31 ideas
4187 | 'There is nothing without a reason why it should be rather than not be' (a generalisation of 'Why?') [Schopenhauer] |
Full Idea: The Principle may be stated as 'There is nothing without a reason why it should be rather than not be', which is a generalisation of the assumption which justifies the question 'Why?', which is the mother of all science. | |
From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.I) | |
A reaction: This faith is the core of philosophy, to be maintained against all defeatists like Wittgenstein and Colin McGinn. Reality must be rational, or we wouldn't be here to think about it. (Maybe!) |
17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
Full Idea: In intuitionist logic double negation elimination fails. After all, proving that there is no proof that there can't be a proof of S is not the same thing as having a proof of S. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
A reaction: I do like people like Colyvan who explain things clearly. All of this difficult stuff is understandable, if only someone makes the effort to explain it properly. |
17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
Full Idea: The intuitionist rejection of double negation elimination undermines the important reductio ad absurdum proof in classical mathematics. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) |
17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
Full Idea: The law of excluded middle (for every proposition P, either P or not-P) must be carefully distinguished from its semantic counterpart bivalence, that every proposition is either true or false. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
A reaction: So excluded middle makes no reference to the actual truth or falsity of P. It merely says P excludes not-P, and vice versa. |
17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
Full Idea: Löwenheim proved that if a first-order sentence has a model at all, it has a countable model. ...Skolem generalised this result to systems of first-order sentences. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) |
17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
Full Idea: A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) | |
A reaction: The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'. |
17928 | Ordinal numbers represent order relations [Colyvan] |
Full Idea: Ordinal numbers represent order relations. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.2.3 n17) |
17923 | Intuitionists only accept a few safe infinities [Colyvan] |
Full Idea: For intuitionists, all but the smallest, most well-behaved infinities are rejected. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
A reaction: The intuitionist idea is to only accept what can be clearly constructed or proved. |
17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
Full Idea: The problem with infinitesimals is that in some places they behaved like real numbers close to zero but in other places they behaved like zero. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.2) | |
A reaction: Colyvan gives an example, of differentiating a polynomial. |
17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
Full Idea: Given Dedekind's reduction of real numbers to sequences of rational numbers, and other known reductions in mathematics, it was tempting to see basic arithmetic as the foundation of mathematics. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.1) | |
A reaction: The reduction is the famous Dedekind 'cut'. Nowadays theorists seem to be more abstract (Category Theory, for example) instead of reductionist. |
17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
Full Idea: Transfinite inductions are inductive proofs that include an extra step to show that if the statement holds for all cases less than some limit ordinal, the statement also holds for the limit ordinal. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1 n11) |
17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
Full Idea: Most mathematical proofs, outside of set theory, do not explicitly state the set theory being employed. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.1) |
17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
Full Idea: Structuralism is able to explain why mathematicians are typically only interested in describing the objects they study up to isomorphism - for that is all there is to describe. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) |
17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
Full Idea: In re structuralism does not posit anything other than the kinds of structures that are in fact found in the world. ...The problem is that the world may not provide rich enough structures for the mathematics. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) | |
A reaction: You can perceive a repeating pattern in the world, without any interest in how far the repetitions extend. |
4192 | All necessity arises from causation, which is conditioned; there is no absolute or unconditioned necessity [Schopenhauer] |
Full Idea: Necessity has no meaning other than the irresistible sequence of the effect where the cause is given. All necessity is thus conditioned, and absolute or unconditioned necessity is a contradiction in terms. | |
From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.VIII) | |
A reaction: I.e. there is only natural necessity, and no such thing as metaphysical necessity. But what about logical necessity(e.g. 2+3=5)? I think there may be metaphysical necessity, but we can't know much about it, and we are over-confident in assessing it. |
4190 | All understanding is an immediate apprehension of the causal relation [Schopenhauer] |
Full Idea: All understanding is an immediate apprehension of the causal relation. | |
From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.IV) | |
A reaction: Based, I take it, on Hume. Presumably he means a posteriori understanding, as it hardly fits an understanding of arithmetic. Understanding needs more than just causation. What aspects of causation? |
17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
Full Idea: Those who see probabilities as ratios of frequencies can't use Bayes's Theorem if there is no objective prior probability. Those who accept prior probabilities tend to opt for a subjectivist account, where probabilities are degrees of belief. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.8) | |
A reaction: [compressed] |
17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
