39 ideas
| 14239 | The empty set is usually derived from Separation, but it also seems to need Infinity [Oliver/Smiley] |
| Full Idea: The empty set is usually derived via Zermelo's axiom of separation. But the axiom of separation is conditional: it requires the existence of a set in order to generate others as subsets of it. The original set has to come from the axiom of infinity. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: They charge that this leads to circularity, as Infinity depends on the empty set. |
| 14240 | The empty set is something, not nothing! [Oliver/Smiley] |
| Full Idea: Some authors need to be told loud and clear: if there is an empty set, it is something, not nothing. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: I'm inclined to think of a null set as a pair of brackets, so maybe that puts it into a metalanguage. |
| 14241 | We don't need the empty set to express non-existence, as there are other ways to do that [Oliver/Smiley] |
| Full Idea: The empty set is said to be useful to express non-existence, but saying 'there are no Us', or ¬∃xUx are no less concise, and certainly less roundabout. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) |
| 14242 | Maybe we can treat the empty set symbol as just meaning an empty term [Oliver/Smiley] |
| Full Idea: Suppose we introduce Ω not as a term standing for a supposed empty set, but as a paradigm of an empty term, not standing for anything. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: This proposal, which they go on to explore, seems to mean that Ω (i.e. the traditional empty set symbol) is no longer part of set theory but is part of semantics. |
| 14243 | The unit set may be needed to express intersections that leave a single member [Oliver/Smiley] |
| Full Idea: Thomason says with no unit sets we couldn't call {1,2}∩{2,3} a set - but so what? Why shouldn't the intersection be the number 2? However, we then have to distinguish three different cases of intersection (common subset or member, or disjoint). | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 2.2) |
| 14234 | If you only refer to objects one at a time, you need sets in order to refer to a plurality [Oliver/Smiley] |
| Full Idea: A 'singularist', who refers to objects one at a time, must resort to the language of sets in order to replace plural reference to members ('Henry VIII's wives') by singular reference to a set ('the set of Henry VIII's wives'). | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
| A reaction: A simple and illuminating point about the motivation for plural reference. Null sets and singletons give me the creeps, so I would personally prefer to avoid set theory when dealing with ontology. |
| 14237 | We can use plural language to refer to the set theory domain, to avoid calling it a 'set' [Oliver/Smiley] |
| Full Idea: Plurals earn their keep in set theory, to answer Skolem's remark that 'in order to treat of 'sets', we must begin with 'domains' that are constituted in a certain way'. We can speak in the plural of 'the objects', not a 'domain' of objects. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
| A reaction: [Skolem 1922:291 in van Heijenoort] Zermelo has said that the domain cannot be a set, because every set belongs to it. |
| 14245 | Logical truths are true no matter what exists - but predicate calculus insists that something exists [Oliver/Smiley] |
| Full Idea: Logical truths should be true no matter what exists, so true even if nothing exists. The classical predicate calculus, however, makes it logically true that something exists. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) |
| 14246 | If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley] |
| Full Idea: If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) | |
| A reaction: Love it! Of course, they are using 'objects' in the rather Fregean sense of genuine abstract entities. I don't see why fictionalism shouldn't allow maths to be wholly 'pure', although we have invented fictions which actually have application. |
| 14247 | Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley] |
| Full Idea: Identifying numbers with sets may mean one of three quite different things: 1) the sets represent the numbers, or ii) they are the numbers, or iii) they replace the numbers. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2) | |
| A reaction: Option one sounds the most plausible to me. I will take numbers to be patterns embedded in nature, and sets are one way of presenting them in shorthand form, in order to bring out what is repeated. |
| 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') |
| 3029 | Stilpo said if Athena is a daughter of Zeus, then a statue is only the child of a sculptor, and so is not a god [Stilpo, by Diog. Laertius] |
| Full Idea: Stilpo asked a man whether Athena is the daughter of Zeus, and when he said yes, said,"But this statue of Athena by Phidias is the child of Phidias, so it is not a god." | |
| From: report of Stilpo (fragments/reports [c.330 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.10.5 |