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Set-theoretic Naturalism as the Foundation of Mathematics

The deep belief in the axiomatic method in set-theory was devastated by Godel thirty years after Russell had destroyed Cantorʹs intuitive set-theory. Paul Cohenʹs article in 1963 was a landmark in the history of the subject. The problem is: how are the independent questions of axiomatic set - theory, to be understood and resolved? Along the way I have met naive realism, classical realism of Frege and Godel and finally I have advanced the scientific realism. The scientific realism is combined with naturalism, but not with Maddyʹs type of it. The existence and nature of mathematical entities (such as sets) would be clear by comparing them with medium sized physical objects and theoretical unobservable entities of physics. In the case of the unobservable entities of physics we admit that they exist as far as our best scientific thoery tells us they do. In the same way to show that there are universals we do not try to give a pre - scientific philosophical argument, but rather we just show that our hest scientific theories cannot do without them. We reject the cartesian demand for certainty beyond the scientific knowledge. We renounce any first philosophy --any extra - scientific challenge with the radical skepticism it engenders. The same goes for semantics. There is no point of view prior or superior to our language. What we want id theory of have our language works --a theory which would be a chapter of our scientific world - view. We stand within our current best theory of the world and ask how our beliefs connect us with the world. It cannot he something extra - scientific. Maddy says that mathematics is its own and only arbiter. But astrology and the contradictory systems of theology could be claimed to be their own arbiter. What distinguishes mathematics from these systems is its application in the physical world. Mathematics is interlocked with physics. To disentangle it from the natural sciences is to reduce it to a wall paper pattern. To my mind, Maddyʹs "naturalism" is not at all naturalism. She divorces mathematics from natural science and insists on the a priori nature of mathmatical truth. But, our web of belief faces the tribunal of experience as a whole. We may nat factor out the spicial contribution of mathematics. Maddy accords mathematics such a separable status. Were she to succeed, any a priorist foundationalist would be in her debt. She says: "Mathematics is not answerable to any extra - mathematical tribunal". Now any theologian may say: "Theology is not answerable to any extra - theological tribunal" or an astrologist could, in a similar way, say: "Astrology is not answerable to any extra - astrological tribunal".