30 ideas
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
| A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
| 13655 | The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate [Putnam, by Shapiro] |
| Full Idea: Putnam claims that the Löwenheim-Skolem theorems indicate that there is no 'fact of the matter' whether all sets are constructible. | |
| From: report of Hilary Putnam (Models and Reality [1977]) by Stewart Shapiro - Foundations without Foundationalism | |
| A reaction: [He refers to the 4th and 5th pages of Putnam's article] Shapiro offers (p.109) a critique of Putnam's proposal. |
| 9915 | V = L just says all sets are constructible [Putnam] |
| Full Idea: V = L just says all sets are constructible. L is the class of all constructible sets, and V is the universe of all sets. | |
| From: Hilary Putnam (Models and Reality [1977], p.425) |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
| A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
| 9913 | The Löwenheim-Skolem Theorem is close to an antinomy in philosophy of language [Putnam] |
| Full Idea: The Löwenheim-Skolem Theorem says that a satisfiable first-order theory (in a countable language) has a countable model. ..I argue that this is not a logical antinomy, but close to one in philosophy of language. | |
| From: Hilary Putnam (Models and Reality [1977], p.421) | |
| A reaction: See the rest of this paper for where he takes us on this. |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
| From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
| A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
| 9914 | It is unfashionable, but most mathematical intuitions come from nature [Putnam] |
| Full Idea: Experience with nature is undoubtedly the source of our most basic 'mathematical intuitions', even if it is unfashionable to say so. | |
| From: Hilary Putnam (Models and Reality [1977], p.424) | |
| A reaction: Correct. I find it quite bewildering how Frege has managed to so discredit all empirical and psychological approaches to mathematics that it has become a heresy to say such things. |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
| A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
| 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! |
| 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. |
| 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. |