66 ideas
| 20771 | Six parts: dialectic, rhetoric, ethics, politics, physics, theology [Cleanthes, by Diog. Laertius] |
| Full Idea: Cleanthes says there are six parts: dialectic, rhetoric, ethics, politics, physics, and theology. | |
| From: report of Cleanthes (fragments/reports [c.270 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.41 | |
| A reaction: This was a minority view, as most stoics agreed with Zeno and Chrysippus that there are three main topics. Nowadays there is little discussion of the 'parts' of philosophy, but the recent revival of meta-philosophy should encourage it. |
| 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. |
| 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. |
| 18335 | There are five problems which the truth-maker theory might solve [Rami] |
| Full Idea: It is claimed that truth-makers explain universals, or ontological commitment, or commitment to realism, or to the correspondence theory of truth, or to falsify behaviourism or phenomenalism. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 04) | |
| A reaction: [compressed] This expands the view that truth-making is based on its explanatory power, rather than on its intuitive correctness. I take the theory to presuppose realism. I don't believe in universals. It marginalises correspondence. Commitment is good! |
| 18334 | The truth-maker idea is usually justified by its explanatory power, or intuitive appeal [Rami] |
| Full Idea: The two strategies for justifying the truth-maker principle are that it has an explanatory role (for certain philosophical problems and theses), or that it captures the best philosophical intuition of the situation. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 04) | |
| A reaction: I would go for 'intuitive', but not in the sense of a pure intuition, but with 'intuitive' as a shorthand for overall coherence. To me the appeal of truth-maker is its place in a naturalistic view of reality. I love explanation, but not here. |
| 18339 | The truth-making relation can be one-to-one, or many-to-many [Rami] |
| Full Idea: The truth-making relation can be one-to-one, or many-many. In the latter case, different truths may have the same truth-maker, and one truth may have different truth-makers. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 05) | |
| A reaction: 'There is at least one cat' obviously has many possible truth-makers. Many statements will be made true by the mere existence of a particular cat (such as 'there is an animal in the room' and 'there is a cat in the room'). Many-many wins? |
| 18333 | Central idea: truths need truthmakers; and possibly all truths have them, and makers entail truths [Rami] |
| Full Idea: The main full-blooded truth-maker principle is that x is true iff there is a y that is its truth-maker. This implies the principles that if x is true x has a truth-maker, and the principle that if x has a truth-maker then x is true. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 03) | |
| A reaction: [compressed] Rami calls the second principle 'maximalism' and the third principle 'purism'. To reject maximalism is to hold a more restricted version of truth-makers. That is, the claim is that lots of truths have truth-makers. |
| 18342 | Most theorists say that truth-makers necessitate their truths [Rami] |
| Full Idea: Most truth-maker theorists regard the necessitation of a truth by a truth-maker as a necessary condition of truth-making. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 07) | |
| A reaction: It seems to me that reality is crammed full of potential truth-makers, but not crammed full of truths. If there is no thinking in the universe, then there are no truths. If that is false, then what sort of weird beast is a 'truth'? |
| 18340 | It seems best to assume different kinds of truth-maker, such as objects, facts, tropes, or events [Rami] |
| Full Idea: Truthmaker anti-monism holds the view that there are truth-makers of different kinds. For example, objects, facts, tropes or events can all be regarded as truthmakers. Objects seem right for existential truths but not others, so anti-monism seems best. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 05) | |
| A reaction: Presumably we need to identify the different types of truth (analytic, synthetic, general, particular...), and only then ask what truth-makers there are for the different types. To presuppose one type of truthmaker would be crazy. |
| 18341 | Truth-makers seem to be states of affairs (plus optional individuals), or individuals and properties [Rami] |
| Full Idea: As truth-makers, some theorists only accept states of affairs, some only accept individuals and states of affairs, and some only accept individuals and particular properties. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 06) | |
| A reaction: It seems to me rash to opt for one of these. Truths come in wide-ranging and subtly different types, and the truth-makers probably have a similar range. Any one of these theories will almost certainly quickly succumb to a counterexample. |
| 18346 | 'Truth supervenes on being' only gives necessary (not sufficient) conditions for contingent truths [Rami] |
