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10044
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Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element [Russell/Whitehead, by Shapiro]
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Full Idea:
Russell adduces two reasons against the extensional view of classes, namely the existence of the null class (which cannot very well be a collection), and the unit classes (which would have to be identical with their single elements).
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Structure and Ontology p.459
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A reaction:
Gödel believes in the reality of classes. I have great sympathy with Russell, when people start to claim that sets are not just conveniences to help us think about things, but actual abstract entities. Is the singleton of my pencil is on this table?
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21707
|
Russell unusually saw logic as 'interpreted' (though very general, and neutral) [Russell/Whitehead, by Linsky,B]
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Full Idea:
Russell did not view logic as an uninterpreted calculus awaiting interpretations [the modern view]. Rather, logic is a single 'interpreted' body of a priori truths, of propositions rather than sentence forms - but maximally general and topic neutral.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 1
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A reaction:
This is the view which Wittgenstein challenged, saying logic is just conventional. Linsky claims that Russell's logicism is much more plausible, once you understand his view of logic.
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8683
|
Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Russell/Whitehead, by Friend]
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Full Idea:
Unlike Frege, Russell and Whitehead were not realists about mathematical objects, and whereas Frege thought that only arithmetic and analysis are branches of logic, they think the vast majority of mathematics (including geometry) is essentially logical.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.1
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A reaction:
If, in essence, Descartes reduced geometry to algebra (by inventing co-ordinates), then geometry ought to be included. It is characteristic of Russell's hubris to want to embrace everything.
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10093
|
The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman]
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Full Idea:
Russell and Whitehead's ramified theory of types worked not with sets, but with propositional functions (similar to Frege's concepts), with a more restrictive assignment of variables, insisting that bound, as well as free, variables be of lower type.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.3
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A reaction:
I don't fully understand this (and no one seems much interested any more), but I think variables are a key notion, and there is something interesting going on here. I am intrigued by ordinary language which behaves like variables.
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8691
|
The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead]
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Full Idea:
The Russell/Whitehead type theory reduces mathematics to a consistent founding discipline, but is criticised for not really being logic. They could not prove the existence of infinite sets, and introduced a non-logical 'axiom of reducibility'.
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From:
comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.6
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A reaction:
To have reduced most of mathematics to a founding discipline sounds like quite an achievement, and its failure to be based in pure logic doesn't sound too bad. However, it seems to reduce some maths to just other maths.
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10305
|
In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead]
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Full Idea:
In the system of 'Principia Mathematica', it is not only the axioms of infinity and reducibility which go beyond pure logic, but also the initial conception of a universal domain of individuals and of a domain of predicates.
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From:
comment on B Russell/AN Whitehead (Principia Mathematica [1913], p.267) by Paul Bernays - On Platonism in Mathematics p.267
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A reaction:
This sort of criticism seems to be the real collapse of the logicist programme, rather than Russell's paradox, or Gödel's Incompleteness Theorems. It just became impossible to stick strictly to logic in the reduction of arithmetic.
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8746
|
To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro]
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Full Idea:
Russell insisted on the vicious circle principle, and thus rejected impredicative definitions, which resulted in an unwieldy ramified type theory, with the ad hoc axiom of reducibility. Ramsey's simpler theory was impredicative and avoided the axiom.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Thinking About Mathematics 5.2
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A reaction:
Nowadays the theory of types seems to have been given up, possibly because it has no real attraction if it lacks the strict character which Russell aspired to.
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12033
|
An object is identical with itself, and no different indiscernible object can share that [Russell/Whitehead, by Adams,RM]
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Full Idea:
Trivially, the Identity of Indiscernibles says that two individuals, Castor and Pollux, cannot have all properties in common. For Castor must have the properties of being identical with Castor and not being identical with Pollux, which Pollux can't share.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913], I p.57) by Robert Merrihew Adams - Primitive Thisness and Primitive Identity 2
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A reaction:
I suspect that either the property of being identical with itself is quite vacuous, or it is parasytic on primitive identity, or it is the criterion which is actually used to define identity. Either way, I don't find this claim very illuminating.
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10040
|
Russell showed, through the paradoxes, that our basic logical intuitions are self-contradictory [Russell/Whitehead, by Gödel]
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Full Idea:
By analyzing the paradoxes to which Cantor's set theory had led, ..Russell brought to light the amazing fact that our logical intuitions (concerning such notions as truth, concept, being, class) are self-contradictory.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.452
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A reaction:
The main intuition that failed was, I take it, that every concept has an extension, that is, there are always objects which will or could fall under the concept.
