48 ideas
| 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. |
| 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) |
| 1556 | By nature people are close to one another, but culture drives them apart [Hippias] |
| Full Idea: I regard you all as relatives - by nature, not by convention. By nature like is akin to like, but convention is a tyrant over humankind and often constrains people to act contrary to nature. | |
| From: Hippias (fragments/reports [c.430 BCE]), quoted by Plato - Protagoras 337c8 |
| 17405 | If a theory can be fudged, so can observations [Scerri] |
| Full Idea: A theorist may have designed his theory to fit the facts, but is it not equally possible for observers to be influenced by a theory in their report of experimental facts? | |
| From: Eric R. Scerri (The Periodic Table [2007], 05 'Power') | |
| A reaction: This is in reply to Lipton's claim that prediction is better than accommodation because of the 'fudging' problem. The reply is that you might fudge to achieve a prediction. If it was correct, that wouldn't avoid the charge of fudging. |
| 17397 | The periodic system is the big counterexample to Kuhn's theory of revolutionary science [Scerri] |
| Full Idea: The history of the periodic system appears to be the supreme counterexample to Kuhn's thesis, whereby scientific developments proceed in a sudden, revolutionary fashion. | |
| From: Eric R. Scerri (The Periodic Table [2007], 03 'Rapid') | |
| A reaction: What is lovely about the periodic table is that it seems so wonderfully right, and hence no revolution has ever been needed. The big theories of physics and cosmology are much more precarious. |
| 17393 | Scientists eventually seek underlying explanations for every pattern [Scerri] |
| Full Idea: Whenever scientists are presented with a useful pattern or system of classification, it is only a matter of time before the begin to ask whether there may be some underlying explanation for the pattern. | |
| From: Eric R. Scerri (The Periodic Table [2007], Intro 'Evol') | |
| A reaction: Music to my ears, against the idea that the sole aim of science is accurately describe the patterns. |
| 17403 | The periodic table suggests accommodation to facts rates above prediction [Scerri] |
| Full Idea: Rather than proving the value of prediction, the development and acceptance of the periodic table may give us a powerful illustration of the importance of accommodation, that is, the ability of a new scientific theory to explain already known facts. | |
| From: Eric R. Scerri (The Periodic Table [2007], 05 'Intro') | |
| A reaction: The original table made famous predictions, but also just as many wrong ones (Scerri:143), and Scerri thinks this aspect has been overrated. |
| 17394 | Natural kinds are what are differentiated by nature, and not just by us [Scerri] |
| Full Idea: Natural kinds are realistic scientific entities that are differentiated by nature itself rather than by our human attempts at classification. | |
| From: Eric R. Scerri (The Periodic Table [2007], Intro 'Evol') |
| 17421 | If elements are natural kinds, might the groups of the periodic table also be natural kinds? [Scerri] |
| Full Idea: Elements defined by their atomic numbers are frequently assumed to represent 'natural kinds' in chemistry. ...The question arises as to whether groups of elements appearing in the periodic table might also represent natural kinds. | |
| From: Eric R. Scerri (The Periodic Table [2007], 10 'Elements') | |
| A reaction: Scerri says the distinction is not as sharp as that between the elements. As a realist, he believes there is 'one ideal periodic classification', which would then make the periods into kinds. |
| 17396 | The colour of gold is best explained by relativistic effects due to fast-moving inner-shell electrons [Scerri] |
| Full Idea: Many seemingly mundane properties of elements such as the characteristic color of gold ....can best be explained by relativistic effects due to fast-moving inner-shell electrons. | |
| From: Eric R. Scerri (The Periodic Table [2007], 01 'Under') | |
| A reaction: John Locke - I wish you were reading this! That we could work out the hidden facts of gold, and thereby explain and predict the surface properties we experience, is exactly what Locke thought to be forever impossible. |
| 17420 | The stability of nuclei can be estimated through their binding energy [Scerri] |
| Full Idea: The stability of nuclei can be estimated through their binding energy, a quantity given by the difference between their masses and the masses of their constituent particles. | |
| From: Eric R. Scerri (The Periodic Table [2007], 10 'Stabil') |
| 17411 | If all elements are multiples of one (of hydrogen), that suggests once again that matter is unified [Scerri] |
