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Single Idea 18392

[filed under theme 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units ]

Full Idea

Classes, because they have a particular cardinality, are essentially a certain number of ones, things that, within the particular class, are each taken as a unit.

Gist of Idea

Classes have cardinalities, so their members must all be treated as units

Source

David M. Armstrong (Truth and Truthmakers [2004], 09.1)

Book Ref

Armstrong,D.M.: 'Truth and Truthmakers' [CUP 2004], p.114

A Reaction

[Singletons are exceptions] So units are basic to set theory, which is the foundations of technical analytic philosophy (as well as, for many, of mathematics). If you can't treat something as a unit, it won't go into set theory. Vagueness...

Related Idea

Idea 18396 The set theory brackets { } assert that the member is a unit [Armstrong]

The 133 ideas from David M. Armstrong

8505 Refusal to explain why different tokens are of the same type is to be an ostrich [Armstrong]
8506 Particulars and properties are distinguishable, but too close to speak of a relation [Armstrong]
8507 Some think of reality as made of things; I prefer facts or states of affairs [Armstrong]
3900 Maybe experience is not essential to perception, but only to the causing of beliefs [Armstrong, by Scruton]
4253 Externalism says knowledge involves a natural relation between the belief state and what makes it true [Armstrong]
15542 All possibilities are recombinations of properties in the actual world [Armstrong, by Lewis]
15547 Negative existentials have 'totality facts' as truthmakers [Armstrong, by Lewis]
5690 A mental state without belief refutes self-intimation; a belief with no state refutes infallibility [Armstrong, by Shoemaker]
5493 If pains are defined causally, and research shows that the causal role is physical, then pains are physical [Armstrong, by Lycan]
4600 Armstrong and Lewis see functionalism as an identity of the function and its realiser [Armstrong, by Heil]
6498 Armstrong suggests secondary qualities are blurred primary qualities [Armstrong, by Robinson,H]
14330 To be realists about dispositions, we can only discuss them through their categorical basis [Armstrong]
11003 The best version of reductionist actualism around is Armstrong's combinatorial account [Armstrong, by Read]
4032 The problem of universals is how many particulars can all be of the same 'type' [Armstrong]
7429 Causal Functionalism says mental states are apt for producing behaviour [Armstrong]
7432 One mental role might be filled by a variety of physical types [Armstrong]
7434 Behaviourism is false, but mind is definable as the cause of behaviour [Armstrong]
7431 The identity of mental states with physical properties is contingent, because the laws of nature are contingent [Armstrong]
7436 The manifestations of a disposition need never actually exist [Armstrong]
7437 Consciousness and experience of qualities are not the same [Armstrong]
7438 A causal theory of mentality would be improved by a teleological element [Armstrong]
7440 Secondary qualities are microscopic primary qualities of physical things [Armstrong]
5492 How can essences generate the right powers to vary with distance between objects? [Armstrong]
4036 What matters is not how many entities we postulate, but how many kinds of entities [Armstrong, by Mellor/Oliver]
8529 Deniers of properties and relations rely on either predicates or on classes [Armstrong]
8532 Resemblances must be in certain 'respects', and they seem awfully like properties [Armstrong]
4034 Whether we apply 'cold' or 'hot' to an object is quite separate from its change of temperature [Armstrong]
8535 To the claim that every predicate has a property, start by eliminating failure of application of predicate [Armstrong]
8533 Predicates need ontological correlates to ensure that they apply [Armstrong]
4035 There must be some explanation of why certain predicates are applicable to certain objects [Armstrong]
8530 Change of temperature in objects is quite independent of the predicates 'hot' and 'cold' [Armstrong]
8536 We want to know what constituents of objects are grounds for the application of predicates [Armstrong]
8531 In most sets there is no property common to all the members [Armstrong]
8540 The introduction of sparse properties avoids the regularity theory's problem with 'grue' [Armstrong]
8541 Regularities theories are poor on causal connections, counterfactuals and probability [Armstrong]
8537 Tropes fall into classes, because exact similarity is symmetrical and transitive [Armstrong]
8538 Trope theory needs extra commitments, to symmetry and non-transitivity, unless resemblance is exact [Armstrong]
