Combining Texts

All the ideas for 'fragments/reports', 'On Formally Undecidable Propositions' and 'The Metaphysic of Abstract Particulars'

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33 ideas

3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine]
     Full Idea: Gödel's proof wrought an abrupt turn in the philosophy of mathematics. We had supposed that truth, in mathematics, consisted in provability.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Willard Quine - Forward to Gödel's Unpublished
     A reaction: This explains the crisis in the early 1930s, which Tarski's theory appeared to solve.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M]
     Full Idea: Gödel's incompleteness results of 1931 show that all axiom systems precise enough to satisfy Hilbert's conception are necessarily incomplete.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1215
     A reaction: [Hallett italicises 'necessarily'] Hilbert axioms have to be recursive - that is, everything in the system must track back to them.
5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
Relations need terms, so they must be second-order entities based on first-order tropes [Campbell,K]
     Full Idea: Because there cannot be relations without terms, in a meta-physic that makes first-order tropes the terms of all relations, relational tropes must belong to a second, derivative order.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §8)
     A reaction: The admission that there could be a 'derivative order' may lead to trouble for trope theory. Ostrich Nominalists could say that properties themselves are derivative second-order abstractions from indivisible particulars. Russell makes them first-order.
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner]
     Full Idea: The inherent limitations of the axiomatic method were first brought to light by the incompleteness theorems.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Koellner - On the Question of Absolute Undecidability 1.1
5. Theory of Logic / K. Features of Logics / 2. Consistency
Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P]
     Full Idea: Second Incompleteness Theorem: roughly, nice theories that include enough basic arithmetic can't prove their own consistency.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.5
     A reaction: On the face of it, this sounds less surprising than the First Theorem. Philosophers have often noticed that it seems unlikely that you could use reason to prove reason, as when Descartes just relies on 'clear and distinct ideas'.
5. Theory of Logic / K. Features of Logics / 3. Soundness
If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh]
     Full Idea: Gödel showed PA cannot be proved consistent from with PA. But 'reflection principles' can be added, which are axioms partially expressing the soundness of PA, by asserting what is provable. A Global Reflection Principle asserts full soundness.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Halbach,V/Leigh,G.E. - Axiomatic Theories of Truth (2013 ver) 1.2
     A reaction: The authors point out that this needs a truth predicate within the language, so disquotational truth won't do, and there is a motivation for an axiomatic theory of truth.
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel]
     Full Idea: Where Gödel's First Theorem sabotages logicist ambitions, the Second Theorem sabotages Hilbert's Programme.
     From: comment on Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 36
     A reaction: Neo-logicism (Crispin Wright etc.) has a strategy for evading the First Theorem.
The undecidable sentence can be decided at a 'higher' level in the system [Gödel]
     Full Idea: My undecidable arithmetical sentence ...is not at all absolutely undecidable; rather, one can always pass to 'higher' systems in which the sentence in question is decidable.
     From: Kurt Gödel (On Formally Undecidable Propositions [1931]), quoted by Peter Koellner - On the Question of Absolute Undecidability 1.1
     A reaction: [a 1931 MS] He says the reals are 'higher' than the naturals, and the axioms of set theory are higher still. The addition of a truth predicate is part of what makes the sentence become decidable.
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman]
     Full Idea: Gödel's far-reaching work on the nature of logic and formal systems reveals that there can be no single consistent theory from which all mathematical truths can be derived.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.8
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
     Full Idea: Gödel's theorem states that either arithmetic is incomplete, or it is inconsistent.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Georges Rey - Contemporary Philosophy of Mind 8.7
First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P]
     Full Idea: First Incompleteness Theorem: any properly axiomatised and consistent theory of basic arithmetic must remain incomplete, whatever our efforts to complete it by throwing further axioms into the mix.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.2
     A reaction: This is because it is always possible to formulate a well-formed sentence which is not provable within the theory.
Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman]
     Full Idea: The vast continent of arithmetical truth cannot be brought into systematic order by laying down a fixed set of axioms and rules of inference from which every true mathematical statement can be formally derived. For some this was a shocking revelation.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by E Nagel / JR Newman - Gödel's Proof VII.C
     A reaction: Good news for philosophy, I'd say. The truth cannot be worked out by mechanical procedures, so it needs the subtle and intuitive intelligence of your proper philosopher (Parmenides is the role model) to actually understand reality.
Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna]
     Full Idea: Gödel's Second Incompleteness Theorem says that true unprovable sentences are clearly semantic consequences of the axioms in the sense that they are necessarily true if the axioms are true. So semantic consequence outruns provability.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Robert Hanna - Rationality and Logic 5.3
First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman]
     Full Idea: First Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S is syntactically incomplete.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
     A reaction: Gödel found a single sentence, effectively saying 'I am unprovable in S', which is neither provable nor refutable in S.
Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman]
     Full Idea: Second Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S cannot prove its own consistency
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
     A reaction: This seems much less surprising than the First Theorem (though it derives from it). It was always kind of obvious that you couldn't use reason to prove that reason works (see, for example, the Cartesian Circle).
There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P]
     Full Idea: The original Gödel construction gives us a sentence that a theory shows is true if and only if it satisfies the condition of being unprovable-in-that-theory.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 20.5
'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg]
     Full Idea: An approximation of Gödel's Theorem imagines a statement 'This system of mathematics can't prove this statement true'. If the system proves the statement, then it can't prove it. If the statement can't prove the statement, clearly it still can't prove it.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15
     A reaction: Gödel's contribution to this simple idea seems to be a demonstration that formal arithmetic is capable of expressing such a statement.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro]
     Full Idea: Gödel defended impredicative definitions on grounds of ontological realism. From that perspective, an impredicative definition is a description of an existing entity with reference to other existing entities.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Stewart Shapiro - Thinking About Mathematics 5.3
     A reaction: This is why constructivists must be absolutely precise about definition, where realists only have to do their best. Compare building a car with painting a landscape.
7. Existence / B. Change in Existence / 4. Events / c. Reduction of events
Events are trope-sequences, in which tropes replace one another [Campbell,K]
     Full Idea: Events are widely acknowledged to be particulars, but they are plainly not ordinary concrete particulars. They are best viewed as trope-sequences, in which one condition gives way to another. They are changes in which tropes replace one another.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §3)
     A reaction: If nothing exists except bundles of tropes, it is worth asking WHY one trope would replace another. Some tropes are active (i.e. they are best described as 'powers').
8. Modes of Existence / B. Properties / 13. Tropes / a. Nature of tropes
Two red cloths are separate instances of redness, because you can dye one of them blue [Campbell,K]
     Full Idea: If we have two cloths of the very same shade of redness, we can show there are two cloths by burning one and leaving the other unaffected; we show there are two cases of redness in the same way: dye one blue, leaving the other unaffected.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §1)
     A reaction: This has to be one of the basic facts of the problem accepted by everyone. If you dye half of one of the pieces, was the original red therefore one instance or two? Has it become two? How many red tropes are there in a red cloth?
Red could only recur in a variety of objects if it was many, which makes them particulars [Campbell,K]
     Full Idea: If there are a varied group of red objects, the only element that recurs is the colour. But it must be the colour as a particular (a 'trope') that is involved in the recurrence, for only particulars can be many in the way required for recurrence.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §1)
     A reaction: This claim seems to depend on the presupposition that rednesses are countable things, but it is tricky trying to count the number of blue tropes in the sky.
Tropes solve the Companionship Difficulty, since the resemblance is only between abstract particulars [Campbell,K]
     Full Idea: The 'companionship difficulty' cannot arise if the members of the resemblance class are tropes rather than whole concrete particulars. The instances of having a heart, as abstract particulars, are quite different from instances of having a kidney.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §6)
     A reaction: The companionship difficulty seems worst if you base your account of properties just on being members of a class. Any talk of resemblance eventually has to talk about 'respects' of resemblance. Is a trope a respect? Is a mode an object?
Tropes solve the Imperfect Community problem, as they can only resemble in one respect [Campbell,K]
     Full Idea: The 'problem of imperfect community' cannot arise where our resemblance sets are sets of tropes. Tropes, by their very nature and mode of differentiation can only resemble in one respect.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §6)
     A reaction: You arrive at very different accounts of what resemblance means according to how you express the problem verbally. We can only find a solution through thinking which transcends language. Heresy!
Trope theory makes space central to reality, as tropes must have a shape and size [Campbell,K]
     Full Idea: The metaphysics of abstract particulars gives a central place to space, or space-time, as the frame of the world. ...Tropes are, of their essence, regional, which carries with it the essential presence of shape and size in any trope occurrence.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §7)
     A reaction: Trope theory has a problem with Aristotle's example (Idea 557) of what happens when white is mixed with white. Do two tropes become one trope if you paint on a second coat of white? How can particulars merge? How can abstractions merge?
