37 ideas
| 18261 | A simplification which is complete constitutes a definition [Kant] |
| Full Idea: By dissection I can make the concept distinct only by making the marks it contains clear. That is what analysis does. If this analysis is complete ...and in addition there are not so many marks, then it is precise and so constitutes a definition. | |
| From: Immanuel Kant (Wiener Logik [1795], p.455), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 1 'Conc' | |
| A reaction: I think Aristotle would approve of this. We need to grasp that a philosophical definition is quite different from a lexicographical definition. 'Completeness' may involve quite a lot. |
| 23623 | Predicativism says only predicated sets exist [Hossack] |
| Full Idea: Predicativists doubt the existence of sets with no predicative definition. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 02.3) | |
| A reaction: This would imply that sets which encounter paradoxes when they try to be predicative do not therefore exist. Surely you can have a set of random objects which don't fall under a single predicate? |
| 23624 | The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack] |
| Full Idea: The iterative conception justifies Power Set, but cannot justify a satisfactory theory of von Neumann ordinals, so ZFC appropriates Replacement from NBG set theory. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9) | |
| A reaction: The modern approach to axioms, where we want to prove something so we just add an axiom that does the job. |
| 23625 | Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack] |
| Full Idea: The limitation of size conception of sets justifies the axiom of Replacement, but cannot justify Power Set, so NBG set theory appropriates the Power Set axiom from ZFC. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9) | |
| A reaction: Which suggests that the Power Set axiom is not as indispensable as it at first appears to be. |
| 22275 | Logic gives us the necessary rules which show us how we ought to think [Kant] |
| Full Idea: In logic the question is not one of contingent but of necessary rules, not how to think, but how we ought to think. | |
| From: Immanuel Kant (Wiener Logik [1795], p.16), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 02 'Trans' | |
| A reaction: Presumably it aspires to the objectivity of a single correct account of how we all ought to think. I'm sympathetic to that, rather than modern cultural relativism about reason. Logic is rooted in nature, not in arbitrary convention. |
| 23628 | The connective 'and' can have an order-sensitive meaning, as 'and then' [Hossack] |
| Full Idea: The sentence connective 'and' also has an order-sensitive meaning, when it means something like 'and then'. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.4) | |
| A reaction: This is support the idea that orders are a feature of reality, just as much as possible concatenation. Relational predicates, he says, refer to series rather than to individuals. Nice point. |
| 23627 | 'Before' and 'after' are not two relations, but one relation with two orders [Hossack] |
| Full Idea: The reason the two predicates 'before' and 'after' are needed is not to express different relations, but to indicate its order. Since there can be difference of order without difference of relation, the nature of relations is not the source of order. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.3) | |
| A reaction: This point is to refute Russell's 1903 claim that order arises from the nature of relations. Hossack claims that it is ordered series which are basic. I'm inclined to agree with him. |
| 23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack] |
| Full Idea: The transfinite ordinal numbers are important in the theory of proofs, and essential in the theory of recursive functions and computability. Mathematics would be incomplete without them. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.1) | |
| A reaction: Hossack offers this as proof that the numbers are not human conceptual creations, but must exist beyond the range of our intellects. Hm. |
| 23621 | Numbers are properties, not sets (because numbers are magnitudes) [Hossack] |
| Full Idea: I propose that numbers are properties, not sets. Magnitudes are a kind of property, and numbers are magnitudes. …Natural numbers are properties of pluralities, positive reals of continua, and ordinals of series. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], Intro) | |
| A reaction: Interesting! Since time can have a magnitude (three weeks) just as liquids can (three litres), it is not clear that there is a single natural property we can label 'magnitude'. Anything we can manage to measure has a magnitude. |
| 23622 | We can only mentally construct potential infinities, but maths needs actual infinities [Hossack] |
| Full Idea: Numbers cannot be mental objects constructed by our own minds: there exists at most a potential infinity of mental constructions, whereas the axioms of mathematics require an actual infinity of numbers. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], Intro 2) | |
| A reaction: Doubt this, but don't know enough to refute it. Actual infinities were a fairly late addition to maths, I think. I would think treating fictional complete infinities as real would be sufficient for the job. Like journeys which include imagined roads. |
