56 ideas
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| Full Idea: Definition provides us with the means for converting our intuitions into mathematically usable concepts. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-1) |
| 17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
| Full Idea: When you have proved something you know not only that it is true, but why it must be true. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-2) | |
| A reaction: Note the word 'must'. Presumably both the grounding and the necessitation of the truth are revealed. |
| 17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
| Full Idea: Set theory cannot be an axiomatic theory, because the very notion of an axiomatic theory makes no sense without it. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.413-2) | |
| A reaction: This will come as a surprise to Penelope Maddy, who battles with ways to accept the set theory axioms as the foundation of mathematics. Mayberry says that the basic set theory required is much more simple and intuitive. |
| 17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
| Full Idea: We can give a semi-categorical axiomatisation of set-theory (all that remains undetermined is the size of the set of urelements and the length of the sequence of ordinals). The system is second-order in formalisation. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.413-2) | |
| A reaction: I gather this means the models may not be isomorphic to one another (because they differ in size), but can be shown to isomorphic to some third ingredient. I think. Mayberry says this shows there is no such thing as non-Cantorian set theory. |
| 17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
| Full Idea: The (misnamed!) Axiom of Infinity expresses Cantor's fundamental assumption that the species of natural numbers is finite in size. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) |
| 17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
| Full Idea: The idea of 'generating' sets is only a metaphor - the existence of the hierarchy is established without appealing to such dubious notions. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) | |
| A reaction: Presumably there can be a 'dependence' or 'determination' relation which does not involve actual generation. |
| 17803 | Limitation of size is part of the very conception of a set [Mayberry] |
| Full Idea: Our very notion of a set is that of an extensional plurality limited in size. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.415-2) |
| 17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
| Full Idea: In the mainstream tradition of modern logic, beginning with Boole, Peirce and Schröder, descending through Löwenheim and Skolem to reach maturity with Tarski and his school ...saw logic as a branch of mathematics. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.410-1) | |
| A reaction: [The lesser tradition, of Frege and Russell, says mathematics is a branch of logic]. Mayberry says the Fregean tradition 'has almost died out'. |
| 17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
| Full Idea: First-order logic is very weak, but therein lies its strength. Its principle tools (Compactness, Completeness, Löwenheim-Skolem Theorems) can be established only because it is too weak to axiomatize either arithmetic or analysis. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.411-2) | |
| A reaction: He adds the proviso that this is 'unless we are dealing with structures on whose size we have placed an explicit, finite bound' (p.412-1). |
| 17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
| Full Idea: Second-order logic is a powerful tool of definition: by means of it alone we can capture mathematical structure up to isomorphism using simple axiom systems. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
| Full Idea: The 'logica magna' [of the Fregean tradition] has quantifiers ranging over a fixed domain, namely everything there is. In the Boolean tradition the domains differ from interpretation to interpretation. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.410-2) | |
| A reaction: Modal logic displays both approaches, with different systems for global and local domains. |
| 17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
| Full Idea: No logic which can axiomatize real analysis can have the Löwenheim-Skolem property. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
| Full Idea: The central dogma of the axiomatic method is this: isomorphic structures are mathematically indistinguishable in their essential properties. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.406-2) | |
| A reaction: Hence it is not that we have to settle for the success of a system 'up to isomorphism', since that was the original aim. The structures must differ in their non-essential properties, or they would be the same system. |
| 17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
| Full Idea: The purpose of a 'classificatory' axiomatic theory is to single out an otherwise disparate species of structures by fixing certain features of morphology. ...The aim is to single out common features. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.406-2) |
| 17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
| Full Idea: The purpose of what I am calling 'eliminatory' axiomatic theories is precisely to eliminate from mathematics those peculiar ideal and abstract objects that, on the traditional view, constitute its subject matter. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-1) | |
| A reaction: A very interesting idea. I have a natural antipathy to 'abstract objects', because they really mess up what could otherwise be a very tidy ontology. What he describes might be better called 'ignoring' axioms. The objects may 'exist', but who cares? |
| 17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
| Full Idea: No logic which can axiomatise arithmetic can be compact or complete. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) | |
| A reaction: I take this to be because there are new truths in the transfinite level (as well as the problem of incompleteness). |
| 17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
| Full Idea: We eliminate the real numbers by giving an axiomatic definition of the species of complete ordered fields. These axioms are categorical (mutually isomorphic), and thus are mathematically indistinguishable. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.408-2) | |
| A reaction: Hence my clever mathematical friend says that it is a terrible misunderstanding to think that mathematics is about numbers. Mayberry says the reals are one ordered field, but mathematics now studies all ordered fields together. |
| 17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
| Full Idea: The abstract objects of modern mathematics, the real numbers, were invented by the mathematicians of the seventeenth century in order to simplify and to generalize the Greek science of quantity. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-2) |
| 17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
