43 ideas
| 13886 | Later Frege held that definitions must fix a function's value for every possible argument [Frege, by Wright,C] |
| Full Idea: Frege later became fastidious about definitions, and demanded that they must provide for every possible case, and that no function is properly determined unless its value is fixed for every conceivable object as argument. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xiv | |
| A reaction: Presumably definitions come in degrees of completeness, but it seems harsh to describe a desire for the perfect definition as 'fastidious', especially if we are talking about mathematics, rather than defining 'happiness'. |
| 9845 | We can't define a word by defining an expression containing it, as the remaining parts are a problem [Frege] |
| Full Idea: Given the reference (bedeutung) of an expression and a part of it, obviously the reference of the remaining part is not always determined. So we may not define a symbol or word by defining an expression in which it occurs, whose remaining parts are known | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §66) | |
| A reaction: Dummett cites this as Frege's rejection of contextual definitions, which he had employed in the Grundlagen. I take it not so much that they are wrong, as that Frege decided to set the bar a bit higher. |
| 10019 | Only what is logically complex can be defined; what is simple must be pointed to [Frege] |
| Full Idea: Only what is logically complex can be defined; what is simple can only be pointed to. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §180), quoted by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.137 | |
| A reaction: Frege presumably has in mind his treasured abstract objects, such as cardinal numbers. It is hard to see how you could 'point to' anything in the phenomenal world that had atomic simplicity. Hodes calls this a 'desperate Kantian move'. |
| 15647 | Truth definitions don't produce a good theory, because they go beyond your current language [Halbach] |
| Full Idea: It is far from clear that a definition of truth can lead to a philosophically satisfactory theory of truth. Tarski's theorem on the undefinability of the truth predicate needs resources beyond those of the language for which it is being defined. | |
| From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1) | |
| A reaction: The idea is that you need a 'metalanguage' for the definition. If I say 'p' is a true sentence in language 'L', I am not making that observation from within language L. The dream is a theory confined to the object language. |
| 15649 | In semantic theories of truth, the predicate is in an object-language, and the definition in a metalanguage [Halbach] |
| Full Idea: In semantic theories of truth (Tarski or Kripke), a truth predicate is defined for an object-language. This definition is carried out in a metalanguage, which is typically taken to include set theory or another strong theory or expressive language. | |
| From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1) | |
| A reaction: Presumably the metalanguage includes set theory because that connects it with mathematics, and enables it to be formally rigorous. Tarski showed, in his undefinability theorem, that the meta-language must have increased resources. |
| 15655 | Should axiomatic truth be 'conservative' - not proving anything apart from implications of the axioms? [Halbach] |
| Full Idea: If truth is not explanatory, truth axioms should not allow proof of new theorems not involving the truth predicate. It is hence said that axiomatic truth should be 'conservative' - not implying further sentences beyond what the axioms can prove. | |
| From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1.3) | |
| A reaction: [compressed] |
| 15654 | If truth is defined it can be eliminated, whereas axiomatic truth has various commitments [Halbach] |
| Full Idea: If truth can be explicitly defined, it can be eliminated, whereas an axiomatized notion of truth may bring all kinds of commitments. | |
| From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1.3) | |
| A reaction: The general principle that anything which can be defined can be eliminated (in an abstract theory, presumably, not in nature!) raises interesting questions about how many true theories there are which are all equivalent to one another. |
| 15650 | Axiomatic theories of truth need a weak logical framework, and not a strong metatheory [Halbach] |
| Full Idea: Axiomatic theories of truth can be presented within very weak logical frameworks which require very few resources, and avoid the need for a strong metalanguage and metatheory. | |
| From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1) |
| 15648 | Instead of a truth definition, add a primitive truth predicate, and axioms for how it works [Halbach] |
