55 ideas
| 3656 | The greatest good for a state is true philosophers [Descartes] |
| Full Idea: The greatest good which can exist in a state is to have true philosophers. | |
| From: René Descartes (Principles of Philosophy [1646], Pref) | |
| A reaction: …because they understand true reality, especially the Good. |
| 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) |
| 16744 | All powers can be explained by obvious features like size, shape and motion of matter [Descartes] |
| Full Idea: There are no powers in stones and plants that are not so mysterious that they cannot be explained …from principles that are known to all and admitted by all, namely the shape, size, position, and motion of particles of matter. | |
| From: René Descartes (Principles of Philosophy [1646], IV.187), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 23.6 | |
| A reaction: This is an invocation of 'categorical' properties, against dispositions. I take this to be quite wrong. The explanation goes the other way. What supports the structures; what drives the motion; what initiates anything? |
| 5016 | Five universals: genus, species, difference, property, accident [Descartes] |
| Full Idea: The five commonly enumerated universals are: genus, species, difference, property and accident. | |
| From: René Descartes (Principles of Philosophy [1646], I.59) | |
| A reaction: Interestingly, this seems to be Descartes passing on his medieval Aristotelian inheritance, in which things are defined by placing them in a class, and then noting what distinguishes them within that class. |
| 5015 | A universal is a single idea applied to individual things that are similar to one another [Descartes] |
| Full Idea: Universals arise solely from the fact that we avail ourselves of one idea in order to think of all individual things that have a certain similitude. When we understand under the same name all the objects represented by this idea, that name is universal. | |
| From: René Descartes (Principles of Philosophy [1646], I.59) | |
| A reaction: Judging by the boldness of the pronouncement, it looks as if Descartes hasn't recognised the complexity of the problem. How do we spot a 'similarity', especially an abstraction like 'tool' or 'useful'? This sounds like Descartes trying to avoid Platonism. |
| 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). |
| 16630 | If we perceive an attribute, we infer the existence of some substance [Descartes] |
| Full Idea: Based on perceiving the presence of some attribute, we conclude there must also be present an existing thing or substance to which it can be attributed. | |
| From: René Descartes (Principles of Philosophy [1646], I.52), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 08.1 | |
| A reaction: A rainbow might be a tricky case. This illustrates the persistent belief in substances, even among philosophers who embraced the new corpuscular and mechanistic view of matter. |
| 5013 | A substance needs nothing else in order to exist [Descartes] |
| Full Idea: By substance we can understand nothing else than a thing which so exists that it needs no other thing in order to exist. | |
| From: René Descartes (Principles of Philosophy [1646], I.51) | |
| A reaction: Properties, of course, are the things which have dependent existence. Can properties be reduced to substances (e.g. by adopting a materialist theory of mind)? Note that Descartes does not think that substances depend on God for existence. |
| 16633 | A substance has one principal property which is its nature and essence [Descartes] |
| Full Idea: Each substance has one principal property that constitutes its nature and essence, to which all its other properties are referred. Extension in length, breadth, and depth constitutes the nature of corporeal substance; and thought of thinking substances. | |
| From: René Descartes (Principles of Philosophy [1646], I.53), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 08.3 | |
| A reaction: Property is likely to be 'propria', which is a property distinctive of some thing, not just any old modern property. This is quite a strikingly original view of the nature of essence. Descartes despised 'substantial forms'. |
| 3658 | Total doubt can't include your existence while doubting [Descartes] |
| Full Idea: He who decides to doubt everything cannot nevertheless doubt that he exists while he doubts. | |
| From: René Descartes (Principles of Philosophy [1646], Pref) |
| 5005 | I think, therefore I am, because for a thinking thing to not exist is a contradiction [Descartes] |
| Full Idea: There is a contradiction in conceiving that what thinks does not (at the same time as it thinks) exist. Hence this conclusion I think, therefore I am, is the first and most certain that occurs to one who philosophises in an orderly way. | |
| From: René Descartes (Principles of Philosophy [1646], I.07) | |
| A reaction: The classic statement of his argument. The significance here is that it seems to have the structure of an argument, as it involves 'philosophising', which leads to a 'contradiction', and hence to the famous conclusion. It is not just intuitive. |
| 5006 | 'Thought' is all our conscious awareness, including feeling as well as understanding [Descartes] |
| Full Idea: By the word 'thought' I understand everything we are conscious of as operating in us. And that is why not only understanding, willing, imagining, but also feeling, are here the same thing as thinking. | |
| From: René Descartes (Principles of Philosophy [1646], I.09) | |
| A reaction: There is a bit of tension here between Descartes' correct need to include feeling in thought for his Cogito argument, and his tendency to dismiss animal consciousness, on the grounds that they only sense things, and don't make judgements. |
| 5012 | 'Nothing comes from nothing' is an eternal truth found within the mind [Descartes] |
| Full Idea: The proposition 'nothing comes from nothing' is not to be considered as an existing thing, or the mode of a thing, but as a certain eternal truth which has its seat in our mind and is a common notion or axiom. | |
| From: René Descartes (Principles of Philosophy [1646], I.49) | |
| A reaction: There is a tension here, in his assertion that it is 'eternal', but 'not existing'. How does one distinguish an innate idea from an innate truth? 'Eternal' sounds like an external guarantee of truth, but being 'in our mind' sounds less reliable. |
| 5004 | We can know basic Principles without further knowledge, but not the other way round [Descartes] |
| Full Idea: It is on the Principles, or first causes, that the knowledge of other things depends, so the Principles can be known without these last, but the other things cannot reciprocally be known without the Principles. | |
| From: René Descartes (Principles of Philosophy [1646], Pref) | |
