49 ideas
| 1502 | Parmenides was much more cautious about accepting ideas than his predecessors [Simplicius on Parmenides] |
| Full Idea: Parmenides would not agree with anything unless it seemed necessary, whereas his predecessors used to come up with unsubstantiated assertions. | |
| From: comment on Parmenides (fragments/reports [c.474 BCE], A28) by Simplicius - On Aristotle's 'Physics' 9.116.2- | |
| A reaction: from Eudemus |
| 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) |
| 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. |
| 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. |
| 448 | No necessity could produce Being either later or earlier, so it must exist absolutely or not at all [Parmenides] |
| Full Idea: What necessity impelled Being, if it did spring from nothing, to be produced later or earlier? Thus it must be absolutely, or not at all. | |
| From: Parmenides (fragments/reports [c.474 BCE], B08 ll.?), quoted by Simplicius - On Aristotle's 'Physics' 9.145.1- |
| 447 | Being must be eternal and uncreated, and hence it is timeless [Parmenides] |
| Full Idea: Being has no coming-to-be and no destruction, for it is whole of limb, without motion, and without end. And it never was, nor will be, because it is now, a whole all together, one, continuous; for what creation of it will you look for? | |
| From: Parmenides (fragments/reports [c.474 BCE], B08 ll.?), quoted by Simplicius - On Aristotle's 'Physics' 9.145.1- |
| 449 | Being is not divisible, since it is all alike [Parmenides] |
| Full Idea: Being is not divisible, since it is all alike. | |
| From: Parmenides (fragments/reports [c.474 BCE], B08 ll.?), quoted by Simplicius - On Aristotle's 'Physics' 9.145.1- |
| 1503 | There is no such thing as nothing [Parmenides] |
| Full Idea: There is no such thing as nothing. | |
| From: Parmenides (fragments/reports [c.474 BCE], B06), quoted by Simplicius - On Aristotle's 'Physics' 9.86.27- |
| 445 | The realm of necessary non-existence cannot be explored, because it is unknowable [Parmenides] |
| Full Idea: The other way of enquiry, that IT IS NOT, and IT is bound NOT TO BE, cannot be explored, for you could neither recognise nor express that which IS NOT. | |
| From: Parmenides (fragments/reports [c.474 BCE], B02), quoted by Simplicius - On Aristotle's 'Physics' 9.116.28- |
| 21820 | Parmenides at least saw Being as the same as Nous, and separate from the sensed realm [Parmenides, by Plotinus] |
| Full Idea: Parmenides made some approach to the doctrine of Plato in identifying Being with Intellectual-Principle [Nous] while separating Real Being from the realm of sense. | |
| From: report of Parmenides (fragments/reports [c.474 BCE]) by Plotinus - The Enneads 5.1.08 | |
| A reaction: The point is that for Parmenides the One is the essence of Being, but for platonists there is something prior to and higher than Being. For Plato it is the Good; for Plotinus it is a revised (non-Being) concept of the One. |
| 452 | All our concepts of change and permanence are just names, not the truth [Parmenides] |
| Full Idea: All things that mortals have established, believing in their truth, are just a name: Becoming and Perishing, Being and Not-Being, and change of position, and alteration of bright colour. | |
| From: Parmenides (fragments/reports [c.474 BCE], B08 ll.?), quoted by Simplicius - On Aristotle's 'Physics' 9.145.1- |
| 1504 | Something must be unchanging to make recognition and knowledge possible [Aristotle on Parmenides] |
| Full Idea: Parmenides and Melissus were the first to appreciate that there must be unchanging entities, if recognition and knowledge are to exist. | |
| From: comment on Parmenides (fragments/reports [c.474 BCE], A25) by Aristotle - On the Heavens 298b14 |
| 444 | The first way of enquiry involves necessary existence [Parmenides] |
| Full Idea: The first way of enquiry is the one that IT IS, and it is not possible for IT NOT TO BE, which is the way of credibility, for it follows truth. | |
| From: Parmenides (fragments/reports [c.474 BCE], B02), quoted by Simplicius - On Aristotle's 'Physics' 9.116.28- | |
| A reaction: also Proclus 'Timeus' |
