85 ideas
| 18495 | The best philosophers I know are the best people I know [Heil] |
| Full Idea: Philosophers are not invariably the best people, but the best philosophers I know are the best people I know. | |
| From: John Heil (The Universe as We Find It [2012], Pref) | |
| A reaction: How very nicely expressed. I have often thought the same about lovers of literature, but been horribly disappointed by some of them. On the whole I have found philosophy-lovers to be slightly superior to literature-lovers! |
| 18494 | Using a technical vocabulary actually prevents discussion of the presuppositions [Heil] |
| Full Idea: Sharing a technical vocabulary is to share a tidy collection of assumptions. Reliance on that vocabulary serves to foreclose discussion of those assumptions. | |
| From: John Heil (The Universe as We Find It [2012], Pref) | |
| A reaction: Love it! I am endlessly frustrated by papers that launch into a discussion using a terminology that is riddled with dubious prior assumptions. And that includes common terms like 'property', as well as obscure neologisms. |
| 18506 | Questions of explanation should not be confused with metaphyics [Heil] |
| Full Idea: There is an unfortunate tendency to conflate epistemological issues bearing on explanation with issues in metaphysics. | |
| From: John Heil (The Universe as We Find It [2012], 01.2) | |
| A reaction: This is where Heil and I part ways. I just don't believe in the utterly pure metaphysics which he thinks we can do. Our drive to explain moulds our vision of reality, say I. |
| 18535 | Without abstraction we couldn't think systematically [Heil] |
| Full Idea: A capacity for abstraction is central to our capacity to think about the universe systematically. | |
| From: John Heil (The Universe as We Find It [2012], 09.7) | |
| A reaction: This strikes me as obvious. We pick out the similarities, and then discuss them, as separate from their bearers. We explain why things have features in common. Some would just say systematic thinking needs universals, but that's less good. |
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| Full Idea: An 'impredicative' definition is one that uses the terms being defined in order to give the definition; in some way the definition is then circular. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], Glossary) | |
| A reaction: There has been a big controversy in the philosophy of mathematics over these. Shapiro gives the definition of 'village idiot' (which probably mentions 'village') as an example. |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| Full Idea: In classical logic definitions are thought of as revealing our attempts to refer to objects, ...but for intuitionist or constructivist logics, if our definitions do not uniquely characterize an object, we are not entitled to discuss the object. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.4) | |
| A reaction: In defining a chess piece we are obviously creating. In defining a 'tree' we are trying to respond to fact, but the borderlines are vague. Philosophical life would be easier if we were allowed a mixture of creation and fact - so let's have that. |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| Full Idea: Reductio ad absurdum arguments are ones that start by denying what one wants to prove. We then prove a contradiction from this 'denied' idea and more reasonable ideas in one's theory, showing that we were wrong in denying what we wanted to prove. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is a mathematical definition, which rests on logical contradiction, but in ordinary life (and philosophy) it would be enough to show that denial led to absurdity, rather than actual contradiction. |
| 18534 | Truth relates truthbearers to truthmakers [Heil] |
| Full Idea: Truth is a relation between a truthbearer and a truthmaker. | |
| From: John Heil (The Universe as We Find It [2012], 08.02) | |
| A reaction: This implies that all truths have truthmakers, which is fairly controversial. Heil himself denies it! |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| Full Idea: For the anti-realist, truth belongs to us, it is our servant, and as such, it must be 'epistemically constrained'. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1) | |
| A reaction: Put as clearly as this, it strikes me as being utterly and spectacularly wrong, a complete failure to grasp the elementary meaning of a concept etc. etc. If we aren't the servants of truth then we jolly we ought to be. Truth is above us. |
| 18531 | Philosophers of the past took the truthmaking idea for granted [Heil] |
| Full Idea: For millenia, philosophers operated with an implicit conception of truthmaking, a conception that remained unarticulated only because it was part of the very fabric of philosophy. | |
| From: John Heil (The Universe as We Find It [2012], 07.2) | |
| A reaction: Presumably it is an advance that we have brought it out into the open, and subjected it to critical study. Does Heil want us to return to it being unquestioned? I like truthmaking, but that can't be right. |
| 18509 | Not all truths need truthmakers - mathematics and logic seem to be just true [Heil] |
| Full Idea: I do not subscribe to the thesis that every truth requires a truthmaker. Mathematical truths and truths of logic are compatible with any way the universe could be. | |
| From: John Heil (The Universe as We Find It [2012], 01.5) | |