| Full Idea: The axiomatic approach does not presuppose that truth can be defined. Instead, a formal language is expanded by a new primitive predicate of truth, and axioms for that predicate are then laid down. | |
| From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1) | |
| A reaction: Idea 15647 explains why Halbach thinks the definition route is no good. |
| 15656 | Deflationists say truth merely serves to express infinite conjunctions [Halbach] |
| Full Idea: According to many deflationists, truth serves merely the purpose of expressing infinite conjunctions. | |
| From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1.3) | |
| A reaction: That is, it asserts sentences that are too numerous to express individually. It also seems, on a deflationist view, to serve for anaphoric reference to sentences, such as 'what she just said is true'. |
| 15657 | To prove the consistency of set theory, we must go beyond set theory [Halbach] |
| Full Idea: The consistency of set theory cannot be established without assumptions transcending set theory. | |
| From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 2.1) |
| 15652 | We can use truth instead of ontologically loaded second-order comprehension assumptions about properties [Halbach] |
| Full Idea: The reduction of 2nd-order theories (of properties or sets) to axiomatic theories of truth may be conceived as a form of reductive nominalism, replacing existence assumptions (for comprehension axioms) by ontologically innocent truth assumptions. | |
| From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1.1) | |
| A reaction: I like this very much, as weeding properties out of logic (without weeding them out of the world). So-called properties in logic are too abundant, so there is a misfit with their role in science. |
| 15651 | Instead of saying x has a property, we can say a formula is true of x - as long as we have 'true' [Halbach] |
| Full Idea: Quantification over (certain) properties can be mimicked in a language with a truth predicate by quantifying over formulas. Instead of saying that Tom has the property of being a poor philosopher, we can say 'x is a poor philosopher' is true of Tom. | |
| From: Volker Halbach (Axiomatic Theories of Truth (2005 ver) [2005], 1.1) | |
| A reaction: I love this, and think it is very important. He talks of 'mimicking' properties, but I see it as philosophers mistakenly attributing properties, when actually what they were doing is asserting truths involving certain predicates. |
| 9886 | Cardinals say how many, and reals give measurements compared to a unit quantity [Frege] |
| Full Idea: The cardinals and the reals are completely disjoint domains. The cardinal numbers answer the question 'How many objects of a given kind are there?', but the real numbers are for measurement, saying how large a quantity is compared to a unit quantity. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §157), quoted by Michael Dummett - Frege philosophy of mathematics Ch.19 | |
| A reaction: We might say that cardinals are digital and reals are analogue. Frege is unusual in totally separating them. They map onto one another, after all. Cardinals look like special cases of reals. Reals are dreams about the gaps between cardinals. |
| 9889 | Real numbers are ratios of quantities [Frege, by Dummett] |
| Full Idea: Frege fixed on construing real numbers as ratios of quantities (in agreement with Newton). | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege philosophy of mathematics Ch.20 | |
| A reaction: If 3/4 is the same real number as 6/8, which is the correct ratio? Why doesn't the square root of 9/16 also express it? Why should irrationals be so utterly different from rationals? In what sense are they both 'numbers'? |
| 10553 | A number is a class of classes of the same cardinality [Frege, by Dummett] |
| Full Idea: For Frege, in 'Grundgesetze', a number is a class of classes of the same cardinality. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 |
| 10020 | Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege] |
| Full Idea: The inconsistency of Grundgesetze was only a minor flaw. Its fundamental flaw was its inability to account for the way in which the senses of number terms are determined. It leaves the reference-magnetic nature of the standard numberer a mystery. | |
| From: comment on Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.139 | |
| A reaction: A point also made by Hofweber. As a logician, Frege was only concerned with the inferential role of number terms, and he felt he had captured their logical form, but it is when you come to look at numbers in natural language that he seem in trouble. |
| 9887 | Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett] |
| Full Idea: Frege's three main objections to radical formalism are that it cannot account for the application of mathematics, that it confuses a formal theory with its metatheory, and it cannot explain an infinite sequence. | |
