31 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. |
| 8623 | Proof reveals the interdependence of truths, as well as showing their certainty [Euclid, by Frege] |
| Full Idea: Euclid gives proofs of many things which anyone would concede to him without question. ...The aim of proof is not merely to place the truth of a proposition beyond doubt, but also to afford us insight into the dependence of truths upon one another. | |
| From: report of Euclid (Elements of Geometry [c.290 BCE]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §02 | |
| A reaction: This connects nicely with Shoemaker's view of analysis (Idea 8559), which I will adopt as my general view. I've always thought of philosophy as the aspiration to wisdom through the cartography of concepts. |
| 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? |
| 13907 | If you pick an arbitrary triangle, things proved of it are true of all triangles [Euclid, by Lemmon] |
| Full Idea: Euclid begins proofs about all triangles with 'let ABC be a triangle', but ABC is not a proper name. It names an arbitrarily selected triangle, and if that has a property, then we can conclude that all triangles have the property. | |
| From: report of Euclid (Elements of Geometry [c.290 BCE]) by E.J. Lemmon - Beginning Logic 3.2 | |
| A reaction: Lemmon adds the proviso that there must be no hidden assumptions about the triangle we have selected. You must generalise the properties too. Pick a triangle, any triangle, say one with three angles of 60 degrees; now generalise from it. |
| 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. |
| 6297 | Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik] |
| Full Idea: Euclid's geometry is a synthetic geometry; Descartes supplied an analytic version of Euclid's geometry, and we now have analytic versions of the early non-Euclidean geometries. | |
| From: report of Euclid (Elements of Geometry [c.290 BCE]) by Michael D. Resnik - Maths as a Science of Patterns One.4 | |
| A reaction: I take it that the original Euclidean axioms were observations about the nature of space, but Descartes turned them into a set of pure interlocking definitions which could still function if space ceased to exist. |
| 9603 | An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR] |
| Full Idea: Assume a largest prime, then multiply the primes together and add one. The new number isn't prime, because we assumed a largest prime; but it can't be divided by a prime, because the remainder is one. So only a larger prime could divide it. Contradiction. | |
| From: report of Euclid (Elements of Geometry [c.290 BCE]) by James Robert Brown - Philosophy of Mathematics Ch.1 | |
| A reaction: Not only a very elegant mathematical argument, but a model for how much modern logic proceeds, by assuming that the proposition is false, and then deducing a contradiction from it. |
| 9894 | A unit is that according to which each existing thing is said to be one [Euclid] |
| Full Idea: A unit is that according to which each existing thing is said to be one. | |
| From: Euclid (Elements of Geometry [c.290 BCE], 7 Def 1) | |
| A reaction: See Frege's 'Grundlagen' §29-44 for a sustained critique of this. Frege is good, but there must be something right about the Euclid idea. If I count stone, paper and scissors as three, each must first qualify to be counted as one. Psychology creeps in. |
| 8738 | Postulate 2 says a line can be extended continuously [Euclid, by Shapiro] |
| Full Idea: Euclid's Postulate 2 says the geometer can 'produce a finite straight line continuously in a straight line'. | |
| From: report of Euclid (Elements of Geometry [c.290 BCE]) by Stewart Shapiro - Thinking About Mathematics 4.2 | |
| A reaction: The point being that this takes infinity for granted, especially if you start counting how many points there are on the line. The Einstein idea that it might eventually come round and hit you on the back of the head would have charmed Euclid. |
| 22278 | Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid] |
| Full Idea: Euclid's axioms were insufficient to derive all the theorems of geometry: at various points in his proofs he appealed to properties that are obvious from the diagrams but do not follow from the stated axioms. | |
| From: comment on Euclid (Elements of Geometry [c.290 BCE]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 03 'aim' | |
| A reaction: I suppose if the axioms of a system are based on self-evidence, this would licence an appeal to self-evidence elsewhere in the system. Only pedants insist on writing down what is obvious to everyone! |
| 8673 | Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend] |
| Full Idea: Euclid's fifth 'parallel' postulate says if there is an infinite straight line and a point, then there is only one straight line through the point which won't intersect the first line. This axiom is independent of Euclid's first four (agreed) axioms. | |
| From: report of Euclid (Elements of Geometry [c.290 BCE]) by Michèle Friend - Introducing the Philosophy of Mathematics 2.2 | |
| A reaction: This postulate was challenged in the nineteenth century, which was a major landmark in the development of modern relativist views of knowledge. |
| 10250 | Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid] |
| Full Idea: Euclid gives no principle of continuity, which would sanction an inference that if a line goes from the outside of a circle to the inside of circle, then it must intersect the circle at some point. | |
| From: comment on Euclid (Elements of Geometry [c.290 BCE]) by Stewart Shapiro - Philosophy of Mathematics 6.1 n2 | |
| A reaction: Cantor and Dedekind began to contemplate discontinuous lines. |
| 10302 | Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays] |
| Full Idea: Euclid postulates: One can join two points by a straight line; Hilbert states the axiom: Given any two points, there exists a straight line on which both are situated. | |
| From: report of Euclid (Elements of Geometry [c.290 BCE]) by Paul Bernays - On Platonism in Mathematics p.259 |
| 14157 | Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid] |
| Full Idea: In descriptive geometry the first 26 propositions of Euclid hold. In projective geometry the 1st, 7th, 16th and 17th require modification (as a straight line is not a closed series). Those after 26 depend on the postulate of parallels, so aren't assumed. | |
| From: comment on Euclid (Elements of Geometry [c.290 BCE]) by Bertrand Russell - The Principles of Mathematics §388 |
| 1600 | Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik] |
| Full Idea: The best known example of Euclid's 'common notions' is "If equals are subtracted from equals the remainders are equal". These can be called axioms, and are what "the man who is to learn anything whatever must have". | |
| From: report of Euclid (Elements of Geometry [c.290 BCE], 72a17) by David Roochnik - The Tragedy of Reason p.149 |
| 12887 | A whole must have one characteristic, an internal relation, and a structure [Rescher/Oppenheim] |
| Full Idea: A whole must possess an attribute peculiar to and characteristic of it as a whole; there must be a characteristic relation of dependence between the parts; and the whole must have some structure which gives it characteristics. | |
| From: Rescher,N/Oppenheim,P (Logical Analysis of Gestalt Concepts [1955], p.90), quoted by Peter Simons - Parts 9.2 | |
| A reaction: Simons says these are basically sensible conditions, and tries to fill them out. They seem a pretty good start, and I must resist the temptation to rush to borderline cases. |
| 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. |
| 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…. |