Mathematical fictionalism

Mathematical fictionalism is a nominalist (anti-realist) view of mathematics. In this interpretation, mathematical statements such as ‘3 is prime’ or ‘a square has four equal angles’ should be interpreted at face value and are therefore false because abstract mathematical objects such as the number 3 or a square do not really exist. Adherents to the fictionalist view often agree with adherents to the Platonist view that mathematical statements and theories purport to make truth claims about abstract mathematical objects. However, fictionalists reject the existence of abstract objects because the Platonist assertion of their existence is not metaphysically well grounded. Therefore, fictionalists conclude that mathematical theories are not objectively true in the same way that statements like 'the moon is grey' are.

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Arguments

Summary of Balaguer's proof:

1. Mathematical statements should be read at face value as making truth claims about the nature of certain objects. Furthermore, if such statements are true, then the objects about which the statements are made must actually exist. For example, the statement '3 is prime' should be read as making a truth claim about the nature of the number 3 and if the statement is literally true, then the number 3 must actually exist.
2. It follows that if statements like '3 is prime' are t[...]
3. However, there are no such things as abstract objects and by extension there is no such thing as a mathematical object like the number 3 about which truth claims can be made.
4. Therefore, statements like '3 is prime' are not true.

History

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Reception

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The modern field of mathematical fictionalism was brought to fame in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of Newtonian mechanics with no reference to numbers or functions at all. He started with the "betweenness" of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields. 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.

Having shown how to do science without using numbers, Field proceeded to rehabilitate mathematics as a kind of useful fiction. He showed that mathematical physics is a conservative extension of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from Field's system), so that mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like "2 + 2 = 4" is just as fictitious as "Sherlock Holmes lived at 221B Baker Street"—but both are true according to the relevant fictions.

Another fictionalist, Mary Leng, expresses the perspective succinctly by dismissing any seeming connection between mathematics and the physical world as "a happy coincidence". This rejection separates fictionalism from other forms of anti-realism, which see mathematics itself as artificial but still bounded or fitted to reality in some way.

By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about fiction in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments of second-order logic to carry out his reduction, and because the statement of conservativity seems to require quantification over abstract models or deductions.^{[citation needed]}

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References

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