40 ideas
| 1606 | You have to be a Platonist to debate about reality, so every philosopher is a Platonist [Roochnik] |
| Full Idea: Everyone who enters into a debate about reality automatically becomes a Platonist. Since such debates are the essence of philosophy, every philosopher is a Platonist. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.199) | |
| A reaction: This is correct |
| 1595 | Philosophy aims to satisfy the chief human desire - the articulation of beauty itself [Roochnik] |
| Full Idea: Philosophy, the attempt to articulate the vision of beauty itself, is the attempt to satisfy the highest human desire. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.120) | |
| A reaction: A million miles away from modern philosophy, but still an ideal to be taken seriously. |
| 1571 | 'Logos' ranges from thought/reasoning, to words, to rational structures outside thought [Roochnik] |
| Full Idea: Logos can mean i) a thought or reasoning, ii) the word which expresses a thought, iii) a rational structure outside human thought. These meanings give 'logos' an extraordinary range. | |
| From: David Roochnik (The Tragedy of Reason [1990], Intro. 12) |
| 1572 | In the seventeenth century the only acceptable form of logos was technical knowledge [Roochnik] |
| Full Idea: In the seventeenth century only a certain type of logos was deemed legitimate, namely that identified with technical knowledge (or 'techné'). | |
| From: David Roochnik (The Tragedy of Reason [1990], Intro. 15) |
| 1573 | The hallmark of a person with logos is that they give reasons why one opinion is superior to another [Roochnik] |
| Full Idea: What is supposed to identify the person of logos from the one without is the commitment to giving reasons explaining why one opinion is superior to another. | |
| From: David Roochnik (The Tragedy of Reason [1990], Intro. 17) |
| 1592 | Logos cannot refute the relativist, and so must admit that it too is a matter of desire (for truth and agreement) [Roochnik] |
| Full Idea: Logos cannot refute the radical, consistent and self-conscious relativist. Therefore it must admit that, like the relativist, it itself is essentially a matter of desire. It wants to say what is right and wrong, true and false, and for others to agree. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.108) |
| 1593 | Human desire has an ordered structure, with logos at the pinnacle [Roochnik] |
| Full Idea: Human desire has an ordered structure, with logos at the pinnacle. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.109) |
| 1603 | Logos is not unconditionally good, but good if there is another person willing to engage with it [Roochnik] |
| Full Idea: Logos is not unconditionally good, but good contingent on there being some other person (out there) who is willing to talk with logos, to approach it even as an opponent. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.175) |
| 1598 | We prefer reason or poetry according to whether basics are intelligible or not [Roochnik] |
| Full Idea: Is the arché (basis) intelligible, or is it chaos? Upon this question hinges all, for answering it determines whether poetry or logos is the form of human speech that best does justice to the world. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.139) |
| 1584 | Modern science, by aiming for clarity about the external world, has abandoned rationality in the human world [Roochnik] |
| Full Idea: The modern scientific world view, with all its hope for clarity and precision, has a flipside, …which is its abandonment of rationality in the world of human significance. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.74) |
| 1591 | Unfortunately for reason, argument can't be used to establish the value of argument [Roochnik] |
| Full Idea: Unfortunately for the logos there is no argument that can, without begging the question, establish the goodness of argumentation. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.106) |
| 1599 | Attempts to suspend all presuppositions are hopeless, because a common ground must be agreed for the process [Roochnik] |
| Full Idea: To debate about suspending all our presuppositions requires a common ground which, upon being established, immediately renders the debate superfluous. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.144) |
| 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) |
| 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) |
| 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. |
| 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) |
| 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. |
| 1605 | Reality can be viewed neutrally, or as an object of desire [Roochnik] |
| Full Idea: There are two extremes: the Aristotelian views reality simply as reality, and the sophist or poet view reality only as an object of desire. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.199) |
| 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) |
| 1577 | Relativism is a disease which destroys the possibility of rational debate [Roochnik] |
| Full Idea: Relativism is disease, is pollution, for it negates the efficacy of logos. It destroys the possibility of a complete rational debate of fundamental questions. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.41) |
| 1457 | Morality requires a minimum commitment to the self [Rashdall] |
| Full Idea: A bare minimum of metaphysical belief about the self is found to be absolutely presupposed in the very idea of morality. | |
| From: Hastings Rashdall (Theory of Good and Evil [1907], II.III.I.4) | |
| A reaction: This may not be true of virtue theory, where we could have a whole creature which lacked any sense of personhood, but yet had clear virtues and vices in its social functioning. Even if choices are central to morality, that might not need a self. |
| 1578 | If relativism is the correct account of human values, then rhetoric is more important than reasoning [Roochnik] |
| Full Idea: If relativism offers an accurate description of human values, then rhetoric replaces logos as the most fundamental human activity. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.47) | |
| A reaction: Or putting it another way, logos (reason) becomes meaningless. I suppose, though, that a relativist can conduct conditional reasoning (but must belief in some rules of reason). |
| 1596 | Reasoning aims not at the understanding of objects, but at the desire to give beautiful speeches [Roochnik] |
| Full Idea: Logos originates not in a cognitive capacity for the apprehension of objects, but in the desire to give birth to beautiful speeches. | |
| From: David Roochnik (The Tragedy of Reason [1990], p.124) | |
| A reaction: It is hard for us to grasp this, but it might be quite life-enhancing if we could return to that old way of thought. |
| 6674 | All moral judgements ultimately concern the value of ends [Rashdall] |
| Full Idea: All moral judgements are ultimately judgements as to the value of ends. | |
| From: Hastings Rashdall (Theory of Good and Evil [1907], VII.I) | |
| A reaction: I am increasingly struck by this, especially when observing that it is the great gap in Kant's theory. For some odd reason, he gives being rational the highest possible value. Why? Nietzsche is good on this. 'Eudaimonia' seems a good start, to me. |
| 6673 | Ideal Utilitarianism is teleological but non-hedonistic; the aim is an ideal end, which includes pleasure [Rashdall] |
| Full Idea: My view, called Ideal Utilitarianism, combines the utilitarian principle that Ethics must be teleological with a non-hedonistic view of ethical ends; actions are right or wrong as they produce an ideal end, which includes, but is not limited to, pleasure. | |
| From: Hastings Rashdall (Theory of Good and Evil [1907], VII.I) | |
| A reaction: I certainly think that if you are going to be a consequentialist, then it is ridiculous to limit the end to pleasure, as it is an 'open question' as to whether we judge pleasures or pains to be good or bad. I am fond of beauty, goodness and truth, myself. |
| 1458 | Conduct is only reasonable or unreasonable if the world is governed by reason [Rashdall] |
| Full Idea: Absolutely reasonable or unreasonable conduct could not exist in a world which was not itself the product of reason or governed by its dictates. | |
| From: Hastings Rashdall (Theory of Good and Evil [1907], II.III.I.4) |
| 1459 | Absolute moral ideals can't exist in human minds or material things, so their acceptance implies a greater Mind [Rashdall, by PG] |
| Full Idea: An absolute moral ideal cannot exist in material things, or in the minds of individual people, so belief in it requires belief in a Mind which contains the ideal and is its source. | |
| From: report of Hastings Rashdall (Theory of Good and Evil [1907], II.III.I.4) by PG - Db (ideas) |