44 ideas
| 24032 | Clever scholars can obscure things which are obvious even to peasants [Descartes] |
| Full Idea: Scholars are usually ingenious enough to find ways of spreading darkness even in things which are obvious by themselves, and which the peasants are not ignorant of. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 12) | |
| A reaction: Wonderful! I see it everywhere in philosophy. It is usually the result of finding ingenious and surprising grounds for scepticism. The amazing thing is not their lovely arguments, but that fools then take their conclusions seriously. Modus tollens. |
| 24033 | Most scholastic disputes concern words, where agreeing on meanings would settle them [Descartes] |
| Full Idea: The questions on which scholars argue are almost always questions of word. …If philosophers were agreed on the meaning of words, almost all their controversies would cease. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 13) | |
| A reaction: He has a low opinion of 'scholars'! It isn't that difficult to agree on the meanings of key words, in a given context. The aim isn't to get rid of the problems, but to focus on the real problems. Some words contain problems. |
| 24024 | The secret of the method is to recognise which thing in a series is the simplest [Descartes] |
| Full Idea: It is necessary, in a series of objects, to recognise which is the simplest thing, and how all the others depart from it. This rule contains the whole secret of the method. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 06) | |
| A reaction: This is an appealing thought, though deciding the criteria for 'simplest' looks tough. Are electrons, for example, simple? Is a person a simple basic thing? |
| 24018 | One truth leads us to another [Descartes] |
| Full Idea: One truth discovered helps us to discover another. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 01) | |
| A reaction: I take this to be one of the key ingredients of objectivity. People who know very little have almost no chance of objectivity. A mind full of falsehoods also blocks it. |
| 22289 | Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Dedekind, by Potter] |
| Full Idea: Dedkind gave a rigorous proof of the principle of definition by recursion, permitting recursive definitions of addition and multiplication, and hence proofs of the familiar arithmetical laws. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 13 'Deriv' |
| 10183 | An infinite set maps into its own proper subset [Dedekind, by Reck/Price] |
| Full Idea: A set is 'Dedekind-infinite' iff there exists a one-to-one function that maps a set into a proper subset of itself. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], §64) by E Reck / M Price - Structures and Structuralism in Phil of Maths n 7 | |
| A reaction: Sounds as if it is only infinite if it is contradictory, or doesn't know how big it is! |
| 22288 | We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter] |
| Full Idea: Dedekind had an interesting proof of the Axiom of Infinity. He held that I have an a priori grasp of the idea of my self, and that every idea I can form the idea of that idea. Hence there are infinitely many objects available to me a priori. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], no. 66) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 12 'Numb' | |
| A reaction: Who said that Descartes' Cogito was of no use? Frege endorsed this, as long as the ideas are objective and not subjective. |
| 10706 | Dedekind originally thought more in terms of mereology than of sets [Dedekind, by Potter] |
| Full Idea: Dedekind plainly had fusions, not collections, in mind when he avoided the empty set and used the same symbol for membership and inclusion - two tell-tale signs of a mereological conception. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], 2-3) by Michael Potter - Set Theory and Its Philosophy 02.1 | |
| A reaction: Potter suggests that mathematicians were torn between mereology and sets, and eventually opted whole-heartedly for sets. Maybe this is only because set theory was axiomatised by Zermelo some years before Lezniewski got to mereology. |
| 9823 | Numbers are free creations of the human mind, to understand differences [Dedekind] |
| Full Idea: Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], Pref) | |
| A reaction: Does this fit real numbers and complex numbers, as well as natural numbers? Frege was concerned by the lack of objectivity in this sort of view. What sort of arithmetic might the Martians have created? Numbers register sameness too. |
| 10090 | Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman] |
| Full Idea: It was primarily Dedekind's accomplishment to define the integers, rationals and reals, taking only the system of natural numbers for granted. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by A.George / D.J.Velleman - Philosophies of Mathematics Intro |
| 7524 | Order, not quantity, is central to defining numbers [Dedekind, by Monk] |
| Full Idea: Dedekind said that the notion of order, rather than that of quantity, is the central notion in the definition of number. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4 | |
| A reaction: Compare Aristotle's nice question in Idea 646. My intuition is that quantity comes first, because I'm not sure HOW you could count, if you didn't think you were changing the quantity each time. Why does counting go in THAT particular order? Cf. Idea 8661. |
| 17452 | Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck] |
