Frege’s argument for the thesis that some simple numerical identities are objectively true relies heavily on the fact that such identities allow for the application of natural numbers in representing and reasoning about reality, especially the non-mathematical parts of reality. Second, for at least the vast majority of spatio-temporal theories, the obtaining of this non-accidental, systematic relationship is underwritten by causal interaction between the subject matter of the theory in question and human brains. That such a response to this Londoner’s claim would be appropriate is obvious. ii. It is easy to see the relationship between naturalism, confirmation holism, and the first premise of the QPIA. Such beliefs are about this realm and so are true when, and only when, they are appropriately related to this realm.
i. Suppose a collection of entities is indispensable to one of our best scientific theories. Occam, like the nominalists of today, saw in universals simply an expedient lexicographical convention, with no extra-linguistic reality. John Burgess’ qualms about metaphysical accounts of mathematics are broader than this. Mathematical platonism refers to a collection of metaphysical accounts of mathematics. Michael Resnik [1981, p. 530], for example, writes: In mathematics, I claim, we do not have objects with an “internal” composition arranged in structures, we only have structures. We can regiment our notation.
At least three classes of accounts fall into this category. Frege, Quine, and “full-blooded platonism” offer the three most promising responses to this challenge. Moreover, Thesis 2 and the Schematic Reference Thesis distinguish the requirements on mathematical reference (and, consequently, truth) from the requirements on reference to (and, consequently, truth concerning) spatio-temporal entities. His test case was the problem of providing a denotation for imaginary terms like bi and complex expressions like a + bi. This proves Propositions 2 and 3 above and thereby the falsity of creationism. Existence is only freedom from formal contradiction. Michael Resnik [1997] goes one step further and argues that, within his Quinean framework, the distinction between the abstract and the concrete cannot be drawn in a meaningful way. On the Necessary Existence of Numbers. Finally there is a third and still higher degree of platonism which goes beyond the classical definition of the real numbers. (See the PM entry "Predicativism" (http://planetmath.org/Predicativism), for a discussion of impredicative definitions and their elimination.). Yet nowhere is it developed in systematic detail. Two dogmas of empiricism. As illustrated in section 3 above, the logico-inferential components of beliefs and statements about spatio-temporal entities have specific, unique spatio-temporal entities or collections of spatio-temporal entities as their referents. On the basis of this function-argument understanding of logical structure, Frege incorporated two categories of linguistic expression into his begriffsschift: those that are saturated and those that are not.
We now describe a template of an argument for the existence ofmathematical objects.
with concrete, most importantly spatio-temporal, items (for example, electrons, planets, particular copies of novels and performances of pieces of music, etc. Plato is supposed to have advanced the theory that it is our grasp of the concept of wisdom that secures our understanding of the meaning of the instantiations "Athena is wise", "Socrates is wise", etc. This book contains Shapiro’s development and defense of a platonist structuralism. Nominalism Reconsidered, in, Cole, Julian 2008. Mark Colyvan [2001] uses his discussion of the QPIA—in particular, the abovementioned similarities between mathematical and theoretical entities—to motivate skepticism about the necessity of mathematical truths and the necessary existence of mathematical entities. Next ask, “Is the fundamental question legitimate?” That is, should platonists feel the need to answer it? Yet this conception of that relationship is the one that proponents of the epistemological challenge ascribe to platonists. All that matters is that those choices meet the constraints placed on them by the sequent in question. Let us address these topics. For example, if the set of numbers S=1,…,n is given, there are nn functions which map to each element of S an element of S. But if S is infinite we allow the conception of a function fixed by an infinite number of independent determinations, which maps an integer to an integer and consider the totality of such functions as being well defined. v. Many numerical identity statements (for example, the one mentioned in (i) are true. The linguistic expressions on each side of identity statements are singular terms. Now, the domain of complex numbers is uncountable. In light of this, what is needed is a mechanism for assessing whether the truth of some theory or part of some theory commits us to the existence of a particular collection of entities.
Thoughts with mental objects as components are not objective. An excellent everyday example of a structure is a baseball defense (abstractly construed); such positions as pitcher and shortstop are the places of this structure. Indeed, the argument is given its first methodical treatment in Hilary Putnam’s Philosophy of Logic [1971]. Applying these observations to the claim “pure mathematical beliefs and statements are about the mathematical realm, and so are true when, and only when, they are appropriately related to this realm,” we find that it maintains that constituents of a mathematical realm are the semantic values of the logico-inferential components of pure mathematical beliefs and statements.
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