27 ideas
| 9193 | ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett] |
| Full Idea: ZF set theory is a first-order axiomatization. Variables range over sets, there are no second-order variables, and primitive predicates are just 'equals' and 'member of'. The axiom of extensionality says sets with the same members are identical. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 7) | |
| A reaction: If the eleven members of the cricket team are the same as the eleven members of the hockey team, is the cricket team the same as the hockey team? Our cricket team is better than our hockey team, so different predicates apply to them. |
| 9194 | The main alternative to ZF is one which includes looser classes as well as sets [Dummett] |
| Full Idea: The main alternative to ZF is two-sorted theories, with some variables ranging over classes. Classes have more generous existence assumptions: there is a universal class, containing all sets, and a class containing all ordinals. Classes are not members. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 7.1.1) | |
| A reaction: My intuition is to prefer strict systems when it comes to logical theories. The whole point is precision. Otherwise we could just think about things, and skip all this difficult symbolic stuff. |
| 9195 | Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett] |
| Full Idea: It must not be concluded from the rejection of excluded middle that intuitionistic logic operates with three values: true, false, and neither true nor false. It does not make use of true and false, but only with a construction being a proof. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 8.1) | |
| A reaction: This just sounds like verificationism to me, with all its problems. It seems to make speculative statements meaningless, which can't be right. Realism has lots of propositions which are assumed to be true or false, but also unknowable. |
| 9186 | First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett] |
| Full Idea: First-order logic is distinguished by generalizations (quantification) only over objects: second-order logic admits generalizations or quantification over properties or kinds of objects, and over relations between them, and functions defined over them. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1) | |
| A reaction: Second-order logic was introduced by Frege, but is (interestingly) rejected by Quine, because of the ontological commitments involved. I remain unconvinced that quantification entails ontological commitment, so I'm happy. |
| 9187 | Logical truths and inference are characterized either syntactically or semantically [Dummett] |
| Full Idea: There are two ways of characterizing logical truths and correct inference. Proof-theoretic or syntactic characterizations, if the formalization admits of proof or derivation; and model-theoretic or semantic versions, being true in all interpretations. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1) | |
| A reaction: Dummett calls this distinction 'fundamental'. The second one involves truth, and hence meaning, where the first one just responds to rules. ..But how can you have a notion of correctly following a rule, without a notion of truth? |
| 9191 | Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett] |
| Full Idea: It can be argued that the notion of ordinal numbers is more fundamental than that of cardinals. To count objects, we must count them in sequence. ..The theory of ordinals forms the substratum of Cantor's theory of cardinals. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 5) | |
| A reaction: Depends what you mean by 'fundamental'. I would take cardinality to be psychologically prior ('that is a lot of sheep'). You can't order people by height without first acquiring some people with differing heights. I vote for cardinals. |
| 9192 | The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett] |
| Full Idea: The number 4 cannot be characterized solely by its position in a system, because it has different positions in the system of natural numbers and that of the positive whole numbers, whereas these systems have the very same structure. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 6.1) | |
| A reaction: Dummett seems to think this is fairly decisive against structuralism. There is also the structure of the real numbers. We will solve this by saying that the wholes are abstracted from the naturals, which are abstracted from the reals. Job done. |
| 20624 | Work degrades into heat, but not vice versa [Close] |
| Full Idea: William Thomson, Lord Kelvin, declared (in 1865) the second law of thermodynamics: mechanical work inevitably tends to degrade into heat, but not vice versa. | |
| From: Frank Close (Theories of Everything [2017], 3 'Perpetual') | |
