51 ideas
20801 | A wise man's chief strength is not being tricked; nothing is worse than error, frivolity or rashness [Zeno of Citium, by Cicero] |
Full Idea: Zeno held that the wise man's chief strength is that he is careful not to be tricked, and sees to it that he is not deceived; for nothing is more alien to the conception that we have of the seriousness of the wise man than error, frivolity or rashness. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by M. Tullius Cicero - Academica II.66 | |
A reaction: I presume that this concerns being deceived by other people, and also being deceived by evidence. I suggest that the greatest ability of the wise person is the accurate assessment of evidence. |
1771 | When shown seven versions of the mowing argument, he paid twice the asking price for them [Zeno of Citium, by Diog. Laertius] |
Full Idea: When shown seven species of dialectic in the mowing argument, he asked the price, and when told 'a hundred drachmas', he gave two hundred, so devoted was he to learning. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.Ze.20 | |
A reaction: Wonderful. I have a watertight proof that pleasure is not the good, which I will auction on the internet. |
20770 | Philosophy has three parts, studying nature, character, and rational discourse [Zeno of Citium, by Diog. Laertius] |
Full Idea: They say that philosophical theory is tripartite. For one part of it concerns nature [i.e. physics], another concerns character [i.e. ethics], and another concerns rational discourse [i.e. logic] | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.39 | |
A reaction: Surely 'nature' included biology, and shouldn't be glossed as 'physics'? And I presume that 'rational discourse' is 'logos', rather than 'logic'. Interesting to see that ethics just is the study of character (and not of good and bad actions). |
6022 | Someone who says 'it is day' proposes it is day, and it is true if it is day [Zeno of Citium, by Diog. Laertius] |
Full Idea: Someone who says 'It is day' seems to propose that it is day; if, then, it is day, the proposition advanced comes out true, but if not, it comes out false. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.65 | |
A reaction: Those who find Tarski's theory annoyingly vacuous should note that the ancient Stoics thought the same point worth making. They seem to have clearly favoured some minimal account of truth, according to this. |
17765 | Propositional language can only relate statements as the same or as different [Walicki] |
Full Idea: Propositional language is very rudimentary and has limited powers of expression. The only relation between various statements it can handle is that of identity and difference. As are all the same, but Bs can be different from As. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 7 Intro) | |
A reaction: [second sentence a paraphrase] In predicate logic you could represent two statements as being the same except for one element (an object or predicate or relation or quantifier). |
17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
Full Idea: A proof of the consistency of propositional logic was given by Emil Post in 1921. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2.1) |
17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
Full Idea: Boolean connectives are interpreted as functions on the set {1,0}. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 5.1) | |
A reaction: 1 and 0 are normally taken to be true (T) and false (F). Thus the functions output various combinations of true and false, which are truth tables. |
17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
Full Idea: The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) |
17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
Full Idea: Without the assumption of the empty set, one would often have to take special precautions for the case where a set happened to contain no elements. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) | |
A reaction: Compare the introduction of the concept 'zero', where special precautions are therefore required. ...But other special precautions are needed without zero. Either he pays us, or we pay him, or ...er. Intersecting sets need the empty set. |
17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level. |
17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
Full Idea: In order to construct precise and valid patterns of arguments one has to determine their 'building blocks'. One has to identify the basic terms, their kinds and means of combination. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], History Intro) | |
A reaction: A deceptively simple and important idea. All explanation requires patterns and levels, and it is the idea of building blocks which makes such things possible. It is right at the centre of our grasp of everything. |
17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
Full Idea: A specification of a domain of objects, and of the rules for interpreting the symbols of a logical language in this domain such that all the theorems of the logical theory are true is said to be a 'model' of the theory. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.1.3) | |
A reaction: The basic ideas of this emerged 1915-30, but it needed Tarski's account of truth to really get it going. |
17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
Full Idea: The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2) | |
A reaction: Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh????? |
17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
Full Idea: Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
Full Idea: Axiomatic systems, their primitive terms and proofs, are purely syntactic, that is, do not presuppose any interpretation. ...[142] They never address the world directly, but address a possible semantic model which formally represents the world. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
