36 ideas
| 4642 | No fact can be real and no proposition true unless there is a Sufficient Reason (even if we can't know it) [Leibniz] |
| Full Idea: The principle of sufficient reason says no fact can be real or existing and no proposition can be true unless there is a sufficient reason why it should be thus and not otherwise, even though in most cases these reasons cannot be known to us. | |
| From: Gottfried Leibniz (Monadology [1716], §32) | |
| A reaction: I think of this as my earliest philosophical perception, a childish rebellion against being told that there was 'no reason' for something. My intuition tells me that it is correct, and the foundation of ontology and truth. Don't ask me to justify it! |
| 2115 | Everything in the universe is interconnected, so potentially a mind could know everything [Leibniz] |
| Full Idea: Every body is sensitive to everything in the universe, so that one who saw everything could read in each body what is happening everywhere, and even what has happened and will happen. | |
| From: Gottfried Leibniz (Monadology [1716], §61) |
| 10482 | The logic of ZF is classical first-order predicate logic with identity [Boolos] |
| Full Idea: The logic of ZF Set Theory is classical first-order predicate logic with identity. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.121) | |
| A reaction: This logic seems to be unable to deal with very large cardinals, precisely those that are implied by set theory, so there is some sort of major problem hovering here. Boolos is fairly neutral. |
| 10492 | A few axioms of set theory 'force themselves on us', but most of them don't [Boolos] |
| Full Idea: Maybe the axioms of extensionality and the pair set axiom 'force themselves on us' (Gödel's phrase), but I am not convinced about the axioms of infinity, union, power or replacement. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.130) | |
| A reaction: Boolos is perfectly happy with basic set theory, but rather dubious when very large cardinals come into the picture. |
| 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. |
| 10485 | Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos] |
| Full Idea: The naïve view of set theory (that any zero or more things form a set) is natural, but inconsistent: the things that do not belong to themselves are some things that do not form a set. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.127) | |
| A reaction: As clear a summary of Russell's Paradox as you could ever hope for. |
| 10484 | The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos] |
| Full Idea: According to the iterative conception, every set is formed at some stage. There is a relation among stages, 'earlier than', which is transitive. A set is formed at a stage if and only if its members are all formed before that stage. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.126) | |
| A reaction: He gives examples of the early stages, and says the conception is supposed to 'justify' Zermelo set theory. It is also supposed to make the axioms 'natural', rather than just being selected for convenience. And it is consistent. |
| 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. |
| 2111 | Falsehood involves a contradiction, and truth is contradictory of falsehood [Leibniz] |
| Full Idea: We judge to be false that which involves a contradiction, and true that which is opposed or contradictory to the false. | |
| From: Gottfried Leibniz (Monadology [1716], §31) |
| 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) |
| 10491 | Infinite natural numbers is as obvious as infinite sentences in English [Boolos] |
| Full Idea: The existence of infinitely many natural numbers seems to me no more troubling than that of infinitely many computer programs or sentences of English. There is, for example, no longest sentence, since any number of 'very's can be inserted. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: If you really resisted an infinity of natural numbers, presumably you would also resist an actual infinity of 'very's. The fact that it is unclear what could ever stop a process doesn't guarantee that the process is actually endless. |
| 10483 | Mathematics and science do not require very high orders of infinity [Boolos] |
| Full Idea: To the best of my knowledge nothing in mathematics or science requires the existence of very high orders of infinity. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.122) | |
| A reaction: He is referring to particular high orders of infinity implied by set theory. Personally I want to wield Ockham's Razor. Is being implied by set theory a sufficient reason to accept such outrageous entities into our ontology? |
| 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. |
| 10490 | Mathematics isn't surprising, given that we experience many objects as abstract [Boolos] |
| Full Idea: It is no surprise that we should be able to reason mathematically about many of the things we experience, for they are already 'abstract'. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: He has just given a list of exemplary abstract objects (Idea 10489), but I think there is a more interesting idea here - that our experience of actual physical objects is to some extent abstract, as soon as it is conceptualised. |
| 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. |
