12105
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Our knowledge starts in theology, passes through metaphysics, and ends in positivism [Comte]
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Full Idea:
Our principal conceptions, each branch of our knowledge, passes in succession through three different theoretical states: the theological or fictitious state, the metaphysical or abstract state, and the scientific or positive state.
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From:
Auguste Comte (Intro to Positive Philosophy [1830], Ch.1)
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A reaction:
See Idea 5077 for the abstraction step. The idea that there is a 'law' here, as Comte thinks, is daft, but something of what he describes is undeniable. I suspect, though, that science rests on abstractions, so the last part is wrong.
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12106
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Positivism gives up absolute truth, and seeks phenomenal laws, by reason and observation [Comte]
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Full Idea:
In the positive state, the human mind, recognizing the impossibility of obtaining absolute truth, gives up the search for hidden and final causes. It endeavours to discover, by well-combined reasoning and observation, the actual laws of phenomena.
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From:
Auguste Comte (Intro to Positive Philosophy [1830], Ch.1)
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A reaction:
[compressed] Positivism attempted to turn the Humean regularity view of laws into a semi-religion. It is striking how pessimistic Comte was (as was Hume) about the chances of science revealing deep explanations. He would be astoundeds.
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10170
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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.
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10175
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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.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
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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.
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10164
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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.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
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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'!
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10167
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Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
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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.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
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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.
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10169
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Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
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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.
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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 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?
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10179
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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.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6)
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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.
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10182
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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.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9)
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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?
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10168
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Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
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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.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3)
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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.
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10178
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Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
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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.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
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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.
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10177
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Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
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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.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
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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.
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20585
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If an experience machine gives you any experience you want, should you hook up for life? [Nozick]
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Full Idea:
Suppose there were an experience machine that would give you any experience you desired ...such as writing a great novel, or making a friend, or reading an interesting book. ...Should you plug into this machine for life?
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From:
Robert Nozick (Anarchy,State, and Utopia [1974], 3 'Experience')
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A reaction:
A classic though experiment which crystalises a major problem with hedonistic utilitarianism. My addition is a machine which maximises the pleasure of my family and friends, to save me the bother of doing it.
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18643
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A minimal state should protect, but a state forcing us to do more is unjustified [Nozick]
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Full Idea:
A minimal state, limited to the narrow functions of protection against force, theft, fraud, enforcement of contracts, and so on, is justified; any more extensive state will violate persons' rights not to be forced to do certain things, and is unjustified.
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From:
Robert Nozick (Anarchy,State, and Utopia [1974], Pref)
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A reaction:
This has some plausibility for a huge modern state, where we don't know one another, but it would be a ridiculous attitude in a traditional village.
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18642
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Individual rights are so strong that the state and its officials must be very limited in power [Nozick]
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Full Idea:
Individuals have rights, and there are things no person or group may do to them (without violating their rights). So strong and far-reaching are these rights that they raise the question of what, if anything, the state and its officials may do.
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From:
Robert Nozick (Anarchy,State, and Utopia [1974], Pref)
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A reaction:
This claim appears to be an axiom, but I'm not sure that the notion of 'rights' make any sense unless someone is granting the rights, where the someone is either a strong individual, or the community (perhaps represented by the state).
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18644
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States can't enforce mutual aid on citizens, or interfere for their own good [Nozick]
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Full Idea:
A state may not use its coercive apparatus for the purposes of getting some citizens to aid others, or in order to prohibit activities to people for their own good or protection.
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From:
Robert Nozick (Anarchy,State, and Utopia [1974], Pref)
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A reaction:
You certainly can't apply these principles to children, so becoming an 'adult' seems to be a very profound step in Nozick's account. At what age must we stop interfering with people for their own good. If the state is prohibited, are neighbours also?
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22661
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My Anarchy, State and Utopia neglected our formal social ties and concerns [Nozick on Nozick]
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Full Idea:
The political philosophy represented in Anarchy, State and Utopia ignored the importance of joint and official symbolic statement and expression of our social ties and concern, and hence (I have written) is inadequate.
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From:
comment on Robert Nozick (Anarchy,State, and Utopia [1974], p.32) by Robert Nozick - The Nature of Rationality p.32
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A reaction:
In other words, it was far too individualistic, and neglected community, even though it has become the sacred text for libertarian individualism. Do any Nozick fans care about this recantation?
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18641
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If people hold things legitimately, just distribution is simply the result of free exchanges [Nozick, by Kymlicka]
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Full Idea:
If we assume that everyone is entitled to the goods they currently possess (their 'holdings'), then a just distribution is simply whatever distribution results from people's free exchanges.
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From:
report of Robert Nozick (Anarchy,State, and Utopia [1974]) by Will Kymlicka - Contemporary Political Philosophy (1st edn) 4.1.b
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A reaction:
If people's current 'legitimate' holdings are hugely unequal, it seems very unlikely that the ensuing exchanges will be 'free' in the way that Nozick envisages.
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20539
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Property is legitimate by initial acquisition, voluntary transfer, or rectification of injustice [Nozick, by Swift]
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Full Idea:
Nozick identified three ways in which people can acquire a legitimate property holding: initial acquisition, voluntary transfer, and rectification (of unjust transfers).
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From:
report of Robert Nozick (Anarchy,State, and Utopia [1974]) by Adam Swift - Political Philosophy (3rd ed) 1 'Nozick'
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A reaction:
I think it is a delusion to look for justice in the ownership of property. You can't claim justice for buying property if the money to do it was acquired unjustly. And what rights over those who live on the land come with the 'ownership'?
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18646
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How did the private property get started? If violence was involved, we can redistribute it [Kymlicka on Nozick]
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Full Idea:
How did these natural resources, which were not initially owned by anyone, come to be part of someone's private property? ...The fact that the initial acquisition often involved force means there is no moral objection to redistributing existing wealth.
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From:
comment on Robert Nozick (Anarchy,State, and Utopia [1974]) by Will Kymlicka - Contemporary Political Philosophy (1st edn) 4.2.b
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A reaction:
[He cites G.A. Cphen 1988 for the second point] Put like this, Nozick's theory just looks like the sort of propaganda which is typically put out by the winners. Is there an implicit threat of violent resistance in his advocacy of individual rights?
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21737
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Unowned things may be permanently acquired, if it doesn't worsen the position of other people [Nozick]
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Full Idea:
One may acquire a permanent bequeathable property right in a previously unowned thing, as long as the position of others no longer at liberty to use the thing is not thereby worsened.
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From:
Robert Nozick (Anarchy,State, and Utopia [1974], p.178), quoted by G.A. Cohen - Are Freedom and Equality Compatible? 2
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A reaction:
Cohen attacks this vigorously. His main point is that Nozick has a very narrow view of what the acquisition should be compared with. There are many alternatives. Does being made unable to improve something 'worsen' a person's condition?
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21738
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Maybe land was originally collectively owned, rather than unowned? [Cohen,GA on Nozick]
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Full Idea:
Why should we not regard land as originally collectively owned rather than, as Nozick takes for granted, owned by no one?
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From:
comment on Robert Nozick (Anarchy,State, and Utopia [1974], p.178) by G.A. Cohen - Are Freedom and Equality Compatible? 2
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A reaction:
Did native Americans and Australians collectively own the land? Lots of peoples, I suspect, don't privately own anything, because the very concept has never occured to them (and they have no legal system).
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