57 ideas
23890 | For Plato true wisdom is supernatural [Plato, by Weil] |
Full Idea: It is evident that Plato regards true wisdom as something supernatural. | |
From: report of Plato (works [c.375 BCE]) by Simone Weil - God in Plato p.61 | |
A reaction: Taken literally, I assume this is wrong, but we can empathise with the thought. Wisdom has the feeling of rising above the level of mere knowledge, to achieve the overview I associate with philosophy. |
3060 | Plato never mentions Democritus, and wished to burn his books [Plato, by Diog. Laertius] |
Full Idea: Plato, who mentions nearly all the ancient philosophers, nowhere speaks of Democritus; he wished to burn all of his books, but was persuaded that it was futile. | |
From: report of Plato (works [c.375 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.7.8 |
23891 | Two contradictories force us to find a relation which will correlate them [Plato, by Weil] |
Full Idea: Where contradictions appear there is a correlation of contraries, which is relation. If a contradiction is imposed on the intelligence, it is forced to think of a relation to transform the contradiction into a correlation, which draws the soul higher. | |
From: report of Plato (works [c.375 BCE]) by Simone Weil - God in Plato p.70 | |
A reaction: A much better account of the dialectic than anything I have yet seen in Hegel. For the first time I see some sense in it. A contradiction is not a falsehood, and it must be addressed rather than side-stepped. A kink in the system, that needs ironing. |
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'. |
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. |
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) |
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. |
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) |
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) |
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. |
14502 | Plato's idea of 'structure' tends to be mathematically expressed [Plato, by Koslicki] |
Full Idea: 'Structure' tends to be characterized by Plato as something that is mathematically expressed. | |
From: report of Plato (works [c.375 BCE]) by Kathrin Koslicki - The Structure of Objects V.3 iv | |
A reaction: [Koslicki is drawing on Verity Harte here] |
3039 | When Diogenes said he could only see objects but not their forms, Plato said it was because he had eyes but no intellect [Plato, by Diog. Laertius] |
Full Idea: When Diogenes told Plato he saw tables and cups, but not 'tableness' and 'cupness', Plato replied that this was because Diogenes had eyes but no intellect. | |
From: report of Plato (works [c.375 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 06.2.6 |
20906 | Platonists argue for the indivisible triangle-in-itself [Plato, by Aristotle] |
Full Idea: The Platonists, on the basis of purely logical arguments, posit the existence of an indivisible 'triangle in itself'. | |
From: report of Plato (works [c.375 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 316a15 | |
A reaction: A helpful confirmation that geometrical figures really are among the Forms (bearing in mind that numbers are not, because they contain one another). What shape is the Form of the triangle? |
17948 | Plato's Forms meant that the sophists only taught the appearance of wisdom and virtue [Plato, by Nehamas] |
Full Idea: Plato's theory of Forms allowed him to claim that the sophists and other opponents were trapped in the world of appearance. What they therefore taught was only apparent wisdom and virtue. | |
From: report of Plato (works [c.375 BCE]) by Alexander Nehamas - Eristic,Antilogic,Sophistic,Dialectic p.118 |
556 | If there is one Form for both the Form and its participants, they must have something in common [Aristotle on Plato] |
Full Idea: If there is the same Form for the Forms and for their participants, then they must have something in common. | |
From: comment on Plato (works [c.375 BCE]) by Aristotle - Metaphysics 991a |
563 | If gods are like men, they are just eternal men; similarly, Forms must differ from particulars [Aristotle on Plato] |
Full Idea: We say there is the form of man, horse and health, but nothing else, making the same mistake as those who say that there are gods but that they are in the form of men. They just posit eternal men, and here we are not positing forms but eternal sensibles. | |
From: comment on Plato (works [c.375 BCE]) by Aristotle - Metaphysics 997b |
565 | The Forms cannot be changeless if they are in changing things [Aristotle on Plato] |
Full Idea: The Forms could not be changeless if they were in changing things. | |
From: comment on Plato (works [c.375 BCE]) by Aristotle - Metaphysics 998a |
557 | A Form is a cause of things only in the way that white mixed with white is a cause [Aristotle on Plato] |
Full Idea: A Form is a cause of things only in the way that white mixed with white is a cause. | |
From: comment on Plato (works [c.375 BCE]) by Aristotle - Metaphysics 991a |
