30 ideas
| 21406 | Because there is only one human reason, there can only be one true philosophy from principles [Kant] |
| Full Idea: Considered objectively, there can only be one human reason, there cannot be many philosophies; in other words, there can only be one true philosophy from principles, in however many conflicting ways men have philosophised about the same proposition. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], Pref) | |
| A reaction: An idea that embodies the Enlightenment ideal. I like the idea that there is one true philosophy, because there is only one world. Kant is talking of philosophy 'from principles', which means his transendental idealism. |
| 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. |
| 16007 | I assume existence, rather than reasoning towards it [Kierkegaard] |
| Full Idea: I always reason from existence, not towards existence. | |
| From: Søren Kierkegaard (Philosophical Fragments [1844], p.40) | |
| A reaction: Kierkegaard's important premise to help show that theistic proofs for God's existence don't actually prove existence, but develop the content of a conception. [SY] |
| 16013 | Nothing necessary can come into existence, since it already 'is' [Kierkegaard] |
| Full Idea: Can the necessary come into existence? That is a change, and everything that comes into existence demonstrates that it is not necessary. The necessary already 'is'. | |
| From: Søren Kierkegaard (Philosophical Fragments [1844], p.74) | |
| A reaction: [SY] |
| 21081 | We are equipped with the a priori intuitions needed for the concept of right [Kant] |
| Full Idea: Reason has taken care that the understanding is as fully equipped as possible with a priori intuitions for the construction of the concept of right. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], Intro E) | |
| A reaction: A priori intuitions are not the same as innate knowledge or innate concepts, but they must require some sort of inbuilt inner resources. Further evidence that Kant is a rationalist philosopher (if we were unsure). |
| 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. |
| 21082 | A power-based state of nature may not be unjust, but there is no justice without competent judges [Kant] |
| Full Idea: The state of nature need not be a state of injustice merely because those who live in it treat one another in terms of power. But it is devoid of justice, for if a dispute over right occurs in it, there is no competent judge to give valid decisions. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], §44) | |
| A reaction: Could you not achieve justice by means of personal violence? Might not a revered older person have been accepted as a judge? |
| 21089 | Monarchs have the highest power; autocrats have complete power [Kant] |
| Full Idea: A monarch has the highest power, while an autocrat or absolute ruler is one who has all the power. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], §51) | |
| A reaction: If society is strictly hierarchical (like an army) then the monarch also has all the power. At the other extreme the one holding the highest power may have very little power, because so many others have their share of the power. |
| 21086 | Hereditary nobility has not been earned, and probably won't be earned [Kant] |
| Full Idea: A hereditary nobility is a distinction bestowed before it is earned, and since it gives no ground for hoping that it will be earned, it is wholly unreal and fanciful. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], §49 Gen D) | |
| A reaction: As the controller of the region of a country, a hereditary noble is the embodiment of a ruling family, which is a well established way of running things. Daft, perhaps, but there are probably worse ways of doing it. Single combat, for example. |
| 21080 | Actions are right if the maxim respects universal mutual freedoms [Kant] |
| Full Idea: Every action which by itself or by its maxim enables the freedom of each individual's will to co-exist with the freedom of everyone else in accordance with a universal law is right. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], Intro C) | |
| A reaction: This idea shows the moral basis for Kant's liberalism in politics. If all individuals acted without contact or reference to other individuals (a race of hermits) then that would appear to be optimum moral right, by this standard. |
| 21083 | Women have no role in politics [Kant] |
| Full Idea: Women in general …have no civil personality. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], §46) | |
| A reaction: In case you were wondering. This is five years after Mary Wollstonecraft's book. |
| 21407 | Equality is not being bound in ways you cannot bind others [Kant] |
| Full Idea: Our innate equality is independence from being bound by others to more than one can in turn bind them. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], Div B) | |
| A reaction: This doesn't seem to capture the whole concept. The two of us may be unequally oppressed by a third. We are unequal with the third, but also with one another, though with no binding relationships. |
| 21084 | In the contract people lose their rights, but immediately regain them, in the new commonwealth [Kant] |
| Full Idea: By the original contract all members of the people give up their external freedom in order to receive it back at once as members of a commonweath. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], §47) | |
| A reaction: This tries to give the impression that absolutely nothing is lost in the original alienation of rights. It is probably better to say that you give up one set of freedoms, which are replaced by a different (and presumably superior) set. |
| 21090 | If someone has largely made something, then they own it [Kant] |
| Full Idea: Whatever someone has himself substantially made is his own undisputed property. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], §55) | |
| A reaction: To this extent Kant offers clear agreement with Locke about a self-evident property right. Ownership of land is the controversial bit. |
| 21087 | Human life is pointless without justice [Kant] |
| Full Idea: If justice perishes, there is no further point in men living on earth. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], §49 Gen E) | |
| A reaction: I suspect that human life is also pointless if it only involves justice, and nothing else worthwhile. Are there other things so good that we might sacrifice justice to achieve them? How about maximal utilitarian happiness? |
| 21088 | Justice asserts the death penalty for murder, from a priori laws [Kant] |
| Full Idea: All murderers …must suffer the death penalty. This is what justice, as the idea of judicial power, wills in accordance with universal laws of a priori origin. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], §49 Gen E) | |
| A reaction: Illustration of how giving a principle an a priori origin puts it beyond dispute. Kant is adamant that mercy mustn't interfere with the enactment of justice. And Kant obviously rejects any consequentialist approach. Remind me what is wrong with murder? |
| 21085 | The church has a political role, by offering a supreme power over people [Kant] |
| Full Idea: The church [as opposed to religion] fulfils a genuine political necessity, for it enables the people to regard themselves as subjects of an invisible supreme power to which they must pay homage. | |
| From: Immanuel Kant (Metaphysics of Morals I: Doctrine of Right [1797], §49 Gen C) | |
| A reaction: I'm sure I remember Marx putting a different spin on this point… This idea captures the conservative attitude to established religion, at least in the UK. |