92 ideas
| 5486 | Essentialism says metaphysics can't be done by analysing unreliable language [Ellis] |
| Full Idea: The new essentialism leads to a turning away from semantic analysis as a fundamental tool for the pursuit of metaphysical aims, ..since there is no reason to think that the language we speak accurately reflects the kind of world we live in. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: The last part of that strikes me as false. We have every reason to think that a lot of our language very accurately reflects reality. It had better, because we have no plan B. We should analyse our best concepts, but not outdated, culture-laden ones. |
| 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. |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| Full Idea: While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: [The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc. |
| 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. |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do? |
| 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. |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| Full Idea: In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types. |
| 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)). |
| 10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
| Full Idea: 'Analysis' is the theory of the real numbers. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: 'Analysis' began with the infinitesimal calculus, which later built on the concept of 'limit'. A continuum of numbers seems to be required to make that work. |
| 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? |
| 10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
| Full Idea: The difficulties for a nominalistic mereological approach to arithmetic is that an infinity of physical objects are needed (space-time points? strokes?), and it must define functions, such as 'successor'. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: Many ontologically austere accounts of arithmetic are faced with the problem of infinity. The obvious non-platonist response seems to be a modal or if-then approach. To postulate infinite abstract or physical entities so that we can add 3 and 2 is mad. |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| Full Idea: A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'! |
| 10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
| Full Idea: The merits of basing an account of mathematics on set theory are that it allows for a comprehensive unified treatment of many otherwise separate branches of mathematics, and that all assumption, including existence, are explicit in the axioms. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I am forming the impression that set-theory provides one rather good model (maybe the best available) for mathematics, but that doesn't mean that mathematics is set-theory. The best map of a landscape isn't a landscape. |
| 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.... |
| 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
| Full Idea: Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent. |
| 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
| Full Idea: Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight? |
| 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
| Full Idea: The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6) | |
| A reaction: This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start. |
| 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
| Full Idea: According to 'pattern' structuralism, what we study are not the various particular isomorphic models of arithmetic, but something in addition to them: a corresponding pattern. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §7) | |
| A reaction: Put like that, we have to feel a temptation to wield Ockham's Razor. It's bad enough trying to give the structure of all the isomorphic models, without seeking an even more abstract account of underlying patterns. But patterns connect to minds.. |
| 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
| Full Idea: There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9) | |
| A reaction: I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind? |
| 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
| Full Idea: Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3) | |
| A reaction: [very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations. |
| 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
| Full Idea: It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: [compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent. |
| 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. |
| 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
| Full Idea: Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator. |
| 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
| Full Idea: Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra. |
| 10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
| Full Idea: Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity. |
| 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). |
| 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. |
| 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. |
| 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. |
| 5468 | Properties are 'dispositional', or 'categorical' (the latter as 'block' or 'intrinsic' structures) [Ellis, by PG] |
| Full Idea: 'Dispositional' properties involve behaviour, and 'categorical properties' are structures in two or more dimensions. 'Block' structures (e.g. molecules) depend on other things, and 'instrinsic' structures (e.g. fields) involve no separate parts. | |
| From: report of Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.4) by PG - Db (ideas) | |