9607 | The greatest discovery in human thought is Plato's discovery of abstract objects [Brown,JR on Plato] |
Full Idea: The greatest discovery in the history of human thought is Plato's discovery of abstract objects. | |
From: comment on Plato (works [c.375 BCE]) by James Robert Brown - Philosophy of Mathematics Ch. 2 | |
A reaction: Compare Idea 2860! Given the diametrically opposed views, it is clearly likely that Plato's central view is the most important idea in the history of human thought, even if it is wrong. |
13263 | We can grasp whole things in science, because they have a mathematics and a teleology [Plato, by Koslicki] |
Full Idea: Due to the mathematical nature of structure and the teleological cause underlying the creation of Platonic wholes, these wholes are intelligible, and are in fact the proper objects of science. | |
From: report of Plato (works [c.375 BCE]) by Kathrin Koslicki - The Structure of Objects 5.3 | |
A reaction: I like this idea, because it pays attention to the connection between how we conceive objects to be, and how we are able to think about objects. Only examining these two together enables us to grasp metaphysics. |
13261 | Plato sees an object's structure as expressible in mathematics [Plato, by Koslicki] |
Full Idea: The 'structure' of an object tends to be characterised by Plato as something that is mathematically expressible. | |
From: report of Plato (works [c.375 BCE]) by Kathrin Koslicki - The Structure of Objects 5.3 | |
A reaction: This seems to be pure Pythagoreanism (see Idea 644). Plato is pursuing Pythagoras's research programme, of trying to find mathematics buried in every aspect of reality. |
13265 | Plato was less concerned than Aristotle with the source of unity in a complex object [Plato, by Koslicki] |
Full Idea: Plato was less concerned than Aristotle with the project of how to account, in completely general terms, for the source of unity within a mereologically complex object. | |
From: report of Plato (works [c.375 BCE]) by Kathrin Koslicki - The Structure of Objects 5.5 | |
A reaction: Plato seems to have simply asserted that some sort of harmony held things together. Aristotles puts the forms [eidos] within objects, rather than external, so he has to give a fuller account of what is going on in an object. He never managed it! |
593 | Plato's holds that there are three substances: Forms, mathematical entities, and perceptible bodies [Plato, by Aristotle] |
Full Idea: Plato's doctrine was that the Forms and mathematicals are two substances and that the third substance is that of perceptible bodies. | |
From: report of Plato (works [c.375 BCE]) by Aristotle - Metaphysics 1028b |
13260 | Plato says wholes are either containers, or they're atomic, or they don't exist [Plato, by Koslicki] |
Full Idea: Plato considers a 'container' model for wholes (which are disjoint from their parts) [Parm 144e3-], and a 'nihilist' model, in which only wholes are mereological atoms, and a 'bare pluralities' view, in which wholes are not really one at all. | |
From: report of Plato (works [c.375 BCE]) by Kathrin Koslicki - The Structure of Objects 5.2 | |
A reaction: [She cites Verity Harte for this analysis of Plato] The fourth, and best, seems to be that wholes are parts which fall under some unifying force or structure or principle. |
11237 | Only universals have essence [Plato, by Politis] |
Full Idea: Plato argues that only universals have essence. | |
From: report of Plato (works [c.375 BCE]) by Vassilis Politis - Aristotle and the Metaphysics 1.4 |
11238 | Plato and Aristotle take essence to make a thing what it is [Plato, by Politis] |
Full Idea: Plato and Aristotle have a shared general conception of essence: the essence of a thing is what that thing is simply in virtue of itself and in virtue of being the very thing it is. It answers the question 'What is this very thing?' | |
From: report of Plato (works [c.375 BCE]) by Vassilis Politis - Aristotle and the Metaphysics 1.4 |
17085 | A good explanation totally rules out the opposite explanation (so Forms are required) [Plato, by Ruben] |
Full Idea: For Plato, an acceptable explanation is one such that there is no possibility of there being the opposite explanation at all, and he thought that only explanations in terms of the Forms, but never physical explanations, could meet this requirement. | |
From: report of Plato (works [c.375 BCE]) by David-Hillel Ruben - Explaining Explanation Ch 2 | |
A reaction: [Republic 436c is cited] |
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? |
1651 | Plato wanted to somehow control and purify the passions [Vlastos on Plato] |
Full Idea: Plato put high on his agenda a project which did not figure in Socrates' programme at all: the hygienic conditioning of the passions. This cannot be an intellectual process, as argument cannot touch them. | |
From: comment on Plato (works [c.375 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.88 | |