| 12747 | Monads are not extended, but have a kind of situation in extension [Leibniz] |
| Full Idea: Even if monads are not extended, they nonetheless have a certain kind of situation in extension. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1703.06.20), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 8 | |
| A reaction: This is the kind of metaphysical mess you get into if you start from the wrong premisses (in this case, a dualism of the spiritual and the material). Later (Garber p.359) he says they are situated because they 'preside' over a mass. |
| 12748 | Only monads are substances, and bodies are collections of them [Leibniz] |
| Full Idea: A monad alone is a substance; a body is substances not a substance. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1704.01.21), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 8 | |
| A reaction: So how many monads in a drop of urine, as Voltaire bluntly wondered. I take the Cartesian dualism (without interaction) that ran through Leibniz's career to be the source of most of his metaphysical problems. In late career it went badly wrong. |
| 13184 | The division of nature into matter makes distinct appearances, and that presupposes substances [Leibniz] |
| Full Idea: If there were no divisions of matter in nature, there would be no things that are different; just the mere possibility of things. It is the actual division into masses that really produces things that appear distinct, which presupposes simple substances. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1704 or 1705) | |
| A reaction: This shows Leibniz to be a straightforward realist about the physical world, and certainly not an 'idealist', despite the mind-like character of monads. I take this to be an argument for reality from best explanation, which is all that's available. |
| 13188 | The only indications of reality are agreement among phenomena, and their agreement with necessities [Leibniz] |
| Full Idea: We don't have, nor should we hope for, any mark of reality in phenomena, but the fact that they agree with one another and with eternal truths. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1706.01.19) | |
| A reaction: Elsewhere he says that divisions in appearance imply divisions in matter. Now he adds two further arguments in favour of realism, but admits that nothing conclusive is available. Quite right. |
| 12752 | Only unities have any reality [Leibniz] |
| Full Idea: There is no reality in anything except the reality of unities. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1704.06.30), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 9 | |
| A reaction: This seems to leave indeterminate stuff like air and water with no reality, as nicely discussed by Henry Laycock. Do we just force unities on the world because that is the only way our minds can cope with it? |
| 13187 | In actual things nothing is indefinite [Leibniz] |
| Full Idea: In actual things nothing is indefinite. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1706.01.19) | |
| A reaction: This seems to be the germ of the controversial modern view of Williamson, that vagueness is entirely epistemic, and that the facts of nature are entirely definite. Thus there is a tallest short giraffe, which I find a bit hard to grasp. |
| 19383 | A man's distant wife dying is a real change in him [Leibniz] |
| Full Idea: No one can become a widower in India because of the death of his wife in Europe unless a real change occurs in him. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], GP ii 240), quoted by Richard T.W. Arthur - Leibniz 7 'Nominalist' | |
| A reaction: This is Leibniz heroically denying so-called 'Cambridge Change'. It is hard to see how a widower is changed if he has not yet heard the bad news. But his situation in life has changed. Compare eudaimonia, which you can lose without realising it. |
| 13179 | A complete monad is a substance with primitive active and passive power [Leibniz] |
| Full Idea: What I take to be the indivisible or complete monad is the substance endowed with primitive power, active and passive, like the 'I' or something similar. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1703.06.20) | |
| A reaction: I love powers, so I really like this quotation. By this date even Garber thinks that he has more or less arrived at his mature view of monads. I used to think monads were mad, but I now think he is closing in on the right answer - sort of. |
| 12749 | Derivate forces are in phenomena, but primitive forces are in the internal strivings of substances [Leibniz] |
| Full Idea: I relegate derivative forces to the phenomena, but I think that it is clear that primitive forces can be nothing other than the internal strivings of simple substances. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1705.01), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 8 | |
| A reaction: I like 'internal strivings', which sounds to me like the Will to Power (Idea 7140). There seems to be an epistemological challenge in trying to disentangle the derivative forces from the primitive ones. |
| 12722 | Thought terminates in force, rather than extension [Leibniz] |
