41 ideas
18194 | 'Forcing' can produce new models of ZFC from old models [Maddy] |
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) |
18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
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) |
18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
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) |
18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
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) |
18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
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) |
18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
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'. |
17447 | Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck] |
Full Idea: In Parsons's demonstrative model of counting, '1' means the first, and counting says 'the first, the second, the third', where one is supposed to 'tag' each object exactly once, and report how many by converting the last ordinal into a cardinal. | |
From: report of Charles Parsons (Frege's Theory of Numbers [1965]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 3 | |
A reaction: This sounds good. Counting seems to rely on that fact that numbers can be both ordinals and cardinals. You don't 'convert' at the end, though, because all the way you mean 'this cardinality in this order'. |
18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
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. |
18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
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? |
18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
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. |
18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
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. |
18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
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. |
18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
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). |
18182 | The extension of concepts is not important to me [Maddy] |
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. |
18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy] |
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. |
18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
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)? |
18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
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. |
18185 | Unified set theory gives a final court of appeal for mathematics [Maddy] |
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. |
18183 | Set theory brings mathematics into one arena, where interrelations become clearer [Maddy] |
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. |
18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
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. |
18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
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. |
18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
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) |
18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
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) |
18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
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) |
18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
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. |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
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. |
12056 | An ancestral relation is either direct or transitively indirect [Wiggins] |
Full Idea: x bears to y the 'ancestral' of the relation R just if either x bears R to y, or x bears R to some w that bears R to y, or x bears R to some w that bears R to some z that bears R to y, or..... | |
From: David Wiggins (Substance [1995], 4.10.1) | |
A reaction: A concept invented by Frege (1879). |
12050 | Substances contain a source of change or principle of activity [Wiggins] |
Full Idea: Substances are things that have a source of change or principle of activity within them. | |
From: David Wiggins (Substance [1995], 4.4.1) | |
A reaction: A vey significant concession. I think we can talk of 'essences' and 'powers', and drop talk of 'substances'. 'Powers' is a much better word, because it immediately pushes the active ingredient to the forefront. |
12052 | We never single out just 'this', but always 'this something-or-other' [Wiggins] |
Full Idea: What is singled out is never a bare this or that, but this or that something or other. | |
From: David Wiggins (Substance [1995], 4.5.1) | |
A reaction: I like, in ontological speculation, to contemplate the problem of the baffling archaeological find. 'This thing I have dug up - what the hell IS it?'. Wiggins is contemptuous of the term 'thisness', and the idea of bare particulars. |
12055 | Sortal predications are answers to the question 'what is x?' [Wiggins] |
Full Idea: Predications which answer the question 'what is x?' are often called 'sortal predications' in present-day philosophy. | |
From: David Wiggins (Substance [1995], 4.10.1) | |
A reaction: The word 'sortal' comes from Locke. Wiggins is the guru of 'sortal essentialism'. I just can't believe that in answer to the question 'what really is David Wiggins?' that he would be happy with a sequence of categorisations. |
12059 | A river may change constantly, but not in respect of being a river [Wiggins] |
Full Idea: To say that the river is changing constantly in every respect is not to say that it is changing in respect of being a river. | |
From: David Wiggins (Substance [1995], 4.11.2) | |
A reaction: Can't a river become a lake, or a mere stream? Wiggins's proposal does not help with the problem of a river which sometimes dries up (as my local river sometimes does). At what point do we decide it is no longer a river? |