| A reaction: The basis of entropy, which makes time an essential part of physics. Might this be the single most important fact about the physical world? |
| 20623 | First Law: energy can change form, but is conserved overall [Close] |
| Full Idea: The first law of thermodynamics : energy can be changed from one form to another, but is always conserved overall. | |
| From: Frank Close (Theories of Everything [2017], 3 'Perpetual') | |
| A reaction: So we have no idea what energy is, but we know it's conserved. (Daniel Bernoulli showed the greater the mean energy, the higher the temperature. James Joule showed the quantitative equivalence of heat and work p.26-7) |
| 20625 | Third Law: total order and minimum entropy only occurs at absolute zero [Close] |
| Full Idea: The third law of thermodynamics says that a hypothetical state of total order and minimum entropy can be attained only at the absolute zero temperature, minus 273 degrees Celsius. | |
| From: Frank Close (Theories of Everything [2017], 3 'Arrow') | |
| A reaction: If temperature is energetic movement of atoms (or whatever), then obviously zero movement is the coldest it can get. So is absolute zero an energy state, or an absence of energy? I have no idea what 'total order' means. |
| 20622 | All motions are relative and ambiguous, but acceleration is the same in all inertial frames [Close] |
| Full Idea: There is no absolute state of rest; only relative motions are unambiguous. Contrast this with acceleration, however, which has the same magnitude in all inertial frames. | |
| From: Frank Close (Theories of Everything [2017], 3 'Newton's') | |
| A reaction: It seems important to remember this, before we start trumpeting about the whole of physics being relative. ....But see Idea 20634! |
| 20628 | The electric and magnetic are tightly linked, and viewed according to your own motion [Close] |
| Full Idea: Electric and magnetic phenomena are profoundly intertwined; what you interpret as electric or magnetic thus depends on your own motion. | |
| From: Frank Close (Theories of Everything [2017], 3 'Light!') | |
| A reaction: This sounds like an earlier version of special relativity. |
| 20635 | The general relativity equations relate curvature in space-time to density of energy-momentum [Close] |
| Full Idea: The essence of general relativity relates 'curvature in space-time' on one side of the equation to the 'density of momentum and energy' on the other. ...In full, Einstein required ten equations of this type. | |
| From: Frank Close (Theories of Everything [2017], 5 'Gravity') | |
| A reaction: Momentum involves mass, and energy is equivalent to mass (e=mc^2). |
| 20642 | Photon exchange drives the electro-magnetic force [Close] |
| Full Idea: The exchange of photons drives the electro-magnetic force. | |
| From: Frank Close (Theories of Everything [2017], 6 'Superstrings') | |
| A reaction: So light, which we just think of as what is visible, is a mere side-effect of the engine room of nature - the core mechanism of the whole electro-magnetic field. |
| 20627 | Electric fields have four basic laws (two by Gauss, one by Ampère, one by Faraday) [Close] |
| Full Idea: Four basic laws of electric and magnetic fields: Gauss's Law (about the flux produced by a field), Gauss's law of magnets (there can be no monopoles), Ampère's Law (fields on surfaces), and Farday's Law (accelerated magnets produce fields). | |
| From: Frank Close (Theories of Everything [2017], 3 'Light!') | |
| A reaction: [Highly compressed, for an overview. Close explains them] |
| 20630 | Light isn't just emitted in quanta called photons - light is photons [Close] |
| Full Idea: Planck had assumed that light is emitted in quanta called photons. Einstein went further - light is photons. | |
| From: Frank Close (Theories of Everything [2017], 3 'Light!') | |
| A reaction: The point is that light travels as entities which are photons, rather than the emissions being quantized packets of some other stuff. |
| 20637 | In general relativity the energy and momentum of photons subjects them to gravity [Close] |
| Full Idea: In Einstein's general theory, gravity acts also on energy and momentum, not simply on mass. For example, massless photons of light feel the gravitational attraction of the Sun and can be deflected. | |
| From: Frank Close (Theories of Everything [2017], 5 'Planck') | |
| A reaction: Ah, a puzzle solved. How come massless photons are bent by gravity? |
| 20629 | Electro-magnetic waves travel at light speed - so light is electromagnetism! [Close] |
| Full Idea: Faradays' measurements predicted the speed of electro-magnetic waves, which happened to be the speed of light, so Maxwell made an inspired leap: light is an electromagnetic wave! | |
| From: Frank Close (Theories of Everything [2017], 3 'Light!') | |
