1989 | Structure and Ontology |
146 | p.60 | 9626 | A structure is an abstraction, focussing on relationships, and ignoring other features |
1991 | Foundations without Foundationalism |
p.225 | 15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else |
Pref | p.-17 | 13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed |
Pref | p.-17 | 13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology |
Pref | p.-16 | 13626 | Semantic consequence is ineffective in second-order logic |
Pref | p.-15 | 13627 | There is no 'correct' logic for natural languages |
Pref | p.-14 | 13628 | We can live well without completeness in logic |
Pref | p.-13 | 13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? |
Pref | p.-12 | 13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures |
Pref | p.-9 | 13631 | Are sets part of logic, or part of mathematics? |
1.1 | p.3 | 13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness |
1.1 | p.5 | 13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} |
1.1 | p.6 | 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' |
1.1 | p.8 | 13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence |
1.2.1 | p.12 | 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation |
1.2.1 | p.12 | 13637 | If a logic is incomplete, its semantic consequence relation is not effective |
1.3 | p.16 | 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects |
1.3 | p.19 | 13640 | Russell's paradox shows that there are classes which are not iterative sets |
2.1 | p.26 | 13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals |
2.3.1 | p.36 | 13642 | Logic is the ideal for learning new propositions on the basis of others |
2.5 | p.43 | 13643 | Aristotelian logic is complete |
2.5.1 | p.44 | 13644 | Semantics for models uses set-theory |
3.3 | p.73 | 13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables |
3.3 | p.73 | 13650 | Henkin semantics has separate variables ranging over the relations and over the functions |
4.1 | p.79 | 13646 | Compactness is derived from soundness and completeness |
4.1 | p.80 | 13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity |
4.1 | p.80 | 13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics |
4.1 | p.80 | 13647 | Choice is essential for proving downward Löwenheim-Skolem |
4.2 | p.85 | 13651 | A set is 'transitive' if contains every member of each of its members |
5 n28 | p.132 | 13657 | First-order arithmetic can't even represent basic number theory |
5.1.2 | p.105 | 13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set |
5.1.3 | p.106 | 13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element |
5.1.4 | p.109 | 13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains |
5.3.3 | p.123 | 13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic |
6.5 | p.158 | 13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes |
6.5 | p.158 | 13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models |
6.5 | p.158 | 13661 | A language is 'semantically effective' if its logical truths are recursively enumerable |
6.5 | p.159 | 13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable |
7.1 | p.173 | 13662 | First-order logic was an afterthought in the development of modern logic |
7.1 | p.174 | 13663 | Some reject formal properties if they are not defined, or defined impredicatively |
7.1 | p.176 | 13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions |
7.1 | p.177 | 13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets |
7.2 | p.178 | 13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order |
7.2.1 | p.180 | 13668 | Bernays (1918) formulated and proved the completeness of propositional logic |
7.2.2 | p.182 | 13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? |
7.3 | p.196 | 13670 | Categoricity can't be reached in a first-order language |
9.1 | p.238 | 13673 | The notion of finitude is actually built into first-order languages |
9.1.4 | p.243 | 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models |
9.1.4 | p.245 | 13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails |
9.1.4 | p.246 | 13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are |
9.3 | p.251 | 13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals |
1997 | Philosophy of Mathematics |
p.258 | 10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? |
Intro | p.4 | 10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) |
Intro | p.5 | 10202 | Natural numbers just need an initial object, successors, and an induction principle |
Intro | p.5 | 10201 | Virtually all of mathematics can be modeled in set theory |
Intro | p.11 | 10203 | We apprehend small, finite mathematical structures by abstraction from patterns |
Intro | p.13 | 10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) |
Intro | p.13 | 10204 | An 'implicit definition' gives a direct description of the relations of an entity |
Intro | p.16 | 10206 | Modal operators are usually treated as quantifiers |
Intro | p.17 | 10207 | Anti-realists reject set theory |
1 | p.24 | 10208 | Axiom of Choice: some function has a value for every set in a given set |
1 | p.24 | 10209 | A function is just an arbitrary correspondence between collections |
1 | p.25 | 10210 | If mathematical objects are accepted, then a number of standard principles will follow |
2.5 | p.53 | 10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts |
2.5 | p.55 | 10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' |
3 | p.42 | 10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' |
3.1 | p.72 | 10215 | Platonists claim we can state the essence of a number without reference to the others |
3.1 | p.74 | 10217 | We can apprehend structures by focusing on or ignoring features of patterns |
3.1 | p.76 | 10218 | Baseball positions and chess pieces depend entirely on context |
3.3 | p.84 | 10220 | Because one structure exemplifies several systems, a structure is a one-over-many |
3.3 | p.85 | 10221 | Is there is no more to structures than the systems that exemplify them? |
3.3 | p.87 | 10222 | Mathematical foundations may not be sets; categories are a popular rival |
3.4 | p.95 | 10223 | There is no 'structure of all structures', just as there is no set of all sets |
