69 ideas
| 17663 | If you know what it is, investigation is pointless. If you don't, investigation is impossible [Armstrong] |
| Full Idea: Paradox of Analysis:if we ask what sort of thing an X is, then either we know what an X is or we do not. If we know then there is no need to ask the question. If we do not know then there is no way to begin the investigation. It's pointless or impossible | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.2) | |
| A reaction: [G.E. Moore is the source of this, somewhere] Plato worried that to get to know something you must already know it. Solving this requires the concept of a 'benign' circularity. |
| 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. |
| 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? |
| 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. |
| 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. |
| 17688 | Negative facts are supervenient on positive facts, suggesting they are positive facts [Armstrong] |
| Full Idea: Negative facts appear to be supervenient upon the positive facts, which suggests that they are nothing more than the positive facts. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 10.3) |
| 17691 | Nothing is genuinely related to itself [Armstrong] |
| Full Idea: I believe that nothing is genuinely related to itself. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 10.7) |
| 17679 | All instances of some property are strictly identical [Armstrong] |
| Full Idea: A property ...is something which is strictly identical, strictly the same, in all its different instances. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.2) | |
| A reaction: Some is gravitation one property, or an infinity of properties, for each of its values? What is the same between objects of different mass. I sort of believe in all the masses, but I'm not sure what 'mass' is. Abstraction, say I. |
| 12677 | Armstrong holds that all basic properties are categorical [Armstrong, by Ellis] |
| Full Idea: I am against Armstrong's strong categoricalism, that is, the thesis that all basic properties are categorical. | |
| From: report of David M. Armstrong (What is a Law of Nature? [1983]) by Brian Ellis - The Metaphysics of Scientific Realism 3 | |
| A reaction: I certainly agree with this, as I cannot see where the power would come from to get the whole thing off the ground. Armstrong depends on universals to necessitate what happens, which I find very peculiar. |
| 17666 | Actualism means that ontology cannot contain what is merely physically possible [Armstrong] |
| Full Idea: Actualism ...debars us from admitting into our ontology the merely possible, not only the merely logically possible, but also the merely physically possible. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.3) | |
| A reaction: This is the big metaphysical question for fans (like myself) of 'powers' in nature. Armstrong declares himself an Actualist. I take it as obvious that the actual world contains powers, but how are we to characterise them? |
| 17667 | Dispositions exist, but their truth-makers are actual or categorical properties [Armstrong] |
| Full Idea: It is not denied that statements attributing dispositions and/or powers to objects are often true. But the truth-makers or ontological ground for such statements must always be found in the actual, or categorical, properties of the objects involved. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.3) | |
| A reaction: This is the big debate in the topic of powers. I love powers, but you always think there must be 'something' which has the power. Could reality entirely consist of powers? See Fetzer. |
| 17687 | If everything is powers there is a vicious regress, as powers are defined by more powers [Armstrong] |
| Full Idea: I believe reducing all universals to powers is involved in vicious regress. The power is what it is by the sort of actualisations it gives rise to in suitable sorts of circumstances. But they themselves can be nothing but powers... | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 08.3) | |
| A reaction: [compressed wording] I don't see this problem. Anything postulated as fundamental is going to be baffling. Why are categorical properties superior to powers? Postulate basic powers (or basic empowered stuff), then build up. |
| 17678 | Universals are just the repeatable features of a world [Armstrong] |
| Full Idea: Universals can be brought into the spatio-temporal world, becoming simply the repeatable features of that world. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.2) | |
| A reaction: I wish Armstrong wouldn't use the word 'universal', which has so much historical baggage. The world obviously has repeatable features, but does that mean that our ontology must include things called 'features'? Hm. |
| 17669 | Realist regularity theories of laws need universals, to pick out the same phenomena [Armstrong] |
| Full Idea: A Realistic version of a Regularity theory of laws will have to postulate universals. How else will it be possible to say that the different instances of a certain uniformity are all instances of objectively the same phenomenon? | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 02.4) | |
| A reaction: I disagree. We may (or may not) need properties, but they can be have a range. We just need stable language. We use one word 'red', even when the shade of redness varies. Non-realists presumably refer to sense-data. |
| 17677 | Past, present and future must be equally real if universals are instantiated [Armstrong] |
| Full Idea: Past, present and future I take to be all and equally real. A universal need not be instantiated now. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.2) | |
| A reaction: This is the price you must pay for saying that you only believe in universals which are instantiated. |
| 15442 | Universals are abstractions from their particular instances [Armstrong, by Lewis] |
| Full Idea: Armstrong takes universals generally, and structural universals along with the rest, to be abstractions from their particular instances. | |
