46 ideas
| 2056 | Philosophers are always switching direction to something more interesting [Plato] |
| Full Idea: Philosophers are always ready to change direction, if a topic crops up which is more attractive than the one to hand. | |
| From: Plato (Theaetetus [c.368 BCE], 172d) | |
| A reaction: Which sounds trivial, but it may be what God does. |
| 2086 | Understanding mainly involves knowing the elements, not their combinations [Plato] |
| Full Idea: A perfect grasp of any subject depends far more on knowing elements than on knowing complexes. | |
| From: Plato (Theaetetus [c.368 BCE], 206b) |
| 2083 | Either a syllable is its letters (making parts as knowable as whole) or it isn't (meaning it has no parts) [Plato] |
| Full Idea: Either a syllable is not the same as its letters, in which case it cannot have the letters as parts of itself, or it is the same as its letters, in which case these basic elements are just as knowable as it is. | |
| From: Plato (Theaetetus [c.368 BCE], 205b) |
| 2082 | A rational account is essentially a weaving together of things with names [Plato] |
| Full Idea: Just as primary elements are woven together, so their names may be woven together to produce a spoken account, because an account is essentially a weaving together of names. | |
| From: Plato (Theaetetus [c.368 BCE], 202b) | |
| A reaction: If justification requires 'logos', and logos is a 'weaving together of names', then Plato might be taken as endorsing the coherence account of justification. Or do the two 'weavings' correspond? |
| 2052 | Eristic discussion is aggressive, but dialectic aims to help one's companions in discussion [Plato] |
| Full Idea: Eristic discussions involve as many tricks and traps as possible, but dialectical discussions involve being serious and correcting the interlocutor's mistakes only when they are his own fault or the result of past conditioning. | |
| From: Plato (Theaetetus [c.368 BCE], 167e) |
| 15854 | A primary element has only a name, and no logos, but complexes have an account, by weaving the names [Plato] |
| Full Idea: A primary element cannot be expressed in an account; it can only be named, for a name is all that it has. But with the things composed of these ...just as the elements are woven together, so the names can woven to become an account. | |
| From: Plato (Theaetetus [c.368 BCE], 202b01-3) | |
| A reaction: This is the beginning of what I see as Aristotle's metaphysics, as derived from his epistemology, that is, ontology is what explains, and what we can give an account [logos] of. Aristotle treats this under 'definitions'. |
| 10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
| Full Idea: In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: Hence people now standardly accept ZFC, rather than just ZF. |
| 10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
| Full Idea: Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it. |
| 10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
| Full Idea: The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory. |
| 10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
| Full Idea: The Trichotomy Principle (any number is less, equal to, or greater than, another number) turned out to be equivalent to the Axiom of Choice. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: [He credits Sierpinski (1918) with this discovery] |
| 10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
| Full Idea: The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) |
| 10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
| Full Idea: A structure is said to be a 'model' of an axiom system if each of its axioms is true in the structure (e.g. Euclidean or non-Euclidean geometry). 'Model theory' concerns which structures are models of a given language and axiom system. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This strikes me as the most interesting aspect of mathematical logic, since it concerns the ways in which syntactic proof-systems actually connect with reality. Tarski is the central theoretician here, and his theory of truth is the key. |
| 10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
| Full Idea: In the late 1950s Tarski and Vaught defined and established basic properties of the relation of elementary equivalence between two structures, which holds when they make true exactly the same first-order sentences. This is fundamental to model theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This is isomorphism, which clarifies what a model is by giving identity conditions between two models. Note that it is 'first-order', and presumably founded on classical logic. |
| 10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
| Full Idea: The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory. |
| 10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
| Full Idea: Before Tarski's work in the 1930s, the main results in model theory were the Löwenheim-Skolem Theorem, and Gödel's establishment in 1929 of the completeness of the axioms and rules for the classical first-order predicate (or quantificational) calculus. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
| Full Idea: Completeness is when, if a sentences holds in every model of a theory, then it is logically derivable from that theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
| Full Idea: 'Recursion theory' is the subject of what can and cannot be solved by computing machines | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Ch.9) | |
| A reaction: This because 'recursion' will grind out a result step-by-step, as long as the steps will 'halt' eventually. |
| 10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
