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Single Idea 10560

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory ]

Full Idea

Set-theoretic imperialists think that it must be possible to represent every mathematical object as a set.

Gist of Idea

Set-theoretic imperialists think sets can represent every mathematical object

Source

Kit Fine (Replies on 'Limits of Abstraction' [2005], 1)

Book Ref

-: 'Philosophical Studies' [-], p.367

The 229 ideas from Kit Fine

12302 Definitions formed an abstract hierarchy for Aristotle, as sets do for us [Fine,K]
14267 There is no distinctive idea of constitution, because you can't say constitution begins and ends [Fine,K]
14264 Is there a plausible Aristotelian notion of constitution, applicable to both physical and non-physical? [Fine,K]
14265 The components of abstract definitions could play the same role as matter for physical objects [Fine,K]
14266 Aristotle sees hierarchies in definitions using genus and differentia (as we see them in sets) [Fine,K]
14268 Maybe bottom-up grounding shows constitution, and top-down grounding shows essence [Fine,K]
9146 After abstraction all numbers seem identical, so only 0 and 1 will exist! [Fine,K]
9148 I think of variables as objects rather than as signs [Fine,K]
9149 To obtain the number 2 by abstraction, we only want to abstract the distinctness of a pair of objects [Fine,K]
9150 We should define abstraction in general, with number abstraction taken as a special case [Fine,K]
9152 If green is abstracted from a thing, it is only seen as a type if it is common to many things [Fine,K]
12295 3-D says things are stretched in space but not in time, and entire at a time but not at a location [Fine,K]
12296 4-D says things are stretched in space and in time, and not entire at a time or at a location [Fine,K]
18882 You can ask when the wedding was, but not (usually) when the bride was [Fine,K, by Simons]
12297 Three-dimensionalist can accept temporal parts, as things enduring only for an instant [Fine,K]
12298 Genuine motion, rather than variation of position, requires the 'entire presence' of the object [Fine,K]
10935 An essential property of something must be bound up with what it is to be that thing [Fine,K, by Rami]
10936 Essential properties are part of an object's 'definition' [Fine,K, by Rami]
16537 Essence as necessary properties produces a profusion of essential properties [Fine,K, by Lowe]
11152 Essences are either taken as real definitions, or as necessary properties [Fine,K]
11151 An object is dependent if its essence prevents it from existing without some other object [Fine,K]
11157 Modern philosophy has largely abandoned real definitions, apart from sortals [Fine,K]
11159 My account shows how the concept works, rather than giving an analysis [Fine,K]
11160 Simple modal essentialism refers to necessary properties of an object [Fine,K]
11158 Essentialist claims can be formulated more clearly with quantified modal logic [Fine,K]
11161 Essentially having a property is naturally expressed as 'the property it must have to be what it is' [Fine,K]
11162 Socrates is necessarily distinct from the Eiffel Tower, but that is not part of his essence [Fine,K]
11163 The nature of singleton Socrates has him as a member, but not vice versa [Fine,K]
11164 It is not part of the essence of Socrates that a huge array of necessary truths should hold [Fine,K]
11165 If Socrates lacks necessary existence, then his nature cannot require his parents' existence [Fine,K]
11168 Metaphysical necessities are true in virtue of the nature of all objects [Fine,K]
11166 The subject of a proposition need not be the source of its necessity [Fine,K]
11167 Metaphysical necessity is a special case of essence, not vice versa [Fine,K]
11169 Conceptual necessities rest on the nature of all concepts [Fine,K]
11170 Analytic truth may only be true in virtue of the meanings of certain terms [Fine,K]
11172 The meaning of 'bachelor' is irrelevant to the meaning of 'unmarried man' [Fine,K]
11171 Defining a term and giving the essence of an object don't just resemble - they are the same [Fine,K]
17271 Is there metaphysical explanation (as well as causal), involving a constitutive form of determination? [Fine,K]
17273 Each basic modality has its 'own' explanatory relation [Fine,K]
17272 2+2=4 is necessary if it is snowing, but not true in virtue of the fact that it is snowing [Fine,K]
17276 If you say one thing causes another, that leaves open that the 'other' has its own distinct reality [Fine,K]
17274 Philosophical explanation is largely by ground (just as cause is used in science) [Fine,K]