| From: report of David M. Armstrong (What is a Law of Nature? [1983], p.83-4) by David Lewis - Against Structural Universals 'The pictorial' | |
| A reaction: To me, 'abstracted' implies a process of human psychology, a way of thinking about the instances. I don't see how there can be an 'abstracted' relation which is a part of the external world. That makes his laws of nature human creations. |
| 17686 | Universals are abstractions from states of affairs [Armstrong] |
| Full Idea: Universals are abstractions from states of affairs. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 7) | |
| A reaction: I'm getting confused about Armstrong's commitments. He bases his whole theory on the existence of universals (repeatable features), but now says those are 'abstracted' from something else. Abstracted by us? |
| 17668 | It is likely that particulars can be individuated by unique conjunctions of properties [Armstrong] |
| Full Idea: For each particular it is likely that there exists at least one individuating conjunction of properties, that is, a conjunction of properties such that the particular instantiates this conjunction and nothing else does. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 02.3) | |
| A reaction: Armstrong commits to a famous Leibniz view, but I don't see his grounds for it. There is nothing incoherent about nature churning out perfect replicas of things, such as quarks and electrons. Would we care if two pens were perfectly identical? |
| 17680 | The identity of a thing with itself can be ruled out as a pseudo-property [Armstrong] |
| Full Idea: There is reason to rule out as pseudo-properties such things as the identity of a thing with itself. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.2) | |
| A reaction: Good on you, David. |
| 17693 | The necessary/contingent distinction may need to recognise possibilities as real [Armstrong] |
| Full Idea: It may be that the necessary/contingent distinction is tied to a metaphysics which recognises possibility as a real something wider than actuality. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 11.2) | |
| A reaction: Armstrong responds by trying to give an account of possibility in terms of 'combinations' from actuality. I think powers offer a much better strategy. |
| 17685 | Induction aims at 'all Fs', but abduction aims at hidden or theoretical entities [Armstrong] |
| Full Idea: Many philosophers of science have distinguished between 'simple induction' - the argument from observed Fs to all Fs - and the argument to hidden or theoretical entities (Peirce's 'abduction'). | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.7) | |
| A reaction: 'Abduction' is (roughly) the same is inference to the best explanation, of which I am a great fan. |
| 17683 | Science suggests that the predicate 'grue' is not a genuine single universal [Armstrong] |
| Full Idea: It is plausible to say, on the basis of total science, that 'grue' is a predicate to which no genuine, that is, unitary, universal corresponds. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.7) |
| 17675 | Unlike 'green', the 'grue' predicate involves a time and a change [Armstrong] |
| Full Idea: The predicate 'grue' involves essential reference to a particular time, which 'green' does not. Also on the 'grue' hypothesis a change occurs in emeralds in a way that change does not occur on the 'green' hypothesis. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 04.5) | |
| A reaction: I'm inclined to think that comparing 'grue' with 'green' is a category mistake. 'Grue' is a behaviour. Armstrong says this is no objection, because Goodman's argument is purely formal. |
| 17674 | The raven paradox has three disjuncts, confirmed by confirming any one of them [Armstrong] |
| Full Idea: We could rewrite the generalisation as For all x, ((x is a raven and x is black) v (x is not a raven and x is black) v (x is not a raven and x is not black)). Instances of any one of the three disjuncts will do as confirmation. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 04.3) | |
| A reaction: A nice clarification. |
| 17672 | A good reason for something (the smoke) is not an explanation of it (the fire) [Armstrong] |
| Full Idea: A good reason for P is not necessarily an explanation of P. The presence of smoke is a good reason for thinking that fire is present. But it is not an explanation of the presence of fire. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 04.2) | |
| A reaction: This may be an equivocation on 'the reason for'. Smoke is a reason for thinking there is a fire, but no one would propose it as a reason for the fire. If the reason for the fire was arson, that would seem to explain it as well. |
| 17684 | To explain observations by a regular law is to explain the observations by the observations [Armstrong] |
| Full Idea: Given the Regularity theory, the explanatory element seems to vanish. For to say that all the observed Fs are Gs because all the Fs are Gs involves explaining the observations in terms of themselves. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.7) | |
| A reaction: This point cries out, it is so obvious (once spotted). Tigers are ferocious because all tigers are ferocious (see?). |
| 17676 | Best explanations explain the most by means of the least [Armstrong] |
| Full Idea: The best explanation explains the most by means of the least. Explanation unifies. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 05.4) | |
| A reaction: To get unification, you need to cite the diversity of what is explained, and not the mere quantity. The force of gravity unifies because it applies to such a diversity of things. |
| 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. |
| 17664 | Each subject has an appropriate level of abstraction [Armstrong] |
| Full Idea: To every subject, its appropriate level of abstraction. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.2) | |
| A reaction: Mathematics rises through many levels of abstraction. Economics can be very concrete or very abstract. It think it is clearer to talk of being 'general', rather than 'abstract'. |
