Combining Texts

All the ideas for 'Against Coherence', 'Set Theory and Its Philosophy' and 'How there could be a private language'

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21 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter]
     Full Idea: Set theory has three roles: as a means of taming the infinite, as a supplier of the subject-matter of mathematics, and as a source of its modes of reasoning.
     From: Michael Potter (Set Theory and Its Philosophy [2004], Intro 1)
     A reaction: These all seem to be connected with mathematics, but there is also ontological interest in set theory. Potter emphasises that his second role does not entail a commitment to sets 'being' numbers.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
Usually the only reason given for accepting the empty set is convenience [Potter]
     Full Idea: It is rare to find any direct reason given for believing that the empty set exists, except for variants of Dedekind's argument from convenience.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 04.3)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
Infinity: There is at least one limit level [Potter]
     Full Idea: Axiom of Infinity: There is at least one limit level.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 04.9)
     A reaction: A 'limit ordinal' is one which has successors, but no predecessors. The axiom just says there is at least one infinity.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
Nowadays we derive our conception of collections from the dependence between them [Potter]
     Full Idea: It is only quite recently that the idea has emerged of deriving our conception of collections from a relation of dependence between them.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.2)
     A reaction: This is the 'iterative' view of sets, which he traces back to Gödel's 'What is Cantor's Continuum Problem?'
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter]
     Full Idea: We group under the heading 'limitation of size' those principles which classify properties as collectivizing or not according to how many objects there are with the property.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 13.5)
     A reaction: The idea was floated by Cantor, toyed with by Russell (1906), and advocated by von Neumann. The thought is simply that paradoxes start to appear when sets become enormous.
4. Formal Logic / G. Formal Mereology / 1. Mereology
Mereology elides the distinction between the cards in a pack and the suits [Potter]
     Full Idea: Mereology tends to elide the distinction between the cards in a pack and the suits.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1)
     A reaction: The example is a favourite of Frege's. Potter is giving a reason why mathematicians opted for set theory. I'm not clear, though, why a pack cannot have either 4 parts or 52 parts. Parts can 'fall under a concept' (such as 'legs'). I'm puzzled.
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
We can formalize second-order formation rules, but not inference rules [Potter]
     Full Idea: In second-order logic only the formation rules are completely formalizable, not the inference rules.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 01.2)
     A reaction: He cites Gödel's First Incompleteness theorem for this.
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter]
     Full Idea: A 'supposition' axiomatic theory is as concerned with truth as a 'realist' one (with undefined terms), but the truths are conditional. Satisfying the axioms is satisfying the theorem. This is if-thenism, or implicationism, or eliminative structuralism.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 01.1)
     A reaction: Aha! I had failed to make the connection between if-thenism and eliminative structuralism (of which I am rather fond). I think I am an if-thenist (not about all truth, but about provable truth).
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
If set theory didn't found mathematics, it is still needed to count infinite sets [Potter]
     Full Idea: Even if set theory's role as a foundation for mathematics turned out to be wholly illusory, it would earn its keep through the calculus it provides for counting infinite sets.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.8)
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter]
     Full Idea: It is a remarkable fact that all the arithmetical properties of the natural numbers can be derived from such a small number of assumptions (as the Peano Axioms).
     From: Michael Potter (Set Theory and Its Philosophy [2004], 05.2)
     A reaction: If one were to defend essentialism about arithmetic, this would be grist to their mill. I'm just saying.
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
A relation is a set consisting entirely of ordered pairs [Potter]
     Full Idea: A set is called a 'relation' if every element of it is an ordered pair.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 04.7)
     A reaction: This is the modern extensional view of relations. For 'to the left of', you just list all the things that are to the left, with the things they are to the left of. But just listing the ordered pairs won't necessarily reveal how they are related.
9. Objects / B. Unity of Objects / 2. Substance / b. Need for substance
If dependence is well-founded, with no infinite backward chains, this implies substances [Potter]
     Full Idea: The argument that the relation of dependence is well-founded ...is a version of the classical arguments for substance. ..Any conceptual scheme which genuinely represents a world cannot contain infinite backward chains of meaning.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3)
     A reaction: Thus the iterative conception of set may imply a notion of substance, and Barwise's radical attempt to ditch the Axiom of Foundation (Idea 13039) was a radical attempt to get rid of 'substances'. Potter cites Wittgenstein as a fan of substances here.
