133 ideas
19693 | There is practical wisdom (for action), and theoretical wisdom (for deep understanding) [Aristotle, by Whitcomb] |
Full Idea: Aristotle takes wisdom to come in two forms, the practical and the theoretical, the former of which is good judgement about how to act, and the latter of which is deep knowledge or understanding. | |
From: report of Aristotle (works [c.330 BCE]) by Dennis Whitcomb - Wisdom Intro | |
A reaction: The interesting question is then whether the two are connected. One might be thoroughly 'sensible' about action, without counting as 'wise', which seems to require a broader view of what is being done. Whitcomb endorses Aristotle on this idea. |
1575 | For Aristotle logos is essentially the ability to talk rationally about questions of value [Roochnik on Aristotle] |
Full Idea: For Aristotle logos is the ability to speak rationally about, with the hope of attaining knowledge, questions of value. | |
From: comment on Aristotle (works [c.330 BCE]) by David Roochnik - The Tragedy of Reason p.26 |
1589 | Aristotle is the supreme optimist about the ability of logos to explain nature [Roochnik on Aristotle] |
Full Idea: Aristotle is the great theoretician who articulates a vision of a world in which natural and stable structures can be rationally discovered. His is the most optimistic and richest view of the possibilities of logos | |
From: comment on Aristotle (works [c.330 BCE]) by David Roochnik - The Tragedy of Reason p.95 |
10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
Full Idea: I take 'coherence' to be a primitive, intuitive notion, not reduced to something formal, and so I do not venture a rigorous definition of it. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8) | |
A reaction: I agree strongly with this. Best to talk of 'the space of reasons', or some such. Rationality extends far beyond what can be formally defined. Coherence is the last court of appeal in rational thought. |
8200 | Aristotelian definitions aim to give the essential properties of the thing defined [Aristotle, by Quine] |
Full Idea: A real definition, according to the Aristotelian tradition, gives the essence of the kind of thing defined. Man is defined as a rational animal, and thus rationality and animality are of the essence of each of us. | |
From: report of Aristotle (works [c.330 BCE]) by Willard Quine - Vagaries of Definition p.51 | |
A reaction: Compare Idea 4385. Personally I prefer the Aristotelian approach, but we may have to say 'We cannot identify the essence of x, and so x cannot be defined'. Compare 'his mood was hard to define' with 'his mood was hostile'. |
4385 | Aristotelian definition involves first stating the genus, then the differentia of the thing [Aristotle, by Urmson] |
Full Idea: For Aristotle, to give a definition one must first state the genus and then the differentia of the kind of thing to be defined. | |
From: report of Aristotle (works [c.330 BCE]) by J.O. Urmson - Aristotle's Doctrine of the Mean p.157 | |
A reaction: Presumably a modern definition would just be a list of properties, but Aristotle seeks the substance. How does he define a genus? - by placing it in a further genus? |
10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
Full Idea: An 'implicit definition' characterizes a structure or class of structures by giving a direct description of the relations that hold among the places of the structure. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
A reaction: This might also be thought of as a 'functional definition', since it seems to say what the structure or entity does, rather than give the intrinsic characteristics that make its relations and actions possible. |
10206 | Modal operators are usually treated as quantifiers [Shapiro] |
Full Idea: It is common now, and throughout the history of philosophy, to interpret modal operators as quantifiers. This is an analysis of modality in terms of ontology. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) |
10073 | There cannot be a set theory which is complete [Smith,P] |
Full Idea: By Gödel's First Incompleteness Theorem, there cannot be a negation-complete set theory. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 01.3) | |
A reaction: This means that we can never prove all the truths of a system of set theory. |
10208 | Axiom of Choice: some function has a value for every set in a given set [Shapiro] |
Full Idea: One version of the Axiom of Choice says that for every set A of nonempty sets, there is a function whose domain is A and whose value, for every a ∈ A, is a member of a. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 1) |
10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
Full Idea: If the Axiom of Choice says we can choose one member from each of a set of non-empty sets and put the chosen elements together in a set, this licenses the constructor to do an infinite amount of choosing. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3) | |
A reaction: This is one reason why the Axiom was originally controversial, and still is for many philosophers. |
10207 | Anti-realists reject set theory [Shapiro] |
Full Idea: Anti-realists reject set theory. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
A reaction: That is, anti-realists about mathematical objects. I would have thought that one could accept an account of sets as (say) fictions, which provided interesting models of mathematics etc. |
13282 | Aristotle relativises the notion of wholeness to different measures [Aristotle, by Koslicki] |
Full Idea: Aristotle proposes to relativise unity and plurality, so that a single object can be both one (indivisible) and many (divisible) simultaneously, without contradiction, relative to different measures. Wholeness has degrees, with the strength of the unity. | |
From: report of Aristotle (works [c.330 BCE]) by Kathrin Koslicki - The Structure of Objects 7.2.12 | |
A reaction: [see Koslicki's account of Aristotle for details] As always, the Aristotelian approach looks by far the most promising. Simplistic mechanical accounts of how parts make wholes aren't going to work. We must include the conventional and conceptual bit. |
10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
Full Idea: Going second-order in arithmetic enables us to prove new first-order arithmetical sentences that we couldn't prove before. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 23.4) | |
A reaction: The wages of Satan, perhaps. We can prove things about objects by proving things about their properties and sets and functions. Smith says this fact goes all the way up the hierarchy. |
10259 | The two standard explanations of consequence are semantic (in models) and deductive [Shapiro] |
Full Idea: The two best historical explanations of consequence are the semantic (model-theoretic), and the deductive versions. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.2) | |
A reaction: Shapiro points out the fictionalists are in trouble here, because the first involves commitment to sets, and the second to the existence of deductions. |
10257 | Intuitionism only sanctions modus ponens if all three components are proved [Shapiro] |
Full Idea: In some intuitionist semantics modus ponens is not sanctioned. At any given time there is likely to be a conditional such that it and its antecedent have been proved, but nobody has bothered to prove the consequent. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.7) | |
A reaction: [He cites Heyting] This is a bit baffling. In what sense can 'it' (i.e. the conditional implication) have been 'proved' if the consequent doesn't immediately follow? Proving both propositions seems to make the conditional redundant. |
