265 ideas
2136 | Philosophers become as divine and orderly as possible, by studying divinity and order [Plato] |
23767 | The winds of the discussion should decide its destination [Plato] |
13876 | The syntactic category is primary, and the ontological category is derivative [Frege, by Wright,C] |
23682 | It would be absurd to be precise about the small things, but only vague about the big things [Plato] |
8415 | Never lose sight of the distinction between concept and object [Frege] |
9841 | Frege was the first to give linguistic answers to non-linguistic questions [Frege, by Dummett] |
9840 | Frege initiated linguistic philosophy, studying number through the sense of sentences [Frege, by Dummett] |
15948 | Frege developed formal systems to avoid unnoticed assumptions [Frege, by Lavine] |
10804 | Thoughts have a natural order, to which human thinking is drawn [Frege, by Yablo] |
9832 | Frege sees no 'intersubjective' category, between objective and subjective [Dummett on Frege] |
8414 | Keep the psychological and subjective separate from the logical and objective [Frege] |
2151 | Dialectic is the only method of inquiry which uproots the things which it takes for granted [Plato] |
2154 | The ability to take an overview is the distinguishing mark of a dialectician [Plato] |
4011 | For Plato, rationality is a vision of and love of a cosmic rational order [Plato, by Taylor,C] |
2093 | You must never go against what you actually believe [Plato] |
2130 | People often merely practice eristic instead of dialectic, because they don't analyse the subject-matter [Plato] |
9844 | Originally Frege liked contextual definitions, but later preferred them fully explicit [Frege, by Dummett] |
9822 | Nothing should be defined in terms of that to which it is conceptually prior [Frege, by Dummett] |
17495 | Proof aims to remove doubts, but also to show the interdependence of truths [Frege] |
8632 | You can't transfer external properties unchanged to apply to ideas [Frege] |
13881 | We need to grasp not number-objects, but the states of affairs which make number statements true [Frege, by Wright,C] |
2145 | In mathematics certain things have to be accepted without further explanation [Plato] |
9154 | Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence [Frege, by Burge] |
9157 | The null set is only defensible if it is the extension of an empty concept [Frege, by Burge] |
9835 | It is because a concept can be empty that there is such a thing as the empty class [Frege, by Dummett] |
9854 | We can introduce new objects, as equivalence classes of objects already known [Frege, by Dummett] |
9883 | Frege introduced the standard device, of defining logical objects with equivalence classes [Frege, by Dummett] |
18104 | Frege, unlike Russell, has infinite individuals because numbers are individuals [Frege, by Bostock] |
9834 | A class is, for Frege, the extension of a concept [Frege, by Dummett] |
8645 | Convert "Jupiter has four moons" into "the number of Jupiter's moons is four" [Frege] |
16891 | Despite Gödel, Frege's epistemic ordering of all the truths is still plausible [Frege, by Burge] |
16906 | The primitive simples of arithmetic are the essence, determining the subject, and its boundaries [Frege, by Jeshion] |
14236 | Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed [Oliver/Smiley on Frege] |
22294 | We can show that a concept is consistent by producing something which falls under it [Frege] |
17624 | To understand axioms you must grasp their logical power and priority [Frege, by Burge] |
8726 | Geometry can lead the mind upwards to truth and philosophy [Plato] |
8640 | We cannot define numbers from the idea of a series, because numbers must precede that [Frege] |
9838 | Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Frege, by Dummett] |
9564 | For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Frege, by Chihara] |
10551 | If objects exist because they fall under a concept, 0 is the object under which no objects fall [Frege, by Dummett] |
8653 | Nought is the number belonging to the concept 'not identical with itself' [Frege] |
8636 | We can say 'a and b are F' if F is 'wise', but not if it is 'one' [Frege] |
8654 | One is the Number which belongs to the concept "identical with 0" [Frege] |
8641 | You can abstract concepts from the moon, but the number one is not among them [Frege] |
9989 | Units can be equal without being identical [Tait on Frege] |
17429 | Frege says only concepts which isolate and avoid arbitrary division can give units [Frege, by Koslicki] |
17427 | Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Frege, by Koslicki] |
17437 | Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Frege, by Koslicki] |
17438 | Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Frege, by Koslicki] |
17426 | A concept creating a unit must isolate and unify what falls under it [Frege] |
17428 | Frege says counting is determining what number belongs to a given concept [Frege, by Koslicki] |
15916 | Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Frege, by Lavine] |
