66 ideas
6947 | Metaphysics does not rest on facts, but on what we are inclined to believe [Peirce] |
Full Idea: Metaphysical systems have not usually rested upon any observed facts, or not in any great degree. They are chiefly adopted because their fundamental propositions seem 'agreeable to reason', which means that which we find ourselves inclined to believe. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.15) | |
A reaction: This leads to Peirce's key claim - that we should allow our beliefs to be formed by something outside of ourselves. I don't share Peirce's contempt for metaphysics, which I take to be about the most abstract presuppositions of our ordinary beliefs. |
6937 | Reason aims to discover the unknown by thinking about the known [Peirce] |
Full Idea: The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p. 7) | |
A reaction: I defy anyone to come up with a better definition of reasoning than that. The emphasis is on knowledge rather than truth, which you would expect from a pragmatist. …Actually the definition doesn't cover conditional reasoning terribly well. |
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. |
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. |
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. |
21492 | Realism is basic to the scientific method [Peirce] |
Full Idea: The fundamental hypothesis of the method of science is this: There are real things, whose characters are entirely independent of our opinion of them. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877]), quoted by Albert Atkin - Peirce 3 'method' | |
A reaction: He admits later that this is only a commitment and not a fact. It seems to me that when you combine this idea with the huge success of science, the denial of realism is crazy. Philosophy has a lot to answer for. |
6949 | If someone doubted reality, they would not actually feel dissatisfaction [Peirce] |
Full Idea: Nobody can really doubt that there are Reals, for, if he did, doubt would not be a source of dissatisfaction. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.19) | |
A reaction: This rests on Peirce's view that all that really matters is a sense of genuine dissatisfaction, rather than a theoretical idea. So even at the end of Meditation One, Descartes isn't actually worried about whether his furniture exists. |
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). |
15083 | The fundamental case of logical necessity is the valid conclusion of an inference [McFetridge, by Hale] |
Full Idea: McFetridge's conception of logical necessity is one which sees the concept as receiving its fundamental exemplification in the connection between the premiss and conclusion of a deductively valid inference. | |
From: report of Ian McFetridge (Logical Necessity: Some Issues [1986]) by Bob Hale - Absolute Necessities 2 | |
A reaction: This would mean that p could be logically necessary but false (if it was a valid argument from false premisses). What if it was a valid inference in a dodgy logical system (including 'tonk', for example)? |
15084 | In the McFetridge view, logical necessity means a consequent must be true if the antecedent is [McFetridge, by Hale] |
Full Idea: McFetridge's view proves that if the conditional corresponding to a valid inference is logically necessary, then there is no sense in which it is possible that its antecedent be true but its consequent false. ..This result generalises to any statement. | |
From: report of Ian McFetridge (Logical Necessity: Some Issues [1986]) by Bob Hale - Absolute Necessities 2 | |
A reaction: I am becoming puzzled by Hale's assertion that logical necessity is 'absolute', while resting his case on a conditional. Are we interested in the necessity of the inference, or the necessity of the consequent? |
12181 | Traditionally, logical necessity is the strongest, and entails any other necessities [McFetridge] |
Full Idea: The traditional crucial assumption is that logical necessity is the strongest notion of necessity. If it is logically necessary that p, then it is necessary that p in any other use of the notion of necessity there may be (physically, practically etc.). | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
A reaction: Sounds right. We might say it is physically necessary simply because it is logically necessary, and even that it is metaphysically necessary because it is logically necessary (required by logic). Logical possibility is hence the weakest kind? |
12184 | Logical necessity overrules all other necessities [McFetridge] |
Full Idea: If it is logically necessary that if p then q, then there is no other sense of 'necessary' in which it is not necessary that if p then q. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
A reaction: The thesis which McFetridge proposes to defend. The obvious rival would be metaphysical necessity, and the rival claim would presumably be that things are only logically necessary if that is entailed by a metaphysical necessity. Metaphysics drives logic. |
12180 | Logical necessity requires that a valid argument be necessary [McFetridge] |