17278 We can only explain how a reduction is possible if we accept the concept of ground [Fine,K]
17280 Ground is best understood as a sentence operator, rather than a relation between predicates [Fine,K]
17281 If grounding is a relation it must be between entities of the same type, preferably between facts [Fine,K]
17275 Realist metaphysics concerns what is real; naive metaphysics concerns natures of things [Fine,K]
17277 If mind supervenes on the physical, it may also explain the physical (and not vice versa) [Fine,K]
17279 Even a three-dimensionalist might identify temporal parts, in their thinking [Fine,K]
17282 Truths need not always have their source in what exists [Fine,K]
17283 If the truth-making relation is modal, then modal truths will be grounded in anything [Fine,K]
17284 An immediate ground is the next lower level, which gives the concept of a hierarchy [Fine,K]
17285 'Strict' ground moves down the explanations, but 'weak' ground can move sideways [Fine,K]
17287 Facts, such as redness and roundness of a ball, can be 'fused' into one fact [Fine,K]
17286 Logical consequence is verification by a possible world within a truth-set [Fine,K]
17291 We explain by identity (what it is), or by truth (how things are) [Fine,K]
17288 We learn grounding from what is grounded, not what does the grounding [Fine,K]
17290 Only metaphysical grounding must be explained by essence [Fine,K]
17289 Every necessary truth is grounded in the nature of something [Fine,K]
9200 Empiricists suspect modal notions: either it happens or it doesn't; it is just regularities. [Fine,K]
9202 Objects, as well as sentences, can have logical form [Fine,K]
9205 The three basic types of necessity are metaphysical, natural and normative [Fine,K]
9206 We must distinguish between the identity or essence of an object, and its necessary features [Fine,K]
9208 Philosophers with a new concept are like children with a new toy [Fine,K]
9209 Metaphysical necessity may be 'whatever the circumstance', or 'regardless of circumstances' [Fine,K]
9207 If sentence content is all worlds where it is true, all necessary truths have the same content! [Fine,K]
9210 Possible objects are abstract; actual concrete objects are possible; so abstract/concrete are compatible [Fine,K]
9211 A non-standard realism, with no privileged standpoint, might challenge its absoluteness or coherence [Fine,K]
9222 The objects and truths of mathematics are imperative procedures for their construction [Fine,K]
9223 My Proceduralism has one simple rule, and four complex rules [Fine,K]
9224 Proceduralism offers a version of logicism with no axioms, or objects, or ontological commitment [Fine,K]
9142 Fine considers abstraction as reconceptualization, to produce new senses by analysing given senses [Fine,K, by Cook/Ebert]
9143 Implicit definitions must be satisfiable, creative definitions introduce things, contextual definitions build on things [Fine,K, by Cook/Ebert]
9144 Fine's 'procedural postulationism' uses creative definitions, but avoids abstract ontology [Fine,K, by Cook/Ebert]
10135 We can abstract from concepts (e.g. to number) and from objects (e.g. to direction) [Fine,K]
10137 Abstractionism can be regarded as an alternative to set theory [Fine,K]
10136 Points in Euclidean space are abstract objects, but not introduced by abstraction [Fine,K]
10138 An object is the abstract of a concept with respect to a relation on concepts [Fine,K]
10141 Many different kinds of mathematical objects can be regarded as forms of abstraction [Fine,K]
10143 'Creative definitions' do not presuppose the existence of the objects defined [Fine,K]
10144 Postulationism says avoid abstract objects by giving procedures that produce truth [Fine,K]
10145 Abstracts cannot be identified with sets [Fine,K]
9560 S5 provides the correct logic for necessity in the broadly logical sense [Fine,K]
15063 Some sentences depend for their truth on worldly circumstances, and others do not [Fine,K]
15064 Proper necessary truths hold whatever the circumstances; transcendent truths regardless of circumstances [Fine,K]
15065 What it is is fixed prior to existence or the object's worldly features [Fine,K]
15066 B-theorists say tensed sentences have an unfilled argument-place for a time [Fine,K]
15067 A-theorists tend to reject the tensed/tenseless distinction [Fine,K]
15068 The actual world is a totality of facts, so we also think of possible worlds as totalities [Fine,K]
15069 Possible worlds may be more limited, to how things might actually turn out [Fine,K]
15070 It is the nature of Socrates to be a man, so necessarily he is a man [Fine,K]
