41 ideas
| 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. |
| 4913 | Brain lesions can erase whole categories of perception, suggesting they are hard-wired [Carter,R] |
| Full Idea: The discovery that a single brain lesion can erase all knowledge of man-made artefacts, or all knowledge of animals, suggests that these categories somehow hard-wired into the brain - that we all have a set of 'memory pigeonholes'. | |
| From: Rita Carter (Mapping the Mind [1998], p.190) | |
| A reaction: Presumably something can become 'hard-wired' through experience, rather than from birth. The whole idea of 'hard-wired' seems misleading about the brain. What matters is that the brain physically constructs categories. |
| 4910 | Sense organs don't discriminate; they reduce various inputs to the same electrical pulses [Carter,R] |
| Full Idea: Despite their variety, each sense organ translates its stimulus into electrical pulses; rather than discriminating one type of input from another, the sense organs actually make them more alike. | |
| From: Rita Carter (Mapping the Mind [1998], p.174) | |
| A reaction: An illuminating observation, which modern 'naïve realists' should bear in mind. Secondary qualities are entirely unrelated to the nature of the input, and are merely 'what the brain decides to make of it'. Discrimination is in our neurons. |
| 4911 | The recognition sequence is: classify, name, locate, associate, feel [Carter,R, by PG] |
| Full Idea: The sequence of events in the brain for perceptual recognition is first identifying a rough class for the object, then a name, then a location, then some associations, and finally an emotion. | |
| From: report of Rita Carter (Mapping the Mind [1998], p.181) by PG - Db (ideas) | |
| A reaction: This seems to be one of those places where neuro-science trumps philosophy. You can't argue with empirical research, so philosophical theories had better adapt themselves to this sequence. The big modern discovery is the place of emotion in recognition. |
| 4919 | There seems to be no dividing line between a memory and a thought [Carter,R] |
| Full Idea: It has become clear from research that there is no clear dividing line between a memory and a thought. | |
| From: Rita Carter (Mapping the Mind [1998], p.308) | |
| A reaction: This always struck me as an obvious criticism of Descartes, when he claimed that memory was not an essential part of the 'thinking thing'. How can you think or understand without memory of the different phases of your thoughts? No memory, no mind! |
| 3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
| Full Idea: Anaxarchus said that he was not even sure that he knew nothing. | |
| From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1 |
| 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. |
| 4908 | No one knows if animals are conscious [Carter,R] |
| Full Idea: No one knows if animals are conscious. | |
| From: Rita Carter (Mapping the Mind [1998], p.155) | |
| A reaction: This is a report from the front line of brain research, and should be born in mind when over-confident people make pronouncements about this topic. It strikes me as important to grasp that animals MIGHT not be conscious. |
| 4902 | Pain doesn't have one brain location, but is linked to attention and emotion [Carter,R] |
| Full Idea: Scans show there is no such thing as a pain centre; pain springs mainly from the activation of areas associated with attention and emotion. | |
| From: Rita Carter (Mapping the Mind [1998], p. 12) | |
| A reaction: Most brain research points to the complex multi-layered nature of experiences that were traditionally considered simple. We can be distracted from a pain, and an enormous number of factors can affect our degree of dislike of a given pain. |
| 4904 | Proper brains appear at seven weeks, and neonates have as many neurons as adults do [Carter,R] |
| Full Idea: The main sections of the brain, including the cerebral cortex, are visible within seven weeks of conception, and by the time the child is born the brain contains as many neurons - about 100 billion - as it will have as an adult. | |
| From: Rita Carter (Mapping the Mind [1998], p. 17) | |
| A reaction: Of interest in the abortion debate, and also in thinking about personal identity. However, it seems clear that the number of connections, rather than neurons, is what really matters. A small infant may well lack personal identity. |
| 4915 | In primates, brain size correlates closely with size of social group [Carter,R] |
| Full Idea: Brain size in primates is closely associated with the size of the social group in which the animal lives. | |
| From: Rita Carter (Mapping the Mind [1998], p.257) | |
| A reaction: Intriguing. Humans can have huge social groups because of language, which suggests a chicken-or-egg question. Language, intelligence and size of social group must have expanded together in humans. |
| 4917 | Consciousness involves awareness, perception, self-awareness, attention and reflection [Carter,R] |
| Full Idea: Awareness, perception, self-awareness, attention and reflection are all separate components of consciousness, and the quality of our experience varies according to which and how many of them are present. | |
| From: Rita Carter (Mapping the Mind [1998], p.300) | |
| A reaction: Philosophers like to emphasise 'qualia' and 'intentionality'. This remark slices the cake differently. 'Attention' is interesting, dividing consciousness into two areas, with some experience fading away into the darkness. Hume denied self-awareness. |
| 4916 | There is enormous evidence that consciousness arises in the frontal lobes of the brain [Carter,R] |