| 7644 | The monad idea incomprehensibly spiritualises matter, instead of materialising soul [La Mettrie on Leibniz] |
| Full Idea: The Leibnizians with their monads have constructed an incomprehensible hypothesis. They have spiritualized matter rather than materialising the soul. | |
| From: comment on Gottfried Leibniz (Monadology [1716]) by Julien Offray de La Mettrie - Machine Man p.3 | |
| A reaction: I agree with La Mettrie. This disagreement shows, I think, how important the problem of interaction between mind and body was in the century after Descartes. Drastic action seemed needed to bridge the gap, one way or the other. |
| 11857 | He replaced Aristotelian continuants with monads [Leibniz, by Wiggins] |
| Full Idea: In the end Leibniz dethroned Aristotelian continuants, seen as imperfect from his point of view, in favour of monads. | |
| From: report of Gottfried Leibniz (Monadology [1716]) by David Wiggins - Sameness and Substance Renewed 3.1 | |
| A reaction: I take the 'continuants' to be either the 'ultimate subject of predication' (in 'Categories'), or 'essences' (in 'Metaphysics'). Since monads seem to be mental (presumably to explain the powers of things), this strikes me as a bit mad. |
| 7843 | Is a drop of urine really an infinity of thinking monads? [Voltaire on Leibniz] |
| Full Idea: Can you really maintain that a drop of urine is an infinity of monads, and that each one of these has ideas, however obscure, of the entire universe? | |
| From: comment on Gottfried Leibniz (Monadology [1716]) by Francois-Marie Voltaire - works Vol 22:434 | |
| A reaction: Monads are a bit like Christian theology - if you meet them cold they seem totally ridiculous, but if you meet them after ten years of careful preliminary study they make (apparently) complete sense. Defenders of panpsychism presumably like them. |
| 12751 | It is unclear in 'Monadology' how extended bodies relate to mind-like monads. [Garber on Leibniz] |
| Full Idea: It is never clear in the 'Monadologie' how exactly the world of extended bodies is related to the world of simple substances, the world of non-extended and mind-like monads. | |
| From: comment on Gottfried Leibniz (Monadology [1716]) by Daniel Garber - Leibniz:Body,Substance,Monad 9 | |
| A reaction: Leibniz was always going to hit the interaction problem, as soon as he started giving an increasingly spiritual account of what a substance, and hence marginalising the 'force' which had held centre-stage earlier on. Presumably they are 'parallel'. |
| 19363 | Changes in a monad come from an internal principle, and the diversity within its substance [Leibniz] |
| Full Idea: A monad's natural changes come from an internal principle, ...but there must be diversity in that which changes, which produces the specification and variety of substances. | |
| From: Gottfried Leibniz (Monadology [1716], §11-12) | |
| A reaction: You don't have to like monads to like this generalisation (and Perkins says Leibniz had a genius for generalisations). Metaphysics must give an account of change. Succeeding time-slices etc explain nothing. Principle and substance must meet. |
| 19352 | A 'monad' has basic perception and appetite; a 'soul' has distinct perception and memory [Leibniz] |
| Full Idea: The general name 'monad' or 'entelechy' may suffice for those substances which have nothing but perception and appetition; the name 'souls' may be reserved for those having perception that is more distinct and accompanied by memory. | |
| From: Gottfried Leibniz (Monadology [1716], §19) | |
| A reaction: It is basic to the study of Leibniz that you don't think monads are full-blown consciousnesses. He isn't really a panpsychist, because the level of mental activity is so minimal. There seem to be degrees of monadhood. |
| 10488 | It is lunacy to think we only see ink-marks, and not word-types [Boolos] |
| Full Idea: It's a kind of lunacy to think that sound scientific philosophy demands that we think that we see ink-tracks but not words, i.e. word-types. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.128) | |
| A reaction: This seems to link him with Armstrong's mockery of 'ostrich nominalism'. There seems to be some ambiguity with the word 'see' in this disagreement. When we look at very ancient scratches on stones, why don't we always 'see' if it is words? |
| 10487 | I am a fan of abstract objects, and confident of their existence [Boolos] |
| Full Idea: I am rather a fan of abstract objects, and confident of their existence. Smaller numbers, sets and functions don't offend my sense of reality. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.128) | |
| A reaction: The great Boolos is rather hard to disagree with, but I disagree. Logicians love abstract objects, indeed they would almost be out of a job without them. It seems to me they smuggle them into our ontology by redefining either 'object' or 'exists'. |
| 10489 | We deal with abstract objects all the time: software, poems, mistakes, triangles.. [Boolos] |
| Full Idea: We twentieth century city dwellers deal with abstract objects all the time, such as bank balances, radio programs, software, newspaper articles, poems, mistakes, triangles. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: I find this claim to be totally question-begging, and typical of a logician. The word 'object' gets horribly stretched in these discussions. We can create concepts which have all the logical properties of objects. Maybe they just 'subsist'? |
