88 ideas
| 19588 | The highest aim of philosophy is to combine all philosophies into a unity [Novalis] |
| Full Idea: He attains the maximum of a philosopher who combines all philosophies into a single philosophy | |
| From: Novalis (Logological Fragments II [1798], 31) | |
| A reaction: I have found the epigraph for my big book! Recently a few narrowly analytical philosophers have attempted big books about everything (Sider, Heil, Chalmers), and they get a huge round of applause from me. |
| 19598 | Philosophy relies on our whole system of learning, and can thus never be complete [Novalis] |
| Full Idea: Now all learning is connected - thus philosophy will never be complete. Only in the complete system of all learning will philosophy be truly visible. | |
| From: Novalis (Logological Fragments II [1798], 39) | |
| A reaction: Philosophy is evidently the unifying subject, which reveals the point of all the other subjects. It matches my maxim that 'science is the servant of philosophy'. |
| 19586 | Philosophers feed on problems, hoping they are digestible, and spiced with paradox [Novalis] |
| Full Idea: The philosopher lives on problems as the human being does on food. An insoluble problem is an indigestible food. What spice is to food, the paradoxical is to problems. | |
| From: Novalis (Logological Fragments II [1798], 09) | |
| A reaction: Novalis would presumably have disliked Hegel's dialectic, where the best food seems to be the indigestible. |
| 19587 | Philosophy aims to produce a priori an absolute and artistic world system [Novalis] |
| Full Idea: Philosophy ...is the art of producing all our conceptions according to an absolute, artistic idea and of developing the thought of a world system a priori out of the depths of our spirit. | |
| From: Novalis (Logological Fragments II [1798], 19) | |
| A reaction: A lovely statement of the dream of building world systems by pure thought - embodying perfectly the view of philosophy despised by logical positivists and modern logical metaphysicians. The Novalis view will never die! I like 'artistic'. |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 19597 | Logic (the theory of relations) should be applied to mathematics [Novalis] |
| Full Idea: Ought not logic, the theory of relations, be applied to mathematics? | |
| From: Novalis (Logological Fragments II [1798], 38) | |
| A reaction: Bolzano was 19 when his was written. I presume Novalis would have been excited by set theory (even though he was a hyper-romantic). |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| Full Idea: Cantor (in his exploration of infinities) pushed the bounds of conceivability further than anyone before him. To discover what is conceivable, we have to enquire into the concept. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics 6.5 | |
| A reaction: This remark comes during a discussion of Husserl's phenomenology. Intuitionists challenge Cantor's claim, and restrict what is conceivable to what is provable. Does possibility depend on conceivability? |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 21167 | Gravity is unusual, in that it always attracts and never repels [New Sci.] |
| Full Idea: Gravity is an odd sort of force, not least because it only ever works one way. Electromagnetism attracts and repels, but with gravity there are only positive masses always attract. | |
| From: New Scientist writers (Why the Universe Exists [2017], 05) | |
| A reaction: This leads to speculation about anti-gravity, but there is no current evidence for it. |
| 21176 | In the Big Bang general relativity fails, because gravity is too powerful [New Sci.] |
| Full Idea: At the origin of the universe gravity becomes so powerful that general relativity breaks down, giving infinity for every answer. | |
| From: New Scientist writers (Why the Universe Exists [2017], 09) |
| 21147 | Quantum electrodynamics incorporates special relativity and quantum mechanics [New Sci.] |
| Full Idea: The theory of electromagnetism that incorporates both special relativity and quantum mechanics is quantum electrodynamics (QED). It was developed by Dirac and others, and perfected in the 1940s. The field is a collection of quanta. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: This builds on Maxwell's earlier classical theory. QED is said to be the best theory in all of physics. |
| 21155 | Photons have zero rest mass, so virtual photons have infinite range [New Sci.] |
