25 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. |
| 21812 | Being is the product of pure intellect [Plotinus] |
| Full Idea: Intellectual-Principle [Nous] by its intellective act establishes Being. | |
| From: Plotinus (The Enneads [c.245], 5.1.04) | |
| A reaction: This is a surprising view - that there is something which is prior to Being - but I take it to be Plotinus giving primacy to Plato's Form of the Good (a pure ideal), ahead of the One of Parmenides (which is Being). |
| 21817 | The One does not exist, but is the source of all existence [Plotinus] |
| Full Idea: The First is no member of existence, but can be the source of all. | |
| From: Plotinus (The Enneads [c.245], 5.1.07) | |
| A reaction: The First is the One, and this explicitly denies that it has Being. This answers the self-predication problem of Forms. Plato thought the Form of the Beautiful was beautiful, but it can't be (because of the regress). The source of existence can't exist. |
| 21824 | The One is a principle which transcends Being [Plotinus] |
| Full Idea: There exists a principle which transcends Being; this the One. | |
| From: Plotinus (The Enneads [c.245], 5.1.10) | |
| A reaction: The idea that the One transcends Being is the distinctive Plotinus doctrine. He defends the view that this was also the view of Anaxagoras, Empedocles and Plato. |
| 21813 | Number determines individual being [Plotinus] |
| Full Idea: Number is the determinant of individual being. | |
| From: Plotinus (The Enneads [c.245], 5.1.05) | |
| A reaction: You might have thought that number was the consequence of the individualities (or units) within being, but not so. You can't get more platonic than saying that the idealised numbers are the source of the particular units. |
| 5506 | If soul was like body, its parts would be separate, without communication [Plotinus] |
| Full Idea: If the soul had the nature of the body, it would have isolated members each unaware of the condition of the other;..there would be a particular soul as a distinct entity to each local experience, so a multiplicity of souls would administer an individual. | |
| From: Plotinus (The Enneads [c.245], 4.2.2), quoted by R Martin / J Barresi - Introduction to 'Personal Identity' p.15 | |
| A reaction: Of course, the modern 'modularity of mind' theory does suggest that we are run by a team, but a central co-ordinator is required, with a full communication network across the modules. |
| 21827 | The movement of Soul is continuous, but we are only aware of the parts of it that are sensed [Plotinus] |
| Full Idea: The Soul maintains its unfailing movement; for not all that passes in the soul is, by that fact, perceptible; we know just as much as impinges on the faculty of the sense. | |
| From: Plotinus (The Enneads [c.245], 5.1.12) | |
| A reaction: This is a straightforward argument in favour of an unconscious aspect to the mind - and a rather good argument too. No one thinks that our minds ever stop working, even in sleep. |
| 21828 | A person is the whole of their soul [Plotinus] |
| Full Idea: Man is not merely a part (the higher part) of the Soul but the total. | |
| From: Plotinus (The Enneads [c.245], 5.1.12) | |
| A reaction: The soul is psuche, which includes the vegetative soul. The higher part is normally taken to be reason. This seems pretty close to John Locke's view of the matter. |
| 21809 | Our soul has the same ideal nature as the oldest god, and is honourable above the body [Plotinus] |
| Full Idea: Our own soul is of that same ideal nature [as the oldest god of them all], so that to consider it, purified, freed from all accruement, is to recognise in ourselves which we have found soul to be, honourable above the body. For what is body but earth? | |
| From: Plotinus (The Enneads [c.245], 5.1.02) | |
| A reaction: The strongest versions of substance dualism are religious in character, because the separateness of the mind elevates us above the grubby physical character of the world. I'm with Nietzsche on this one - this view is actually harmful to us. |
| 21825 | The soul is outside of all of space, and has no connection to the bodily order [Plotinus] |
| Full Idea: We may not seek any point in space in which to seat the soul; it must be set outside of all space; its distinct quality, its separateness, its immateriality, demand that it be a thing alone, untouched by all of the bodily order. | |