| A reaction: This is an essentialist approach to properties, and sounds correct to me. The crucial preliminary step to understanding properties is to eliminate secondary qualities (e.g. colour), which are not properties at all, and cause confusion. |
| 5469 | The passive view of nature says categorical properties are basic, but others say dispositions [Ellis] |
| Full Idea: 'Categorical realism' is the most widely accepted theory of dispositional properties, because passivists can accept it, ..that is, that dispositions supervene on categorical properties; ..the opposite would imply nature is active and reactive. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.4) | |
| A reaction: Essentialists believe 'the opposite' - i.e. that dispositions are fundamental, and hence that the essence of nature is active. See 5468 for explanations of the distinctions. I am with the essentialists on this one. |
| 5456 | Redness is not a property as it is not mind-independent [Ellis] |
| Full Idea: Redness is not a property, because it has no mind-independent existence. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: Well said. Secondary qualities are routinely cited in discussions of properties, and they shouldn't be. Redness causes nothing to happen in the physical world, unless a consciousness experiences it. |
| 5481 | Properties have powers; they aren't just ways for logicians to classify objects [Ellis] |
| Full Idea: One cannot think of a property as just a set of objects in a domain (as Fregean logicians do), as though the property has no powers, but is just a way of classifying objects. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: I agree. It is sometimes suggested that properties are what 'individuate' objects, but how could they do that if they didn't have some power? If properties are known by their causal role, why do they have that causal role? |
| 5458 | Nearly all fundamental properties of physics are dispositional [Ellis] |
| Full Idea: With few, if any, exceptions, the fundamental properties of physical theory are dispositional properties of the things that have them. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: He is denying that they are passive (as Locke saw primary qualities), and says they are actively causal, or else capacities or propensities. Sounds right to me. |
| 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
| Full Idea: One way for a nominalist to reject appeal to all abstract objects, including sets, is to only appeal to nominalistically acceptable objects, including mereological sums. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I'm suddenly thinking that this looks very interesting and might be the way to go. The issue seems to be whether mereological sums should be seen as constrained by nature, or whether they are unrestricted. See Mereology in Ontology...|Intrinsic Identity. |
| 5443 | Kripke and others have made essentialism once again respectable [Ellis] |
| Full Idea: The revival of essentialism owes much to the work of Saul Kripke and Hilary Putnam, who made belief in essences once again respectable, with Harré and Madden arguing that there were real causal powers in nature. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Intro) | |
| A reaction: It seems to me important to separate two stages of this: 1) causation results from essences, and 2) essences can never change. The first seems persuasive to me. For the second, see METAPHYSICS/IDENTITY/COUNTERPARTS. |
| 5444 | 'Individual essences' fix a particular individual, and 'kind essences' fix the kind it belongs to [Ellis] |
| Full Idea: The new essentialism retains Aristotelian ideas about essential properties, but it distinguishes more clearly between 'individual essences' and 'kind essences'; the former define a particular individual, the latter what kind it belongs to. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.1) | |
| A reaction: This might actually come into conflict with Aristotle, who seems to think that my personal essence is largely a human nature I share with everyone else. The new distinction is trying to keep the Kantian individual on the stage. |
| 5462 | Essential properties are usually quantitatively determinate [Ellis] |
| Full Idea: Most of the essential properties of things are quantitatively determinate properties. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: This makes the essential nature of the world very much the province of science, which deals in quantities and equations. Essentialists must deal with mental events, as well as basic physics. |
| 5448 | 'Real essence' makes it what it is; 'nominal essence' makes us categorise it a certain way [Ellis] |
| Full Idea: The 'real essence' of a thing is that set of its properties or structures in virtue of which it is a thing of that kind; its 'nominal essence' is the properties or structures in virtue of which it is described as a thing of that kind. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.1) | |
| A reaction: I like this distinction, because it is the kind made by realists like me who are fighting to make philosophers keep their epistemology and their ontology separate. |
| 5477 | One thing can look like something else, without being the something else [Ellis] |
| Full Idea: In considering questions of real possibility, it is important to keep the distinction between what a thing is and what it looks like clearly in mind. There is a possible world containing a horse that could then look like a cow, but it wouldn't BE a horse. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.6) | |
| A reaction: This is an interesting test assertion of the notion that there are essences (although Ellis does not allow that animals actually have essences - how could you, given evolution?). His point is a good one. |
