41 ideas
| 23853 | Truth is not a object we love - it is the radiant manifestation of reality [Weil] |
| Full Idea: Love of truth is not a correct form of expression. Truth is not an object of love. It is not an object at all. …Truth is the radiant manifestation of reality. | |
| From: Simone Weil (The Need for Roots [1943], III 'Growing') | |
| A reaction: Wow! Love that one! |
| 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) |
| 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). |
| 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) |
| 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) |
| 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) |
| 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) |
| 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. |
| 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. |
| 23855 | Creation produced a network or web of determinations [Weil] |
| Full Idea: What is sovereign in this world is determinateness, limit. Eternal Wisdom imprisons this universe in a network, a web of determinations. | |
| From: Simone Weil (The Need for Roots [1943], III 'Growth') | |
| A reaction: Love this, because I take 'determination' to be the defining relationship in ontology. It covers both physical causation and abstract necessities. |
| 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) |
| 23848 | The aesthete's treatment of beauty as amusement is sacreligious; beauty should nourish [Weil] |
| Full Idea: The aesthete's point of view is sacreligious, not only in matters of religion but even in those of art. It consists in amusing oneself with beauty by handling it and looking at it. Beauty is something to be eaten: it is a food. | |
| From: Simone Weil (The Need for Roots [1943], II 'Country') | |
| A reaction: She is endorsing the 'food' view against the 'handling' view. Beauty should nourish, she says. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 23854 | Beauty is the proof of what is good [Weil] |
| Full Idea: When the subject in question is the good, beauty is a rigorous and positive proof. | |
| From: Simone Weil (The Need for Roots [1943], III 'Growing') | |
| A reaction: Purest platonism! It is incomprehensible to say 'this thing is evil, but it is beautiful'. But there are plenty of things which strike me as beautiful, without connecting that in any way to moral goodness. |
| 23837 | Respect is our only obligation, which can only be expressed through deeds, not words [Weil] |
| Full Idea: Humans have only one obligation: respect. The obligation is only performed if the respect is effectively expressed in a real, not a fictitious, way; and this can only be done through the medium of Man's earthly needs. | |
| From: Simone Weil (The Need for Roots [1943], I 'Needs') | |
| A reaction: She says man's 'eternal destiny' imposes this obligation. I read this as saying that you should not imagine that you treat people respectfully if you are merely polite to them. Col. Pickering and Eliza Doolittle! Respect is the supreme virtue. |
| 23844 | The most important human need is to have multiple roots [Weil] |
| Full Idea: To be rooted is perhaps the most important and least recognised need of the human soul. …Every human being needs to have multiple roots. | |
| From: Simone Weil (The Need for Roots [1943], II 'Uprootedness') | |
| A reaction: Agree. I think we are just like trees, in that we need roots to grow well, and plenty of space to fully flourish. Identifying those roots is the main task of parents and teachers. |
| 23838 | The need for order stands above all others, and is understood via the other needs [Weil] |
| Full Idea: Order is the first need of all; it evens stands above all needs properly so-called. To be able to conceive it we must know what the other needs are. | |
| From: Simone Weil (The Need for Roots [1943], I 'Order') | |
| A reaction: This may be music to conservative ears, but you should examine Weil's other ideas to see what she has in mind. |
| 23836 | Obligations only bind individuals, not collectives [Weil] |
| Full Idea: Obligations are only binding on human beings. There are no obligations for collectivities, as such. | |
| From: Simone Weil (The Need for Roots [1943], I 'Needs') | |
| A reaction: I take it that 'as such' excludes the institutions created by collectivities, such as parliaments and courts. A nomadic tribe seems to have no duties, as a tribe, apart from mutual obligations among its members. Does this excuse crimes by the tribe? |
| 23843 | Even the poorest should feel collective ownership, and participation in grand display [Weil] |
| Full Idea: Participation in collective possessions is important. Where real civic life exists, each feels he has a personal ownership in the public monuments, gardens, ceremonial pomp and circumstances; sumptuousness is thus place within the reach of the poorest. | |
| From: Simone Weil (The Need for Roots [1943], I 'Collective') | |
| A reaction: OK with gardens. Dubious about fobbing the poor off with pomp. Monuments are a modern controversy, when they turn out to commemorate slavery and colonial conquest. I agree with her basic thought. |
| 23840 | A citizen should be able to understand the whole of society [Weil] |
| Full Idea: A man needs to be able to encompass in thought the entire range of activity of the social organism to which he belongs. | |
| From: Simone Weil (The Need for Roots [1943], I 'Responsibility') | |
| A reaction: She is urging the active involvement of citizens in decision making - for which they need appropriate knowledge. |
| 23846 | Culture is an instrument for creating an ongoing succession of teachers [Weil] |
| Full Idea: Culture - as we know it - is an instrument manipulated by teachers for manufacturing more teachers, who, in their turn, will manufacture still more teachers. | |
| From: Simone Weil (The Need for Roots [1943], II 'Towns') | |
| A reaction: Lot of truth in this. We tend to view our greatest successes in students who become academics and teachers. Culture is very much seen as something which must be 'transmitted' to each new generation. |
| 23839 | A lifelong head of society should only be a symbol, not a ruler [Weil] |
| Full Idea: Wherever a man is placed for life at the head of a social organism, he ought to be a symbol and not a ruler, as is the case with the King of England. | |
| From: Simone Weil (The Need for Roots [1943], I 'Obedience') | |
| A reaction: Nice to hear a radical French thinker endorsing an ancient British tradition! She may not be endorsing a lifelong head of state. Lifelong rulers are the main agents of totalitarianism. |
| 23842 | Party politics in a democracy can't avoid an anti-democratic party [Weil] |
| Full Idea: A democracy where public life is made up of strife between political parties is incapable of preventing the formation of a party whose avowed aim is the overthrow of that democracy. | |
| From: Simone Weil (The Need for Roots [1943], I 'Opinion') | |
| A reaction: We have seen this around 2020 in the USA and the UK. Freedom is compulsory? Weil hates political parties (as did Rousseau). |
| 23847 | Socialism tends to make a proletariat of the whole population [Weil] |
| Full Idea: What is called Socialism tends to force everybody without distinction into the proletarian condition. | |
| From: Simone Weil (The Need for Roots [1943], II 'Towns') | |
| A reaction: For example, Weil favours maximising private house ownership, rather than communally owned housing. She is describing wholesale nationalisation. I would incline towards nationalisation only of all basic central services. |
| 23845 | The capitalists neglect the people and the nation, and even their own interests [Weil] |
| Full Idea: The capitalists have betrayed their calling by criminally neglecting not only the interests of the people, not only those of the nation, but even their own. | |
| From: Simone Weil (The Need for Roots [1943], II 'Towns') | |
| A reaction: It is certainly true that the dedicated capitalist has little loyalty either to the people or to the nation. She doesn't spell out their failure of self-interest. I guess it produces a way of life they don't really want, deep down. |
| 23841 | By making money the sole human measure, inequality has become universal [Weil] |
| Full Idea: By making money the sole, or almost the sole, motive of all actions, the sole, or almost the sole, measure of all things, the poison of inequality has been introduced everywhere. | |
| From: Simone Weil (The Need for Roots [1943], I 'Equality') | |
| A reaction: Presumably this dates right back to the invention of money, and then increases with the endless rise of capitalism. |
| 23835 | People have duties, and only have rights because of the obligations of others to them [Weil] |
| Full Idea: A right is effectual only in relation to its corresponding obligation, springing not from the individual who possesses it, but from others who consider themselves under an obligation to him. In isolation a man only has duties, and only others have rights. | |
| From: Simone Weil (The Need for Roots [1943], I 'Needs') | |
| A reaction: This seems correct, and obviously refutes the idea that people have intrinsic natural rights. However, it may be our sense of what nature requires which gives rise to the obligations we feel towards others. |
| 23852 | To punish people we must ourselves be innocent - but that undermines the desire to punish [Weil] |
| Full Idea: In order to have the right to punish the guilty, we ought first of all to purify ourselves of their crimes. …But once this is accomplished we shall no longer feel the least desire to punish, or as little as possible and with extreme sorrow. | |
| From: Simone Weil (The Need for Roots [1943], III 'Growing') | |
| A reaction: Elsewhere she endorses punishment, as a social necessity, and a redemption for the wicked. This idea looks like a bit of a change of heart. She may be thinking of Jesus on the mote in someone's eye. |
| 23850 | The soldier-civilian distinction should be abolished; every citizen is committed to a war [Weil] |
| Full Idea: The distinction between soldiers and civilians, which the pressure of circumstances has already almost obliterated, should be entirely abolished. Every individual in the population owes his country the whole of his strength, resources and life itself. | |
| From: Simone Weil (The Need for Roots [1943], II 'Nation') | |
| A reaction: Written in London in 1943. The year carpet bombing seriously escalated. The facts of warfare can change the ethics. |
| 23851 | Education is essentially motivation [Weil] |
| Full Idea: Education - whether its object be children or adults, individuals or an entire people, or even oneself - consists in creating motives. | |
| From: Simone Weil (The Need for Roots [1943], III 'Growing') | |
| A reaction: I can't disagree. Intellectual motivation is simply what we find interesting, and there is no formula for that. A teacher can teach a good session, and only 5% of the pupils find it interesting. A bad session could be life-changing for one student. |
| 23849 | Religion should quietly suffuse all human life with its light [Weil] |
| Full Idea: The proper function of religion is to suffuse with its light all secular life, public or private, without in any way dominating it. | |
| From: Simone Weil (The Need for Roots [1943], II 'Nation') | |
| A reaction: Even for the non-religious there is something attractive about some view of the world which 'suffuses our lives with light'. It probably describes medieval Christendom, but that contained an awful lot of darkness. |