99 ideas
| 12274 | Begin examination with basics, and subdivide till you can go no further [Aristotle] |
| Full Idea: The examination must be carried on and begin from the primary classes and then go on step by step until further division is impossible. | |
| From: Aristotle (Topics [c.331 BCE], 109b17) | |
| A reaction: This is a good slogan for the analytic approach to thought. I take Aristotle (or possibly Socrates) to be the father of analysis, not Frege (though see Idea 9840). (He may be thinking of the tableau method of proof). |
| 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. |
| 12260 | Dialectic starts from generally accepted opinions [Aristotle] |
| Full Idea: Reasoning is dialectical which reasons from generally accepted opinions. | |
| From: Aristotle (Topics [c.331 BCE], 100a30) | |
| A reaction: This is right at the heart of Aristotle's philosophical method, and Greek thinking generally. There are nice modern debates about 'folk' understanding, derived from science (e.g. quantum theory) which suggest that starting from normal views is a bad idea. |
| 12291 | There can't be one definition of two things, or two definitions of the same thing [Aristotle] |
| Full Idea: There cannot possibly be one definition of two things, or two definitions of the same thing. | |
| From: Aristotle (Topics [c.331 BCE], 154a11) | |
| A reaction: The second half of this is much bolder and more controversial, and plenty of modern thinkers would flatly reject it. Are definitions contextual, that is, designed for some specific human purpose. Must definitions be of causes? |
| 12292 | Definitions are easily destroyed, since they can contain very many assertions [Aristotle] |
| Full Idea: A definition is the easiest of all things to destroy; for, since it contains many assertions, the opportunities which it offers are very numerous, and the more abundant the material, the more quickly the reasoning can set to work. | |
| From: Aristotle (Topics [c.331 BCE], 155a03) | |
| A reaction: I quote this to show that Aristotle expected many definitions to be very long affairs (maybe even of book length?) |
| 12272 | We describe the essence of a particular thing by means of its differentiae [Aristotle] |
| Full Idea: We usually isolate the appropriate description of the essence of a particular thing by means of the differentiae which are peculiar to it. | |
| From: Aristotle (Topics [c.331 BCE], 108b05) | |
| A reaction: I take this to be important for showing the definition is more than mere categorisation. A good definition homes in the particular, by gradually narrowing down the differentiae. |
| 12279 | The differentia indicate the qualities, but not the essence [Aristotle] |
| Full Idea: No differentia indicates the essence [ti estin], but rather some quality, such as 'pedestrian' or 'biped'. | |
| From: Aristotle (Topics [c.331 BCE], 122b17) | |
| A reaction: We must disentangle this, since essence is what is definable, and definition seems to give us the essence, and yet it appears that definition only requires genus and differentia. Differentiae seem to be both generic and fine-grained. See Idea 12280! |
| 12283 | In definitions the first term to be assigned ought to be the genus [Aristotle] |
| Full Idea: In definitions the first term to be assigned ought to be the genus. | |
| From: Aristotle (Topics [c.331 BCE], 132a12) | |
| A reaction: We mustn't be deluded into thinking that nothing else is required. I take the increasing refinement of differentiae to be where the real action is. The genus gives you 70% of the explanation. |
| 12289 | The genera and the differentiae are part of the essence [Aristotle] |
| Full Idea: The genera and the differentiae are predicated in the category of essence. | |
| From: Aristotle (Topics [c.331 BCE], 153a19) | |
| A reaction: The definition is words, and the essence is real, so our best definition might not fully attain to the essence. Aristotle has us reaching out to the world through our definitions. |
| 12261 | Differentia are generic, and belong with genus [Aristotle] |
| Full Idea: The differentia, being generic in character, should be ranged with the genus. | |
| From: Aristotle (Topics [c.331 BCE], 101b18) | |
| A reaction: This does not mean that naming the differentia amounts to mere classification. I presume we can only state individual differences by using a language which is crammed full of universals. |
| 12263 | 'Genus' is part of the essence shared among several things [Aristotle] |
| Full Idea: A 'genus' is that which is predicated in the category of essence of several things which differ in kind. | |
| From: Aristotle (Topics [c.331 BCE], 102a32) | |
| A reaction: Hence a genus is likely to be expressed by a universal, a one-over-many. A particular will be a highly individual collection of various genera, but what ensures the uniqueness of each thing, if they are indiscernible? |