| Full Idea: Dedekind and Cantor said the cardinals may be defined in terms of the ordinals: The cardinal number of a set S is the least ordinal onto whose predecessors the members of S can be mapped one-one. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 5 |
| 14131 | Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell] |
| Full Idea: Dedekind's ordinals are not essentially either ordinals or cardinals, but the members of any progression whatever. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - The Principles of Mathematics §243 | |
| A reaction: This is part of Russell's objection to Dedekind's structuralism. The question is always why these beautiful structures should actually be considered as numbers. I say, unlike Russell, that the connection to counting is crucial. |
| 14437 | Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell] |
| Full Idea: Dedekind set up the axiom that the gap in his 'cut' must always be filled …The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - Introduction to Mathematical Philosophy VII | |
| A reaction: This remark of Russell's is famous, and much quoted in other contexts, but I have seen the modern comment that it is grossly unfair to Dedekind. |
| 18094 | Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock] |
| Full Idea: One view, favoured by Dedekind, is that the cut postulates a real number for each cut in the rationals; it does not identify real numbers with cuts. ....A view favoured by later logicists is simply to identify a real number with a cut. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by David Bostock - Philosophy of Mathematics 4.4 | |
| A reaction: Dedekind is the patriarch of structuralism about mathematics, so he has little interest in the existenc of 'objects'. |
| 24036 | I can only see the proportion of two to three if there is a common measure - their unity [Descartes] |
| Full Idea: I do not recognise what the proportion of magnitude is between two and three, unless I consider a third term, namely unity, which is the common measure of the one and the other. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 14) | |
| A reaction: A striking defence of the concept of the need for the unit in arithmetic. To say 'three is half as big again', you must be discussing the same size of 'half' in each instance. |
| 24035 | Unity is something shared by many things, so in that respect they are equals [Descartes] |
| Full Idea: Unity is that common nature in which all things that are compared with each other must participate equally. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 14) | |
| A reaction: A lovely explanation of the concept of 'units' for counting. Fregeans hate units, but we Grecian thinkers love them. |
| 9824 | In counting we see the human ability to relate, correspond and represent [Dedekind] |
| Full Idea: If we scrutinize closely what is done in counting an aggregate of things, we see the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, without which no thinking is possible. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], Pref) | |
| A reaction: I don't suppose it occurred to Dedekind that he was reasserting Hume's observation about the fundamental psychology of thought. Is the origin of our numerical ability of philosophical interest? |
| 9826 | A system S is said to be infinite when it is similar to a proper part of itself [Dedekind] |
| Full Idea: A system S is said to be infinite when it is similar to a proper part of itself. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], V.64) |
| 13508 | Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD] |
| Full Idea: Dedekind's natural numbers: an object is in a set (0 is a number), a function sends the set one-one into itself (numbers have unique successors), the object isn't a value of the function (it isn't a successor), plus induction. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by William D. Hart - The Evolution of Logic 5 | |
| A reaction: Hart notes that since this refers to sets of individuals, it is a second-order account of numbers, what we now call 'Second-Order Peano Arithmetic'. |
| 18096 | Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock] |
| Full Idea: Dedekind's idea is that the set of natural numbers has zero as a member, and also has as a member the successor of each of its members, and it is the smallest set satisfying this condition. It is the intersection of all sets satisfying the condition. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by David Bostock - Philosophy of Mathematics 4.4 |
| 18841 | Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind] |
| Full Idea: It is Dedekind's categoricity result that convinces most of us that he has articulated our implicit conception of the natural numbers, since it entitles us to speak of 'the' domain (in the singular, up to isomorphism) of natural numbers. | |
| From: comment on Richard Dedekind (Nature and Meaning of Numbers [1888]) by Ian Rumfitt - The Boundary Stones of Thought 9.1 | |
| A reaction: The main rival is set theory, but that has an endlessly expanding domain. He points out that Dedekind needs second-order logic to achieve categoricity. Rumfitt says one could also add to the 1st-order version that successor is an ancestral relation. |
| 14130 | Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell] |
| Full Idea: Dedekind proves mathematical induction, while Peano regards it as an axiom, ...and Peano's method has the advantage of simplicity, and a clearer separation between the particular and the general propositions of arithmetic. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - The Principles of Mathematics §241 |