15071 Tensed and tenseless sentences state two sorts of fact, which belong to two different 'realms' of reality [Fine,K]
15072 Bottom level facts are subject to time and world, middle to world but not time, and top to neither [Fine,K]
15074 We would understand identity between objects, even if their existence was impossible [Fine,K]
15073 Self-identity should have two components, its existence, and its neutral identity with itself [Fine,K]
15076 Essential features of an object have no relation to how things actually are [Fine,K]
15075 Modal features are not part of entities, because they are accounted for by the entity [Fine,K]
15078 There are levels of existence, as well as reality; objects exist at the lowest level in which they can function [Fine,K]
15077 It is said that in the A-theory, all existents and objects must be tensed, as well as the sentences [Fine,K]
14217 The 'standard' view of relations is that they hold of several objects in a given order [Fine,K]
14216 The 'positionalist' view of relations says the number of places is fixed, but not the order [Fine,K]
14218 A block on top of another contains one relation, not both 'on top of' and 'beneath' [Fine,K]
14219 Language imposes a direction on a road which is not really part of the road [Fine,K]
14220 Explain biased relations as orderings of the unbiased, or the unbiased as permutation classes of the biased? [Fine,K]
14252 We should understand identity in terms of the propositions it renders true [Fine,K]
14251 A natural modal account of dependence says x depends on y if y must exist when x does [Fine,K]
14253 An object's 'being' isn't existence; there's more to an object than existence, and its nature doesn't include existence [Fine,K]
14250 Metaphysics deals with the existence of things and with the nature of things [Fine,K]
14255 We understand things through their dependency relations [Fine,K]
14254 Dependency is the real counterpart of one term defining another [Fine,K]
14257 An object depends on another if the second cannot be eliminated from the first's definition [Fine,K]
14256 How do we distinguish basic from derived esssences? [Fine,K]
14260 An object only essentially has a property if that property follows from every definition of the object [Fine,K]
14258 Maybe some things have essential relationships as well as essential properties [Fine,K]
14261 There is 'weak' dependence in one definition, and 'strong' dependence in all the definitions [Fine,K]
14259 Maybe two objects might require simultaneous real definitions, as with two simultaneous terms [Fine,K]
10527 An abstraction principle should not 'inflate', producing more abstractions than objects [Fine,K]
10528 Definitions concern how we should speak, not how things are [Fine,K]
10529 If Hume's Principle can define numbers, we needn't worry about its truth [Fine,K]
10530 Hume's Principle is either adequate for number but fails to define properly, or vice versa [Fine,K]
9213 The actual world is a possible world, so we can't define possible worlds as 'what might have been' [Fine,K]
9212 Possible states of affairs are not propositions; a proposition can't be a state of affairs! [Fine,K]
16755 The possible Aristotelian view that forms are real and active principles is clearly wrong [Fine,K, by Pasnau]
12211 It is plausible that x^2 = -1 had no solutions before complex numbers were 'introduced' [Fine,K]
12209 The indispensability argument shows that nature is non-numerical, not the denial of numbers [Fine,K]
12212 Just as we introduced complex numbers, so we introduced sums and temporal parts [Fine,K]
12214 'Exists' is a predicate, not a quantifier; 'electrons exist' is like 'electrons spin' [Fine,K]
12213 Ontological claims are often universal, and not a matter of existential quantification [Fine,K]
12215 The existence of numbers is not a matter of identities, but of constituents of the world [Fine,K]
12216 Real objects are those which figure in the facts that constitute reality [Fine,K]
12218 Being real and being fundamental are separate; Thales's water might be real and divisible [Fine,K]
12217 For ontology we need, not internal or external views, but a view from outside reality [Fine,K]
15007 If you make 'grounding' fundamental, you have to mention some non-fundamental notions [Sider on Fine,K]
15046 Reality is a primitive metaphysical concept, which cannot be understood in other terms [Fine,K]
15047 What is real can only be settled in terms of 'ground' [Fine,K]
15048 In metaphysics, reality is regarded as either 'factual', or as 'fundamental' [Fine,K]
15006 Something is grounded when it holds, and is explained, and necessitated by something else [Fine,K, by Sider]
