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Quine's empiricist platonism Quine adopted an empiricist stance: he believed the justification of mathematical truths was wholly empirical. On the other hand - and despite their apparent incompatibility - he argued for the inclusion of platonic objects into mathematical ontology. Quine's position is often seen as having three main strands:

1. Confirmational holism

2. Naturalism

3. The indispensability argument And sometimes the additional strand is added: set-theoretic reductionism.

Confirmational holism Confirmational holism is the claim that no claim of theoretical science can be confirmed or refuted in isolation, but only as a part of a system of hypotheses. "Mathematics - not uninterpreted mathematics but genuine set theory, logic, number theory...and so on - is best looked upon as an integral part of science, on a par with physics, etc...in which mathematics is said to achieve its applications". Following Michael Resnik, we can see this view as coming from?

The observation that statements of theoretical science often require other 'auxilliary' hypotheses in order to imply observational claims. The logical point that if certain hypotheses H imply an observation O only in the presence of auxiliary hypotheses A, then the falsity of O implies the falsity of (H&A) and not of H alone.

Quine applies this to mathematics as well: mathematical principles share in the empirical content of systems of hypotheses containing them. When we test certain physical hypotheses we are also (as part of our auxiliary hypotheses) testing mathematics: because the empirical evidence justifies mathematics-in-science, it justifies mathematics. Quine expressed this view by the metaphor of the 'web of belief'. He thought of our knowledge or beliefs as forming a kind of 'fabric', which "only impinges on experience along the edges". Thus, mathematics and universal physical laws are very close to the centre, far away from direct justification by experience - but nonetheless they are empirically justified. It is an important consequence of this view that any statement is empirically revisable. Any statement can be held true come what may by making drastic enough changes elsewhere in the web; conversely, even statements close to the periphery can be held true in the face of experience by pleading hallucination or amending logical laws. For Quine, even logic is revisable - if only slowly, as any change to the system would require testing to see if the modifications had affected previously-sound hypotheses. 1

Michael Resnik interprets this revisionism as a "methodological code to which scientists subscribe". The claim therefore amounts to saying that from a logical point of view none of our beliefs are immune to revision. Absent further specification, revising our hypotheses or our auxiliary beliefs are both logically consistent. Quine buttressed these considerations by his rejection of the analytic-synthetic distinction. Although the exact interpretation is still controversial, in 'Two Dogmas of Empiricism' Quine argues that the notion of 'analytic' is problematic. He bases this on the impossibility he finds in trying spell out in any meaningful, non-circular way the notion of 'analytic' or 'true in virtue of meaning' or any similar definition. Instead of a sharp dividing line between analytic and synthetic, then, Quine imagines a continuum: the difference in justification is a matter of degree, not of type. We might be sceptical about this idea, but it is at least easy to introduce a certain level of doubt into the notion of analyticity. Consider the canonical 'all bachelors are married'. Some philosophers have pointed out this concept relies on more than one criterion for its application: there are vague boundaries to both 'bachelor' and 'unmarried'. (For example: does 'bachelor' or 'married' include men in civil partnerships?) Whether or not these vague notions actually match up is not as clear as it first appears.

Objections to Holism Nature of mathematical truths Objection: are mathematical truths really known a posteriori? It seems that basic arithmetical knowledge is not refutable by experience, for example. Reply 1: Oracle example. Logie/Matt example: if we come to discover an empirical response every time we make a certain kind of mistake, then even when we have made an apparently correct inference, the occurrence of that response can defeat justifiable belief in the inference. Apparently a priori truths can be defeasible by experience - for example Descartes' a priori (he claimed) belief in substance- and mind-body dualism. Could take basic arithmetical claims to be irrefutable in some way akin to a strict finist's approach. Reply 2: note that we are generally very bad in predicting how we would act in outlandish circumstances. Cannot lean too much on the unimaginability of a proposition's falsification to argue it would never be falsified. Relatedly: Objection: we have no plausible account of the 'obviousness' of elementary mathematics. The axioms of arithmetic do not seem to be on a par with those of particle physics, for example. (Charles Parsons makes this objection; Maddy attempts to fill this gap). For example, '1+1=2' is obvious in a way that '1+1=2 is a part of our best science' is not, so the claim owes no credibility to being included in our best science.

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