64 ideas
| 16325 | Analysis rests on natural language, but its ideal is a framework which revises language [Halbach] |
| Full Idea: For me, although the enterprise of philosophical analysis is driven by natural language, its goal is not a linguistic analysis of English but rather an expressively strong framework that may at best be seen as a revision of English. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 12) | |
| A reaction: I agree, but the problem is that there are different ideals for the revision, which may be in conflict. Logicians, mathematicians, metaphysicians, scientists, moralists and aestheticians are queueing up to improve in their own way. |
| 6601 | Science rules the globe because of colonising power, not inherent rationality [Feyerabend] |
| Full Idea: Science now reigns supreme all over the globe; but the reason was not insight in its 'inherent rationality' but power play (the colonising nations imposed their way of living) and the need for weapons. | |
| From: Paul Feyerabend (Against Method [1975], 3), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5 | |
| A reaction: A nice clear statement of ridiculous relativism about science. What gave the colonisers their power if it was not more accurate knowledge of how to manipulate nature? |
| 16292 | An explicit definition enables the elimination of what is defined [Halbach] |
| Full Idea: Explicit definitions allow for a complete elimination of the defined notion (at least in extensional contexts). | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |
| A reaction: If the context isn't extensional (concerning the things themselves) then we could define one description of it, but be unable to eliminate it under another description. Elimination is no the aim of an Aristotelian definition. Halbach refers to truth. |
| 16307 | Don't trust analogies; they are no more than a guideline [Halbach] |
| Full Idea: Arguments from analogy are to be distrusted: at best they can serve as heuristics. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 4) |
| 16330 | Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth [Halbach] |
| Full Idea: Truth-value 'gluts' correspond to a so-called dialethic conception of truth; excluding gluts and admitting only 'gaps' leads to a conception of what is usually called 'partial' truth. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 15.2) | |
| A reaction: Talk of 'gaps' and 'gluts' seem to be the neatest way of categorising views of truth. I want a theory with no gaps or gluts. |
| 16339 | Truth axioms prove objects exist, so truth doesn't seem to be a logical notion [Halbach] |
| Full Idea: Two typed disquotation sentences, truth axioms of TB, suffice for proving that there at least two objects. Hence truth is not a logical notion if one expects logical notions to be ontologically neutral. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 21.2) |
| 16324 | Any definition of truth requires a metalanguage [Halbach] |
| Full Idea: It is plain that the distinction between object and metalanguage is required for the definability of truth. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 11) | |
| A reaction: Halbach's axiomatic approach has given up on definability, and therefore it can seek to abandon the metalanguage and examine 'type-free' theories. |
| 16293 | Traditional definitions of truth often make it more obscure, rather than less [Halbach] |
| Full Idea: A common complaint against traditional definitional theories of truth is that it is far from clear that the definiens is not more in need of clarification than the definiendum (that is, the notion of truth). | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |
| A reaction: He refers to concepts like 'correspondence', 'facts', 'coherence' or 'utility', which are said to be trickier to understand than 'true'. I suspect that philosophers like Halbach confuse 'clear' with 'precise'. Coherence is quite clear, but imprecise. |
| 16301 | If people have big doubts about truth, a definition might give it more credibility [Halbach] |
| Full Idea: If one were wondering whether truth should be considered a legitimate notion at all, a definition might be useful in dispersing doubts about its legitimacy. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 3) | |
| A reaction: Halbach is proposing to skip definitions, and try to give rules for using 'true' instead, but he doesn't rule out definitions. A definition of 'knowledge' or 'virtue' or 'democracy' might equally give those credibility. |
| 16297 | Semantic theories avoid Tarski's Theorem by sticking to a sublanguage [Halbach] |
| Full Idea: In semantic theories (e.g.Tarski's or Kripke's), a definition evades Tarski's Theorem by restricting the possible instances in the schema T[φ]↔φ to sentences of a proper sublanguage of the language formulating the equivalences. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |
| A reaction: The schema says if it's true it's affirmable, and if it's affirmable it's true. The Liar Paradox is a key reason for imposing this restriction. |
| 16337 | Disquotational truth theories are short of deductive power [Halbach] |
| Full Idea: The problem of restricted deductive power has haunted disquotational theories of truth (…because they can't prove generalisations). | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 19.5) |
| 16322 | CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA [Halbach] |
| Full Idea: Compositional Truth CT proves the consistency of Peano arithmetic, which is not provable in Peano arithmetic by Gödel's second incompleteness theorem. Hence the theory CT is not conservative over Peano arithmetic. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8.6) |
| 16294 | Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage [Halbach] |