| Full Idea: The thesis that 'truth supervenes on being' (with or without possible worlds) offers only a necessary condition for the truth of contingent propositions, whereas the standard truth-maker theory offers necessary and sufficient conditions. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 09) | |
| A reaction: The point, I suppose, is that the change in being might be irrelevant to the proposition in question, so any old change in being will not ensure a change in the truth of the proposition. Again we ask - but what is this truth about? |
| 18345 | 'Truth supervenes on being' avoids entities as truth-makers for negative truths [Rami] |
| Full Idea: The important advantage of 'truth supervenes on being' is that it can be applied to positive and negative contingent truths, without postulating any entities that are responsible for the truth of negative truths. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 09) | |
| A reaction: [For this reason, Lewis favours a possible worlds version of the theory] I fear that it solves that problem by making the truth-maker theory so broad-brush that it not longer says very much, apart from committing it to naturalism. |
| 18343 | Maybe a truth-maker also works for the entailments of the given truth [Rami] |
| Full Idea: The 'entailment principle' for truth-makers says that if x is a truth-maker for y, and y entails z, then x is a truth-maker for z. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 08) | |
| A reaction: I think the correct locution is that 'x is a potential truth-maker for z' (should anyone every formulate z, which in most cases they never will, since the entailments of y are probably infinite). Merricks would ask 'but are y and z about the same thing?'. |
| 18338 | Truth-making is usually internalist, but the correspondence theory is externalist [Rami] |
| Full Idea: Most truth-maker theorists are internalists about the truth-maker relation. ...But the correspondence theory makes truth an external relation to some portion of reality. So a truth-maker internalist should not claim to be a narrow correspondence theorist. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 05) | |
| A reaction: [wording rearranged] Like many of Rami's distinctions in this article, this feels simplistic. Sharp distinctions can only be made using sharp vocabulary, and there isn't much of that around in philosophy! |
| 18337 | Correspondence theories assume that truth is a representation relation [Rami] |
| Full Idea: One guiding intuition concerning a correspondence theory of truth says that the relation that accounts for the truth of a truth-bearer is some kind of representation relation. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 05) | |
| A reaction: I unfashionably cling on to some sort of correspondence theory. The paradigm case is of a non-linguistic animal which forms correct or incorrect views about its environment. Truth is a relation, not a property. I see the truth in a bad representation. |
| 18347 | Deflationist truth is an infinitely disjunctive property [Rami] |
| Full Idea: According to the moderate deflationist truth is an infinitely disjunctive property. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 10) | |
| A reaction: [He cites Horwich 1998] That is, I presume, that truth is embodied in an infinity of propositions of the form '"p" is true iff p'. |
| 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. |
| 18350 | Truth-maker theorists should probably reject the converse Barcan formula [Rami] |
| Full Idea: There are good reasons for the truth-maker theorist to reject the converse Barcan formula. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], note 16) | |
| A reaction: In the text (p.15) Rami cites the inference from 'necessarily everything exists' to 'everything exists necessarily'. [See Williamson 1999] |
| 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. |
| 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. |
| 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. |
| 18336 | Internal relations depend either on the existence of the relata, or on their properties [Rami] |
| Full Idea: An internal relation is 'existential' if x and y relate in that way whenever they both exist. An internal relation is 'qualitative' if x and y relate in that way whenever they have certain intrinsic properties. | |
| From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 05) | |
| A reaction: [compressed - Rami likes to write these things in fashionable quasi-algebra, but I have a strong prejudice in this database for expressing ideas in English; call me old-fashioned] The distinction strikes me as simplistic. I would involve dispositions. |
| 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). |
| 10938 | The extremes of essentialism are that all properties are essential, or only very trivial ones [Rami] |
| Full Idea: It would be natural to label one extreme view 'maximal essentialism' - that all of an object's properties are essential - and the other extreme 'minimal' - that only trivial properties such as self-identity of being either F or not-F are essential. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008]) | |