8. Modes of Existence / E. Nominalism / 2. Resemblance Nominalism
Nominalism has the problem that without humans nothing would resemble anything else [Campbell,K]
     Full Idea: The objection to nominalism is its consequence that if there were no human race (or other living things), nothing would be like anything else.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §6)
     A reaction: Anti-realists will be unflustered by this difficulty. Personally it strikes me as obvious that some aspects of resemblance are part of reality which we did not contribute. This I take to be a contingent fact, founded on the existence of natural kinds.
9. Objects / A. Existence of Objects / 1. Physical Objects
Tropes are basic particulars, so concrete particulars are collections of co-located tropes [Campbell,K]
     Full Idea: If tropes are basic particulars, then concrete particulars count as dependent realities. They are collections of co-located tropes, depending on these tropes as a fleet does upon its component ships.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §2)
     A reaction: If I sail my yacht through a fleet, do I become part of it? Presumably trope theory could avoid a bundle view of objects. A bare substratum could be a magnet which attracts tropes.
Bundles must be unique, so the Identity of Indiscernibles is a necessity - which it isn't! [Campbell,K]
     Full Idea: Each individual is distinct from each other individual, so the bundle account of objects requires each bundle to be different from every other bundle. So the Identity of Indiscernibles must be a necessary truth, which, unfortunately, it is not.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §5)
     A reaction: Clearly the Identity of Indiscernibles is not a necessary truth (consider just two identical spheres). Location and time must enter into it. Could we not add a further individuation requirement to the necessary existence of a bundle? (Quinton)
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
Two pure spheres in non-absolute space are identical but indiscernible [Campbell,K]
     Full Idea: The Identity of Indiscernibles is not a necessary truth. It fails in possible worlds where there are two identical spheres in a non-absolute space, or worlds without beginning or end where events are exactly cyclically repeated.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §5)
     A reaction: The principle was always very suspect, and these seem nice counterexamples. As so often, epistemology and ontology had become muddled.
17. Mind and Body / C. Functionalism / 2. Machine Functionalism
Basic logic can be done by syntax, with no semantics [Gödel, by Rey]
     Full Idea: Gödel in his completeness theorem for first-order logic showed that a certain set of syntactically specifiable rules was adequate to capture all first-order valid arguments. No semantics (e.g. reference, truth, validity) was necessary.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Georges Rey - Contemporary Philosophy of Mind 8.2
     A reaction: This implies that a logic machine is possible, but we shouldn't raise our hopes for proper rationality. Validity can be shown for purely algebraic arguments, but rationality requires truth as well as validity, and that needs propositions and semantics.
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
Abstractions come before the mind by concentrating on a part of what is presented [Campbell,K]
     Full Idea: An item is abstract if it is got before the mind by an act of abstraction, that is, by concentrating attention on some, but not all, of what is presented.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §1)
     A reaction: I think this point is incredibly important. Pure Fregean semantics tries to leave out the psychological component, and yet all the problems in semantics concern various sorts of abstraction. Imagination is the focus of the whole operation.
21. Aesthetics / C. Artistic Issues / 7. Art and Morality
Musical performance can reveal a range of virtues [Damon of Ath.]
     Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice.
     From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where?
26. Natural Theory / C. Causation / 8. Particular Causation / b. Causal relata
Causal conditions are particular abstract instances of properties, which makes them tropes [Campbell,K]
     Full Idea: The conditions in causal statements are usually particular cases of properties. A collapse results from the weakness of this cable (not any other). This is specific to a time and place; it is an abstract particular. It is, in short, a trope.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §3)
     A reaction: The fan of universals could counter this by saying that the collapse results from this unique combination of universals. Resemblance nominalist can equally build an account on the coincidence of certain types of concrete particulars.
Davidson can't explain causation entirely by events, because conditions are also involved [Campbell,K]
     Full Idea: Not all singular causal statements are of Davidson's event-event type. Many involve conditions, so there are condition-event (weakness/collapse), event-condition (explosion/movement), and condition-condition (hot/warming) causal connections.
     From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §3)
     A reaction: Fans of Davidson need to reduce conditions to events. The problem of individuation keeps raising its head. Davidson makes it depend on description. Kim looks good, because events, and presumably conditions, reduce to something small and precise.