10253 | Either logic determines objects, or objects determine logic, or they are separate [Shapiro] |
Full Idea: Ontology does not depend on language and logic if either one has the objects determining the logic, or the objects are independent of the logic. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.4) | |
A reaction: I favour the first option. I think we should seek an account of how logic grows from our understanding of the physical world. If this cannot be established, I shall invent a new Mad Logic, and use it for all my future reasoning, with (I trust) impunity. |
10251 | The law of excluded middle might be seen as a principle of omniscience [Shapiro] |
Full Idea: The law of excluded middle might be seen as a principle of omniscience. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3) | |
A reaction: [E.Bishop 1967 is cited] Put that way, you can see why a lot of people (such as intuitionists in mathematics) might begin to doubt it. |
4730 | For Aristotle, the subject-predicate structure of Greek reflected a substance-accident structure of reality [Aristotle, by O'Grady] |
Full Idea: Aristotle apparently believed that the subject-predicate structure of Greek reflected the substance-accident nature of reality. | |
From: report of Aristotle (works [c.330 BCE]) by Paul O'Grady - Relativism Ch.4 | |
A reaction: We need not assume that Aristotle is wrong. It is a chicken-and-egg. There is something obvious about subject-predicate language, if one assumes that unified objects are part of nature, and not just conventional. |
10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro] |
Full Idea: To some extent, every truth-functional connective differs from its counterpart in ordinary language. Classical conjunction, for example, is timeless, whereas the word 'and' often is not. 'Socrates runs and Socrates stops' cannot be reversed. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 3) | |
A reaction: Shapiro suggests two interpretations: either the classical connectives are revealing the deeper structure of ordinary language, or else they are a simplification of it. |
10075 | A 'partial function' maps only some elements to another set [Smith,P] |
Full Idea: A 'partial function' is one which maps only some elements of a domain to elements in another set. For example, the reciprocal function 1/x is not defined for x=0. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1 n1) |
10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
Full Idea: If a function f maps the argument a back to a itself, so that f(a) = a, then a is said to be a 'fixed point' for f. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 20.5) |
10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
Full Idea: The modern extensional notion of function is just an arbitrary correspondence between collections. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 1) | |
A reaction: Shapiro links this with the idea that a set is just an arbitrary collection. These minimalist concepts seem like a reaction to a general failure to come up with a more useful and common sense definition. |
10074 | A 'total function' maps every element to one element in another set [Smith,P] |
Full Idea: A 'total function' is one which maps every element of a domain to exactly one corresponding value in another set. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) |
10605 | Two functions are the same if they have the same extension [Smith,P] |
Full Idea: We count two functions as being the same if they have the same extension, i.e. if they pair up arguments with values in the same way. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 11.3) | |
A reaction: So there's only one way to skin a cat in mathematical logic. |
10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
Full Idea: The 'range' of a function is the set of elements in the output set that are values of the function for elements in the original set. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
A reaction: In other words, the range is the set of values that were created by the function. |
10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
Full Idea: The so-called Comprehension Schema ∃X∀x(Xx ↔ φ(x)) says that there is a property which is had by just those things which satisfy the condition φ. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 22.3) |
10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
Full Idea: 'Theorem': given a derivation of the sentence φ from the axioms of the theory T using the background logical proof system, we will say that φ is a 'theorem' of the theory. Standard abbreviation is T |- φ. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
10268 | Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro] |
Full Idea: Maybe plural quantifiers should themselves be understood in terms of classes (or sets). | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.4) | |
A reaction: [Shapiro credits Resnik for this criticism] |
10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
Full Idea: A 'natural deduction system' will have no logical axioms but may rules of inference. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 09.1) | |
A reaction: He contrasts this with 'Hilbert-style systems', which have many axioms but few rules. Natural deduction uses many assumptions which are then discharged, and so tree-systems are good for representing it. |
10613 | No nice theory can define truth for its own language [Smith,P] |
Full Idea: No nice theory can define truth for its own language. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 21.5) | |
A reaction: This leads on to Tarski's account of truth. |
10235 | A sentence is 'satisfiable' if it has a model [Shapiro] |
Full Idea: Normally, to say that a sentence Φ is 'satisfiable' is to say that there exists a model of Φ. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8) | |
A reaction: Nothing is said about whether the model is impressive, or founded on good axioms. Tarski builds his account of truth from this initial notion of satisfaction. |
10240 | Model theory deals with relations, reference and extensions [Shapiro] |
Full Idea: Model theory determines only the relations between truth conditions, the reference of singular terms, the extensions of predicates, and the extensions of the logical terminology. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9) |
10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
Full Idea: The central notion of model theory is the relation of 'satisfaction', sometimes called 'truth in a model'. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9) |
10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
Full Idea: A 'surjective' function is 'onto' - the whole of the output set results from the function being applied to elements of the original set. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) |
10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
Full Idea: An 'injective' function is 'one-to-one' - each element of the output set results from a different element of the original set. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
A reaction: That is, two different original elements cannot lead to the same output element. |
10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
Full Idea: A 'bijective' function has 'one-to-one correspondence' - it is both surjective and injective, so that every element in each of the original and the output sets has a matching element in the other. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