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21725
|
The multiple relations theory says assertions about propositions are about their ingredients [Russell/Whitehead, by Linsky,B]
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Full Idea:
The multiple relations theory of judgement proposes that assertions about propositions are dependent upon genuine facts involving belief and other attitude relations, subjects of those attitudes, and the constituents of the belief.
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From:
report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 7.2
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A reaction:
This seems to require a commitment to universals (especially relations) with which we can be directly acquainted. I prefer propositions, but as mental entities, not platonic entities.
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23453
|
Propositions as objects of judgement don't exist, because we judge several objects, not one [Russell/Whitehead]
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Full Idea:
A 'proposition', in the sense in which a proposition is supposed to be the object of a judgement, is a false abstraction, because a judgement has several objects, not one.
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From:
B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by Michael Morris - Guidebook to Wittgenstein's Tractatus 2E
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A reaction:
This is the rejection of the 'Russellian' theory of propositions, in favour of his multiple-relations theory of judgement. But why don't the related objects add up to a proposition about a state of affairs?
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7903
|
The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]
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Full Idea:
The six perfections are of giving, morality, patience, vigour, meditation, and wisdom.
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|
From:
Nagarjuna (Mahaprajnaparamitashastra [c.120], 88)
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A reaction:
What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate').
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12759
|
There are atoms of substance, but no atoms of bulk or extension [Leibniz]
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Full Idea:
Although there are atoms of substance, namely monads, which lack parts, there are no atoms of bulk [moles], that is, atoms of the least possible extension, nor are there any ultimate elements, since a continuum cannot be composed out of points.
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|
From:
Gottfried Leibniz (On Nature Itself (De Ipsa Natura) [1698], §11)
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A reaction:
Leibniz has a constant battle for the rest of his career to explain what these 'atoms of substance' are, since they have location but no extension, they are self-sufficient yet generate force, and are non-physical but interact with matter.
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12718
|
Secondary matter is active and complete; primary matter is passive and incomplete [Leibniz]
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Full Idea:
I understand matter as either secondary or primary. Secondary matter is, indeed, a complete substance, but it is not merely passive; primary matter is merely passive, but it is not a complete substance. So we must add a soul or form...
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|
From:
Gottfried Leibniz (On Nature Itself (De Ipsa Natura) [1698], §12), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 4
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A reaction:
It sounds as if primary matter is redundant, but Garber suggests that secondary matter is just the combination of primary matter with form.
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11854
|
If there is some trace of God in things, that would explain their natural force [Leibniz]
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Full Idea:
If the law of God does indeed leave some vestige of him expressed in things...then it must be granted that there is a certain efficacy residing in things, a form or force such as we usually designate by the name of nature, from which the phenomena follow.
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|
From:
Gottfried Leibniz (On Nature Itself (De Ipsa Natura) [1698], §06)
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A reaction:
I wouldn't rate this as a very promising theory of powers, but it seems to me important that Leibniz recognises the innate power in things as needing explanation. If you remove divine power, you are left with unexplained intrinsic powers.
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12758
|
It is plausible to think substances contain the same immanent force seen in our free will [Leibniz]
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Full Idea:
If we attribute an inherent force to our mind, a force acting immanently, then nothing forbids us to suppose that the same force would be found in other souls or forms, or, if you prefer, in the nature of substances.
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|
From:
Gottfried Leibniz (On Nature Itself (De Ipsa Natura) [1698], §10)
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A reaction:
This is the kind of bizarre idea that you are driven to, once you start thinking that God must have a will outside nature, and then that we have the same thing. Why shouldn't such a thing pop up all over the place? Only Leibniz spots the slippery slope.
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19408
|
To say that nature or the one universal substance is God is a pernicious doctrine [Leibniz]
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Full Idea:
To say that nature itself or the substance of all things is God is a pernicious doctrine, recently introduced into the world or renewed by a subtle or profane author.
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|
From:
Gottfried Leibniz (On Nature Itself (De Ipsa Natura) [1698], 8)
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|
A reaction:
The dastardly profane author is, of course, Spinoza, whom Leibniz had met in 1676. The doctrine may be pernicious to religious orthodoxy, but to me it is rather baffling, since in my understanding nature and God have almost nothing in common.
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