scholarly journals Hilbert, logicism, and mathematical existence

Synthese ◽  
2008 ◽  
Vol 170 (1) ◽  
pp. 33-70 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Mathematical Existence

Mathematical Existence

2005 ◽  
Vol 11 (3) ◽  
pp. 351-376 ◽  
Author(s):  
Penelope Maddy
Keyword(s):  
Set Theory ◽  
Elliptic Functions ◽  
Mathematical Truth ◽  
Mathematical Methods ◽  
Explicit Representations ◽  
Mathematical Existence ◽  
Synthetic Approaches ◽  
Areas Of Interest ◽  
Compare And Contrast ◽  
Excluded Middle

Despite some discomfort with this grandly philosophical topic, I do in fact hope to address a venerable pair of philosophical chestnuts: mathematical truth and existence. My plan is to set out three possible stands on these issues, for an exercise in compare and contrast. A word of warning, though, to philosophical purists (and perhaps of comfort to more mathematical readers): I will explore these philosophical positions with an eye to their interconnections with some concrete issues of set theoretic method.Let me begin with a brief look at what to count as ‘philosophy’. To some extent, this is a matter of usage, and mathematicians sometimes classify as ‘philosophical’ any considerations other than outright proofs. So, for example, discussions of the propriety of particular mathematical methods would fall under this heading: should we prefer analytic or synthetic approaches in geometry? Should elliptic functions be treated in terms of explicit representations (as in Weierstrass) or geometrically (as in Riemann)? Should we allow impredicative definitions? Should we restrict ourselves to a logic without bivalence or the law of the excluded middle? Also included in this category would be the trains of thought that shaped our central concepts: should a function always be defined by a formula? Should a group be required to have an inverse for every element? Should ideal divisors be defined contextually or explicitly, treated computationally or abstractly? In addition, there are more general questions concerning mathematical values, aims and goals: Should we strive for powerful theories or low-risk theories? How much stress should be placed on the fact or promise of physical applications? How important are interconnections between the various branches of mathematics? These philosophical questions of method naturally include several peculiar to set theory: should set theorists focus their efforts on drawing consequences for areas of interest to mathematicians outside mathematical logic? Should exploration of the standard axioms of ZFC be preferred to the exploration and exploitation of new axioms? How should axioms for set theory be chosen? What would a solution to the Continuum Problem look like?

MATHEMATICAL EXISTENCE: A REORIENTATION

1974 ◽  
Vol s1-11 (1-2) ◽  
pp. 5-8
Author(s):  
J. Fang
Keyword(s):  
Mathematical Existence

Auxiliary Lines And Ratios

1967 ◽  
Vol 60 (2) ◽  
pp. 109-114
Author(s):  
Donald W. Stover
Keyword(s):  
Testing Situation ◽  
Mathematical Existence ◽  
Auxiliary Lines ◽  
Time Limitations

This depressing query, dashing the hopes engendered by eagerly waving hand and bright expression, is just one of those small, recurring punches teachers learn to roll with. Its occurrence may be forecast with confidence when geometry students encounter their first proof in which the figure contains auxiliary lines. In this instance, the answer is also predictable. Few teachers, taking into account the pressures and time limitations of a testing situation, dare require their students to select from the infinitude of mathematical existence an unrehearsed assertion of the form “There exists a line….”

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