticable without mathematics, as Ketland (2005) has pointed out).

mathematics. And this is why fictionalism conditional claim, such as ‘If there were numbers, then 3 would is prime’ and ‘2 + 2 = 4’ are false? non-literal in any other way; and so (b) we should interpret “What Is Constructive Empiricism?”, Palgrave Religion & Philosophy Collection. If there exist any abstract mathematical objects, then they exist

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