Combining Texts

All the ideas for 'General Draft', 'Maths as a Science of Patterns' and 'Elements of Set Theory'

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18 ideas

1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / a. Philosophy as worldly
Philosophy is homesickness - the urge to be at home everywhere [Novalis]
     Full Idea: Philosophy is actually homesickness - the urge to be everywhere at home.
     From: Novalis (General Draft [1799], 45)
     A reaction: The idea of home [heimat] is powerful in German culture. The point of romanticism was seen as largely concerning restless souls like Byron and his heroes, who do not feel at home. Hence ironic detachment.
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
Axioms are often affirmed simply because they produce results which have been accepted [Resnik]
     Full Idea: Many axioms have been proposed, not on the grounds that they can be directly known, but rather because they produce a desired body of previously recognised results.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.5.1)
     A reaction: This is the perennial problem with axioms - whether we start from them, or whether we deduce them after the event. There is nothing wrong with that, just as we might infer the existence of quarks because of their results.
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]
     Full Idea: To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B.
     From: Herbert B. Enderton (Elements of Set Theory [1977], 1:04)
     A reaction: This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case.
The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton]
     Full Idea: The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}; hence it can be proved that <u,v> = <x,y> iff u = x and v = y (given by Kuratowski in 1921). ...The definition is somewhat arbitrary, and others could be used.
     From: Herbert B. Enderton (Elements of Set Theory [1977], 3:36)
     A reaction: This looks to me like one of those regular cases where the formal definitions capture all the logical behaviour of the concept that are required for inference, while failing to fully capture the concept for ordinary conversation.
A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton]
     Full Idea: A 'linear ordering' (or 'total ordering') on A is a binary relation R meeting two conditions: R is transitive (of xRy and yRz, the xRz), and R satisfies trichotomy (either xRy or x=y or yRx).
     From: Herbert B. Enderton (Elements of Set Theory [1977], 3:62)
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton]
     Full Idea: Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ. A man with an empty container is better off than a man with nothing.
     From: Herbert B. Enderton (Elements of Set Theory [1977], 1.03)
The empty set may look pointless, but many sets can be constructed from it [Enderton]
     Full Idea: It might be thought at first that the empty set would be a rather useless or even frivolous set to mention, but from the empty set by various set-theoretic operations a surprising array of sets will be constructed.
     From: Herbert B. Enderton (Elements of Set Theory [1977], 1:02)
     A reaction: This nicely sums up the ontological commitments of mathematics - that we will accept absolutely anything, as long as we can have some fun with it. Sets are an abstraction from reality, and the empty set is the very idea of that abstraction.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
The singleton is defined using the pairing axiom (as {x,x}) [Enderton]
     Full Idea: Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}.
     From: Herbert B. Enderton (Elements of Set Theory [1977], 2:19)
     A reaction: An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter!
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton]
     Full Idea: It was observed by several people that for a satisfactory theory of ordinal numbers, Zermelo's axioms required strengthening. The Axiom of Replacement was proposed by Fraenkel and others, giving rise to the Zermelo-Fraenkel (ZF) axioms.
     From: Herbert B. Enderton (Elements of Set Theory [1977], 1:15)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
We can only define functions if Choice tells us which items are involved [Enderton]
     Full Idea: For functions, we know that for any y there exists an appropriate x, but we can't yet form a function H, as we have no way of defining one particular choice of x. Hence we need the axiom of choice.
     From: Herbert B. Enderton (Elements of Set Theory [1977], 3:48)
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematical realism says that maths exists, is largely true, and is independent of proofs [Resnik]
     Full Idea: Mathematical realism is the doctrine that mathematical objects exist, that much contemporary mathematics is true, and that the existence and truth in question is independent of our constructions, beliefs and proofs.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.12.9)
     A reaction: As thus defined, I would call myself a mathematical realist, but everyone must hesitate a little at the word 'exist' and ask, how does it exist? What is it 'made of'? To say that it exists in the way that patterns exist strikes me as very helpful.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik]
     Full Idea: In maths the primary subject-matter is not mathematical objects but structures in which they are arranged; our constants and quantifiers denote atoms, structureless points, or positions in structures; they have no identity outside a structure or pattern.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.1)
     A reaction: This seems to me a very promising idea for the understanding of mathematics. All mathematicians acknowledge that the recognition of patterns is basic to the subject. Even animals recognise patterns. It is natural to invent a language of patterns.
Sets are positions in patterns [Resnik]
     Full Idea: On my view, sets are positions in certain patterns.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.5)
     A reaction: I have always found the ontology of a 'set' puzzling, because they seem to depend on prior reasons why something is a member of a given set, which cannot always be random. It is hard to explain sets without mentioning properties.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik]
     Full Idea: An objection is that structuralism fails to explain why certain mathematical patterns are unified wholes while others are not; for instance, some think that an ontological account of mathematics must explain why a triangle is not a 'random' set of points.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.4)
     A reaction: This is an indication that we are not just saying that we recognise patterns in nature, but that we also 'see' various underlying characteristics of the patterns. The obvious suggestion is that we see meta-patterns.
There are too many mathematical objects for them all to be mental or physical [Resnik]
     Full Idea: If we take mathematics at its word, there are too many mathematical objects for it to be plausible that they are all mental or physical objects.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1)
     A reaction: No one, of course, has ever claimed that they are, but this is a good starting point for assessing the ontology of mathematics. We are going to need 'rules', which can deduce the multitudinous mathematical objects from a small ontology.
Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik]
     Full Idea: I argue that mathematical knowledge has its roots in pattern recognition and representation, and that manipulating representations of patterns provides the connection between the mathematical proof and mathematical truth.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1)
     A reaction: The suggestion that patterns are at the basis of the ontology of mathematics is the most illuminating thought I have encountered in the area. It immediately opens up the possibility of maths being an entirely empirical subject.
Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik]
     Full Idea: Of the equivalence relationships which occur between patterns, congruence is the strongest, equivalence the next, and mutual occurrence the weakest. None of these is identity, which would require the same position.
     From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.3)
     A reaction: This gives some indication of how an account of mathematics as a science of patterns might be built up. Presumably the recognition of these 'degrees of strength' cannot be straightforward observation, but will need an a priori component?
15. Nature of Minds / C. Capacities of Minds / 6. Idealisation
Desire for perfection is an illness, if it turns against what is imperfect [Novalis]
     Full Idea: An absolute drive toward perfection and completeness is an illness, as soon as it shows itself to be destructive and averse toward the imperfect, the incomplete.
     From: Novalis (General Draft [1799], 33)
     A reaction: Deep and true! Novalis seems to be a particularist - hanging on to the fine detail of life, rather than being immersed in the theory. These are the philosophers who also turn to literature.