| From: Plotinus (The Enneads [c.245], 5.1.10) | |
| A reaction: You can't get more dualist than that. He doesn't seem bothered about the interaction problem. He likens such influence to the radiation of the sun, rather than to physical movement. |
| 21826 | The Soul reasons about the Right, so there must be some permanent Right about which it reasons [Plotinus] |
| Full Idea: Since there is a Soul which reasons upon the right and good - for reasoning is an enquiry into the rightness and goodness of this rather than that - there must exist some pemanent Right, the source and foundation of this reasoning in our soul. | |
| From: Plotinus (The Enneads [c.245], 5.1.11) | |
| A reaction: This is pretty close the Kant's concept of 'the moral order within me', and Plotinus even sees it as rational. Presumably this right is 'permanent' because the revelatlons of reason about it are necessary truths. |
| 6922 | Ecstasy is for the neo-Platonist the highest psychological state of man [Plotinus, by Feuerbach] |
| Full Idea: Ecstasy or rapture is for the neo-Platonist the highest psychological state of man. | |
| From: report of Plotinus (The Enneads [c.245]) by Ludwig Feuerbach - Principles of Philosophy of the Future §29 | |
| A reaction: See Bernini's statue of St Theresa. Personally I find this very unappealing because of its utter irrationality, but what is the 'highest' human psychological state? Doing mental arithmetic? Doing what is morally right? Dignity under pressure? |
| 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 |
| 21816 | Soul is the logos of Nous, just as Nous is the logos of the One [Plotinus] |
| Full Idea: The soul is an utterance [logos] and act of the Intellectual-Principle [Nous], as that is an utterance and act of the One. | |
| From: Plotinus (The Enneads [c.245], 5.1.06) | |
| A reaction: Being only comes into the picture at the secondary Nous stage. Nous is the closest to the modern concept of God. |
| 21815 | Because the One is immobile, it must create by radiation, light the sun producing light [Plotinus] |
| Full Idea: Given this immobility of the Supreme ...what happened then? It must be a circumradiation, which may be compared to the brilliant light encircling the sun and ceaselessly generating from that unchanging substance, | |
| From: Plotinus (The Enneads [c.245], 5.1.06) | |
| A reaction: This is the answer given to the problem raised in Idea 21814. The sun produces energy, without apparent movement. Not an answer that will satisfy a physicist, but an interesting answer. |
| 21814 | How can multiple existence arise from the unified One? [Plotinus] |
| Full Idea: The problem endlessly debated is how, from such a unity as we have declared the One to be, does anything at all come into substantial existence, any multiplicity, dyad or number? | |
| From: Plotinus (The Enneads [c.245], 5.1.06) | |
| A reaction: This was precisely Aristotle's objection to the One of Parmenides, and especially the problem of the source of movement (which Plotinus also notices). |
| 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. |
| 21808 | Soul is author of all of life, and of the stars, and it gives them law and movement [Plotinus] |
| Full Idea: Soul is the author of all living things, ...it has breathed life into them all, whatever is nourished by earth and sea, the divine stars in the sky; ...it is the principle distinct from all of these to which it gives law and movement and life. | |
| From: Plotinus (The Enneads [c.245], 5.1.02) | |
| A reaction: This seems to derive from Anaxagoras, who is mentioned by Plotinus. The soul he refers to his not the same as our concept of God. Note the word 'law', which I am guessing is nomos. Not, I think, modern laws of nature, but closer to guidelines. |
| 21811 | Even the soul is secondary to the Intellectual-Principle [Nous], of which soul is an utterance [Plotinus] |
| Full Idea: Soul, for all the worth we have shown to belong to it, is yet a secondary, an image of the Intellectual-Principle [Nous]; reason uttered is an image of reason stored within the soul, and similarly soul is an utterance of the Intellectual-Principle. | |
| From: Plotinus (The Enneads [c.245], 5.1.03) | |
| A reaction: It then turns out that Nous is secondary to the One, so there is a hierarchy of Being (which only enters at the Nous stage). |