|
10170
|
While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price]
|
|
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.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)
|
|
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.
|
|
10751
|
Second-order logic needs the sets, and its consequence has epistemological problems [Rossberg]
|
|
Full Idea:
Second-order logic raises doubts because of its ontological commitment to the set-theoretic hierarchy, and the allegedly problematic epistemic status of the second-order consequence relation.
|
|
From:
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §1)
|
|
A reaction:
The 'epistemic' problem is whether you can know the truths, given that the logic is incomplete, and so they cannot all be proved. Rossberg defends second-order logic against the second problem. A third problem is that it may be mathematics.
|
|
10753
|
Logical consequence is intuitively semantic, and captured by model theory [Rossberg]
|
|
Full Idea:
Logical consequence is intuitively taken to be a semantic notion, ...and it is therefore the formal semantics, i.e. the model theory, that captures logical consequence.
|
|
From:
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
|
|
A reaction:
If you come at the issue from normal speech, this seems right, but if you start thinking about the necessity of logical consequence, that formal rules and proof-theory seem to be the foundation.
|
|
10752
|
Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true [Rossberg]
|
|
Full Idea:
Deductive consequence, written Γ|-S, is loosely read as 'the sentence S can be deduced from the sentences Γ', and semantic consequence Γ|=S says 'all models that make Γ true make S true as well'.
|
|
From:
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
|
|
A reaction:
We might read |= as 'true in the same model as'. What is the relation, though, between the LHS and the RHS? They seem to be mutually related to some model, but not directly to one another.
|
|
10175
|
Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price]
|
|
Full Idea:
In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
|
|
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.
|
|
10756
|
A model is a domain, and an interpretation assigning objects, predicates, relations etc. [Rossberg]
|
|
Full Idea:
A standard model is a set of objects called the 'domain', and an interpretation function, assigning objects in the domain to names, subsets to predicate letters, subsets of the Cartesian product of the domain with itself to binary relation symbols etc.
|
|
From:
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
|
|
A reaction:
The model actually specifies which objects have which predicates, and which objects are in which relations. Tarski's account of truth in terms of 'satisfaction' seems to be just a description of those pre-decided facts.
|
|
10758
|
If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg]
|
|
Full Idea:
A mathematical theory is 'categorical' if, and only if, all of its models are isomorphic. Such a theory then essentially has just one model, the standard one.
|
|
From:
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
|
|
A reaction:
So the term 'categorical' is gradually replacing the much-used phrase 'up to isomorphism'.
|
|
10164
|
Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
|
|
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.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
|
|
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'!
|
|
10167
|
Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
|
|
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.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
|
|
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.
|
|
10169
|
Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
|
|
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.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)
|
|
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?
|
|
10179
|
There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
|
|
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.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6)
|
|
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.
|
|
10182
|
There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
|
|
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.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9)
|
|
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?
|
|
10168
|
Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
|
|
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.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3)
|
|
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.
|
|
10178
|
Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
|
|
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.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
|
|
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.
|
|
10177
|
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
|
|
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.
|
|
From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
|
|
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.
|
|
8326
|
Science has shown that causal relations are just transfers of energy or momentum [Fair, by Sosa/Tooley]
|
|
Full Idea:
Basic causal relations can, as a consequence of our scientific knowledge, be identified with certain physicalistic [sic] relations between objects that can be characterized in terms of transference of either energy or momentum between objects.
|
|
From:
report of David Fair (Causation and the Flow of Energy [1979]) by E Sosa / M Tooley - Introduction to 'Causation' §1
|
|
A reaction:
Presumably a transfer of momentum is a transfer of energy. If only anyone had the foggiest idea what energy actually is, we'd be doing well. What is energy made of? 'No identity without substance', I say. I like Fair's idea.
|
|
10379
|
Fair shifted his view to talk of counterfactuals about energy flow [Fair, by Schaffer,J]
|
|
Full Idea:
Fair, who originated the energy flow view of causation, moved to a view that understands connection in terms of counterfactuals about energy flow.
|
|
From:
report of David Fair (Causation and the Flow of Energy [1979]) by Jonathan Schaffer - The Metaphysics of Causation 2.1.2
|
|
A reaction:
David Fair was a pupil of David Lewis, the king of the counterfactual view. To me that sounds like a disappointing move, but it is hard to think that a mere flow of energy through space would amount to causation. Cause must work back from an effect.
|