Full Idea: Every member of an ordinal is itself an ordinal, and every ordinal is a transitive set (its members are also its subsets; a member of a member of an ordinal is also a member of the ordinal). | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
Full Idea: The collection of ordinals is defined inductively: Basis: the empty set is an ordinal; Ind: for an ordinal x, the union with its singleton is also an ordinal; and any arbitrary (possibly infinite) union of ordinals is an ordinal. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
A reaction: [symbolism translated into English] Walicki says they are called 'ordinal numbers', but are in fact a set. |
17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
Full Idea: We can form infinite ordinals by taking unions of ordinals. We can thus form 'limit ordinals', which have no immediate predecessor. ω is the first (the union of all finite ordinals), ω + ω = sω is second, 3ω the third.... | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
Full Idea: There may be several ordinals for the same cardinality. ...Two ordinals can represent different ways of well-ordering the same number (aleph-0) of elements. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
A reaction: This only applies to infinite ordinals and cardinals. For the finite, the two coincide. In infinite arithmetic the rules are different. |
17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
Full Idea: An ordinal can be defined as a transitive set of transitive sets, or else, as a transitive set totally ordered by set inclusion. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
7555 | Zeno achieved the statement of the problems of infinitesimals, infinity and continuity [Russell on Zeno of Citium] |
Full Idea: Zeno was concerned with three increasingly abstract problems of motion: the infinitesimal, the infinite, and continuity; to state the problems is perhaps the hardest part of the philosophical task, and this was done by Zeno. | |
From: comment on Zeno (Citium) (fragments/reports [c.294 BCE]) by Bertrand Russell - Mathematics and the Metaphysicians p.81 | |
A reaction: A very nice tribute, and a beautiful clarification of what Zeno was concerned with. |
17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
Full Idea: Since non-Euclidean geometry preserves all Euclid's postulates except the fifth one, all the theorems derived without the use of the fifth postulate remain valid. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
17754 | Inductive proof depends on the choice of the ordering [Walicki] |
Full Idea: Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1) | |
A reaction: There has to be an well-founded ordering for inductive proofs to be possible. |
20860 | Whatever participates in substance exists [Zeno of Citium, by Stobaeus] |
Full Idea: Zeno says that whatever participates in substance exists. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by John Stobaeus - Anthology 2.05a | |
A reaction: This seems Aristotelian, implying that only objects exist. Unformed stuff would not normally qualify as a 'substance'. So does mud exist? See the ideas of Henry Laycock. |
14217 | The 'standard' view of relations is that they hold of several objects in a given order [Fine,K] |
Full Idea: The 'standard' view of relations, held by philosophers and logicians alike, is that we may meaningfully talk of a relation holding of several objects in a given order (which works for examples like 'loves' and 'between'). | |
From: Kit Fine (Neutral Relations [2000], Intro) | |
A reaction: The point of Fine's paper is that there are many relations for which this model seems to fail. |
14216 | The 'positionalist' view of relations says the number of places is fixed, but not the order [Fine,K] |
Full Idea: The 'positionalist' view of relations is that each relation is taken to be endowed with a given number of argument places, or positions, in no specified order. [...The argument-places are specific entities, such as 'lover' and 'beloved'] | |
From: Kit Fine (Neutral Relations [2000], Intro) | |
A reaction: Fine offers this as an alternative to the 'standard' view of relations, in which the order of the objects matters. He then adds, and favours, the 'anti-positionalist' view, where there are not even a fixed number of places. |
14218 | A block on top of another contains one relation, not both 'on top of' and 'beneath' [Fine,K] |
Full Idea: If block a is on block b, it is hard to see how this state of affairs might consist of both 'on top of' and 'beneath'. Surely if the state is a genuine relational complex, there must be a single relation for these relata? | |
From: Kit Fine (Neutral Relations [2000], 1) | |
A reaction: He has already shown that if such relations imply their converses, then that gives you two separate relations. He goes on to observe that you cannot pick one of the two as correct, because of symmetry. He later offers the 'vertical placement' relation. |
14219 | Language imposes a direction on a road which is not really part of the road [Fine,K] |
Full Idea: Roads in the directional sense (A-to-B or B-to-A) are merely roads in the adirectional sense up which a direction has been imposed. | |
From: Kit Fine (Neutral Relations [2000], 1) | |
A reaction: This is Fine's linguistic objection to the standard view of relations. It is undeniable that language imposes an order where it may not exist ('Bob and Jane play tennis'), and this fact is very significant in discussing relations. |
14220 | Explain biased relations as orderings of the unbiased, or the unbiased as permutation classes of the biased? [Fine,K] |
Full Idea: A 'biased' relation can be taken to be the result of imposing ordering on the argument-places of an unbiased relation, ..or we can take an unbiased relation to be a 'permutation class' of biased relations. This is a familiar metaphysic predicament. | |
From: Kit Fine (Neutral Relations [2000], 3) | |