| Full Idea: I believe that our thought is completed and terminated more in the notion of the dynamic [i.e. force] than in that of extension. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], G II 170), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 4 | |
| A reaction: Presenting this as the place where 'our thought' is 'terminated' seems to place it as mainly having a role in explanation, rather than in speculative metaphysics. |
| 19379 | The law of the series, which determines future states of a substance, is what individuates it [Leibniz] |
| Full Idea: That there should be a persistent law of the series, which involves the future states of that which we conceive to be the same, is exactly what I say constitutes it as the same substance. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1704), quoted by Richard T.W. Arthur - Leibniz 4 'Applying' | |
| A reaction: The 'law of the series' is a bit dubious, but it is reasonable to say that a substance is individuated by its coherent progress of change over time. Disjointed change would imply an absence of substance. The law of the series is called 'primitive force'. |
| 13182 | Changeable accidents are modifications of unchanging essences [Leibniz] |
| Full Idea: Everything accidental or changeable ought to be a modification of something essential or perpetual. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1704.06.30) | |
| A reaction: Clear evidence that Leibniz is very much a traditional Aristotelian essentialist, and not as modal logicians tend to characterise him, as a super-essentialist who thinks all properties are essential. They are necessary for identity, but that's different. |
| 13178 | Things in different locations are different because they 'express' those locations [Leibniz] |
| Full Idea: Things that differ in place must express their place, that is, they must express the things surrounding, and thus they must be distinguished not only by place, that is, not by an extrinsic denomination alone, as is commonly thought. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1703.06.20) | |
| A reaction: This is an unusual view, which has some attractions, as it enables the relations of a thing to individuate it, while maintaining that this is a real difference in character. |
| 19411 | In nature there aren't even two identical straight lines, so no two bodies are alike [Leibniz] |
| Full Idea: In nature any straight line you may take is individually different from any other straight line you may find. Accordingly, it cannot come about that two bodies are perfectly equal and alike. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1703.06.20) | |
| A reaction: Leibniz was very good at persuasive examples! It remains unclear, though, why he takes the Identity of Indiscernibles to be a necessary truth, when he seems to have only observed it from experience. This is counter to his other principles. |
| 19412 | If two bodies only seem to differ in their position, those different environments will matter [Leibniz] |
| Full Idea: If two bodies differ only in their position, their individual relations to the environment must be taken into account, so that more is involved in their distinguishability than just position. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1703.06.20) | |
| A reaction: This seems to allow that two bodies could be intrinsically type-identical (though differing in extrinsic features), which is contrary to his normal view. I suppose a different location in the gravitational field will make an intrinsic difference. |
| 18260 | If we knew what we know, we would be astonished [Kant] |
| Full Idea: If we only know what we know ...we would be astonished by the treasures contained in our knowledge. | |
| From: Immanuel Kant (Wiener Logik [1795], p.843), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 1 'Conc' | |
| A reaction: Nice remark. He doesn't require immediat recall of knowledge. You can't be required to know that you know something. That doesn't imply externalism, though. I believe in securely founded internal knowledge which is hard to recall. |
| 19410 | Scientific truths are supported by mutual agreement, as well as agreement with the phenomena [Leibniz] |
| Full Idea: Among the most powerful indications of truth belongs the fact that scientific propositions agree with one another as well as with phenomena. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1699.03.24/04.03) | |
| A reaction: I take this to be the case not only with science, but with all other truths. Leibniz is particularly keen on the interconnectedness of things, so coherence justification suits him especially well. But surely all scientists embrace this idea? |
| 13183 | Primitive forces are internal strivings of substances, acting according to their internal laws [Leibniz] |
| Full Idea: Primitive forces can be nothing but the internal strivings [tendentia] of simple substances, striving by means of which they pass from perception to perception in accordance with a certain law of their nature. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1704 or 1705) | |
| A reaction: 'Perception' sounds a bit crazy, but he usually qualifies that sort of remark by saying that it is an 'analogy' with conscious willing souls. The 'internal strivings of substances' is a nice phrase for the basic powers in nature where explanations stop. |