| A reaction: He makes that sound like a knock-down argument, but I'm not convinced. I see logic and mathematics as growing out of nature, though that is a very unfashionable view. I'm almost ashamed of it. But I'm not giving it up. See Carrie Jenkins. |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| Full Idea: In the classical or realist view of logic the meaning of abstract symbols for logical connectives is given by the truth-tables for the symbol. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007]) | |
| A reaction: Presumably this is realist because it connects them to 'truth', but only if that involves a fairly 'realist' view of truth. You could, of course, translate 'true' and 'false' in the table to empty (formalist) symbols such a 0 and 1. Logic is electronics. |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| Full Idea: In intuitionist logic, if we do not know that we do not know A, it does not follow that we know A, so the inference (and, in general, double negation elimination) is not intuitionistically valid. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: That inference had better not be valid in any logic! I am unaware of not knowing the birthday of someone I have never heard of. Propositional attitudes such as 'know' are notoriously difficult to explain in formal logic. |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| Full Idea: Free logic is especially designed to help regiment our reasoning about fictional objects, or nonexistent objects of some sort. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.7) | |
| A reaction: This makes it sound marginal, but I wonder whether existential commitment shouldn't be eliminated from all logic. Why do fictional objects need a different logic? What logic should we use for Robin Hood, if we aren't sure whether or not he is real? |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| Full Idea: A 'subset' of A is a set containing only members of A, and a 'proper subset' is one that does not contain all the members of A. Note that the empty set is a subset of every set, but it is not a member of every set. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Is it the same empty set in each case? 'No pens' is a subset of 'pens', but is it a subset of 'paper'? Idea 8219 should be borne in mind when discussing such things, though I am not saying I agree with it. |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| Full Idea: The 'powerset' of a set is a set made up of all the subsets of a set. For example, the powerset of {3,7,9} is {null, {3}, {7}, {9}, {3,7}, {3,9}, {7,9}, {3,7,9}}. Taking the powerset of an infinite set gets us from one infinite cardinality to the next. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Note that the null (empty) set occurs once, but not in the combinations. I begin to have queasy sympathies with the constructivist view of mathematics at this point, since no one has the time, space or energy to 'take' an infinite powerset. |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| Full Idea: As a realist choice of what is basic in mathematics, set theory is rather clever, because it only makes a very simple ontological claim: that, independent of us, there exists the empty set. The whole hierarchy of finite and infinite sets then follows. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: Even so, for non-logicians the existence of the empty set is rather counterintuitive. "There was nobody on the road, so I overtook him". See Ideas 7035 and 8322. You might work back to the empty set, but how do you start from it? |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| Full Idea: Two sets are the same size if they can be placed in one-to-one correspondence. But even numbers have one-to-one correspondence with the natural numbers. So a set is infinite if it has one-one correspondence with a proper subset. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Dedekind's definition. We can match 1 with 2, 2 with 4, 3 with 6, 4 with 8, etc. Logicians seem happy to give as a definition anything which fixes the target uniquely, even if it doesn't give the essence. See Frege on 0 and 1, Ideas 8653/4. |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| Full Idea: Zermelo-Fraenkel and Gödel-Bernays set theory differ over the notions of ordinal construction and over the notion of class, among other things. Then there are optional axioms which can be attached, such as the axiom of choice and the axiom of infinity. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.6) | |
| A reaction: This summarises the reasons why we cannot just talk about 'set theory' as if it was a single concept. The philosophical interest I would take to be found in disentangling the ontological commitments of each version. |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| Full Idea: The law of excluded middle is purely syntactic: it says for any well-formed formula A, either A or not-A. It is not a semantic law; it does not say that either A is true or A is false. The semantic version (true or false) is the law of bivalence. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: No wonder these two are confusing, sufficiently so for a lot of professional philosophers to blur the distinction. Presumably the 'or' is exclusive. So A-and-not-A is a contradiction; but how do you explain a contradiction without mentioning truth? |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| Full Idea: In the intuitionist version of quantification, the universal quantifier (normally read as "all") is understood as "we have a procedure for checking every" or "we have checked every". | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.5) | |
