|
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.
|
|
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.
|
|
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.
|
|
14193
|
'Substance theorists' take modal properties as primitive, without structure, just falling under a sortal [Paul,LA]
|
|
Full Idea:
Some deep essentialists resist the need to explain the structure under de re modal properties, taking them as primitive. One version (which we can call 'substance theory') takes them to fall under a sortal concept, with no further explanation.
|
|
From:
L.A. Paul (In Defense of Essentialism [2006], §1)
|
|
A reaction:
A very helpful identification of what Wiggins stands for, and why I disagree with him. The whole point of essences is to provide a notion that fits in with sciences, which means they must have an explanatory role, which needs structures.
|
|
14195
|
If an object's sort determines its properties, we need to ask what determines its sort [Paul,LA]
|
|
Full Idea:
If the substance essentialist holds that the sort an object belongs to determines its de re modal properties (rather than the other way round), then he needs to give an (ontological, not conceptual) explanation of what determines an object's sort.
|
|
From:
L.A. Paul (In Defense of Essentialism [2006], §1)
|
|
A reaction:
See Idea 14193 for 'substance essentialism'. I find it quite incredible that anyone could think that a thing's sort could determine its properties, rather than the other way round. Even if sortals are conventional, they are not arbitrary.
|
|
14196
|
Substance essentialism says an object is multiple, as falling under various different sortals [Paul,LA]
|
|
Full Idea:
The explanation of material constitution given by substance essentialism is that there are multiple objects. A person is essentially human-shaped (falling under the human sort), while their hunk of tissue is accidentally human-shaped (as tissue).
|
|
From:
L.A. Paul (In Defense of Essentialism [2006], §1)
|
|
A reaction:
At this point sortal essentialism begins to look crazy. Persons are dubious examples (with sneaky dualism involved). A bronze statue is essentially harder to dent than a clay one, because of its bronze. If you remake it of clay, it isn't the same statue.
|
|
14190
|
Deep essentialist objects have intrinsic properties that fix their nature; the shallow version makes it contextual [Paul,LA]
|
|
Full Idea:
Essentialism says that objects have their properties essentially. 'Deep' essentialists take the (nontrivial) essential properties of an object to determine its nature. 'Shallow' essentialists substitute context-dependent truths for the independent ones.
|
|
From:
L.A. Paul (In Defense of Essentialism [2006], Intro)
|
|
A reaction:
If the deep essence determines a things nature, we should not need to say 'nontrivial'. This is my bete noire, the confusion of essential properties with necessary ones, where necessary properties (or predicates, at least) can indeed be trivial.
|
|
14189
|
'Modal realists' believe in many concrete worlds, 'actualists' in just this world, 'ersatzists' in abstract other worlds [Paul,LA]
|
|
Full Idea:
A 'modal realist' believes that there are many concrete worlds, while the 'actualist' believes in only one concrete world, the actual world. The 'ersatzist' is an actualist who takes nonactual possible worlds and their contents to be abstracta.
|
|
From:
L.A. Paul (In Defense of Essentialism [2006], Intro)
|
|
A reaction:
My view is something like that modal realism is wrong, and actualism is right, and possible worlds (if they really are that useful) are convenient abstract fictions, constructed (if we have any sense) out of the real possibilities in the actual world.
|