A reaction: Note that 'injective' is also one-to-one, but only in the one direction. |
10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
Full Idea: Set theorists often point out that the set-theoretical hierarchy contains as many isomorphism types as possible; that is the point of the theory. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8) | |
A reaction: Hence there are a huge number of models for any theory, which are then reduced to the one we want at the level of isomorphism. |
10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
Full Idea: No object-language theory determines its ontology by itself. The best possible is that all models are isomorphic, in which case the ontology is determined 'up to isomorphism', but only if the domain is finite, or it is stronger than first-order. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 2.5) | |
A reaction: This seems highly significant when ontological claims are being made, and is good support for Shapiro's claim that the structures matter, not the objects. There is a parallel in Tarksi's notion of truth-in-all-models. [The Skolem Paradox is the problem] |
10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
Full Idea: The Löwenheim-Skolem theorems (which apply to first-order formal theories) show that any theory with an infinite model has a model of every infinite cardinality. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8) | |
A reaction: This aspect of the theorems is the Skolem Paradox. Shapiro argues that in first-order this infinity of models for arithmetic must be accepted, but he defends second-order model theory, where 'standard' models can be selected. |
10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
Full Idea: If everything that a theory proves must be true, then it is a 'sound' theory. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) |
10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
Full Idea: Soundness is normally a matter of having true axioms and a truth-preserving proof system. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) | |
A reaction: The only exception I can think of is if a theory consisted of nothing but the axioms. |
10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
Full Idea: A theory is 'sound' iff every theorem of it is true (i.e. true on the interpretation built into its language). Soundness is normally a matter of having true axioms and a truth-preserving proof system. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
Full Idea: Logicians say that a theory T is '(negation) complete' if, for every sentence φ in the language of the theory, either φ or ¬φ is deducible in T's proof system. If this were the case, then truth could be equated with provability. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) | |
A reaction: The word 'negation' seems to be a recent addition to the concept. Presumable it might be the case that φ can always be proved, but not ¬φ. |
10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
Full Idea: A theory is 'negation complete' if it decides every sentence of its language (either the sentence, or its negation). | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
Full Idea: There is an annoying double-use of 'complete': a logic may be semantically complete, but there may be an incomplete theory expressed in it. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
Full Idea: There are two routes to Incompleteness results. One goes via the semantic assumption that we are dealing with sound theories, using a result about what they can express. The other uses the syntactic notion of consistency, with stronger notions of proof. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 18.1) |
10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
Full Idea: An 'effectively decidable' (or 'computable') algorithm will be step-by-small-step, with no need for intuition, or for independent sources, with no random methods, possible for a dumb computer, and terminates in finite steps. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.2) | |
A reaction: [a compressed paragraph] |
10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
Full Idea: A theory is 'decidable' iff there is a mechanical procedure for determining whether any sentence of its language can be proved. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) | |
A reaction: Note that it doesn't actually have to be proved. The theorems of the theory are all effectively decidable. |
10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
Full Idea: Any consistent, axiomatized, negation-complete formal theory is decidable. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 03.6) |
10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
Full Idea: A set is 'enumerable' iff either the set is empty, or there is a surjective function to the set from the set of natural numbers, so that the set is in the range of that function. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.3) |
10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
Full Idea: A finite set of finitely specifiable objects is always effectively enumerable (for example, the prime numbers). | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.4) |
10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
Full Idea: The set of ordered pairs of natural numbers (i,j) is effectively enumerable, as proven by listing them in an array (across: <0,0>, <0,1>, <0,2> ..., and down: <0,0>, <1,0>, <2,0>...), and then zig-zagging. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.5) |
10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
Full Idea: The theorems of any properly axiomatized theory can be effectively enumerated. However, the truths of any sufficiently expressive arithmetic can't be effectively enumerated. Hence the theorems and truths of arithmetic cannot be the same. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 05 Intro) |
10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
Full Idea: A set is 'effectively enumerable' if an (idealised) computer could be programmed to generate a list of its members such that any member will eventually be mentioned (even if the list is empty, or without end, or contains repetitions). | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.4) |
10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
Full Idea: Whether a property is 'expressible' in a given theory depends on the richness of the theory's language. Whether the property can be 'captured' (or 'represented') by the theory depends on the richness of the axioms and proof system. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 04.7) |
10201 | Virtually all of mathematics can be modeled in set theory [Shapiro] |
Full Idea: It is well known that virtually every field of mathematics can be reduced to, or modelled in, set theory. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
A reaction: The word 'virtually' is tantalising. The fact that something can be 'modeled' in set theory doesn't mean it IS set theory. Most weather can be modeled in a computer. |
10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
Full Idea: For prime numbers we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))). That is, the only way to multiply two numbers and a get a prime is if one of them is 1. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 04.5) |
10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