10034 | The number of natural numbers is not a natural number [Frege, by George/Velleman] |
16883 | Arithmetical statements can't be axioms, because they are provable [Frege, by Burge] |
17460 | A statement of number contains a predication about a concept [Frege] |
16890 | Frege's problem is explaining the particularity of numbers by general laws [Frege, by Burge] |
8630 | Individual numbers are best derived from the number one, and increase by one [Frege] |
11029 | 'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt on Frege] |
10013 | Numerical statements have first-order logical form, so must refer to objects [Frege, by Hodes] |
18181 | The Number for F is the extension of 'equal to F' (or maybe just F itself) [Frege] |
18103 | Numbers are objects because they partake in identity statements [Frege, by Bostock] |
10625 | Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright on Frege] |
13871 | Frege claims that numbers are objects, as opposed to them being Fregean concepts [Frege, by Wright,C] |
13872 | Numbers are second-level, ascribing properties to concepts rather than to objects [Frege, by Wright,C] |
9816 | For Frege, successor was a relation, not a function [Frege, by Dummett] |
9953 | Numbers are more than just 'second-level concepts', since existence is also one [Frege, by George/Velleman] |
9954 | "Number of x's such that ..x.." is a functional expression, yielding a name when completed [Frege, by George/Velleman] |
10139 | Frege gives an incoherent account of extensions resulting from abstraction [Fine,K on Frege] |
10028 | For Frege the number of F's is a collection of first-level concepts [Frege, by George/Velleman] |
17636 | A cardinal number may be defined as a class of similar classes [Frege, by Russell] |
10029 | Numbers need to be objects, to define the extension of the concept of each successor to n [Frege, by George/Velleman] |
9973 | The number of F's is the extension of the second level concept 'is equipollent with F' [Frege, by Tait] |
16500 | Frege showed that numbers attach to concepts, not to objects [Frege, by Wiggins] |
9990 | Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Frege, by Tait] |
7738 | Zero is defined using 'is not self-identical', and one by using the concept of zero [Frege, by Weiner] |
23456 | Frege said logical predication implies classes, which are arithmetical objects [Frege, by Morris,M] |
13887 | Frege started with contextual definition, but then switched to explicit extensional definition [Frege, by Wright,C] |
13897 | Each number, except 0, is the number of the concept of all of its predecessors [Frege, by Wright,C] |
9856 | Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett on Frege] |
9902 | Frege's incorrect view is that a number is an equivalence class [Benacerraf on Frege] |
17814 | The natural number n is the set of n-membered sets [Frege, by Yourgrau] |
17819 | A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau on Frege] |
17820 | If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau on Frege] |
9956 | 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [Frege, by George/Velleman] |
13527 | Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC [Frege, by Wolf,RS] |
22292 | Hume's Principle fails to implicitly define numbers, because of the Julius Caesar [Frege, by Potter] |
17442 | Frege thinks number is fundamentally bound up with one-one correspondence [Frege, by Heck] |
11030 | The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt on Frege] |
10030 | 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [Frege, by George/Velleman] |
8690 | From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Frege, by Friend] |
10219 | Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Frege, by Shapiro] |
13889 | Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Frege, by Wright,C] |
18142 | One-one correlations imply normal arithmetic, but don't explain our concept of a number [Frege, by Bostock] |
9046 | Our definition will not tell us whether or not Julius Caesar is a number [Frege] |
16896 | If numbers can be derived from logic, then set theory is superfluous [Frege, by Burge] |
8639 | If numbers are supposed to be patterns, each number can have many patterns [Frege] |
9863 | We aim for elevated discussion of pure numbers, not attaching them to physical objects [Plato] |
9864 | In pure numbers, all ones are equal, with no internal parts [Plato] |
8727 | Geometry is not an activity, but the study of unchanging knowledge [Plato] |
13874 | Numbers seem to be objects because they exactly fit the inference patterns for identities [Frege] |
13875 | Frege's platonism proposes that objects are what singular terms refer to [Frege, by Wright,C] |
7731 | How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Frege, by Weiner] |
7737 | Identities refer to objects, so numbers must be objects [Frege, by Weiner] |
8635 | Numbers are not physical, and not ideas - they are objective and non-sensible [Frege] |