Full Idea: There will be a legitimate notion of 'logical' necessity only if there is a notion of necessity which attaches to the claim, concerning a deductively valid argument, that if the premisses are true then so is the conclusion. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
A reaction: He quotes Aristotle's Idea 11148 in support. Is this resting a stronger idea on a weaker one? Or is it the wrong way round? We endorse validity because we see the necessity; we don't endorse necessity because we see 'validity'. |
12183 | It is only logical necessity if there is absolutely no sense in which it could be false [McFetridge] |
Full Idea: Is there any sense in which, despite an ascription of necessity to p, it is held that not-p is possible? If there is, then the original claim then it was necessary is not a claim of 'logical' necessity (which is the strongest necessity). | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
A reaction: See Idea 12181, which leads up to this proposed "test" for logical necessity. McFetridge has already put epistemic ('for all I know') possibility to one side. □p→¬◊¬p is the standard reading of necessity. His word 'sense' bears the burden. |
12192 | The mark of logical necessity is deduction from any suppositions whatever [McFetridge] |
Full Idea: The manifestation of the belief that a mode of inference is logically necessarily truth-preserving is the preparedness to employ that mode of inference in reasoning from any set of suppositions whatsoever. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §4) | |
A reaction: He rests this on the idea of 'cotenability' of the two sides of a counterfactual (in Mill, Goodman and Lewis). There seems, at first blush, to be a problem of the relevance of the presuppositions. |
12182 | We assert epistemic possibility without commitment to logical possibility [McFetridge] |
Full Idea: Time- and person-relative epistemic possibility can be asserted even when logical possibility cannot, such as undecided mathematical propositions. 'It may be that p' just comes to 'For all I know, not-p'. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
A reaction: If it is possible 'for all I know', then it could be actual for all I know, and if we accept that it might be actual, we could hardly deny that it is logically possible. Logical and epistemic possibilities of mathematical p stand or fall together. |
12187 | Objectual modal realists believe in possible worlds; non-objectual ones rest it on the actual world [McFetridge] |
Full Idea: The 'objectual modal realist' holds that what makes modal beliefs true are certain modal objects, typically 'possible worlds'. ..The 'non-objectual modal realist' says modal judgements are made true by how things stand with respect to this world. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §2) | |
A reaction: I am an enthusiastic 'non-objectual modal realist'. I accept the argument that real possible worlds have no relevance to the actual world, and explain nothing (see Jubien). The possibilities reside in the 'powers' of this world. See Molnar on powers. |
12186 | Modal realists hold that necessities and possibilities are part of the totality of facts [McFetridge] |
Full Idea: The 'modal realist' holds that part of the totality of what is the case, the totality of facts, are such things as that certain events could have happened, certain propositions are necessarily true, if this happened then that would have been the case. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §2) | |
A reaction: I am an enthusiastic modal realist. If the aim of philosophy is 'to understand' (and I take that to be the master idea of the subject) then no understanding is possible which excludes the possibilities and necessities in things. |
6941 | We are entirely satisfied with a firm belief, even if it is false [Peirce] |
Full Idea: As soon as a firm belief is reached we are entirely satisfied, whether the belief be true or false. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.10) | |
A reaction: This does not deny that the truth or falsehood of a belief is independent of whether we are satisfied with it. It is making a fair point, though, about why we believe things, and it can't be because of truth, because we don't know how to ensure that. |
6940 | The feeling of belief shows a habit which will determine our actions [Peirce] |
Full Idea: The feeling of believing is a more or less sure indication of there being established in our nature some habit which will determine our actions. Doubt never has such an effect. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.10) | |
A reaction: It is one thing to assert this fairly accurate observation, and another to assert that this is the essence or definition of a belief. Perhaps it is the purpose of belief, without being the phenomenological essence of it. We act in states of uncertainty. |
6942 | We want true beliefs, but obviously we think our beliefs are true [Peirce] |
Full Idea: We seek for a belief that we shall think to be true; but we think each one of our beliefs to be true, and, indeed, it is mere tautology to say so. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