Full Idea: It has been proved (by Tarski) that the real numbers R is a complete theory. But this means that while the real numbers contain the natural numbers, the pure theory of real numbers doesn't contain the theory of natural numbers. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 18.2) |
10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro] |
Full Idea: Real numbers are either Cauchy sequences of rational numbers (interpreted as pairs of integers), or else real numbers can be thought of as Dedekind cuts, certain sets of rational numbers. So π is a Dedekind cut, or an equivalence class of sequences. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 2.5) | |
A reaction: This question is parallel to the question of whether natural numbers are Zermelo sets or Von Neumann sets. The famous problem is that there seems no way of deciding. Hence, for Shapiro, we are looking at models, not actual objects. |
18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro] |
Full Idea: There is no more to understanding the real-number structure than knowing how to use the language of analysis. .. One learns the axioms of the implicit definition. ...These determine the realtionships between real numbers. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9) | |
A reaction: This, of course, is the structuralist view of such things, which isn't really interested in the intrinsic nature of anything, but only in its relations. The slogan that 'meaning is use' seems to be in the background. |
18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
Full Idea: A Dedekind Cut is a division of rationals into two set (A1,A2) where every member of A1 is less than every member of A2. If n is the largest A1 or the smallest A2, the cut is produced by n. Some cuts aren't produced by rationals. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.4) |
10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
Full Idea: The truths of arithmetic are just the true equations involving particular numbers, and universally quantified versions of such equations. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 27.7) | |
A reaction: Must each equation be universally quantified? Why can't we just universally quantify over the whole system? |
10236 | There is no grounding for mathematics that is more secure than mathematics [Shapiro] |
Full Idea: We cannot ground mathematics in any domain or theory that is more secure than mathematics itself. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8) | |
A reaction: This pronouncement comes after a hundred years of hard work, notably by Gödel, so we'd better believe it. It might explain why Putnam rejects the idea that mathematics needs 'foundations'. Personally I'm prepare to found it in countable objects. |
10256 | For intuitionists, proof is inherently informal [Shapiro] |
Full Idea: For intuitionists, proof is inherently informal. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.7) | |
A reaction: This thought is quite appealing, so I may have to take intuitionism more seriously. It connects with my view of coherence, which I take to be a notion far too complex for precise definition. However, we don't want 'proof' to just mean 'persuasive'. |
10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
Full Idea: All numbers are related to zero by the ancestral of the successor relation. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 23.5) | |
A reaction: The successor relation only ties a number to the previous one, not to the whole series. Ancestrals are a higher level of abstraction. |
10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
Full Idea: The number of Fs is the 'successor' of the number of Gs if there is an object which is an F, and the remaining things that are F but not identical to the object are equinumerous with the Gs. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 14.1) |
10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
Full Idea: Baby Arithmetic is negation complete, so it can prove every claim (or its negation) that it can express, but it is expressively extremely impoverished. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 08.3) |
10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
Full Idea: Baby Arithmetic 'knows' the addition of particular numbers and multiplication, but can't express general facts about numbers, because it lacks quantification. It has a constant '0', a function 'S', and functions '+' and 'x', and identity and negation. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 08.1) |
10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
Full Idea: Robinson Arithmetic (Q) is not negation complete | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 08.4) |
10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
Full Idea: We can beef up Baby Arithmetic into Robinson Arithmetic (referred to as 'Q'), by restoring quantifiers and variables. It has seven generalised axioms, plus standard first-order logic. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 08.3) |
10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
Full Idea: The sequence of natural numbers starts from zero, and each number has just one immediate successor; the sequence continues without end, never circling back on itself, and there are no 'stray' numbers, lurking outside the sequence. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) | |
A reaction: These are the characteristics of the natural numbers which have to be pinned down by any axiom system, such as Peano's, or any more modern axiomatic structures. We are in the territory of Gödel's theorems. |
10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
Full Idea: The natural-number structure is a pattern common to any system of objects that has a distinguished initial object and a successor relation that satisfies the induction principle | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
A reaction: If you started your number system with 5, and successors were only odd numbers, something would have gone wrong, so a bit more seems to be needed. How do we decided whether the initial object is 0, 1 or 2? |
10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
Full Idea: If the logic of arithmetic doesn't have second-order quantifiers to range over properties of numbers, how can it handle induction? | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 10.1) |
10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
Full Idea: Putting multiplication together with addition and successor in the language of arithmetic produces incompleteness. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 10.7) | |
A reaction: His 'Baby Arithmetic' has all three and is complete, but lacks quantification (p.51) |
10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
Full Idea: Multiplication in itself isn't is intractable. In 1929 Skolem showed a complete theory for a first-order language with multiplication but lacking addition (or successor). Multiplication together with addition and successor produces incompleteness. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 10.7 n8) |
10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
Full Idea: Originally, the focus of geometry was space - matter and extension - and the subject matter of arithmetic was quantity. Geometry concerned the continuous, whereas arithmetic concerned the discrete. Mathematics left these roots in the nineteenth century. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
A reaction: Mathematicians can do what they like, but I don't think philosophers of mathematics should lose sight of these two roots. It would be odd if the true nature of mathematics had nothing whatever to do with its origin. |