| 12285 | The definition is peculiar to one thing, not common to many [Aristotle] |
| Full Idea: The definition ought to be peculiar to one thing, not common to many. | |
| From: Aristotle (Topics [c.331 BCE], 149b24) | |
| A reaction: I take this to be very important, against those who think that definition is no more than mere categorisation. To explain, you must get down to the level of the individual. We must explain that uniquely docile tiger. |
| 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) |
| 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. |
| 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. |
| 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. |
| 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. |
| 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] |
| 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) |
| 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] |
| 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. |
| 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. |
| 11261 | Puzzles arise when reasoning seems equal on both sides [Aristotle] |
| Full Idea: The equality of opposite reasonings is the cause of aporia; for it is when we reason on both [sides of a question] and it appears to us that everything can come about either way, that we are in a state of aporia about which of the two ways to take up. | |
| From: Aristotle (Topics [c.331 BCE], 145b17), quoted by Vassilis Politis - Aristotle and the Metaphysics 3.1 | |
| A reaction: Other philosophers give up on the subject in this situation, but I love Aristotle because he takes this to be the place where philosophy begins. |
| 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. |
| 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) |
| 12273 | Unit is the starting point of number [Aristotle] |
| Full Idea: They say that the unit [monada] is the starting point of number (and the point the starting-point of a line). | |
| From: Aristotle (Topics [c.331 BCE], 108b30) | |
| A reaction: Yes, despite Frege's objections in the early part of the 'Grundlagen' (1884). I take arithmetic to be rooted in counting, despite all abstract definitions of number by Frege and Dedekind. Identity gives the unit, which is countable. See also Topics 141b9 |
| 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'. |
| 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? |
| 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. |
| 12267 | There are ten categories: essence, quantity, quality, relation, place, time, position, state, activity, passivity [Aristotle] |
| Full Idea: The four main types of predicates fall into ten categories: essence, quantity, quality, relation, place, time, position, state, activity, passivity. | |
| From: Aristotle (Topics [c.331 BCE], 103b20) | |
| A reaction: These are the standard ten categories of Aristotle. He is notable for the divisions not being sharp, and ten being a rough total. He is well aware of the limits of precision in such matters. |
| 12282 | An individual property has to exist (in past, present or future) [Aristotle] |
| Full Idea: If it does not at present exist, or, if it has not existed in the past, or if it is not going to exist in the future, it will not be a property [idion] at all. | |
| From: Aristotle (Topics [c.331 BCE], 129a27) | |
| A reaction: This seems to cramp our style in counterfactual discussion. Can't we even mention an individual property if we believe that it will never exist. Utopian political discussion will have to cease! |
| 12264 | An 'accident' is something which may possibly either belong or not belong to a thing [Aristotle] |
| Full Idea: An 'accident' [sumbebekos] is something which may possibly either belong or not belong to any one and the self-same thing, such as 'sitting posture' or 'whiteness'. This is the best definition, because it tells us the essential meaning of the term itself. | |
| From: Aristotle (Topics [c.331 BCE], 102b07) | |
| A reaction: Thus a car could be red, or not red. Accidents are contingent. It does not follow that necessary properties are essential (see Idea 12262). There are accidents [sumbebekos], propria [idion] and essences [to ti en einai]. |
| 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. |
| 12280 | Genus gives the essence better than the differentiae do [Aristotle] |
| Full Idea: In assigning the essence [ti estin], it is more appropriate to state the genus than the differentiae; for he who describes 'man' as an 'animal' indicates his essence better than he who describes him as 'pedestrian'. | |
| From: Aristotle (Topics [c.331 BCE], 128a24) | |
| A reaction: See Idea 12279. This idea is only part of the story. My reading of this is simply that assigning a genus gives more information. We learn more about him when we say he is a man than when we say he is Socrates. |
| 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). |
| 13269 | In the case of a house the parts can exist without the whole, so parts are not the whole [Aristotle] |