| Full Idea: Photons, the field quanta of the electromagnetic force, have zero rest mass, so virtual photons can exist indefinitely and travel any distance, meaning the electromagnetic force has an infinite range. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) |
| 21161 | In the standard model all the fundamental force fields merge at extremely high energies [New Sci.] |
| Full Idea: The standard model says that the fields of all fundamental forces should merge at extremely high energies, meaning there is also a unified, high-energy field out there. | |
| From: New Scientist writers (Why the Universe Exists [2017], 03) | |
| A reaction: Not quite sure what 'out there' means. This idea is linked to the quest for dark energy. Is this unified phenomenon only found near the Big Bang? |
| 21146 | Electrons move fast, so are subject to special relativity [New Sci.] |
| Full Idea: Electrons in atoms move at high speeds, so they are subject to the special theory of relativity. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: Presumably this implies a frame of reference, and defining velocities relative to other electrons. Plus time-dilation? |
| 21148 | The strong force is repulsive at short distances, strong at medium, and fades at long [New Sci.] |
| Full Idea: Experiments show that the nuclear binding force does not follow the inverse square law, but is repulsive at the shortest distances, then attractive, then fades away rapidly as distance increases further. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: So how does it know when to be strong? Magnetism doesn't vary according to distance, and light obeys the inverse square law, because everything is decided at the output. - See 21151 for an explanation. It interacts after departure. |
| 21151 | Gluons, the particles carrying the strong force, interact because of their colour charge [New Sci.] |
| Full Idea: In QCD the particles that carry the strong force are called gluons. ...Gluons carry their own colour charges, so they can interact with each other (unlike photons) via the strong nuclear force (which limits the range of the force). | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: So the force varies in strength with distance because the degree of separation among the spreading gluons varies? The force has one range, which is squashed when close, effective at medium, and loses touch with distance? |
| 21152 | The strong force binds quarks tight, and the nucleus more weakly [New Sci.] |
| Full Idea: The strong force holds quarks together within protons and neutrons, and residual effects of the strong force bind protons (whch repel one another) and neutrons together in nuclei. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: So the force is much stronger between quarks (which can't escape), and only 'residual' in the nucleus, which must be why smashing nuclei open is fairly easy, but smashin protons open needs higher energies. |
| 21143 | Quarks in threes can build hadrons with spin ½ or with spin 3/2 [New Sci.] |
| Full Idea: Quarks in threes can build hadrons with spin ½ (proton, duu; neutron, ddu; lambda, dus), or with spin 3/2 (omega-minus, sss). | |
| From: New Scientist writers (Why the Universe Exists [2017], 01) |
| 21142 | Classifying hadrons revealed two symmetry patterns, produced by three basic elements [New Sci.] |
| Full Idea: Classifying hadrons according to charge, strangeness and spin revealed patterns of eight and ten particles (SU(3) symmetery). The mathematics then showed that these are built from a basic group of only three members. | |
| From: New Scientist writers (Why the Universe Exists [2017], 01) |
| 21150 | Three different colours of quark (as in the proton) can cancel out to give no colour [New Sci.] |
| Full Idea: Just as mixing three colours of light gives white, so the three colour charges of quarks can add up to give no colour. This is what happens in the proton, which always contains one blue-charge quark, one red and one green. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) |
| 21145 | The four fundamental forces (gravity, electromagnetism, weak and strong) are the effects of particles [New Sci.] |
| Full Idea: There are four fundamental forces: gravity, electromagnetism, and the weak and strong nuclear forces. Particle physics has so far failed to encompass the force of gravity. The forces that shape our world are themselves the effect of particles. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: Philosophers must take note of the fact that forces are the effects of particles. Common sense pictures forces imposed on particles, like throwing a tennis ball, but the particles are actually the sources of force. The gravitino is speculative. |
