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10170
|
While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price]
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|
Full Idea:
While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models.
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|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)
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|
A reaction:
[The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc.
|
|
18784
|
In classical logic the connectives can be related elegantly, as in De Morgan's laws [Mares]
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|
Full Idea:
Among the virtues of classical logic is the fact that the connectives are related to one another in elegant ways that often involved negation. For example, De Morgan's Laws, which involve negation, disjunction and conjunction.
|
|
From:
Edwin D. Mares (Negation [2014], 2.2)
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|
A reaction:
Mares says these enable us to take disjunction or conjunction as primitive, and then define one in terms of the other, using negation as the tool.
|
|
10175
|
Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price]
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|
Full Idea:
In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
|
|
A reaction:
It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types.
|
|
10164
|
Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
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|
Full Idea:
A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
|
|
A reaction:
This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'!
|
|
10167
|
Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
|
|
Full Idea:
Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
|
|
A reaction:
In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent.
|
|
10169
|
Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
|
|
Full Idea:
Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)
|
|
A reaction:
The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight?
|
|
10179
|
There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
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|
Full Idea:
The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6)
|
|
A reaction:
This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start.
|
|
10182
|
There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
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|
Full Idea:
There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9)
|
|
A reaction:
I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind?
|
|
10168
|
Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
|
|
Full Idea:
Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3)
|
|
A reaction:
[very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations.
|
|
10178
|
Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
|
|
Full Idea:
It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
|
|
A reaction:
[compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent.
|
|
10177
|
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
|
|
Full Idea:
Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
|
|
A reaction:
I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra.
|
|
18788
|
For intuitionists there are not numbers and sets, but processes of counting and collecting [Mares]
|
|
Full Idea:
For the intuitionist, talk of mathematical objects is rather misleading. For them, there really isn't anything that we should call the natural numbers, but instead there is counting. What intuitionists study are processes, such as counting and collecting.
|
|
From:
Edwin D. Mares (Negation [2014], 5.1)
|
|
A reaction:
That is the first time I have seen mathematical intuitionism described in a way that made it seem attractive. One might compare it to a metaphysics based on processes. Apparently intuitionists struggle with infinite sets and real numbers.
|
|
10359
|
In causation there are three problems of relata, and three metaphysical problems [Schaffer,J]
|
|
Full Idea:
The questions about causation concern their relata (in space-time, how fine-grained, how many?) and the metaphysics (distinguish causal sequences from others, the direction of causation, selecting causes among pre-conditions?).
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], Intro)
|
|
A reaction:
A very nice map (which has got me thinking about restructuring this database). I can't think of a better way to do philosophy than this (let's hear it for analysis - but the greatest role models for the approach are Aristotle and Aquinas).
|
|
10372
|
Causation may not be transitive; the last event may follow from the first, but not be caused by it [Schaffer,J]
|
|
Full Idea:
It is not clear whether causation is transitive. For example, if a boulder roll's towards a hiker's head, causing the hiker to duck, which causes the hiker to survive, it does not seem that the rolling boulder causes the survival of the hiker.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 1.2)
|
|
A reaction:
Maybe survival is not an event or an effect. How many times have I survived in my life? We could, though, say that the hiker strained a muscle as he or she ducked. But then it is unclear whether the boulder caused the muscle-strain.
|
|
10374
|
There are at least ten theories about causal connections [Schaffer,J]
|
|
Full Idea:
Theories of causal connection are: nomological subsumption, statistical correlation, counterfactual dependence, agential manipulability, contiguous change, energy flow, physical processes, property transference, primitivism and eliminativism.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 1.3.1)
|
|
A reaction:
Schaffer reduces these to probability and process. I prefer the latter. The first two are wrong, the third right but superficial, the fourth wrong, the fifth, sixth and seventh on the right lines, the eighth wrong, the ninth tempting, and the last wrong.
|
|
10377
|
Causation may not be a process, if a crucial part of the process is 'disconnected' [Schaffer,J]
|
|
Full Idea:
One problem case for the process view of causation is 'disconnection'. If a brick breaks a window by being fired from a catapult, a latch is released which was preventing the catapult from firing, so the 'process' is just internal to the catapult.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.1)
|
|
A reaction:
Schaffer says the normal reply is to deny that the catch-releasing is genuinely causal. I would have thought we should go more fine-grained, and identify linked components of the causal process.
|
|
10375
|
At least four rivals have challenged the view that causal direction is time direction [Schaffer,J]
|
|
Full Idea:
The traditional view that the direction of causation is the direction of time has been challenged, by the direction of forking, by overdetermination, by independence, and by manipulation, which all seem to be one-directional features.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 1.3.1)
|
|
A reaction:
Personally I incline to the view that time is prior, and fixes the direction of causation. I'm not sure that 'backward causation' can be stated coherently, even if it is metaphysically or naturally possible.
|
|
10389
|
Causal order must be temporal, or else causes could be blocked, and time couldn't be explained [Schaffer,J]
|
|
Full Idea:
Reasons for causal order being temporal order are that otherwise the effect might occur but the cause then get prevented, ..and that they must be the same, because the temporal order can only be analysed in terms of the causal order.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 2.2)
|
|
A reaction:
If one took both time and causation as primitive, then the second argument would be void. The first argument, though, sounds pretty overwhelming to me.