| A reaction: A particularly strong assertion of foundationalism, as it says that not only must the foundations exist, but also we must actually know them. This sounds false, as elementary knowledge then seems to require far too much sophistication. |
| 5014 | We can understand thinking occuring without imagination or sensation [Descartes] |
| Full Idea: We can understand thinking without imagination or sensation, as is quite clear to anyone who attends to the matter. | |
| From: René Descartes (Principles of Philosophy [1646], I.53) | |
| A reaction: We may certainly take it that Descartes means if it is understandable then it is logically possible. To believe that thinking could occur without imagination strikes me as an astonishing error. I take imagination to be more central than understanding. |
| 5017 | In thinking we shut ourselves off from other substances, showing our identity and separateness [Descartes] |
| Full Idea: Because each one of us understands what he thinks, and that in thinking he can shut himself off from every other substance, we may conclude that each of us is really distinct from every other thinking substance and from corporeal substance. | |
| From: René Descartes (Principles of Philosophy [1646], I.60) | |
| A reaction: This seems to be a novel argument which requires elucidation. I can 'shut myself off from every other substance'? If I shut myself off from thinking about food, does that mean hunger is not part of me? Or convince yourself that you don't have a brother? |
| 5010 | Our free will is so self-evident to us that it must be a basic innate idea [Descartes] |
| Full Idea: It is so evident that we are possessed of a free will that can give or withhold its assent, that this may be counted as one of the first and most common notions found innately in us. | |
| From: René Descartes (Principles of Philosophy [1646], I.39) | |
| A reaction: It seems to me plausible to say that we have an innate conception of our own will (our ability to make decisions), though Hume says we only learn about the will from experience, but the idea that it is absolutely 'free' might never cross our minds. |
| 5011 | There are two ultimate classes of existence: thinking substance and extended substance [Descartes] |
| Full Idea: I observe two ultimate classes of things: intellectual or thinking things, pertaining to the mind or to thinking substance, and material things, pertaining to extended substance or to body. | |
| From: René Descartes (Principles of Philosophy [1646], I.48) | |
| A reaction: This is clear confirmation that Descartes believed the mind is a substance, rather than an insubstantial world of thinking. It leaves open the possibility of a different theory: that mind is not a substance, but is a Platonic adjunct to reality. |
| 5018 | Even if tightly united, mind and body are different, as God could separate them [Descartes] |
| Full Idea: Even if we suppose God had united a body and a soul so closely that they couldn't be closer, and made a single thing out of the two, they would still remain distinct, because God has the power of separating them, or conserving out without the other. | |
| From: René Descartes (Principles of Philosophy [1646], I.60) | |
| A reaction: If Descartes lost his belief in God (after discussing existence with Kant) would he cease to be a dualist? This quotation seems to be close to conceding a mind-body relationship more like supervenience than interaction. |
| 5007 | Most errors of judgement result from an inaccurate perception of the facts [Descartes] |
| Full Idea: What usually misleads us is that we very frequently form a judgement although we do not have an accurate perception of what we judge. | |
| From: René Descartes (Principles of Philosophy [1646], I.33) | |
| A reaction: This seems to me a generally accurate observation, particularly in the making of moral judgements (which was probably not what Descartes was considering). The implication is that judgements are to a large extent forced by our perceptions. |
| 5009 | We do not praise the acts of an efficient automaton, as their acts are necessary [Descartes] |
| Full Idea: We do not praise automata, although they respond exactly to the movements they were designed to produce, since their actions are performed necessarily | |
| From: René Descartes (Principles of Philosophy [1646], I.37) | |
| A reaction: I say we attribute responsibility when we perceive something like a 'person' as causing them. We don't blame small animals, because there is 'no one at home', but we blame children as they develop a full character and identity. We can ignore free will. |
| 5008 | The greatest perfection of man is to act by free will, and thus merit praise or blame [Descartes] |
| Full Idea: That the will should extend widely accords with its nature, and it is the greatest perfection in man to be able to act by its means, that is, freely, and by so doing we are in peculiar way masters of our actions, and thereby merit praise or blame. | |
| From: René Descartes (Principles of Philosophy [1646], I.37) | |
| A reaction: This seems to me to be a deep-rooted and false understanding which philosophy has inherited from theology. It doesn't strike me that there must an absolute 'buck-stop' to make us responsible. Why is it better for a decision to appear out of nowhere? |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |
| 15987 | Physics only needs geometry or abstract mathematics, which can explain and demonstrate everything [Descartes] |
| Full Idea: I do not accept or desire any other principle in physics than in geometry or abstract mathematics, because all the phenomena of nature may be explained by their means, and sure demonstrations can be given of them. | |
| From: René Descartes (Principles of Philosophy [1646], 2.64), quoted by Peter Alexander - Ideas, Qualities and Corpuscles 7 | |
| A reaction: This is his famous and rather extreme view, which might be described as hyper-pythagoreanism (by adding geometry to numbers). It seems to leave out matter, forces and activity. |
| 12730 | We will not try to understand natural or divine ends, or final causes [Descartes] |
|
Full Idea:
We will not seek for the reason of natural things from the end which God or nature has set before him in their creation |
|
| From: René Descartes (Principles of Philosophy [1646], §28) | |
| A reaction: Teleology is more relevant to biology than to the other sciences, and it is hard to understand an eye without a notion of 'what it is for'. Planetary motion reveals nothing about purposes. If you demand a purpose, it becomes more baffling. |
| 16601 | Matter is not hard, heavy or coloured, but merely extended in space [Descartes] |
| Full Idea: The nature of matter, or body viewed as a whole, consists not in its being something which is hard, heavy, or colored, or which in any other way affects the senses, but only in its being a thing extended in length, breadth and depth. | |
| From: René Descartes (Principles of Philosophy [1646], 2.4), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 04.5 |