10 ideas
| 8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
| Full Idea: Maddy dispenses with pure sets, by sketching a strong set theory in which everything is either a physical object or a set of sets of ...physical objects. Eventually a physiological story of perception will extend to sets of physical objects. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3 | |
| A reaction: This doesn't seem to find many supporters, but if we accept the perception of resemblances as innate (as in Hume and Quine), it is isn't adding much to see that we intrinsically see things in groups. |
| 10718 | A natural number is a property of sets [Maddy, by Oliver] |
| Full Idea: Maddy takes a natural number to be a certain property of sui generis sets, the property of having a certain number of members. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990], 3 §2) by Alex Oliver - The Metaphysics of Properties | |
| A reaction: [I believe Maddy has shifted since then] Presumably this will make room for zero and infinities as natural numbers. Personally I want my natural numbers to count things. |
| 8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
| Full Idea: Maddy says that intuition alone does not support very much mathematics; more importantly, a naturalist cannot accept intuition at face value, but must ask why we are justified in relying on intuition. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3 | |
| A reaction: It depends what you mean by 'intuition', but I identify with her second objection, that every faculty must ultimately be subject to criticism, which seems to point to a fairly rationalist view of things. |
| 17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
| Full Idea: Maddy proposes that we can know (some) mind-independent mathematical truths through knowing about sets, and that we can obtain knowledge of sets through experience. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Carrie Jenkins - Grounding Concepts 6.5 | |
| A reaction: Maddy has since backed off from this, and now tries to merely defend 'objectivity' about sets (2011:114). My amateurish view is that she is overrating the importance of sets, which merely model mathematics. Look at category theory. |
| 5040 | Necessary truths can be analysed into original truths; contingent truths are infinitely analysable [Leibniz] |
| Full Idea: Derivative truths are of two sorts: some are analysed into original truths, others admit of an infinite process of analysis. The former are necessary, the latter are contingent. | |
| From: Gottfried Leibniz (On Freedom [1689], p.108) | |
| A reaction: An intriguing proposal. Hume would presumably see contingent truths as being analysed until you reach 'impressions'. Analysis of necessary truths soon comes to the blinding light of what is obvious, but analysis of contingency never gets there. |
| 13159 | Only God sees contingent truths a priori [Leibniz] |
| Full Idea: Only God sees contingent truths a priori. | |
| From: Gottfried Leibniz (On Freedom [1689], p.95) | |
| A reaction: This because everything is interconnected, and the whole picture must be seen to understand a contingent truth. |
| 5039 | If non-existents are possible, their existence would replace what now exists, which cannot therefore be necessary [Leibniz] |
| Full Idea: If certain possibles never exist, then existing things are not always necessary; otherwise it would be impossible for other things to exist instead of them, and so all things that never exist would be impossible. | |
| From: Gottfried Leibniz (On Freedom [1689], p.106) | |
| A reaction: A neat argument, though it is not self-evident that when possibles came into existence they would have to replace what is already there. Can't something be possible, but only in another world, because this one is already booked? |
| 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 |
| 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. |
| 5041 | God does everything in a perfect way, and never acts contrary to reason [Leibniz] |
| Full Idea: We can regard it as certain that everything is done by God in the most perfect way, that he does nothing which is contrary to reason. | |
| From: Gottfried Leibniz (On Freedom [1689], p.109) | |
| A reaction: The famous optimism which Voltaire laughed at in 'Candide'. I can't help thinking that there is an ideal of God being ABOVE reason. We reason, and give reasons, because we are unsure, and life is a struggle. The highest ideal is mystically self-evident. |