A reaction: If, as I do, you like to define belief as 'commitment to truth', Peirce makes a rather startling observation. You are rendered unable to ask whether your beliefs are true, because you have defined them as true. Nice point… |
6943 | A mere question does not stimulate a struggle for belief; there must be a real doubt [Peirce] |
Full Idea: The mere putting of a proposition into the interrogative form does not stimulate the mind to any struggle after belief; there must be a real and living doubt. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
A reaction: This the attractive aspect of Peirce's pragmatism, that he is always focusing on real life rather than abstract theory or pure logic. |
6598 | We need our beliefs to be determined by some external inhuman permanency [Peirce] |
Full Idea: It is necessary that a method should be found by which our beliefs be determined by nothing human, but by some external permanency - by something upon which our thinking has no effect. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877]), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5 | |
A reaction: This very sensible and interesting remark hovers somewhere between empiricism and pragmatism. Fogelin very persuasively builds his account of knowledge on it. The key point is that we hardly ever choose what to believe. See Idea 2454. |
6944 | Demonstration does not rest on first principles of reason or sensation, but on freedom from actual doubt [Peirce] |
Full Idea: It is a common idea that demonstration must rest on indubitable propositions, either first principles of a general nature, or first sensations; but actual demonstration is completely satisfactory if it starts from propositions free from all actual doubt. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
A reaction: Another nice example of Peirce focusing on the practical business of thinking, rather than abstract theory. I agree with this approach, that explanation and proof do not aim at perfection and indubitability, but at what satisfies a critical mind. |
6948 | Doubts should be satisfied by some external permanency upon which thinking has no effect [Peirce] |
Full Idea: To satisfy our doubts it is necessary that a method should be found by which our beliefs may be determined by nothing human, but by some external permanency - by something upon which our thinking has no effect. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.18) | |
A reaction: This may be the single most important idea in pragmatism and in the philosophy of science. See Fodor on experiments (Idea 2455). Put the question to nature. The essential aim is to be passive in our beliefs - just let reality form them. |
6945 | Once doubt ceases, there is no point in continuing to argue [Peirce] |
Full Idea: Some people seem to love to argue a point after all the world is fully convinced of it. But no further advance can be made. When doubt ceases, mental action on the subject comes to an end; and, if it did go on, it would be without purpose. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
A reaction: This is the way Peirce's pragmatism, which deals with how real thinking actually works (rather than abstract logic), deals with scepticism. However, there is a borderline where almost everyone is satisfied, but the very wise person remains sceptical. |
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. |
6939 | What is true of one piece of copper is true of another (unlike brass) [Peirce] |
Full Idea: The guiding principle is that what is true of one piece of copper is true of another; such a guiding principle with regard to copper would be much safer than with regard to many other substances - brass, for example. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p. 8) | |
A reaction: Peirce is so beautifully simple and sensible. This gives the essential notion of a natural kind, and is a key notion in our whole understanding of physical reality. |
6938 | Natural selection might well fill an animal's mind with pleasing thoughts rather than true ones [Peirce] |
Full Idea: It is probably of more advantage to an animal to have his mind filled with pleasing and encouraging visions, independently of their truth; and thus, upon unpractical subjects, natural selection might occasion a fallacious tendency of thought. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p. 8) | |
A reaction: Note that this is a pragmatist saying that a set of beliefs might work fine but be untrue. So Peirce does not have the highly relativistic notion of truth of some later pragmatists. Good for him. Note the early date to be thinking about Darwin. |
6946 | If death is annihilation, belief in heaven is a cheap pleasure with no disappointment [Peirce] |
Full Idea: If death is annihilation, then the man who believes that he will certainly go straight to heaven when he dies, provided he have fulfilled certain simple observances in this life, has a cheap pleasure which will not be followed by the least disappointment. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.12) | |
A reaction: This is a nicely wicked summary of one side of Pascal's options. All the problems of the argument are built into Peirce's word "cheap". Peirce goes on to talk about ostriches burying their heads. |