A reaction: 'Biased' relations such as 'on top of' have an ordering to their places, but 'unbiased' relations such as 'vertical placement' do not. This is a nice question in the metaphysics of grounding relations between key concepts. |
17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
Full Idea: The link between time and modality was severed by Duns Scotus, who proposed a notion of possibility based purely on the notion of semantic consistency. 'Possible' means for him logically possible, that is, not involving contradiction. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], History B.4) |
21397 | Perception an open hand, a fist is 'grasping', and holding that fist is knowledge [Zeno of Citium, by Long] |
Full Idea: Zeno said perceptions starts like an open hand; then the assent by our governing-principle is partly closing the hand; then full 'grasping' is like making a fist; and finally knowledge is grasping the fist with the other hand. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by A.A. Long - Hellenistic Philosophy 4.3.1 | |
A reaction: [In Cicero, Acad 2.145] It sounds as if full knowledge requires meta-cognition - knowing that you know. |
20799 | A grasp by the senses is true, because it leaves nothing out, and so nature endorses it [Zeno of Citium, by Cicero] |
Full Idea: He thought that a grasp made by the senses was true and reliable, …because it left out nothing about the object that could be grasped, and because nature had provided this grasp as a standard of knowledge, and a basis for understanding nature itself. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by M. Tullius Cicero - Academica I.42 | |
A reaction: Sounds like Williamson's 'knowledge first' claim - that the basic epistemic state is knowledge, which we have when everything is working normally. I like Zeno's idea that a 'grasp' leaves nothing out about the object. Compare nature with Descartes' God. |
20797 | If a grasped perception cannot be shaken by argument, it is 'knowledge' [Zeno of Citium, by Cicero] |
Full Idea: What had been grasped by sense-perception, he called this itself a 'sense-perception', and if it was grasped in such a way that it could not be shaken by argument he called it 'knowledge'. And between knowledge and ignorance he placed the 'grasp'. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by M. Tullius Cicero - Academica I.41 | |
A reaction: This seems to say that a grasped perception is knowledge if there is no defeater. |
21398 | A presentation is true if we judge that no false presentation could appear like it [Zeno of Citium, by Cicero] |
Full Idea: I possess a standard enabling me to judge presentations to be true when they have a character of a sort that false ones could not have. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by M. Tullius Cicero - Academica II.18.58 | |
A reaction: [This is a spokesman in Cicero for the early Stoic view] No sceptic will accept this, but it is pretty much how I operate. If you see something weird, like a leopard wandering wild in Hampshire, you believe it once you have eliminated possible deceptions. |
1770 | When a slave said 'It was fated that I should steal', Zeno replied 'Yes, and that you should be beaten' [Zeno of Citium, by Diog. Laertius] |
Full Idea: When a slave who was being beaten for theft said, 'It was fated that I should steal', Zeno replied, 'Yes, and that you should be beaten.' | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.Ze.19 |
3799 | A dog tied to a cart either chooses to follow and is pulled, or it is just pulled [Zeno of Citium, by Hippolytus] |
Full Idea: Zeno and Chrysippus say everything is fated with the following model: when a dog is tied to a cart, if it wants to follow it is pulled and follows, making its spontaneous act coincide with necessity, but if it does not want to follow it will be compelled. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Hippolytus - Refutation of All Heresies §1.21 | |
A reaction: A nice example, but it is important to keep the distinction clear between freedom and free will. The dog lacks freedom as it is dragged along, but it is still free to will that it is asleep in its kennel. |
21402 | Incorporeal substances can't do anything, and can't be acted upon either [Zeno of Citium, by Cicero] |
Full Idea: Zeno held that an incorporeal substance was incapable of any activity, whereas anything capable of acting, or being acted upon in any way, could not be incorporeal. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by M. Tullius Cicero - Academica I.11.39 | |
A reaction: This is substance dualism kicked into the long grass by Zeno, long before Descartes defended dualism, and was swiftly met with exactly the same response. The interaction problem. |
20816 | A body is required for anything to have causal relations [Zeno of Citium, by Cicero] |
Full Idea: Zeno held (contrary to Xenocrates and others) that it was impossible for anything to be effected that lacked a body, and indeed that whatever effected something or was affected by something must be body. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by M. Tullius Cicero - Academica I.39 | |
A reaction: This seems to make stoics thoroughgoing physicalists, although they consider the mind to be made of refined fire, rather than of flesh. |
1773 | A sentence always has signification, but a word by itself never does [Zeno of Citium, by Diog. Laertius] |
Full Idea: A sentence is always significative of something, but a word by itself has no signification. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.Ze.28 | |
A reaction: This is the Fregean dogma. Words obviously can signify, but that is said to be parasitic on their use in sentences. It feels like a false dichotomy to me. Much sentence meaning is compositional. |
1774 | Since we are essentially rational animals, living according to reason is living according to nature [Zeno of Citium, by Diog. Laertius] |
