green numbers give full details | back to texts | unexpand these ideas
| 18194 | '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) |
| 18195 | 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) |
| 18191 | 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) |
| 18193 | 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) |
| 18169 | 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) |
| 18168 | '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'. |
| 18190 | 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. |
| 18171 | 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? |
| 18172 | 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. |
| 18196 | 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. |
| 18175 | 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. |
| 18187 | 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). |
| 18182 | 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. |
| 18177 | 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. |
| 18164 | 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)? |
| 18163 | 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) |
| 18185 | 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. |
| 18183 | 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. |
| 18186 | 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. |
| 18184 | 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. |
| 18188 | 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. |
| 18204 | 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) |
| 18207 | 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) |
| 18167 | 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. |
| 18205 | 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. |
| 18206 | 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? |