| 21153 | The weak force explains beta decay, and the change of type by quarks and leptons [New Sci.] |
| Full Idea: The beta decay of the neutron (into a proton, an electron and an antineutrino) can be described in terms of the weak force, which is 10,000 times weaker than the strong force. It allows the quarks and leptons to change from one type to another. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: This seems to make it the key source of radioactivity. Perhaps it should be called the Force of Change? |
| 21154 | Three particles enable the weak force: W+ and W- are charged, and Z° is not [New Sci.] |
| Full Idea: The quantum field theory of the weak force needs three carrier particles. The W+ and W- are electrically charged, and enable the weak force to change the charge of a particle. The Z° is uncharged, and mediates weak interactions with no charge change. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) |
| 21156 | The weak force particles are heavy, so the force has a short range [New Sci.] |
| Full Idea: The W and Z particles are heavy, and so cannot travel far from their parents. The weak force therefore has a very short range. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) |
| 21164 | Why do the charges of the very different proton and electron perfectly match up? [New Sci.] |
| Full Idea: Why do the proton and electron charges mirror each other so perfectly when they are such different particles? | |
| From: New Scientist writers (Why the Universe Exists [2017], 04) | |
| A reaction: We seem to have reached a common stage in science, where we have a wonderful descriptive model (the Standard Model), but we cannot explain why what is modelled is the way it is. |
| 21170 | The Standard Model cannot explain dark energy, survival of matter, gravity, or force strength [New Sci.] |
| Full Idea: The standard model cannot explain dark matter, or dark energy (which is causing expansion to accelerate). It cannot explain how matter survived annihilation with anti-matter in the Big Bang, or explain gravity. The strength of each force is unexplained. | |
| From: New Scientist writers (Why the Universe Exists [2017], 06) | |
| A reaction: [compressed] P.141 adds that the model has to be manipulated to keep the Higgs mass low enough. |
| 21140 | Spin is a built-in ration of angular momentum [New Sci.] |
| Full Idea: Spin is a built-in ration of angular momentum. | |
| From: New Scientist writers (Why the Universe Exists [2017], 01) | |
| A reaction: As an outsider all I can do is collect descriptions of such properties from the experts. The experts appear to be happy with the numbers inserted in the equations. |
| 21149 | Quarks have red, green or blue colour charge (akin to electric charge) [New Sci.] |
| Full Idea: Quarks have a property akin to electric charge, called their colour charge. It comes in three varieties, red, green and blue. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) |
| 21158 | Fermions, with spin ½, are antisocial, and cannot share quantum states [New Sci.] |
| Full Idea: Particles with half-integer spin, such as electrons, protons or quarks (all spin ½) have an asymmetry in their wavefunction that makes them antisocial. These particles (Fermions) cannot share a quantum state. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: This is said to explain the complexity of matter, with carbon an especially good example. |
| 21165 | Spin is akin to rotation, and is easily measured in a magnetic field [New Sci.] |
| Full Idea: Spin is a quantum-mechanical property of a particle akin to rotation about its own axis. Particles of different spins respond to magnetic fields in different ways, so it is a relatively easy thing to measure. | |
| From: New Scientist writers (Why the Universe Exists [2017], 04) | |
| A reaction: I wish I knew what 'akin to' meant. Maybe particles are not rigid bodies, so they cannot spin in the way a top can? It must be an electro-magnetic property. Idea 21166 says spin has two possible directions. |
| 21157 | Particles are spread out, with wave-like properties, and higher energy shortens the wavelength [New Sci.] |