| Full Idea: The work of Moseley and others rehabilitated Prout's hypothesis that all elements were composites of hydrogen, being exact multiples of 1. ..This revitalized some philososophical notions of the unity of all matter, criticised by Mendeleev and others. | |
| From: Eric R. Scerri (The Periodic Table [2007], 06 'Philos') |
| 17407 | The electron is the main source of chemical properties [Scerri] |
| Full Idea: It is the electron that is mainly responsible for the chemical properties of the elements. | |
| From: Eric R. Scerri (The Periodic Table [2007], 06 'Intro') |
| 17415 | A big chemistry idea is that covalent bonds are shared electrons, not transfer of electrons [Scerri] |
| Full Idea: One of the most influential ideas in modern chemistry is of a covalent bond as a shared pair of electrons (not as transfer of electrons and the formation of ionic bonds). | |
| From: Eric R. Scerri (The Periodic Table [2007], 08 'Intro') | |
| A reaction: Gilbert Newton Lewis was responsible for this. |
| 17392 | How can poisonous elements survive in the nutritious compound they compose? [Scerri] |
| Full Idea: A central mystery of chemistry is how the elements survive in the compounds they form. For example, how can poisonous grey metal sodium combine with green poisonous gas chlorine, to make salt, which is non-poisonous and essential for life? | |
| From: Eric R. Scerri (The Periodic Table [2007], Intro 'Elem') | |
| A reaction: A very nice question which had never occurred to me. If our digestive system pulled the sodium apart from the chlorine, we would die. |
| 17391 | Periodicity and bonding are the two big ideas in chemistry [Scerri] |
| Full Idea: The two big ideas in chemistry are chemical periodicity and chemical bonding, and they are deeply interconnected. | |
| From: Eric R. Scerri (The Periodic Table [2007], Intro 'Per') |
| 17404 | Chemistry does not work from general principles, but by careful induction from large amounts of data [Scerri] |
| Full Idea: Unlike in physics, chemical reasoning does not generally proceed unambiguously from general principles. It is a more inductive science in which large amounts of observational data must be carefully weighed. | |
| From: Eric R. Scerri (The Periodic Table [2007], 05 'Mendel') | |
| A reaction: This is why essentialist thinking was important for Mendeleev, because it kept his focus on the core facts beneath the messy and incomplete data. |
| 17409 | Does radioactivity show that only physics can explain chemistry? [Scerri] |
| Full Idea: Some authors believe that the interpretation of the properties of the elements passed from chemistry to physics as a result of the discovery of radioactivity. ...I believe this view to be overly reductionist. | |
| From: Eric R. Scerri (The Periodic Table [2007], 06 'Radio') | |
| A reaction: It is all a matter of the explanations, and how far down they have to go. If most non-radiocative chemistry doesn't need to mention the physics, then chemistry is largely autonomous. |
| 17418 | It is now thought that all the elements have literally evolved from hydrogen [Scerri] |
| Full Idea: The elements are now believed to have literally evolved from hydrogen by various mechanisms. | |
| From: Eric R. Scerri (The Periodic Table [2007], 10 'Evol) |
| 17398 | 19th C views said elements survived abstractly in compounds, but also as 'material ingredients' [Scerri] |
| Full Idea: In the 19th century abstract elements were believed to be permanent and responsible for observed properties in compounds, but (departing from Aristotle) they were also 'material ingredients', thus linking the metaphysical and material realm. | |
| From: Eric R. Scerri (The Periodic Table [2007], 04 'Nature') | |
| A reaction: I'm not sure I can make sense of this gulf between the metaphysical and the material realm, so this was an account heading for disaster. |
| 17406 | Moseley, using X-rays, showed that atomic number ordered better than atomic weight [Scerri] |
| Full Idea: By using X-rays, Henry Moseley later discovered that a better ordering principle for the periodic system is atomic numbers rather than atomic weight, by subjecting many different elements to bombardment. | |
| From: Eric R. Scerri (The Periodic Table [2007], 06 'Intro') | |
| A reaction: Moseley was killed in the First World War at the age of 26. It is interesting that they more or less worked out the whole table, before they discovered the best principle on which to found it. |
| 17408 | Some suggested basing the new periodic table on isotopes, not elements [Scerri] |
| Full Idea: Some chemists even suggested that the periodic table would have to be abandoned in favor of a classification system that included a separate place for every single isotope. | |