Full Idea: Mathematics can demonstrate structural similarities between systems (e.g. missing population periods and the gaps in the rings of Saturn). | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) | |
A reaction: [Colyvan expounds the details of his two examples] It is these sorts of results that get people enthusiastic about the mathematics embedded in nature. A misunderstanding, I think. |
17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
Full Idea: Mathematics can show that under a broad range of conditions, something initially surprising must occur (e.g. the hexagonal structure of honeycomb). | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) |
17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
Full Idea: Another style of proof often cited as unexplanatory are brute-force methods such as proof by cases (or proof by exhaustion). | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) |
17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
Full Idea: One kind of proof that is thought to be unexplanatory is the 'reductio' proof. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
A reaction: Presumably you generate a contradiction, but are given no indication of why the contradiction has arisen? Tracking back might reveal the source of the problem? Colyvan thinks reductio can be explanatory. |
17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
Full Idea: It might be argued that any proof by induction is revealing the explanation of the theorem, namely, that it holds by virtue of the structure of the natural numbers. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
A reaction: This is because induction characterises the natural numbers, in the Peano Axioms. |
17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
Full Idea: The proof of the four-colour theorem raises questions about whether a 'proof' that no one understands is a proof. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.6) | |
A reaction: The point is that the theorem (that you can colour countries on a map with just four colours) was proved with the help of a computer. |
3488 | Freud treats the unconscious as intentional and hence mental [Freud, by Searle] |
Full Idea: Freud thinks that our unconscious mental states exist as occurrent intrinsic intentional states even when unconscious. Their ontology is that of the mental, even when they are unconscious. | |
From: report of Sigmund Freud (works [1900]) by John Searle - The Rediscovery of the Mind Ch. 7.V | |
A reaction: Searle states this view in order to attack it. Whether such states are labelled as 'mental' seems uninteresting. Whether unconscious states can be intentional is crucial, and modern scientific understanding of the brain strongly suggest they can. |
17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
Full Idea: One type of generalisation in mathematics extends a system to go beyond what is was originally set up for; another kind involves abstracting away from some details in order to capture similarities between different systems. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.2) |
4191 | What we know in ourselves is not a knower but a will [Schopenhauer] |
Full Idea: What we know in ourselves is never what knows, but what wills, the will. | |
From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.VII) | |
A reaction: An interesting slant on Hume's scepticism about personal identity. Hume was hunting for a thing-which-experiences. If he had sought his will, he might have spotted it. |
5689 | Freud and others have shown that we don't know our own beliefs, feelings, motive and attitudes [Freud, by Shoemaker] |
Full Idea: Freud persuaded many that beliefs, wishes and feelings are sometimes unconscious, and even sceptics about Freud acknowledge that there is self-deception about motive and attitudes. | |
From: report of Sigmund Freud (works [1900]) by Sydney Shoemaker - Introspection p.396 | |
A reaction: This seems to me obviously correct. The traditional notion is that the consciousness is the mind, but now it seems obvious that consciousness is only one part of the mind, and maybe even a peripheral (epiphenomenal) part of it. |
21368 | The knot of the world is the use of 'I' to refer to both willing and knowing [Schopenhauer] |
Full Idea: The identity of the subject of willing with that of knowing by virtue whereof ...the word 'I' includes and indicates both, is the knot of the world, and hence inexplicable. | |
From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], p.211-2), quoted by Christopher Janaway - Schopenhauer 4 'Self' | |
A reaction: I'm struggling to see this as a deep mystery. If we look objectively at animals and ask 'what is their brain for?' the answer seems obvious. This may be a case of everything looking mysterious after a philosopher has stared at it for a while. |
23950 | Freud said passions are pressures of some flowing hydraulic quantity [Freud, by Solomon] |
Full Idea: Freud argued that the passions in general …were the pressures of a yet unknown 'quantity' (which he simply designated 'Q'). He first thought this flowed through neurones, …and always couched the idea in the language of hydraulics. | |
From: report of Sigmund Freud (works [1900]) by Robert C. Solomon - The Passions 3.4 | |
A reaction: This is the main target of Solomon's criticism, because its imagery has become so widespread. It leads to talk of suppressing emotions, or sublimating them. However, it is not too different from Nietzsche's 'drives' or 'will to power'. |
22344 | Freud is pessimistic about human nature; it is ambivalent motive and fantasy, rather than reason [Freud, by Murdoch] |
Full Idea: Freud takes a thoroughly pessimistic view of human nature. ...Introspection reveals only the deep tissue of ambivalent motive, and fantasy is a stronger force than reason. Objectivity and unselfishness are not natural to human beings. | |
From: report of Sigmund Freud (works [1900], II) by Iris Murdoch - The Sovereignty of Good II | |
A reaction: Interesting. His view seems to have coloured the whole of modern culture, reinforced by the hideous irrationality of the Nazis. Adorno and Horkheimer attacking the Enlightenment was the last step in that process. |
4189 | Time may be defined as the possibility of mutually exclusive conditions of the same thing [Schopenhauer] |
Full Idea: Time may be defined as the possibility of mutually exclusive conditions of the same thing. | |
From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.IV) | |
A reaction: An off-beat philosophical view of the question. Sounds more like a consequence of time than its essential nature. |