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Ideas of Penelope Maddy, by Text

[American, b.1950, Professor of Logic and Philosophy of Science at the University of California, Irvine.]

1981 Sets and Numbers
I p.345 If mathematical objects exist, how can we know them, and which objects are they?
     Full Idea: The popular challenges to platonism in philosophy of mathematics are epistemological (how are we able to interact with these objects in appropriate ways) and ontological (if numbers are sets, which sets are they).
     From: Penelope Maddy (Sets and Numbers [1981], I)
     A reaction: These objections refer to Benacerraf's two famous papers - 1965 for the ontology, and 1973 for the epistemology. Though he relied too much on causal accounts of knowledge in 1973, I'm with him all the way.
II p.347 Set theory (unlike the Peano postulates) can explain why multiplication is commutative
     Full Idea: If you wonder why multiplication is commutative, you could prove it from the Peano postulates, but the proof offers little towards an answer. In set theory Cartesian products match 1-1, and n.m dots when turned on its side has m.n dots, which explains it.
     From: Penelope Maddy (Sets and Numbers [1981], II)
     A reaction: 'Turning on its side' sounds more fundamental than formal set theory. I'm a fan of explanation as taking you to the heart of the problem. I suspect the world, rather than set theory, explains the commutativity.
II p.347 The master science is physical objects divided into sets
     Full Idea: The master science can be thought of as the theory of sets with the entire range of physical objects as ur-elements.
     From: Penelope Maddy (Sets and Numbers [1981], II)
     A reaction: This sounds like Quine's view, since we have to add sets to our naturalistic ontology of objects. It seems to involve unrestricted mereology to create normal objects.
III p.347 Standardly, numbers are said to be sets, which is neat ontology and epistemology
     Full Idea: The standard account of the relationship between numbers and sets is that numbers simply are certain sets. This has the advantage of ontological economy, and allows numbers to be brought within the epistemology of sets.
     From: Penelope Maddy (Sets and Numbers [1981], III)
     A reaction: Maddy votes for numbers being properties of sets, rather than the sets themselves. See Yourgrau's critique.
III p.349 Numbers are properties of sets, just as lengths are properties of physical objects
     Full Idea: I propose that ...numbers are properties of sets, analogous, for example, to lengths, which are properties of physical objects.
     From: Penelope Maddy (Sets and Numbers [1981], III)
     A reaction: Are lengths properties of physical objects? A hole in the ground can have a length. A gap can have a length. Pure space seems to contain lengths. A set seems much more abstract than its members.
III p.349 Sets exist where their elements are, but numbers are more like universals
     Full Idea: A set of things is located where the aggregate of those things is located, ...but a number is simultaneously located at many different places (10 in my hand, and a baseball team) ...so numbers seem more like universals than particulars.
     From: Penelope Maddy (Sets and Numbers [1981], III)
     A reaction: My gut feeling is that Maddy's master idea (of naturalising sets by building them from ur-elements of natural objects) won't work. Sets can work fine in total abstraction from nature.
IV p.350 Number words are unusual as adjectives; we don't say 'is five', and numbers always come first
     Full Idea: Number words are not like normal adjectives. For example, number words don't occur in 'is (are)...' contexts except artificially, and they must appear before all other adjectives, and so on.
     From: Penelope Maddy (Sets and Numbers [1981], IV)
     A reaction: [She is citing Benacerraf's arguments]
V p.353 Number theory doesn't 'reduce' to set theory, because sets have number properties
     Full Idea: I am not suggesting a reduction of number theory to set theory ...There are only sets with number properties; number theory is part of the theory of finite sets.
     From: Penelope Maddy (Sets and Numbers [1981], V)
1988 Believing the Axioms I
0 p.482 New axioms are being sought, to determine the size of the continuum
     Full Idea: In current set theory, the search is on for new axioms to determine the size of the continuum.
     From: Penelope Maddy (Believing the Axioms I [1988], 0)
     A reaction: This sounds the wrong way round. Presumably we seek axioms that fix everything else about set theory, and then check to see what continuum results. Otherwise we could just pick our continuum, by picking our axioms.
1.1 p.484 The Axiom of Extensionality seems to be analytic
     Full Idea: Most writers agree that if any sense can be made of the distinction between analytic and synthetic, then the Axiom of Extensionality should be counted as analytic.