8539 Universals are required to give a satisfactory account of the laws of nature [Armstrong]
15754 Without properties we would be unable to express the laws of nature [Armstrong]
15753 Essences might support Resemblance Nominalism, but they are too coarse and ill-defined [Armstrong]
9478 Even if all properties are categorical, they may be denoted by dispositional predicates [Armstrong, by Bird]
7024 Properties are universals, which are always instantiated [Armstrong, by Heil]
10729 Universals explain resemblance and causal power [Armstrong, by Oliver]
15544 If what is actual might have been impossible, we need S4 modal logic [Armstrong, by Lewis]
10728 A thing's self-identity can't be a universal, since we can know it a priori [Armstrong, by Oliver]
4031 It doesn't follow that because there is a predicate there must therefore exist a property [Armstrong]
10024 The type-token distinction is the universal-particular distinction [Armstrong, by Hodes]
18467 Truth-making can't be entailment, because truthmakers are portions of reality [Armstrong]
18468 Armstrong says truthmakers necessitate their truth, where 'necessitate' is a primitive relation [Armstrong, by MacBride]
18368 For all being, there is a potential proposition which expresses its existence and nature [Armstrong]
18370 A realm of abstract propositions is causally inert, so has no explanatory value [Armstrong]
18371 The class of similar things is much too big a truthmaker for the feature of a particular [Armstrong]
18372 We need properties, as minimal truthmakers for the truths about objects [Armstrong]
18373 If tropes are non-transferable, then they necessarily belong to their particular substance [Armstrong]
18374 Truthmaking needs states of affairs, to unite particulars with tropes or universals. [Armstrong]
18375 General truths are a type of negative truth, saying there are no more ravens than black ones [Armstrong]
18377 Negative truths have as truthmakers all states of affairs relevant to the truth [Armstrong]
18378 Length is a 'determinable' property, and one mile is one its 'determinates' [Armstrong]
18379 The determinates of a determinable must be incompatible with each other [Armstrong]
18380 Negative causations supervene on positive causations plus their laws? [Armstrong]
18381 Necessitating general truthmakers must also specify their limits [Armstrong]
18382 The nature of arctic animals is truthmaker for the absence of penguins there [Armstrong]
18384 One truthmaker will do for a contingent truth and for its contradictory [Armstrong]
18385 Logical atomism builds on the simple properties, but are they the only possible properties? [Armstrong]
18386 What is the truthmaker for 'it is possible that there could have been nothing'? [Armstrong]
18387 The truthmakers for possible unicorns are the elements in their combination [Armstrong]
18388 Possible worlds don't fix necessities; intrinsic necessities imply the extension in worlds [Armstrong]
18389 When entities contain entities, or overlap with them, there is 'partial' identity [Armstrong]
18390 All metaphysical discussion should be guided by a quest for truthmakers [Armstrong]
18393 For 'there is a class with no members' we don't need the null set as truthmaker [Armstrong]
18392 Classes have cardinalities, so their members must all be treated as units [Armstrong]
18391 'Naturalism' says only the world of space-time exists [Armstrong]
18394 In mathematics, truthmakers are possible instantiations of structures [Armstrong]
18396 The set theory brackets { } assert that the member is a unit [Armstrong]
18397 Powers must result in some non-powers, or there would only be potential without result [Armstrong]
18399 How does the power of gravity know the distance it acts over? [Armstrong]
18400 Properties are not powers - they just have powers [Armstrong]
18401 The pure present moment is too brief to be experienced [Armstrong]
4437 'Mereological Nominalism' sees whiteness as a huge white object consisting of all the white things [Armstrong]
4438 'Mereological Nominalism' may work for whiteness, but it doesn't seem to work for squareness [Armstrong]
4434 'Class Nominalism' says that properties or kinds are merely membership of a set (e.g. of white things) [Armstrong]
4435 'Class Nominalism' cannot explain co-extensive properties, or sets with random members [Armstrong]
4436 'Class Nominalism' may explain properties if we stick to 'natural' sets, and ignore random ones [Armstrong]
4431 'Predicate Nominalism' says that a 'universal' property is just a predicate applied to lots of things [Armstrong]