Full Idea: As reason is given to rational animals according to a more perfect principle, it follows that to live correctly according to reason, is properly predicated of those who live according to nature. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.Ze.52 | |
A reaction: This is the key idea for understanding what the stoics meant by 'live according to nature'. The modern idea of rationality doesn't extend to 'perfect principles', however. |
20841 | Zeno said live in agreement with nature, which accords with virtue [Zeno of Citium, by Diog. Laertius] |
Full Idea: Zeno first (in his book On Human Nature) said that the goal was to live in agreement with nature, which is to live according to virtue. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.87 | |
A reaction: The main idea seems to be Aristotelian - that the study of human nature reveals what our virtues are, and following them is what nature requires. Nature is taken to be profoundly rational. |
20863 | The goal is to 'live in agreement', according to one rational consistent principle [Zeno of Citium, by Stobaeus] |
Full Idea: Zeno says the goal of life is 'living in agreement', which means living according to a single and consonant rational principle, since those who live in conflict are unhappy. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by John Stobaeus - Anthology 2.06a | |
A reaction: If there is a 'single' principle, is it possible to state it? To live by consistent principles sets the bar incredibly high, as any professional philosopher can tell you. |
2662 | Zeno saw virtue as a splendid state, not just a source of splendid action [Zeno of Citium, by Cicero] |
Full Idea: Zeno held that not merely the exercise of virtue, as his predecessors held, but the mere state of virtue is in itself a splendid thing, although nobody possesses virtue without continuously exercising it. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by M. Tullius Cicero - Academica I.10.38 |
21395 | One of Zeno's books was 'That Which is Appropriate' [Zeno of Citium, by Long] |
Full Idea: Zeno of Citium wrote a (lost) book entitled 'That Which is Appropriate'. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by A.A. Long - Hellenistic Philosophy 4.1 | |
A reaction: I cite this because I take it to be about what in Aristotle called 'the mean' - to emphasise that the mean is not what is average, or midway between the extremes, but what is a balanced response to each situation |
5964 | Zeno says there are four main virtues, which are inseparable but distinct [Zeno of Citium, by Plutarch] |
Full Idea: Zeno (like Plato) admits a plurality of specifically different virtues, namely prudence, courage, sobriety, justice, which he takes to be inseparable but yet distinct and different from one another. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Plutarch - 70: Stoic Self-contradictions 1034c | |
A reaction: In fact, the virtues are 'supervenient' on one another, which is the doctrine of the unity of virtue. Zeno is not a pluralist in the way Aristotle is - who says there are other goods apart from the virtues. |
20822 | There is no void in the cosmos, but indefinite void outside it [Zeno of Citium, by Ps-Plutarch] |
Full Idea: Zeno and his followers say that there is no void within the cosmos but an indefinite void outside it. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Pseudo-Plutarch - On the Doctrine of the Philosophers 884a | |
A reaction: Only atomists (such as Epicureans) need void within the cosmos, as space within which atoms can move. What would they make of modern 'fields'? Posidonius later said there was sufficient, but not infinite, void. |
20811 | Since the cosmos produces what is alive and rational, it too must be alive and rational [Zeno of Citium] |
Full Idea: Nothing which lacks life and reason can produce from itself something which is alive and rational; but the cosmos can produce from itself things which are alive and rational; therefore the cosmos is alive and rational. | |
From: Zeno (Citium) (fragments/reports [c.294 BCE]), quoted by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') 2.22 | |
A reaction: Eggs and sperm don't seem to be rational, but I don't suppose they count. I note that this is presented as a formal proof, when actually it is just an evaluation of evidence. Logic as rhetoric, I would say. |
2648 | Things are more perfect if they have reason; nothing is more perfect than the universe, so it must have reason [Zeno of Citium] |
Full Idea: That which has reason is more perfect than that which has not. But there is nothing more perfect than the universe; therefore the universe is a rational being. | |
From: Zeno (Citium) (fragments/reports [c.294 BCE]), quoted by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') II.20 |
20810 | Rational is better than non-rational; the cosmos is supreme, so it is rational [Zeno of Citium] |
Full Idea: That which is rational is better than that which is not rational; but there is nothing better than the cosmos; therefore, the cosmos is rational. | |
From: Zeno (Citium) (fragments/reports [c.294 BCE]), quoted by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') 2.21 | |
A reaction: This looks awfully like Anselm's ontological argument to me. The cosmos was the greatest thing that Zeno could conceive. |
2649 | If tuneful flutes grew on olive trees, you would assume the olive had some knowledge of the flute [Zeno of Citium] |
Full Idea: If flutes playing tunes were to grow on olive trees, would you not infer that the olive must have some knowledge of the flute? | |
From: Zeno (Citium) (fragments/reports [c.294 BCE]), quoted by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') II.22 |
20807 | The cosmos and heavens are the substance of god [Zeno of Citium, by Diog. Laertius] |
Full Idea: Zeno says that the entire cosmos and the heaven are the substance of god. | |
From: report of Zeno (Citium) (fragments/reports [c.294 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.148 |