37 ideas
| 6161 | Structuralism is neo-Kantian idealism, with language playing the role of categories of understanding [Rowlands] |
| Full Idea: Structuralism is a form of neo-Kantian idealism, in which the job of creating Kant's phenomenal world has been taken over by language instead of forms of sensibility and categories of the understanding. | |
| From: Mark Rowlands (Externalism [2003], Ch.3) | |
| A reaction: A helpful connection, which explains my aversion to any attempt at understanding the world simply by analysing language, either in its ordinary usage, or in its underlying logical form. |
| 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. |
| 6163 | If bivalence is rejected, then excluded middle must also be rejected [Rowlands] |
| Full Idea: If you reject the principle of bivalence (that a proposition is either determinately true or false), then statements are also not subject to the Law of Excluded Middle (P or not-P). | |
| From: Mark Rowlands (Externalism [2003], Ch.3) | |
| A reaction: I think Rowlands is wrong about this. Excluded Middle could be purely syntacti, or its semantics could be 'True or Not-True'. Only bivalent excluded middle introduces 'True or False'. Compare Idea 4752. |
| 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. |
| 6155 | Supervenience is a one-way relation of dependence or determination between properties [Rowlands] |
| Full Idea: Supervenience is essentially a one-way relation of dependence or determination, …which holds, in the first instance, between properties. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: This definition immediately shows why supervenient properties are in danger of being epiphenomenal (i.e. causally irrelevant). Carefully thought about the notion of a 'one-way' relation will, I think, make it more obscure rather than clearer. |
| 6154 | It is argued that wholes possess modal and counterfactual properties that parts lack [Rowlands] |
| Full Idea: Some have argued that a mereological whole should not be identified with the sum of its parts on the grounds that the former possess certain properties - specifically modal and (perhaps) counterfactual properties - that the latter lacks. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: I am not convinced that modal and counterfactual claims should count as properties. If my pen is heated it melts (a property), but if my pen were intelligent it could do philosophy. Intelligence is a property, but the situation isn't. |
| 6157 | Tokens are dated, concrete particulars; types are their general properties or kinds [Rowlands] |
| Full Idea: Tokens are dated, concrete, particular occurrences or instances; types are the general properties that these occurrences exemplify or the kinds to which they belong. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: It might be said that types are sets, of which tokens are the members. The question of 'general properties' raises the question of whether universals must exist to make kinds possible. |
| 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) |
| 6159 | Strong idealism is the sort of mess produced by a Cartesian separation of mind and world [Rowlands] |
| Full Idea: Neo-Kantian idealism, and the excesses of recent versions of it, are precisely the sort of mess one can get oneself into through an uncritical acceptance of the dichotomizing of mind and world along Cartesian internalist lines. | |
| From: Mark Rowlands (Externalism [2003], Ch.3) | |
| A reaction: I am unconvinced that internalism about the mind (that its contents can be defined without reference to anything external) leads to this disastrous split. We don't have to abandon the links between an internal mind and the world. |
| 6152 | Minds are rational, conscious, subjective, self-knowing, free, meaningful and self-aware [Rowlands] |
| Full Idea: The apparent features of mind which are not obviously physical include: rationality, thought, consciousness, subjectivity, infallible first-person knowledge, freedom, meaning and self-awareness. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: A helpful list, some of which can be challenged. Ryle challenges first-person infallibility. Hume challenges self-awareness. Quine challenges meaning. Lots of people (e.g. Spinoza) challenge freedom. The Churchlands seem to challenge consciousness. |
| 6173 | Content externalism implies that we do not have privileged access to our own minds [Rowlands] |
| Full Idea: Content externalism threatens the idea of first-person authority in all its forms, and does so because it calls into question the idea that the access we have to our own mental states is privileged in the way required for such authority. | |
| From: Mark Rowlands (Externalism [2003], Ch.7) | |
| A reaction: I am inclined to respond by saying that since we clearly have privileged access to our own minds, that means there must be something wrong with content externalism. |
| 6174 | If someone is secretly transported to Twin Earth, others know their thoughts better than they do [Rowlands] |
| Full Idea: If someone knew that a thinker had, without realising it, been transported to Twin Earth, they would almost certainly be a higher authority on the content of the thinker's thoughts than would the thinker. | |
| From: Mark Rowlands (Externalism [2003], Ch.8) | |
| A reaction: They would certainly be a higher authority on the truth of the thinker's thoughts, but only in the way that you might think I hold a diamond when I know it is a club. If the thinker believes it is H2O, the fact that it isn't is irrelevant to content. |
| 6158 | Supervenience of mental and physical properties often comes with token-identity of mental and physical particulars [Rowlands] |
| Full Idea: One often finds a supervenience thesis concerning the relation between mental and physical properties combined with a token identity theory concerning the relation between mental and physical particulars. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: This brings out the important clarifying point that supervenience is said to be between properties, not substances. The point is that supervenience will always cry out for an explanation, preferably a sensible one. |
| 6168 | The content of a thought is just the meaning of a sentence [Rowlands] |
| Full Idea: The content of the thought that the sky is blue is simply the meaning of the sentence "The sky is blue". | |
| From: Mark Rowlands (Externalism [2003], Ch.5) | |
| A reaction: This seems to imply that it is logically impossible for a non-language-speaker, such as a chimpanzee, to think that the sky is the same colour as the water. If we allow propositions, we might be able to keep meanings without the sentences. |
| 6167 | Action is bodily movement caused by intentional states [Rowlands] |
| Full Idea: An action is a bodily movement that is caused by intentional states such as beliefs, desires and so on. | |
| From: Mark Rowlands (Externalism [2003], Ch.5) | |
| A reaction: A useful definition, and clearly one that has no truck with attempts at giving behaviourist definitions of action. The definition of a 'moral action' needs to be built on this one. Particular types of belief and desire, presumably. |
| 6177 | Moral intuition seems unevenly distributed between people [Rowlands] |
| Full Idea: The faculty of moral intuition seems to be unevenly distributed between people. | |
| From: Mark Rowlands (Externalism [2003], Ch.11) | |
| A reaction: This would be a good argument if it was thought that the source of moral intuitions was divine, but people vary enormously in their intuitions about maths, about character, about danger. If you believe in any intuition at all, you must accept its variety. |
| 6156 | The 17th century reintroduced atoms as mathematical modes of Euclidean space [Rowlands] |
| Full Idea: The seventeenth century revolution reintroduced the classical concept of the atom in somewhat new attire as an essentially mathematical entity whose primary qualities could be precisely quantified as modes or aspects of Euclidean space. | |
| From: Mark Rowlands (Externalism [2003], Ch.2) | |
| A reaction: Obviously this very abstract view of atoms didn't last, once they began to identify specific physical atoms, such as oxygen. This view fits in with Newton's use of pure (abstract) points such as the 'centre of gravity'. |
| 6170 | Natural kinds are defined by their real essence, as in gold having atomic number 79 [Rowlands] |
| Full Idea: Part of what it means to be a natural kind is that they are defined by a real essence, a constitution that marks them out as the substance they are (as water is essentially H2O, and gold essentially has atomic number 79). | |
| From: Mark Rowlands (Externalism [2003], Ch.6) | |
| A reaction: A 'real essence' would be the opposite of a 'conventional essence', which is just a human way of seeing things. |
| 11945 | In addition to laws, God must also create appropriate natures for things [Leibniz] |
| Full Idea: It isn't sufficient to say that God has made a general law, for in addition to the decree there has also to be a natural way of carrying it out. It is necessary, that is, that what happens should be explicable in terms of the God-given nature of things. | |
| From: Gottfried Leibniz (Letter to the Editor about Bayle [1698], p.205) | |
| A reaction: Thus Leibniz is an ancestor of scientific essentialism, but he was too frightened to take the next step, which is to see that once God has endowed the natures, he doesn't need to wield his laws as well. The natures will do the whole job. |
| 19348 | All that is real in motion is the force or power which produces change [Leibniz] |
| Full Idea: As for motion, what is real in it is force or power; that is to say, what there is in the present state which carries with it a change in the future. The rest is only phenomena and relations. | |
| From: Gottfried Leibniz (Letter to the Editor about Bayle [1698], §13) | |
| A reaction: This presumably contradicts Newton's concept of inertia, which allows constant motion without force. I always like a reference to powers. What is 'kinetic energy' in this context? |
| 6178 | It is common to see the value of nature in one feature, such as life, diversity, or integrity [Rowlands] |
| Full Idea: In recent environmental philosophy it is common to see the value of nature identified with one or another natural feature of the environment: life, diversity, ecosystemic integrity and so on. | |
| From: Mark Rowlands (Externalism [2003], Ch.11) | |
| A reaction: This thought seems to be asking for the Open Question argument. What is so good about life, or diversity? Our strongest intuition must be that the survival of the ecosystem, and whatever makes that possible, is the highest value. |