59 ideas
| 4697 | There has been a distinct 'Social Turn' in recent philosophy, like the earlier 'Linguistic Turn' [O'Grady] |
| Full Idea: The Social Turn is as defining a characteristic of contemporary philosophy as the Linguistic Turn has been of the earlier twentieth century period. | |
| From: Paul O'Grady (Relativism [2002], Ch.1) | |
| A reaction: A helpful observation. It ties in with externalism about concepts (Twin Earth), impossibility of Private Language, and externalism about knowledge. |
| 4731 | Good reasoning will avoid contradiction, enhance coherence, not ignore evidence, and maximise evidence [O'Grady] |
| Full Idea: The four basic principles of rationality are 1) avoid contradiction, 2) enhance coherence, 3) avoid ignoring evidence, and 4) maximise evidence. | |
| From: Paul O'Grady (Relativism [2002], Ch.5) | |
| A reaction: I like this, and can't think of any additions. 'Coherence' is the vaguest of the conditions. Maximising evidence is still the driving force of science, even if it does sound quaintly positivist. |
| 9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
| Full Idea: The standard requirement of definitions involves 'eliminability' (any defined terms must be replaceable by primitives) and 'non-creativity' (proofs of theorems should not depend on the definition). | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 7) | |
| A reaction: [He cites Russell and Whitehead as a source for this view] This is the austere view of the mathematician or logician. But almost every abstract concept that we use was actually defined in a creative way. |
| 4735 | Just as maps must simplify their subject matter, so thought has to be reductionist about reality [O'Grady] |
| Full Idea: A map that is identical in all respects with that which is mapped is just useless. So reductionism is not just a good thing - it is essential to thought. | |
| From: Paul O'Grady (Relativism [2002], Ch.6) | |
| A reaction: A useful warning, when thinking about truth. It is folly to want your thoughts to exactly correspond to reality. I want to understand the world, but not if it requires being the world. |
| 4703 | The epistemic theory of truth presents it as 'that which is licensed by our best theory of reality' [O'Grady] |
| Full Idea: The epistemic theory of truth presents it as 'that which is licensed by our best theory of reality'. | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: Dangerous nonsense. This leaves truth shifting as our theories change, it leads to different truths in different cultures, and no palpable falsehood in ignorant cultures. Don't give it house-room. |
| 4701 | To say a relative truth is inexpressible in other frameworks is 'weak', while saying it is false is 'strong' [O'Grady] |
| Full Idea: Weak alethic relativism holds that while a statement may be true in one framework, it is inexpressible in another. Strong alethic relativism is where a sentence is true relative to one framework, but false relative to another. | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: The weak version will be Kuhn's 'incommensurability' of scientific theories, while the strong version will be full Protagorean relativism, saying all beliefs are true. |
| 9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
| Full Idea: The set-theory account of infinity doesn't just say that we can keep on counting, but that the natural numbers are an actual infinite set. This is necessary to make sense of the powerset of ω, as the set of all its subsets, and thus even bigger. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5) | |
| A reaction: I don't personally find this to be sufficient reason to commit myself to the existence of actual infinities. In fact I have growing doubts about the whole role of set theory in philosophy of mathematics. Shows how much I know. |
| 9613 | Naïve set theory assumed that there is a set for every condition [Brown,JR] |
| Full Idea: In the early versions of set theory ('naïve' set theory), the axiom of comprehension assumed that for any condition there is a set of objects satisfying that condition (so P(x)↔x∈{x:P(x)}), but this led directly to Russell's Paradox. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
| A reaction: How rarely any philosophers state this problem clearly (as Brown does here). This is incredibly important for our understanding of how we classify the world. I'm tempted to just ignore Russell, and treat sets in a natural and sensible way. |
| 9615 | Nowadays conditions are only defined on existing sets [Brown,JR] |
| Full Idea: In current set theory Russell's Paradox is avoided by saying that a condition can only be defined on already existing sets. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
| A reaction: A response to Idea 9613. This leaves us with no account of how sets are created, so we have the modern notion that absolutely any grouping of daft things is a perfectly good set. The logicians seem to have hijacked common sense. |
| 9617 | The 'iterative' view says sets start with the empty set and build up [Brown,JR] |
| Full Idea: The modern 'iterative' concept of a set starts with the empty set φ (or unsetted individuals), then uses set-forming operations (characterized by the axioms) to build up ever more complex sets. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
