This paper discusses the usages of variables in mathematics, especially those of variables in polynomials and propositional variables. As the conclusion, variables in polynomials and propositional variables take a role of transcendental elements and that of generators. In this sense, the point in common between these variables and letters in identities is that we can substitute such letters with any elements. In addition, this paper shows the usages of unknowns in equations,coordinate variables and probabilistic variables with the references of past literatures, and discusses the usages of dependent variables and independent variables as a future work.

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    In the first half of this essay, I try to trace the development of Dedekindʼs theory of numbers from his inaugural lecture to his monumental theory of natural numbers, in which Dedekind complementarily employed his genetic and axiomatic methods. Further I briefly mention Fregean logicism. In the second half, I am concerned with the emergence of meta-mathematical inquiries into the completeness theorems proved by Post,Bernays, Gödel, and with Gödel's anticipation of his Incompleteness Theorem.

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    We first raise an issue on a view that Gödel’s completeness theorem assures that first-order logic gives an adequate formalization of the concept of proofs in ordinary mathematics. Then we discuss philosophical backgrounds of Henkin’s proof of the completeness theorem and their effects on the change of interpretations of first-order and second-order quantifiers. Lastly, we analyze the meanings of formal consistency statements and their relation to Hilbert’s program, and we claim that the relationship between the concepts of formal and informal proofs is closely connected to interpretations of quantifiers.

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    Conceptualism is the view that perceptual content is fully conceptual. One of the most promising arguments against conceptualism is the so-called “learning argument.” Recently, Adina L. Roskies has developed this argument by insisting that, in order to explain the formation of demonstrative concepts adequately, we must accept a version of nonconceptualism. My aim in this paper is twofold: first, to show that Roskiesʼ argument is not fully convincing, and second, to argue that we can provide a conceptualist account of the formation of demonstrative concepts on the basis of a version of nativism.

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    In this paper, I address the problem of what the poverty of the stimulus argument tells us about innateness and what not. This argument has played an important role in the nativism/empiricism debate, but the connection between this argument and the concept of innateness is not clear. Given this, I clarify the structure of this argument, and consider what this argument requires the characterization of innateness (not) to be.

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    In this paper, I reconsider the theory of punctuated equilibria and argue that the essential idea of the theory originated from the biostratigraphic way of seeing the fossil record. It must be noted that biostratigraphers have recognized and exploited the longterm morphological stability following a geologically abrupt origin for a long time. I suggest that the empirical success of biostratigraphy supports the pattern hypothesis of punctuated equilibria. Reexamining the controversy over the theory of punctuated equilibria, I conclude that the theory was really important not because it changed the explanans (evolutionary mechanisms) but because it changed the explanandum (macroevolutionary patterns that need to be explained).

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    In this paper, I attempt to illuminate some unique and significant features of inferentialism: A pragmatist theory of meaning developed by Brandom (1994). Inferentialism is unique in that it explains not only the concept of meaning, but also that of representation, thoroughly in the pragmatist sprit. I argue that inferentialists are enabled to do this by their choice to use normative concepts, rather than the standard, causal-dispositional ones, at the base level of their pragmatics. After briefly surveying those criticisms directed at this normative basis of inferentialism, I point out one significant advantage that inferentialism can have in the attempt to dissolve an apparent incompatibility problem between content externalism and the a priori understanding of meaning.

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