| A reaction: It seems better to describe this as 'verificationist' (or, as Dummett prefers, 'justificationist'). Intuition suggests an ability to 'see' beyond the evidence. It strikes me as bizarre to say that you can't discuss things you can't check. |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| Full Idea: The realist meets the Burali-Forti paradox by saying that all the ordinals are a 'class', not a set. A proper class is what we discuss when we say "all" the so-and-sos when they cannot be reached by normal set-construction. Grammar is their only limit. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This strategy would be useful for Class Nominalism, which tries to define properties in terms of classes, but gets tangled in paradoxes. But why bother with strict sets if easy-going classes will do just as well? Descartes's Dream: everything is rational. |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| Full Idea: The Burali-Forti paradox says that if ordinals are defined by 'gathering' all their predecessors with the empty set, then is the set of all ordinals an ordinal? It is created the same way, so it should be a further member of this 'complete' set! | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is an example (along with Russell's more famous paradox) of the problems that began to appear in set theory in the early twentieth century. See Idea 8675 for a modern solution. |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| Full Idea: The set of 'integers' is all of the negative natural numbers, and zero, together with the positive natural numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Zero always looks like a misfit at this party. Credit and debit explain positive and negative nicely, but what is the difference between having no money, and money being irrelevant? I can be 'broke', but can the North Pole be broke? |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| Full Idea: The 'rational' numbers are all those that can be represented in the form m/n (i.e. as fractions), where m and n are natural numbers different from zero. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Pythagoreans needed numbers to stop there, in order to represent the whole of reality numerically. See irrational numbers for the ensuing disaster. How can a universe with a finite number of particles contain numbers that are not 'rational'? |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| Full Idea: A number is 'irrational' just in case it cannot be represented as a fraction. An irrational number has an infinite non-repeating decimal expansion. Famous examples are pi and e. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: There must be an infinite number of irrational numbers. You could, for example, take the expansion of pi, and change just one digit to produce a new irrational number, and pi has an infinity of digits to tinker with. |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| Full Idea: The natural numbers are quite primitive, and are what we first learn about. The order of objects (the 'ordinals') is one level of abstraction up from the natural numbers: we impose an order on objects. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: Note the talk of 'levels of abstraction'. So is there a first level of abstraction? Dedekind disagrees with Friend (Idea 7524). I would say that natural numbers are abstracted from something, but I'm not sure what. See Structuralism in maths. |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| Full Idea: The 'cardinal' numbers answer the question 'How many?'; the order of presentation of the objects being counted as immaterial. Def: the cardinality of a set is the number of members of the set. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: If one asks whether cardinals or ordinals are logically prior (see Ideas 7524 and 8661), I am inclined to answer 'neither'. Presenting them as answers to the questions 'how many?' and 'which comes first?' is illuminating. |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| Full Idea: The set of 'real' numbers, which consists of the rational numbers and the irrational numbers together, represents "the continuum", since it is like a smooth line which has no gaps (unlike the rational numbers, which have the irrationals missing). | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: The Continuum is the perfect abstract object, because a series of abstractions has arrived at a vast limit in its nature. It still has dizzying infinities contained within it, and at either end of the line. It makes you feel humble. |
| 18518 | Infinite numbers are qualitatively different - they are not just very large numbers [Heil] |
| Full Idea: It is a mistake to think of an infinite number as a very large number. Infinite numbers differ qualitatively from finite numbers. | |
| From: John Heil (The Universe as We Find It [2012], 03.5) | |
| A reaction: He cites Dedekind's idea that a proper subset of an infinite number can match one-one with the number. Respectable numbers don't behave in this disgraceful fashion. This should be on the wall of every seminar on philosophy of mathematics. |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| Full Idea: After the multiples of omega, we can successively raise omega to powers of omega, and after that is done an infinite number of times we arrive at a new limit ordinal, which is called 'epsilon'. We have an infinite number of infinite ordinals. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: When most people are dumbstruck by the idea of a single infinity, Cantor unleashes an infinity of infinities, which must be the highest into the stratosphere of abstract thought that any human being has ever gone. |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| Full Idea: The first 'limit ordinal' is called 'omega', which is ordinal because it is greater than other numbers, but it has no immediate predecessor. But it has successors, and after all of those we come to twice-omega, which is the next limit ordinal. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: This is the gateway to Cantor's paradise of infinities, which Hilbert loved and defended. Who could resist the pleasure of being totally boggled (like Aristotle) by a concept such as infinity, only to have someone draw a map of it? See 8663 for sequel. |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| Full Idea: Since between any two rational numbers there is an infinite number of rational numbers, we could consider that we have infinity in three dimensions: positive numbers, negative numbers, and the 'depth' of infinite numbers between any rational numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: This is before we even reach Cantor's staggering infinities (Ideas 8662 and 8663), which presumably reside at the outer reaches of all three of these dimensions of infinity. The 'deep' infinities come from fractions with huge denominators. |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| Full Idea: Successful competing founding disciplines in mathematics include: the various set theories, type theory, category theory, model theory and topology. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: Or none of the above? Set theories are very popular. Type theory is, apparently, discredited. Shapiro has a version of structuralism based on model theory (which sound promising). Topology is the one that intrigues me... |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| Full Idea: Most of mathematics can be faithfully redescribed by classical (realist) set theory. More precisely, we can translate other mathematical theories - such as group theory, analysis, calculus, arithmetic, geometry and so on - into the language of set theory. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is why most mathematicians seem to regard set theory as foundational. We could also translate football matches into the language of atomic physics. |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| Full Idea: There is no interest for the mathematician in studying the number 8 in isolation from the other numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: This is a crucial and simple point (arising during a discussion of Shapiro's structuralism). Most things are interesting in themselves, as well as for their relationships, but mathematical 'objects' just are relationships. |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| Full Idea: Structuralists give a historical account of why the 'same' number occupies different structures. Numbers are equivalent rather than identical. 8 is the immediate predecessor of 9 in the whole numbers, but in the rationals 9 has no predecessor. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: I don't become a different person if I move from a detached house to a terraced house. This suggests that 8 can't be entirely defined by its relations, and yet it is hard to see what its intrinsic nature could be, apart from the units which compose it. |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| Full Idea: Structuralists disagree over whether objects in structures are 'ante rem' (before reality, existing independently of whether the objects exist) or 'in re' (in reality, grounded in the real world, usually in our theories of physics). | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: Shapiro holds the first view, Hellman and Resnik the second. The first view sounds too platonist and ontologically extravagant; the second sounds too contingent and limited. The correct account is somewhere in abstractions from the real. |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| Full Idea: According to the structuralist, mathematicians study the concepts (objects of study) such as variable, greater, real, add, similar, infinite set, which are one level of abstraction up from prima facie base objects such as numbers, shapes and lines. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1) | |
| A reaction: This still seems to imply an ontology in which numbers, shapes and lines exist. I would have thought you could eliminate the 'base objects', and just say that the concepts are one level of abstraction up from the physical world. |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| Full Idea: Structuralism says we study whole structures: objects together with their predicates, relations that bear between them, and functions that take us from one domain of objects to a range of other objects. The objects can even be eliminated. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1) | |
| A reaction: The unity of object and predicate is a Quinean idea. The idea that objects are inessential is the dramatic move. To me the proposal has very strong intuitive appeal. 'Eight' is meaningless out of context. Ordinality precedes cardinality? Ideas 7524/8661. |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| Full Idea: In the 'in re' version of mathematical structuralism, pattern-spotting is the process of abstraction. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: This might work for non-mathematical abstraction as well, if we are allowed to spot patterns within sensual experience, and patterns within abstractions. Properties are causal patterns in the world? No - properties cause patterns. |
| 18500 | How could structures be mathematical truthmakers? Maths is just true, without truthmakers [Heil] |