8652 | Numbers are objects, because they can take the definite article, and can't be plurals [Frege] |
17816 | Frege's logicism aimed at removing the reliance of arithmetic on intuition [Frege, by Yourgrau] |
9861 | The same thing is both one and an unlimited number at the same time [Plato] |
8633 | There is no physical difference between two boots and one pair of boots [Frege] |
9951 | It appears that numbers are adjectives, but they don't apply to a single object [Frege, by George/Velleman] |
9952 | Numerical adjectives are of the same second-level type as the existential quantifier [Frege, by George/Velleman] |
11031 | 'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt on Frege] |
8637 | The number 'one' can't be a property, if any object can be viewed as one or not one [Frege] |
9999 | For science, we can translate adjectival numbers into noun form [Frege] |
7739 | Arithmetic is analytic [Frege, by Weiner] |
9945 | Logicism shows that no empirical truths are needed to justify arithmetic [Frege, by George/Velleman] |
8782 | Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Frege, by Hale/Wright] |
13608 | Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Frege, by Bostock] |
16905 | Arithmetic must be based on logic, because of its total generality [Frege, by Jeshion] |
5658 | Numbers are definable in terms of mapping items which fall under concepts [Frege, by Scruton] |
8655 | Arithmetic is analytic and a priori, and thus it is part of logic [Frege] |
10831 | Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine on Frege] |
13864 | Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C on Frege] |
10033 | Why should the existence of pure logic entail the existence of objects? [George/Velleman on Frege] |
10010 | Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes on Frege] |
9631 | Formalism fails to recognise types of symbols, and also meta-games [Frege, by Brown,JR] |
9875 | Frege was completing Bolzano's work, of expelling intuition from number theory and analysis [Frege, by Dummett] |
8642 | Abstraction from things produces concepts, and numbers are in the concepts [Frege] |
8621 | Mental states are irrelevant to mathematics, because they are vague and fluctuating [Frege] |
8643 | Affirmation of existence is just denial of zero [Frege] |
9862 | To become rational, philosophers must rise from becoming into being [Plato] |
21818 | Being depends on the Good, which is not itself being, but superior to being [Plato] |
8911 | If abstracta are non-mental, quarks are abstracta, and yet chess and God's thoughts are mental [Rosen on Frege] |
8634 | The equator is imaginary, but not fictitious; thought is needed to recognise it [Frege] |
2061 | The best things (gods, healthy bodies, good souls) are least liable to change [Plato] |
17443 | Many of us find Frege's claim that truths depend on one another an obscure idea [Heck on Frege] |
17445 | Parallelism is intuitive, so it is more fundamental than sameness of direction [Frege, by Heck] |
10539 | Frege refers to 'concrete' objects, but they are no different in principle from abstract ones [Frege, by Dummett] |
6562 | Plato's reality has unchanging Parmenidean forms, and Heraclitean flux [Plato, by Fogelin] |
17431 | Vagueness is incomplete definition [Frege, by Koslicki] |
13879 | For Frege, ontological questions are to be settled by reference to syntactic structures [Frege, by Wright,C] |
10642 | Second-order quantifiers are committed to concepts, as first-order commits to objects [Frege, by Linnebo] |
10032 | 'Ancestral' relations are derived by iterating back from a given relation [Frege, by George/Velleman] |
10606 | Frege treats properties as a kind of function, and maybe a property is its characteristic function [Frege, by Smith,P] |
2142 | The plurality of beautiful things must belong to a single class, because they have a single particular character [Plato] |
5094 | Plato's Forms are said to have no location in space [Plato, by Aristotle] |
12043 | Forms are not universals, as they don't cover every general term [Plato, by Annas] |
2159 | Craftsmen making furniture refer to the form, but no one manufactures the form of furniture [Plato] |
17 | A Form applies to a set of particular things with the same name [Plato] |
12122 | Plato mistakenly thought forms were totally abstracted away from matter [Bacon on Plato] |
5574 | Plato's Forms not only do not come from the senses, but they are beyond possibility of sensing [Plato, by Kant] |
8647 | Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates [Frege] |
10309 | Frege says singular terms denote objects, numerals are singular terms, so numbers exist [Frege, by Hale] |
10550 | Frege establishes abstract objects independently from concrete ones, by falling under a concept [Frege, by Dummett] |
8785 | For Frege, objects just are what singular terms refer to [Frege, by Hale/Wright] |
10278 | Without concepts we would not have any objects [Frege, by Shapiro] |
17432 | Frege's universe comes already divided into objects [Frege, by Koslicki] |