| From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §86-137) by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: The application is because we don't design maths randomly, but to be useful. The third objection might be dealt with by potential infinities (from formal rules). The second objection sounds promising. |
| 8751 | Only applicability raises arithmetic from a game to a science [Frege] |
| Full Idea: It is applicability alone which elevates arithmetic from a game to the rank of a science. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §91), quoted by Stewart Shapiro - Thinking About Mathematics 6.1.2 | |
| A reaction: This is the basic objection to Formalism. It invites the question of why it is applicable, which platonists like Frege don't seem to answer (though Plato himself has reality modelled on the Forms). This is why I like structuralism. |
| 9891 | The first demand of logic is of a sharp boundary [Frege] |
| Full Idea: The first demand of logic is of a sharp boundary. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §160), quoted by Michael Dummett - Frege philosophy of mathematics Ch.22 | |
| A reaction: Nothing I have read about vagueness has made me doubt Frege's view of this, although precisification might allow you to do logic with vague concepts without having to finally settle where the actual boundaries are. |
| 23308 | Reasoning relates to understanding as time does to eternity [Boethius, by Sorabji] |
| Full Idea: Boethius says that reasoning [ratiocinatio] is related to intellectual understanding [intellectus] as time to eternity, involving as it does movement from one stage to another. | |
| From: report of Boethius (The Consolations of Philosophy [c.520], 4, prose 6) by Richard Sorabji - Rationality 'Shifting' | |
| A reaction: This gives true understanding a quasi-religious aura, as befits a subject which is truly consoling. |
| 5771 | Knowledge of present events doesn't make them necessary, so future events are no different [Boethius] |
| Full Idea: Just as the knowledge of present things imposes no necessity on what is happening, so foreknowledge imposes no necessity on what is going to happen. | |
| From: Boethius (The Consolations of Philosophy [c.520], V.IV) | |
| A reaction: This, I think, is the key idea if you are looking for a theological answer to the theological problem of free will. Don't think of God as seeing the future 'now'. God is outside time, and so only observes all of history just as we observe the present. |
| 5767 | Rational natures require free will, in order to have power of judgement [Boethius] |
| Full Idea: There is freedom of the will, for it would be impossible for any rational nature to exist without it. Whatever by nature has the use of reason has the power of judgement to decide each matter. | |
| From: Boethius (The Consolations of Philosophy [c.520], V.II) | |
| A reaction: A view taken up by Aquinas (Idea 1849) and Kant (Idea 3740). The 'power of judgement' pinpoints the core of rationality, and it is not clear how a robot could fulfil such a power, if it lacked consciousness. Does a machine 'judge' barcodes? |
| 5768 | God's universal foreknowledge seems opposed to free will [Boethius] |
| Full Idea: God's universal foreknowledge and freedom of the will seem clean contrary and opposite. | |
| From: Boethius (The Consolations of Philosophy [c.520], V.III) | |
| A reaction: The original source of the great theological and philosophical anguish over free will. The problem is anything which fixes future facts, be it oracular knowledge or scientific prediction. Personally I think free will was an invention by religions. |
| 5769 | Does foreknowledge cause necessity, or necessity cause foreknowledge? [Boethius] |
| Full Idea: Does foreknowledge of the future cause the necessity of events, or necessity cause the foreknowledge? | |
| From: Boethius (The Consolations of Philosophy [c.520], V.III) | |
| A reaction: An intriguing question, though not one that bothers me. I don't understand how foreknowledge causes necessity, unless God's vision of the future is a kind of 'freezing ray'. Even the gods must bow to necessity (Idea 3016). |
| 9890 | The modern account of real numbers detaches a ratio from its geometrical origins [Frege] |
| Full Idea: From geometry we retain the interpretation of a real number as a ratio of quantities or measurement-number; but in more recent times we detach it from geometrical quantities, and from all particular types of quantity. | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §159), quoted by Michael Dummett - Frege philosophy of mathematics | |
| A reaction: Dummett glosses the 'recent' version as by Cantor and Dedekind in 1872. This use of 'detach' seems to me startlingly like the sort of psychological abstractionism which Frege was so desperate to avoid. |
