Combining Texts

All the ideas for 'Locke on Human Understanding', 'works' and 'Must We Believe in Set Theory?'

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18 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
The logic of ZF is classical first-order predicate logic with identity [Boolos]
     Full Idea: The logic of ZF Set Theory is classical first-order predicate logic with identity.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.121)
     A reaction: This logic seems to be unable to deal with very large cardinals, precisely those that are implied by set theory, so there is some sort of major problem hovering here. Boolos is fairly neutral.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
A few axioms of set theory 'force themselves on us', but most of them don't [Boolos]
     Full Idea: Maybe the axioms of extensionality and the pair set axiom 'force themselves on us' (Gödel's phrase), but I am not convinced about the axioms of infinity, union, power or replacement.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.130)
     A reaction: Boolos is perfectly happy with basic set theory, but rather dubious when very large cardinals come into the picture.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos]
     Full Idea: The naïve view of set theory (that any zero or more things form a set) is natural, but inconsistent: the things that do not belong to themselves are some things that do not form a set.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.127)
     A reaction: As clear a summary of Russell's Paradox as you could ever hope for.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos]
     Full Idea: According to the iterative conception, every set is formed at some stage. There is a relation among stages, 'earlier than', which is transitive. A set is formed at a stage if and only if its members are all formed before that stage.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.126)
     A reaction: He gives examples of the early stages, and says the conception is supposed to 'justify' Zermelo set theory. It is also supposed to make the axioms 'natural', rather than just being selected for convenience. And it is consistent.
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
De Morgan introduced a 'universe of discourse', to replace Boole's universe of 'all things' [De Morgan, by Walicki]
     Full Idea: In 1846 De Morgan introduced the enormously influential notion of a possibly arbitrary and stipulated 'universe of discourse'. It replaced Boole's original - and metaphysically a bit suspect - universe of 'all things'.
     From: report of Augustus De Morgan (works [1846]) by Michal Walicki - Introduction to Mathematical Logic History D.1.1
     A reaction: This not only brings formal logic under control, but also reflects normal talk, because there is always an explicit or implicit domain of discourse when we talk. Of virtually any conversation, you can say what it is 'about'.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Infinite natural numbers is as obvious as infinite sentences in English [Boolos]
     Full Idea: The existence of infinitely many natural numbers seems to me no more troubling than that of infinitely many computer programs or sentences of English. There is, for example, no longest sentence, since any number of 'very's can be inserted.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.129)
     A reaction: If you really resisted an infinity of natural numbers, presumably you would also resist an actual infinity of 'very's. The fact that it is unclear what could ever stop a process doesn't guarantee that the process is actually endless.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
Mathematics and science do not require very high orders of infinity [Boolos]
     Full Idea: To the best of my knowledge nothing in mathematics or science requires the existence of very high orders of infinity.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.122)
     A reaction: He is referring to particular high orders of infinity implied by set theory. Personally I want to wield Ockham's Razor. Is being implied by set theory a sufficient reason to accept such outrageous entities into our ontology?
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
Mathematics isn't surprising, given that we experience many objects as abstract [Boolos]
     Full Idea: It is no surprise that we should be able to reason mathematically about many of the things we experience, for they are already 'abstract'.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.129)
     A reaction: He has just given a list of exemplary abstract objects (Idea 10489), but I think there is a more interesting idea here - that our experience of actual physical objects is to some extent abstract, as soon as it is conceptualised.
8. Modes of Existence / D. Universals / 1. Universals
It is lunacy to think we only see ink-marks, and not word-types [Boolos]
     Full Idea: It's a kind of lunacy to think that sound scientific philosophy demands that we think that we see ink-tracks but not words, i.e. word-types.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.128)
     A reaction: This seems to link him with Armstrong's mockery of 'ostrich nominalism'. There seems to be some ambiguity with the word 'see' in this disagreement. When we look at very ancient scratches on stones, why don't we always 'see' if it is words?
8. Modes of Existence / E. Nominalism / 2. Resemblance Nominalism
Two things can only resemble one another in some respect, and that may reintroduce a universal [Lowe]
     Full Idea: A problem for resemblance nominalism is that in saying that two particulars 'resemble' one another, it is necessary to specify in what respect they do so (e.g. colour, shape, size), and this threatens to reintroduce what appears to be talk of universals.
     From: E.J. Lowe (Locke on Human Understanding [1995], Ch.7)
     A reaction: We see resemblance between faces instantly, long before we can specify the 'respects' of the resemblance. This supports the Humean hard-wired view of resemblance, rather than some appeal to Platonic universals.
