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    When W. V. O. Quine wrote “Two Dogmas of Empiricism”, he confronted philosophers with the nature of mathematics, logic and science. For him, all of these disciplines are inter-related forming a very big web of theoretical dependency on each other. There is no separation between analytic and synthetic judgments, there can only be a whole body of “knowledge”. Scientific theories and mathematics are posits like the early Greek gods of Homer. The so-called a priori disciplines can be changed for empirical reasons. A clear example of this is the way logic was revised by quantum phenomena, and mathematics was revised by the general theory of relativity. Of course, Quine says: “Posited objects can be real. As I also wrote elsewhere, to call a posit a posit is not to patronize it”.⁶  But posits are posits regardless of whether they are real or not.

    We should leave behind intensional notions, such as the platonist conception of meaning. “Meanings are what essences become, when they are detached from an object and wedded to the word”.⁷ Science, on the other hand, is just extensional, it does not deal with intensions. Since physics is the north which guides philosophy in its research, we should be content with that. Logic and mathematics, though posits, are important tools for science; and with the exception of some problems in set theory, mathematics can be pretty reliable. In the long run, meanings cannot establish the difference between analytic and synthetic in any case. The only meanings we should pay attention to are the ones which can account for sensible experience.

    In this book I shall attempt to challenge this aspect of “Two Dogmas of Empiricism” as a way to start dealing with the all-permeating problem of the underdetermination of science. Many still stipulate and support the idea that logic and mathematics can be changed in order to account for raw sensory-data. One of them has been Hilary Putnam, who shows that the general theory of relativity refuted euclidean geometry when it adopted a non-euclidean view of space-time. This way of thinking seems to be supported by some epistemologists, and is generally accepted among philosophers of science. For example, in Donald Gillies' excellent demystification of the Duhem-Quine Thesis, he says that Duhem rejected non-euclidean geometry and accepted euclidean geometry as a common-sense way of providing physical foundations. Gillies points out correctly that Duhem's belief is not true, but Gillies agrees with Quine when he says that non-euclidean geometry superseded euclidean geometry, and that Duhem's common-sense foundation of physics is false.⁸ He also quotes Duhem saying that Aristotle formulated logic in almost its final form, when we all know that Frege, Peano and Russell clearly superseded classic logic, and that we can also look at Brower's intuitionistic logic and quantum logic as proofs that logic itself can also be changed. From here he says that it “seems reasonable to extend the holistic thesis to include logic as well as to allow the possibility of altering logical laws as well as scientific laws to explain recalcitrant observations”.⁹ He later develops a new version of the Duhem-Quine Thesis which allows logical and mathematical revision in light of recalcitrant experience.¹⁰

    Philosophers of science in general are not friends of supposing other entities besides physical entities. Perhaps they can, like Popper, accept a kind of semi-platonism, or a cultural realm, but not the independent existence of meanings, truth values, mathematical objects such as cardinal numbers, ordinal numbers, sets, and the true and false formal relationships between them. In fregean terms, almost no philosopher of science is a “third realm” lover. From a scientific point of view, it can even be seen as a posit of unnecessary entities (Occam's Razor). For example, like Mario Bunge, they even reject third realms either in platonist or popperian forms; it is for them a kind of medieval concept which is not needed.¹¹ For many philosophers of science, the more naturalistic the epistemology, the better! So, apparently, there is no space for platonism with respect to natural science.

    I hope to show that one sort of platonism is the only way to understand science well and adequately. In fact, it is the only way to make epistemological sense on the a priori and its relationship with the a posteriori. A naturalistic view of logic and mathematics is not satisfactory to understand the real relation between formal science and natural science.

    This book is intended to refute many antiplatonist claims that logic and mathematics can both be changed on the basis of a recalcitrant sensory experience. For that, it is necessary to refute Quine's claim that there can be no distinction between analytic and synthetic judgments. To make platonism a viable option for philosophers, a particular platonist philosophy of mathematics will be adopted, namely that of Edmund Husserl. His views on mathematics have been neglected by many analytic philosophers. His philosophy happens to be practically the only platonist proposal that provides an adequate epistemology of mathematics, one which is actually very close to twenty-first century mathematics. At the same time, some of the most important objections to platonism will be addressed.

    Finally, once the nature of formal science and its epistemology are understood, there will be a thorough refutation of two philosophical statements presented by Quine and Putnam, which will be called, following Jerrold Katz, the “Quine-Putnam Theses”. These refutations will provide an adequate basis to understand the real relationship between science, logic and mathematics.



Footnotes:

⁶Quine, 1953, p. viii.  [Return to Text]

⁷Quine, 1953, p. 22.  [Return to Text]

⁸Gillies, 1993, pp. 114-115.  [Return to Text]

⁹Gillies, 1993, p. 115.  [Return to Text]

¹⁰Gillies, 1993, pp. 115-116.  [Return to Text]

¹¹Bunge, 1997, pp. 66, 76.  [Return to Text]