16022 | The idea of a criterion of identity was introduced by Frege [Frege, by Noonan] |
11100 | Frege's algorithm of identity is the law of putting equals for equals [Frege, by Quine] |
12153 | Geach denies Frege's view, that 'being the same F' splits into being the same and being F [Perry on Frege] |
9853 | Identity between objects is not a consequence of identity, but part of what 'identity' means [Frege, by Dummett] |
490 | Everything happens by reason and necessity [Leucippus] |
2133 | Knowledge must be of the permanent unchanging nature of things [Plato] |
17623 | To understand a thought you must understand its logical structure [Frege, by Burge] |
9158 | For Frege a priori knowledge derives from general principles, so numbers can't be primitive [Frege] |
8657 | Mathematicians just accept self-evidence, whether it is logical or intuitive [Frege] |
9352 | An a priori truth is one derived from general laws which do not require proof [Frege] |
16889 | A truth is a priori if it can be proved entirely from general unproven laws [Frege] |
2514 | Frege tried to explain synthetic a priori truths by expanding the concept of analyticity [Frege, by Katz] |
2162 | If theory and practice conflict, the best part of the mind accepts theory, so the other part is of lower grade [Plato] |
16900 | Intuitions cannot be communicated [Frege, by Burge] |
2140 | True belief without knowledge is like blind people on the right road [Plato] |
16903 | Justifications show the ordering of truths, and the foundation is what is self-evident [Frege, by Jeshion] |
8624 | Induction is merely psychological, with a principle that it can actually establish laws [Frege] |
8626 | In science one observation can create high probability, while a thousand might prove nothing [Frege] |
8648 | Ideas are not spatial, and don't have distances between them [Frege] |
2096 | Is the function of the mind management, authority and planning - or is it one's whole way of life? [Plato] |
6009 | Psychic conflict is clear if appetite is close to the body and reason fairly separate [Plato, by Modrak] |
6041 | There is a third element to the mind - spirit - lying between reason and appetite [Plato] |
2127 | The mind has parts, because we have inner conflicts [Plato] |
1737 | The soul seems to have an infinity of parts [Aristotle on Plato] |
8620 | Thought is the same everywhere, and the laws of thought do not vary [Frege] |
9870 | Early Frege takes the extensions of concepts for granted [Frege, by Dummett] |
13878 | Concepts are, precisely, the references of predicates [Frege, by Wright,C] |
7736 | A concept is a non-psychological one-place function asserting something of an object [Frege, by Weiner] |
17430 | Fregean concepts have precise boundaries and universal applicability [Frege, by Koslicki] |
8622 | Psychological accounts of concepts are subjective, and ultimately destroy truth [Frege] |
8651 | A concept is a possible predicate of a singular judgement [Frege] |
9846 | Defining 'direction' by parallelism doesn't tell you whether direction is a line [Dummett on Frege] |
9976 | Frege accepts abstraction to the concept of all sets equipollent to a given one [Tait on Frege] |
10803 | Frege himself abstracts away from tone and color [Yablo on Frege] |
9988 | If we abstract 'from' two cats, the units are not black or white, or cats [Tait on Frege] |
9855 | Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects [Frege, by Dummett] |
10802 | Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first [Yablo on Frege] |
10525 | Frege put the idea of abstraction on a rigorous footing [Frege, by Fine,K] |
10526 | Fregean abstraction creates concepts which are equivalences between initial items [Frege, by Fine,K] |
10556 | We create new abstract concepts by carving up the content in a different way [Frege] |
9882 | You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett on Frege] |
9881 | From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements [Frege, by Dummett] |
8646 | Words in isolation seem to have ideas as meanings, but words have meaning in propositions [Frege] |
7732 | Never ask for the meaning of a word in isolation, but only in the context of a proposition [Frege] |
9370 | A statement is analytic if substitution of synonyms can make it a logical truth [Frege, by Boghossian] |
8743 | Frege considered analyticity to be an epistemic concept [Frege, by Shapiro] |
20295 | All analytic truths can become logical truths, by substituting definitions or synonyms [Frege, by Rey] |
2515 | Frege fails to give a concept of analyticity, so he fails to explain synthetic a priori truth that way [Katz on Frege] |
5945 | The 'Republic' is a great work of rhetorical theory [Lawson-Tancred on Plato] |
23316 | For Plato and Aristotle there is no will; there is only rational desire for what is seen as good [Plato, by Frede,M] |
16 | We avoid evil either through a natural aversion, or because we have acquired knowledge [Plato] |
16565 | Without the surface decoration, poetry shows only appearances and nothing of what is real [Plato] |