| Full Idea: In the case of a house, where the process of compounding the parts is obvious, though the parts exist, there is no reason why the whole should not be non-existent, and so the parts are not the same as the whole. | |
| From: Aristotle (Topics [c.331 BCE], 150a19) | |
| A reaction: Compare buying a piece of furniture, and being surprised to discover, when it is delivered, that it is self-assembly. This idea is a simple refutation of the claims of classical mereology, that wholes are just some parts. Aristotle uses modal claims. |
| 12284 | Everything that is has one single essence [Aristotle] |
| Full Idea: Everything that is has one single essence [en esti to einai]. | |
| From: Aristotle (Topics [c.331 BCE], 141a36) | |
| A reaction: Does this include vague objects, and abstract 'objects'? Sceptics might ask what grounds this claim. Does Dr Jeckyll have two essences? |
| 12262 | An 'idion' belongs uniquely to a thing, but is not part of its essence [Aristotle] |
| Full Idea: A property [idion] is something which does not show the essence of a thing but belongs to it alone. ...No one calls anything a property which can possibly belong to something else. | |
| From: Aristotle (Topics [c.331 BCE], 102a18) | |
| A reaction: [See Charlotte Witt 106 on this] 'Property' is clearly a bad translation for such an individual item. Witt uses 'proprium', which is a necessary but nonessential property of something. Necessity is NOT the hallmark of essence. See Idea 12266. |
| 12290 | Destruction is dissolution of essence [Aristotle] |
| Full Idea: Destruction is a dissolution of essence. | |
| From: Aristotle (Topics [c.331 BCE], 153b30) | |
| A reaction: [plucked from context!] I can't think of a better way to define destruction, in order to distinguish it from damage. A vase is destroyed when its essential function cannot be recovered. |
| 12286 | If two things are the same, they must have the same source and origin [Aristotle] |
| Full Idea: When things are absolutely the same, their coming-into-being and destruction are also the same and so are the agents of their production and destruction. | |
| From: Aristotle (Topics [c.331 BCE], 152a02) | |
| A reaction: Thus Queen Elizabeth II has to be the result of that particular birth, and from those particular parents, as Kripke says? The inverse may not be true. Do twins have a single origin? Things that fission and then re-fuse differently? etc |
| 12266 | 'Same' is mainly for names or definitions, but also for propria, and for accidents [Aristotle] |
| Full Idea: 'The same' is employed in several senses: its principal sense is for same name or same definition; a second sense occurs when sameness is applied to a property [idiu]; a third sense is applied to an accident. | |
| From: Aristotle (Topics [c.331 BCE], 103a24-33) | |
| A reaction: [compressed] 'Property' is better translated as 'proprium' - a property unique to a particular thing, but not essential - see Idea 12262. Things are made up of essence, propria and accidents, and three ways of being 'the same' are the result. |
| 12287 | Two identical things have the same accidents, they are the same; if the accidents differ, they're different [Aristotle] |
| Full Idea: If two things are the same then any accident of one must also be an accident of the other, and, if one of them is an accident of something else, so must the other be also. For, if there is any discrepancy on these points, obviously they are not the same. | |
| From: Aristotle (Topics [c.331 BCE], 152a36) | |
| A reaction: So what is always called 'Leibniz's Law' should actually be 'Aristotle's Law'! I can't see anything missing from the Aristotle version, but then, since most people think it is pretty obvious, you would expect the great stater of the obvious to get it. |
| 12288 | Numerical sameness and generic sameness are not the same [Aristotle] |
| Full Idea: Things which are the same specifically or generically are not necessarily the same or cannot possibly be the same numerically. | |
| From: Aristotle (Topics [c.331 BCE], 152b32) | |
| A reaction: See also Idea 12266. This looks to me to be a pretty precise anticipation of Peirce's type/token distinction, but without the terminology. It is reassuring that Aristotle spotted it, as that makes it more likely to be a genuine distinction. |
| 12259 | Reasoning is when some results follow necessarily from certain claims [Aristotle] |
| Full Idea: Reasoning [sullogismos] is a discussion in which, certain things having been laid down, something other than these things necessarily results through them. | |
| From: Aristotle (Topics [c.331 BCE], 100a25) | |
| A reaction: This is cited as the standard statement of the nature of logical necessity. One might challenge either the very word 'necessary', or the exact sense of the word employed here. Is it, in fact, metaphysical, or merely analytic? |