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
I am a fan of abstract objects, and confident of their existence [Boolos]
     Full Idea: I am rather a fan of abstract objects, and confident of their existence. Smaller numbers, sets and functions don't offend my sense of reality.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.128)
     A reaction: The great Boolos is rather hard to disagree with, but I disagree. Logicians love abstract objects, indeed they would almost be out of a job without them. It seems to me they smuggle them into our ontology by redefining either 'object' or 'exists'.
9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
We deal with abstract objects all the time: software, poems, mistakes, triangles.. [Boolos]
     Full Idea: We twentieth century city dwellers deal with abstract objects all the time, such as bank balances, radio programs, software, newspaper articles, poems, mistakes, triangles.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.129)
     A reaction: I find this claim to be totally question-begging, and typical of a logician. The word 'object' gets horribly stretched in these discussions. We can create concepts which have all the logical properties of objects. Maybe they just 'subsist'?
9. Objects / B. Unity of Objects / 2. Substance / d. Substance defined
On substances, Leibniz emphasises unity, Spinoza independence, Locke relations to qualities [Lowe]
     Full Idea: Later philosophers emphasised different strands of Aristotle's concept of substances: Leibniz (in his theory of monads) emphasised their unity; Spinoza emphasised their ontological independence; Locke emphasised their role in relation to qualities.
     From: E.J. Lowe (Locke on Human Understanding [1995], Ch.4)
     A reaction: Note that this Aristotelian idea had not been jettisoned in the late seventeenth century, unlike other Aristotelianisms. I think it is only with the success of atomism in chemistry that the idea of substance is forced to recede.
12. Knowledge Sources / B. Perception / 6. Inference in Perception
Perception is a mode of belief-acquisition, and does not involve sensation [Lowe]
     Full Idea: According to one school of thought, perception is simply a mode of belief-acquisition,and there is no reason to suppose that any element of sensation is literally involved in perception.
     From: E.J. Lowe (Locke on Human Understanding [1995], Ch.3)
     A reaction: Blindsight would be an obvious supporting case for this view. I think this point is crucial in understanding what is wrong with Jackson's 'knowledge argument' (involving Mary, see Idea 7377). Sensation gives knowledge, so it can't be knowledge.
12. Knowledge Sources / B. Perception / 7. Causal Perception
Science requires a causal theory - perception of an object must be an experience caused by the object [Lowe]
     Full Idea: Only a causal theory of perception will respect the facts of physiology and physics ...meaning a theory which maintains that for a subject to perceive a physical object the subject should enjoy some appropriate perceptual experience caused by the object.
     From: E.J. Lowe (Locke on Human Understanding [1995], Ch.3)
     A reaction: If I hallucinate an object, then presumably I am not allowed to say that I 'perceive' it, but that seems to make the causal theory an idle tautology. If we are in virtual reality then there aren't any objects.
16. Persons / D. Continuity of the Self / 1. Identity and the Self
Personal identity is a problem across time (diachronic) and at an instant (synchronic) [Lowe]
     Full Idea: There is the question of the identity of a person over or across time ('diachronic' personal identity), and there is also the question of what makes for personal identity at a time ('synchronic' personal identity).
     From: E.J. Lowe (Locke on Human Understanding [1995], Ch.5)
     A reaction: This seems to me to be the first and most important distinction in the philosophy of personal identity, and they regularly get run together. Locke, for example, has an account of synchronic identity, which is often ignored. It applies to objects too.
18. Thought / B. Mechanics of Thought / 4. Language of Thought
Mentalese isn't a language, because it isn't conventional, or a means of public communication [Lowe]
     Full Idea: 'Mentalese' would be neither conventional nor a means of public communication so that even to call it a language is seriously misleading.
     From: E.J. Lowe (Locke on Human Understanding [1995], Ch.7)
     A reaction: It is, however, supposed to contain symbolic representations which are then used as tokens for computation, so it seems close to a language, if (for example) symbolic logic or mathematics were accepted as languages. But who understands it?
19. Language / A. Nature of Meaning / 2. Meaning as Mental
If meaning is mental pictures, explain "the cat (or dog!) is NOT on the mat" [Lowe]
     Full Idea: If meaning is a private mental picture, what does 'the cat is NOT on the mat' mean, and how does it differ from 'the dog is not on the mat?'.
     From: E.J. Lowe (Locke on Human Understanding [1995], Ch.7)
     A reaction: Not insurmountable. We picture an empty mat, combined with a cat (or whatever) located somewhere else. A mental 'picture' of something shouldn't be contrued as a single image in a neat black frame.