| 5479 | Scientific essentialists say science should define the limits of the possible [Ellis] |
| Full Idea: Scientific essentialists hold that one of the primary aims of science is to define the limits of the possible. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.6) | |
| A reaction: I'm not sure working scientists will go along with that, but I like the claim that philosophy is very much part of the same enterprise as practical science (and NOT subservient to it!). I think of metaphysics as very high level physics. |
| 5483 | Essentialists deny possible worlds, and say possibilities are what is compatible with the actual world [Ellis] |
| Full Idea: Essentialists are modal realists; ..what is really possible, they say, is what is compatible with the natures of things in this world (and this does not commit them to the existence of any world other than the actual world). | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: This introduces something like 'compatibilities' into our ontology. That must rest on some kind of idea of a 'natural contradiction'. We can discuss the possibilities resulting from essences, but what are the possible variations in the essences? |
| 5447 | Metaphysical necessities are true in virtue of the essences of things [Ellis] |
| Full Idea: Metaphysical necessities are propositions that are true in virtue of the essences of things. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.1) | |
| A reaction: I am cautious about this. It sounds like huge Leibnizian metaphysical claims riding in on the back of a rather sensible new view of the laws of science. How can we justify equating natural necessity with metaphysical necessity? |
| 5476 | Essentialists say natural laws are in a new category: necessary a posteriori [Ellis] |
| Full Idea: Essentialists do not accept the standard position, which says necessity is a priori, and contingency is a posteriori. They have a radically new category: the necessary a posteriori. The laws of nature are, for example, both necessary and a posteriori. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.6) | |
| A reaction: Based on Kripke. I'm cautious about this. Presumably God, who would know the essences, could therefore infer the laws a priori. The laws may follow of necessity from the essences, but the essences can't be known a posteriori to be necessary. |
| 5478 | Imagination tests what is possible for all we know, not true possibility [Ellis] |
| Full Idea: The imaginability test of possibility confuses what is really or metaphysically possible with what is only epistemically possible. ..The latter is just what is possible for all we know. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.6) |
| 5482 | Possible worlds realism is only needed to give truth conditions for modals and conditionals [Ellis] |
| Full Idea: The main trouble with possible worlds realism is that the only reason anyone has, or ever could have, to believe in other possible worlds (other than this one) is that they are needed, apparently, to provide truth conditions for modals and conditionals. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: This attacks Lewis. Ellis makes this sound like a trivial technicality, but if our metaphysics is going to make sense it must cover modals and conditionals. What do they actually mean? Lewis has a theory, at least. |
| 5453 | Essentialists mostly accept the primary/secondary qualities distinction [Ellis] |
| Full Idea: Essentialists mostly accept the distinction between primary and secondary qualities, ..where the primary qualities of things are those that are intrinsic to the objects that have them. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: One reason I favour essentialism is because I have always thought that the primary/secondary distinction was a key to understanding the world. 'Primary' gets at the ontology, 'secondary' shows us the epistemology. |
| 5466 | Primary qualities are number, figure, size, texture, motion, configuration, impenetrability and (?) mass [Ellis] |
| Full Idea: For Boyle, Locke and Newton, the qualities inherent in bodies were just the primary qualities, namely number, figure, size, texture, motion and configuration of parts, impenetrability and, perhaps, body (or mass). | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.4) | |
| A reaction: It is nice to have a list. Ellis goes on to say these are too passive, and urges dispositions as primary. Even so, the original seventeenth century insight seems to me a brilliant step forward in our understanding of the world. |
| 5485 | Emeralds are naturally green, and only an external force could turn them blue [Ellis] |
| Full Idea: Emeralds cannot all turn blue in 2050 (as Nelson Goodman envisaged), because to do so they would have to have an extrinsically variable nature. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: I was never very impressed by the 'grue' problem, probably for this reason, but also because Goodman probably thought predicates and properties are the same thing, which they aren't (Idea 5457). |
| 5484 | Essentialists don't infer from some to all, but from essences to necessary behaviour [Ellis] |
| Full Idea: For essentialists the problem of induction reduces to discovering what natural kinds there are, and identifying their essential problems and structures. We then know how they must behave in any world, and there is no inference from some to all. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: The obvious question is how you would determine the essences if you are not allowed to infer 'from some to all'. Personally I don't see induction as a problem, because it is self-evidently rational in a stable world. Hume was right to recommend caution. |
| 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. |
| 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). |