| 11846 | If we abstract the difference between two houses, they don't become the same house [Frege] |
| Full Idea: If abstracting from the difference between my house and my neighbour's, I were to regard both houses as mine, the defect of the abstraction would soon be made clear. It may, though, be possible to obtain a concept by means of abstraction... | |
| From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §99) | |
| A reaction: Note the important concession at the end, which shows Frege could never deny the abstraction process, despite all the modern protests by Geach and Dummett that he totally rejected it. |
| 5762 | The wicked want goodness, so they would not be wicked if they obtained it [Boethius] |
| Full Idea: If the wicked obtained what they want - that is goodness - they could not be wicked. | |
| From: Boethius (The Consolations of Philosophy [c.520], IV.II) | |
| A reaction: This is a nice paradox which arises from Boethius being, like Socrates, an intellectualist. The question is whether the wicked want the good de re or de dicto. If they wanted to good de re (as its true self) they would obviously not be wicked. |
| 5770 | Rewards and punishments are not deserved if they don't arise from free movement of the mind [Boethius] |
| Full Idea: If there is no free will, then in vain is reward offered to the good and punishment to the bad, because they have not been deserved by any free and willed movement of the mind. | |
| From: Boethius (The Consolations of Philosophy [c.520], V.III) | |
| A reaction: I just don't see why decisions have to come out of nowhere in order to have any merit. People are different from natural forces, because the former can be persuaded by reasons. A moral agent is a mechanism which decides according to reasons. |
| 5764 | When people fall into wickedness they lose their human nature [Boethius] |
| Full Idea: When people fall into wickedness they lose their human nature. | |
| From: Boethius (The Consolations of Philosophy [c.520], IV.III) | |
| A reaction: This is a view I find quite sympathetic, but which is a million miles from the modern view. Today's paper showed a picture of a famous criminal holding a machine gun and a baby. We seem to delight in the idea that human nature is partly wicked. |
| 5756 | Happiness is a good which once obtained leaves nothing more to be desired [Boethius] |
| Full Idea: Happiness is a good which once obtained leaves nothing more to be desired. | |
| From: Boethius (The Consolations of Philosophy [c.520], III.I) | |
| A reaction: This sounds like the ancient 'eudaimonism' of Socrates and Aristotle, which might not be entirely compatible with orthodox Christianity. It is not true, though, that happy people lack ambition. To be happy, an unfilfilled aim may be needed. |
| 5763 | The bad seek the good through desire, but the good through virtue, which is more natural [Boethius] |
| Full Idea: The supreme good is the goal of good men and bad men alike, and the good seek it by means of a natural activity - the exercise of virtue - while the bad strive to acquire it by means of their desires, which is not a natural way of obtaining the good. | |
| From: Boethius (The Consolations of Philosophy [c.520], IV.II) | |
| A reaction: Interesting here is the slightly surprising claim that the pursuit of virtue is 'natural', implying that the mere pursuit of desire is not. Doesn't nature have to be restrained to achieve the good? Boethius is in the tradition of Aristotle and stoicism. |
| 5759 | Varied aims cannot be good because they differ, but only become good when they unify [Boethius] |
| Full Idea: The various things that men pursue are not perfect and good, because they differ from one another; ..when they differ they are not good, but when they begin to be one they become good, so it is through the acquisition of unity that these things are good. | |
| From: Boethius (The Consolations of Philosophy [c.520], III.XI) | |
| A reaction: This is a criticism of Aristotle's pluralism about the good(s) for man. Boethius' thought is appealing, and ties in with the Socratic notion that the virtues might be unified in some way. I think it is right that true virtues merge together, ideally. |
| 5754 | You can't control someone's free mind, only their body and possessions [Boethius] |
| Full Idea: The only way one man can exercise power over another is over his body and what is inferior to it, his possessions. You cannot impose anything on a free mind. | |
| From: Boethius (The Consolations of Philosophy [c.520], II.VI) | |
| A reaction: Written, of course, in prison. Boethius had not met hypnotism, or mind-controlling drugs, or invasive brain surgery. He hadn't read '1984'. He hadn't seen 'The Ipcress File'. (In fact, he should have got out more…) |