| From: Eric R. Scerri (The Periodic Table [2007], 06 'Intro') | |
| A reaction: The extreme case is tin, which has 21 isotopes, so is tin a fundamental, or is each of the isotopes a fundamental? Does there have to be a right answer to that? All tin isotopes basically react in the same way, so we stick with the elements table. |
| 17412 | Elements are placed in the table by the number of positive charges - the atomic number [Scerri] |
| Full Idea: The serial number of an element in the periodic table, its atomic number, corresponds to the number of positive charges in the atom. | |
| From: Eric R. Scerri (The Periodic Table [2007], 07 'Models') | |
| A reaction: Note that this is a feature of the nucleus, despite that fact that the electrons decide the chemical properties. A nice model for Locke's views on essentialism. |
| 17413 | Elements in the table are grouped by having the same number of outer-shell electrons [Scerri] |
| Full Idea: The modern notion is that atoms fall into the same group of the periodic table if they possess the same numbers of outer-shell electrons. | |
| From: Eric R. Scerri (The Periodic Table [2007], 07 'Quantum') | |
| A reaction: Scerri goes on to raise questions about this, on p.242. By this principle helium should be an alkaline earth element, but it isn't. |
| 17416 | Orthodoxy says the periodic table is explained by quantum mechanics [Scerri] |
| Full Idea: The prevailing reductionist climate implies that quantum mechanics inevitably provides a more fundamental explanation for the periodic system. | |
| From: Eric R. Scerri (The Periodic Table [2007], 08 'Concl') | |
| A reaction: Scerri has argued that chemists did much better than physicists in working out how the outer electron shells of atoms worked, by induction from data, rather than inference from basic principles. |
| 17414 | Pauli explained the electron shells, but not the lengths of the periods in the table [Scerri] |
| Full Idea: Pauli explained the maximum number of electrons successive shells can accommodate, ...but it does not explain the lengths of the periods, which is the really crucial property of the periodic table. | |
| From: Eric R. Scerri (The Periodic Table [2007], 07 'Pauli') | |
| A reaction: Paulis' Exclusion Principle says no two electrons in an atom can have the same set of four quantum numbers. He added 'spin' as a fourth number. It means 'electrons cannot be distinguished' (243). Scerri says the big problem is still not fully explained. |
| 17410 | Moseley showed the elements progress in units, and thereby clearly identified the gaps [Scerri] |
| Full Idea: Moseley's work showed that the successive elements in the periodic table have an atomic number greater by one unit. The gaps could then be identified definitively, as 43, 61, 72, 75, 85, 87, and 91. | |
| From: Eric R. Scerri (The Periodic Table [2007], 06 'Henry') | |
| A reaction: [compressed] |
| 17395 | Elements were ordered by equivalent weight; later by atomic weight; finally by atomic number [Scerri] |
| Full Idea: Historically, the ordering of elements across periods was determined by equivalent weight, then later by atomic weight, and eventually by atomic number. | |
| From: Eric R. Scerri (The Periodic Table [2007], 01 'React') | |
| A reaction: So they used to be ordered by quantities (measured by real numbers), but eventually were ordered by unit items (counted by natural numbers). There need to be distinct protons (unified) to be counted. |
| 17422 | The best classification needs the deepest and most general principles of the atoms [Scerri] |
| Full Idea: An optimal classification can be obtained by identifying the deepest and most general principles that govern the atoms of the elements. | |
| From: Eric R. Scerri (The Periodic Table [2007], 10 'Continuum') | |
| A reaction: He adds (p.286) that the best system will add the 'greatest degree of regularity' to these best principles. |
| 17417 | To explain the table, quantum mechanics still needs to explain order of shell filling [Scerri] |
| Full Idea: The order of shell filling has not yet been deduced from first principles, and this issue cannot be avoided if one is to really ask whether quantum mechanics explains the periodic system in a fundamental manner. | |
| From: Eric R. Scerri (The Periodic Table [2007], 09 'From') |
| 17419 | Since 99.96% of the universe is hydrogen and helium, the periodic table hardly matters [Scerri] |
| Full Idea: All the elements other than hydrogen and helium make up just 0.04% of the universe. Seen from this perspective, the periodic table appears to rather insignificant. | |
| From: Eric R. Scerri (The Periodic Table [2007], 10 'Astro') |