| Full Idea: I do not understand how structures could serve as truthmakers for mathematical truths, ...Mathematical truths are not true in virtue of any way the universe is. ...Mathematical truths hold, whatever ways the universe is. | |
| From: John Heil (The Universe as We Find It [2012], 08.08) | |
| A reaction: I like the idea of enquiring about truthmakers for mathematical truths (and my view is more empirical than Heil's), but I think it may be a misunderstanding to think that structures are intended as truthmakers. Mathematics just IS structures? |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| Full Idea: The main philosophical problem with the position of platonism or realism is the epistemic problem: of explaining what perception or intuition consists in; how it is possible that we should accurately detect whatever it is we are realists about. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.5) | |
| A reaction: The best bet, I suppose, is that the mind directly perceives concepts just as eyes perceive the physical (see Idea 8679), but it strikes me as implausible. If we have to come up with a special mental faculty for an area of knowledge, we are in trouble. |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| Full Idea: Central to naturalism about mathematics are 'indispensability arguments', to the effect that some part of mathematics is indispensable to our best physical theory, and therefore we ought to take that part of mathematics to be true. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.1) | |
| A reaction: Quine and Putnam hold this view; Field challenges it. It has the odd consequence that the dispensable parts (if they can be identified!) do not need to be treated as true (even though they might follow logically from the dispensable parts!). Wrong! |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| Full Idea: There are not enough constraints in the Formalist view of mathematics, so there is no way to select a direction for trying to develop mathematics. There is no part of mathematics that is more important than another. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.6) | |
| A reaction: One might reply that an area of maths could be 'important' if lots of other areas depended on it, and big developments would ripple big changes through the interior of the subject. Formalism does, though, seem to reduce maths to a game. |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| Full Idea: Too much of mathematics is rejected by the constructivist. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1) | |
| A reaction: This was Hilbert's view. This seems to be generally true of verificationism. My favourite example is that legitimate speculations can be labelled as meaningless. |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| Full Idea: An intuitionist typically retains bivalence, but rejects the law of excluded middle. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: The idea would be to say that only T and F are available as truth-values, but failing to be T does not ensure being F, but merely not-T. 'Unproven' is not-T, but may not be F. |
| 18539 | Our categories lack the neat arrangement needed for reduction [Heil] |
| Full Idea: Categories we use to describe and explain our universe do not line up in the neat way reductive schemes require. | |
| From: John Heil (The Universe as We Find It [2012], 13.2) | |
| A reaction: He takes reduction to be largely a relation between our categories, rather than between entities, so he is bound to get this result. He may be right. |
| 18505 | Fundamental ontology aims at the preconditions for any true theory [Heil] |
| Full Idea: Fundamental ontology is in the business of telling us what the universe must be like if any theory is true. | |
| From: John Heil (The Universe as We Find It [2012], 01.1) | |
| A reaction: Heil is good at stating simple ideas simply. This seems to be a bold claim, but I think I agree with it. |
| 18499 | Our quantifications only reveal the truths we accept; the ontology and truthmakers are another matter [Heil] |
| Full Idea: Looking at what you quantify over reveals, at most, truths to which you are committed. What the ontology is, what the truthmakers are for these truths, is another matter, one tackled, if at all, only in the pursuit of fundamental physics. | |
| From: John Heil (The Universe as We Find It [2012], 08.08) | |
| A reaction: Exactly right. Nouns don't guarantee objects, verbs don't guarantee processes. If you want to know my ontological commitments, ask me about them! Don't infer them from the sentences I hold true, because they need interpreting. |
| 18512 | Ontology aims to give the fundamental categories of being [Heil] |
| Full Idea: The task of ontology is to spell out the fundamental categories of being. | |
| From: John Heil (The Universe as We Find It [2012], 02.5) | |
| A reaction: This is the aspiration of 'pure' metaphysics, which I don't quite believe in. There is too much convention involved, on the one hand, and physics on the other. |
| 18508 | Most philosophers now (absurdly) believe that relations fully exist [Heil] |
| Full Idea: It is a measure of how far we have fallen that so few philosophers nowadays see any difficulty at all in the idea that relations have full ontological standing. | |
| From: John Heil (The Universe as We Find It [2012], 01.4) | |
| A reaction: We have 'fallen' because medieval metaphysicians didn't believe it. Russell seems to have started, and the tendency to derive ontology from logic has secured the belief in relations. How else can you be allowed to write aRb? I agree with Heil. |
| 18532 | If causal relations are power manifestations, that makes them internal relations [Heil] |
| Full Idea: If causal relations are the manifesting of powers, then causal relations would appear to be a species of internal relation. | |
| From: John Heil (The Universe as We Find It [2012], 07.4) | |
| A reaction: The point being that any relations formed are entirely dependent on the internal powers of the relata. Sounds right. There are also non-causal relations, of course. |
| 18510 | We need properties to explain how the world works [Heil] |
| Full Idea: When a tomato depresses a scale, it does so in virtue of its mass - how it is masswise - and not in virtue of its colour or shape. Were we barred from saying such things, we would be unable to formulate truths about the fundamental things. | |
| From: John Heil (The Universe as We Find It [2012], 02.3) | |
| A reaction: It doesn't follow that we have an ontological commitment to properties, but we certainly need to point out the obvious fact that things being one way rather than another makes a difference to what happens. |
| 18522 | Categorical properties were introduced by philosophers as actual properties, not if-then properties [Heil] |
| Full Idea: Categorical properties were introduced originally by philosophers bent on distinguishing properties possessed 'categorically', that is, actually, by objects from mere if-then, conditional properties, mere potentialities. | |
| From: John Heil (The Universe as We Find It [2012], 04.3) | |
| A reaction: He cites Ryle on dispositions in support. It is questionable whether it is a clear or useful distinction. Heil says the new distinction foreclosed the older more active view of properties. |
| 18513 | Emergent properties will need emergent substances to bear them [Heil] |
| Full Idea: If you are going to have emergent fundamental properties, you are going to need emergent fundamental substances as bearers of those properties. | |
| From: John Heil (The Universe as We Find It [2012], 02.6) | |
| A reaction: Presumably the theory of emergent properties (which I take to be nonsense, in its hardcore form) says that the substance is unchanged, but the property is new. Or else the bundle gives collective birth to a new member. Search me. |
| 18540 | Predicates only match properties at the level of fundamentals [Heil] |
| Full Idea: Only when you get to fundamental physics, do predicates begin to line up with properties. | |
| From: John Heil (The Universe as We Find It [2012], 13.2) | |
| A reaction: A nice thought. I assume the actual properties of daily reality only connect to our predicates in very sloppy ways. I suppose our fundamental predicates have to converge on the actual properties, because the fog clears. Sort of. |
| 18533 | In Fa, F may not be a property of a, but a determinable, satisfied by some determinate [Heil] |
| Full Idea: It may be that F applies truly to a because F is a determinable predicate satisfied by a's possession of a property answering to a determinate of that determinable predicate. | |
| From: John Heil (The Universe as We Find It [2012], 08.01) | |
| A reaction: Heil aims to break the commitment of predicates to the existence of properties. The point is that there is no property 'coloured' to correspond to 'a is coloured'. Red might be the determinate that does the job. Nice. |
| 18511 | Properties have causal roles which sets can't possibly have [Heil] |
| Full Idea: Properties are central to the universe's causal order in a way that sets could not possibly be. | |
| From: John Heil (The Universe as We Find It [2012], 02.3) | |
| A reaction: The idea that properties actually are sets is just ridiculous. It may be that you can treat them as sets and get by quite well. The sets can be subsumed into descriptions of causal processes (or something). |
| 18523 | Are all properties powers, or are there also qualities, or do qualities have the powers? [Heil] |
| Full Idea: Some philosophers who embrace properties as powers hold that every property is a power (Bird), or that some properties are qualities and some are powers (Ellis; Molnar). The latter include powers which are 'grounded in' qualities (Mumford). | |
| From: John Heil (The Universe as We Find It [2012], 04.4) | |
| A reaction: I don't like Heil's emphasis on 'qualities', which seems to imply their phenomenal rather than their real aspect. I'm inclined to favour the all-powers view, but can't answer the question 'but what HAS these powers?' Stuff is intrinsically powerful. |
| 18524 | Properties are both qualitative and dispositional - they are powerful qualities [Heil] |
| Full Idea: In my account of properties they are at once qualitative and dispositional: properties are powerful qualities. | |
| From: John Heil (The Universe as We Find It [2012], 05.1) | |
| A reaction: I have never managed to understand what Heil means by 'qualities'. Is he talking about the phenomenal aspects of powers? Does he mean categorical properties. I can't find an ontological space for his things to slot into. |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| Full Idea: What the mathematician labels an 'object' in her discipline, is called 'a place in a structure' by the structuralist. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.5) | |
| A reaction: This is a strategy for dispersing the idea of an object in the world of thought, parallel to attempts to eliminate them from physical ontology (e.g. Idea 614). |
| 18498 | Abstract objects wouldn't be very popular without the implicit idea of truthmakers [Heil] |