| 19409 | Soul represents body, but soul remains unchanged, while body continuously changes [Leibniz] |
| Full Idea: The essence of the soul is to represent bodies. ...The soul and the idea of the body do not signify the same thing. For the soul remains one and the same, while the idea of the body perpetually changes as the body itself changes. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1699.03.24/04.03) | |
| A reaction: This seems to rest on the Cartesian Ego, as the essence of mind which does not change. And yet elsewhere he describes the Ego as a mere abstraction from introspected mental life. |
| 11873 | Our notions may be formed from concepts, but concepts are formed from things [Leibniz] |
| Full Idea: You assert that the notion of substance is formed from concepts, and not from things. But are not concepts themselves formed from things? | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1699.06.23), quoted by David Wiggins - Sameness and Substance Renewed 5.7 | |
| A reaction: A nice remark, which is true even of highly abstruse, abstract or fanciful concepts. You are still left with the question of how far away from reality you have moved when you construct things from your reality-based concepts. |
| 13186 | Universals are just abstractions by concealing some of the circumstances [Leibniz] |
| Full Idea: In forming universals the soul only abstracts certain circumstances by concealing innumerable others. ..A spherical body complete in all respects is nowhere in nature; the soul forms such a notion by concealing aberrations. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1704 or 1705) | |
| A reaction: This is Leibniz's affirmation of traditional 'abstraction by ignoring', which everyone seems to have believed in before Frege, and which I personally think is simply correct, even though it is deeply unfashionable and I keep it to myself. |
| 13185 | Even if extension is impenetrable, this still offers no explanation for motion and its laws [Leibniz] |
| Full Idea: Even if we grant impenetrability is added to extension, nothing complete is brought about, nothing from which a reason for motion, and especially the laws of motion, can be given. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1704 or 1705) | |
| A reaction: When it comes to the reasons for the so-called 'laws of nature', scientists give up, because they've only got mathematical descriptions, whereas the philosopher won't give up (even though, embarassingly, the evidence is running a bit thin). |
| 13177 | An entelechy is a law of the series of its event within some entity [Leibniz] |
| Full Idea: I recognize a primitive entelechy in the active force found in motion, something analogous to the soul, whose nature consists in a certain law of the same series of changes. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1699.03.24) | |
| A reaction: This is his 'law-of-the-series', which is a speculative attempt to pin down the character of the active essence of things which gives rise to activity. The law of such activity is within the things themselves, as scientific essentialists claim. |
| 13093 | The only permanence in things, constituting their substance, is a law of continuity [Leibniz] |
| Full Idea: Nothing is permanent in things except the law itself, which involves a continuous succession ...The fact that a certain law persists ...is the very fact that constitutes the same substance. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1704) | |
| A reaction: Aristotle and Leibniz are the very clear ancestors of modern scientific essentialism. I've left out a few inconvenient bits, about containing 'the whole universe', and containing all 'future states'. For Leibniz, laws are entirely rooted in things. |
| 13096 | The force behind motion is like a soul, with its own laws of continual change [Leibniz] |
| Full Idea: I recognise, in the active force which exerts itself through motion, the primitive entelechy or in a word, something analogous to the soul, whose nature consists in a certain perpetual law of the same series of changes through which it runs unhindered. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1699), quoted by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 6.1.3 | |
| A reaction: This is a hugely metaphysical account of force, contrasting with Newton's largely mathematical account. He very often says that force is 'analogous' to the soul, rather than that it actually is a soul. He never quite believes that monads are real minds. |
| 13180 | Space is the order of coexisting possibles [Leibniz] |
| Full Idea: Extension is the order of coexisting possibles. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1703.06.20) | |
| A reaction: [In his next letter he uses the word 'space' instead of 'extension'] This is a rather startling different and modal definition of space. Cf Idea 13181. |
| 13181 | Time is the order of inconsistent possibilities [Leibniz] |
| Full Idea: Time is the order of inconsistent possibilities. | |
| From: Gottfried Leibniz (Letters to Burcher De Volder [1706], 1703.06.20) | |
| A reaction: Cf. Idea 13180. This sounds wonderfully bold and interesting, but I can't make much sense of it. One might say it is 'an' order for such things, but 'the' order is weird. |