| A reaction: The only sets in our system will be those we can construct, rather than anything accepted intuitively. It is more about building an elaborate machine that works than about giving a good model of reality. |
| 9642 | A flock of birds is not a set, because a set cannot go anywhere [Brown,JR] |
| Full Idea: Neither a flock of birds nor a pack of wolves is strictly a set, since a flock can fly south, and a pack can be on the prowl, whereas sets go nowhere and menace no one. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 7) | |
| A reaction: To say that the pack menaced you would presumably be to commit the fallacy of composition. Doesn't the number 64 have properties which its set-theoretic elements (whatever we decide they are) will lack? |
| 4705 | Logical relativism appears if we allow more than one legitimate logical system [O'Grady] |
| Full Idea: Logical relativism emerges if one defends the existence of two or more rival systems that one may legitimately choose between, or move back and forth between. | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: All my instincts rebel against this possibility. All of Aristotle's and Kant's philosophy would be rendered meaningless. Obviously you can create artificial logics (like games), but I believe there is a truth logic. (Pathetic, isn't it?) |
| 4700 | A third value for truth might be "indeterminate", or a point on a scale between 'true' and 'false' [O'Grady] |
| Full Idea: Suggestions for a third value for truth are "indeterminate", or a scale running from "true", through "mostly true", "mainly true", "half true", "mainly false", "mostly false", to "false", or maybe even "0.56 true". | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: Anything on a sliding scale sounds wrong, as it seems to be paracitic on an underlying fixed idea of 'true'. "Indeterminate", though, seems just right for the truth of predictions ('sea-fight tomorrow'). |
| 9605 | If a proposition is false, then its negation is true [Brown,JR] |
| Full Idea: The law of excluded middle says if a proposition is false, then its negation is true | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1) | |
| A reaction: Surely that is the best statement of the law? How do you write that down? ¬(P)→¬P? No, because it is a semantic claim, not a syntactic claim, so a truth table captures it. Semantic claims are bigger than syntactic claims. |
| 4704 | Wittgenstein reduced Russell's five primitive logical symbols to a mere one [O'Grady] |
| Full Idea: While Russell and Whitehead used five primitive logical symbols in their system, Wittgenstein suggested in his 'Tractatus' that this be reduced to one. | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: This certainly captures why Russell was so impressed by him. In retrospect what looked like progress presumably now looks like the beginning of the collapse of the enterprise. |
| 9649 | Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR] |
| Full Idea: The three views one could adopt concerning axioms are that they are self-evident truths, or that they are arbitrary stipulations, or that they are fallible attempts to describe how things are. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch.10) | |
| A reaction: Presumably modern platonists like the third version, with others choosing the second, and hardly anyone now having the confidence to embrace the first. |
| 9638 | Berry's Paradox finds a contradiction in the naming of huge numbers [Brown,JR] |
| Full Idea: Berry's Paradox refers to 'the least integer not namable in fewer than nineteen syllables' - a paradox because it has just been named in eighteen syllables. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5) | |
| A reaction: Apparently George Boolos used this quirky idea as a basis for a new and more streamlined proof of Gödel's Theorem. Don't tell me you don't find that impressive. |
| 9604 | Mathematics is the only place where we are sure we are right [Brown,JR] |
| Full Idea: Mathematics seems to be the one and only place where we humans can be absolutely sure that we got it right. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1) | |
| A reaction: Apart from death and taxes, that is. Personally I am more certain of the keyboard I am typing on than I am of Pythagoras's Theorem, but the experts seem pretty confident about the number stuff. |
| 9622 | 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR] |
| Full Idea: 'There are two apples' can be recast as 'x is an apple and y is an apple, and x isn't y, and if z is an apple it is the same as x or y', which makes no appeal at all to mathematics. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
| A reaction: He cites this as the basis of Hartry Field's claim that science can be done without numbers. The logic is ∃x∃y∀z(Ax&Ay&(x¬=y)&(Az→z=x∨z=y)). |
| 9648 | π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR] |
| Full Idea: The number π is not only irrational, but it is also (unlike √2) a 'transcendental' number, because it is not the solution of an algebraic equation. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch.10) | |
| A reaction: So is that a superficial property, or a profound one? Answers on a post card. |
| 9621 | Mathematics represents the world through structurally similar models. [Brown,JR] |
| Full Idea: Mathematics hooks onto the world by providing representations in the form of structurally similar models. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