15050 Reduction might be producing a sentence which gets closer to the logical form [Fine,K]
15051 Reduction might be semantic, where a reduced sentence is understood through its reduction [Fine,K]
15052 Reduction is modal, if the reductions necessarily entail the truth of the target sentence [Fine,K]
15054 'Quietist' says abandon metaphysics because answers are unattainable (as in Kant's noumenon) [Fine,K]
15053 If metaphysics can't be settled, it hardly matters whether it makes sense [Fine,K]
15056 The notion of reduction (unlike that of 'ground') implies the unreality of what is reduced [Fine,K]
15055 Grounding relations are best expressed as relations between sentences [Fine,K]
15057 Ultimate explanations are in 'grounds', which account for other truths, which hold in virtue of the grounding [Fine,K]
15058 A proposition ingredient is 'essential' if changing it would change the truth-value [Fine,K]
15059 Grounding is an explanation of truth, and needs all the virtues of good explanations [Fine,K]
15061 Although colour depends on us, we can describe the world that way if it picks out fundamentals [Fine,K]
15060 Why should what is explanatorily basic be therefore more real? [Fine,K]
12220 Is it the sentence-token or the sentence-type that has a logical form? [Fine,K]
12222 Substitutional quantification is referential quantification over expressions [Fine,K]
10569 If you ask what F the second-order quantifier quantifies over, you treat it as first-order [Fine,K]
10560 Set-theoretic imperialists think sets can represent every mathematical object [Fine,K]
10565 There is no stage at which we can take all the sets to have been generated [Fine,K]
10564 We might combine the axioms of set theory with the axioms of mereology [Fine,K]
10563 A generative conception of abstracts proposes stages, based on concepts of previous objects [Fine,K]
10561 Abstraction-theoretic imperialists think Fregean abstracts can represent every mathematical object [Fine,K]
10562 We can combine ZF sets with abstracts as urelements [Fine,K]
10567 We can create objects from conditions, rather than from concepts [Fine,K]
10571 Concern for rigour can get in the way of understanding phenomena [Fine,K]
10573 Dedekind cuts lead to the bizarre idea that there are many different number 1's [Fine,K]
10575 Why should a Dedekind cut correspond to a number? [Fine,K]
10574 Unless we know whether 0 is identical with the null set, we create confusions [Fine,K]
10570 Assigning an entity to each predicate in semantics is largely a technical convenience [Fine,K]
10568 Logicists say mathematics can be derived from definitions, and can be known that way [Fine,K]
14530 The role of semantic necessity in semantics is like metaphysical necessity in metaphysics [Fine,K, by Hale/Hoffmann,A]
14618 Semantics is either an assignment of semantic values, or a theory of truth [Fine,K]
14619 The Quinean doubt: are semantics and facts separate, and do analytic sentences have no factual part? [Fine,K]
14621 Semantics is a body of semantic requirements, not semantic truths or assigned values [Fine,K]
14622 Referential semantics (unlike Fregeanism) allows objects themselves in to semantic requirements [Fine,K]
14620 Theories in logic are sentences closed under consequence, but in truth discussions theories have axioms [Fine,K]
15587 That two utterances say the same thing may not be intrinsic to them, but involve their relationships [Fine,K]
15589 The two main theories are Holism (which is inferential), and Representational (which is atomistic) [Fine,K]
15588 You cannot determine the full content from a thought's intrinsic character, as relations are involved [Fine,K]
15590 It seemed that Frege gave the syntax for variables, and Tarski the semantics, and that was that [Fine,K]
15591 In separate expressions variables seem identical in role, but in the same expression they aren't [Fine,K]
15592 The usual Tarskian interpretation of variables is to specify their range of values [Fine,K]
15593 Variables can be viewed as special terms - functions taking assignments into individuals [Fine,K]
15595 The 'algebraic' account of variables reduces quantification to the algebra of its component parts [Fine,K]
15594 'Instantial' accounts of variables say we grasp arbitrary instances from their use in quantification [Fine,K]
15596 The standard aim of semantics is to assign a semantic value to each expression [Fine,K]
15598 We should pursue semantic facts as stated by truths in theories (and not put the theories first!) [Fine,K]
