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**1 - 6**of**6**### Large Cardinals and Determinacy

, 2011

"... The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measura ..."

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The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measurable?”), cardinal arithmetic (“Does Cantor’s Continuum Hypothesis hold?”), combinatorics(“DoesSuslin’sHypotheseshold?”), andgrouptheory (“Is there a Whitehead group?”). These developments gave rise to two conflicting positions. The first position—which we shall call pluralism—maintains that the independence results largely undermine the enterprise of set theory as an objective enterprise. On this view, although there are practical reasons that one might give in favour of one set of axioms over another—say, that it is more useful for a given task—, there are no theoretical reasons that can be given; and, moreover, this either implies or is a consequence of the fact—depending on the variant of the view, in particular, whether it places realism before reason,

### VISUALIZATION OF ORDINALS

, 2007

"... We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. ..."

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We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics.

### A First Glance at Non-Restrictiveness

, 1999

"... Maddy's notion of restrictiveness has many problematic aspects, one of them being that it is almost impossible to show that a theory is not restrictive. In this note the author addresses a crucial question of Martin Goldstern and shows some positive aspects of Maddy's notion. ..."

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Maddy's notion of restrictiveness has many problematic aspects, one of them being that it is almost impossible to show that a theory is not restrictive. In this note the author addresses a crucial question of Martin Goldstern and shows some positive aspects of Maddy's notion.

### Towards a theory of mathematical argument

"... Abstract. In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, an ..."

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Abstract. In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumen-tation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘deriva-tion indicator ’ view of proofs because it places derivations – which may be thought to invoke formal logic – at the center of mathematical justificatory practice. However, when the notion of ‘derivation ’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.