     From: Penelope Maddy (Believing the Axioms I [1988], 1.1)
     A reaction: [Boolos is the source of the idea] In other words Extensionality is not worth discussing, because it simply tells you what the world 'set' means, and there is no room for discussion about that. The set/class called 'humans' varies in size.
1.1 p.484 Extensional sets are clearer, simpler, unique and expressive
     Full Idea: The extensional view of sets is preferable because it is simpler, clearer, and more convenient, because it individuates uniquely, and because it can simulate intensional notions when the need arises.
     From: Penelope Maddy (Believing the Axioms I [1988], 1.1)
     A reaction: [She cites Fraenkel, Bar-Hillet and Levy for this] The difficulty seems to be whether the extensional notion captures our ordinary intuitive notion of what constitutes a group of things, since that needs flexible size and some sort of unity.
1.3 p.485 The Iterative Conception says everything appears at a stage, derived from the preceding appearances
     Full Idea: The Iterative Conception (Zermelo 1930) says everything appears at some stage. Given two objects a and b, let A and B be the stages at which they first appear. Suppose B is after A. Then the pair set of a and b appears at the immediate stage after B.
     From: Penelope Maddy (Believing the Axioms I [1988], 1.3)
     A reaction: Presumably this all happens in 'logical time' (a nice phrase I have just invented!). I suppose we might say that the existence of the paired set is 'forced' by the preceding sets. No transcendental inferences in this story?
1.3 p.485 Limitation of Size is a vague intuition that over-large sets may generate paradoxes
     Full Idea: The 'limitation of size' is a vague intuition, based on the idea that being too large may generate the paradoxes.
     From: Penelope Maddy (Believing the Axioms I [1988], 1.3)
     A reaction: This is an intriguing idea to be found right at the centre of what is supposed to be an incredibly rigorous system.
1.5 p.486 The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics
     Full Idea: The Axiom of Infinity is a simple statement of Cantor's great breakthrough. His bold hypothesis that a collection of elements that had lurked in the background of mathematics could be infinite launched modern mathematics.
     From: Penelope Maddy (Believing the Axioms I [1988], 1.5)
     A reaction: It also embodies one of those many points where mathematics seems to depart from common sense - but then most subjects depart from common sense when they get more sophisticated. Look what happened to art.
1.5 p.486 Infinite sets are essential for giving an account of the real numbers
     Full Idea: If one is interested in analysis then infinite sets are indispensable since even the notion of a real number cannot be developed by means of finite sets alone.
     From: Penelope Maddy (Believing the Axioms I [1988], 1.5)
     A reaction: [Maddy is citing Fraenkel, Bar-Hillel and Levy] So Cantor's great breakthrough (Idea 13021) actually follows from the earlier acceptance of the real numbers, so that's where the departure from common sense started.
1.6 p.486 The Power Set Axiom is needed for, and supported by, accounts of the continuum
     Full Idea: The Power Set Axiom is indispensable for a set-theoretic account of the continuum, ...and in so far as those attempts are successful, then the power-set principle gains some confirmatory support.
     From: Penelope Maddy (Believing the Axioms I [1988], 1.6)
     A reaction: The continuum is, of course, notoriously problematic. Have we created an extra problem in our attempts at solving the first one?
1.7 p.487 Efforts to prove the Axiom of Choice have failed
     Full Idea: Jordain made consistent and ill-starred efforts to prove the Axiom of Choice.
     From: Penelope Maddy (Believing the Axioms I [1988], 1.7)
     A reaction: This would appear to be the fate of most axioms. You would presumably have to use a different system from the one you are engaged with to achieve your proof.
1.7 p.488 Modern views say the Choice set exists, even if it can't be constructed
     Full Idea: Resistance to the Axiom of Choice centred on opposition between existence and construction. Modern set theory thrives on a realistic approach which says the choice set exists, regardless of whether it can be defined, constructed, or given by a rule.
     From: Penelope Maddy (Believing the Axioms I [1988], 1.7)
     A reaction: This seems to be a key case for the ontology that lies at the heart of theory. Choice seems to be an invaluable tool for proofs, so it won't go away, so admit it to the ontology. Hm. So the tools of thought have existence?
1.7 p.488 A large array of theorems depend on the Axiom of Choice
     Full Idea: Many theorems depend on the Axiom of Choice, including that a countable union of sets is countable, and results in analysis, topology, abstract algebra and mathematical logic.