| A reaction: This is Brown's conclusion. It needs notions of mapping, one-to-one correspondence, and similarity. I like the idea of a 'model', as used in both logic and mathematics, and children's hobbies. The mind is a model-making machine. |
| 9646 | There is no limit to how many ways something can be proved in mathematics [Brown,JR] |
| Full Idea: I'm tempted to say that mathematics is so rich that there are indefinitely many ways to prove anything - verbal/symbolic derivations and pictures are just two. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 9) | |
| A reaction: Brown has been defending pictures as a form of proof. I wonder how long his list would be, if we challenged him to give more details? Some people have very low standards of proof. |
| 9647 | Computers played an essential role in proving the four-colour theorem of maps [Brown,JR] |
| Full Idea: The celebrity of the famous proof in 1976 of the four-colour theorem of maps is that a computer played an essential role in the proof. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch.10) | |
| A reaction: The problem concerns the reliability of the computers, but then all the people who check a traditional proof might also be unreliable. Quis custodet custodies? |
| 9643 | Set theory may represent all of mathematics, without actually being mathematics [Brown,JR] |
| Full Idea: Maybe all of mathematics can be represented in set theory, but we should not think that mathematics is set theory. Functions can be represented as order pairs, but perhaps that is not what functions really are. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 7) | |
| A reaction: This seems to me to be the correct view of the situation. If 2 is represented as {φ,{φ}}, why is that asymmetrical? The first digit seems to be the senior and original partner, but how could the digits of 2 differ from one another? |
| 9644 | When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR] |
| Full Idea: The basic definition of a graph can be given in set-theoretic terms,...but then what could an unlabelled graph be? | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 7) | |
| A reaction: An unlabelled graph will at least need a verbal description for it to have any significance at all. My daily mood-swings look like this.... |
| 9625 | To see a structure in something, we must already have the idea of the structure [Brown,JR] |
| Full Idea: Epistemology is a big worry for structuralists. ..To conjecture that something has a particular structure, we must already have conceived of the idea of the structure itself; we cannot be discovering structures by conjecturing them. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
| A reaction: This has to be a crucial area of discussion. Do we have our heads full of abstract structures before we look out of the window? Externalism about the mind is important here; mind and world are not utterly distinct things. |
| 9628 | Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR] |
| Full Idea: Set theory is at the very heart of mathematics; it may even be all there is to mathematics. The notion of set, however, seems quite contrary to the spirit of structuralism. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
| A reaction: So much the worse for sets, I say. You can, for example, define ordinality in terms of sets, but that is no good if ordinality is basic to the nature of numbers, rather than a later addition. |
| 9606 | The irrationality of root-2 was achieved by intellect, not experience [Brown,JR] |
| Full Idea: We could not discover irrational numbers by physical measurement. The discovery of the irrationality of the square root of two was an intellectual achievement, not at all connected to sense experience. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1) | |
| A reaction: Brown declares himself a platonist, and this is clearly a key argument for him, and rather a good one. Hm. I'll get back to you on this one... |
| 9612 | There is an infinity of mathematical objects, so they can't be physical [Brown,JR] |
| Full Idea: A simple argument makes it clear that all mathematical arguments are abstract: there are infinitely many numbers, but only a finite number of physical entities, so most mathematical objects are non-physical. The best assumption is that they all are. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
| A reaction: This, it seems to me, is where constructivists score well (cf. Idea 9608). I don't have an infinity of bricks to build an infinity of houses, but I can imagine that the bricks just keep coming if I need them. Imagination is what is unbounded. |
| 9610 | Numbers are not abstracted from particulars, because each number is a particular [Brown,JR] |
| Full Idea: Numbers are not 'abstract' (in the old sense, of universals abstracted from particulars), since each of the integers is a unique individual, a particular, not a universal. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
| A reaction: An interesting observation which I have not seen directly stated before. Compare Idea 645. I suspect that numbers should be thought of as higher-order abstractions, which don't behave like normal universals (i.e. they're not distributed). |
| 9620 | Empiricists base numbers on objects, Platonists base them on properties [Brown,JR] |