| 8924 | Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride] |
| Full Idea: Dedekind is the philosopher-mathematician with whom the structuralist conception originates. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], §3 n13) by Fraser MacBride - Structuralism Reconsidered | |
| A reaction: Hellman says the idea grew naturally out of modern mathematics, and cites Hilbert's belief that furniture would do as mathematical objects. |
| 9153 | Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K] |
| Full Idea: Dedekindian abstraction says mathematical objects are 'positions' in a model, while Cantorian abstraction says they are the result of abstracting on structurally similar objects. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §6 | |
| A reaction: The key debate among structuralists seems to be whether or not they are committed to 'objects'. Fine rejects the 'austere' version, which says that objects have no properties. Either version of structuralism can have abstraction as its basis. |
| 24029 | Among the simples are the graspable negations, such as rest and instants [Descartes] |
| Full Idea: Among the simple things, we must also place their negation and deprivation, insofar as they fall under out intelligence, because the idea of nothingness, of the instant, of rest, is no less true an idea than that of existence, of duration, of motion. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 12) | |
| A reaction: He sees the 'simple' things as the foundation of all knowledge, because they are self-evident. Not sure about 'no less true', since the specific nothings are parasitic on the somethings. |
| 9825 | A thing is completely determined by all that can be thought concerning it [Dedekind] |
| Full Idea: A thing (an object of our thought) is completely determined by all that can be affirmed or thought concerning it. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], I.1) | |
| A reaction: How could you justify this as an observation? Why can't there be unthinkable things (even by God)? Presumably Dedekind is offering a stipulative definition, but we may then be confusing epistemology with ontology. |
| 24030 | 3+4=7 is necessary because we cannot conceive of seven without including three and four [Descartes] |
| Full Idea: When I say that four and three make seven, this connection is necessary, because one cannot conceive the number seven distinctly without including in it in a confused way the number four and the number three. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 12) | |
| A reaction: This seems to make the truths of arithmetic conceptual, and hence analytic. |
| 24019 | If we accept mere probabilities as true we undermine our existing knowledge [Descartes] |
| Full Idea: It is better never to study than to be unable to distinguish the true from the false, and be obliged to accept as certain what is doubtful. One risks losing the knowledge one already has. Hence we reject all those knowledges which are only probable. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 02) | |
| A reaction: This is usually seen nowadays (and I agree) that this is a false dichotomy. Knowledge can't be all-or-nothing. We should accept probabilities as probable, not as knowledge. Probability became a science after Descartes. |
| 24031 | When Socrates doubts, he know he doubts, and that truth is possible [Descartes] |
| Full Idea: If Socrates says he doubts everything, it necessarily follows that he at least understands that he doubts, and that he knows that something can be true or false: for these are notions that necessarily accompany doubt. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 12) | |
| A reaction: An early commitment to the Cogito. But note that the inescapable commitment is not just to his existence, but also to his own reasoning, and his own commitment, and to the possibility of truth. Many, many things are undeniable. |
| 24020 | We all see intuitively that we exist, where intuition is attentive, clear and distinct rational understanding [Descartes] |
| Full Idea: By intuition I mean the conception of an attentive mind, so distinct and clear that it has no doubt about what it understands, …a conception that is borne of the sole light of reason. Thus everyone can see intuitively that he exists. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 03) | |
| A reaction: By 'intuition' he means self-evident certainty, whereas my concept is of a judgement of which I am reasonably confident, but without sufficient grounds for certainty. This is an early assertion of the Cogito, with a clear statement of its grounding. |
| 24025 | Clear and distinct truths must be known all at once (unlike deductions) [Descartes] |
| Full Idea: We require two conditions for intuition, namely that the proposition appear clear and distinct, and then that it be understood all at once and not successively. Deduction, on the other hand, implies a certain movement of the mind. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 11) | |
| A reaction: A nice distinction. Presumably with deduction you grasp each step clearly, and then the inference and conclusion, and you can then forget the previous steps because you have something secure. |
| 24022 | Our souls possess divine seeds of knowledge, which can bear spontaneous fruit [Descartes] |
| Full Idea: The human soul possesses something divine in which are deposited the first seeds of useful knowledge, which, in spite of the negligence and embarrassment of poorly done studies, bear spontaneous fruit. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 04) | |
