48 ideas
| 17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
| Full Idea: In intuitionist logic double negation elimination fails. After all, proving that there is no proof that there can't be a proof of S is not the same thing as having a proof of S. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: I do like people like Colyvan who explain things clearly. All of this difficult stuff is understandable, if only someone makes the effort to explain it properly. |
| 17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
| Full Idea: The intuitionist rejection of double negation elimination undermines the important reductio ad absurdum proof in classical mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) |
| 17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
| Full Idea: The law of excluded middle (for every proposition P, either P or not-P) must be carefully distinguished from its semantic counterpart bivalence, that every proposition is either true or false. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: So excluded middle makes no reference to the actual truth or falsity of P. It merely says P excludes not-P, and vice versa. |
| 17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
| Full Idea: Löwenheim proved that if a first-order sentence has a model at all, it has a countable model. ...Skolem generalised this result to systems of first-order sentences. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) |
| 17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
| Full Idea: A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) | |
| A reaction: The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'. |
| 17928 | Ordinal numbers represent order relations [Colyvan] |
| Full Idea: Ordinal numbers represent order relations. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.2.3 n17) |
| 17923 | Intuitionists only accept a few safe infinities [Colyvan] |
| Full Idea: For intuitionists, all but the smallest, most well-behaved infinities are rejected. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: The intuitionist idea is to only accept what can be clearly constructed or proved. |
| 17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
| Full Idea: The problem with infinitesimals is that in some places they behaved like real numbers close to zero but in other places they behaved like zero. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.2) | |
| A reaction: Colyvan gives an example, of differentiating a polynomial. |
| 17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
| Full Idea: Given Dedekind's reduction of real numbers to sequences of rational numbers, and other known reductions in mathematics, it was tempting to see basic arithmetic as the foundation of mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.1) | |
| A reaction: The reduction is the famous Dedekind 'cut'. Nowadays theorists seem to be more abstract (Category Theory, for example) instead of reductionist. |
| 17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
| Full Idea: Transfinite inductions are inductive proofs that include an extra step to show that if the statement holds for all cases less than some limit ordinal, the statement also holds for the limit ordinal. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1 n11) |
| 17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
| Full Idea: Most mathematical proofs, outside of set theory, do not explicitly state the set theory being employed. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.1) |
| 17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
| Full Idea: Structuralism is able to explain why mathematicians are typically only interested in describing the objects they study up to isomorphism - for that is all there is to describe. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) |
| 17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
| Full Idea: In re structuralism does not posit anything other than the kinds of structures that are in fact found in the world. ...The problem is that the world may not provide rich enough structures for the mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) | |
| A reaction: You can perceive a repeating pattern in the world, without any interest in how far the repetitions extend. |
| 21982 | I only wish I had such eyes as to see Nobody! It's as much as I can do to see real people. [Carroll,L] |
| Full Idea: "I see nobody on the road," said Alice. - "I only wish I had such eyes," the King remarked. ..."To be able to see Nobody! ...Why, it's as much as I can do to see real people." | |
| From: Lewis Carroll (C.Dodgson) (Through the Looking Glass [1886], p.189), quoted by A.W. Moore - The Evolution of Modern Metaphysics 07.7 | |
| A reaction: [Moore quotes this, inevitably, in a chapter on Hegel] This may be a better candidate for the birth of philosophy of language than Frege's Groundwork. |
| 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. |
| 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. |
| 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. |
| 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? |
| 17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
| Full Idea: Those who see probabilities as ratios of frequencies can't use Bayes's Theorem if there is no objective prior probability. Those who accept prior probabilities tend to opt for a subjectivist account, where probabilities are degrees of belief. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.8) | |
| A reaction: [compressed] |
| 17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
| Full Idea: Mathematics can demonstrate structural similarities between systems (e.g. missing population periods and the gaps in the rings of Saturn). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) | |
| A reaction: [Colyvan expounds the details of his two examples] It is these sorts of results that get people enthusiastic about the mathematics embedded in nature. A misunderstanding, I think. |
| 17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
| Full Idea: Mathematics can show that under a broad range of conditions, something initially surprising must occur (e.g. the hexagonal structure of honeycomb). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) |
| 17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
| Full Idea: Another style of proof often cited as unexplanatory are brute-force methods such as proof by cases (or proof by exhaustion). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) |
| 17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
| Full Idea: One kind of proof that is thought to be unexplanatory is the 'reductio' proof. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
| A reaction: Presumably you generate a contradiction, but are given no indication of why the contradiction has arisen? Tracking back might reveal the source of the problem? Colyvan thinks reductio can be explanatory. |
| 17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
| Full Idea: It might be argued that any proof by induction is revealing the explanation of the theorem, namely, that it holds by virtue of the structure of the natural numbers. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
| A reaction: This is because induction characterises the natural numbers, in the Peano Axioms. |
| 17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
| Full Idea: The proof of the four-colour theorem raises questions about whether a 'proof' that no one understands is a proof. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.6) | |
| A reaction: The point is that the theorem (that you can colour countries on a map with just four colours) was proved with the help of a computer. |
| 17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
| Full Idea: One type of generalisation in mathematics extends a system to go beyond what is was originally set up for; another kind involves abstracting away from some details in order to capture similarities between different systems. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.2) |
| 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. |