| Full Idea: Perhaps, instead of objects, numbers are associated with properties of objects. Basing them on objects is strongly empiricist and uses first-order logic, whereas the latter view is somewhat Platonistic, and uses second-order logic. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4) | |
| A reaction: I don't seem to have a view on this. You can count tomatoes, or you can count red objects, or even 'instances of red'. Numbers refer to whatever can be individuated. No individuation, no arithmetic. (It's also Hume v Armstrong on laws on nature). |
| 9639 | Does some mathematics depend entirely on notation? [Brown,JR] |
| Full Idea: Are there mathematical properties which can only be discovered using a particular notation? | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 6) | |
| A reaction: If so, this would seem to be a serious difficulty for platonists. Brown has just been exploring the mathematical theory of knots. |
| 9629 | For nomalists there are no numbers, only numerals [Brown,JR] |
| Full Idea: For the instinctive nominalist in mathematics, there are no numbers, only numerals. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5) | |
| A reaction: Maybe. A numeral is a specific sign, sometimes in a specific natural language, so this seems to miss the fact that cardinality etc are features of reality, not just conventions. |
| 9630 | The most brilliant formalist was Hilbert [Brown,JR] |
| Full Idea: In mathematics, the most brilliant formalist of all was Hilbert | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5) | |
| A reaction: He seems to have developed his fully formalist views later in his career. See Mathematics|Basis of Mathematic|Formalism in our thematic section. Kreisel denies that Hilbert was a true formalist. |
| 9608 | There are no constructions for many highly desirable results in mathematics [Brown,JR] |
| Full Idea: Constuctivists link truth with constructive proof, but necessarily lack constructions for many highly desirable results of classical mathematics, making their account of mathematical truth rather implausible. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
| A reaction: The tricky word here is 'desirable', which is an odd criterion for mathematical truth. Nevertheless this sounds like a good objection. How flexible might the concept of a 'construction' be? |
| 9645 | Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR] |
| Full Idea: If we define p as '3 if Goldbach's Conjecture is true' and '5 if Goldbach's Conjecture is false', it seems that p must be a prime number, but, amazingly, constructivists would not accept this without a proof of Goldbach's Conjecture. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 8) | |
| A reaction: A very similar argument structure to Schrödinger's Cat. This seems (as Brown implies) to be a devastating knock-down argument, but I'll keep an open mind for now. |
| 9619 | David's 'Napoleon' is about something concrete and something abstract [Brown,JR] |
| Full Idea: David's painting of Napoleon (on a white horse) is a 'picture' of Napoleon, and a 'symbol' of leadership, courage, adventure. It manages to be about something concrete and something abstract. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 3) | |
| A reaction: This strikes me as the germ of an extremely important idea - that abstraction is involved in our perception of the concrete, so that they are not two entirely separate realms. Seeing 'as' involves abstraction. |
| 4711 | Anti-realists say our theories (such as wave-particle duality) give reality incompatible properties [O'Grady] |
| Full Idea: The anti-realist says we have theories about the world that are incompatible with each other, and irreducible to each other. They often cite wave-particle duality, which postulate incompatible properties to reality. | |
| From: Paul O'Grady (Relativism [2002], Ch.3) | |
| A reaction: Most physicists, of course, hate this duality, precisely because they can't conceive how the two properties could be real. I say realism comes first, and the theories must try to accommodate that assumption. |
| 4698 | What counts as a fact partly depends on the availability of human concepts to describe them [O'Grady] |
| Full Idea: What counts as a fact partly depends on human input, such as the availability of concepts to describe such facts. | |
| From: Paul O'Grady (Relativism [2002], Ch.1) | |
| A reaction: The point must be taken. I am happy to generalise about 'The Facts', meaning 'whatever is the case', but the individuation of specific facts is bound to hit the current problem. |
| 4715 | We may say that objects have intrinsic identity conditions, but still allow multiple accounts of them [O'Grady] |
| Full Idea: Those defending the claim that objects exist with identity conditions not imposed by us, do not have to say that there is just one account of those objects possible. | |
| From: Paul O'Grady (Relativism [2002], Ch.3) | |
| A reaction: This seems right, but the test question is whether the mind of God contains a single unified theory/account. Are multiple accounts the result of human inadequacy? Yes, I surmise. |
| 4719 | Maybe developments in logic and geometry have shown that the a priori may be relative [O'Grady] |
