41 ideas
| 10702 | Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter] |
| Full Idea: Set theory has three roles: as a means of taming the infinite, as a supplier of the subject-matter of mathematics, and as a source of its modes of reasoning. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], Intro 1) | |
| A reaction: These all seem to be connected with mathematics, but there is also ontological interest in set theory. Potter emphasises that his second role does not entail a commitment to sets 'being' numbers. |
| 10713 | Usually the only reason given for accepting the empty set is convenience [Potter] |
| Full Idea: It is rare to find any direct reason given for believing that the empty set exists, except for variants of Dedekind's argument from convenience. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 04.3) |
| 13044 | Infinity: There is at least one limit level [Potter] |
| Full Idea: Axiom of Infinity: There is at least one limit level. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 04.9) | |
| A reaction: A 'limit ordinal' is one which has successors, but no predecessors. The axiom just says there is at least one infinity. |
| 10708 | Nowadays we derive our conception of collections from the dependence between them [Potter] |
| Full Idea: It is only quite recently that the idea has emerged of deriving our conception of collections from a relation of dependence between them. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.2) | |
| A reaction: This is the 'iterative' view of sets, which he traces back to Gödel's 'What is Cantor's Continuum Problem?' |
| 13546 | The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter] |
| Full Idea: We group under the heading 'limitation of size' those principles which classify properties as collectivizing or not according to how many objects there are with the property. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 13.5) | |
| A reaction: The idea was floated by Cantor, toyed with by Russell (1906), and advocated by von Neumann. The thought is simply that paradoxes start to appear when sets become enormous. |
| 10707 | Mereology elides the distinction between the cards in a pack and the suits [Potter] |
| Full Idea: Mereology tends to elide the distinction between the cards in a pack and the suits. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1) | |
| A reaction: The example is a favourite of Frege's. Potter is giving a reason why mathematicians opted for set theory. I'm not clear, though, why a pack cannot have either 4 parts or 52 parts. Parts can 'fall under a concept' (such as 'legs'). I'm puzzled. |
| 10704 | We can formalize second-order formation rules, but not inference rules [Potter] |
| Full Idea: In second-order logic only the formation rules are completely formalizable, not the inference rules. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 01.2) | |
| A reaction: He cites Gödel's First Incompleteness theorem for this. |
| 10703 | Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter] |
| Full Idea: A 'supposition' axiomatic theory is as concerned with truth as a 'realist' one (with undefined terms), but the truths are conditional. Satisfying the axioms is satisfying the theorem. This is if-thenism, or implicationism, or eliminative structuralism. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 01.1) | |
| A reaction: Aha! I had failed to make the connection between if-thenism and eliminative structuralism (of which I am rather fond). I think I am an if-thenist (not about all truth, but about provable truth). |
| 10712 | If set theory didn't found mathematics, it is still needed to count infinite sets [Potter] |
| Full Idea: Even if set theory's role as a foundation for mathematics turned out to be wholly illusory, it would earn its keep through the calculus it provides for counting infinite sets. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.8) |
| 17882 | It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter] |
| Full Idea: It is a remarkable fact that all the arithmetical properties of the natural numbers can be derived from such a small number of assumptions (as the Peano Axioms). | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 05.2) | |
| A reaction: If one were to defend essentialism about arithmetic, this would be grist to their mill. I'm just saying. |
| 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. |
| 13043 | A relation is a set consisting entirely of ordered pairs [Potter] |
| Full Idea: A set is called a 'relation' if every element of it is an ordered pair. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 04.7) | |
| A reaction: This is the modern extensional view of relations. For 'to the left of', you just list all the things that are to the left, with the things they are to the left of. But just listing the ordered pairs won't necessarily reveal how they are related. |
| 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'. |
| 13042 | If dependence is well-founded, with no infinite backward chains, this implies substances [Potter] |
| Full Idea: The argument that the relation of dependence is well-founded ...is a version of the classical arguments for substance. ..Any conceptual scheme which genuinely represents a world cannot contain infinite backward chains of meaning. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3) | |
| A reaction: Thus the iterative conception of set may imply a notion of substance, and Barwise's radical attempt to ditch the Axiom of Foundation (Idea 13039) was a radical attempt to get rid of 'substances'. Potter cites Wittgenstein as a fan of substances here. |
| 13041 | Collections have fixed members, but fusions can be carved in innumerable ways [Potter] |
| Full Idea: A collection has a determinate number of members, whereas a fusion may be carved up into parts in various equally valid (although perhaps not equally interesting) ways. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1) | |
| A reaction: This seems to sum up both the attraction and the weakness of mereology. If you doubt the natural identity of so-called 'objects', then maybe classical mereology is the way to go. |
| 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. |
| 10709 | Priority is a modality, arising from collections and members [Potter] |
| Full Idea: We must conclude that priority is a modality distinct from that of time or necessity, a modality arising in some way out of the manner in which a collection is constituted from its members. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3) | |
| A reaction: He is referring to the 'iterative' view of sets, and cites Aristotle 'Metaphysics' 1019a1-4 as background. |
| 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? |
| 22200 | If you eliminate the impossible, the truth will remain, even if it is weird [Conan Doyle] |
| Full Idea: When you have eliminated the impossible, whatever remains, however improbable, must be the truth. | |
| From: Arthur Conan Doyle (The Sign of Four [1890], Ch. 6) | |
| A reaction: A beautiful statement, by Sherlock Holmes, of Eliminative Induction. It is obviously not true, of course. Many options may still face you after you have eliminated what is actually impossible. |
| 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. |