|
5651
|
Traditional views of truth are tautologies, and truth is empty without a subject [Kierkegaard, by Scruton]
|
|
Full Idea:
Kierkegaard developed the idea of 'truth as subjectivity'; the traditional conceptions of truth - correspondence or coherence - he regarded as equally empty, not because false, but because tautologous; truth ceases to be empty when related to a subject.
|
|
From:
report of Søren Kierkegaard (Either/Or: a fragment of life [1843]) by Roger Scruton - Short History of Modern Philosophy Ch.13
|
|
A reaction:
It strikes me that the correspondence theory of truth also involves a subject. If you become too obsessed with the subject, you lose the concept of truth. You need a concept of the non-subject too. Truth concerns the contents of thought.
|
|
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.
|
|
22090
|
For me time stands still, and I with it [Kierkegaard, by Carlisle]
|
|
Full Idea:
Time flows, life is a stream, people say, and so on. I do not notice it. Time stands still, and I with it.
|
|
From:
report of Søren Kierkegaard (Either/Or: a fragment of life [1843], I:26) by Clare Carlisle - Kierkegaard: a guide for the perplexed 3
|
|
A reaction:
This is from the spokesman for the aesthetic option in life, which is largely pleasure-seeking. No real choices ever occur.
|
|
5650
|
Reason is just abstractions, so our essence needs a subjective 'leap of faith' [Kierkegaard, by Scruton]
|
|
Full Idea:
For Kierkegaard, reason, which produces only abstractions, negates our individual essence; this essence is subjectivity, and subjectivity exists only in the 'leap of faith', whereby the individual casts in his lot with eternity.
|
|
From:
report of Søren Kierkegaard (Either/Or: a fragment of life [1843]) by Roger Scruton - Short History of Modern Philosophy Ch.13
|
|
A reaction:
Interesting, but this strikes me as a confusion of reason and logic. A logical life would indeed be a sort of death, and need faith as an escape, but a broad view of the rational life includes emotion, imagination and laughter. Blind faith is disaster.
|
|
22095
|
There are aesthetic, ethical and religious subjectivity [Kierkegaard, by Carlisle]
|
|
Full Idea:
Kierkegaard distinguishes three main types of subjectivity: aesthetic, ethical and religious. But are these types of people, or different phases of one person's life?
|
|
From:
report of Søren Kierkegaard (Either/Or: a fragment of life [1843]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 4
|
|
A reaction:
His picture of the religious mode holds no appeal for me. I also can't accept that the aesthetic and the moral are somewho distinct. People may discover they have slipped into one of these modes, but no one chooses them, do they?
|
|
20747
|
What matters is not right choice, but energy, earnestness and pathos in the choosing [Kierkegaard]
|
|
Full Idea:
In making a choice, it is not so much a question of choosing the right way as of the energy, the earnestness, and the pathos with which one chooses.
|
|
From:
Søren Kierkegaard (Either/Or: a fragment of life [1843], p.106), quoted by Kevin Aho - Existentialism: an introduction 2 'Phenomenology'
|
|
A reaction:
I'm struggling to identify with the experience he is describing. I can't imagine a more quintessentially existentialist remark than this. Reference to 'energy' in choosing strikes me as very romantic. Is 'the way not taken' crucial (in 'pathos')?
|