| Full Idea: A weaker form of relativism holds that developments in logic, in maths and in geometry have shown how a relativised notion of the a priori is possible. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: This is non-Euclidean geometry, and multiple formalisations of logic. Personally I don't believe it. You can expand these subjects, and pursue whimsical speculations, but I have faith in their stable natural core. Neo-Platonism. |
| 4720 | Sense-data are only safe from scepticism if they are primitive and unconceptualised [O'Grady] |
| Full Idea: The reason sense-data were immune from doubt was because they were so primitive; they were unstructured and below the level of conceptualisation. Once they were given structure and conceptualised, they were no longer safe from sceptical challenge. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: The question of whether sense-data are conceptualised doesn't have to be all-or-nothing. As concepts creep in, so does scepticism, but so what? Sensible philosophers live with scepticism, like a mad aunt in the attic. |
| 4722 | Modern epistemology centres on debates about foundations, and about external justification [O'Grady] |
| Full Idea: The two dichotomies which set the agenda in contemporary epistemology are the foundationalist-coherentist debate, and the internalist-externalist debate. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: Helpful. Roughly, foundationalists are often externalists (if they are empiricists), and coherentists are often internalists (esp. if they are rationalists). An eccentric combination would make a good PhD. |
| 4724 | Internalists say the reasons for belief must be available to the subject, and externalists deny this [O'Grady] |
| Full Idea: Internalism about justification says that the reasons one has for a belief must be in some sense available to the knowing subject, ..while externalism holds that it is possible for a person to have a justified belief without having access to the reason. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: It strikes me that internalists are talking about the believer being justified, and externalists talk about the belief being justified. I'm with the internalists. If this means cats don't know much, so much the worse for cats. |
| 4723 | Coherence involves support from explanation and evidence, and also probability and confirmation [O'Grady] |
| Full Idea: Coherentist justification is more than absence of contradictions, and will involve issues like explanatory support and evidential support, and perhaps issues about probability and confirmation too. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: Something like this is obviously essential. Is the notion of 'relevance' also needed (e.g. to avoid the raven paradox of induction)? Coherence of justification will combine with correspondence for truth. |
| 4709 | Ontological relativists are anti-realists, who deny that our theories carve nature at the joints [O'Grady] |
| Full Idea: Ontological relativists are anti-realists in the strong sense; they hold as meaningless the view that our theories carve nature at the joints. | |
| From: Paul O'Grady (Relativism [2002], Ch.3) | |
| A reaction: This pinpoints my disagreement with such relativism, as it seems obvious to me that nature has 'joints', and that we would agree with any sensible alien about lots of things. |
| 4725 | Contextualism says that knowledge is relative to its context; 'empty' depends on your interests [O'Grady] |
| Full Idea: Contextualist about knowledge say that "to know" means different things in different context. For example, a warehouse may be empty for a furniture owner, but not for a bacteriologist or a physicist. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: There is obviously some truth in this, but we might say that 'empty' is a secondary quality, or that 'empty for furniture' is not relative. We needn't accept relativism here. |
| 4732 | One may understand a realm of ideas, but be unable to judge their rationality or truth [O'Grady] |
| Full Idea: It is possible to conceive of one understanding the meaning of a realm of ideas, but holding that one cannot judge as to the truth or rationality of the claims made in it. | |
| From: Paul O'Grady (Relativism [2002], Ch.5) | |
| A reaction: I think Davidson gives good grounds for challenging this, by doubt whether one 'conceptual scheme' can know another without grasping its rationality and truth-conditions. |
| 13165 | Geometrical proofs do not show causes, as when we prove a triangle contains two right angles [Proclus] |
| Full Idea: Geometry does not ask 'why?' ..When from the exterior angle equalling two opposite interior angles it is shown that the interior angles make two right angles, this is not a causal demonstration. With no exterior angle they still equal two right angles. | |
| From: Proclus (Commentary on Euclid's 'Elements' [c.452], p.161-2), quoted by Paolo Mancosu - Explanation in Mathematics §5 | |
| A reaction: A very nice example. It is hard to imagine how one might demonstrate the cause of the angles making two right angles. If you walk, turn left x°, then turn left y°, then turn left z°, and x+y+z=180°, you end up going in the original direction. |
| 9611 | 'Abstract' nowadays means outside space and time, not concrete, not physical [Brown,JR] |
| Full Idea: The current usage of 'abstract' simply means outside space and time, not concrete, not physical. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
