40 ideas
| 4643 | The Principle of Sufficient Reason does not presuppose that all explanations will be causal explanations [Baggini /Fosl] |
| Full Idea: The Principle of Sufficient Reason does not presuppose that all explanations will be causal explanations. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §3.28) | |
| A reaction: This sounds a reasonable note of caution, but doesn't carry much weight unless some type of non-causal reason can be envisaged. God's free will? Our free will? The laws of causation? |
| 4633 | You cannot rationally deny the principle of non-contradiction, because all reasoning requires it [Baggini /Fosl] |
| Full Idea: Anyone who denies the principle of non-contradiction simultaneously affirms it; it cannot be rationally criticised, because it is presupposed by all rationality. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §1.12) | |
| A reaction: Nietzsche certainly wasn't afraid to ask why we should reject something because it is a contradiction. The 'logic of personal advantage' might allow logical contradictions. |
| 4635 | Dialectic aims at unified truth, unlike analysis, which divides into parts [Baggini /Fosl] |
| Full Idea: Dialectic can be said to aim at wholeness or unity, while 'analytic' thinking divides that with which it deals into parts. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §2.03) | |
| A reaction: I don't accept this division (linked here to Hegel). I am a fan of analysis, as practised by Aristotle, but it is like dismantling an engine to identify and clean the parts, before reassembling it more efficiently. |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| Full Idea: While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: [The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc. |
| 4632 | 'Natural' systems of deduction are based on normal rational practice, rather than on axioms [Baggini /Fosl] |
| Full Idea: A 'natural' system of deduction does not posit any axioms, but looks instead for its formulae to the practices of ordinary rationality. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §1.09) | |
| A reaction: Presumably there is some middle ground, where we attempt to infer the axioms of normal practice, and then build a strict system on them. We must be allowed to criticise 'normal' rationality, I hope. |
| 4631 | In ideal circumstances, an axiom should be such that no rational agent could possibly object to its use [Baggini /Fosl] |
| Full Idea: In ideal circumstances, an axiom should be such that no rational agent could possibly object to its use. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §1.09) | |
| A reaction: Yes, but the trouble is that all our notions of 'rational' (giving reasons, being consistent) break down when we look at unsupported axioms. In what sense is something rational if it is self-evident? |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do? |
| 4638 | The principle of bivalence distorts reality, as when claiming that a person is or is not 'thin' [Baggini /Fosl] |
| Full Idea: Forcing everything into the straightjacket of bivalence seriously distorts the world. The problem is most acute in the case of vague concepts, such as thinness. It is not straightforwardly true or false that a person is thin. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §3.03) | |
| A reaction: Can't argue with that. Can we divide all our concepts into either bivalent or vague? Presumably both propositions and concepts could be bivalent. |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| Full Idea: In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types. |
| 10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
| Full Idea: 'Analysis' is the theory of the real numbers. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: 'Analysis' began with the infinitesimal calculus, which later built on the concept of 'limit'. A continuum of numbers seems to be required to make that work. |
| 10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
| Full Idea: The difficulties for a nominalistic mereological approach to arithmetic is that an infinity of physical objects are needed (space-time points? strokes?), and it must define functions, such as 'successor'. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: Many ontologically austere accounts of arithmetic are faced with the problem of infinity. The obvious non-platonist response seems to be a modal or if-then approach. To postulate infinite abstract or physical entities so that we can add 3 and 2 is mad. |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| Full Idea: A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'! |
| 10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
| Full Idea: The merits of basing an account of mathematics on set theory are that it allows for a comprehensive unified treatment of many otherwise separate branches of mathematics, and that all assumption, including existence, are explicit in the axioms. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I am forming the impression that set-theory provides one rather good model (maybe the best available) for mathematics, but that doesn't mean that mathematics is set-theory. The best map of a landscape isn't a landscape. |
| 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
| Full Idea: Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent. |
| 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
| Full Idea: Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight? |
| 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
| Full Idea: The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6) | |
| A reaction: This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start. |
| 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
