33 ideas
2098 | The principle of sufficient reason is needed if we are to proceed from maths to physics [Leibniz] |
Full Idea: In order to proceed from mathematics to physics the principle of sufficient reason is necessary, that nothing happens without there being a reason why it should be thus rather than otherwise. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], §2) |
2104 | No reason could limit the quantity of matter, so there is no limit [Leibniz] |
Full Idea: There is no possible reason which could limit the quantity of matter; therefore there cannot in fact be any such limitation. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 4.21) |
3646 | There is always a reason why things are thus rather than otherwise [Leibniz] |
Full Idea: Nothing happens without a sufficient reason why it should be thus rather than otherwise. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 3.2) |
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. |
19385 | All simply substances are in harmony, because they all represent the one universe [Leibniz] |
Full Idea: All simple substances will always have a harmony among themselves, because they always represent the same universe. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], V §91), quoted by Richard T.W. Arthur - Leibniz | |
A reaction: We can accept that the universe itself does not contain contradictions (how could it), but it is a leap of faith to say that all monads represent the universe well enough to avoid contradictions. Maps can contradict one another. |
21346 | The ratio between two lines can't be a feature of one, and cannot be in both [Leibniz] |
Full Idea: If the ratio of two lines L and M is conceived as abstracted from them both, without considering which is the subject and which the object, which will then be the subject? We cannot say both, for then we should have an accident in two subjects. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 5th Paper, §47), quoted by John Heil - Relations 'External' | |
A reaction: [compressed] Leibniz is rejecting external relations as having any status in ontology. It looks like a mistake (originating in Aristotle) to try to shoehorn the ontology of relations into the substance-properties framework. |
10197 | An immanent universal is wholly present in more than one place [Zimmerman,DW] |
Full Idea: An immanent universal will routinely be 'at some distance from itself', in the sense that it is wholly present in more than one place. | |
From: Dean W. Zimmerman (Distinct Indiscernibles and the Bundle Theory [1997], p.306) | |
A reaction: This is the Aristotelian view, which sounds distinctly implausible in this formulation. Though I suppose redness is wholly present in a tomato, in the way that fourness is wholly present in the Horsemen of the Apocalypse. How many rednesses are there? |
10198 | If only two indiscernible electrons exist, future differences must still be possible [Zimmerman,DW] |
Full Idea: If nothing existed except two electrons, which are indiscernible, it remains possible that differences will emerge later. Even if this universe has eternal symmetry, such differences are still logically, metaphysically, physically and causally possible. | |
From: Dean W. Zimmerman (Distinct Indiscernibles and the Bundle Theory [1997], p.306) | |
A reaction: The question then is whether the two electrons have hidden properties that make differences possible. Zimmerman assumes that 'laws' of an indeterministic kind will do the job. I doubt that. Can differences be discerned after the event? |
10199 | Discernible differences at different times may just be in counterparts [Zimmerman,DW] |
Full Idea: Possible differences which may later become discernible could be treated as differences in a counterpart, which is similar to, but not identical with, the original object. | |
From: Dean W. Zimmerman (Distinct Indiscernibles and the Bundle Theory [1997], p.307) | |
A reaction: [compressed] This is a reply to Idea 10198, which implies that two things could never be indiscernible over time, because of their different possibilities. One must then decide issues about rigid designation and counterparts. |
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. |
2105 | Things are infinitely subdivisible and contain new worlds, which atoms would make impossible [Leibniz] |
Full Idea: The least corpuscle is actually subdivided ad infinitum and contains a world of new created things, which this universe would lack if this corpuscle were an atom. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 4.PS) |
2106 | The only simple things are monads, with no parts or extension [Leibniz] |
Full Idea: According to me there is nothing simple except true monads, which have no parts and no extensions. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 5.24) |
2102 | Atomism is irrational because it suggests that two atoms can be indistinguishable [Leibniz] |
Full Idea: There are no two individuals indiscernible from one another - leaves, or drops of water, for example. This is an argument against atoms, which, like the void, are opposed to the principles of a true metaphysic. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 4.04) |
20965 | Leibniz upheld conservations of momentum and energy [Leibniz, by Papineau] |
Full Idea: In place of Descartes's conservation of 'quantity of motion', Leibniz upheld both the conservation of linear momentum and the conservation of kinetic energy. | |
From: report of Gottfried Leibniz (Letters to Samuel Clarke [1716], 5th paper) by David Papineau - Thinking about Consciousness App 2 | |
A reaction: The point is that momentum involves velocity (which includes direction) rather than speed. Leibniz more or less invented the concept of 'energy' ('vis viva'). Papineau says these two leave no room for causation by mental substance. |
2103 | The idea that the universe could be moved forward with no other change is just a fantasy [Leibniz] |
Full Idea: To say that God could cause the universe to move forward in a straight line or otherwise without changing it in any other way is another fanciful supposition. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 4.14) |
2100 | Space and time are purely relative [Leibniz] |
Full Idea: I have more than once stated that I held space to be something purely relative, like time. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 3.4) |
2107 | No time exists except instants, and instants are not even a part of time, so time does not exist [Leibniz] |
Full Idea: How could a thing exist, no part of which ever exists? In the case of time, nothing exists but instants, and an instant is not even a part of time. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 5.49) |
2101 | If everything in the universe happened a year earlier, there would be no discernible difference [Leibniz] |
Full Idea: To ask why God did not make everything a year sooner would be reasonable if time were something apart from temporal things, but time is just the succession of things, which remains the same if the universe is created a year sooner. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], 3.6) |
22894 | If time were absolute that would make God's existence dependent on it [Leibniz, by Bardon] |
Full Idea: Leibniz argues that if time is a thing in itself, and God is 'in' time, then God would be dependent for His existence on the existence of time. | |
From: report of Gottfried Leibniz (Letters to Samuel Clarke [1716]) by Adrian Bardon - Brief History of the Philosophy of Time 3 'Newton' | |
A reaction: Hence Leibniz says time is merely relations between events. Not sure what he thinks an event is. What is God made of? Is there some divine matter upon which God's existence must depend? |
2099 | The existence of God, and all metaphysics, follows from the Principle of Sufficient Reason [Leibniz] |
Full Idea: By this principle alone, that there must be a sufficient reason why things are thus rather than otherwise, I prove the existence of the Divinity, and all the rest of metaphysics or natural theology. | |
From: Gottfried Leibniz (Letters to Samuel Clarke [1716], §2) |