| 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? |
| 12271 | Induction is the progress from particulars to universals [Aristotle] |
| Full Idea: Induction is the progress from particulars to universals; if the skilled pilot is the best pilot and the skilled charioteer the best charioteer, then, in general, the skilled man is the best man in any particular sphere. | |
| From: Aristotle (Topics [c.331 BCE], 105a15) | |
| A reaction: It is a bit unclear whether we are deriving universal concepts, or merely general truths. Need general truths be absolute or necessary truths? Presumably occasionally the best person is not the most skilled, as in playing a musical instrument. |
| 12293 | We say 'so in cases of this kind', but how do you decide what is 'of this kind'? [Aristotle] |
| Full Idea: When it is necessary to establish the universal, people use the expression 'So in all cases of this kind'; but it is one of the most difficult tasks to define which of the terms proposed are 'of this kind' and which are not. | |
| From: Aristotle (Topics [c.331 BCE], 157a25) | |
| A reaction: It is particularly hard if induction is expressed as the search for universals, since the kind presumably is the universal, so the universal must be known before the induction can apply, which really is the most frightful nuisance for truth-seekers. |
| 15308 | Science is the reduction of diverse forces and powers to a smaller number that explain them [Kant] |
| Full Idea: All natural philosophy consists in the reduction of given forces apparently diverse to a smaller number of forces and powers sufficient for the explication of the actions of the former. | |
| From: Immanuel Kant (Metaphysical Foundations of Natural Science [1786], 534) | |
| A reaction: I'm beginning to think science is just tracking of complex forces and powers back to fundamental forces and powers. In which case, that is the analysis Kant is talking of. The standard model of physics would thrill him to bits. |
| 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. |
| 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. |
| 12277 | Friendship is preferable to money, since its excess is preferable [Aristotle] |
| Full Idea: Friendship is preferable to money; for excess of friendship is preferable to excess of money. | |
| From: Aristotle (Topics [c.331 BCE], 118b07) | |
| A reaction: Compare Idea 12276, which gives a different criterion for choosing between virtues. This idea is an interesting qualification of the doctrine of the mean. |
| 12276 | Justice and self-control are better than courage, because they are always useful [Aristotle] |
| Full Idea: Justice [dikaiosune] and self-control [sophrosune] are preferable to courage, for the first two are always useful, but courage only sometimes. | |
| From: Aristotle (Topics [c.331 BCE], 117a36) | |
| A reaction: One could challenge his criterion. What of something which is absolutely vital on occasions, against something which is very mildly useful all the time? You may survive without justice, but not without courage. Compare Idea 12277. |
| 12275 | We value friendship just for its own sake [Aristotle] |
| Full Idea: We value friendship for its own sake, even if we are not likely to get anything else from it. | |
| From: Aristotle (Topics [c.331 BCE], 117a03) | |
| A reaction: In 'Ethics' he distinguishes some friendships which don't meet this requirement. Presumably true friendships survive all vicissitudes (except betrayal), but that makes such things fairly rare. |
| 12281 | Man is intrinsically a civilized animal [Aristotle] |
| Full Idea: It is an essential [kath' auto] property of man to be 'by nature a civilized animal'. | |
| From: Aristotle (Topics [c.331 BCE], 128b17) | |
| A reaction: I take this, along with man being intrinsically rational, to be the foundation of Aristotelian ethics. Given that we are civilized, self-evident criteria emerge for how to be good at it. A good person is, above all, a good citizen. |
| 12265 | All water is the same, because of a certain similarity [Aristotle] |
| Full Idea: Any water is said to be specifically the same as any other water because it has a certain similarity to it. | |
| From: Aristotle (Topics [c.331 BCE], 103a20) | |
| A reaction: (Cf. Idea 8153) It take this to be the hallmark of a natural kind, and we should not lose sight of it in the midst of discussions about rigid designation and essential identity. Tigers are only a natural kind insofar as they are indistinguishable. |
| 12278 | 'Being' and 'oneness' are predicated of everything which exists [Aristotle] |
| Full Idea: 'Being' and 'oneness' are predicated of everything which exists. | |
| From: Aristotle (Topics [c.331 BCE], 121a18) | |
| A reaction: Is 'oneness' predicated of water? So existence always was a predicate, it seems, until Kant told us it wasn't. That existence is a quantifier, not a predicate, seems to be up for question again these days. |