52 ideas
| 3879 | Philosophy aims to provide a theory of everything [Scruton] |
| Full Idea: Philosophy studies everything: it tries to provide a theory of the whole of things. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 1.2) | |
| A reaction: Good, but you can't avoid value-judgements about which things are important; philosophers place more value on moral theories than on theories about glacier movement. There is a tension in philosophy between human and eternal concerns. |
| 3891 | If p entails q, then p is sufficient for q, and q is necessary for p [Scruton] |
| Full Idea: If p entails q, then p is sufficient for q, and q is necessary for p. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 15.7) |
| 3894 | We may define 'good' correctly, but then ask whether the application of the definition is good [Scruton] |
| Full Idea: The 'open question' argument is clearly invalid. A question remains open just so long as our ignorance permits. …It may be an open question whether promoting happiness is good, even though this is what 'good' means. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 20.1) | |
| A reaction: A nice objection. Like small children, we can keep asking questions forever. Whether there is a question to be asked about a thing is not a property of that thing, but of us who ask it. |
| 3883 | A true proposition is consistent with every other true proposition [Scruton] |
| Full Idea: A true proposition is consistent with every other true proposition: no truth is contradicted by another. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 9.1) | |
| A reaction: Interesting. It resembles the rule that if you always tell the truth you don't need to remember what you said. Close to the heart of the concept of truth. Coherence and correspondence. |
| 3884 | The pragmatist does not really have a theory of truth [Scruton] |
| Full Idea: The pragmatist does not really have a theory of truth. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 9.4) |
| 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) |
| 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. |
| 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) |
| 3907 | Could you be intellectually acquainted with numbers, but unable to count objects? [Scruton] |
| Full Idea: Could someone have a perfect intellectual acquaintance with numbers, but be incapable of counting a flock of sheep? | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 26.6) |
| 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. |
| 3908 | If maths contains unprovable truths, then maths cannot be reduced to a set of proofs [Scruton] |
| Full Idea: If there can be unprovable truths of mathematics, then mathematics cannot be reduced to the proofs whereby we construct it. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 26.7) |
| 3906 | If possible worlds are needed to define properties, maybe we should abandon properties [Scruton] |
| Full Idea: If the only way of defining properties involves quantifying over possible worlds, this could be taken as another reason for abandoning properties altogether. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 26.4) |
| 3888 | Hume assumes that necessity can only be de dicto, not de re [Scruton] |
| Full Idea: It was one of the assumptions of Hume's empiricism that all necessities are de dicto: i.e. they are artefacts of language. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 13.5) |
| 3903 | The conceivable can't be a test of the possible, if there are things which are possible but inconceivable [Scruton] |
| Full Idea: If there are things which are possible but inconceivable, we must abandon the view, which has had a considerable following since Descartes, that the conceivable is a test of the possible. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 25) |
| 3897 | Epistemology is about the justification of belief, not the definition of knowledge [Scruton] |
| Full Idea: In my view the concept of knowledge is of no very great interest in epistemology, which actually concerns the justification of belief. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 22) | |
| A reaction: I think this is an excellent thought. I see knowledge as slippery, and partially contextual, and I don't care whether someone precisely 'knows' something. I just want to know why they believe it. |
| 3881 | In the Cogito argument consciousness develops into self-consciousness [Scruton] |
| Full Idea: In the course of the argument the first person has acquired a character; he is not merely conscious, but self-conscious. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 4) |
| 3887 | Maybe our knowledge of truth and causation is synthetic a priori [Scruton] |
| Full Idea: 'Every event has a cause' and 'truth is correspondence to facts' are candidates for being synthetic a priori knowledge. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 13.2) |
| 3901 | Touch only seems to reveal primary qualities [Scruton] |
| Full Idea: Touch seems to deliver a purely primary-quality account of the world. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 24) | |
| A reaction: Interesting, though a little over-confident. It seems occasionally possible for touch to be an illusion. |
| 3885 | We only conceive of primary qualities as attached to secondary qualities [Scruton] |
| Full Idea: Bradley argued that we cannot conceive of primary qualities except as attached to secondary qualities. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 10.1) |