| Full Idea: A huge volume of evidence suggests that consciousness emerges from the activity of the cerebral cortex, and in particular from the frontal lobes. | |
| From: Rita Carter (Mapping the Mind [1998], p.298) | |
| A reaction: Dualists must face up to this, and even many physicalists have a rather vague notion about the location of awareness, but we are clearly homing in very precise physical substances which have consciousness as a feature. |
| 4905 | Normal babies seem to have overlapping sense experiences [Carter,R] |
| Full Idea: Connections in a baby's brain probably give the infant the experience of 'seeing' sounds and 'hearing' colours - which occasionally continues into adulthood, where it is known as 'synaesthesia'. | |
| From: Rita Carter (Mapping the Mind [1998], p. 19) | |
| A reaction: A fact to remember when discussing secondary qualities, and the relativism involved in the way we perceive the world. If you have done your philosophy right, you shouldn't be surprised by this discovery. |
| 4918 | In blindsight V1 (normal vision) is inactive, but V5 (movement) lights up [Carter,R] |
| Full Idea: Scans show that a sub-section of the visual cortex called V5 - the area that registers movement - lights up during blindsight, even though V1 - the primary sensory area that is essential for normal sight - is not active. | |
| From: Rita Carter (Mapping the Mind [1998], p.307) | |
| A reaction: The whole point of blindsight is to make us realise that vision involves not one module, but a whole team of them. The inference is that V1 involves consciousness, but other areas of the visual cortex don't. |
| 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) |
| 4912 | Out-of-body experiences may be due to temporary loss of proprioception [Carter,R] |
| Full Idea: Out-of-body experiences may be due to temporary loss of proprioception. | |
| From: Rita Carter (Mapping the Mind [1998], p.187) | |
| A reaction: This is only a speculation, but it is an effect which can be caused by brain injury, and dualists should face the possibility that this evidence (prized by many dualists) can have a physical explanation. |
| 4903 | Scans of brains doing similar tasks produce very similar patterns of activation [Carter,R] |
| Full Idea: The pattern of brain activation during, say, a word retrieval task is usually similar enough among the dozen or so participants who typically take part in such studies for their scans to be overlaid and still show a clear pattern. | |
| From: Rita Carter (Mapping the Mind [1998], p. 17) | |
| A reaction: This doesn't surprise me, though it could be interpreted as supporting type-type identity, or as supporting functionalism. Armstrong and Lewis endorse a sort of reductive functionalism which would fit this observation. |
| 4920 | Thinking takes place on the upper side of the prefrontal cortex [Carter,R] |
| Full Idea: The nuts and bolts of thinking - holding ideas in mind and manipulating them - takes place on the upper side of the prefrontal cortex. | |
| From: Rita Carter (Mapping the Mind [1998], p.312) | |
| A reaction: Keep this firmly in view! Imagine that the skull is transparent, and brain activity moves in waves of colour. Dualism would, in those circumstances, never have even occurred to anyone. |
| 4906 | Babies show highly emotional brain events, but may well be unaware of them [Carter,R] |
| Full Idea: Babies show emotion dramatically, but the areas of the brain that in adults are linked to the conscious experience of emotions are not active in newborn babies. Such emotions may therefore be unconscious. | |
| From: Rita Carter (Mapping the Mind [1998], p. 19) | |
| A reaction: Traditionally, 'unconscious emotion' is a contradiction, but I think we should accept this new evidence and rethink the nature of mind. Not only might emotion be non-conscious, but we should even consider that rational thinking could be too. |
| 4909 | The only way we can control our emotions is by manipulating the outside world that influences them [Carter,R] |
| Full Idea: We try to manipulate our emotions all the time, but all we are doing is arranging the outside world so it triggers certain emotions - we cannot control our reactions directly. | |
| From: Rita Carter (Mapping the Mind [1998], p.155) | |
| A reaction: This seems to me to throw a very illuminating light on a huge amount of human behaviour, such as going to the cinema or listening to music. The romantic movement encouraged direct internal manipulation. Compare sex fantasies with viewing pornography. |
| 4914 | A frog will starve to death surrounded by dead flies [Carter,R] |
| Full Idea: A frog will starve to death surrounded by dead flies. | |
| From: Rita Carter (Mapping the Mind [1998], p.195) | |
| A reaction: A nice warning against assuming that rationality is operating when a frog feels hungry and 'decides' to have lunch. We should take comfort from the fact that humans are NOT this stupid, and philosophers should try to accurately describe our gift. |
| 4907 | The 'locus coeruleus' is one of several candidates for the brain's 'pleasure centre' [Carter,R] |
| Full Idea: Noradrenaline is an excitatory chemical that induces physical and mental arousal and heightens mood. Production is centred in an area of the brain called the locus coeruleus, which is one of several candidates for the brain's 'pleasure' centre. | |
| From: Rita Carter (Mapping the Mind [1998], p. 30) | |
| A reaction: It seems to me very morally desirable that people understand facts of this kind, so that they can be more objective about pleasure. Pleasure is one cog in the machine that makes a person, not the essence of human life. |