| 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. |
| 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). |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 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. |
| 17692 | We can't deduce the phenomena from the One [Armstrong] |
| Full Idea: No serious and principled deduction of the phenomena from the One has ever been given, or looks likely to be given. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 11) | |
| A reaction: This seems to pick out the best reason why hardly anybody (apart from Jonathan Schaffer) takes the One seriously. |
| 17689 | Absences might be effects, but surely not causes? [Armstrong] |
| Full Idea: Lacks and absences could perhaps by thought of as effects, but we ought to be deeply reluctant to think of them as causes. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 10.4) | |
| A reaction: Odd. So we allow that they exist (as effects), but then deny that they have any causal powers? |
| 17682 | A universe couldn't consist of mere laws [Armstrong] |
| Full Idea: A universe could hardly consist of laws and nothing else. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.4) | |
| A reaction: Hm. Discuss. How does a universe come into existence, if there are no laws to guide its creation? |
| 17662 | Science depends on laws of nature to study unobserved times and spaces [Armstrong] |
| Full Idea: The scientist trying to establish the geography and history of the unobserved portion of the universe must depend upon what he takes to be the laws of the universe. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.1) | |
| A reaction: This does seem to be the prime reason why we wish to invoke 'laws', but we could just as well say that we have to rely on induction. Spot patterns, then expect more of the same. Spot necessities? Mathematics is very valuable here, of course. |
| 17690 | Oaken conditional laws, Iron universal laws, and Steel necessary laws [Armstrong, by PG] |
| Full Idea: Three degress of law: 1) 'Oaken laws' where all Fs that aren't Hs are Gs; 2) 'Iron' laws where all Fs are Gs; and 3) 'Steel' laws where all Fs must be Gs. | |
| From: report of David M. Armstrong (What is a Law of Nature? [1983], 10.4) by PG - Db (ideas) | |
| A reaction: [My summary of Armstrong's distinction] One response is to say that all laws are actually Oaken - see Mumfor and Mumford/Lill Anjum. It's all ceteris paribus. |
| 17670 | Newton's First Law refers to bodies not acted upon by a force, but there may be no such body [Armstrong] |
| Full Idea: Newton's First Law of Motion tells us what happens to a body which is not acted upon by a force. Yet it may be that the antecedent of the law is never instantiated. It may be that every body that there is, is acted upon by some force. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 02.7) |
| 8582 | Regularities are lawful if a second-order universal unites two first-order universals [Armstrong, by Lewis] |
| Full Idea: Armstrong's theory holds that what makes certain regularities lawful are second-order states of affairs N(F,G) in which the two ordinary first-order universals F and G are related by a certain dyadic second-order universal N. | |
| From: report of David M. Armstrong (What is a Law of Nature? [1983]) by David Lewis - New work for a theory of universals 'Laws and C' | |
| A reaction: [see Lewis's footnote] I take the view (from Shoemaker and Ellis) that laws of nature are just plain regularities which arise from the hierarchy of natural kinds. We don't need a commitment to 'universals'. |
| 17671 | A naive regularity view says if it never occurs then it is impossible [Armstrong] |
| Full Idea: It is a Humean uniformity that no race of ravens is white-feathered. Hence, if the Naive Regularity analysis of law is correct, it is a law that no race of ravens is white-feathered, that is, such a race is physically impossible. A most unwelcome result. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 02.6) | |
| A reaction: Chapters 2-4 of Armstrong are a storming attack on the regularity view of laws of nature, and this idea is particularly nice. Laws must refer to what could happen, not what happens to happen. |
| 17681 | The laws of nature link properties with properties [Armstrong] |
| Full Idea: There is an utterly natural idea that the laws of nature link properties with properties. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.3) | |
| A reaction: Put it this way: given that properties are expressions of invariant powers, the interaction of two properties will (ceteris paribus) be invariant, and laws are just invariances in natural behaviour. |
| 16246 | Rather than take necessitation between universals as primitive, just make laws primitive [Maudlin on Armstrong] |
| Full Idea: My own view is simple: the laws of nature ought to be accepted as ontologically primitive. …They are preferable in point of familiarity to such necessitation relations between universals. | |
| From: comment on David M. Armstrong (What is a Law of Nature? [1983]) by Tim Maudlin - The Metaphysics within Physics 1.4 | |
| A reaction: I think you make natures of things primitive, and reduce laws to regularities and universals to resemblances. Job done. Natures are even more 'familiar' as primitives than laws are. |
| 9480 | Armstrong has an unclear notion of contingent necessitation, which can't necessitate anything [Bird on Armstrong] |
| Full Idea: The two criticisms levelled against Armstrong are that it is unclear what his relation of contingent necessitation is, and that it is unclear how it is able to necessitate anything. | |
| From: comment on David M. Armstrong (What is a Law of Nature? [1983]) by Alexander Bird - Nature's Metaphysics 3.1.2 | |
| A reaction: I suppose someone has to explore the middle ground between the mere contingencies of Humean regularities and the strong necessities of scientific essentialism. The area doesn't, however, look promising. |