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**1 - 3**of**3**### Conceptions of the Continuum

"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."

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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical

### Frequently Asked Questions in Mathematics

"... Algebra. McGraw-Hill, 1975. This subsection of the FAQ is Copyright (c) 1994, 1995 Hans de Vreught. Send comments and or corrections relating to this part to J.P.M.deVreught@cs.tudelft.nl 13 Chapter 3 Number Theory 3.1 Fermat's Last Theorem 3.1.1 History of Fermat's Last Theorem Pie ..."

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Algebra. McGraw-Hill, 1975. This subsection of the FAQ is Copyright (c) 1994, 1995 Hans de Vreught. Send comments and or corrections relating to this part to J.P.M.deVreught@cs.tudelft.nl 13 Chapter 3 Number Theory 3.1 Fermat's Last Theorem 3.1.1 History of Fermat's Last Theorem Pierre de Fermat (1601-1665) was a lawyer and amateur mathematician. In about 1637, he annotated his copy (now lost) of Bachet's translation of Diophantus' Arithmetika with the following statement: Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. In English, and using modern terminology, the paragraph above reads as: There are no positive integers such that x^n + y^n = z^n for n > 2. I've found a remarkable proof of this fact, but there is not enough space in the margin [of the ...