| Full Idea: Particles obeying the laws of quantum mechanics have wave-like properties - moving as a quantum wave-function, spread out in space, with wavelengths that get shorter as their energy increases. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: Thus X-rays are dangerous, but long wave radio is not. De Broglie's equation. |
| 21163 | The mass of protons and neutrinos is mostly binding energy, not the quarks [New Sci.] |
| Full Idea: Most of a proton's or neutrino's mass is contained in the interaction energies of a 'sea' of quarks, antiquarks and gluons that bind them. ...You might feel solid, but in fact you're 99 per cent binding energy. | |
| From: New Scientist writers (Why the Universe Exists [2017], 04) | |
| A reaction: This is because energy is equivalent to mass (although gluons are said to have energy but no mass - puzzled by that). This is a fact which needs a bit of time to digest. Once you've grasped we are full of space, you still have understood it. |
| 21168 | Gravitional mass turns out to be the same as inertial mass [New Sci.] |
| Full Idea: There are two types of mass: gravitational mass quantifies how strongly an object feels gravity, while inertial mass quantifies an object's resistance to acceleration. There proven equality is at the heart of General Relativity. | |
| From: New Scientist writers (Why the Universe Exists [2017], 05) | |
| A reaction: It had never occurred to me that these two values might come apart. Doesn't their identical values demonstrate that they are in fact the same thing? Sounds like Hesperus/Phosphorus to me. The book calls it 'mysterious'. |
| 21138 | Neutrons are slightly heavier than protons, and decay into them by emitting an electron [New Sci.] |
| Full Idea: The proton (938.3 MeV) is lighter than the neutron (939.6 MeV) and does not decay, but the heavier neutron can change into a proton by emitting an electron. (If you gather a bucketful of neutrons, after ten minutes only half of them would be left). | |
| From: New Scientist writers (Why the Universe Exists [2017], 01) | |
| A reaction: Protons are more or less eternal, but some theories have them decaying after billions of years. Smashing protons together is a popular pastime for physicists. |
| 21144 | Top, bottom, charm and strange quarks quickly decay into up and down [New Sci.] |
| Full Idea: Quarks can change from one variety to another, and the top, bottom, charm and strange quarks all rapidly decay to the up and down quarks of everyday life. | |
| From: New Scientist writers (Why the Universe Exists [2017], 01) | |
| A reaction: Hence the universe is largely composed of up and down quarks and electrons. The other quarks seem to be more important in the early universe. |
| 21141 | Neutrinos were proposed as the missing energy in neutron beta decay [New Sci.] |
| Full Idea: When a neutron decays into a proton and an electron (one example of beta decay), the energy of the two particles adds up to less than the starting energy of the neutron. Pauli and Fermi concluded that a neutrino (an electron antineutrino) is emitted. | |
| From: New Scientist writers (Why the Universe Exists [2017], 01) | |
| A reaction: I'm wondering how much they could infer about the nature of the new particle (which was only confirmed 26 years later). |
| 21169 | Only neutrinos spin anticlockwise [New Sci.] |
| Full Idea: Neutrinos are the only particles that seem just to spin anticlockwise. | |
| From: New Scientist writers (Why the Universe Exists [2017], 06) | |
| A reaction: See 21166. Anti-neutrino spin is the opposite way. Which way up do you hold the neutrino when pronouncing that it is 'anticlockwise? |
| 21166 | Standard antineutrinos have opposite spin and opposite lepton number [New Sci.] |
| Full Idea: In the conventional standard model neutrinos have antiparticles - which spin in the opposite direction, and have the opposite lepton number. | |
| From: New Scientist writers (Why the Universe Exists [2017], 05) |
| 21171 | The symmetry of unified electromagnetic and weak forces was broken by the Higgs field [New Sci.] |
| Full Idea: In the very early hot universe the electromagnetic and weak nuclear forces were one. The early emergence of the Higgs field led to electroweak symmetry breaking. The W and Z bosons grew fat, and the photon raced away mass-free. | |
| From: New Scientist writers (Why the Universe Exists [2017], 07) |
| 21175 | String theory might be tested by colliding strings to make bigger 'stringballs' [New Sci.] |
| Full Idea: A future accelerator might create 'stringballs', when two strings slam into one another and, rather than combining to form a stretched string, make a tangled ball. Finding them would prove string theory. | |