| A reaction: Personally I don't accept the trivial ones as being in any way describable as 'properties'. The maximal view destroys any useful notion of essence. Leibniz is a minority holder of the maximal view. I would defend a middle way. |
| 10940 | An 'individual essence' is possessed uniquely by a particular object [Rami] |
| Full Idea: An 'individual essence' is a property that in addition to being essential is also unique to the object, in the sense that it is not possible that something distinct from that object possesses that property. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008], §5) | |
| A reaction: She cites a 'haecceity' (or mere bare identity) as a trivial example of an individual essence. |
| 10939 | 'Sortal essentialism' says being a particular kind is what is essential [Rami] |
| Full Idea: According to 'sortal essentialism', an object could not have been of a radically different kind than it in fact is. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008], §4) | |
| A reaction: This strikes me as thoroughly wrong. Things belong in kinds because of their properties. Could you remove all the contingent features of a tiger, leaving it as merely 'a tiger', despite being totally unrecognisable? |
| 10934 | Unlosable properties are not the same as essential properties [Rami] |
| Full Idea: It is easy to confuse the notion of an essential property that a thing could not lack, with a property it could not lose. My having spent Christmas 2007 in Tennessee is a non-essential property I could not lose. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008], §1) | |
| A reaction: The idea that having spent Christmas in Tennessee is a property I find quite bewildering. Is my not having spent my Christmas in Tennessee one of my properties? I suspect that real unlosable properties are essential ones. |
| 10933 | Physical possibility is part of metaphysical possibility which is part of logical possibility [Rami] |
| Full Idea: The usual view is that 'physical possibilities' are a natural subset of the 'metaphysical possibilities', which in turn are a subset of the 'logical possibilities'. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008], §1) | |
| A reaction: [She cites Fine 2002 for an opposing view] I prefer 'natural' to 'physical', leaving it open where the borders of the natural lie. I take 'metaphysical' possibility to be 'in all naturally possible worlds'. So is a round square a logical possibility? |
| 10932 | If it is possible 'for all I know' then it is 'epistemically possible' [Rami] |
| Full Idea: There is 'epistemic possibility' when it is 'for all I know'. That is, P is epistemically possible for agent A just in case P is consistent with what A knows. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008], §1) | |
| A reaction: Two problems: maybe 'we' know, and A knows we know, but A doesn't know. And maybe someone knows, but we are not sure about that, which seems to introduce a modal element into the knowing. If someone knows it's impossible, it's impossible. |
| 6028 | Bodies interact with other bodies, and cuts cause pain, and shame causes blushing, so the soul is a body [Cleanthes, by Nemesius] |
| Full Idea: Cleanthes says no incorporeal interacts with a body, but one body interacts with another body; the soul interacts with the body when it is sick and being cut, and the body feels shame and fear, and turns red or pale, so the soul is a body. | |
| From: report of Cleanthes (fragments/reports [c.270 BCE]) by Nemesius - De Natura Hominis 78,7 | |
| A reaction: This is precisely the interaction problem with dualism, or, as we might now say, the problem of mental causation. The standard Stoic view is that the soul is a sort of rarefied fire, which disperses at death. |
| 20831 | The soul suffers when the body hurts, creates redness from shame, and pallor from fear [Cleanthes] |
| Full Idea: Nothing incorporeal shares an experience with a body …but the soul suffers with the body when it is ill and when it is cut, and the body suffers with the soul - when the soul is ashamed the body turns red, and pale when the soul is frightened. | |
| From: Cleanthes (fragments/reports [c.270 BCE]), quoted by Nemesius - De Natura Hominis 2 | |
| A reaction: Aha - my favourite example of the corporeal nature of the mind - blushing! It is the conscious content of the thought which brings blood to the cheeks. |
| 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. |
| 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. |
| 5993 | The ascending scale of living creatures requires a perfect being [Cleanthes, by Tieleman] |
| Full Idea: Cleanthes tried to prove the existence of God, arguing that the ascending scale of living creatures requires there to be a perfect being. | |
| From: report of Cleanthes (fragments/reports [c.270 BCE]) by Teun L. Tieleman - Cleanthes | |
| A reaction: Not a very good argument. Even if you accept its basic claim, it is not clear what has to exist. A perfect tree? If the being transcends the physical (in order to achieve perfection), does it cease to be a 'being'? |