12063 | Sortal classification becomes science, with cross reference clarifying individuals [Wiggins] |
Full Idea: The sense of the sortal term under which we pick out an individual expands into the scientific account of things of that kind, where the account clarifies what is at issue in questions of sameness and difference of specimens of that kind. | |
From: David Wiggins (Substance [1995], 4.13.1) | |
A reaction: This is how the sortal approach is supposed to deal with individuals. So the placid tiger reveals much by falling under 'tiger', and a crucial extra bit by falling under 'placid'. See Idea 12053 for problems with this proposal. |
12051 | If the kinds are divided realistically, they fall into substances [Wiggins] |
Full Idea: Substance are what the world is articulated into when the segments of kinds corresponds to the real divisions in reality. | |
From: David Wiggins (Substance [1995], 4.5.1) | |
A reaction: This is very helpful in clarifying Wiggins's very obscurely expressed views. He appears to be saying that if we divide the sheep from the goats correctly, we reveal sheep-substance and goat-substance (one substance per species). Crazy! |
12053 | 'Human being' is a better answer to 'what is it?' than 'poet', as the latter comes in degrees [Wiggins] |
Full Idea: One person can be more or less of a poet than another, so 'poet' is not a conclusory answer to the question 'What is it that is singled out here?' 'Poet' rides on the back of the answer 'human being'. | |
From: David Wiggins (Substance [1995], 4.5.1) | |
A reaction: So apparently one must assign a natural kind, and not just a class. Wiggins lacks science fiction imagination. In the genetic salad of the far future, being a poet may be more definitive than being a human being. See Idea 12063. |
12054 | Secondary substances correctly divide primary substances by activity-principles and relations [Wiggins] |
Full Idea: A system of secondary substances with a claim to separate reality into its genuine primary substances must arise from an understanding of a set of principles of activity on the basis of which identities can be glossed in terms of determinate relations. | |
From: David Wiggins (Substance [1995], 4.5.1) | |
A reaction: I translate this as saying that individual essences are categorised according to principles which explain behaviour and relations. I'm increasingly bewildered by the 'secondary substances' Wiggins got from 'Categories', and loves so much. |
12047 | We refer to persisting substances, in perception and in thought, and they aid understanding [Wiggins] |
Full Idea: A substance is a persisting and somehow basic object of reference that is there to be discovered in perception and thought, an object whose claim to be recognized as a real entity is a claim on our aspirations to understand the world. | |
From: David Wiggins (Substance [1995], 4.1) | |
A reaction: A lot of components are assigned by Wiggins to the concept, and the tricky job, inititiated by Aristotle, is to fit all the pieces together nicely. Personally I am wondering if the acceptance of 'essences' implies dropping 'substances'. |
12057 | Matter underlies things, composes things, and brings them to be [Wiggins] |
Full Idea: Matter ex hypothesi is what ultimately underlies (to huperkeimenon) a thing; it is that from which something comes to be and which remains as a non-coincidental component in the thing's make-up. | |
From: David Wiggins (Substance [1995], 192a30) | |
A reaction: This is an interesting prelude to the much more comprehensive discussion of matter in Metaphysics, where he crucially adds the notion of 'form', and gives it priority over the underlying matter. |
12064 | The category of substance is more important for epistemology than for ontology [Wiggins] |
Full Idea: For us the importance of the category of substance, if it has any importance, is not so much ontological as relative to our epistemological circumstances and the conditions under which we have to undertake inquiry. | |
From: David Wiggins (Substance [1995], 4.13.2) | |
A reaction: This seems to be a rather significant concession. Wiggins has revived the notion of substance in recent times, but he is not quite adding it to the furniture of the world. Personally I increasingly think we can dump it, in ontology and epistemology. |
12049 | Naming the secondary substance provides a mass of general information [Wiggins] |
Full Idea: Answering 'what is it?' with the secondary substance identifies an object with a class of continuants which survive certain changes, come into being in certain ways, are qualified in certain ways, behave in certain ways, and cease to be in certain ways. | |
From: David Wiggins (Substance [1995], 4.3.3) | |
A reaction: Thus the priority of this sort of answer is that a huge range of explanations immediately flow from it. I take the explanation to be prior, and the primary substance to be prior, since secondary substance is inductively derived from it. |
12065 | Seeing a group of soldiers as an army is irresistible, in ontology and explanation [Wiggins] |
Full Idea: It seems mandatory to an observer of soldiers to give 'the final touch of unity' to their aggregate entity (the army). ...Similar claims arise with the ontological and explanatory claims of other corporate entities. | |
From: David Wiggins (Substance [1995], 4.13.3) | |
A reaction: Wiggins must say (following Leibniz Essays II.xxiv,1) that we add the unity, but I take the view that an army has powers, and hence offers explanations, which are lacking in a merely group of disparate soldiers. So an army has an essence and identity. |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
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? |