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Ideas for 'Commentary on 'De Anima'', 'Introducing the Philosophy of Mathematics' and 'Science without Numbers'

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33 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
In Field's version of science, space-time points replace real numbers [Field,H, by Szabó]
     Full Idea: Field's nominalist version of science develops a version of Newtonian gravitational theory, where no quantifiers range over mathematical entities, and space-time points and regions play the role of surrogates for real numbers.
     From: report of Hartry Field (Science without Numbers [1980]) by Zoltán Gendler Szabó - Nominalism 5.1
     A reaction: This seems to be a very artificial contrivance, but Field has launched a programme for rewriting science so that numbers can be omitted. All of this is Field's rebellion against the Indispensability Argument for mathematics. I sympathise.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
The 'integers' are the positive and negative natural numbers, plus zero [Friend]
     Full Idea: The set of 'integers' is all of the negative natural numbers, and zero, together with the positive natural numbers.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
     A reaction: Zero always looks like a misfit at this party. Credit and debit explain positive and negative nicely, but what is the difference between having no money, and money being irrelevant? I can be 'broke', but can the North Pole be broke?
The 'rational' numbers are those representable as fractions [Friend]
     Full Idea: The 'rational' numbers are all those that can be represented in the form m/n (i.e. as fractions), where m and n are natural numbers different from zero.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
     A reaction: Pythagoreans needed numbers to stop there, in order to represent the whole of reality numerically. See irrational numbers for the ensuing disaster. How can a universe with a finite number of particles contain numbers that are not 'rational'?
A number is 'irrational' if it cannot be represented as a fraction [Friend]
     Full Idea: A number is 'irrational' just in case it cannot be represented as a fraction. An irrational number has an infinite non-repeating decimal expansion. Famous examples are pi and e.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
     A reaction: There must be an infinite number of irrational numbers. You could, for example, take the expansion of pi, and change just one digit to produce a new irrational number, and pi has an infinity of digits to tinker with.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]
     Full Idea: The natural numbers are quite primitive, and are what we first learn about. The order of objects (the 'ordinals') is one level of abstraction up from the natural numbers: we impose an order on objects.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4)
     A reaction: Note the talk of 'levels of abstraction'. So is there a first level of abstraction? Dedekind disagrees with Friend (Idea 7524). I would say that natural numbers are abstracted from something, but I'm not sure what. See Structuralism in maths.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
Cardinal numbers answer 'how many?', with the order being irrelevant [Friend]
     Full Idea: The 'cardinal' numbers answer the question 'How many?'; the order of presentation of the objects being counted as immaterial. Def: the cardinality of a set is the number of members of the set.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
     A reaction: If one asks whether cardinals or ordinals are logically prior (see Ideas 7524 and 8661), I am inclined to answer 'neither'. Presenting them as answers to the questions 'how many?' and 'which comes first?' is illuminating.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend]
     Full Idea: The set of 'real' numbers, which consists of the rational numbers and the irrational numbers together, represents "the continuum", since it is like a smooth line which has no gaps (unlike the rational numbers, which have the irrationals missing).
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
     A reaction: The Continuum is the perfect abstract object, because a series of abstractions has arrived at a vast limit in its nature. It still has dizzying infinities contained within it, and at either end of the line. It makes you feel humble.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
Raising omega to successive powers of omega reveal an infinity of infinities [Friend]
     Full Idea: After the multiples of omega, we can successively raise omega to powers of omega, and after that is done an infinite number of times we arrive at a new limit ordinal, which is called 'epsilon'. We have an infinite number of infinite ordinals.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4)
     A reaction: When most people are dumbstruck by the idea of a single infinity, Cantor unleashes an infinity of infinities, which must be the highest into the stratosphere of abstract thought that any human being has ever gone.
The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
     Full Idea: The first 'limit ordinal' is called 'omega', which is ordinal because it is greater than other numbers, but it has no immediate predecessor. But it has successors, and after all of those we come to twice-omega, which is the next limit ordinal.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4)
     A reaction: This is the gateway to Cantor's paradise of infinities, which Hilbert loved and defended. Who could resist the pleasure of being totally boggled (like Aristotle) by a concept such as infinity, only to have someone draw a map of it? See 8663 for sequel.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
Between any two rational numbers there is an infinite number of rational numbers [Friend]
     Full Idea: Since between any two rational numbers there is an infinite number of rational numbers, we could consider that we have infinity in three dimensions: positive numbers, negative numbers, and the 'depth' of infinite numbers between any rational numbers.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
     A reaction: This is before we even reach Cantor's staggering infinities (Ideas 8662 and 8663), which presumably reside at the outer reaches of all three of these dimensions of infinity. The 'deep' infinities come from fractions with huge denominators.
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Is mathematics based on sets, types, categories, models or topology? [Friend]
     Full Idea: Successful competing founding disciplines in mathematics include: the various set theories, type theory, category theory, model theory and topology.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3)
     A reaction: Or none of the above? Set theories are very popular. Type theory is, apparently, discredited. Shapiro has a version of structuralism based on model theory (which sound promising). Topology is the one that intrigues me...
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H]
     Full Idea: There are two approaches to axiomatising geometry. The 'metric' approach uses a function which maps a pair of points into the real numbers. The 'synthetic' approach is that of Euclid and Hilbert, which does without real numbers and functions.
     From: Hartry Field (Science without Numbers [1980], 5)
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Most mathematical theories can be translated into the language of set theory [Friend]
     Full Idea: Most of mathematics can be faithfully redescribed by classical (realist) set theory. More precisely, we can translate other mathematical theories - such as group theory, analysis, calculus, arithmetic, geometry and so on - into the language of set theory.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3)
     A reaction: This is why most mathematicians seem to regard set theory as foundational. We could also translate football matches into the language of atomic physics.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
The number 8 in isolation from the other numbers is of no interest [Friend]
     Full Idea: There is no interest for the mathematician in studying the number 8 in isolation from the other numbers.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4)
     A reaction: This is a crucial and simple point (arising during a discussion of Shapiro's structuralism). Most things are interesting in themselves, as well as for their relationships, but mathematical 'objects' just are relationships.
In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend]
     Full Idea: Structuralists give a historical account of why the 'same' number occupies different structures. Numbers are equivalent rather than identical. 8 is the immediate predecessor of 9 in the whole numbers, but in the rationals 9 has no predecessor.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4)
     A reaction: I don't become a different person if I move from a detached house to a terraced house. This suggests that 8 can't be entirely defined by its relations, and yet it is hard to see what its intrinsic nature could be, apart from the units which compose it.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend]
     Full Idea: Structuralists disagree over whether objects in structures are 'ante rem' (before reality, existing independently of whether the objects exist) or 'in re' (in reality, grounded in the real world, usually in our theories of physics).
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4)
     A reaction: Shapiro holds the first view, Hellman and Resnik the second. The first view sounds too platonist and ontologically extravagant; the second sounds too contingent and limited. The correct account is somewhere in abstractions from the real.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend]
     Full Idea: According to the structuralist, mathematicians study the concepts (objects of study) such as variable, greater, real, add, similar, infinite set, which are one level of abstraction up from prima facie base objects such as numbers, shapes and lines.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1)
     A reaction: This still seems to imply an ontology in which numbers, shapes and lines exist. I would have thought you could eliminate the 'base objects', and just say that the concepts are one level of abstraction up from the physical world.
Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend]
     Full Idea: Structuralism says we study whole structures: objects together with their predicates, relations that bear between them, and functions that take us from one domain of objects to a range of other objects. The objects can even be eliminated.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1)
     A reaction: The unity of object and predicate is a Quinean idea. The idea that objects are inessential is the dramatic move. To me the proposal has very strong intuitive appeal. 'Eight' is meaningless out of context. Ordinality precedes cardinality? Ideas 7524/8661.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]
     Full Idea: In the 'in re' version of mathematical structuralism, pattern-spotting is the process of abstraction.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4)
     A reaction: This might work for non-mathematical abstraction as well, if we are allowed to spot patterns within sensual experience, and patterns within abstractions. Properties are causal patterns in the world? No - properties cause patterns.
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H]
     Full Idea: There is one and only one serious argument for the existence of mathematical entities, and that is the Indispensability Argument of Putnam and Quine.
     From: Hartry Field (Science without Numbers [1980], p.5), quoted by Stewart Shapiro - Thinking About Mathematics 9.1
     A reaction: Personally I don't believe (and nor does Field) that this gives a good enough reason to believe in such things. Quine (who likes 'desert landscapes' in ontology) ends up believing that sets are real because of his argument. Not for me.
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend]
     Full Idea: The main philosophical problem with the position of platonism or realism is the epistemic problem: of explaining what perception or intuition consists in; how it is possible that we should accurately detect whatever it is we are realists about.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.5)
     A reaction: The best bet, I suppose, is that the mind directly perceives concepts just as eyes perceive the physical (see Idea 8679), but it strikes me as implausible. If we have to come up with a special mental faculty for an area of knowledge, we are in trouble.
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H]
     Full Idea: The most popular approach of nominalistically inclined philosophers is to try to reinterpret mathematics, so that its terms and quantifiers only make reference to, say, physical objects, or linguistic expressions, or mental constructions.
     From: Hartry Field (Science without Numbers [1980], Prelim)
     A reaction: I am keen on naturalism and empiricism, but only referring to physical objects is a non-starter. I think I favour constructions, derived from the experience of patterns, and abstracted, idealised and generalised. Field says application is the problem.
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro]
     Full Idea: Field argues that to account for the applicability of mathematics, we need to assume little more than the possibility of the mathematics, not its truth.
     From: report of Hartry Field (Science without Numbers [1980]) by Stewart Shapiro - Philosophy of Mathematics 7.2
     A reaction: Very persuasive. We can apply chess to real military situations, provided that chess isn't self-contradictory (or even naturally impossible?).
Hilbert explains geometry, by non-numerical facts about space [Field,H]
     Full Idea: Facts about geometric laws receive satisfying explanations, by the intrinsic facts about physical space, i.e. those laid down without reference to numbers in Hilbert's axioms.
     From: Hartry Field (Science without Numbers [1980], 3)
     A reaction: Hilbert's axioms mention points, betweenness, segment-congruence and angle-congruence (Field 25-26). Field cites arithmetic and geometry (as well as Newtonian mechanics) as not being dependent on number.
Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H]
     Full Idea: Field needs the notion of logical consequence in second-order logic, but (since this is not recursively axiomatizable) this is a semantical notion, which involves the idea of 'true in all models', a set-theoretic idea if there ever was one.
     From: comment on Hartry Field (Science without Numbers [1980], Ch.4) by James Robert Brown - Philosophy of Mathematics
     A reaction: Brown here summarises a group of critics. Field was arguing for modern nominalism, that actual numbers could (in principle) be written out of the story, as useful fictions. Popper's attempt to dump induction seemed to need induction.
Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend]
     Full Idea: Central to naturalism about mathematics are 'indispensability arguments', to the effect that some part of mathematics is indispensable to our best physical theory, and therefore we ought to take that part of mathematics to be true.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.1)
     A reaction: Quine and Putnam hold this view; Field challenges it. It has the odd consequence that the dispensable parts (if they can be identified!) do not need to be treated as true (even though they might follow logically from the dispensable parts!). Wrong!
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H]
     Full Idea: No clear explanation of the idea that the conclusion was 'implicitly contained in' the premises was ever given, and I do not believe that any clear explanation is possible.
     From: Hartry Field (Science without Numbers [1980], 1)
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]
     Full Idea: There are not enough constraints in the Formalist view of mathematics, so there is no way to select a direction for trying to develop mathematics. There is no part of mathematics that is more important than another.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.6)
     A reaction: One might reply that an area of maths could be 'important' if lots of other areas depended on it, and big developments would ripple big changes through the interior of the subject. Formalism does, though, seem to reduce maths to a game.
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
Abstractions can form useful counterparts to concrete statements [Field,H]
     Full Idea: Abstract entities are useful because we can use them to formulate abstract counterparts of concrete statements.
     From: Hartry Field (Science without Numbers [1980], 3)
     A reaction: He defends the abstract statements as short cuts. If the concrete statements were 'true', then it seems likely that the abstract counterparts will also be true, which is not what fictionalism claims.
Mathematics is only empirical as regards which theory is useful [Field,H]
     Full Idea: Mathematics is in a sense empirical, but only in the rather Pickwickian sense that is an empirical question as to which mathematical theory is useful.
     From: Hartry Field (Science without Numbers [1980], 1)
     A reaction: Field wants mathematics to be fictions, and not to be truths. But can he give an account of 'useful' that does not imply truth? Only in a rather dubiously pragmatist way. A novel is not useful.
Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H]
     Full Idea: Why regard the axioms of standard mathematics as truths, rather than as fictions that for a variety of reasons mathematicians have become interested in?
     From: Hartry Field (Science without Numbers [1980], p.viii)
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Constructivism rejects too much mathematics [Friend]
     Full Idea: Too much of mathematics is rejected by the constructivist.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1)
     A reaction: This was Hilbert's view. This seems to be generally true of verificationism. My favourite example is that legitimate speculations can be labelled as meaningless.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Intuitionists typically retain bivalence but reject the law of excluded middle [Friend]
     Full Idea: An intuitionist typically retains bivalence, but rejects the law of excluded middle.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2)
     A reaction: The idea would be to say that only T and F are available as truth-values, but failing to be T does not ensure being F, but merely not-T. 'Unproven' is not-T, but may not be F.