| Full Idea: It would be difficult to understand the popularity of 'abstract entities' - numbers, sets, propositions - in the absence of an implicit acknowledgement of the importance of truthmakers. | |
| From: John Heil (The Universe as We Find It [2012], 08.07) | |
| A reaction: I love Idea 18496, because it leads us towards a better account of modality, but dislike this one because it reveals that the truthmaking idea has led us to a very poor theory. Truthmaking is a good question, but not much of an answer? |
| 18507 | Substances bear properties, so must be simple, and not consist of further substances [Heil] |
| Full Idea: Substances, as property bearers, must be simple; substances of necessity lack constituents that are themselves substances. | |
| From: John Heil (The Universe as We Find It [2012], 01.3) | |
| A reaction: How can he think that this is a truth of pure metaphysics? A crowd has properties because we think of it as a simple substance, not because it actually is one. Can properties have properties? Are tree and leaf both substances? |
| 18515 | Spatial parts are just regions, but objects depend on and are made up of substantial parts [Heil] |
| Full Idea: An object is not made up of its spatial parts: spatial parts are regions of some object. ...Complex objects, wholes, are made up of, and so depend on, their substantial parts. | |
| From: John Heil (The Universe as We Find It [2012], 03.1) | |
| A reaction: Presumably objects also 'depend on' their spatial parts, so I am not convinced that we have a sharp distinction here. |
| 18516 | A 'gunky' universe would literally have no parts at all [Heil] |
| Full Idea: Blancmange 'gunky' universes are not just universes with an endless number of parts. Rather a blancmange universe is a universe with no simple parts, no parts themselves lacking parts. | |
| From: John Heil (The Universe as We Find It [2012], 03.3) | |
| A reaction: Hm. Lewis seemed to think it was parts all the way down. Is gunk homogeneous stuff, or what is endlessly subdividable, or an infinite shrinking of parts? We demand clarity. |
| 18514 | Many wholes can survive replacement of their parts [Heil] |
| Full Idea: A whole - or some wholes - might be thought to survive gradual replacement of its parts, perhaps, but not their elimination. | |
| From: John Heil (The Universe as We Find It [2012], 03.1) | |
| A reaction: You can't casually replace the precious golden parts of a statue with cheap lead ones. It depends on whether the parts matter. Nevertheless this is a really important idea in metaphysics. It enables the s=Ship of Theseus to survive some change. |
| 18517 | Dunes depend on sand grains, but line segments depend on the whole line [Heil] |
| Full Idea: A sand dune depends on the individual grains of sand that make it up. In an important sense, however, a line's segments depend on the line rather than it on them. | |
| From: John Heil (The Universe as We Find It [2012], 03.4) | |
| A reaction: The illustrations are not clear cut. As you cut off segments of the line, you reduce its length. Heil is hoping for something neat here, but I don't think he has quite got. The difficulty of trying to do pure metaphysics! |
| 18502 | If basic physics has natures, then why not reality itself? That would then found the deepest necessities [Heil] |
| Full Idea: If electrons and gravitational fields have definite natures, why not reality itself? And if reality has a nature, if this makes sense, then reality grounds the deepest necessities of all. | |
| From: John Heil (The Universe as We Find It [2012], 08.09) | |
| A reaction: Nice speculation! Scientists and verificationists seem to cry 'foul!' when philosophers offer such wild speculations, but I say that's exactly what we pay them do. I'm not sure whether I understand reality having its own nature, though! |
| 18496 | If possible worlds are just fictions, they can't be truthmakers for modal judgements [Heil] |
| Full Idea: If the other possible worlds are merely useful fictions, we are left wondering what the truthmakers for all those modal judgements might be. | |
| From: John Heil (The Universe as We Find It [2012], 08.07) | |
| A reaction: I suddenly see that this is the train of thought that led me to believe in real powers and dispositions, and which retrospectively led me to love the truthmaker idea. Even real Lewisian worlds don't seem adequate as truthmakers here. |
| 18525 | Mental abstraction does not make what is abstracted mind-dependent [Heil] |
| Full Idea: Talk of abstraction and 'partial consideration' (Locke) does not make what is abstracted mind-dependent. In abstracting, you attend to what is there to be considered. | |
| From: John Heil (The Universe as We Find It [2012], 05.7) | |
| A reaction: Quite so. The point is to focus on aspects of reality. Does anyone seriously doubt that reality has 'aspects'? |
| 18504 | Only particulars exist, and generality is our mode of presentation [Heil] |
| Full Idea: Existing things are particular, and generality is a feature of our ways of representing the universe. | |
| From: John Heil (The Universe as We Find It [2012], 01.1) | |
| A reaction: This is right, and expressed with beautiful simplicity. How could anyone disagree with this? But they do! |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| Full Idea: In the hierarchy of reduction, when we investigate questions in biology, we have to assume the laws of chemistry but not of economics. We could never find a law of biology that contradicted something in physics or in chemistry. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.1) | |
