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Single Idea 9876

[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique ]

Full Idea

The two frequent modern objects to logicism are that set theory is not part of logic, or that it is of no interest to 'reduce' a mathematical theory to another, more complex, one.

Gist of Idea

Set theory isn't part of logic, and why reduce to something more complex?

Source

Michael Dummett (Frege philosophy of mathematics [1991], Ch.18)

Book Ref

Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.224

A Reaction

Dummett says these are irrelevant (see context). The first one seems a good objection. The second one less so, because whether something is 'complex' is a quite different issue from whether it is ontologically more fundamental.

The 132 ideas from Michael Dummett

16953 Relative possibility one way may be impossible coming back, so it isn't symmetrical [Dummett]
16952 If something is only possible relative to another possibility, the possibility relation is not transitive [Dummett]
16951 It was realised that possible worlds covered all modal logics, if they had a structure [Dummett]
16954 Generalised talk of 'natural kinds' is unfortunate, as they vary too much [Dummett]
16956 To explain generosity in a person, you must understand a generous action [Dummett]
16957 Possible worlds aren't how the world might be, but how a world might be, given some possibility [Dummett]
16959 If possible worlds have no structure (S5) they are equal, and it is hard to deny them reality [Dummett]
16960 If possibilitiy is relative, that might make accessibility non-transitive, and T the correct system [Dummett]
16958 In S4 the actual world has a special place [Dummett]
15939 For intuitionists it is constructed proofs (which take time) which make statements true [Dummett]
18832 Mathematical statements and entities that result from an infinite process must lack a truth-value [Dummett]
15938 Platonists ruin infinity, which is precisely a growing structure which is never completed [Dummett]
9181 The causal theory of reference can't distinguish just hearing a name from knowing its use [Dummett]
9182 Ancient names like 'Obadiah' depend on tradition, not on where the name originated [Dummett]
19168 Concepts only have a 'functional character', because they map to truth values, not objects [Dummett, by Davidson]
10320 If a genuine singular term needs a criterion of identity, we must exclude abstract nouns [Dummett, by Hale]
10547 Abstract objects can never be confronted, and need verbal phrases for reference [Dummett]
10531 There is a modern philosophical notion of 'object', first introduced by Frege [Dummett]
10555 If we can intuitively apprehend abstract objects, this makes them observable and causally active [Dummett]
10543 Abstract objects must have names that fall within the range of some functional expression [Dummett]
10549 Since abstract objects cannot be picked out, we must rely on identity statements [Dummett]
10516 A realistic view of reference is possible for concrete objects, but not for abstract objects [Dummett, by Hale]
10537 The ordered pairs <x,y> can be reduced to the class of sets of the form {{x},{x,y}} [Dummett]
10542 To associate a cardinal with each set, we need the Axiom of Choice to find a representative [Dummett]
17621 What matters in mathematics is its objectivity, not the existence of the objects [Dummett]
10554 Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett]
10552 Intuitionism says that totality of numbers is only potential, but is still determinate [Dummett]
10540 We can't say that light is concrete but radio waves abstract [Dummett]
10515 Ostension is possible for concreta; abstracta can only be referred to via other objects [Dummett, by Hale]
10544 The concrete/abstract distinction seems crude: in which category is the Mistral? [Dummett]
10546 We don't need a sharp concrete/abstract distinction [Dummett]
10548 The context principle for names rules out a special philosophical sense for 'existence' [Dummett]
10281 The objects we recognise the world as containing depends on the structure of our language [Dummett]
10532 We can understand universals by studying predication [Dummett]
10534 'Nominalism' used to mean denial of universals, but now means denial of abstract objects [Dummett]
10541 Concrete objects such as sounds and smells may not be possible objects of ostension [Dummett]
10545 Abstract objects may not cause changes, but they can be the subject of change [Dummett]
9993 There is no reason why abstraction by equivalence classes should be called 'logical' [Dummett, by Tait]
9848 Content is replaceable if identical, so replaceability can't define identity [Dummett, by Dummett]
18257 Why should the limit of measurement be points, not intervals? [Dummett]
18255 Addition of quantities is prior to ordering, as shown in cyclic domains like angles [Dummett]
9896 A prime number is one which is measured by a unit alone [Dummett]
9895 A number is a multitude composed of units [Dummett]
9820 In classical logic, logical truths are valid formulas; in higher-order logics they are purely logical [Dummett]
9829 The identity of a number may be fixed by something outside structure - by counting [Dummett]
9828 Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1 [Dummett]
9836 Fregean semantics assumes a domain articulated into individual objects [Dummett]
9833 To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too [Dummett]
9842 Frege introduced criteria for identity, but thought defining identity was circular [Dummett]
9847 A contextual definition permits the elimination of the expression by a substitution [Dummett]
9852 We understand 'there are as many nuts as apples' as easily by pairing them as by counting them [Dummett]
9849 Maybe a concept is 'prior' to another if it can be defined without the second concept [Dummett]
9850 An argument for conceptual priority is greater simplicity in explanation [Dummett]
9857 We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus' [Dummett]
9859 It is absurd to deny the Equator, on the grounds that it lacks causal powers [Dummett]
9860 'We've crossed the Equator' has truth-conditions, so accept the Equator - and it's an object [Dummett]
9858 Abstract objects nowadays are those which are objective but not actual [Dummett]
9869 Realism is just the application of two-valued semantics to sentences [Dummett]
9872 Abstract objects need the context principle, since they can't be encountered directly [Dummett]
9873 Abstract terms are acceptable as long as we know how they function linguistically [Dummett]