| 5457 | Predicates assert properties, values, denials, relations, conventions, existence and fabrications [Ellis, by PG] |
| Full Idea: As well as properties, predicates can assert evaluation, denial, relations, conventions, existence or fabrication. | |
| From: report of Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) by PG - Db (ideas) | |
| A reaction: This seems important, in order to disentangle our ontological commitments from our language, which was a confusion that ran throughout twentieth-century philosophy. A property is a real thing in the world, not a linguistic convention. |
| 5488 | Regularity theories of causation cannot give an account of human agency [Ellis] |
| Full Idea: A Humean theory of causation (as observed regularities) makes it very difficult for anyone even to suggest a plausible theory of human agency. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: I'm not quite sure what a 'theory' of human agency would look like. Hume himself said we only get to understand our mental powers from repeated experience (Idea 2220). How do we learn about the essence of our own will? |
| 5489 | Humans have variable dispositions, and also power to change their dispositions [Ellis] |
| Full Idea: It seems that human beings not only have variable dispositional properties, as most complex systems have, but also meta-powers: powers to change their own dispositional properties. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: This seems to me a key to how we act, and also to morality. 'What dispositions do you want to have?' is the central question of virtue theory. Humans are essentially multi-level thinkers. Irony is the window into the soul. |
| 5490 | Essentialism fits in with Darwinism, but not with extreme politics of left or right [Ellis] |
| Full Idea: The extremes of left and right in politics have much more reason than Darwinists to be threatened by the 'new essentialism', because it must reinstate the concept of human nature. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: The point being that political extremes go against the grain of our nature. Personally I am favour of essentialism, and human nature. I notice that Steven Pinker is now defending human nature, from a background of linguistics and psychology. |
| 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. |
| 5472 | Natural kinds are of objects/substances, or events/processes, or intrinsic natures [Ellis] |
| Full Idea: Natural kinds appear to be of objects or substances, or of events or processes, or of the intrinsic nature of things; hence there should be laws of nature specific to each of these categories. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.5) | |
| A reaction: It is nice to see someone actually discussing what sort of natural kinds there are, instead of getting bogged down in how natural kinds terms get their meaning or reference. Ellis recognises that 'intrinsic nature' needs some discussion. |
| 5471 | Essentialism says natural kinds are fundamental to nature, and determine the laws [Ellis] |
| Full Idea: According to essentialists, the world is wholly structured at the most fundamental level into natural kinds, and the laws of nature are all determined by those kinds. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.5) | |
| A reaction: I am a fan of this view, despite being cautious about claims that natural kinds have necessary identity. Why are the essences active? That is the old Greek puzzle about the origin of movement. And why are natural kinds stable? |
| 5446 | For essentialists two members of a natural kind must be identical [Ellis] |
| Full Idea: Modern essentialists would insist that any two members of the same natural kind must be identical in all essential respects. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.1) | |
| A reaction: For this reason, animals no longer qualify as natural kinds, but electrons, gold atoms, and water molecules do. My sticking point is when anyone asserts that an electron necessarily has (say) its mass. Why no close counterpart of electrons? |
| 5480 | The whole of our world is a natural kind, so all worlds like it necessarily have the same laws [Ellis] |
| Full Idea: It is plausible to suppose that the world is an instance of a natural kind, ..and what is naturally necessary in our world is what must be true in any world of the same natural kind. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.6) | |
| A reaction: This is putting an awful lot of metaphysical weight on the concept of a 'natural kind', so it had better be a secure one. If we accept that natural laws necessarily follow from essences, why shouldn't the whole of our world have an essence, as water does? |
| 5445 | Essentialists regard inanimate objects as genuine causal agents [Ellis] |
| Full Idea: Essentialist suppose that the inanimate objects of nature are genuine causal agents: things capable of acting or interacting. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Intro) | |
| A reaction: I have no idea how one might demonstrate such a fact, even though it seems to stare us in the face. This is where science bumps into philosophy. I find myself intuitively taking the essentialist side quite strongly. |
| 5463 | Essentialists believe causation is necessary, resulting from dispositions and circumstances [Ellis] |
| Full Idea: Essentialists believe elementary causal relations involve necessary connections between events, namely between the displays of dispositional properties and the circumstances that give rise to them. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.4) | |
| A reaction: I like essentialism, but I feel a Humean caution about talk of 'natural necessity'. Let's just say that causation seems to be entirely the result of the nature of how things are. How things could be is a large topic for little mites like us. |