| From: New Scientist writers (Why the Universe Exists [2017], 08) | |
| A reaction: This is the only possible test for string theory which I have seen suggested. How do you 'slam strings together'? |
| 21177 | String theory offers a quantum theory of gravity, by describing the graviton [New Sci.] |
| Full Idea: String theory works as a quantum theory of gravity because string vibrations can describe gravitons, the hypothetical carriers of the gravitational force. | |
| From: New Scientist writers (Why the Universe Exists [2017], 09) | |
| A reaction: Presumably the main aim of a quantum theory of gravity is to include gravitons within particle theory. This idea has to be a main attraction of string theory. Compare Idea 21166. |
| 21179 | Supersymmetric string theory can be expressed using loop quantum gravity [New Sci.] |
| Full Idea: String theory, together with its supersymmetric particles, has recently been rewritten in the space-time described by loop quantum gravity (which says that space-time ust be made from finite chunks). | |
| From: New Scientist writers (Why the Universe Exists [2017], 09) |
| 21178 | String theory is now part of 11-dimensional M-Theory, involving p-branes [New Sci.] |
| Full Idea: String theory has now been incorporated into Ed Witten's M-Theory, which is a mathematical framework that lives in 11-dimensional space-time, involving higher-dimensional objects called p-branes, of which strings are a special case. | |
| From: New Scientist writers (Why the Universe Exists [2017], 09) |
| 21173 | Supersymmetry says particles and superpartners were unities, but then split [New Sci.] |
| Full Idea: The key to supersymmetry is that in the high-energy soup of the early universe, particles and their superpartners were indistinguishable. Each pair existed as single massless entities. With expansion and cooling this supersymmetry broke down. | |
| From: New Scientist writers (Why the Universe Exists [2017], 08) |
| 21159 | Supersymmetry has extra heavy bosons and heavy fermions [New Sci.] |
| Full Idea: Supersymmetry posits heavy boson partners for all fermions, and heavy fermions for all bosons. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: The main Fermions are electron, proton and quark. Do extra bosons imply extra forces? Peter Higgs favours supersymmetry. |
| 21162 | Only supersymmetry offers to incorporate gravity into the scheme [New Sci.] |
| Full Idea: Peter Higgs says he is a fan of supersymmetry, largely because it seems to be the only route by which gravity can be brought into the scheme. | |
| From: New Scientist writers (Why the Universe Exists [2017], 03) | |
| A reaction: Peter Higgs proposed the Higgs boson (now discovered). This seems a very good reason to favour supersymmetry. A grand unified theory that left out gravity doesn't seem to be unified quite grandly enough. |
| 21172 | The evidence for supersymmetry keeps failing to appear [New Sci.] |
| Full Idea: The old front-runner theory, supersymmetry, has fallen from grace as the Large Hadron Collider keeps failing to find it. | |
| From: New Scientist writers (Why the Universe Exists [2017], 07) |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 21160 | The Higgs field means even low energy space is not empty [New Sci.] |
| Full Idea: The point about the Higgs field is that even the lowest-energy state of space is not empty. | |
| From: New Scientist writers (Why the Universe Exists [2017], 02) | |
| A reaction: So where is the Higgs field located? Even if there is no utterly empty space, the concept of location implies a concept of space more basic than the fields (about 16, I gather) which occupy it. You can't describe movement without a concept of location. |
| 21174 | Dark matter must have mass, to produce gravity, and no electric charge, to not reflect light [New Sci.] |
| Full Idea: Whatever dark matter is made of, it must have mass to feel and generate gravity; but no electric charge, so it does not interact with light. The leading candidate has been the weakly interacting massive particle (WIMP), much heavier than a proton. | |
| From: New Scientist writers (Why the Universe Exists [2017], 08) | |
| A reaction: Note that it must 'generate' gravity. The idea of a law of gravity which is externally imposed on matter is long dead. Heavy WIMPs have not yet been detected. |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |