40 ideas
| 5035 | The two basics of reasoning are contradiction and sufficient reason [Leibniz] |
| Full Idea: The two first principles of reasoning are: the principle of contradiction, and the principle of the need for giving a reason. | |
| From: Gottfried Leibniz (A Specimen of Discoveries [1686], p.75) | |
| A reaction: Could animals have any reasoning ability (say, in solving a physical problem)? Leibniz's criteria both require language. Note the overlapping of the principle of sufficient reason (there IS a reason) with the contractual idea of GIVING reasons. |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 16588 | I prefer a lack of form to mean non-existence, than to think of some quasi-existence [Augustine] |
| Full Idea: I sooner judged that what lacks all form does not exist, than thought of as something in between form and nothing, neither formed nor nothing, unformed and next to nothing. | |
| From: Augustine (Confessions [c.398], XII.6), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 03.1 | |
| A reaction: Scholastics were struck by the contrast between this remark, and the remark of Averroes (Idea 16587) that prime matter was halfway existence. Their two great authorities disagreed! This sort of thing stimulated the revival of metaphysics. |
| 22979 | Three main questions seem to be whether a thing is, what it is, and what sort it is [Augustine] |
| Full Idea: I am told that I can ask three sorts of questions - whether a thing is, what it is, and what sort it is. | |
| From: Augustine (Confessions [c.398], X.10) | |
| A reaction: This seems to be a very Aristotelian approach. I am pleased to see that what it is and what sort it is are not conflated. The first one must be its individual essence, and the second its generic essence. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 22981 | Mind and memory are the same, as shown in 'bear it in mind' or 'it slipped from mind' [Augustine] |
| Full Idea: The mind and the memory are one and the same. We even call the memory the mind, for when we tell a person to remember something, we tell them to 'bear this in mind', and when we forget something 'it slipped out of my mind'. | |
| From: Augustine (Confessions [c.398], X.14) | |
| A reaction: This idea has become familiar in modern neuroscience, I think, presumably because we do not find distinct types of neurons for consciousness and for memory. |
| 22980 | Memory contains innumerable principles of maths, as well as past sense experiences [Augustine] |
| Full Idea: The memory contains the innumerable principles and laws of numbers and dimensions. None of these can have been conveyed to me by the bodily senses. | |
| From: Augustine (Confessions [c.398], X.12) | |
| A reaction: Even if you have a fairly empirical view of the sources of mathematics (a view with which I sympathise), it must by admitted that our endless extrapolations from the sources also reside in memory. So we remember thoughts as well as experiences. |
| 22983 | We would avoid remembering sorrow or fear if that triggered the emotions afresh [Augustine] |
| Full Idea: If we had to experience sorrow or fear every time that we mentioned these emotions, no one would be willing to speak of them. | |
| From: Augustine (Confessions [c.398], X.14) | |
| A reaction: Remembering the death of a loved one can trigger fresh grief, but remembering their dangerous illness from which they recovered no longer contains the feeling of fear. |
| 22977 | I can distinguish different smells even when I am not experiencing them [Augustine] |
| Full Idea: I can distinguish the scent of lilies from that of violets, even though there is no scent at all in my nostrils. | |
| From: Augustine (Confessions [c.398], X.08) | |
| A reaction: Augustine has a nice introspective account of how we experience memory, and identifies lots of puzzling features. I know I can identify the smell of vinegar, but I can't bring it to mind, the way I can the appearance of roses. |
| 22982 | Why does joy in my mind make me happy, but joy in my memory doesn't? [Augustine] |
| Full Idea: How can it be that my mind can be happy because of the joy that is in it, and yet my memory is not sad by reason of the sadness that is in it? | |
| From: Augustine (Confessions [c.398], X.14) | |
| A reaction: This seems to contradict his thought in Idea 22981, that memory and mind are the same. Recall seems to be a part of consciousness which is not fully wired up to the rest of the mind. |