| A reaction: This makes clear the religious underpinning which is required for his commitment to such useful innate ideas (such as basic geometry) |
| 24034 | If someone had only seen the basic colours, they could deduce the others from resemblance [Descartes] |
| Full Idea: Let there be a man who has sometimes seen the fundamental colours, and never the intermediate and mixed colours; it may be that by a sort of deduction he will represent those he has not seen, by their resemblance to the others. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 14) | |
| A reaction: Thus Descartes solved Hume's shade of blue problem, by means of 'a sort of deduction' from resemblance, where Hume was paralysed by his need to actually experience it. Dogmatic empiricism is a false doctrine! |
| 24021 | The method starts with clear intuitions, followed by a process of deduction [Descartes] |
| Full Idea: If the method shows clearly how we must use intuition to avoid mistaking the false for the true, and how deduction must operate to lead us to the knowledge of all things, it will be complete in my opinion. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 04) | |
| A reaction: A perfect statement of his foundationalist view. It needs a clear and distinct basis, and the steps of building must be strictly logical. Of course, most of our knowledge relies on induction, rather than deduction. |
| 24027 | Nerves and movement originate in the brain, where imagination moves them [Descartes] |
| Full Idea: The motive power or the nerves themselves originate in the brain, which contains the imagination, which moves them in a thousand ways, as the common sense is moved by the external sense. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 12) | |
| A reaction: This sounds a lot more physicalist than his later explicit dualism in Meditations. Even in that work the famous passage on the ship's pilot acknowledged tight integration of mind and brain. |
| 24026 | Our four knowledge faculties are intelligence, imagination, the senses, and memory [Descartes] |
| Full Idea: There are four faculties in us which we can use to know: intelligence, imagination, the senses, and memory. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 12) | |
| A reaction: Philosophers have to attribute faculties to the mind, even if the psychologists and neuroscientists won't accept them. We must infer the sources of our modes of understanding. He is cautious about imagination. |
| 24028 | The force by which we know things is spiritual, and quite distinct from the body [Descartes] |
| Full Idea: This force by which we properly know objects is purely spiritual, and is no less distinct from the body than is the blood from the bones. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 12) | |
| A reaction: This firmly contradicts any physicalism I thought I detected in Idea 24027! He uses the word 'spiritual' of the mind here, which I don't think he uses in later writings. |
| 9189 | Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dedekind, by Dummett] |
| Full Idea: By applying the operation of abstraction to a system of objects isomorphic to the natural numbers, Dedekind believed that we obtained the abstract system of natural numbers, each member having only properties consequent upon its position. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Michael Dummett - The Philosophy of Mathematics | |
| A reaction: Dummett is scornful of the abstractionism. He cites Benacerraf as a modern non-abstractionist follower of Dedekind's view. There seems to be a suspicion of circularity in it. How many objects will you abstract from to get seven? |
| 9827 | We derive the natural numbers, by neglecting everything of a system except distinctness and order [Dedekind] |
| Full Idea: If in an infinite system, set in order, we neglect the special character of the elements, simply retaining their distinguishability and their order-relations to one another, then the elements are the natural numbers, created by the human mind. | |
| From: Richard Dedekind (Nature and Meaning of Numbers [1888], VI.73) | |
| A reaction: [compressed] This is the classic abstractionist view of the origin of number, but with the added feature that the order is first imposed, so that ordinals remain after the abstraction. This, of course, sounds a bit circular, as well as subjective. |
| 9979 | Dedekind has a conception of abstraction which is not psychologistic [Dedekind, by Tait] |
| Full Idea: Dedekind's conception is psychologistic only if that is the only way to understand the abstraction that is involved, which it is not. | |
| From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by William W. Tait - Frege versus Cantor and Dedekind IV | |
| A reaction: This is a very important suggestion, implying that we can retain some notion of abstractionism, while jettisoning the hated subjective character of private psychologism, which seems to undermine truth and logic. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 24023 | All the sciences searching for order and measure are related to mathematics [Descartes] |
| Full Idea: I have discovered that all the sciences which have as their aim the search for order and measure are related to mathematics. | |
| From: René Descartes (Rules for the Direction of the Mind [1628], 04) | |
| A reaction: Note that he sound a more cautious note than Galileo's famous remark. It leaves room for biology to still be a science, even when it fails to be mathematical. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |