38 ideas
| 2098 | The principle of sufficient reason is needed if we are to proceed from maths to physics [Leibniz] |
| Full Idea: In order to proceed from mathematics to physics the principle of sufficient reason is necessary, that nothing happens without there being a reason why it should be thus rather than otherwise. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], §2) |
| 3646 | There is always a reason why things are thus rather than otherwise [Leibniz] |
| Full Idea: Nothing happens without a sufficient reason why it should be thus rather than otherwise. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 3.2) |
| 2104 | No reason could limit the quantity of matter, so there is no limit [Leibniz] |
| Full Idea: There is no possible reason which could limit the quantity of matter; therefore there cannot in fact be any such limitation. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 4.21) |
| 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) |
| 8717 | Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend] |
| Full Idea: Hilbert wanted to derive ideal mathematics from the secure, paradox-free, finite mathematics (known as 'Hilbert's Programme'). ...Note that for the realist consistency is not something we need to prove; it is a precondition of thought. | |
| From: report of David Hilbert (works [1900], 6.7) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: I am an intuitive realist, though I am not so sure about that on cautious reflection. Compare the claims that there are reasons or causes for everything. Reality cannot contain contradicitions (can it?). Contradictions would be our fault. |
| 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. |
| 10113 | The grounding of mathematics is 'in the beginning was the sign' [Hilbert] |
| Full Idea: The solid philosophical attitude that I think is required for the grounding of pure mathematics is this: In the beginning was the sign. | |
| From: David Hilbert (works [1900]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: Why did people invent those particular signs? Presumably they were meant to designate something, in the world or in our experience. |
| 10115 | Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman] |
| Full Idea: Hilbert replaced a semantic construal of inconsistency (that the theory entails a statement that is necessarily false) by a syntactic one (that the theory formally derives the statement (0 =1 ∧ 0 not-= 1). | |
| From: report of David Hilbert (works [1900]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: Finding one particular clash will pinpoint the notion of inconsistency, but it doesn't seem to define what it means, since the concept has very wide application. |
| 10116 | Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [Hilbert, by George/Velleman] |
| Full Idea: Hilbert's project was to establish the consistency of classical mathematics using just finitary means, to convince all parties that no contradictions will follow from employing the infinitary notions and reasoning. | |
| From: report of David Hilbert (works [1900]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: This is the project which was badly torpedoed by Gödel's Second Incompleteness Theorem. |
| 19385 | All simply substances are in harmony, because they all represent the one universe [Leibniz] |
| Full Idea: All simple substances will always have a harmony among themselves, because they always represent the same universe. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], V §91), quoted by Richard T.W. Arthur - Leibniz | |
| A reaction: We can accept that the universe itself does not contain contradictions (how could it), but it is a leap of faith to say that all monads represent the universe well enough to avoid contradictions. Maps can contradict one another. |
| 21346 | The ratio between two lines can't be a feature of one, and cannot be in both [Leibniz] |
| Full Idea: If the ratio of two lines L and M is conceived as abstracted from them both, without considering which is the subject and which the object, which will then be the subject? We cannot say both, for then we should have an accident in two subjects. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 5th Paper, §47), quoted by John Heil - Relations 'External' | |
| A reaction: [compressed] Leibniz is rejecting external relations as having any status in ontology. It looks like a mistake (originating in Aristotle) to try to shoehorn the ontology of relations into the substance-properties framework. |
| 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) |
| 2105 | Things are infinitely subdivisible and contain new worlds, which atoms would make impossible [Leibniz] |
| Full Idea: The least corpuscle is actually subdivided ad infinitum and contains a world of new created things, which this universe would lack if this corpuscle were an atom. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 4.PS) |
| 2106 | The only simple things are monads, with no parts or extension [Leibniz] |
| Full Idea: According to me there is nothing simple except true monads, which have no parts and no extensions. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 5.24) |
| 2102 | Atomism is irrational because it suggests that two atoms can be indistinguishable [Leibniz] |
| Full Idea: There are no two individuals indiscernible from one another - leaves, or drops of water, for example. This is an argument against atoms, which, like the void, are opposed to the principles of a true metaphysic. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 4.04) |
| 20965 | Leibniz upheld conservations of momentum and energy [Leibniz, by Papineau] |
| Full Idea: In place of Descartes's conservation of 'quantity of motion', Leibniz upheld both the conservation of linear momentum and the conservation of kinetic energy. | |
| From: report of Gottfried Leibniz (Letters to Samuel Clarke [1716], 5th paper) by David Papineau - Thinking about Consciousness App 2 | |
| A reaction: The point is that momentum involves velocity (which includes direction) rather than speed. Leibniz more or less invented the concept of 'energy' ('vis viva'). Papineau says these two leave no room for causation by mental substance. |
| 2103 | The idea that the universe could be moved forward with no other change is just a fantasy [Leibniz] |
| Full Idea: To say that God could cause the universe to move forward in a straight line or otherwise without changing it in any other way is another fanciful supposition. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 4.14) |
| 2100 | Space and time are purely relative [Leibniz] |
| Full Idea: I have more than once stated that I held space to be something purely relative, like time. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 3.4) |
| 2107 | No time exists except instants, and instants are not even a part of time, so time does not exist [Leibniz] |
| Full Idea: How could a thing exist, no part of which ever exists? In the case of time, nothing exists but instants, and an instant is not even a part of time. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 5.49) |
| 2101 | If everything in the universe happened a year earlier, there would be no discernible difference [Leibniz] |
| Full Idea: To ask why God did not make everything a year sooner would be reasonable if time were something apart from temporal things, but time is just the succession of things, which remains the same if the universe is created a year sooner. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 3.6) |
| 22894 | If time were absolute that would make God's existence dependent on it [Leibniz, by Bardon] |
| Full Idea: Leibniz argues that if time is a thing in itself, and God is 'in' time, then God would be dependent for His existence on the existence of time. | |
| From: report of Gottfried Leibniz (Letters to Samuel Clarke [1716]) by Adrian Bardon - Brief History of the Philosophy of Time 3 'Newton' | |
| A reaction: Hence Leibniz says time is merely relations between events. Not sure what he thinks an event is. What is God made of? Is there some divine matter upon which God's existence must depend? |
| 2099 | The existence of God, and all metaphysics, follows from the Principle of Sufficient Reason [Leibniz] |
| Full Idea: By this principle alone, that there must be a sufficient reason why things are thus rather than otherwise, I prove the existence of the Divinity, and all the rest of metaphysics or natural theology. | |
| From: Gottfried Leibniz (Letters to Samuel Clarke [1716], §2) |