| Full Idea: Choosing an axiomatic approach to truth might well be compatible with the view that truth is definable; the definability of truth is just not presupposed at the outset. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 1) | |
| A reaction: Is it possible that a successful axiomatisation is a successful definition? |
| 16326 | The main semantic theories of truth are Kripke's theory, and revisions semantics [Halbach] |
| Full Idea: Revision semantics is arguably the main competitor of Kripke's theory of truth among semantic truth theories. …In the former one may hope through revision to arrive at better and better models, ..sorting out unsuitable extensions of the truth predicate. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 14) | |
| A reaction: Halbach notes later that Kripke's theory (believe it or not) is considerably simpler than revision semantics. |
| 16311 | To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction' [Halbach] |
| Full Idea: If the clauses of Tarski's definition of truth are turned into axioms (as Davidson proposed) then a primitive binary predicate symbol for satisfaction is needed, as Tarski defined truth in terms of satisfaction. Standard language has a unary predicate. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 5.2) |
| 16318 | Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents [Halbach] |
| Full Idea: In the typed Compositional Truth theory CT, it is compositional because the truth of a sentence depends on the semantic values of the constituents of that sentence. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8) | |
| A reaction: [axioms on p. 65 of Halbach] |
| 16299 | Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory [Halbach] |
| Full Idea: Often syntactic objects are identified with their numerical codes. …Expressions of a countable formal language can be coded in the natural numbers. This allows a theory of truth to use Peano Arithmetic (with its results) as a base theory. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 2) | |
| A reaction: The numbering system is the famous device invented by Gödel for his great proof of incompleteness. This idea is a key to understanding modern analytic philosophy. It is the bridge which means philosophical theories can be treated mathematically. |
| 16340 | Truth axioms need a base theory, because that is where truth issues arise [Halbach] |
| Full Idea: Considering the truth axioms in the absence of a base theory is not very sensible because characteristically truth theoretic reasoning arises from the interplay of the truth axioms with the base theory. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 21.2) | |
| A reaction: The base theory usually seems to be either Peano arithmetic or set theory. We might say that introverted thought (e.g. in infants) has little use for truth; it is when you think about the world that truth becomes a worry. |
| 16305 | We know a complete axiomatisation of truth is not feasible [Halbach] |
| Full Idea: In the light of incompleteness phenomena, one should not expect a categorical axiomatisation of truth to be feasible, but this should not keep one from studying axiomatic theories of truth (or of arithmetic). | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 3) | |
| A reaction: This, of course, is because of Gödel's famous results. It is important to be aware in this field that there cannot be a dream of a final theory, so we are just seeing what can be learned about truth. |
| 16313 | A theory is 'conservative' if it adds no new theorems to its base theory [Halbach, by PG] |
| Full Idea: A truth theory is 'conservative' if the addition of the truth predicate does not add any new theorems to the base theory. | |
| From: report of Volker Halbach (Axiomatic Theories of Truth [2011], 6 Df 6.6) by PG - Db (ideas) | |
| A reaction: Halbach presents the definition more formally, and this is my attempt at getting it into plain English. Halbach uses Peano Arithmetic as his base theory, but set theory is also sometimes used. |
| 16315 | The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals [Halbach] |
| Full Idea: The truth theory TB (Tarski Biconditional) is all the axioms of Peano Arithmetic, including all instances of the induction schema with the truth predicate, plus all the sentences of the form T[φ] ↔ φ. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 7) | |
| A reaction: The biconditional formula is the famous 'snow is white' iff snow is white. The truth of the named sentence is equivalent to asserting the sentence. This is a typed theory of truth, and it is conservative over PA. |
| 16314 | Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free' [Halbach] |
| Full Idea: I sort theories of truth into the large families of 'typed' and 'type-free'. Roughly, typed theories prohibit a truth predicate's application to sentences with occurrences of that predicate, and one cannot prove the truth of sentences containing 'true'. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], II Intro) | |
| A reaction: The problem sentence the typed theories are terrified of is the Liar Sentence. Typing produces a hierarchy of languages, referring down to the languages below them. |
| 16327 | Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth [Halbach] |
| Full Idea: The Friedman-Sheard truth system FS is based on compositional theory CT. The axioms of FS are obtained by relaxing the type restriction on the CT-axioms, and adding rules inferring sentences from their truth, and vice versa. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 15) | |
| A reaction: The rules are called NEC and CONEC by Halbach. The system FSN is FS without the two rules. |
| 16329 | Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts [Halbach] |