9876 Set theory isn't part of logic, and why reduce to something more complex? [Dummett]
9885 The existence of abstract objects is a pseudo-problem [Dummett]
9880 Nominalism assumes unmediated mental contact with objects [Dummett]
9884 The distinction of concrete/abstract, or actual/non-actual, is a scale, not a dichotomy [Dummett]
11066 Deduction is justified by the semantics of its metalanguage [Dummett, by Hanna]
19058 Syntactic consequence is positive, for validity; semantic version is negative, with counterexamples [Dummett]
19059 In standard views you could replace 'true' and 'false' with mere 0 and 1 [Dummett]
19060 Truth-tables are dubious in some cases, and may be a bad way to explain connective meaning [Dummett]
19061 An explanation is often a deduction, but that may well beg the question [Dummett]
19062 Classical two-valued semantics implies that meaning is grasped through truth-conditions [Dummett]
19063 Beth trees show semantics for intuitionistic logic, in terms of how truth has been established [Dummett]
19064 Holism is not a theory of meaning; it is the denial that a theory of meaning is possible [Dummett]
19065 Soundness and completeness proofs test the theory of meaning, rather than the logic theory [Dummett]
19066 Philosophy aims to understand the world, through ordinary experience and science [Dummett]
19067 A successful proof requires recognition of truth at every step [Dummett]
18801 Classical negation is circular, if it relies on knowing negation-conditions from truth-conditions [Dummett]
18073 Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability [Dummett, by Kitcher]
19054 Meaning as use puts use beyond criticism, and needs a holistic view of language [Dummett]
19055 Stating a sentence's truth-conditions is just paraphrasing the sentence [Dummett]
19056 If a sentence is effectively undecidable, we can never know its truth conditions [Dummett]
19057 Classical quantification is an infinite conjunction or disjunction - but you may not know all the instances [Dummett]
9186 First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett]
9187 Logical truths and inference are characterized either syntactically or semantically [Dummett]
9191 Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett]
9192 The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett]
9193 ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett]
9194 The main alternative to ZF is one which includes looser classes as well as sets [Dummett]
9195 Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett]
19052 Natural language 'not' doesn't apply to sentences [Dummett]
19053 Logic would be more natural if negation only referred to predicates [Dummett]
22297 Dummett saw realism as acceptance of bivalence, rather than of mind-independent entities [Dummett, by Potter]
15049 Metaphysical realists are committed to all unambiguous statements being true or not true [Dummett]
8163 Since 'no bird here' and 'no squirrel here' seem the same, we must talk of 'atomic' facts [Dummett]
8161 We know we can state facts, with true statements [Dummett]
8165 To 'abstract from' is a logical process, as opposed to the old mental view [Dummett]
8167 If Presentism is correct, we cannot even say that the present changes [Dummett]
8166 Truth is part of semantics, since valid inference preserves truth [Dummett]
8169 We can't distinguish a proposition from its content [Dummett]
8170 Sentences are the primary semantic units, because they can say something [Dummett]
8168 To know the truth-conditions of a sentence, you must already know the meaning [Dummett]
8176 We could only guess the meanings of 'true' and 'false' when sentences were used [Dummett]
8174 The theories of meaning and understanding are the only routes to an account of thought [Dummett]
8175 A theory of thought will include propositional attitudes as well as propositions [Dummett]
8173 Language can violate bivalence because of non-referring terms or ill-defined predicates [Dummett]
8179 The law of excluded middle is the logical reflection of the principle of bivalence [Dummett]
8180 'That is red or orange' might be considered true, even though 'that is red' and 'that is orange' were not [Dummett]
8181 A justificationist theory of meaning leads to the rejection of classical logic [Dummett]
8182 Verificationism could be realist, if we imagined the verification by a superhuman power [Dummett]
8178 Empirical and a priori knowledge are not distinct, but are extremes of a sliding scale [Dummett]
8183 If truths about the past depend on memories and current evidence, the past will change [Dummett]
8184 Philosophers should not presume reality, but only invoke it when language requires it [Dummett]
8185 We can't make sense of a world not apprehended by a mind [Dummett]
8186 Time is the measure of change, so we can't speak of time before all change [Dummett]
19171 Tarski's truth is like rules for winning games, without saying what 'winning' means [Dummett, by Davidson]
10838 To explain a concept, we need its purpose, not just its rules of usage [Dummett]
10837 It is part of the concept of truth that we aim at making true statements [Dummett]
10839 You can't infer a dog's abstract concepts from its behaviour [Dummett]
10840 We must be able to specify truths in a precise language, like winning moves in a game [Dummett]
8189 Truth-condition theorists must argue use can only be described by appeal to conditions of truth [Dummett]
8191 The truth-conditions theory must get agreement on a conception of truth [Dummett]
8190 Intuitionists rely on the proof of mathematical statements, not their truth [Dummett]
8192 I no longer think what a statement about the past says is just what can justify it [Dummett]
8193 Verification is not an individual but a collective activity [Dummett]
8194 Surely there is no exact single grain that brings a heap into existence [Dummett]
8195 Undecidable statements result from quantifying over infinites, subjunctive conditionals, and the past tense [Dummett]
8198 A 'Cambridge Change' is like saying 'the landscape changes as you travel east' [Dummett]
8199 The existence of a universe without sentience or intelligence is an unintelligible fantasy [Dummett]
8197 Maybe past (which affects us) and future (which we can affect) are both real [Dummett]
8196 The present cannot exist alone as a mere boundary; past and future truths are rendered meaningless [Dummett]
21628 To say reality itself is vague is not properly intelligible [Dummett]
3303 For anti-realists there are no natural distinctions between objects [Dummett, by Benardete,JA]
7334 Anti-realism needs an intuitionist logic with no law of excluded middle [Dummett, by Miller,A]