| 16692 | Divine eternity is the all-at-once and complete possession of unending life [Boethius] |
| Full Idea: Divine eternity is the all-at-once [tota simul] and complete possession of unending life. | |
| From: Boethius (The Consolations of Philosophy [c.520], V.6), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 18.1 | |
| A reaction: This is a famous definition, and 'tota simul' became the phrase used for 'entia successiva', such as a day, or the Olympic Games. |
| 5752 | Where does evil come from if there is a god; where does good come from if there isn't? [Boethius] |
| Full Idea: A philosopher (possibly Epicurus) asked where evil comes from if there is a god, and where good comes from if there isn't. | |
| From: Boethius (The Consolations of Philosophy [c.520], I.IV) | |
| A reaction: A nice question. The best known answer to the first question is 'Satan'. Some would say that in the second case good is impossible, but I would have thought that the only possible answer is 'mankind'. |
| 5758 | God is the good [Boethius] |
| Full Idea: God is the good. | |
| From: Boethius (The Consolations of Philosophy [c.520], III.XI) | |
| A reaction: This summary follows on from the rather dubious discussion in Idea 5757. If God IS the good, it is not clear how God could be usefully described as 'good'. We would know that he was good a priori, without any enquiry into his nature being needed. |
| 5757 | God is the supreme good, so no source of goodness could take precedence over God [Boethius] |
| Full Idea: That which by its own nature is something distinct from supreme good, cannot be supreme good. ..It is impossible for anything to be by nature better than that from which it is derived, so that which is the origin of all things is supreme good. | |
| From: Boethius (The Consolations of Philosophy [c.520], III.X) | |
| A reaction: This is the contortion early Christians got into once they decided God had to be 'supreme' in the moral world (and every other world). Boethius allows a possible external source of all morality, but then has to say that this source is morally inferior. |
| 5760 | The power through which creation remains in existence and motion I call 'God' [Boethius] |
| Full Idea: For this power, whatever it is, through which creation remains in existence and in motion, I use the word which all people use, namely God. | |
| From: Boethius (The Consolations of Philosophy [c.520], III.XII) | |
| A reaction: An interesting caution in the phrase 'whatever it is'. Boethius would have been very open-minded in discussion with modern science about the stability of nature. Personally I reject Boethius' theory, but don't have a better one. Cf Idea 1431. |
| 5753 | The regular events of this life could never be due to chance [Boethius] |
| Full Idea: I could never believe that events of such regularity as we find in this life are due to the haphazards of chance. | |
| From: Boethius (The Consolations of Philosophy [c.520], I.VI) | |
| A reaction: It depends what you mean by 'chance'. Boethius infers a conscious mind, and presumes this to be God, but that is two large and unsupported steps. Modern atheists must acknowledge Boethius' problem. Why is there order? |
| 5765 | The reward of the good is to become gods [Boethius] |
| Full Idea: Goodness is happiness, ..but we agree that those who attain happiness are divine. The reward of the good, then, is to become gods. | |
| From: Boethius (The Consolations of Philosophy [c.520], IV.III) | |
| A reaction: Kant offered a similar argument (see Idea 1455). Most of us are unlikely to agree with the second premise of Boethius' argument. The idea that we might somehow become gods gripped the imagination for the next thousand years. |
| 5761 | God can do anything, but he cannot do evil, so evil must be nothing [Boethius] |
| Full Idea: 'There is nothing that an omnipotent power could not do?' 'No.' 'Then can God do evil?' 'No.' 'So evil is nothing, since that is what He cannot do who can do anthing.' | |
| From: Boethius (The Consolations of Philosophy [c.520], III.XII) | |
| A reaction: A lovely example of the contortions necessary once you insist that God must be 'omnipotent', in some absolute sense of the term. Saying that evil is 'nothing' strikes me as nothing more than a feeble attempt to insult it. |
| 5766 | If you could see the plan of Providence, you would not think there was evil anywhere [Boethius] |
| Full Idea: If you could see the plan of Providence, you would not think there was evil anywhere. | |
| From: Boethius (The Consolations of Philosophy [c.520], IV.VI) | |
| A reaction: This brings out the verificationist in me. See Idea 1467, by Antony Flew. Presumably Boethius would retain his faith as Europe moved horribly from 1939 to 1945, and even if the whole of humanity sank into squalid viciousness. |