| A reaction: This is in contrast to Idea 9609 (the older notion of being abstracted). It seems odd that our ancestors had a theory about where such ideas came from, but modern thinkers have no theory at all. Blame Frege for that. |
| 9569 | The origin of geometry started in sensation, then moved to calculation, and then to reason [Proclus] |
| Full Idea: It is unsurprising that geometry was discovered in the necessity of Nile land measurement, since everything in the world of generation goes from imperfection to perfection. They would naturally pass from sense-perception to calculation, and so to reason. | |
| From: Proclus (Commentary on Euclid's 'Elements' [c.452]), quoted by Charles Chihara - A Structural Account of Mathematics 9.12 n55 | |
| A reaction: The last sentence is the core of my view on abstraction, that it proceeds by moving through levels of abstraction, approaching more and more general truths. |
| 9609 | The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars [Brown,JR] |
| Full Idea: The older sense of 'abstract' applies to universals, where a universal like 'redness' is abstracted from red particulars; it is the one associated with the many. In mathematics, the notion of 'group' or 'vector space' perhaps fits this pattern. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
| A reaction: I am currently investigating whether this 'older' concept is in fact dead. It seems to me that it is needed, as part of cognitive science, and as the crucial link between a materialist metaphysic and the world of ideas. |
| 4710 | Verificationism was attacked by the deniers of the analytic-synthetic distinction, needed for 'facts' [O'Grady] |
| Full Idea: Verificationism came under attack from empiricists who were friendly to the banishment of traditional metaphysics, but unfriendly to the analytic-synthetic distinction, on which the idea of a 'factual statement' depended. | |
| From: Paul O'Grady (Relativism [2002], Ch.3) | |
| A reaction: I don't accept this move because I don't consider the 'facts' to be language-dependent. They are pre-linguistic, they outrun that capacity of our language, and they are available to animals. |
| 9640 | A term can have not only a sense and a reference, but also a 'computational role' [Brown,JR] |
| Full Idea: In addition to the sense and reference of term, there is the 'computational' role. The name '2' has a sense (successor of 1) and a reference (the number 2). But the word 'two' has little computational power, Roman 'II' is better, and '2' is a marvel. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 6) | |
| A reaction: Very interesting, and the point might transfer to natural languages. Synonymous terms carry with them not just different expressive powers, but the capacity to play different roles (e.g. slang and formal terms, gob and mouth). |
| 4717 | If we abandon the analytic-synthetic distinction, scepticism about meaning may be inevitable [O'Grady] |
| Full Idea: There may be no way to avoid scepticism about meaning if you abandon the analytic-synthetic distinction in the way Quine does. | |
| From: Paul O'Grady (Relativism [2002], Ch.3) | |
| A reaction: My suspicion was always that Quine's proposal began the slippery road to hell. It appears to be pragmatists who are most drawn to Quine's idea. The proposal that all my analytic propositions could be treated as synthetic totally baffles me. |
| 4706 | Early Quine says all beliefs could be otherwise, but later he said we would assume mistranslation [O'Grady] |
| Full Idea: In his earlier work, Quine defended the view that no belief (including logic) is in principle unrevisable, but in his later work (1970) he took the conservative view that we would always impute mistranslation rather than deviancy. | |
| From: Paul O'Grady (Relativism [2002], Ch.2) | |
| A reaction: I take it he was influenced by Davidson's 'principle of charity'. He says that if someone asserts 'p and not-p', we would assume a misunderstanding of 'and' or 'not'. |
| 4734 | Cryptographers can recognise that something is a language, without translating it [O'Grady] |
| Full Idea: It makes sense to think that one could recognise that something is a language without necessarily being able to translate it; cryptographers do this all the time. | |
| From: Paul O'Grady (Relativism [2002], Ch.5) | |
| A reaction: Maybe, but cryptographers usually have a lot of context to work with. If we met extraterrestrials if might not be so clear. One can only spot patterns, and crystals have those. |
| 9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR] |
| Full Idea: There seem to be no actual infinites in the physical realm. Given the correctness of atomism, there are no infinitely small things, no infinite divisibility. And General Relativity says that the universe is only finitely large. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5) | |
| A reaction: If time was infinite, you could travel round in a circle forever. An atom has size, so it has a left, middle and right to it. Etc. They seem to be physical, so we will count those too. |
| 4727 | The chief problem for fideists is other fideists who hold contrary ideas [O'Grady] |
| Full Idea: The chief problem for fideists is other fideists who hold contrary ideas. | |
| From: Paul O'Grady (Relativism [2002], Ch.4) | |
| A reaction: The other problem is trying to find grounds for sticking to the object of one's faith, rather than changing from time to time. |