     From: Penelope Maddy (Believing the Axioms I [1988], 1.7)
     A reaction: The modern attitude seems to be to admit anything if it leads to interesting results. It makes you wonder about the modern approach of using mathematics and logic as the cutting edges of ontological thinking.
1990 Realism in Mathematics
p.191 We know mind-independent mathematical truths through sets, which rest on experience
     Full Idea: Maddy proposes that we can know (some) mind-independent mathematical truths through knowing about sets, and that we can obtain knowledge of sets through experience.
     From: report of Penelope Maddy (Realism in Mathematics [1990]) by Carrie Jenkins - Grounding Concepts 6.5
     A reaction: Maddy has since backed off from this, and now tries to merely defend 'objectivity' about sets (2011:114). My amateurish view is that she is overrating the importance of sets, which merely model mathematics. Look at category theory.
p.223 Maddy replaces pure sets with just objects and perceived sets of objects
     Full Idea: Maddy dispenses with pure sets, by sketching a strong set theory in which everything is either a physical object or a set of sets of ...physical objects. Eventually a physiological story of perception will extend to sets of physical objects.
     From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3
     A reaction: This doesn't seem to find many supporters, but if we accept the perception of resemblances as innate (as in Hume and Quine), it is isn't adding much to see that we intrinsically see things in groups.
p.224 Intuition doesn't support much mathematics, and we should question its reliability
     Full Idea: Maddy says that intuition alone does not support very much mathematics; more importantly, a naturalist cannot accept intuition at face value, but must ask why we are justified in relying on intuition.
     From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3
     A reaction: It depends what you mean by 'intuition', but I identify with her second objection, that every faculty must ultimately be subject to criticism, which seems to point to a fairly rationalist view of things.
3 2 p.19 A natural number is a property of sets
     Full Idea: Maddy takes a natural number to be a certain property of sui generis sets, the property of having a certain number of members.
     From: report of Penelope Maddy (Realism in Mathematics [1990], 3 2) by Alex Oliver - The Metaphysics of Properties
     A reaction: [I believe Maddy has shifted since then] Presumably this will make room for zero and infinities as natural numbers. Personally I want my natural numbers to count things.
1997 Naturalism in Mathematics
107 p.117 The extension of concepts is not important to me
     Full Idea: I attach no decisive importance even to bringing in the extension of the concepts at all.
     From: Penelope Maddy (Naturalism in Mathematics [1997], 107)
     A reaction: He almost seems to equate the concept with its extension, but that seems to raise all sorts of questions, about indeterminate and fluctuating extensions.
I Intro p.1 Mathematics rests on the logic of proofs, and on the set theoretic axioms
     Full Idea: Our much loved mathematical knowledge rests on two supports: inexorable deductive logic (the stuff of proof), and the set theoretic axioms.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I Intro)
I.1 p.5 Frege solves the Caesar problem by explicitly defining each number
     Full Idea: To solve the Julius Caesar problem, Frege requires explicit definitions of the numbers, and he proposes his well-known solution: the number of Fs = the extension of the concept 'equinumerous with F' (based on one-one correspondence).
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.1)
     A reaction: Why do there have to be Fs before there can be the corresponding number? If there were no F for 523, would that mean that '523' didn't exist (even if 522 and 524 did exist)?
I.1 p.7 We can get arithmetic directly from HP; Law V was used to get HP from the definition of number
     Full Idea: Recent commentators have noted that Frege's versions of the basic propositions of arithmetic can be derived from Hume's Principle alone, that the fatal Law V is only needed to derive Hume's Principle itself from the definition of number.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.1)
     A reaction: Crispin Wright is the famous exponent of this modern view. Apparently Charles Parsons (1965) first floated the idea.
I.1 p.9 'Propositional functions' are propositions with a variable as subject or predicate
     Full Idea: A 'propositional function' is generated when one of the terms of the proposition is replaced by a variable, as in 'x is wise' or 'Socrates'.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.1)
     A reaction: This implies that you can only have a propositional function if it is derived from a complete proposition. Note that the variable can be in either subject or in predicate position. It extends Frege's account of a concept as 'x is F'.
