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21844
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The history of philosophy is an agent of power: how can you think if you haven't read the great names? [Deleuze]
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Full Idea:
The history of philosophy has always been the agent of power in philosophy, and even in thought. It has played the oppressor's role: how can you think without having read Plato, Descartes, Kant and Heidegger.
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From:
Gilles Deleuze (A Conversation: what is it? What is it for? [1977], I)
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A reaction:
I find it hard to relate to this French 1960s obsession with everybody being oppressed in every conceivable way, so that 'liberation' is the only value that matters. If you ask why liberty is needed, you seem to have missed the point.
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21839
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When I meet objections I just move on; they never contribute anything [Deleuze]
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Full Idea:
Not reflection, and objections are even worse. Every time someone puts an objection to me, I want to say: 'OK, OK, let's get on to something else'. Objections have never contributed anything.
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From:
Gilles Deleuze (A Conversation: what is it? What is it for? [1977], I)
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A reaction:
I know it is heresy in analytic philosophy, but I love this! In analytic seminars you can barely complete your first sentence before someone interrupts. It's like road range - the philosophical mind state is always poised to attack, attack.
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21842
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Don't assess ideas for truth or justice; look for another idea, and establish a relationship with it [Deleuze]
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Full Idea:
You should not try to find whether an idea is just or correct. You should look for a completely different idea, elsewhere, in another area, so that something passes between the two which is neither in one nor the other.
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From:
Gilles Deleuze (A Conversation: what is it? What is it for? [1977], I)
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A reaction:
Neither relativism nor dialectic. Sounds like just having fun with ideas, but a commentator tells me it is a strategy for liberating our thought, following an agenda created by Nietzsche.
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21850
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Dualisms can be undone from within, by tracing connections, and drawing them to a new path [Deleuze]
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Full Idea:
It is always possible to undo dualisms from the inside, by tracing the line of flight which passes between the two terms or the two sets …and which draws both into a non-parallel evolution. At least this does not belong to the dialectic.
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From:
Gilles Deleuze (A Conversation: what is it? What is it for? [1977], II)
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A reaction:
Deleuze disliked Hegel's version of the dialectic. Not clear what he means here, but he is evidently groping for an alternative account of the reasoning process, which is interesting. Deleuze hates rigid dualisms.
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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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21843
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People consist of many undetermined lines, some rigid, some supple, some 'lines of flight' [Deleuze]
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Full Idea:
Things, people, are made up of varied lines, and they do not necessarily know which line they are on or where they should make the line which they are tracing pass; there is a whole geography in people, with rigid lines, supple lines, lines of flight etc.
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From:
Gilles Deleuze (A Conversation: what is it? What is it for? [1977], I)
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A reaction:
An example of Deleuze creating a novel concept, in order to generate a liberating way of seeing our lives. His big focus is on 'lines of flight' (which, I think, are less restrained by local culture than the others).
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21848
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Some lines (of flight) are becomings which escape the system [Deleuze]
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Full Idea:
There are lines which do not amount to the path of a point, which break free from structure - lines of flight, becomings, without future or past, without memory, which resist the binary machine. …The rhizome is all this.
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From:
Gilles Deleuze (A Conversation: what is it? What is it for? [1977], II)
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A reaction:
The binary machine enforces simplistic either/or choices. I assume the 'lines' are to replace the Self, with something much more indeterminate, active and changing.
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