| Full Idea: In 1936 Church showed that Principia Mathematica is undecidable if it is ω-consistent, and a year later Rosser showed that Peano Arithmetic is undecidable, and any consistent extension of it. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int IV) |
| 10216 | We master arithmetic by knowing all the numbers in our soul [Plato] |
| Full Idea: It must surely be true that a man who has completely mastered arithmetic knows all numbers? Because there are pieces of knowledge covering all numbers in his soul. | |
| From: Plato (Theaetetus [c.368 BCE], 198b) | |
| A reaction: This clearly views numbers as objects. Expectation of knowing them all is a bit startling! They also appear to be innate in us, and hence they appear to be Forms. See Aristotle's comment in Idea 645. |
| 2060 | There seem to be two sorts of change: alteration and motion [Plato] |
| Full Idea: There are two kinds of change, I think: alteration and motion. | |
| From: Plato (Theaetetus [c.368 BCE], 181d) |
| 2084 | If a word has no parts and has a single identity, it turns out to be the same kind of thing as a letter [Plato] |
| Full Idea: If a complex or a syllable has no parts and is a single identity, hasn't it turned out to be the same kind of thing as an element or letter? | |
| From: Plato (Theaetetus [c.368 BCE], 205d) |
| 15844 | A sum is that from which nothing is lacking, which is a whole [Plato] |
| Full Idea: But this sum now - isn't it just when there is nothing lacking that it is a sum? Yes, necessarily. And won't this very same thing - that from which nothing is lacking - be a whole? | |
| From: Plato (Theaetetus [c.368 BCE], 205a) | |
| A reaction: This seems to be right, be rather too vague and potentially circular to be of much use. What is the criterion for deciding that nothing is lacking? |
| 15843 | The whole can't be the parts, because it would be all of the parts, which is the whole [Plato] |
| Full Idea: The whole does not consist of parts; for it did, it would be all the parts and so would be the sum. | |
| From: Plato (Theaetetus [c.368 BCE], 204e) | |
| A reaction: That is, 'the whole is the sum of its parts' is a tautology! The claim that 'the whole is more than the sum of its parts' gets into similar trouble. See Verity Harte on this. |
| 2080 | Things are only knowable if a rational account (logos) is possible [Plato] |
| Full Idea: Things which are susceptible to a rational account are knowable. | |
| From: Plato (Theaetetus [c.368 BCE], 201d) |
| 16126 | Expertise is knowledge of the whole by means of the parts [Plato] |
| Full Idea: A man has passed from mere judgment to expert knowledge of the being of a wagon when he has done so in virtue of having gone over the whole by means of the elements. | |
| From: Plato (Theaetetus [c.368 BCE], 207c) | |
| A reaction: Plato is emphasising that the expert must know the hundred parts of a wagon, and not just the half dozen main components, but here the point is to go over the whole via the parts, and not just list the parts. |
| 2050 | It is impossible to believe something which is held to be false [Plato] |
| Full Idea: It is impossible to believe something which is not the case. | |
| From: Plato (Theaetetus [c.368 BCE], 167a) |
| 2076 | How can a belief exist if its object doesn't exist? [Plato] |
| Full Idea: If the object of a belief is what is not, the object of this belief is nothing; but if there is no object to a belief, then that is not belief at all. | |
| From: Plato (Theaetetus [c.368 BCE], 189a) |
| 2045 | Perception is infallible, suggesting that it is knowledge [Plato] |
| Full Idea: Perception is always of something that is, and it is infallible, which suggests that it is knowledge. | |
| From: Plato (Theaetetus [c.368 BCE], 152c) |
| 2067 | Our senses could have been separate, but they converge on one mind [Plato] |
| Full Idea: It would be peculiar if each of us were like a Trojan horse, with a whole bunch of senses sitting inside us, rather than that all these perceptions converge onto a single identity (mind, or whatever one ought to call it). | |
| From: Plato (Theaetetus [c.368 BCE], 184d) |
| 2068 | With what physical faculty do we perceive pairs of opposed abstract qualities? [Plato] |
| Full Idea: With what physical faculty do we perceive being and not-being, similarity and dissimilarity, identity and difference, oneness and many, odd and even and other maths, ….fineness and goodness? | |
| From: Plato (Theaetetus [c.368 BCE], 185d) |
| 2078 | You might mistake eleven for twelve in your senses, but not in your mind [Plato] |
| Full Idea: Sight or touch might make someone take eleven for twelve, but he could never form this mistaken belief about the contents of his mind. | |
| From: Plato (Theaetetus [c.368 BCE], 195e) |
| 2069 | Thought must grasp being itself before truth becomes possible [Plato] |
| Full Idea: If you can't apprehend being you can't apprehend truth, and so a thing could not be known. Therefore knowledge is not located in immediate experience but in thinking about it, since the latter makes it possible to grasp being and truth. | |
| From: Plato (Theaetetus [c.368 BCE], 186c) |
| 2089 | An inadequate rational account would still not justify knowledge [Plato] |
| Full Idea: If you don't know which letters belong together in the right syllables…it is possible for true belief to be accompanied by a rational account and still not be entitled to the name of knowledge. | |
| From: Plato (Theaetetus [c.368 BCE], 208b) | |
| A reaction: In each case of justification there is a 'clinching' stage, for which there is never going to be a strict rule. It might be foundational, but equally it might be massive coherence, or no alternative. |
| 2085 | Parts and wholes are either equally knowable or equally unknowable [Plato] |
