23 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. |
| 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. |
| 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. |
| 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. |
| 20657 | There are 23 core brain functions, with known circuit, transmitters, genes and behaviour [Watson] |
| Full Idea: In 2014 the National Institutes of Mental Health published a list of 23 core brain functions and their associated neural circuitry, neurotransmitters and genes, and the behaviour and emotions that go with them. | |
| From: Peter Watson (Convergence [2016], 16 'Physics') | |
| A reaction: They were interested in the functions behind mental health, but I am interested in the functions behind our belief systems, which might produce a different focus. Sub-functions, perhaps. |
| 20656 | Traditional ideas of the mind were weakened in the 1950s by mind-influencing drugs [Watson] |
| Full Idea: One development in particular in the 1950s helped to discredit the traditional concept of the mind. This was medical drugs that influenced the workings of the brain. | |
| From: Peter Watson (Convergence [2016], 16 'Intro') | |
| A reaction: This explains Ryle's 1949 book, and the Australian physicalists emerging in the late 1950s. Philosophers don't grasp how their subject is responsive to other areas of human knowledge. Of course, opium had always done this. |
| 20655 | Humans have been hunter-gatherers for 99.5% of their existence [Watson] |
| Full Idea: Anthropology shows that the hunter-gathering lifestyle has occupied 99.5 per cent of the time humans have been on earth. | |
| From: Peter Watson (Convergence [2016], 13 'Emergence') | |
| A reaction: If you are trying to understand humanity, you ignore this fact at your peril. Even agriculture is only a tiny part of our history, and that only disappeared as a major human activity (in many nations) in the last hundred years. |
| 22410 | Maybe the unthinkable is a moral category, and considering some options is dishonourable or absurd [Williams,B] |
| Full Idea: One might have the idea that the unthinkable was itself a moral category. ...Regarding certain things even as alternatives is itself something to be regarded as dishonourable or morally absurd. | |
| From: Bernard Williams (A Critique of Utilitarianism [1973], 2) | |
| A reaction: He's very tentative about this, but I think it is a powerful moral idea. See Kekes. He is particularly aiming at utilitarians, who happily assess vile possibilities. |
| 22408 | Consequentialism assumes that situations can be compared [Williams,B] |
| Full Idea: The emphasis on the necessary comparability of situations is a peculiar feature of consequentialism in general. | |
| From: Bernard Williams (A Critique of Utilitarianism [1973], 2) | |
| A reaction: A nice point. Utilitarians might achieve comparison by totting up the happiness in each situation, but once you include the consequences of the consequences the problems are obvious. Was 1789 a good thing? Too early to say. |
| 22411 | For a consequentialist massacring 7 million must be better than massacring 7 million and one [Williams,B] |
| Full Idea: Making the best of a bad job is a consequentialist maxim, and it will have something to say even pn the difference between massacring seven million and massacring seven million and one. | |
| From: Bernard Williams (A Critique of Utilitarianism [1973], 2) | |
| A reaction: If every life counts, the consequentialists have got something right here. Not caring exactly how many were massacred is a sort of callousness (even when the number can't be established). |
| 22409 | We don't have a duty to ensure that others do their duty [Williams,B] |
| Full Idea: If the goodness of the world were to consist in people's fulfilling their obligations, it would by no means follow that one of my obligations was to bring it about that other people kept their obligations | |
| From: Bernard Williams (A Critique of Utilitarianism [1973], 2) | |
| A reaction: If the maxim of my action is 'ensure that everyone does their duty', presumably that can be universalised. Nelson thought so. It just sounds like a hideous world of self-righteous interference. |
| 22407 | Utilitarianism cannot make any serious sense of integrity [Williams,B] |
| Full Idea: Utilitarianism cannot hope to make sense, at any serious level, of integrity. | |
| From: Bernard Williams (A Critique of Utilitarianism [1973], 1) | |
| A reaction: There will be obvious problems with this. 'My whole platoon got killed, but looking on the bright side, I preserved my integrity'. Once a theory commits entirely to one value, it then has no way to make sense of rival values. |
| 20650 | The Uncertainty Principle implies that cause and effect can't be measured [Watson] |
| Full Idea: The Uncertainty Principle implied that in the subatomic world cause and effect could never be measured. | |
| From: Peter Watson (Convergence [2016], 05 'Against') | |
| A reaction: The fact that it can't be measured does not, presumably, entail that it doesn't exist. Physicists seem to ignore causation, rather than denying it. Can causation be real if it only exists at the macro-level, as an emergent phenomenon? |
| 20649 | The interference of light through two slits confirmed that it is waves [Watson] |
| Full Idea: Thomas Young in 1803 confirmed the idea of Huyghens that light is waves, showing how light passing through two slits produces an interference pattern that resembles water waves sluicing through two slits. | |
| From: Peter Watson (Convergence [2016], 04 'Conception') | |
| A reaction: The great puzzle emerges when it also turns out to be quantised particles. |
| 20661 | Electrons rotate in hyrogen atoms 10^13 times per second [Watson] |
| Full Idea: In the hydrogen atom the electron rotates some 10,000 billion times per second. | |
| From: Peter Watson (Convergence [2016], 18 'Evolutionary') | |
| A reaction: That's an awful lot. Is it at the speed of light? |
| 20647 | Quantum theory explains why nature is made up of units, such as elements [Watson] |
| Full Idea: Planck's quantum idea explained so much, including the observation that the chemical world is made up of discrete units - the elements. Discrete elements implied fundamental units of matter that were themselves discrete (as Dalton had said). | |
| From: Peter Watson (Convergence [2016], 4 'Intro') | |
| A reaction: The atomic theory was only finally confirmed by Einstein in 1905. This idea implies that the very lowest level of all must have distinct building blocks, but so far we have got down to 'fields', which seem to be a sort of 'foam'. |
| 20654 | Only four particles are needed for matter: up and down quark, electron, electron-neutrino [Watson] |
| Full Idea: We need twelve particles in the master equation of the standard model, but it is necessary to have only four to build a universe (up and down quarks, the electron and the electron neutrino (or lepton). The existence of the others is 'a bit of a mystery'. | |
| From: Peter Watson (Convergence [2016], 11 'First Three') |
| 20651 | The shape of molecules is important, as well as the atoms and their bonds [Watson] |
| Full Idea: Pauling showed that the architecture - the shape of molecules was relevant (as well as the bonds). This meant that molecules were just as important as atoms in the understanding of matter. Molecules were not just the sum of their parts. | |
| From: Peter Watson (Convergence [2016], 05 'Three') | |
| A reaction: If Aristotle struggled to understand matter, then so should modern philosophers. This involves thermodynamics and chemistry, as well as quantum theory. |
| 20652 | In 1828 the animal substance urea was manufactured from inorganic ingredients [Watson] |
| Full Idea: In 1828 Wöhler, in an iconic experiment, had manufactured an organic substance, urea, hitherto the product solely of animals, out of inorganic materials, and without any interventions of vital force. | |
| From: Peter Watson (Convergence [2016], 06 'Inorganic') | |
| A reaction: For reductionists like me, the gradual explanation of life in inorganic terms is the great role model of explanation. I take it for granted that the human mind will go the same way, despite partisan resistance from a lot of philosophers. |
| 20658 | Information is physical, and living can be seen as replicating and preserving information [Watson] |
| Full Idea: In passing information, physical changes take place, and information is thus physical. On this account, the act of living can be seen as replicating and preserving the information that a living body is comprised of. | |
| From: Peter Watson (Convergence [2016], 17 'Dreams') | |
| A reaction: [He emphasises 'the act' of living, rather than a life] |