display all the ideas for this combination of texts
31 ideas
| 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) |
| 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 Michčle 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. |