| Full Idea: Either a syllable and its letters are equally knowable and expressible in a rational account, or they are both equally unknowable and inexpressible. | |
| From: Plato (Theaetetus [c.368 BCE], 205e) | |
| A reaction: Presumably you could explain the syllable by the letters, but not vice versa, but he must mean that the explanation is worthless without the letters being explained too. So all explanation is worthless? |
| 2091 | Without distinguishing marks, how do I know what my beliefs are about? [Plato] |
| Full Idea: If I only have beliefs about Theaetetus when I don't know his distinguishing mark, how on earth were my beliefs about you rather than anyone else? | |
| From: Plato (Theaetetus [c.368 BCE], 209b) | |
| A reaction: This is a rather intellectualist approach to mental activity. Presumably Theaetetus has lots of distinguishing marks, but they are not conscious. Must Socrates know everything? |
| 2087 | A rational account might be seeing an image of one's belief, like a reflection in a mirror [Plato] |
| Full Idea: A rational account might be forming an image of one's belief, as in a mirror or a pond. | |
| From: Plato (Theaetetus [c.368 BCE], 206d) | |
| A reaction: Not promising, since the image is not going to be clearer than the original, or contain any new information. Maybe it would be clarified by being 'framed', instead of drifting in muddle. |
| 2090 | A rational account involves giving an image, or analysis, or giving a differentiating mark [Plato] |
| Full Idea: A third sort of rational account (after giving an image, or analysing elements) is being able to mention some mark which differentiates the object in question ('the sun is the brightest heavenly body'). | |
| From: Plato (Theaetetus [c.368 BCE], 208c) | |
| A reaction: This is Plato's clearest statement of what would be involved in adding the necessary logos to your true belief. An image of it, or an analysis, or an individuation. How about a cause? |
| 2081 | Maybe primary elements can be named, but not receive a rational account [Plato] |
| Full Idea: Maybe the primary elements of which things are composed are not susceptible to rational accounts. Each of them taken by itself can only be named, but nothing further can be said about it. | |
| From: Plato (Theaetetus [c.368 BCE], 201e) | |
| A reaction: This still seems to be more or less the central issue in philosophy - which things should be treated as 'primitive', and which other things are analysed and explained using the primitive tools? |
| 2088 | A rational account of a wagon would mean knowledge of its hundred parts [Plato] |
| Full Idea: In the case of a wagon, we may only have correct belief, but someone who is able to explain what it is by going through its hundred parts has got hold of a rational account. | |
| From: Plato (Theaetetus [c.368 BCE], 207b) | |
| A reaction: A wonderful example. In science, you know smoking correlates with cancer, but you only know it when you know the mechanism, the causal structure. This may be a general truth. |
| 2047 | What evidence can be brought to show whether we are dreaming or not? [Plato] |
| Full Idea: What evidence could be brought if we were asked at this very moment whether we are asleep and are dreaming all our thoughts? | |
| From: Plato (Theaetetus [c.368 BCE], 158b) |
| 2053 | If you claim that all beliefs are true, that includes beliefs opposed to your own [Plato] |
| Full Idea: To say that everyone believes what is the case, is to concede the truth of the oppositions' beliefs; in other words, the person has to concede that he himself is wrong. | |
| From: Plato (Theaetetus [c.368 BCE], 171a) |
| 2059 | How can a relativist form opinions about what will happen in the future? [Plato] |
| Full Idea: Does a relativist have any authority to decide about things which will happen in the future? | |
| From: Plato (Theaetetus [c.368 BCE], 178c) | |
| A reaction: Nice question! It seems commonsense that such speculations are possible, but without a concept of truth they are ridiculous. |
| 2054 | Clearly some people are superior to others when it comes to medicine [Plato] |
| Full Idea: In medicine, at least, most people are not self-sufficient at prescribing and effecting cures for themselves, and here some people are superior to others. | |
| From: Plato (Theaetetus [c.368 BCE], 171e) |
| 23224 | That all matter thinks is absurd, and would make each part of our bodies a distinct self-consciousness [Bentley] |
| Full Idea: [Belief in thinking matter] leads to monstrous absurdities. …Every stock and stone would be a percipient and rational creature. …every single Atom of our bodies would be a distinct Animal, endued with self-consciousness and personal sensation of its own. | |
| From: Richard Bentley (Matter and Motion Cannot Think [1692], p.14-15), quoted by Matthew Cobb - The Idea of the Brain 2 | |
| A reaction: Sounds correct, though presumably panpsychists don't think the flickers of consciousness in my toenails and hair constitute full-blown persons. I can't imagine what awareness is being claimed for my toenails. |
| 2058 | God must be the epitome of goodness, and we can only approach a divine state by being as good as possible [Plato] |
| Full Idea: It is impossible for God to be immoral and not to be the acme of morality; and the only way any of us can approximate to God is to become as moral as possible. | |
| From: Plato (Theaetetus [c.368 BCE], 176c) |
| 2057 | There must always be some force of evil ranged against good [Plato] |
| Full Idea: The elimination of evil is impossible, Theodorus; there must always be some force ranged against good. | |
| From: Plato (Theaetetus [c.368 BCE], 176a) |