4433 Concept and predicate nominalism miss out some predicates, and may be viciously regressive [Armstrong]
4439 'Resemblance Nominalism' says properties are resemblances between classes of particulars [Armstrong]
4440 'Resemblance Nominalism' finds that in practice the construction of resemblance classes is hard [Armstrong]
4432 'Concept Nominalism' says a 'universal' property is just a mental concept applied to lots of things [Armstrong]
4446 It is claimed that some universals are not exemplified by any particular, so must exist separately [Armstrong]
4445 If properties and relations are particulars, there is still the problem of how to classify and group them [Armstrong]
4444 One moderate nominalist view says that properties and relations exist, but they are particulars [Armstrong]
4448 Should we decide which universals exist a priori (through words), or a posteriori (through science)? [Armstrong]
4442 Most thinkers now reject self-predication (whiteness is NOT white) so there is no Third Man problem [Armstrong]
12677 Armstrong holds that all basic properties are categorical [Armstrong, by Ellis]
16246 Rather than take necessitation between universals as primitive, just make laws primitive [Maudlin on Armstrong]
9480 Armstrong has an unclear notion of contingent necessitation, which can't necessitate anything [Bird on Armstrong]
8582 Regularities are lawful if a second-order universal unites two first-order universals [Armstrong, by Lewis]
17662 Science depends on laws of nature to study unobserved times and spaces [Armstrong]
17664 Each subject has an appropriate level of abstraction [Armstrong]
17663 If you know what it is, investigation is pointless. If you don't, investigation is impossible [Armstrong]
17666 Actualism means that ontology cannot contain what is merely physically possible [Armstrong]
17667 Dispositions exist, but their truth-makers are actual or categorical properties [Armstrong]
17668 It is likely that particulars can be individuated by unique conjunctions of properties [Armstrong]
17669 Realist regularity theories of laws need universals, to pick out the same phenomena [Armstrong]
17671 A naive regularity view says if it never occurs then it is impossible [Armstrong]
17670 Newton's First Law refers to bodies not acted upon by a force, but there may be no such body [Armstrong]
17672 A good reason for something (the smoke) is not an explanation of it (the fire) [Armstrong]
17674 The raven paradox has three disjuncts, confirmed by confirming any one of them [Armstrong]
17675 Unlike 'green', the 'grue' predicate involves a time and a change [Armstrong]
17676 Best explanations explain the most by means of the least [Armstrong]
17680 The identity of a thing with itself can be ruled out as a pseudo-property [Armstrong]
17677 Past, present and future must be equally real if universals are instantiated [Armstrong]
17678 Universals are just the repeatable features of a world [Armstrong]
17679 All instances of some property are strictly identical [Armstrong]
17681 The laws of nature link properties with properties [Armstrong]
17682 A universe couldn't consist of mere laws [Armstrong]
17683 Science suggests that the predicate 'grue' is not a genuine single universal [Armstrong]
17684 To explain observations by a regular law is to explain the observations by the observations [Armstrong]
17685 Induction aims at 'all Fs', but abduction aims at hidden or theoretical entities [Armstrong]
17687 If everything is powers there is a vicious regress, as powers are defined by more powers [Armstrong]
17688 Negative facts are supervenient on positive facts, suggesting they are positive facts [Armstrong]
17689 Absences might be effects, but surely not causes? [Armstrong]
17690 Oaken conditional laws, Iron universal laws, and Steel necessary laws [Armstrong, by PG]
17691 Nothing is genuinely related to itself [Armstrong]
17692 We can't deduce the phenomena from the One [Armstrong]
17693 The necessary/contingent distinction may need to recognise possibilities as real [Armstrong]
17686 Universals are abstractions from states of affairs [Armstrong]
15442 Universals are abstractions from their particular instances [Armstrong, by Lewis]
15550 Properties are contingently existing beings with multiple locations in space and time [Armstrong, by Lewis]
4798 In recent writings, Armstrong makes a direct identification of necessitation with causation [Armstrong, by Psillos]
9497 Without modality, Armstrong falls back on fictionalism to support counterfactual laws [Bird on Armstrong]
4743 The truth-maker for a truth must necessitate that truth [Armstrong]
4742 Correspondence may be one-many or many one, as when either p or q make 'p or q' true [Armstrong]