| Full Idea: The Kripke-Feferman theory KF is an axiomatisation of the fixed points of an operator, that is, of a Kripkean fixed-point semantics with the Strong Kleene evaluation schema with truth-value gluts. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 15.1) |
| 16331 | The KF is much stronger deductively than FS, which relies on classical truth [Halbach] |
| Full Idea: The Kripke-Feferman theory is relatively deductively very strong. In particular, it is much stronger than its competitor FS, which is based on a completely classical notion of truth. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 15.3) |
| 16332 | The KF theory is useful, but it is not a theory containing its own truth predicate [Halbach] |
| Full Idea: KF is useful for explicating Peano arithmetic, but it certainly does not come to close to being a theory that contains its own truth predicate. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 16) | |
| A reaction: Since it is a type-free theory, its main philosophical aspiration was to contain its own truth predicate, so that is bad news (for philosophers). |
| 16320 | Some say deflationism is axioms which are conservative over the base theory [Halbach] |
| Full Idea: Some authors have tried to understand the deflationist claim that truth is not a substantial notion as the claim that a satisfactory axiomatisation of truth should be conservative over the base theory. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8) |
| 16338 | Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge [Halbach] |
| Full Idea: There are two doctrines at the core of deflationism. The first says truth is a device of disquotation used to express generalisations, and the second says truth is a thin notion that contributes nothing to our knowledge of the world | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 21) |
| 16317 | The main problem for deflationists is they can express generalisations, but not prove them [Halbach] |
| Full Idea: The main criticism that deflationist theories based on the disquotation sentences or similar axioms have to meet was raised by Tarski: the disquotation sentences do not allow one to prove generalisations. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 7) |
| 16316 | Deflationists say truth is just for expressing infinite conjunctions or generalisations [Halbach] |
| Full Idea: Deflationists do not hold that truth is completely dispensable. They claim that truth serves the purpose of expressing infinite conjunctions or generalisations. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 7) | |
| A reaction: It is also of obvious value as a shorthand in ordinary conversation, but rigorous accounts can paraphrase that out. 'What he said is true'. 'Pick out the true sentences from p,q,r and s' seems to mean 'affirm some of them'. What does 'affirm' mean? |
| 16319 | Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism [Halbach] |
| Full Idea: Compositional Truth CT and its variants has desirable generalisations among its logical consequences, so they seem to have ousted purely disquotational theories such as TB in the discussion on deflationism. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8) |
| 16335 | In Strong Kleene logic a disjunction just needs one disjunct to be true [Halbach] |
| Full Idea: In Strong Kleene logic a disjunction of two sentences is true if at least one disjunct is true, even when the other disjunct lacks a truth value. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 18) | |
| A reaction: This sounds fine to me. 'Either I'm typing this or Homer had blue eyes' comes out true in any sensible system. |
| 16334 | In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value [Halbach] |
| Full Idea: In Weak Kleene Logic, with truth-value gaps, a sentence is neither true nor false if one of its components lacks a truth value. A line of the truth table shows a gap if there is a gap anywhere in the line, and the other lines are classical. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 18) | |
| A reaction: This will presumably apply even if the connective is 'or', so a disjunction won't be true, even if one disjunct is true, when the other disjunct is unknown. 'Either 2+2=4 or Lot's wife was left-handed' sounds true to me. Odd. |
| 16309 | Every attempt at formal rigour uses some set theory [Halbach] |
| Full Idea: Almost any subject with any formal rigour employs some set theory. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 4.1) | |
| A reaction: This is partly because mathematics is often seen as founded in set theory, and formal rigour tends to be mathematical in character. |
| 16333 | The underestimated costs of giving up classical logic are found in mathematical reasoning [Halbach] |
| Full Idea: The costs of giving up classical logic are easily underestimated, …the price being paid in terms of mathematical reasoning. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 16.2) | |
| A reaction: No one cares much about such costs, until you say they are 'mathematical'. Presumably this is a message to Graham Priest and his pals. |
| 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. |
| 16310 | A theory is some formulae and all of their consequences [Halbach] |
| Full Idea: A theory is a set of formulae closed under first-order logical consequence. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 5.1) |
| 16342 | You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system [Halbach] |
| Full Idea: One cannot just accept that all the theorems of Peano arithmetic are true when one accepts Peano arithmetic as the notion of truth is not available in the language of arithmetic. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) | |
| A reaction: This is given as the reason why Kreisel and Levy (1968) introduced 'reflection principles', which allow you to assert whatever has been proved (with no mention of truth). (I think. The waters are closing over my head). |