3.5 | p.100 | 10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 |
4.1 | p.111 | 10227 | The abstract/concrete boundary now seems blurred, and would need a defence |
4.1 | p.112 | 10228 | Could infinite structures be apprehended by pattern recognition? |
4.1 n1 | p.109 | 10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract |
4.2 | p.113 | 10229 | Simple types can be apprehended through their tokens, via abstraction |
4.2 | p.115 | 10230 | The 4-pattern is the structure common to all collections of four objects |
4.4 | p.96 | 8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics |
4.5 | p.124 | 10231 | Abstract objects might come by abstraction over an equivalence class of base entities |
4.6 | p.127 | 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem |
4.7 | p.131 | 10233 | Platonism must accept that the Peano Axioms could all be false |
4.8 | p.132 | 10234 | Any theory with an infinite model has a model of every infinite cardinality |
4.8 | p.135 | 10235 | A sentence is 'satisfiable' if it has a model |
4.8 | p.135 | 10237 | Coherence is a primitive, intuitive notion, not reduced to something formal |
4.8 | p.135 | 10236 | There is no grounding for mathematics that is more secure than mathematics |
4.8 | p.136 | 10238 | The set-theoretical hierarchy contains as many isomorphism types as possible |
4.9 | p.138 | 18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis |
4.9 | p.139 | 10240 | Model theory deals with relations, reference and extensions |
4.9 | p.139 | 10239 | The central notion of model theory is the relation of 'satisfaction' |
5.2 | p.150 | 10244 | Intuition is an outright hindrance to five-dimensional geometry |
5.3.4 | p.166 | 10248 | Number statements are generalizations about number sequences, and are bound variables |
5.4 | p.171 | 18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational |
5.5 | p.176 | 10249 | The main mathematical structures are algebraic, ordered, and topological |
6.3 | p.187 | 10251 | The law of excluded middle might be seen as a principle of omniscience |
6.3 | p.188 | 10252 | The Axiom of Choice seems to license an infinite amount of choosing |
6.4 | p.192 | 10253 | Either logic determines objects, or objects determine logic, or they are separate |
6.5 | p.194 | 10254 | Can the ideal constructor also destroy objects? |
6.5 | p.194 | 10255 | Presumably nothing can block a possible dynamic operation? |
6.7 | p.205 | 10256 | For intuitionists, proof is inherently informal |
6.7 | p.207 | 10257 | Intuitionism only sanctions modus ponens if all three components are proved |
7.1 | p.216 | 10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models |
7.2 | p.222 | 10259 | The two standard explanations of consequence are semantic (in models) and deductive |
7.2 | p.223 | 10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) |
7.4 | p.233 | 10266 | Why does the 'myth' of possible worlds produce correct modal logic? |
7.4 | p.235 | 10268 | Maybe plural quantifiers should be understood in terms of classes or sets |
7.5 | p.242 | 10270 | The main versions of structuralism are all definitionally equivalent |
8.1 | p.245 | 10272 | The notion of 'object' is at least partially structural and mathematical |
8.2 | p.248 | 10273 | Some structures are exemplified by both abstract and concrete |
8.2 | p.254 | 10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? |
8.3 | p.255 | 10276 | Mathematical structures are defined by axioms, or in set theory |
8.3 | p.255 | 10275 | A blurry border is still a border |
8.4 | p.256 | 10277 | Structuralism blurs the distinction between mathematical and ordinary objects |
8.4 | p.259 | 10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location |
p.74 | p.79 | 9554 | We can focus on relations between objects (like baseballers), ignoring their other features |
2000 | Thinking About Mathematics |
1.1 | p.3 | 8725 | Rationalism tries to apply mathematical methodology to all of knowledge |
1.2 | p.9 | 8730 | 'Impredicative' definitions refer to the thing being described |
1.2 | p.9 | 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects |
2.2.1 | p.26 | 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts |
2.4 | p.42 | 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals |
5.1 | p.113 | 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own |
5.3 | p.128 | 8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities |
6.1.1 | p.142 | 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names |
6.1.2 | p.144 | 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? |
6.2 | p.149 | 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms |
7.1 | p.174 | 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions |
10.1 | p.258 | 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers |
10.1 | p.259 | 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them |
10.2 | p.265 | 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 |
10.2 | p.267 | 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex |
10.3 n7 | p.272 | 8764 | Categories are the best foundation for mathematics |
7.2 n4 | p.181 | 18249 | Cauchy gave a formal definition of a converging sequence. |
2001 | Higher-Order Logic |
2.1 | p.33 | 10290 | Second-order variables also range over properties, sets, relations or functions |
2.1 | p.34 | 10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model |
2.1 | p.34 | 10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them |
2.1 | p.34 | 10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems |
2.1 | p.34 | 10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory |
2.2.1 | p.36 | 10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it |
2.3.2 | p.47 | 10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics |
2.4 | p.49 | 10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic |
2.4 | p.50 | 10298 | Some say that second-order logic is mathematics, not logic |
2.4 | p.51 | 10299 | If the aim of logic is to codify inferences, second-order logic is useless |
2.4 | p.51 | 10300 | Logical consequence can be defined in terms of the logical terminology |
n 3 | p.52 | 10301 | The axiom of choice is controversial, but it could be replaced |