10222 | Mathematical foundations may not be sets; categories are a popular rival [Shapiro] |
Full Idea: Foundationalists (e.g. Quine and Lewis) have shown that mathematics can be rendered in theories other than the iterative hierarchy of sets. A dedicated contingent hold that the category of categories is the proper foundation (e.g. Lawvere). | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3) | |
A reaction: I like the sound of that. The categories are presumably concepts that generate sets. Tricky territory, with Frege's disaster as a horrible warning to be careful. |
10218 | Baseball positions and chess pieces depend entirely on context [Shapiro] |
Full Idea: We cannot imagine a shortstop independent of a baseball infield, or a piece that plays the role of black's queen bishop independent of a chess game. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1) | |
A reaction: This is the basic thought that leads to the structuralist view of things. I must be careful because I like structuralism, but I have attacked the functionalist view in many areas, because it neglects the essences of the functioning entities. |
10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro] |
Full Idea: The even numbers and the natural numbers greater than 4 both exemplify the natural-number structure. In the former, 6 plays the 3 role, and in the latter 8 plays the 3 role. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.5) | |
A reaction: This begins to sound a bit odd. If you count the even numbers, 6 is the third one. I could count pebbles using only evens, but then presumably '6' would just mean '3'; it wouldn't be the actual number 6 acting in a different role, like Laurence Olivier. |
10228 | Could infinite structures be apprehended by pattern recognition? [Shapiro] |
Full Idea: It is contentious, to say the least, to claim that infinite structures are apprehended by pattern recognition. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1) | |
A reaction: It only seems contentious for completed infinities. The idea that the pattern continues in same way seems (pace Wittgenstein) fairly self-evident, just like an arithmetical series. |
10230 | The 4-pattern is the structure common to all collections of four objects [Shapiro] |
Full Idea: The 4-pattern is the structure common to all collections of four objects. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.2) | |
A reaction: This seems open to Frege's objection, that you can have four disparate abstract concepts, or four spatially scattered items of unknown pattern. It certainly isn't a visual pattern, but then if the only detectable pattern is the fourness, it is circular. |
10249 | The main mathematical structures are algebraic, ordered, and topological [Shapiro] |
Full Idea: According to Bourbaki, there are three main types of structure: algebraic structures, such as group, ring, field; order structures, such as partial order, linear order, well-order; topological structures, involving limit, neighbour, continuity, and space. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.5) | |
A reaction: Bourbaki is mentioned as the main champion of structuralism within mathematics. |
10273 | Some structures are exemplified by both abstract and concrete [Shapiro] |
Full Idea: Some structures are exemplified by both systems of abstracta and systems of concreta. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.2) | |
A reaction: It at least seems plausible that one might try to build a physical structure that modelled arithmetic (an abacus might be an instance), so the parallel is feasible. Then to say that the abstract arose from modelling the physical seems equally plausible. |
10276 | Mathematical structures are defined by axioms, or in set theory [Shapiro] |
Full Idea: Mathematical structures are characterised axiomatically (as implicit definitions), or they are defined in set theory. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.3) | |
A reaction: Presumably earlier mathematicians had neither axiomatised their theories, nor expressed them in set theory, but they still had a good working knowledge of the relationships. |
10270 | The main versions of structuralism are all definitionally equivalent [Shapiro] |
Full Idea: Ante rem structuralism, eliminative structuralism formulated over a sufficiently large domain of abstract objects, and modal eliminative structuralism are all definitionally equivalent. Neither is to be ontologically preferred, but the first is clearer. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.5) | |
A reaction: Since Shapiro's ontology is platonist, I would have thought there were pretty obvious grounds for making a choice between that and eliminativm, even if the grounds are intuitive rather than formal. |
10221 | Is there is no more to structures than the systems that exemplify them? [Shapiro] |
Full Idea: The 'in re' view of structures is that there is no more to structures than the systems that exemplify them. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3) | |
A reaction: I say there is more than just the systems, because we can abstract from them to a common structure, but that doesn't commit us to the existence of such a common structure. |
10248 | Number statements are generalizations about number sequences, and are bound variables [Shapiro] |
Full Idea: According to 'in re' structuralism, a statement that appears to be about numbers is a disguised generalization about all natural-number sequences; the numbers are bound variables, not singular terms. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.3.4) | |
A reaction: Any theory of anything which comes out with the thought that 'really it is a variable, not a ...' has my immediate attention and sympathy. |
10220 | Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro] |
Full Idea: Because the same structure can be exemplified by more than one system, a structure is a one-over-many. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3) | |
A reaction: The phrase 'one-over-many' is a classic Greek hallmark of a universal. Cf. Idea 10217, where Shapiro talks of arriving at structures by abstraction, through focusing and ignoring. This sounds more like a creation than a platonic universal. |
10223 | There is no 'structure of all structures', just as there is no set of all sets [Shapiro] |
Full Idea: There is no 'structure of all structures', just as there is no set of all sets. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.4) | |
A reaction: If one cannot abstract from all the structures to a higher level, why should Shapiro have abstracted from the systems/models to get the over-arching structures? |
8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend] |
Full Idea: Shapiro's structuralism champions model theory as the branch of mathematics that best describes mathematics. The essence of mathematical activity is seen as an exercise in comparing mathematical structures to each other. | |
From: report of Stewart Shapiro (Philosophy of Mathematics [1997], 4.4) by Michèle Friend - Introducing the Philosophy of Mathematics | |
A reaction: Note it 'best describes' it, rather than being foundational. Assessing whether propositional logic is complete is given as an example of model theory. That makes model theory a very high-level activity. Does it capture simple arithmetic? |
10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro] |