| A reaction: This spells out the idea that there is a direction of dependence between aspects of the world, though we should be cautious of talking about 'levels' (see Idea 7003). We cannot choose the direction in which reduction must go. |
| 18503 | You can think of tomatoes without grasping what they are [Heil] |
| Full Idea: You can entertain thoughts of things like tomatoes without a grasp of what they are. | |
| From: John Heil (The Universe as We Find It [2012], 08.10) | |
| A reaction: Lowe seemed to think that you had to grasp the generic essence of a tomato before you could think about it, but I agree entirely with Heil. |
| 18537 | Linguistic thought is just as imagistic as non-linguistic thought [Heil] |
| Full Idea: Thinking - ordinary conscious thinking - is imagistic. This is so for 'linguistic' or 'sentential' thoughts as well as for patently non-linguistic thoughts. | |
| From: John Heil (The Universe as We Find It [2012], 12.10) | |
| A reaction: This claim (that linguistic thought is just as imagistic as non-linguistic thought) strikes me as an excellent insight. |
| 18538 | Non-conscious thought may be unlike conscious thought [Heil] |
| Full Idea: Non-conscious thought need not resemble conscious thought occurring out of sight. | |
| From: John Heil (The Universe as We Find It [2012], 12.10) |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| Full Idea: The extensional presentation of a concept is just a list of the objects falling under the concept. In contrast, an intensional presentation of a concept gives a characterization of the concept, which allows us to pick out which objects fall under it. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.4) | |
| A reaction: Logicians seem to favour the extensional view, because (in the standard view) sets are defined simply by their members, so concepts can be explained using sets. I take this to be a mistake. The intensional view seems obviously prior. |
| 18536 | The subject-predicate form reflects reality [Heil] |
| Full Idea: I like to think that the subject-predicate form reflects a fundamental division in reality. | |
| From: John Heil (The Universe as We Find It [2012], 10.1) | |
| A reaction: That is, he defends the idea that there are substances, and powerful qualities pertaining to those substances. I sympathise, but this slogan makes it too simple. |
| 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? |
| 18497 | Many reject 'moral realism' because they can't see any truthmakers for normative judgements [Heil] |
| Full Idea: It is the difficulty in imagining what truthmakers for normative judgements might be that leads many philosophers to find 'moral realism' unappealing. | |
| From: John Heil (The Universe as We Find It [2012], 08.07) | |
| A reaction: I like that a lot. My proposal for metaethics is that it should be built on the concept of a 'value-maker' |
| 18519 | If there were infinite electrons, they could vanish without affecting total mass-energy [Heil] |
| Full Idea: In a universe containing an infinite number of electrons would mass-energy be conserved? ...Electrons could come and go without affecting the total mass-energy. | |
| From: John Heil (The Universe as We Find It [2012], 03.6) | |
| A reaction: This seems to be a very persuasive reason for doubting that the universe contains an infinite number of electrons. In fact I suspect that infinite numbers have no bearing on nature at all. (Actually, I suspect them of being fictions). |
| 18526 | We should focus on actual causings, rather than on laws and causal sequences [Heil] |
| Full Idea: I believe our understanding of causation would benefit from a shift of attention from causal sequences and laws, to instances of causation: 'causings'. | |
| From: John Heil (The Universe as We Find It [2012], 06.1) | |
| A reaction: His aim is to get away from generalities, and focus on the actual operation of powers which is involved. He likes the case of two playing cards propped against one another. I'm on his side. Laws come last in the story, and should not come first. |
| 18527 | Probabilistic causation is not a weak type of cause; it is just a probability of there being a cause [Heil] |
| Full Idea: The label 'probabilistic causation' is misleading. What you have is not a weakened or tentative kind of causing, but a probability of there being a cause. | |
| From: John Heil (The Universe as We Find It [2012], 06.5) | |
| A reaction: The idea of 'probabilistic causation' strikes me as an empty philosophers' concoction, so I agree with Heil. |
| 18520 | Electrons are treated as particles, but they lose their individuality in relations [Heil] |
| Full Idea: Although it is convenient to speak of electrons as particles or elementary substances, when they enter into relations they can 'lose their individuality. Then an electron becomes a kind of 'abstract particular', a way a given system is, a mode. | |
| From: John Heil (The Universe as We Find It [2012], 03.7) | |
| A reaction: Heil rightly warns us against basing our metaphysics on disputed theories of quantum mechanics. |
| 18501 | Maybe the universe is fine-tuned because it had to be, despite plans by God or Nature? [Heil] |
| Full Idea: Maybe the universe is fine-tuned as it is, not because things happened to fall out as they did during and immediately after the Big Bang, or because God so ordained it, but because God or the Big Bang had no choice. | |
| From: John Heil (The Universe as We Find It [2012], 08.09) | |
| A reaction: You'd be hard put to so why it had to be fine-tuned, so this seems to be a nice speculation. Unverifiable but wholly meaningful. Maybe the stuff fine-tunes itself, by mutual interaction. Or it is the result of natural selection (Lee Smolin). |