33 ideas
| 14767 | The demonstrations of the metaphysicians are all moonshine [Peirce] |
| Full Idea: The demonstrations of the metaphysicians are all moonshine. | |
| From: Charles Sanders Peirce (Concerning the Author [1897], p.2) |
| 14764 | I am saturated with the spirit of physical science [Peirce] |
| Full Idea: I am saturated, through and through, with the spirit of the physical sciences. | |
| From: Charles Sanders Peirce (Concerning the Author [1897], p.1) |
| 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) |
| 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) |
| 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. |
| 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) |
| 14768 | Infallibility in science is just a joke [Peirce] |
| Full Idea: Infallibility in scientific matters seems to me irresistibly comical. | |
| From: Charles Sanders Peirce (Concerning the Author [1897], p.3) |
| 14765 | Association of ideas is the best philosophical idea of the prescientific age [Peirce] |
| Full Idea: The doctrine of the association of ideas is, to my thinking, the finest piece of philosophical work of the prescientific ages. | |
| From: Charles Sanders Peirce (Concerning the Author [1897], p.2) |
| 14766 | Duns Scotus offers perhaps the best logic and metaphysics for modern physical science [Peirce] |
| Full Idea: The works of Duns Scotus have strongly influenced me. …His logic and metaphysics, torn away from its medievalism, …will go far toward supplying the philosophy which is best to harmonize with physical science. | |
| From: Charles Sanders Peirce (Concerning the Author [1897], p.2) |
| 21243 | An existing thing is even greater if its non-existence is inconceivable [Anselm] |
| Full Idea: Something can be thought of as existing, which cannot be thought of as not existing, and this is greater than that which cannot be thought of as not existing. | |
| From: Anselm (Proslogion [1090], Ch 3) | |
| A reaction: This is a necessary addition, to single out the concept of God as special. But you really must give reasons for saying God's non-existence is inconceivable. Atheists seem to manage. |
| 21241 | Even the fool can hold 'a being than which none greater exists' in his understanding [Anselm] |
| Full Idea: Even the fool must be convinced that a being than which none greater can be thought exists at least in his understanding, since when he hears this he understands it, and whatever is understood is in the understanding. | |
| From: Anselm (Proslogion [1090], Ch 2) | |
| A reaction: Psalm 14.1: 'The fool hath said in his heart, there is no God'. But how does the fool interpret the words, if he has limited imagination? He might get no further than an attractive film star. He would need prompting to think of a spiritual being. |
| 21242 | If that than which a greater cannot be thought actually exists, that is greater than the mere idea [Anselm] |
| Full Idea: Clearly that than which a greater cannot be thought cannot exist in the understanding alone. For it it is actually in the understanding alone, it can be thought of as existing also in reality, and this is greater. | |
| From: Anselm (Proslogion [1090], Ch 2) | |
| A reaction: The suppressed premise is 'something actually existing is greater than the mere conception of it'. As it stands this is wrong. I can imagine a supreme evil. But see Idea 21243. |
| 21244 | Conceiving a greater being than God leads to absurdity [Anselm] |
| Full Idea: If some mind could think of something better than thou, the creature would rise above the Creator and judge its Creator; but this is altogether absurd. | |
| From: Anselm (Proslogion [1090], Ch 3) | |
| A reaction: An error, revealing a certain desperation. If a greafer being could be conceived than the being so far imagined as God (a necessarily existing being), that being would BE God, by his own argument (and not some arrogant 'creature'). |
| 1421 | A perfection must be independent and unlimited, and the necessary existence of Anselm's second proof gives this [Malcolm on Anselm] |
| Full Idea: Anselm's second proof works, because he sees that necessary existence (or the impossibility of non-existence) really is a perfection. This is because a perfection requires no dependence or limit or impediment. | |
| From: comment on Anselm (Proslogion [1090], Ch 3) by Norman Malcolm - Anselm's Argument Sect II | |
| A reaction: I have the usual problem, that it doesn't seem to follow that the perfect existence of something bestows a perfection. It may be necessary that 'for every large animal there exists a disease'. Satan may exist necessarily. |
| 21245 | The word 'God' can be denied, but understanding shows God must exist [Anselm] |
| Full Idea: We think of a thing when we say the world, and in another way when we think of the very thing itself. In the second sense God cannot be thought of as nonexistent. No one who understands can think God does not exist. | |
| From: Anselm (Proslogion [1090], Ch 4) | |
| A reaction: It seems open to the atheist to claim the exact opposite - that you can commit to God's existence if it is just a word, but understanding shows that God is impossible (perhaps because of contradictions). How to arbitrate? |
| 21246 | Guanilo says a supremely fertile island must exist, just because we can conceive it [Anselm] |
| Full Idea: Guanilo supposes that we imagine an island surpassing all lands in its fertility. We might then say that we cannot doubt that it truly exists is reality, because anyone can conceive it from a verbal description. | |
| From: Anselm (Proslogion [1090], Reply 3) | |
| A reaction: Guanilo was a very naughty monk, who must have had sleepless nights over this. One could further ask whether an island might have necessary existence. Anselm needs 'a being' to be a special category of thing. |
| 21247 | Nonexistence is impossible for the greatest thinkable thing, which has no beginning or end [Anselm] |
| Full Idea: If anyone does think of something a greater than which cannot be thought, then he thinks of something which cannot be thought of as nonexistent, ...for then it could be thought of as having a beginning and an end. And this is impossible. | |
| From: Anselm (Proslogion [1090], Reply 3) | |
| A reaction: A nice idea, but it has a flip side. If the atheist denies God's existence, then it follows that (because no beginning is possible for such a being) the existence of God is impossible. Anselm adds that contingent existents have parts (unlike God). |
| 1420 | Anselm's first proof fails because existence isn't a real predicate, so it can't be a perfection [Malcolm on Anselm] |
| Full Idea: Anselm's first proof fails, because he treats existence as being a perfection, which it isn't, because that would make it a real predicate. | |
| From: comment on Anselm (Proslogion [1090], Ch 2) by Norman Malcolm - Anselm's Argument Sect I | |
| A reaction: Not everyone accepts Kant's claim that existence cannot be a predicate. They all seem to know what a perfection is. Can the Mona Lisa (an object) not be a perfection? Must it be broken down into perfect predicates? |