| Full Idea: According to 'pattern' structuralism, what we study are not the various particular isomorphic models of arithmetic, but something in addition to them: a corresponding pattern. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §7) | |
| A reaction: Put like that, we have to feel a temptation to wield Ockham's Razor. It's bad enough trying to give the structure of all the isomorphic models, without seeking an even more abstract account of underlying patterns. But patterns connect to minds.. |
| 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
| Full Idea: There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9) | |
| A reaction: I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind? |
| 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
| Full Idea: Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3) | |
| A reaction: [very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations. |
| 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
| Full Idea: It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: [compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent. |
| 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
| Full Idea: Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator. |
| 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
| Full Idea: Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra. |
| 10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
| Full Idea: Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity. |
| 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
| Full Idea: One way for a nominalist to reject appeal to all abstract objects, including sets, is to only appeal to nominalistically acceptable objects, including mereological sums. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I'm suddenly thinking that this looks very interesting and might be the way to go. The issue seems to be whether mereological sums should be seen as constrained by nature, or whether they are unrestricted. See Mereology in Ontology...|Intrinsic Identity. |
| 4640 | If identity is based on 'true of X' instead of 'property of X' we get the Masked Man fallacy ('I know X but not Y') [Baggini /Fosl, by PG] |
| Full Idea: The Masked Man fallacy is when Leibniz's Law is taken as 'X and Y are identical if what is true of X is true of Y' (rather than being about properties). Then 'I know X' but 'I don't know Y' (e.g. my friend wearing a mask) would make X and Y non-identical. | |
| From: report of J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §3.17) by PG - Db (ideas) | |
| A reaction: As the book goes on to explain, Descartes is guilty of this when arguing that I necessarily know my mind but not my body, so they are different. Seems to me that Kripke falls into the same trap. |
| 4647 | 'I have the same car as you' is fine; 'I have the same fiancée as you' is not so good [Baggini /Fosl] |
| Full Idea: If you found that I had the same car as you, I don't suppose you would care, but if you found I had the same fiancée as you, you might not be so happy. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §4.17) | |
| A reaction: A very nice illustration of the ambiguity of "same", and hence of identity. 'I had the same thought as you'. 'I have the same DNA as you'. |
| 4639 | Leibniz's Law is about the properties of objects; the Identity of Indiscernibles is about perception of objects [Baggini /Fosl] |
| Full Idea: Leibniz's Law ('if identical, must have same properties') defines identity according to the properties possessed by the object itself, but the Identity of Indiscernibles defines identity in terms of how things are conceived or grasped by the mind. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §3.16) | |
| A reaction: This is the heart of the problem of identity. We realists must fight for Leibniz's Law, and escort the Identity of Indiscernibles to the door. |
| 4646 | Is 'events have causes' analytic a priori, synthetic a posteriori, or synthetic a priori? [Baggini /Fosl] |
| Full Idea: Of the proposition that "all experienced events have causes", Descartes says this is analytic a priori, Hume says it is synthetic a posteriori, and Kant says it is synthetic a priori. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §4.01) | |
| A reaction: I am not sympathetic to Hume on this (though most people think he is right). I prefer the Kantian view, but he makes a very large claim. Something has to be intuitive. |
| 4645 | 'A priori' does not concern how you learn a proposition, but how you show whether it is true or false [Baggini /Fosl] |
| Full Idea: What makes something a priori is not the means by which it came to be known, but the means by which it can be shown to be true or false. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §4.01) | |
| A reaction: Helpful. Kripke in particular has labelled the notion as an epistemological one, but that does imply a method of acquiring it. Clearly I can learn an a priori truth by reading it the newspaper. |
| 4582 | Basic beliefs are self-evident, or sensual, or intuitive, or revealed, or guaranteed [Baggini /Fosl] |
| Full Idea: Sentence are held to be basic because they are self-evident or 'cataleptic' (Stoics), or rooted in sense data (positivists), or grasped by intuition (Platonists), or revealed by God, or grasped by faculties certified by God (Descartes). | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §1.01) | |
| A reaction: These are a bit blurred. Isn't intuition self-evident? Isn't divine guarantee a type of revelation? How about reason, experience or authority? |
| 4644 | A proposition such as 'some swans are purple' cannot be falsified, only verified [Baggini /Fosl] |