A reaction: This is the standard traditional view of any thinker who exaggerates the importance and potential of reason in our lives. |
3324 | Plato's whole philosophy may be based on being duped by reification - a figure of speech [Benardete,JA on Plato] |
Full Idea: Plato is liable to the charge of having been duped by a figure of speech, albeit the most profound of all, the trope of reification. | |
From: comment on Plato (works [c.375 BCE]) by José A. Benardete - Metaphysics: the logical approach Ch.12 | |
A reaction: That might be a plausible account if his view was ridiculous, but given how many powerful friends Plato has, especially in the philosophy of mathematics, we should assume he was cleverer than that. |
7503 | Plato never refers to examining the conscience [Plato, by Foucault] |
Full Idea: Plato never speaks of the examination of conscience - never! | |
From: report of Plato (works [c.375 BCE]) by Michel Foucault - On the Genealogy of Ethics p.276 | |
A reaction: Plato does imply some sort of self-evident direct knowledge about that nature of a healthy soul. Presumably the full-blown concept of conscience is something given from outside, from God. In 'Euthyphro', Plato asserts the primacy of morality (Idea 337). |
2173 | As religion and convention collapsed, Plato sought morals not just in knowledge, but in the soul [Williams,B on Plato] |
Full Idea: Once gods and fate and social expectation were no longer there, Plato felt it necessary to discover ethics inside human nature, not just as ethical knowledge (Socrates' view), but in the structure of the soul. | |
From: comment on Plato (works [c.375 BCE]) by Bernard Williams - Shame and Necessity II - p.43 | |
A reaction: anti Charles Taylor |
9274 | Plato's legacy to European thought was the Good, the Beautiful and the True [Plato, by Gray] |
Full Idea: Plato's legacy to European thought was a trio of capital letters - the Good, the Beautiful and the True. | |
From: report of Plato (works [c.375 BCE]) by John Gray - Straw Dogs 2.8 | |
A reaction: It seems to have been Baumgarten who turned this into a slogan (Idea 8117). Gray says these ideals are lethal, but I identify with them very strongly, and am quite happy to see the good life as an attempt to find the right balance between them. |
94 | Pleasure is better with the addition of intelligence, so pleasure is not the good [Plato, by Aristotle] |
Full Idea: Plato says the life of pleasure is more desirable with the addition of intelligence, and if the combination is better, pleasure is not the good. | |
From: report of Plato (works [c.375 BCE]) by Aristotle - Nicomachean Ethics 1172b27 | |
A reaction: It is obvious why we like pleasure, but not why intelligence makes it 'better'. Maybe it is just because we enjoy intelligence? |
17947 | Plato decided that the virtuous and happy life was the philosophical life [Plato, by Nehamas] |
Full Idea: Plato came to the conclusion that virtue and happiness consist in the life of philosophy itself. | |
From: report of Plato (works [c.375 BCE]) by Alexander Nehamas - Eristic,Antilogic,Sophistic,Dialectic p.117 | |
A reaction: This view is obviously ridiculous, because it largely excludes almost the entire human race, which sees philosophy as a cul-de-sac, even if it is good. But virtue and happiness need some serious thought. |
6015 | Plato, unusually, said that theoretical and practical wisdom are inseparable [Plato, by Kraut] |
Full Idea: Two virtues that are ordinarily kept distinct - theoretical and practical wisdom - are joined by Plato; he thinks that neither one can be fully possessed unless it is combined with the other. | |
From: report of Plato (works [c.375 BCE]) by Richard Kraut - Plato | |
A reaction: I get the impression that this doctrine comes from Socrates, whose position is widely reported as 'intellectualist'. Aristotle certainly held the opposite view. |
2912 | Plato is boring [Nietzsche on Plato] |
Full Idea: Plato is boring. | |
From: comment on Plato (works [c.375 BCE]) by Friedrich Nietzsche - Twilight of the Idols 9.2 |
4784 | Salmon says processes rather than events should be basic in a theory of physical causation [Salmon, by Psillos] |
Full Idea: Salmon argues that processes rather than events should be the basic entities in a theory of physical causation. | |
From: report of Wesley Salmon (Causal Connections [1984]) by Stathis Psillos - Causation and Explanation §4.2 | |
A reaction: It increasingly strikes me that the concept of a 'process' ought to be ontologically basic. Edelman says the mind is a process. An 'event' is too loose, and a 'fact' too vague, and heaven knows what Hume meant by an 'object'. |
1526 | Almost everyone except Plato thinks that time could not have been generated [Plato, by Aristotle] |
Full Idea: With a single exception (Plato) everyone agrees about time - that it is not generated. Democritus says time is an obvious example of something not generated. | |
From: report of Plato (works [c.375 BCE]) by Aristotle - Physics 251b14 |