| 22978 | Memory is so vast that I cannot recognise it as part of my mind [Augustine] |
| Full Idea: The memory is a vast immeasurable sanctuary. It is part of my nature, but I cannot understand all that I am. Hence the mind is too narrow to contain itself entirely. Is the other part outside of itself, and not within it? How then can it be a part? | |
| From: Augustine (Confessions [c.398], X.08) | |
| A reaction: He seems to understand the mind as entirely consisting of consciousness. Nevertheless, this seems to be the first inklings of the modern externalist view of the mind. |
| 22984 | Without memory I could not even speak of myself [Augustine] |
| Full Idea: I do not understand the power of memory that is in myself, although without it I could not even speak of myself. | |
| From: Augustine (Confessions [c.398], X.16) | |
| A reaction: Even if the self is not identical with memory, this idea seems to establish that memory is an essential aspect of the self. This point is neglected by those who see the self as an entity (the 'soul pearl') which persists through all experience. |
| 5982 | If the future does not exist, how can prophets see it? [Augustine] |
| Full Idea: How do prophets see the future, if there is not a future to be seen? | |
| From: Augustine (Confessions [c.398], XI.17) | |
| A reaction: The answer, I suspect, is that prophets can't see the future. The prospect that the future already exists would seem to saboutage human freedom and responsibility, and point to Calvinist predestination, and even fatalism. |
| 5038 | Assume that mind and body follow their own laws, but God has harmonised them [Leibniz] |
| Full Idea: Why not assume that God initially created the soul and body with so much ingenuity that, whilst each follows its own laws and properties and operations, all thing agree most beautifull among themselves? This is the 'hypothesis of concomitance'. | |
| From: Gottfried Leibniz (A Specimen of Discoveries [1686], p.80) | |
| A reaction: They may be in beautifully planned harmony, but how do we know that they are in harmony? Presumably their actions must be compared, and God would even have to harmonise the comparison. Parallelism seems to imply epiphenomenalism or idealism. |
| 22976 | Memories are preserved separately, according to category [Augustine] |
| Full Idea: In memory everything is preserved separately, according to its category. | |
| From: Augustine (Confessions [c.398], X.08) | |
| A reaction: This strikes me as the first seeds of the idea that the mind functions by means of mental files. Our memories of cats are 'close to' or 'linked to' our memories of dogs. |
| 22985 | Everyone wants happiness [Augustine] |
| Full Idea: Surely happiness is what everyone wants, so much so that there can be none who do not want it? | |
| From: Augustine (Confessions [c.398], X.20) | |
| A reaction: His concept of happiness is, of course, religious. Occasionally you meet habitual grumblers about life who give the impression that they are only happy when they are discontented. So happiness is achieving desires, not feeling good? |
| 5984 | Maybe time is an extension of the mind [Augustine] |
| Full Idea: I begin to wonder whether time is an extension of the mind itself. | |
| From: Augustine (Confessions [c.398], XI.26) | |
| A reaction: The observation that the mind creates a 'specious present' (spreading experience out over a short fraction of second) reinforces this. Personally I like David Marshall's proposal that consciousness is entirely memory, which would deny this idea. |
| 22888 | To be aware of time it can only exist in the mind, as memory or anticipation [Augustine, by Bardon] |
| Full Idea: Augustine answers that for us to be aware of time it must exist only in the mind, …and the difference between past and future is just the difference between memory and anticipation. | |
| From: report of Augustine (Confessions [c.398]) by Adrian Bardon - Brief History of the Philosophy of Time 1 'Augustine's' | |
| A reaction: This is an extreme idealist view. Are we to say that the past consists only of what can be remembered, and the future only of what is anticipated? Absurd anti-realism, in my view. Where do his concepts come from, asks Le Poidevin. |
| 5980 | How can ten days ahead be a short time, if it doesn't exist? [Augustine] |
| Full Idea: A short time ago or a short time ahead we might put at ten days, but how can anything which does not exist be either long or short? | |
| From: Augustine (Confessions [c.398], XI.15) | |