| Full Idea: The problem with falsification is that it fails to work with logically particular claims such as 'some swans are purple'. Examining a million swans and finding no purple ones does not falsify the claim, as there might still be a purple swan out there. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §3.29) | |
| A reaction: Isn't it beautiful how unease about a theory (Popper's) slowly crystallises into an incredibly simple and devastating point? Maybe 'some swans are purple' isn't science unless there is a good reason to propose it? |
| 4584 | The problem of induction is how to justify our belief in the uniformity of nature [Baggini /Fosl] |
| Full Idea: At its simplest, the problem of induction can be boiled down to the problem of justifying our belief in the uniformity of nature. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §1.03) | |
| A reaction: An easy solution to the problem of induction: we treat the uniformity of nature as axiomatic, and then induction is all reasoning which is based on that axiom. The axiom is a working hypothesis, which may begin to appear false. Anomalies are hard. |
| 4583 | How can an argument be good induction, but poor deduction? [Baggini /Fosl] |
| Full Idea: The problem of induction is the problem of how an argument can be good reasoning as induction but poor reasoning as deduction. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §1.03) | |
| A reaction: Nicely put, and a good defence of Hume against the charge that he has just muddled induction and deduction. All reasoning, we insist, should be consistent, or it isn't reasoning. |
| 4634 | Abduction aims at simplicity, testability, coherence and comprehensiveness [Baggini /Fosl] |
| Full Idea: There are some 'principles of selection' in abduction: 1) prefer simple explanations, 2) prefer coherent explanations (consistent with what is already held true), 3) prefer theories that make testable predictions, and 4) be comprehensive in scope. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §2.01) | |
| A reaction: Note that these are desirable, but not necessary (pace Ockham and Ayer). I cannot think of anything to add to the list, so I will adopt it. Abduction is the key to rationality. |
| 4637 | To see if an explanation is the best, it is necessary to investigate the alternative explanations [Baggini /Fosl] |
| Full Idea: The only way to be sure we have the best explanation is to investigate the alternatives and see if they are any better. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §3.01) | |
| A reaction: Unavoidable! Since I love 'best explanation', I now seem to be committed to investigation every mad theory that comes up, just in case it is better. I hope I am allowed to reject after a very quick sniff. |
| 5504 | Maybe we should see persons in four dimensions, with stages or time-slices at an instant [Martin/Barresi] |
| Full Idea: Some recent philosophers have argued that we should replace the three-dimensional view of persons with a four-dimensional view according to which only time-slices, or 'stages', of persons exist at short intervals of time. | |
| From: R Martin / J Barresi (Introduction to 'Personal Identity' [2003], p.3) | |
| A reaction: At first glance this seems to neatly eliminate lots of traditional worries. But why would I want to retain my identity, if someone threatened to brainwash me. I also want to disown my inadequate earlier selves. Interesting, though. Lewis. |
| 5503 | Maybe personal identity is not vital in survival, and other continuations would suffice [Martin/Barresi] |
| Full Idea: A modern question is whether personal identity is primarily what matters in survival; that is, people might cease and be continued by others whose continuation the original people would value as much. | |
| From: R Martin / J Barresi (Introduction to 'Personal Identity' [2003], p.3) | |
| A reaction: When put like this, the proposal seems hard to grasp. It only makes sense if you don't really believe in a thing called 'personal identity'. I don't see how you can believe in it without also believing that for you it has central importance. |
| 5502 | Locke's intrinsic view of personal identity has been replaced by an externalist view [Martin/Barresi] |
| Full Idea: In modern times the Lockean intrinsic relations view of personal identity has been superseded by an extrinsic relations view (also called the 'closest-continuer' or 'externalist' view). | |
| From: R Martin / J Barresi (Introduction to 'Personal Identity' [2003], p.1) | |
| A reaction: Sounds sweeping. My suspicion is that there is a raging fashion for externalist views of everything (justification, content etc.), but this will pass. I take Parfit to be the source of the modern views. |
| 4629 | Consistency is the cornerstone of rationality [Baggini /Fosl] |
| Full Idea: Consistency is the cornerstone of rationality. | |
| From: J Baggini / PS Fosl (The Philosopher's Toolkit [2003], §1.06) | |
| A reaction: This is right, and is a cornerstone of Kant's approach to ethics. Rational beings must follow principles - in order to be consistent in their behaviour. 'Consistent' now requires a definition…. |
| 5505 | For Aristotle the psyche perishes with the body (except possibly 'nous') [Martin/Barresi] |
| Full Idea: In Aristotle's view, with the possible exception of 'nous' the psyche and all its parts come into being at the same time as its associated body; it is inseparable from the body, and perishes along with it. | |
| From: R Martin / J Barresi (Introduction to 'Personal Identity' [2003], p.8) | |
| A reaction: It is suggested that he thought there was only one 'nous', which all humans share (p.9). If he wants to claim that one part is immortal, he doesn't have much evidence. If psyche is the form of the body, it is bound to perish. |