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22864
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Philosophy is the study and criticsm of cultural beliefs, to achieve new possibilities [Dewey]
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Full Idea:
Philosophy is criticism of the influential beliefs that underlie culture, tracking them to their generating conditions and results, and considering their mutual compatibility. This terminates in a new perspective, which leads to new possibilities.
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From:
John Dewey (The Later Works (17 vols, ed Boydston) [1930], 6:19), quoted by David Hildebrand - Dewey Intro
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A reaction:
[compressed] This would make quite a good manifesto for French thinkers of the 1960s. Foucault could hardly disagree. An excellent idea.
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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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22873
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Liberalism should improve the system, and not just ameliorate it [Dewey]
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Full Idea:
Liberalism must become radical in the sense that, instead of using social power to ameliorate the evil consequences of the existing system, it shall use social power to change the system.
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From:
John Dewey (The Later Works (17 vols, ed Boydston) [1930], 11:287), quoted by David Hildebrand - Dewey 4 'Dewey'
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A reaction:
Conservative liberals ask what people want, and try to give it to them. Radical liberals ask what people actually need, and try to make it possible. The latter is bound to be a bit paternalistic, but will probably create a better world.
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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
|
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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22869
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Knowledge is either the product of competent enquiry, or it is meaningless [Dewey]
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Full Idea:
Knowledge, as an abstract term, is a name for the product of competent enquiries. Apart from this relation, its meaning is so empty that any content or filling may be arbitrarily poured into it.
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From:
John Dewey (The Later Works (17 vols, ed Boydston) [1930], 12:16), quoted by David Hildebrand - Dewey 2 'Knowledge'
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A reaction:
What is the criterion of 'competent'? Danger of tautology, if competent enquiry is what produces knowledge.
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22867
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The quest for certainty aims for peace, and avoidance of the stress of action [Dewey]
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Full Idea:
The quest for certainty is a quest for a peace which is assured, an object which is unqualified by risk and the shadow of fear which action costs.
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From:
John Dewey (The Later Works (17 vols, ed Boydston) [1930], 4:7), quoted by David Hildebrand - Dewey 2 'Intro'
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A reaction:
This is a characteristic pragmatist account. I think Dewey and Peirce offer us the correct attitude to certainty. It is just not available to us, and can only be a delusion. That doesn't mean we don't know anything, however!
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22866
|
Mind is never isolated, but only exists in its interactions [Dewey]
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Full Idea:
Mind is primarily a verb. ...Mind never denotes anything self-contained, isolated from the world of persons and things, but is always used with respect to situations, events, objects, persons and groups.
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From:
John Dewey (The Later Works (17 vols, ed Boydston) [1930], 10:267), quoted by David Hildebrand - Dewey 1 'emerge'
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A reaction:
I strongly agree with the idea that mind is a process, not a thing. Certain types of solitary introspection don't seem to quite fit his account, but in general he is right.
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