| 3910 | If primary and secondary qualities are distinct, what has the secondary qualities? [Scruton] |
| Full Idea: If primary and secondary qualities are distinct, what do secondary qualities inhere in? | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], Ch.10 n) | |
| A reaction: What is the problem? A pin causes me pain, but I know the pain isn't in the pin. It is the same with colour. It is a mental property, if you like, triggered by a wavelength of radiation. |
| 3899 | The representational theory says perceptual states are intentional states [Scruton] |
| Full Idea: The representational theory is the unsurprising view that perceptual states are intentional, like beliefs, emotions and desires. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 23.3) |
| 3898 | My belief that it will rain tomorrow can't be caused by its raining tomorrow [Scruton] |
| Full Idea: It is impossible that my present belief that it will rain tomorrow is caused by its raining tomorrow. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 22.4) | |
| A reaction: This doesn't demolish a causal account of belief. It would be very surprising if I were to believe it was going to rain tomorrow for no cause whatsoever. That would be irrational. |
| 3880 | Logical positivism avoids scepticism, by closing the gap between evidence and conclusion [Scruton] |
| Full Idea: If the evidence for p is q, and that is the only evidence there is or can be, then 'p' means q. Hence there is no gap between evidence and conclusion, and the sceptical problem does not arise. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 3.2) |
| 3878 | Why should you believe someone who says there are no truths? [Scruton] |
| Full Idea: A writer who says that there are no truths, or that all truth is 'merely relative', is asking you not to believe him. So don't. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 1.1) |
| 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. |
| 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) |
| 3892 | Every event having a cause, and every event being determined by its cause, are not the same [Scruton] |
| Full Idea: To say that every event has a cause is one thing; to say that every event is determined by its cause is quite another thing. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 17.1) |
| 3911 | The very concept of a substance denies the possibility of mutual interaction and dependence [Scruton] |
| Full Idea: It is often held to be a consequence of the rationalist conception of substance, that separate substances cannot interact (since causal interaction is a form of mutual dependence). | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], Ch.16 n) | |
| A reaction: Yes, substances seem incapable of interaction, just as Leibniz argues that perfections could never interact. They are too pure. |
| 3882 | Wittgenstein makes it impossible to build foundations from something that is totally private [Scruton] |
| Full Idea: Wittgenstein's point is that if I search for foundations in what can only be known to me, then the belief that I have discovered those foundations will also fall victim to Descartes' demon. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 5.3) | |
| A reaction: Why should foundations based in wider society or a language community fare any better? Getting a lot of people to agree won't trouble the demon too much. Flat earthers. |
| 3896 | Any social theory of morality has the problem of the 'free rider', who only pretends to join in [Scruton] |
| Full Idea: Any attempt to provide a social justification of morality runs the risk of the 'free rider' - one who pretends to play the game in order to enjoy the fruits of it. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 20.6) |
| 3886 | Membership is the greatest source of obligation [Scruton] |
| Full Idea: Membership is the greatest source of obligation. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 11.2) | |
| A reaction: An interesting and rather Aristotelian idea. The alternative is individual debt or obligation. |
| 3895 | The categorical imperative is not just individual, but can be used for negotiations between strangers [Scruton] |
| Full Idea: The categorical imperative is also an instrument of negotiation and compromise between strangers, through which they can rise out of enmity and confront each other as equals. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 20.6) |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 3890 | 'Cause' used to just mean any valid explanation [Scruton] |
| Full Idea: Traditionally (before Leibniz and Spinoza) the world 'cause' signified any valid explanation. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 14) |
| 3904 | Measuring space requires no movement while I do it [Scruton] |
| Full Idea: I can measure the length of something only if I know that it has not moved between the moment when I locate one end of it and the moment when I locate the other. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 25.3) | |
| A reaction: A nice example of how even simple propositions have many presuppositions. |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |
| 3905 | 'Existence' is not a predicate of 'man', but of the concept of man, saying it has at least one instance [Scruton] |
| Full Idea: When I say that a man exists, Frege argues, I do not predicate existence of a man, but rather of the concept man: I say the concept has at least one instance (and existence is a predicate of predicates). | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 26.2) |