| 7931 | If a substance is just a thing that has properties, it seems to be a characterless non-entity [Leibniz, by Macdonald,C] |
| Full Idea: For Leibniz, to distinguish between a substance and its properties in order to provide a thing or entity in which properties can inhere leads necessarily to the absurd conclusion that the substance itself must be a truly characterless non-entity. | |
| From: report of Gottfried Leibniz (Monadology [1716]) by Cynthia Macdonald - Varieties of Things Ch.3 | |
| A reaction: This is obviously one of the basic thoughts in any discussion of substances. It is why physicists ignore them, and Leibniz opted for a 'bundle' theory. But the alternative seems daft too - free-floating properties, hooked onto one another. |
| 17554 | There must be some internal difference between any two beings in nature [Leibniz] |
| Full Idea: There are never two beings in nature that are perfectly alike, two beings in which it is not possible to discover an internal difference, that is, one founded on an intrinsic denomination. | |
| From: Gottfried Leibniz (Monadology [1716], §09) | |
| A reaction: From this it follows that if two things really are indiscernible, then we must say that they are one thing. He says monads all differ from one another. People certainly do. Leibniz must say this of electrons. How can he know this? |
| 2112 | Truths of reason are known by analysis, and are necessary; facts are contingent, and their opposites possible [Leibniz] |
| Full Idea: There are two kinds of truths: of reasoning and of facts. Truths of reasoning are necessary and their opposites impossible. Facts are contingent and their opposites possible. A necessary truth is known by analysis. | |
| From: Gottfried Leibniz (Monadology [1716], §33) |
| 9344 | Mathematical analysis ends in primitive principles, which cannot be and need not be demonstrated [Leibniz] |
| Full Idea: At the end of the analytical method in mathematics there are simple ideas of which no definition can be given. Moreover there are axioms and postulates, in short, primitive principles, which cannot be demonstrated and do not need demonstration. | |
| From: Gottfried Leibniz (Monadology [1716], §35) | |
| A reaction: My view is that we do not know such principles when we apprehend them in isolation. I would call them 'intuitions'. They only ascend to the status of knowledge when the mathematics is extended and derived from them, and found to work. |
| 2110 | We all expect the sun to rise tomorrow by experience, but astronomers expect it by reason [Leibniz] |
| Full Idea: When we expect it to be day tomorrow, we all behave as empiricists, because until now it has always happened thus. The astronomer alone knows this by reason. | |
| From: Gottfried Leibniz (Monadology [1716], §28) |
| 2109 | Increase a conscious machine to the size of a mill - you still won't see perceptions in it [Leibniz] |
| Full Idea: If a conscious machine were increased in size, one might enter it like a mill, but we should only see the parts impinging on one another; we should not see anything which would explain a perception. | |
| From: Gottfried Leibniz (Monadology [1716], §17) | |
| A reaction: A wonderful image for capturing a widely held intuition. It seems to motivate Colin McGinn's 'Mysterianism'. The trouble is Leibniz didn't think big/small enough. Down at the level of molecules it might become obvious what a perception is. 'Might'. |
| 19362 | We know the 'I' and its contents by abstraction from awareness of necessary truths [Leibniz] |
| Full Idea: It is through the knowledge of necessary truths and through their abstraction that we rise to reflective acts, which enable us to think of that which is called "I" and enable us to consider that this or that is in us. | |
| From: Gottfried Leibniz (Monadology [1716], §30) | |
| A reaction: For Leibniz, necessary truth can only be known a priori. Sense experience won't reveal the self, as Hume observed. We evidently 'abstract' the idea of 'I' from the nature of a priori thought. Animals have no self (or morals) for this reason. |
| 12707 | The true elements are atomic monads [Leibniz] |
| Full Idea: Monads are the true atoms of nature and, in brief, the elements of things. | |
| From: Gottfried Leibniz (Monadology [1716], (opening)), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 2 | |
| A reaction: Thus in one sentence Leibniz gives us a theory of natural elements, and an account of atoms. This kind of speculation got metaphysics a bad name when science unravelled a more accurate picture. The bones must be picked out of Leibniz. |
| 2114 | This is the most perfect possible universe, in its combination of variety with order [Leibniz] |
| Full Idea: From all the possible universes God chooses this one to obtain as much variety as possible, but with the greatest order possible; that is, it is the means of obtaining the greatest perfection possible. | |
| From: Gottfried Leibniz (Monadology [1716], §58) |
| 2113 | God alone (the Necessary Being) has the privilege that He must exist if He is possible [Leibniz] |
| Full Idea: God alone (or the Necessary Being) has the privilege that He must exist if He is possible. | |
| From: Gottfried Leibniz (Monadology [1716], §45) |