Full Idea: According to structuralism, someone who uses small natural numbers in everyday life presupposes an infinite structure. It seems absurd that a child who learns to count his toes applies an infinite structure to reality, and thus presupposes the structure. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.2) | |
A reaction: Shapiro says we can meet this objection by thinking of smaller structures embedded in larger ones, with the child knowing the smaller ones. |
10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro] |
Full Idea: We must distinguish between 'realism in ontology' - that mathematical objects exist - and 'realism in truth-value', which is suggested by the model-theoretic framework - that each well-formed meaningful sentence is non-vacuously either true or false. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
A reaction: My inclination is fairly strongly towards realism of the second kind, but not of the first. A view about the notion of a 'truth-maker' might therefore be required. What do the truths refer to? Answer: not objects, but abstractions from objects. |
10210 | If mathematical objects are accepted, then a number of standard principles will follow [Shapiro] |
Full Idea: One who believes in the independent existence of mathematical objects is likely to accept the law of excluded middle, impredicative definitions, the axiom of choice, extensionality, and arbitrary sets and functions. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 1) | |
A reaction: The underlying thought is that since the objects pre-exist, all of the above simply describe the relations between them, rather than having to actually bring the objects into existence. Personally I would seek a middle ground. |
10215 | Platonists claim we can state the essence of a number without reference to the others [Shapiro] |
Full Idea: The Platonist view may be that one can state the essence of each number, without referring to the other numbers. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1) | |
A reaction: Frege certainly talks this way (in his 'borehole' analogy). Fine, we are asked to spell out the essence of some number, without making reference either to any 'units' composing it, or to any other number adjacent to it or composing it. Reals? |
10233 | Platonism must accept that the Peano Axioms could all be false [Shapiro] |
Full Idea: A traditional Platonist has to face the possibility that all of the Peano Axioms are false. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.7) | |
A reaction: This would be because the objects exist independently, and so the Axioms are a mere human attempt at pinning them down. For the Formalist the axioms create the numbers, and so couldn't be false. This makes me, alas, warm to platonism! |
10244 | Intuition is an outright hindrance to five-dimensional geometry [Shapiro] |
Full Idea: Even if spatial intuition provides a little help in the heuristics of four-dimensional geometry, intuition is an outright hindrance for five-dimensional geometry and beyond. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.2) | |
A reaction: One might respond by saying 'so much the worse for five-dimensional geometry'. One could hardly abolish the subject, though, so the point must be taken. |
10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro] |
Full Idea: For each stone, there is at least one pattern such that the stone is a position in that pattern. The stone can be treated in terms of places-are-objects, or places-are-offices, to be filled with objects drawn from another ontology. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.4) | |
A reaction: I believe this is the story J.S. Mill had in mind. His view was that the structures move off into abstraction, but it is only at the empirical and physical level that we can possibly learn the structures. |
10254 | Can the ideal constructor also destroy objects? [Shapiro] |
Full Idea: Can we assume that the ideal constructor cannot destroy objects? Presumably the ideal constructor does not have an eraser, and the collection of objects is non-reducing over time. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.5) | |
A reaction: A very nice question, which platonists should enjoy. |
10255 | Presumably nothing can block a possible dynamic operation? [Shapiro] |
Full Idea: Presumably within a dynamic system, once the constructor has an operation available, then no activity can preclude the performance of the operation? | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.5) | |
A reaction: There seems to be an interesting assumption in static accounts of mathematics, that all the possible outputs of (say) a function actually exist with a theory. In an actual dynamic account, the constructor may be smitten with lethargy. |
10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro] |
Full Idea: Can we 'discover' whether a deck is really identical with its fifty-two cards, or whether a person is identical with her corresponding time-slices, molecules, or space-time points? This is like Benacerraf's problem about numbers. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997]) | |
A reaction: Shapiro is defending the structuralist view, that each of these is a model of an agreed reality, so we cannot choose a right model if they all satisfy the necessary criteria. |
10227 | The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro] |
Full Idea: The epistemic proposals of ontological realists in mathematics (such as Maddy and Resnik) has resulted in the blurring of the abstract/concrete boundary. ...Perhaps the burden is now on defenders of the boundary. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1) | |
A reaction: As Shapiro says, 'a vague boundary is still a boundary', so we need not be mesmerised by borderline cases. I would defend the boundary, with the concrete just being physical. A chair is physical, but our concept of a chair may already be abstract. |
10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro] |
Full Idea: Mathematicians use the 'abstract/concrete' label differently, with arithmetic being 'concrete' because it is a single structure (up to isomorphism), while group theory is considered more 'abstract'. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1 n1) | |
A reaction: I would say that it is the normal distinction, but they have moved the significant boundary up several levels in the hierarchy of abstraction. |
10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro] |
Full Idea: Fictionalism takes an epistemology of the concrete to be more promising than concrete-and-abstract, but fictionalism requires an epistemology of the actual and possible, secured without the benefits of model theory. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.2) | |
A reaction: The idea that possibilities (logical, natural and metaphysical) should be understood as features of the concrete world has always struck me as appealing, so I have (unlike Shapiro) no intuitive problems with this proposal. |
10277 | Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro] |
Full Idea: One result of the structuralist perspective is a healthy blurring of the distinction between mathematical and ordinary objects. ..'According to the structuralist, physical configurations often instantiate mathematical patterns'. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.4) | |
A reaction: [The quotation is from Penelope Maddy 1988 p.28] This is probably the main reason why I found structuralism interesting, and began to investigate it. |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
Full Idea: The 'ancestral' of a relation is that relation which holds when there is an indefinitely long chain of things having the initial relation. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 23.5) | |