| A reaction: Put that way, it doesn't sound like an 'inspired' leap, because travelling at exactly the same speed seems a pretty good indication that they are the same sort of thing. (But I'm not denying that Maxwell was a special guy!) |
| 20632 | In QED, electro-magnetism exists in quantum states, emitting and absorbing electrons [Close] |
| Full Idea: Dirac created quantum electrodynamics (QED): the universal electro-magnetic field can exist in discreet states of energy (with photons appearing and disappearing by energy excitations. This combined classical ideas, quantum theory and special relativity. | |
| From: Frank Close (Theories of Everything [2017], 3 'Light!') | |
| A reaction: Close says this is the theory of everything in atomic structure, but not in nuclei (which needs QCD and QFD). So if there are lots of other 'fields' (e.g. gravitational, weak, strong, Higgs), how do they all fit together? Do they talk to one another? |
| 20639 | Quantum fields contain continual rapid creation and disappearance [Close] |
| Full Idea: Quantum field theory implies that the vacuum of space is filled with particles and antiparticles which bubble in and out of existence on faster and faster timescales over shorter and shorter distances. | |
| From: Frank Close (Theories of Everything [2017], 6 'Intro') | |
| A reaction: Ponder this sentence until you head aches. Existence, but not as we know it, Jim. Close says calculations in QED about the electron confirm this. |
| 20641 | Electrons get their mass by interaction with the Higgs field [Close] |
| Full Idea: The electron gets its mass by interaction with the ubiquitous Higgs field. | |
| From: Frank Close (Theories of Everything [2017], 6 'Hierarchy') | |
| A reaction: I thought I understood mass until I read this. Is it just wrong to say the mass of a table is the 'amount of stuff' in it? |
| 20631 | Dirac showed how electrons conform to special relativity [Close] |
| Full Idea: In 1928 Paul Dirac discovered the quantum equation that describes the electron and conforms to the requirements special relativity theory. | |
| From: Frank Close (Theories of Everything [2017], 3 'Light!') | |
| A reaction: This sounds like a major step in the unification of physics. Quantum theory and General relativity remain irreconcilable. |
| 20626 | Modern theories of matter are grounded in heat, work and energy [Close] |
| Full Idea: The link between temperature, heat, work and energy is at the root of our historical ability to construct theories of matter, such as Newton's dynamics, while ignoring, and indeed being ignorant of - atomic dimensions. | |
| From: Frank Close (Theories of Everything [2017], 3 'Arrow') | |
| A reaction: That is, presumably, that even when you fill in the atoms, and the standard model of physics, these aspects of matter do the main explaiining (of the behaviour, rather than of the structure). |
| 20633 | The Higgs field is an electroweak plasma - but we don't know what stuff it consists of [Close] |
| Full Idea: In 2012 it was confirmed that we are immersed in an electroweak plasma - the Higgs field. We curently have no knowledge of what this stuff might consist of. | |
| From: Frank Close (Theories of Everything [2017], 4 'Higgs') | |
| A reaction: The second sentence has my full attention. So we don't understand a field properly until we understand the 'stuff' it is made of? So what are all the familiar fields made of? Tell me more! |
| 20640 | Space-time is indeterminate foam over short distances [Close] |
| Full Idea: At very short distances, space-time itself becomes some indeterminate foam. | |
| From: Frank Close (Theories of Everything [2017], 6 'Intro') | |
| A reaction: [see Close for a bit more detail of this weird idea] |
| 9111 | God is not wise, but more-than-wise; God is not good, but more-than-good [William of Ockham] |
| Full Idea: God is not wise, but more-than-wise; God is not good, but more-than-good. | |
| From: William of Ockham (Reportatio [1330], III Q viii) | |
| A reaction: [He is quoting 'Damascene'] I quote this for interest, but I very much doubt whether Damascene or William knew what it meant, and I certainly don't. There seems to have been a politically correct desire to invent super-powers for God. |
| 9112 | We could never form a concept of God's wisdom if we couldn't abstract it from creatures [William of Ockham] |
| Full Idea: What we abstract is said to belong to perfection in so far as it can be predicated of God and can stand for Him. For if such a concept could not be abstracted from a creature, then in this life we could not arrive at a cognition of God's wisdom. | |
| From: William of Ockham (Reportatio [1330], III Q viii) | |
| A reaction: This seems to be the germ of an important argument. Without the ability to abstract from what is experienced, we would not be able to apply general concepts to things which are beyond experience. It is a key idea for empiricism. |