I.1 p.11 Axiom of Reducibility: propositional functions are extensionally predicative
     Full Idea: The Axiom of Reducibility states that every propositional function is extensionally equivalent to some predicative proposition function.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.1)
I.1 p.16 Infinity has degrees, and large cardinals are the heart of set theory
     Full Idea: The stunning discovery that infinity comes in different degrees led to the theory of infinite cardinal numbers, the heart of contemporary set theory.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.1)
     A reaction: It occurs to me that these huge cardinals only exist in set theory. If you took away that prop, they would vanish in a puff.
I.1 p.17 For any cardinal there is always a larger one (so there is no set of all sets)
     Full Idea: By the mid 1890s Cantor was aware that there could be no set of all sets, as its cardinal number would have to be the largest cardinal number, while his own theorem shows that for any cardinal there is a larger.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.1)
     A reaction: There is always a larger cardinal because of the power set axiom. Some people regard that with suspicion.
I.1 n39 p.15 Cantor and Dedekind brought completed infinities into mathematics
     Full Idea: Both Cantor's real number (Cauchy sequences of rationals) and Dedekind's cuts involved regarding infinite items (sequences or sets) as completed and subject to further manipulation, bringing the completed infinite into mathematics unambiguously.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.1 n39)
     A reaction: So it is the arrival of the real numbers which is the culprit for lumbering us with weird completed infinites, which can then be the subject of addition, multiplication and exponentiation. Maybe this was a silly mistake?
I.2 p.23 In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets
     Full Idea: In the ZFC cumulative hierarchy, Frege's candidates for numbers do not exist. For example, new three-element sets are formed at every stage, so there is no stage at which the set of all three-element sets could he formed.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.2)
     A reaction: Ah. This is a very important fact indeed if you are trying to understand contemporary discussions in philosophy of mathematics.
I.2 p.26 Identifying geometric points with real numbers revealed the power of set theory
     Full Idea: The identification of geometric points with real numbers was among the first and most dramatic examples of the power of set theoretic foundations.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.2)
     A reaction: Hence the clear definition of the reals by Dedekind and Cantor was the real trigger for launching set theory.
I.2 p.26 Unified set theory gives a final court of appeal for mathematics
     Full Idea: The single unified area of set theory provides a court of final appeal for questions of mathematical existence and proof.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.2)
     A reaction: Maddy's third benefit of set theory. 'Existence' means being modellable in sets, and 'proof' means being derivable from the axioms. The slightly ad hoc character of the axioms makes this a weaker defence.
I.2 p.26 Set theory brings mathematics into one arena, where interrelations become clearer
     Full Idea: Set theoretic foundations bring all mathematical objects and structures into one arena, allowing relations and interactions between them to be clearly displayed and investigated.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.2)
     A reaction: The first of three benefits of set theory which Maddy lists. The advantages of the one arena seem to be indisputable.
I.2 p.26 Making set theory foundational to mathematics leads to very fruitful axioms
     Full Idea: The set theory axioms developed in producing foundations for mathematics also have strong consequences for existing fields, and produce a theory that is immensely fruitful in its own right.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.2)
     A reaction: [compressed] Second of Maddy's three benefits of set theory. This benefit is more questionable than the first, because the axioms may be invented because of their nice fruit, instead of their accurate account of foundations.
I.2 p.27 The line of rationals has gaps, but set theory provided an ordered continuum
     Full Idea: The structure of a geometric line by rational points left gaps, which were inconsistent with a continuous line. Set theory provided an ordering that contained no gaps. These reals are constructed from rationals, which come from integers and naturals.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.2)
     A reaction: This completes the reduction of geometry to arithmetic and algebra, which was launch 250 years earlier by Descartes.
I.2 p.27 Theorems about limits could only be proved once the real numbers were understood
     Full Idea: Even the fundamental theorems about limits could not [at first] be proved because the reals themselves were not well understood.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.2)
     A reaction: This refers to the period of about 1850 (Weierstrass) to 1880 (Dedekind and Cantor).
I.3 p.51 Completed infinities resulted from giving foundations to calculus
     Full Idea: The line of development that finally led to a coherent foundation for the calculus also led to the explicit introduction of completed infinities: each real number is identified with an infinite collection of rationals.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.3)
     A reaction: Effectively, completed infinities just are the real numbers.
I.3 p.52 Axiom of Infinity: completed infinite collections can be treated mathematically
     Full Idea: The axiom of infinity: that there are infinite sets is to claim that completed infinite collections can be treated mathematically. In its standard contemporary form, the axioms assert the existence of the set of all finite ordinals.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.3)
I.3 p.60 The Axiom of Foundation says every set exists at a level in the set hierarchy
     Full Idea: In the presence of other axioms, the Axiom of Foundation is equivalent to the claim that every set is a member of some Vα.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.3)
I.4 p.66 'Forcing' can produce new models of ZFC from old models
     Full Idea: Cohen's method of 'forcing' produces a new model of ZFC from an old model by appending a carefully chosen 'generic' set.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.4)
I.5 p.73 A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy
     Full Idea: A possible axiom is the Large Cardinal Axiom, which asserts that there are more and more stages in the cumulative hierarchy. Infinity can be seen as the first of these stages, and Replacement pushes further in this direction.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.5)
I.5 p.74 An 'inaccessible' cardinal cannot be reached by union sets or power sets
     Full Idea: An 'inaccessible' cardinal is one that cannot be reached by taking unions of small collections of smaller sets or by taking power sets.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.5)
     A reaction: They were introduced by Hausdorff in 1908.
II.5 p.131 Scientists posit as few entities as possible, but set theorist posit as many as possible
     Full Idea: Crudely, the scientist posits only those entities without which she cannot account for observations, while the set theorist posits as many entities as she can, short of inconsistency.
     From: Penelope Maddy (Naturalism in Mathematics [1997], II.5)
II.6 p.143 The theoretical indispensability of atoms did not at first convince scientists that they were real
     Full Idea: The case of atoms makes it clear that the indispensable appearance of an entity in our best scientific theory is not generally enough to convince scientists that it is real.
     From: Penelope Maddy (Naturalism in Mathematics [1997], II.6)
     A reaction: She refers to the period between Dalton and Einstein, when theories were full of atoms, but there was strong reluctance to actually say that they existed, until the direct evidence was incontrovertable. Nice point.
II.6 p.143 Science idealises the earth's surface, the oceans, continuities, and liquids
     Full Idea: In science we treat the earth's surface as flat, we assume the ocean to be infinitely deep, we use continuous functions for what we know to be quantised, and we take liquids to be continuous despite atomic theory.
     From: Penelope Maddy (Naturalism in Mathematics [1997], II.6)
     A reaction: If fussy people like scientists do this all the time, how much more so must the confused multitude be doing the same thing all day?
II.6 p.152 Maybe applications of continuum mathematics are all idealisations
     Full Idea: It could turn out that all applications of continuum mathematics in natural sciences are actually instances of idealisation.
     From: Penelope Maddy (Naturalism in Mathematics [1997], II.6)
2007 Second Philosophy
III.8 n1 p.299 Henkin semantics is more plausible for plural logic than for second-order logic
     Full Idea: Henkin-style semantics seem to me more plausible for plural logic than for second-order logic.
     From: Penelope Maddy (Second Philosophy [2007], III.8 n1)
     A reaction: Henkin-style semantics are presented by Shapiro as the standard semantics for second-order logic.
2011 Defending the Axioms
1.3 p.31 Hilbert's geometry and Dedekind's real numbers were role models for axiomatization
     Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers.
     From: Penelope Maddy (Defending the Axioms [2011], 1.3)
1.3 p.35 The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres
     Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original.
     From: Penelope Maddy (Defending the Axioms [2011], 1.3)
     A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure.
2.3 p.53 The connection of arithmetic to perception has been idealised away in modern infinitary mathematics
     Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics.
     From: Penelope Maddy (Defending the Axioms [2011], 2.3)
     A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor.
2.4 n40 p.56 Every infinite set of reals is either countable or of the same size as the full set of reals
     Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals.
     From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40)
3.3 p.99 Critics of if-thenism say that not all starting points, even consistent ones, are worth studying
     Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation.
     From: Penelope Maddy (Defending the Axioms [2011], 3.3)
     A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics.
3.4 p.82 Set-theory tracks the contours of mathematical depth and fruitfulness
     Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness.
     From: Penelope Maddy (Defending the Axioms [2011], 3.4)
     A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics.
5.3ii p.129 If two mathematical themes coincide, that suggest a single deep truth
     Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth.
     From: Penelope Maddy (Defending the Axioms [2011], 5.3ii)