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Ideas for
'Frege's Concept of Numbers as Objects', 'Hat-Tricks and Heaps' and 'What Required for Foundation for Maths?'
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35 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
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13861
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Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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13892
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One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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17784
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Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
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17782
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Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
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17781
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Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts
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13867
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Instances of a non-sortal concept can only be counted relative to a sortal concept [Wright,C]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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17799
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Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
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17797
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Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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17775
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If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
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17776
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The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
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17777
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Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
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17804
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Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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17441
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Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are [Wright,C, by Heck]
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13862
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There are five Peano axioms, which can be expressed informally [Wright,C]
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17853
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Number truths are said to be the consequence of PA - but it needs semantic consequence [Wright,C]
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17854
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What facts underpin the truths of the Peano axioms? [Wright,C]
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17792
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1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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17793
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It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
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13894
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Sameness of number is fundamental, not counting, despite children learning that first [Wright,C]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
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10140
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We derive Hume's Law from Law V, then discard the latter in deriving arithmetic [Wright,C, by Fine,K]
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8692
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Frege has a good system if his 'number principle' replaces his basic law V [Wright,C, by Friend]
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17440
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Wright says Hume's Principle is analytic of cardinal numbers, like a definition [Wright,C, by Heck]
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13893
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It is 1-1 correlation of concepts, and not progression, which distinguishes natural number [Wright,C]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
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13888
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If numbers are extensions, Frege must first solve the Caesar problem for extensions [Wright,C]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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17794
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Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
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17802
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We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
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17805
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Set theory is not just another axiomatised part of mathematics [Mayberry]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
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13869
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Number platonism says that natural number is a sortal concept [Wright,C]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
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13870
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We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism [Wright,C]
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
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13873
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Treating numbers adjectivally is treating them as quantifiers [Wright,C]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
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13899
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The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals [Wright,C]
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13896
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The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes [Wright,C]
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7804
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Wright has revived Frege's discredited logicism [Wright,C, by Benardete,JA]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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13863
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Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms [Wright,C]
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13895
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The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems [Wright,C]
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