15599 Cicero/Cicero and Cicero/Tully may differ in relationship, despite being semantically the same [Fine,K]
15600 Referentialist semantics has objects for names, properties for predicates, and propositions for connectives [Fine,K]
15601 Fregeans approach the world through sense, Referentialists through reference [Fine,K]
15602 Mental files are devices for keeping track of basic coordination of objects [Fine,K]
15604 If Cicero=Tully refers to the man twice, then surely Cicero=Cicero does as well? [Fine,K]
15603 I can only represent individuals as the same if I do not already represent them as the same [Fine,K]
15605 I take indexicals such as 'this' and 'that' to be linked to some associated demonstration [Fine,K]
11175 Logical concepts rest on certain inferences, not on facts about implications [Fine,K]
11174 A logical truth is true in virtue of the nature of the logical concepts [Fine,K]
11173 Being a man is a consequence of his essence, not constitutive of it [Fine,K]
11176 The property of Property Abstraction says any suitable condition must imply a property [Fine,K]
11177 Can the essence of an object circularly involve itself, or involve another object? [Fine,K]
11179 If there are alternative definitions, then we have three possibilities for essence [Fine,K]
11178 The essence or definition of an essence involves either a class of properties or a class of propositions [Fine,K]
14262 Formal grounding needs transitivity of grounding, no self-grounding, and the existence of both parties [Fine,K]
14263 Strong Kleene disjunction just needs one true disjunct; Weak needs the other to have some value [Fine,K]
13328 Two sorts of whole have 'rigid embodiment' (timeless parts) or 'variable embodiment' (temporary parts) [Fine,K]
13326 A 'temporary' part is a part at one time, but may not be at another, like a carburetor [Fine,K]
13327 A 'timeless' part just is a part, not a part at some time; some atoms are timeless parts of a water molecule [Fine,K]
13329 An 'aggregative' sum is spread in time, and exists whenever a component exists [Fine,K]
13330 An 'compound' sum is not spread in time, and only exists when all the components exists [Fine,K]
13331 Part and whole contribute asymmetrically to one another, so must differ [Fine,K]
13333 The matter is a relatively unstructured version of the object, like a set without membership structure [Fine,K]
13332 Hierarchical set membership models objects better than the subset or aggregate relations do [Fine,K]
23541 Supervaluation can give no answer to 'who is the last bald man' [Fine,K]
23544 Local indeterminacy concerns a single object, and global indeterminacy covers a range [Fine,K]
23540 Conjoining two indefinites by related sentences seems to produce a contradiction [Fine,K]
23542 Identifying vagueness with ignorance is the common mistake of confusing symptoms with cause [Fine,K]
23539 Classical semantics has referents for names, extensions for predicates, and T or F for sentences [Fine,K]
23543 We identify laws with regularities because we mistakenly identify causes with their symptoms [Fine,K]
23545 We do not have an intelligible concept of a borderline case [Fine,K]
23546 Standardly vagueness involves borderline cases, and a higher standpoint from which they can be seen [Fine,K]
23548 Indeterminacy is in conflict with classical logic [Fine,K]
23547 It seems absurd that there is no identity of any kind between two objects which involve survival [Fine,K]
9766 Study vagueness first by its logic, then by its truth-conditions, and then its metaphysics [Fine,K]
9768 Vagueness is semantic, a deficiency of meaning [Fine,K]
9767 A vague sentence is only true for all ways of making it completely precise [Fine,K]
9770 Logical connectives cease to be truth-functional if vagueness is treated with three values [Fine,K]
9769 Vagueness can be in predicates, names or quantifiers [Fine,K]
9772 Meaning is both actual (determining instances) and potential (possibility of greater precision) [Fine,K]
9771 Logic holding between indefinite sentences is the core of all language [Fine,K]
9773 With the super-truth approach, the classical connectives continue to work [Fine,K]
9774 Borderline cases must be under our control, as capable of greater precision [Fine,K]
9775 Excluded Middle, and classical logic, may fail for vague predicates [Fine,K]
9776 A thing might be vaguely vague, giving us higher-order vagueness [Fine,K]
9214 Unsupported testimony may still be believable [Fine,K]
9216 Each area of enquiry, and its source, has its own distinctive type of necessity [Fine,K]
9215 Causation is easier to disrupt than logic, so metaphysics is part of nature, not vice versa [Fine,K]