2160 | Representation is two steps removed from the truth [Plato] |
2163 | Artists should be excluded from a law-abiding community, because they destroy the rational mind [Plato] |
2135 | Truth is closely related to proportion [Plato] |
2141 | I suggest that we forget about trying to define goodness itself for the time being [Plato] |
1869 | The good cannot be expressed in words, but imprints itself upon the soul [Plato, by Celsus] |
4115 | Plato found that he could only enforce rational moral justification by creating an authoritarian society [Williams,B on Plato] |
4547 | Plato measured the degree of reality by the degree of value [Nietzsche on Plato] |
2094 | A thing's function is what it alone can do, or what it does better than other things [Plato] |
2095 | If something has a function then it has a state of being good [Plato] |
2129 | Goodness is mental health, badness is mental sickness [Plato] |
12 | If we were invisible, would the just man become like the unjust? [Plato] |
2168 | Clever criminals do well at first, but not in the long run [Plato] |
2137 | The main aim is to understand goodness, which gives everything its value and advantage [Plato] |
2139 | Every person, and every activity, aims at the good [Plato] |
4007 | For Plato we abandon honour and pleasure once we see the Good [Plato, by Taylor,C] |
2144 | Goodness makes truth and knowledge possible [Plato] |
2164 | Bad is always destructive, where good preserves and benefits [Plato] |
2143 | Good has the same role in the world of knowledge as the sun has in the physical world [Plato] |
2147 | The sight of goodness leads to all that is fine and true and right [Plato] |
2138 | Pleasure is commonly thought to be the good, though the more ingenious prefer knowledge [Plato] |
2070 | Even people who think pleasure is the good admit that there are bad pleasures [Plato] |
2157 | Nice smells are intensive, have no preceding pain, and no bad after-effect [Plato] |
2134 | Philosophers are concerned with totally non-physical pleasures [Plato] |
2156 | There are three types of pleasure, for reason, for spirit and for appetite [Plato] |
2158 | Pleasure-seekers desperately seek illusory satisfaction, like filling a leaky vessel [Plato] |
2123 | Excessive pleasure deranges people, making the other virtues impossible [Plato] |
2166 | We should behave well even if invisible, for the health of the mind [Plato] |
2097 | Isn't it better to have a reputation for goodness than to actually be good? [Plato] |
19946 | Morality is a compromise, showing restraint, to avoid suffering wrong without compensation [Plato] |
5 | Justice is merely the interests of the stronger party [Plato] |
7 | Surely you don't return a borrowed weapon to a mad friend? [Plato] |
8 | Is right just the interests of the powerful? [Plato] |
15 | Sin first, then sacrifice to the gods from the proceeds [Plato] |
5944 | For Plato, virtue is its own reward [Lawson-Tancred on Plato] |
2155 | True goodness requires mental unity and harmony [Plato] |
2126 | A good community necessarily has wisdom, courage, self-discipline and morality [Plato] |
23562 | If the parts of our soul do their correct work, we will be just people, and will act justly [Plato] |
2092 | Simonides said morality is helping one's friends and harming one's enemies [Plato] |
19889 | People need society because the individual has too many needs [Plato] |
19890 | All exchanges in a community are for mutual benefit [Plato] |
10 | After a taste of mutual harm, men make a legal contract to avoid it [Plato] |
23561 | People doing their jobs properly is the fourth cardinal virtue for a city [Plato] |
2149 | Reluctant rulers make a better and more unified administration [Plato] |
2132 | Only rule by philosophers of integrity can keep a community healthy [Plato] |
2131 | Is there anything better for a community than to produce excellent people? [Plato] |
2148 | To gain knowledge, turn away from the world of change, and focus on true goodness [Plato] |
8619 | To learn something, you must know that you don't know [Frege] |
2152 | Dialectic is the highest and most important part of the curriculum [Plato] |
2153 | Compulsory intellectual work never remains in the mind [Plato] |
8656 | The laws of number are not laws of nature, but are laws of the laws of nature [Frege] |
2630 | If Plato's God is immaterial, he will lack consciousness, wisdom, pleasure and movement, which are essential to him [Cicero on Plato] |
22286 | Existence is not a first-level concept (of God), but a second-level property of concepts [Frege, by Potter] |
8644 | Because existence is a property of concepts the ontological argument for God fails [Frege] |
14 | If the gods are non-existent or indifferent, why bother to deceive them? [Plato] |
2165 | Something is unlikely to be immortal if it is imperfectly made from diverse parts [Plato] |
13 | Is the supreme reward for virtue to be drunk for eternity? [Plato] |
2120 | God is responsible for the good things, but we must look elsewhere for the cause of the bad things [Plato] |