| A reaction: A nice question, which gets at the paradoxical nature of time very nicely. How can it be long, but non-existent? We could break the paradox by concluding '..and therefore time does exist', even though we can't see how. |
| 5979 | If the past is no longer, and the future is not yet, how can they exist? [Augustine] |
| Full Idea: Of the three divisions of time, how can two, the past and the future, be, when the past no longer is, and the future is not yet? | |
| From: Augustine (Confessions [c.398], XI.14) | |
| A reaction: This is the oldest bewilderment about time, which naturally leads us to the thought that time cannot actually 'exist'. The remark implies that at least 'now' is safe, but that also succumbs to paradox pretty quickly. |
| 5981 | The whole of the current year is not present, so how can it exist? [Augustine] |
| Full Idea: We cannot say that the whole of the current year is present, and if the whole of it is not present, the year is not present. | |
| From: Augustine (Confessions [c.398], XI.15) | |
| A reaction: Another nice way of presenting the paradox of time. We are in a particular year, so it has to be real. |
| 5978 | I know what time is, until someone asks me to explain it [Augustine] |
| Full Idea: I know well enough what time is, provided that nobody asks me; but if I am asked what it is and try to explain, I am baffled. | |
| From: Augustine (Confessions [c.398], XI.14) | |
| A reaction: A justly famous remark, even though it adds nothing to our knowledge of time. This sort of thought pushes us towards accepting many things as axiomatic, such as time, space, identity, persons, mind. |
| 5983 | I disagree with the idea that time is nothing but cosmic movement [Augustine] |
| Full Idea: I once heard a learned man say that time is nothing but the movement of the sun and the moon and the stars, but I do not agree. | |
| From: Augustine (Confessions [c.398], XI.22) | |
| A reaction: It is tempting to say that you either take time or movement as axiomatic, and describe one in terms of the other, but you are stuck unable to give the initial statement of the axiom without mentioning the second property you were saving for later. |
| 5977 | Heaven and earth must be created, because they are subject to change [Augustine] |
| Full Idea: The fact that heaven and earth are there proclaims that they were created, for they are subject to change and variation; ..the meaning of change and variation is that something is there which was not there before. | |
| From: Augustine (Confessions [c.398], XI.04) | |
| A reaction: It seems possible that the underlying matter is eternal (as in various conservation laws, such as that of energy), and that all change is in the form rather than the substance. |
| 22887 | If God existed before creation, why would a perfect being desire to change things? [Augustine, by Bardon] |
| Full Idea: If nothing existed by God before creation, then what could have happened to, or within, God that led God to decide to create the universe at that particular moment? Why would an eternal or perfect being want or need to change? | |
| From: report of Augustine (Confessions [c.398]) by Adrian Bardon - Brief History of the Philosophy of Time 1 'Augustine's' | |
| A reaction: I suppose you could reply that change is superior to stasis, but then why did God delay the creation? |
| 5976 | If God is outside time in eternity, can He hear prayers? [Augustine] |
| Full Idea: O Lord, since you are outside time in eternity, are you unaware of the things that I tell you? | |
| From: Augustine (Confessions [c.398], XI.01) | |
| A reaction: This strikes me as the single most difficult and most elusive question about the nature of a supreme divine being. If the being is trapped in time, as we are, it is greatly diminished, and if it is outside, it is hard to see how it could be a participant. |
| 5037 | God doesn't decide that Adam will sin, but that sinful Adam's existence is to be preferred [Leibniz] |
| Full Idea: God does not decide whether Adam should sin, but whether that series of things in which there is an Adam whose perfect individual notion involves sin should nevertheless be preferred to others. | |
| From: Gottfried Leibniz (A Specimen of Discoveries [1686], p.78) | |
| A reaction: Compare whether the person responsible for setting a road speed limit is responsible for subsequent accidents. Leibniz's belief that the world could have been made no better than it is (by an omnipotent being) strikes me as blind faith, not an argument. |