9. Objects / C. Structure of Objects / 8. Parts of Objects / b. Sums of parts
Collections have fixed members, but fusions can be carved in innumerable ways [Potter]
     Full Idea: A collection has a determinate number of members, whereas a fusion may be carved up into parts in various equally valid (although perhaps not equally interesting) ways.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1)
     A reaction: This seems to sum up both the attraction and the weakness of mereology. If you doubt the natural identity of so-called 'objects', then maybe classical mereology is the way to go.
10. Modality / A. Necessity / 1. Types of Modality
Priority is a modality, arising from collections and members [Potter]
     Full Idea: We must conclude that priority is a modality distinct from that of time or necessity, a modality arising in some way out of the manner in which a collection is constituted from its members.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3)
     A reaction: He is referring to the 'iterative' view of sets, and cites Aristotle 'Metaphysics' 1019a1-4 as background.
13. Knowledge Criteria / B. Internal Justification / 5. Coherentism / a. Coherence as justification
Incoherence may be more important for enquiry than coherence [Olsson]
     Full Idea: While coherence may lack the positive role many have assigned to it, ...incoherence plays an important negative role in our enquiries.
     From: Erik J. Olsson (Against Coherence [2005], 10.1)
     A reaction: [He cites Peirce as the main source for this idea] We can hardly by deeply impressed by incoherence if we have no sense of coherence. Incoherence is just one of many markers for theory failure. Missing the target, bad concepts...
Coherence is the capacity to answer objections [Olsson]
     Full Idea: According to Lehrer, coherence should be understood in terms of the capacity to answer objections.
     From: Erik J. Olsson (Against Coherence [2005], 9)
     A reaction: [Keith Lehrer 1990] We can connect this with the Greek requirement of being able to give an account [logos], which is the hallmark of understanding. I take coherence to be the best method of achieving understanding. Any understanding meets Lehrer's test.
13. Knowledge Criteria / B. Internal Justification / 5. Coherentism / c. Coherentism critique
Mere agreement of testimonies is not enough to make truth very likely [Olsson]
     Full Idea: Far from guaranteeing a high likelihood of truth by itself, testimonial agreement can apparently do so only if the circumstances are favourable as regards independence, prior probability, and individual credibility.
     From: Erik J. Olsson (Against Coherence [2005], 1)
     A reaction: This is Olson's main thesis. His targets are C.I.Lewis and Bonjour, who hoped that a mere consensus of evidence would increase verisimilitude. I don't see a problem for coherence in general, since his favourable circumstances are part of it.
Coherence is only needed if the information sources are not fully reliable [Olsson]
     Full Idea: An enquirer who is fortunate enough to have at his or her disposal fully reliable information sources has no use for coherence, the need for which arises only in the context of less than fully reliable informations sources.
     From: Erik J. Olsson (Against Coherence [2005], 2.6.2)
     A reaction: I take this to be entirely false. How do you assess reliability? 'I've seen it with my own eyes'. Why trust your eyes? In what visibility conditions do you begin to doubt your eyes? Why do rational people mistrust their intuitions?
A purely coherent theory cannot be true of the world without some contact with the world [Olsson]
     Full Idea: The Input Objection says a pure coherence theory would seem to allow that a system of beliefs be justified in spite of being utterly out of contact with the world it purports to describe, so long as it is, to a sufficient extent, coherent.
     From: Erik J. Olsson (Against Coherence [2005], 4.1)
     A reaction: Olson seems impressed by this objection, but I don't see how a system could be coherently about the world if it had no known contact with the world. Olson seems to ignore meta-coherence, which evaluates the status of the system being studied.
Extending a system makes it less probable, so extending coherence can't make it more probable [Olsson]
     Full Idea: Any non-trivial extension of a belief system is less probable than the original system, but there are extensions that are more coherent than the original system. Hence more coherence does not imply a higher probability.
     From: Erik J. Olsson (Against Coherence [2005], 6.4)
     A reaction: [Olson cites Klein and Warfield 1994; compressed] The example rightly says the extension could have high internal coherence, but not whether the extension is coherent with the system being extended.
18. Thought / B. Mechanics of Thought / 4. Language of Thought
We must have expressive power BEFORE we learn language [Fodor]
     Full Idea: I am denying that one can learn a language whose expressive power is greater than that of a language that one already knows.
     From: Jerry A. Fodor (How there could be a private language [1975], p.389)
     A reaction: I presume someone who had a native language of limited vocabulary could learn a new language with a vast vocabulary. I can increase my expressive power with a specialist vocabulary (e.g. legal).