| Full Idea: Quantities for Greeks were concrete things - lines, surfaces, solids, times, weights. At the centre of their science of quantity was the beautiful theory of ratio and proportion (...in which the notion of number does not appear!). | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-2) | |
| A reaction: [He credits Eudoxus, and cites Book V of Euclid] |
| 17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
| Full Idea: In Cantor's new vision, the infinite, the genuine infinite, does not disappear, but presents itself in the guise of the absolute, as manifested in the species of all sets or the species of all ordinal numbers. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) |
| 17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
| Full Idea: We may describe Cantor's achievement by saying, not that he tamed the infinite, but that he extended the finite. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) |
| 17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
| Full Idea: If we grant, as surely we must, the central importance of proof and definition, then we must also grant that mathematics not only needs, but in fact has, foundations. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-1) |
| 17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
| Full Idea: The ultimate principles upon which mathematics rests are those to which mathematicians appeal without proof; and the primitive concepts of mathematics ...themselves are grasped directly, if grasped at all, without the mediation of definition. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-1) | |
| A reaction: This begs the question of whether the 'grasping' is purely a priori, or whether it derives from experience. I defend the latter, and Jenkins puts the case well. |
| 17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
| Full Idea: An account of the foundations of mathematics must specify four things: the primitive concepts for use in definitions, the rules governing definitions, the ultimate premises of proofs, and rules allowing advance from premises to conclusions. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-2) |
| 17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
| Full Idea: No axiomatic theory, formal or informal, of first or of higher order can logically play a foundational role in mathematics. ...It is obvious that you cannot use the axiomatic method to explain what the axiomatic method is. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.415-2) |
| 17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
| Full Idea: The sole theoretical interest of first-order Peano arithmetic derives from the fact that it is a first-order reduct of a categorical second-order theory. Its axioms can be proved incomplete only because the second-order theory is categorical. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
| Full Idea: If we did not know that the second-order axioms characterise the natural numbers up to isomorphism, we should have no reason to suppose, a priori, that first-order Peano Arithmetic should be complete. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
| Full Idea: The idea that set theory must simply be identified with first-order Zermelo-Fraenkel is surprisingly widespread. ...The first-order axiomatic theory of sets is clearly inadequate as a foundation of mathematics. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-2) | |
| A reaction: [He is agreeing with a quotation from Skolem]. |
| 17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
| Full Idea: One does not have to translate 'ordinary' mathematics into the Zermelo-Fraenkel system: ordinary mathematics comes embodied in that system. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.415-1) | |
| A reaction: Mayberry seems to be a particular fan of set theory as spelling out the underlying facts of mathematics, though it has to be second-order. |
| 17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
| Full Idea: The fons et origo of all confusion is the view that set theory is just another axiomatic theory and the universe of sets just another mathematical structure. ...The universe of sets ...is the world that all mathematical structures inhabit. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.416-1) |
| 16661 | There are two sorts of category - referring to things, and to circumstances of things [Boethius] |
| Full Idea: Is it not now clear what the difference is between items in the categories? Some serve to refer to a thing, whereas others serve to refer to the circumstances of a thing. | |
| From: Boethius (Concerning the Trinity [c.518], Ch. 4), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.5 |
| 15035 | If universals are not separate, we can isolate them by abstraction [Boethius, by Panaccio] |
| Full Idea: Boethius argued that universals can be successfully isolated by abstraction, even if they do not exist as separate entities in the world. | |
| From: report of Boethius (Second Commentary on 'Isagoge' [c.517]) by Claude Panaccio - Medieval Problem of Universals 'Sources' | |
| A reaction: Personally I rather like this unfashionable view. I can't think of any other plausible explanation, unless it is a less conscious psychological process of labelling. Boethius's idea led to medieval 'immanent realism'. |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| Full Idea: The abstractness of the old fashioned real numbers has been replaced by generality in the modern theory of complete ordered fields. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.408-2) | |
| A reaction: In philosophy, I'm increasingly thinking that we should talk much more of 'generality', and a great deal less about 'universals'. (By which I don't mean that redness is just the set of red things). |
| 14665 | We can call the quality of Plato 'Platonity', and say it is a quality which only he possesses [Boethius] |
| Full Idea: Let the incommunicable property of Plato be called 'Platonity'. For we can call this quality 'Platonity' by a fabricated word, in the way in which we call the quality of man 'humanity'. Therefore this Platonity is one man's alone - Plato's. | |
| From: Boethius (Librium de interpretatione editio secunda [c.516], PL64 462d), quoted by Alvin Plantinga - Actualism and Possible Worlds 5 | |
| A reaction: Plantinga uses this idea to reinstate the old notion of a haecceity, to bestow unshakable identity on things. My interest in the quotation is that the most shocking confusions about properties arose long before the invention of set theory. |
| 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. |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
| From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
| A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |
| 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). |
| 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'. |
| 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. |
| 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. |
| 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. |