37 ideas
| 23027 | Ideals and metaphysics are practical, not imaginative or speculative [Green,TH, by Muirhead] |
| Full Idea: To T.H. Green an ideal was no creation of an idle imagination, metaphysics no mere play of the speculative reason. Ideals were the most solid, and metaphysics the most practical thing about a man. | |
| From: report of T.H. Green (works [1875]) by John H. Muirhead - The Service of the State I | |
| A reaction: This is despite the fact that Green was an idealist in the Hegelian tradition. I like this. I see it not just as ideals having practical guiding influence, but also that ideals themselves arising out of experience. |
| 23030 | Truth is a relation to a whole of organised knowledge in the collection of rational minds [Green,TH, by Muirhead] |
| Full Idea: When we speak of anything as true or false, we do so on the ground of its relation to a whole of organised knowledge existing actually in no human mind, but prefigured in every mind which is possessed of reason. | |
| From: report of T.H. Green (works [1875]) by John H. Muirhead - The Service of the State I n1 | |
| A reaction: This seems to be the super-idealist view of the coherence account of truth. I have no idea what 'prefigured' means here. |
| 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) |
| 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. |
| 4242 | Pure supervenience explains nothing, and is a sign of something fundamental we don't know [Nagel] |
| Full Idea: Pure, unexplained supervenience is never a solution to a problem but a sign that there is something fundamental we don't know. | |
| From: Thomas Nagel (The Psychophysical Nexus [2000], §III) | |
| A reaction: This seems right. It is not a theory or an explanation, merely the observation of a correlation which will require explanation. Why are they correlated? |
| 23044 | All knowledge rests on a fundamental unity between the knower and what is known [Green,TH, by Muirhead] |
| Full Idea: All knowledge is seen on ultimate analysis to rest upon the idea of a fundamental unity between subject and object, between the knower and that which there is to be known. | |
| From: report of T.H. Green (works [1875]) by John H. Muirhead - The Service of the State III | |
| A reaction: I don't really understand this thought, but I think it embodies the essence of Hegelian idealism. If I know a tree in the wood, any 'unity' between us strikes as merely imaginary. If the tree isn't separate, what does 'knowing' it mean? |
| 23034 | The ultimate test for truth is the systematic interdependence in nature [Green,TH, by Muirhead] |
| Full Idea: Systematic interdependence in the world of nature is the ultimate test of truth. | |
| From: report of T.H. Green (works [1875]) by John H. Muirhead - The Service of the State II | |
| A reaction: Green (or Muirhead) drifts between coherence as the nature of truth and coherence as the nature of justification. He it is the 'test' for truth, which was Russell's view. |
| 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) |
| 23032 | What is distinctive of human life is the desire for self-improvement [Green,TH, by Muirhead] |
| Full Idea: All that is distinctively human in the life of man springs not from the desire to possess this or that object, and so far to realise a better, but to be something more and better than he is. | |
| From: report of T.H. Green (works [1875]) by John H. Muirhead - The Service of the State II | |
| A reaction: An example of Victorian optimism, I think. I'm guessing that people who are not motivated by this impulse are not behaving in a way that is 'distinctively human'. That said, this is an interesting aspect of human nature. |
| 23033 | Hedonism offers no satisfaction, because what we desire is self-betterment [Green,TH, by Muirhead] |
| Full Idea: Hedonism failed because it offered as an end of human aspiration an object in which the human spirit, pledged by its own nature to self-betterment, …could never find satisfaction. | |
| From: report of T.H. Green (works [1875]) by John H. Muirhead - The Service of the State II | |
| A reaction: It is always both sad and amusing to see that 150 years ago someone wrote of a doctrine that is still with us that it has 'failed'. Nowadays they try to say the same of physicalism. His objection rests on optimism about humanity. |
| 23045 | Politics is compromises, which seem supported by a social contract, but express the will of no one [Green,TH] |
| Full Idea: Where laws and institutions are apparently the work of deliberate volition, they are in reality the result of a compromise, which while by a kind of social contract it has the acquiescence of all, expresses the will of none. | |
| From: T.H. Green (works [1875]), quoted by John H. Muirhead - The Service of the State III | |
| A reaction: Politicians who claim to be enacting the 'will of the people' (e.g. when they won a referendum 52-48) are simply lying. Committees usually end up enacting one person's will, but often without realising what has happened. |