| 5491 | A general theory of causation is only possible in an area if natural kinds are involved [Ellis] |
| Full Idea: A general theory of causation in an area is possible only if the kinds of entities under investigation can reasonably be assumed to belong to natural kinds. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: Human beings will be a problem, and also different levels of natural kinds (e.g. a chemical and an organism). 'Natural kind' is a very loose concept. He is referring to scientific, rather than philosophical, theories, I presume. |
| 5442 | For 'passivists' behaviour is imposed on things from outside [Ellis] |
| Full Idea: A 'passivist' believes that the tendencies of things to behave as they do can never be inherent in the things themselves; they must always be imposed on them from the outside. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Intro) | |
| A reaction: This is the medieval view, inherited by Newton and Hume, which makes miracles a possibility, and makes the laws of nature contingent. Essentialism disagree. I think I am with the essentialists. |
| 5473 | The laws of nature imitate the hierarchy of natural kinds [Ellis] |
| Full Idea: If the natural kinds are divided into hierarchical categories, then essentialists would expect the laws of nature also to divide up into these categories, with the same hierarchy. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.5) | |
| A reaction: This seems to me a real step forwards in our understanding of nature, and hence a nice example of the contribution which philosophy can make, instead of just physics. |
| 5474 | Laws of nature tend to describe ideal things, or ideal circumstances [Ellis] |
| Full Idea: Most of the propositions we think of as being (or as expressing) genuine laws of nature seem to describe only the behaviour of ideal kinds of things, or of things in ideal circumstances. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.5) | |
| A reaction: Ellis this suggests that this phenomenon is because science aims at broad understanding instead of strict prediction. Do we simplify because we are a bit dim? Or is it because generalisation wouldn't exist without idealisation and abstraction? |
| 5475 | We must explain the necessity, idealisation, ontology and structure of natural laws [Ellis] |
| Full Idea: There are four major problems about the laws of nature: a necessity problem (must they be true?), an idealisation problem (why is this preferable?), an ontological problem (their grounds), and a structural problem (their relationships). | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.5) | |
| A reaction: One might also ask why the laws (or their underlying essences) are the way they are, and not some other way, though the prospects of answering that don't look good. I don't think we should be satisfied with saying all of these questions are hopeless. |
| 5460 | Causal relations cannot be reduced to regularities, as they could occur just once [Ellis] |
| Full Idea: Causal relations cannot be reduced to mere regularities, as Hume supposed, as they could exist as a singular case, even if it never happened on more than one occasions. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: This seems to be the key reason for modern views moving away from Hume. The suspicion is that regularity is a test for or symptom of causation, but we are deeply committed to the real nature of causation being whatever creates the regularities. |
| 5459 | Essentialists say dispositions are basic, rather than supervenient on matter and natural laws [Ellis] |
| Full Idea: Essentialists say that dispositional properties may be fundamental, whereas for a passivist such qualities are not primary, but supervene on the primary qualities of matter, and on the laws of nature. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: I am strongly in favour of this view of nature. Without essentialism, we have laws of nature arising out of a total void (or God), and arbitrarily imposing themselves on matter. What are the 'primary qualities of matter', if not dispositions? |
| 5461 | The essence of uranium is its atomic number and its electron shell [Ellis] |
| Full Idea: The essential properties of uranium are its atomic number, and the common electron shell structure for all uranium atoms. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: For those who deny essences (e.g. Quineans) this is a nice challenge. You might have to add accounts of the essences of the various particles that make up the atoms. There is nothing arbitrary or conventional about what makes something uranium. |
| 5464 | For essentialists, laws of nature are metaphysically necessary, being based on essences of natural kinds [Ellis] |
| Full Idea: Essentialist believe the laws of nature are metaphysically necessary, because anything that belongs to a natural kind is logically required (or is necessarily disposed) to behave as its essential properties dictate. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.4) | |
| A reaction: What a thrillingly large claim. Best approached with caution.. If we say 'essences make laws, and essences are necessary', we might wonder whether a natural kind essence could be SLIGHTLY different (a counterpart) in another world. |
| 5487 | Essentialism requires a clear separation of semantics, epistemology and ontology [Ellis] |
| Full Idea: Scientific essentialism requires that philosophers distinguish clearly between semantic issues, epistemological issues, and ontological issues. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: Music to my ears - but then I think everyone should require that of philosophers, because it where they get themselves most confused. The trouble is that ontology is only obtainable epistemologically, and only expressible semantically. |