|
|
10390
|
Causal order is not temporal, because of time travel, and simultanous, joint or backward causes [Schaffer,J]
|
|
Full Idea:
Reasons for denying that causal order is temporal order are that time travel seems possible, that cause and effect can be simultaneous, because joint effects have temporal order without causal connection, and because backward causation may exist.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 2.2)
|
|
A reaction:
The possibility of time travel and backward causation can clearly be doubted, and certainly can't be grounds for one's whole metaphysics. The other two need careful analysis, but I think they can be answered. Causation is temporal.
|
|
10380
|
Causation is primitive; it is too intractable and central to be reduced; all explanations require it [Schaffer,J]
|
|
Full Idea:
Primitivism arises from our failure to reduce causation, but also from causation being too central to reduce. The probability and process accounts are said to be inevitably circular, as they cannot be understood without reference to causation.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2)
|
|
A reaction:
This is very tempting. The primitive view, though, must deal with the direction problem, which may suggest that time is even more primitive. Can we have a hierarchy of primitiveness? To be alive is to be causal.
|
|
10385
|
If causation is just observables, or part of common sense, or vacuous, it can't be primitive [Schaffer,J]
|
|
Full Idea:
The three main objections to causation being primitive are that causation can't be anything more than what we observe, or that such a primitive is too spooky to be acceptable, or that primitivism leads to elimination of causation.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2)
|
|
A reaction:
[summarised] I don't like the first (Humean) view. I suspect that anything which we finally decide has to be primitive (time, for example) is going to be left looking 'spooky', and I suspect that eliminativism is just Humeanism in disguise.
|
|
10387
|
The notion of causation allows understanding of science, without appearing in equations [Schaffer,J]
|
|
Full Idea:
The concepts of 'event', 'law', 'cause' and 'explanation' are nomic concepts which serve to allow a systematic understanding of science; they do not themselves appear in the equations.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2)
|
|
A reaction:
This is a criticism of Russell's attempt to eliminate causation from science. It shows that there has to be something we can call 'metascience', which is the province of philosophers, since scientists don't have much interest in it.
|
|
10388
|
Causation is utterly essential for numerous philosophical explanations [Schaffer,J]
|
|
Full Idea:
Causation can't be eliminated if it is needed to explain persistence, explanation, disposition, perception, warrant, action, responsibility, mental functional role, conceptual content, and reference. It's elimination would be catastrophic.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2)
|
|
A reaction:
[compressed list] I think I am going to vote for the view that causation is one of the primitives in the metaphysics of nature, so I have to agree with this. Most of the listed items, though, are controversial, so eliminativists are not defeated.
|
|
10384
|
If two different causes are possible in one set of circumstances, causation is primitive [Schaffer,J]
|
|
Full Idea:
Causation seems to be primitive if the same laws and patterns of events might embody three different possible causes, as when two magicians cast the same successful spell, each with a 50% chance of success, and who was successful is unclear.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2)
|
|
A reaction:
I'm cautious when the examples involve magic. It implies that the process that leads to the result will be impossible to observe, but if magic never really happens, then the patterns of events will always be different.
|
|
10386
|
If causation is primitive, it can be experienced in ourselves, or inferred as best explanation [Schaffer,J]
|
|
Full Idea:
The view that causation is primitive can be defended against Humean critics by saying that causation can be directly observed in the will or our bodies, or that it can be inferred as the best explanation.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2)
|
|
A reaction:
I like both views, and have just converted myself to the primitivist view of causation! I can't know the essence of a tree, because I am not a tree, but I can know the essence of causation. The Greek fascination with explaining movement is linked.
|
|
10361
|
Events are fairly course-grained (just saying 'hello'), unlike facts (like saying 'hello' loudly) [Schaffer,J]
|
|
Full Idea:
Events are relatively coarse-grained, unlike facts; so the event of John's saying 'hello' seems to be the same event as John's saying 'hello' loudly, while they seem to be different facts.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 1)
|
|
A reaction:
The example seems good support for facts, since saying 'hello' loudly could have quite different effects from just saying 'hello'. I also incline temperamentally towards a fine-grained account, because it is more reductivist.
|
|
10360
|
Causal relata are events - or facts, features, tropes, states, situations or aspects [Schaffer,J]
|
|
Full Idea:
The standard view make causal relata events (Davidson, Kim, Lewis), but there is considerable support for facts (Bennett, Mellor), and occasional support for features (Dretske), tropes (Campbell), states of affairs (Armstrong), and situations and aspects.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 1)
|
|
A reaction:
An event is presumed to be concrete, while a fact is more abstract (a proposition, perhaps). I'm always drawn to 'processes' (because they are good for discussing the mind), so an event, as a sort of natural process, looks good.
|
|
10368
|
If causal relata must be in nature and fine-grained, neither facts nor events will do [Schaffer,J]
|
|
Full Idea:
Theorists who reject both events and facts as causal relata do so because the relata must be immanent in nature, and thus not facts, but also fine-grained and thus not events.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 1.2)
|
|
A reaction:
Kim, however, offers a fine-grained account of events (as triples), and Bennett individuates them even more finely (as propositions), so events might be saved. Descriptions can be very fine-grained.
|
|
10383
|
The relata of causation (such as events) need properties as explanation, which need causation! [Schaffer,J]
|
|
Full Idea:
The primitivist about causation might say that the notion of an event (or other relata) cannot be understood without reference to causation, because properties themselves are individuated by their causal role.
|
|
From:
Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2)
|
|
A reaction:
Having enthusiastically embraced the causal view of properties (see Shoemaker and Ellis), I suddenly realise that I seem required to embrace primitivism about causation, which I hadn't anticipated! I've no immediate problem with that.
|