A reaction: The standard example is spotting the relation 'ancestor' from the receding relation 'parent'. This is a sort of abstraction derived from a relation which is not equivalent (parenthood being transitive but not reflexive). The idea originated with Frege. |
10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
Full Idea: The very notion of 'object' is at least partially structural and mathematical. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.1) | |
A reaction: [In the context, Shapiro clearly has physical objects in mind] This view seems to fit with Russell's 'relational' view of the physical world, though Russell rejected structuralism in mathematics. I take abstraction to be part of perception. |
10275 | A blurry border is still a border [Shapiro] |
Full Idea: A blurry border is still a border. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.3) | |
A reaction: This remark deserves to be quoted in almost every area of philosophy, against those who attack a concept by focusing on its vague edges. Philosophers should focus on central cases, not borderline cases (though the latter may be of interest). |
13276 | The unmoved mover and the soul show Aristotelian form as the ultimate mereological atom [Aristotle, by Koslicki] |
Full Idea: Aristotle's discussion of the unmoved mover and of the soul confirms the suspicion that form, when it is not thought of as the object represented in a definition, plays the role of the ultimate mereological atom within his system. | |
From: report of Aristotle (works [c.330 BCE]) by Kathrin Koslicki - The Structure of Objects 6.6 | |
A reaction: Aristotle is concerned with which things are 'divisible', and he cites these two examples as indivisible, but they may be too unusual to offer an actual theory of how Aristotle builds up wholes from atoms. He denies atoms in matter. |
13277 | The 'form' is the recipe for building wholes of a particular kind [Aristotle, by Koslicki] |
Full Idea: Thus in Aristotle we may think of an object's formal components as a sort of recipe for how to build wholes of that particular kind. | |
From: report of Aristotle (works [c.330 BCE]) by Kathrin Koslicki - The Structure of Objects 7.2.5 | |
A reaction: In the elusive business of pinning down what Aristotle means by the crucial idea of 'form', this analogy strikes me as being quite illuminating. It would fit DNA in living things, and the design of an artifact. |
10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
Full Idea: For many philosophers the logical notions of possibility and necessity are exceptions to a general scepticism, perhaps because they have been reduced to model theory, via set theory. Thus Φ is logically possible if there is a model that satisfies it. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.1) | |
A reaction: Initially this looks a bit feeble, like an empiricist only believing what they actually see right now, but the modern analytical philosophy project seems to be the extension of logical accounts further and further into what we intuit about modality. |
10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
Full Idea: The fact that the 'myth' of possible worlds happens to produce the correct modal logic is itself a phenomenon in need of explanation. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.4) | |
A reaction: The claim that it produces 'the' correct modal logic seems to beg a lot of questions, given the profusion of modal systems. This is a problem with any sort of metaphysics which invokes fictionalism - what were those particular fictions responding to? |
5991 | For Aristotle, knowledge is of causes, and is theoretical, practical or productive [Aristotle, by Code] |
Full Idea: Aristotle thinks that in general we have knowledge or understanding when we grasp causes, and he distinguishes three fundamental types of knowledge - theoretical, practical and productive. | |
From: report of Aristotle (works [c.330 BCE]) by Alan D. Code - Aristotle | |
A reaction: Productive knowledge we tend to label as 'knowing how'. The centrality of causes for knowledge would get Aristotle nowadays labelled as a 'naturalist'. It is hard to disagree with his three types, though they may overlap. |
11239 | The notion of a priori truth is absent in Aristotle [Aristotle, by Politis] |
Full Idea: The notion of a priori truth is conspicuously absent in Aristotle. | |
From: report of Aristotle (works [c.330 BCE]) by Vassilis Politis - Aristotle and the Metaphysics 1.5 | |
A reaction: Cf. Idea 11240. |
23312 | Aristotle is a rationalist, but reason is slowly acquired through perception and experience [Aristotle, by Frede,M] |
Full Idea: Aristotle is a rationalist …but reason for him is a disposition which we only acquire over time. Its acquisition is made possible primarily by perception and experience. | |
From: report of Aristotle (works [c.330 BCE]) by Michael Frede - Aristotle's Rationalism p.173 | |
A reaction: I would describe this process as the gradual acquisition of the skill of objectivity, which needs the right knowledge and concepts to evaluate new experiences. |
16111 | Aristotle wants to fit common intuitions, and therefore uses language as a guide [Aristotle, by Gill,ML] |
Full Idea: Since Aristotle generally prefers a metaphysical theory that accords with common intuitions, he frequently relies on facts about language to guide his metaphysical claims. | |
From: report of Aristotle (works [c.330 BCE]) by Mary Louise Gill - Aristotle on Substance Ch.5 | |
A reaction: I approve of his procedure. I take intuition to be largely rational justifications too complex for us to enunciate fully, and language embodies folk intuitions in its concepts (especially if the concepts occur in many languages). |
16971 | Plato says sciences are unified around Forms; Aristotle says they're unified around substance [Aristotle, by Moravcsik] |
Full Idea: Plato's unity of science principle states that all - legitimate - sciences are ultimately about the Forms. Aristotle's principle states that all sciences must be, ultimately, about substances, or aspects of substances. | |
From: report of Aristotle (works [c.330 BCE], 1) by Julius Moravcsik - Aristotle on Adequate Explanations 1 |
11243 | Aristotelian explanations are facts, while modern explanations depend on human conceptions [Aristotle, by Politis] |
Full Idea: For Aristotle things which explain (the explanantia) are facts, which should not be associated with the modern view that says explanations are dependent on how we conceive and describe the world (where causes are independent of us). | |
From: report of Aristotle (works [c.330 BCE]) by Vassilis Politis - Aristotle and the Metaphysics 2.1 | |
A reaction: There must be some room in modern thought for the Aristotelian view, if some sort of robust scientific realism is being maintained against the highly linguistic view of philosophy found in the twentieth century. |
3320 | Aristotle's standard analysis of species and genus involves specifying things in terms of something more general [Aristotle, by Benardete,JA] |
Full Idea: The standard Aristotelian doctrine of species and genus in the theory of anything whatever involves specifying what the thing is in terms of something more general. | |
From: report of Aristotle (works [c.330 BCE]) by José A. Benardete - Metaphysics: the logical approach Ch.10 |
12000 | Aristotle regularly says that essential properties explain other significant properties [Aristotle, by Kung] |