| 23050 | The ideal is a society in which all citizens are ladies and gentlemen [Green,TH] |
| Full Idea: With all seriousness and reverence we may hope and pray for a condition of English society in which all honest citizens will recognise themselves and be recognised by each other as gentlemen. | |
| From: T.H. Green (works [1875]), quoted by John H. Muirhead - The Service of the State IV | |
| A reaction: Call me old fashioned but, as long as we expand this to include ladies, I like this thought. Chaucer's knight (in his Prologue) should be our national role model. The true gentleman is an Aristotelian ideal. |
| 23052 | Enfranchisement is an end in itself; it makes a person moral, and gives a basis for respect [Green,TH] |
| Full Idea: Enfranchisement of the people is an end in itself. …Only citizenship makes the moral man; only citizenship gives that respect which is the true basis of the respect for others. | |
| From: T.H. Green (works [1875], iii:436), quoted by John H. Muirhead - The Service of the State IV | |
| A reaction: Should people respect their betters? If so, that is a sort of deferential respect which is different from the mutual respect between equals. That said, I wholly approve of this idea. |
| 23036 | The good is identified by the capacities of its participants [Green,TH, by Muirhead] |
| Full Idea: The modern idea of the good has developed in respect of the range of persons who have the capacity and therefore the right to participate in this good. | |
| From: report of T.H. Green (works [1875]) by John H. Muirhead - The Service of the State II | |
| A reaction: Green is a notable Victorian liberal, starting from an idealist metaphysics. This is an intriguing view of liberal values. The concept of the good should be what suits persons with full capacity. Having the capacity bestows the right of access to it. Hm. |
| 23039 | A true state is only unified and stabilised by acknowledging individuality [Green,TH, by Muirhead] |
| Full Idea: In so far as society commits itself to the principle of individuality of its citizens does it realise the unity and stability that constitute it a true 'State'. | |
| From: report of T.H. Green (works [1875]) by John H. Muirhead - The Service of the State II | |
| A reaction: This asserts the liberal vision of a state, rather than asserting a fact. A state consistently mostly of slaves still seems to be a state, and may achieve a lot. |
| 23038 | People only develop their personality through co-operation with the social whole [Green,TH, by Muirhead] |
| Full Idea: In so far as the individual commits himself to the principle of co-operation in a social whole does he realise his end as individual personality. | |
| From: report of T.H. Green (works [1875]) by John H. Muirhead - The Service of the State II | |
| A reaction: This makes for a very communitarian type of liberalism. The question is whether we create insitutions which suck our free citizens into a communal way of life, or whether that is a matter of their own initiative. |
| 23040 | If something develops, its true nature is embodied in its end [Green,TH] |
| Full Idea: To anyone who understands a process of development, the result being developed is the reality; and it is its ability to become this that the subject undergoing development has its true nature. | |
| From: T.H. Green (works [1875], iii: 224), quoted by John H. Muirhead - The Service of the State II | |
| A reaction: Although this contains the dubious Hegelian idea that development tends towards some 'end', presented as fixed and final, it still seems important that anything accepted as a 'development' is the expression of some natural potential. |
| 23031 | God is the ideal end of the mature mind's final development [Green,TH] |
| Full Idea: God is a subject which is eternally all that the self-conscious subject as developed in time has the possibility of becoming. | |
| From: T.H. Green (works [1875]), quoted by John H. Muirhead - The Service of the State I | |
| A reaction: [Ethics p.197] Reminiscent of Peirce's account of truth, as the ideal end of enquiry. If God is a human ideal, we either limit God, or exaggerate our powers of idealisation. |
| 23041 | God is the realisation of the possibilities of each man's self [Green,TH] |
| Full Idea: God is identical with the self of every man in the sense of being the realisation of its determinate possibilities.…In being conscious of himself man is conscious of God and thus knows that God is, but only in so far as he knows what he himself really is. | |
| From: T.H. Green (works [1875], iii:226-7), quoted by John H. Muirhead - The Service of the State II | |
| A reaction: Does this, by the transitivity of identity, imply the identity of all individual men? Do we all contain identical possibilities, which converge on a unified concept of God? I always take the monotheistic God to far exceed mere human possibilities. |