| 16341 | Normally we only endorse a theory if we believe it to be sound [Halbach] |
| Full Idea: If one endorses a theory, so one might argue, one should also take it to be sound. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) |
| 16344 | Soundness must involve truth; the soundness of PA certainly needs it [Halbach] |
| Full Idea: Soundness seems to be a notion essentially involving truth. At least I do not know how to fully express the soundness of Peano arithmetic without invoking a truth predicate. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) | |
| A reaction: I suppose you could use some alternative locution such as 'assertible' or 'cuddly'. Intuitionists seem a bit vague about the truth end of things. |
| 16347 | Many new paradoxes may await us when we study interactions between frameworks [Halbach] |
| Full Idea: Paradoxes that arise from interaction of predicates such as truth, necessity, knowledge, future and past truths have receive little attention. There may be many unknown paradoxes lurking when we develop frameworks with these intensional notions. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 24.2) | |
| A reaction: Nice. This is a wonderful pointer to new research in the analytic tradition, in which formal problems will gradually iron out our metaphysical framework. |
| 16336 | The liar paradox applies truth to a negated truth (but the conditional will serve equally) [Halbach] |
| Full Idea: An essential feature of the liar paradox is the application of the truth predicate to a sentence with a negated occurrence of the truth predicate, though the negation can be avoided by using the conditional. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 19.3) |
| 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. |
| 16321 | The compactness theorem can prove nonstandard models of PA [Halbach] |
| Full Idea: Nonstandard models of Peano arithmetic are models of PA that are not isomorphic to the standard model. Their existence can be established with the compactness theorem or the adequacy theorem of first-order logic. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8.3) |
| 16343 | The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach] |
| Full Idea: The global reflection principle ∀x(Sent(x) ∧ Bew[PA](x) → Tx) …seems to be the full statement of the soundness claim for Peano arithmetic, as it expresses that all theorems of Peano arithmetic are true. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) | |
| A reaction: That is, an extra principle must be introduced to express the soundness. PA is, of course, not complete. |
| 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. |
| 16312 | To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals [Halbach] |
| Full Idea: For the reduction of Peano Arithmetic to ZF set theory, usually the set of finite von Neumann ordinals is used to represent the non-negative integers. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 6) | |
| A reaction: Halbach makes it clear that this is just one mode of reduction, relative interpretability. |
| 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. |
| 16308 | Set theory was liberated early from types, and recent truth-theories are exploring type-free [Halbach] |
| Full Idea: While set theory was liberated much earlier from type restrictions, interest in type-free theories of truth only developed more recently. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 4) | |
| A reaction: Tarski's theory of truth involves types (or hierarchies). |
| 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. |
| 16345 | That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction [Halbach] |
| Full Idea: The observation that Peano arithmetic is relatively interpretable in ZF set theory is taken by many philosophers to be a reduction of numbers to sets. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 23) | |
| A reaction: Nice! Being able to express something in a different language is not the same as a reduction. Back to the drawing board. What do you really mean by a reduction? If we model something, we don't 'reduce' it to the model. |
| 16346 | Maybe necessity is a predicate, not the usual operator, to make it more like truth [Halbach] |
| Full Idea: Should necessity be treated as a predicate rather than (as in modal logic) as a sentential operator? It is odd to assign different status to necessity and truth, hampering their interaction. That all necessities are true can't be expressed by an operator. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 24.2) | |
| A reaction: [compressed] Halbach and Horsten consistently treat truth as a predicate, but maybe truth is an operator. Making necessity a predicate and not an operator would be a huge upheaval in the world of modal logic. Nice move! |
| 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. |
| 2561 | For Feyerabend the meaning of a term depends on a whole theory [Feyerabend, by Rorty] |
| Full Idea: For Feyerabend the meaning of a term depends on a whole theory containing the term. | |
| From: report of Paul Feyerabend (Against Method [1975]) by Richard Rorty - Philosophy and the Mirror of Nature 6.3 |
| 16298 | We need propositions to ascribe the same beliefs to people with different languages [Halbach] |
| Full Idea: Being able to ascribe the same proposition as a belief to persons who do not have a common language seems to be one of the main reasons to employ propositions. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 2) | |
| A reaction: Propositions concern beliefs, as well as sentence meanings. I would want to say that a dog and I could believe the same thing, and that is a non-linguistic reason to believe in propositions. Maybe 'translation' cuts out the proposition middleman? |