Full Idea: The view that essential properties are those in virtue of which other significant properties of the subjects under investigation can be explained is encountered repeatedly in Aristotle's work. | |
From: report of Aristotle (works [c.330 BCE]) by Joan Kung - Aristotle on Essence and Explanation IV | |
A reaction: What does 'significant' mean here? I take it that the significant properties are the ones which explain the role, function and powers of the object. |
10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
Full Idea: The epistemological account of mathematical structures depends on the size and complexity of the structure, but small, finite structures are apprehended through abstraction via simple pattern recognition. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
A reaction: Yes! This I take to be the reason why John Stuart Mill was not a fool in his discussion of the pebbles. Successive abstractions (and fictions) will then get you to more complex structures. |
23300 | Aristotle and the Stoics denied rationality to animals, while Platonists affirmed it [Aristotle, by Sorabji] |
Full Idea: Aristotle, and also the Stoics, denied rationality to animals. …The Platonists, the Pythagoreans, and some more independent Aristotelians, did grant reason and intellect to animals. | |
From: report of Aristotle (works [c.330 BCE]) by Richard Sorabji - Rationality 'Denial' | |
A reaction: This is not the same as affirming or denying their consciousness. The debate depends on how rationality is conceived. |
10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
Full Idea: Some realists argue that simple types can be apprehended through their tokens, via abstraction. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.2) | |
A reaction: One might rephrase that to say that types are created by abstraction from tokens (and then preserved in language). |
10217 | We can apprehend structures by focusing on or ignoring features of patterns [Shapiro] |
Full Idea: One way to apprehend a particular structure is through a process of pattern recognition, or abstraction. One observes systems in a structure, and focuses attention on the relations among the objects - ignoring features irrelevant to their relations. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1) | |
A reaction: A lovely statement of the classic Aristotelian abstractionist approach of focusing-and-ignoring. But this is made in 1997, long after Frege and Geach ridiculed it. It just won't go away - not if you want a full and unified account of what is going on. |
9554 | We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro] |
Full Idea: One can observe a system and focus attention on the relations among the objects - ignoring those features of the objects not relevant to the system. For example, we can understand a baseball defense system by going to several games. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], p.74), quoted by Charles Chihara - A Structural Account of Mathematics | |
A reaction: This is Shapiro perpetrating precisely the wicked abstractionism which Frege and Geach claim is ridiculous. Frege objects that abstract concepts then become private, but baseball defences are discussed in national newspapers. |
10231 | Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro] |
Full Idea: Perhaps we can introduce abstract objects by abstraction over an equivalence relation on a base class of entities, just as Frege suggested that 'direction' be obtained from parallel lines. ..Properties must be equinumerous, but need not be individuated. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.5) | |
A reaction: [He cites Hale and Wright as the originators of this} It is not entirely clear why this is 'abstraction', rather than just drawing attention to possible groupings of entities. |
11240 | The notion of analytic truth is absent in Aristotle [Aristotle, by Politis] |
Full Idea: The notion of analytic truth is conspicuously absent in Aristotle. | |
From: report of Aristotle (works [c.330 BCE]) by Vassilis Politis - Aristotle and the Metaphysics 1.5 | |
A reaction: Cf. Idea 11239. |
6559 | Aristotle never actually says that man is a rational animal [Aristotle, by Fogelin] |
Full Idea: To the best of my knowledge (and somewhat to my surprise), Aristotle never actually says that man is a rational animal; however, he all but says it. | |
From: report of Aristotle (works [c.330 BCE]) by Robert Fogelin - Walking the Tightrope of Reason Ch.1 | |
A reaction: When I read this I thought that this database would prove Fogelin wrong, but it actually supports him, as I can't find it in Aristotle either. Descartes refers to it in Med.Two. In Idea 5133 Aristotle does say that man is a 'social being'. But 22586! |
11150 | It is the mark of an educated mind to be able to entertain an idea without accepting it [Aristotle] |
Full Idea: It is the mark of an educated mind to be able to entertain an idea without accepting it. | |
From: Aristotle (works [c.330 BCE]) | |
A reaction: The epigraph on a David Chalmers website. A wonderful remark, and it should be on the wall of every beginners' philosophy class. However, while it is in the spirit of Aristotle, it appears to be a misattribution with no ancient provenance. |
3037 | Aristotle said the educated were superior to the uneducated as the living are to the dead [Aristotle, by Diog. Laertius] |
Full Idea: Aristotle was asked how much educated men were superior to those uneducated; "As much," he said, "as the living are to the dead." | |
From: report of Aristotle (works [c.330 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 05.1.11 |
8660 | There are potential infinities (never running out), but actual infinity is incoherent [Aristotle, by Friend] |
Full Idea: Aristotle developed his own distinction between potential infinity (never running out) and actual infinity (there being a collection of an actual infinite number of things, such as places, times, objects). He decided that actual infinity was incoherent. | |
From: report of Aristotle (works [c.330 BCE]) by Michèle Friend - Introducing the Philosophy of Mathematics 1.3 | |
A reaction: Friend argues, plausibly, that this won't do, since potential infinity doesn't make much sense if there is not an actual infinity of things to supply the demand. It seems to just illustrate how boggling and uncongenial infinity was to Aristotle. |
12058 | Aristotle's matter can become any other kind of matter [Aristotle, by Wiggins] |
Full Idea: Aristotle's conception of matter permits any kind of matter to become any other kind of matter. | |
From: report of Aristotle (works [c.330 BCE]) by David Wiggins - Substance 4.11.2 | |
A reaction: This is obviously crucial background information when we read Aristotle on matter. Our 92+ elements, and fixed fundamental particles, gives a quite different picture. Aristotle would discuss form and matter quite differently now. |
22729 | The concepts of gods arose from observing the soul, and the cosmos [Aristotle, by Sext.Empiricus] |
Full Idea: Aristotle said that the conception of gods arose among mankind from two originating causes, namely from events which concern the soul and from celestial phenomena. | |
From: report of Aristotle (works [c.330 BCE], Frag 10) by Sextus Empiricus - Against the Physicists (two books) I.20 | |
A reaction: The cosmos suggests order, and possible creation. What do events of the soul suggest? It doesn't seem to be its non-physical nature, because Aristotle is more of a functionalist. Puzzling. (It says later that gods are like the soul). |