| 450 | Necessity sets limits on being, in order to give it identity [Parmenides] |
| Full Idea: Powerful necessity holds Being in the bonds of a limit, which constrains it round about, because divine law decrees that Being shall not be without boundary. For it is not lacking, but if it were spatially infinite, it would lack everything. | |
| From: Parmenides (fragments/reports [c.474 BCE], B08 ll.?), quoted by Simplicius - On Aristotle's 'Physics' 9.145.1- |
| 451 | Thinking implies existence, because thinking depends on it [Parmenides] |
| Full Idea: To think is the same as the thought that IT IS, for you will not find thinking without Being, on which it depends for its expression. | |
| From: Parmenides (fragments/reports [c.474 BCE], B08 ll.?), quoted by Simplicius - On Aristotle's 'Physics' 9.145.1- |
| 1506 | Parmenides treats perception and intellectual activity as the same [Theophrastus on Parmenides] |
| Full Idea: Parmenides treats perception and intellectual activity as the same. | |
| From: comment on Parmenides (fragments/reports [c.474 BCE], A46) by Theophrastus - On the Senses 3.1 | |
| A reaction: cf Theaetetus pt 1 |
| 3058 | Only reason can prove the truth of facts [Parmenides] |
| Full Idea: Reason alone will prove the truth of facts. | |
| From: Parmenides (fragments/reports [c.474 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 09.3.3 |
| 1556 | By nature people are close to one another, but culture drives them apart [Hippias] |
| Full Idea: I regard you all as relatives - by nature, not by convention. By nature like is akin to like, but convention is a tyrant over humankind and often constrains people to act contrary to nature. | |
| From: Hippias (fragments/reports [c.430 BCE]), quoted by Plato - Protagoras 337c8 |
| 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) |
| 555 | People who say that the cosmos is one forget that they must explain movement [Aristotle on Parmenides] |
| Full Idea: Those who assert that the universe is one and a single nature, when they try to give the causes of generation and destruction, miss out the cause of movement. | |
| From: comment on Parmenides (fragments/reports [c.474 BCE]) by Aristotle - Metaphysics 988b |
| 5081 | There could be movement within one thing, as there is within water [Aristotle on Parmenides] |
| Full Idea: Why does it follow from there being only one thing that it is unmoving, since, for example, water moves internally while remaining one? | |
| From: comment on Parmenides (fragments/reports [c.474 BCE]) by Aristotle - Physics 186a16 | |
| A reaction: One suspects that Parmenides wasn't used to critical questions like this, and would have sharpened up his theory if it had been subjected to criticism. How big was the One? Maybe Aristotle is the real father of philosophy. |
| 1509 | The one can't be divisible, because if it was it could be infinitely divided down to nothing [Parmenides, by Simplicius] |
| Full Idea: Since the one is everywhere alike, then if it is divisible, it will be equally divisible everywhere….so let it be divided everywhere. It is obvious that nothing will remain and the whole will vanish, and so (if it is compound) it is composed of nothing. | |
| From: report of Parmenides (fragments/reports [c.474 BCE]) by Simplicius - On Aristotle's 'Physics' 9.139.5- | |
| A reaction: he is quoting Porphyry |
| 20900 | Defenders of the One say motion needs the void - but that is not part of Being [Parmenides, by Aristotle] |
| Full Idea: Defenders of the One say that there could not be motion without a void, and that void is what does not exist, and that nothing that is not belongs to being. | |
| From: report of Parmenides (fragments/reports [c.474 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 325a26 | |
| A reaction: This is why motion is an illusion, a view also supported by the paradoxes of Zeno of Elea. Aristotle goes on to give Democritus's response to this idea. Parmenides was contemplating 'void', before Democritus got to it. |
| 226 | The one is without any kind of motion [Parmenides] |
| Full Idea: The one is without any kind of motion. | |
