45 ideas
| 2653 | If the parts of the universe are subject to the law of nature, the whole universe must also be subject to it [Cicero] |
| Full Idea: If the parts of the universe are subject to the law of nature, then the universe itself must be subject to this law. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], II.86) |
| 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. |
| 11211 | If a sound conclusion comes from two errors that cancel out, the path of the argument must matter [Rumfitt] |
| Full Idea: If a designated conclusion follows from the premisses, but the argument involves two howlers which cancel each other out, then the moral is that the path an argument takes from premisses to conclusion does matter to its logical evaluation. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], II) | |
| A reaction: The drift of this is that our view of logic should be a little closer to the reasoning of ordinary language, and we should rely a little less on purely formal accounts. |
| 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. |
| 11212 | The sense of a connective comes from primitively obvious rules of inference [Rumfitt] |
| Full Idea: A connective will possess the sense that it has by virtue of its competent users' finding certain rules of inference involving it to be primitively obvious. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], III) | |
| A reaction: Rumfitt cites Peacocke as endorsing this view, which characterises the logical connectives by their rules of usage rather than by their pure semantic value. |
| 11210 | Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt] |
| Full Idea: If 'and' and 'but' really are alike in sense, in what might that likeness consist? Some philosophers of classical logic will reply that they share a sense by virtue of sharing a truth table. | |
| From: Ian Rumfitt ("Yes" and "No" [2000]) | |
| A reaction: This is the standard view which Rumfitt sets out to challenge. |
| 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. |
| 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) |
| 2628 | Why would mind mix with matter if it didn't need it? [Cicero] |
| Full Idea: If the gods have no need of the sensible world, why mix up mind with water and water with mind, if mind can exist by itself without any need of matter? | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], I.24) | |
| A reaction: This question migrates into our puzzles about why a separate mental substance would be produced by evolution. If it is device physical systems use to promote themselves, mental substance is reduced to an inferior and dependent role. |
| 20814 | Eloquence educates, exhorts, comforts, distracts and unites us, and raises us from savagery [Cicero] |
| Full Idea: How wonderful is the power of eloquence! It enables us to learn and to teach. We use it to exhort and persuade, to comfort the unfortunate, to distract the timid and calm the passionate. It unites us in law and society, and raises us from savagery. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], 2.147) | |
| A reaction: [compressed] Cicero would have been well aware of the doubts about rhetoric felt by Socrates (and possibly Plato). Cicero was probably the greatest Roman orator. |
| 11214 | We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt] |
| Full Idea: The standard view is that affirming not-A is more complex than affirming the atomic sentence A itself, with the latter determining its sense. But we could learn 'not' directly, by learning at once how to either affirm A or reject A. | |
| From: Ian Rumfitt ("Yes" and "No" [2000], IV) | |
| A reaction: [compressed] This seems fairly anti-Fregean in spirit, because it looks at the psychology of how we learn 'not' as a way of clarifying what we mean by it, rather than just looking at its logical behaviour (and thus giving it a secondary role). |
| 2640 | We have the death penalty, but still have thousands of robbers [Cicero] |
| Full Idea: We have robbers by the thousand, although they have the penalty of death before their eyes. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], I.86) |
| 2652 | Some regard nature simply as an irrational force that imparts movement [Cicero] |
| Full Idea: Some regard nature as an irrational force which merely imparts a mechanical motion to material bodies. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], II.81) |
| 2645 | Why shouldn't the gods fear their own destruction? [Cicero] |
| Full Idea: Why should the gods not be apprehensive of their own possible dissolution? | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], I.114) |
| 2627 | I wonder whether loss of reverence for the gods would mean the end of all virtue [Cicero] |
| Full Idea: I do not know whether, if our reverence for the gods were lost, we should not also see the end of good faith, of human brotherhood, and even of justice itself, which is the keystone of all the virtues. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], I.3) |
| 2651 | God doesn't obey the laws of nature; they are subject to the law of God [Cicero] |
| Full Idea: God is not subject to obey the laws of nature. It is nature that is subject to the laws of God. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], II.77) |
| 2634 | It seems clear to me that we have an innate idea of the divine [Cicero] |
| Full Idea: Let us take it as agreed that we have a preconception or "an innate idea" (as I have called it) or a prior knowledge of the divine. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], I.44) |
| 2636 | Many primitive people know nothing of the gods [Cicero] |
| Full Idea: There must be many wild and primitive peoples who have no idea of the gods at all. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], I.62) |
| 2647 | It is obvious from order that someone is in charge, as when we visit a gymnasium [Cicero] |
| Full Idea: If one comes into a gymnasium and sees everything properly arranged and carried on in order, one does not imagine these arrangements to be accidental, but infers that there is someone in command whose orders are obeyed. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], II.15) |
| 2650 | If a person cannot feel the power of God when looking at the stars, they are probably incapable of feeling [Cicero] |
| Full Idea: If any man cannot feel the power of God when he looks upon the stars, then I doubt whether he is capable of any feeling at all. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], II.55) |
| 2655 | If the barbarians of Britain saw a complex machine, they would be baffled, but would know it was designed [Cicero] |
| Full Idea: If someone were to take the celestial globe of Posidonius and show it to the people of Britain, would a single one of those barbarians fail to see that it was the product of a conscious intelligence? | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], II.88) |
| 2656 | Chance is no more likely to create the world than spilling lots of letters is likely to create a famous poem [Cicero] |
| Full Idea: If someone thinks chance made the world, he should also think that if an infinite number of the letters of the alphabet were shaken together and poured out on the ground it would be possible for them to spell out the whole 'Annals' of Ennius. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], II.93) |
| 2657 | If everything with regular movement and order is divine, then recurrent illnesses must be divine [Cicero] |
| Full Idea: Are we to find a divinity in every regular movement and in everything which happens in a constant order? If so, we shall have to say that tertian and quartan agues are divine because their course and recurrence is absolutely uniform. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], III.24) |
| 2638 | Either the gods are identical, or one is more beautiful than another [Cicero] |
| Full Idea: Are the gods all exactly the same? If not, then one must be more beautiful than another. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], I.80) |
| 2635 | The gods are happy, so virtuous, so rational, so must have human shape [Cicero] |
| Full Idea: We agree the gods are happy, and no happiness is possible without virtue: there is no virtue without reason: and reason is associated only with the human form: then it must follow that the gods themselves have human shape. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], I.48) |
| 2641 | Why believe in gods if you have never seen them? [Cicero] |
| Full Idea: Did you ever actually see a god? Then why do you believe that gods exist? | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], I.88) |
| 2659 | The lists of good men who have suffered and bad men who have prospered are endless [Cicero] |
| Full Idea: Time would fail me if I tried to list all the good men for whom things have turned out badly. So it would if I tried to mention all the wicked who have prospered. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], III.80) |
| 2658 | The gods blame men for having vices, but they could have given us enough reason to avoid them [Cicero] |
| Full Idea: You gods say that the fault lies in the vices of mankind. But you could have endowed men with reason in a form which would exclude all vice and crime. | |
| From: M. Tullius Cicero (On the Nature of the Gods ('De natura deorum') [c.44 BCE], III.76) |