| From: Parmenides (fragments/reports [c.474 BCE]), quoted by Plato - Parmenides 139a |
| 1505 | Reason sees reality as one, the senses see it as many [Aristotle on Parmenides] |
| Full Idea: Since he is forced to be guided by appearances, he assumes that the one exists from the viewpoint of reason, but that a plurality exists from the viewpoint of the sense, and so he posits two principles and causes - hot and cold. | |
| From: comment on Parmenides (fragments/reports [c.474 BCE], A24) by Aristotle - Metaphysics 986b27- | |
| A reaction: A profound thought. Empiricists emphasies experience, and end up with fragmented reality. Reason explains experience, and in the process sees the world as unities (like objects), though a single unity is going too far. |
| 453 | Reality is symmetrical and balanced, like a sphere, with no reason to be greater one way rather than another [Parmenides] |
| Full Idea: Since there is a spatial limit, it is complete on every side, like the mass of a well-rounded sphere, equally balanced from its centre in every direction; for it is not bound to be at all either greater or less in this direction or that. | |
| From: Parmenides (fragments/reports [c.474 BCE], B08 ll.?), quoted by Simplicius - On Aristotle's 'Physics' 9.145.1- |
| 1792 | He taught that there are two elements, fire the maker, and earth the matter [Parmenides, by Diog. Laertius] |
| Full Idea: He taught that there were two elements, fire and earth; and that one of them occupies the place of the maker, the other that of the matter. | |
| From: report of Parmenides (fragments/reports [c.474 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.Pa.2 |
| 5115 | It is feeble-minded to look for explanations of everything being at rest [Aristotle on Parmenides] |
| Full Idea: For people to ignore the evidence of their senses and look for an explanation for everything being at rest is feeble-minded. | |
| From: comment on Parmenides (fragments/reports [c.474 BCE]) by Aristotle - Physics 253a32 | |
| A reaction: Not exactly an argument, but an interestingly robust assertion of commonsense against dodgy arguments. Aristotle is not exactly an empiricist, but he is on that side of the fence. |
| 13217 | The void can't exist, and without the void there can't be movement or separation [Parmenides, by Aristotle] |
| Full Idea: Some philosophers thought what is must be one and immovable. The void, they say, is not: but unless there is a void what is cannot be moved, nor can it be many, since there is nothing to keep things apart. | |
| From: report of Parmenides (fragments/reports [c.474 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 325a06 | |
| A reaction: Somehow this doesn't seem very persuasive any more! I suppose we would distinguish various degrees of void, and assert the existence of sufficient void to allow movement and separation. We must surely agree that total nothingness doesn't exist. |
| 22918 | What could have triggered the beginning [of time and being]? [Parmenides] |
| Full Idea: What need would have aroused it later or sooner, starting from nothing to come into being? | |
| From: Parmenides (fragments/reports [c.474 BCE]), quoted by Robin Le Poidevin - Travels in Four Dimensions 02 'Everything' | |
| A reaction: [Barnes 1982:178] This remains an excellent question. The last I heard was a 'quantum fluctuation', but that seems to be an event, which therefore needs time. |
| 1794 | He was the first to discover the identity of the Morning and Evening Stars [Parmenides, by Diog. Laertius] |
| Full Idea: He appears to have been the first to discover that Hesperus and Lucifer were the same star. | |
| From: report of Parmenides (fragments/reports [c.474 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.Pa.3 | |
| A reaction: This is the famous example used by Frege to discuss reference and meaning. |
| 1791 | He was the first person to say the earth is spherical [Parmenides, by Diog. Laertius] |
| Full Idea: He was the first person who asserted that the earth was of a spherical form. | |
| From: report of Parmenides (fragments/reports [c.474 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.Pa.2 |