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Single Idea 10236

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics ]

Full Idea

We cannot ground mathematics in any domain or theory that is more secure than mathematics itself.

Gist of Idea

There is no grounding for mathematics that is more secure than mathematics

Source

Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)

Book Ref

Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.135

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.

The 142 ideas from Stewart Shapiro

15944 Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine]
13629 Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro]
13626 Semantic consequence is ineffective in second-order logic [Shapiro]
13624 The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro]
13627 There is no 'correct' logic for natural languages [Shapiro]
13631 Are sets part of logic, or part of mathematics? [Shapiro]
13628 We can live well without completeness in logic [Shapiro]
13630 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
13625 Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
13635 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro]
13633 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro]
13634 Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro]
13632 Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro]
13637 If a logic is incomplete, its semantic consequence relation is not effective [Shapiro]
13636 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
13640 Russell's paradox shows that there are classes which are not iterative sets [Shapiro]
13638 Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro]
13641 Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
13642 Logic is the ideal for learning new propositions on the basis of others [Shapiro]
13643 Aristotelian logic is complete [Shapiro]
13644 Semantics for models uses set-theory [Shapiro]
13645 In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]
13650 Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]
13647 Choice is essential for proving downward Löwenheim-Skolem [Shapiro]
13649 Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro]
13646 Compactness is derived from soundness and completeness [Shapiro]
13648 The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
13651 A set is 'transitive' if contains every member of each of its members [Shapiro]
13657 First-order arithmetic can't even represent basic number theory [Shapiro]
13652 The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
13653 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro]
13654 It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro]
13656 Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
13660 Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro]
13661 A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro]
13658 Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
13659 Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
13662 First-order logic was an afterthought in the development of modern logic [Shapiro]
13666 Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro]
13664 Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
13663 Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
13667 Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro]
13668 Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro]
13669 Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro]
13670 Categoricity can't be reached in a first-order language [Shapiro]
13673 The notion of finitude is actually built into first-order languages [Shapiro]
13674 We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro]
13675 Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
13676 Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
13677 Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
10294 Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
10290 Second-order variables also range over properties, sets, relations or functions [Shapiro]
10588 First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
10591 Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]
10296 The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
10299 If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
10300 Logical consequence can be defined in terms of the logical terminology [Shapiro]
10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
10298 Some say that second-order logic is mathematics, not logic [Shapiro]
10301 The axiom of choice is controversial, but it could be replaced [Shapiro]
10279 Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro]
10203 We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro]
10200 We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
10202 Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
10205 Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
10207 Anti-realists reject set theory [Shapiro]
10206 Modal operators are usually treated as quantifiers [Shapiro]
10204 An 'implicit definition' gives a direct description of the relations of an entity [Shapiro]
10201 Virtually all of mathematics can be modeled in set theory [Shapiro]
10208 Axiom of Choice: some function has a value for every set in a given set [Shapiro]
10209 A function is just an arbitrary correspondence between collections [Shapiro]
10210 If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
10213 Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
10214 Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
10212 Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro]
10218 Baseball positions and chess pieces depend entirely on context [Shapiro]
10215 Platonists claim we can state the essence of a number without reference to the others [Shapiro]
10217 We can apprehend structures by focusing on or ignoring features of patterns [Shapiro]
10221 Is there is no more to structures than the systems that exemplify them? [Shapiro]
10222 Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
10220 Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
10223 There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
10224 The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
10228 Could infinite structures be apprehended by pattern recognition? [Shapiro]
10227 The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro]
10226 Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro]
10229 Simple types can be apprehended through their tokens, via abstraction [Shapiro]
10230 The 4-pattern is the structure common to all collections of four objects [Shapiro]
8703 Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
10231 Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro]
10233 Platonism must accept that the Peano Axioms could all be false [Shapiro]
10236 There is no grounding for mathematics that is more secure than mathematics [Shapiro]
10235 A sentence is 'satisfiable' if it has a model [Shapiro]
10238 The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
10234 Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
10237 Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro]
10239 The central notion of model theory is the relation of 'satisfaction' [Shapiro]
10240 Model theory deals with relations, reference and extensions [Shapiro]
18243 Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
10244 Intuition is an outright hindrance to five-dimensional geometry [Shapiro]
10248 Number statements are generalizations about number sequences, and are bound variables [Shapiro]
18245 Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
10249 The main mathematical structures are algebraic, ordered, and topological [Shapiro]
10251 The law of excluded middle might be seen as a principle of omniscience [Shapiro]
10252 The Axiom of Choice seems to license an infinite amount of choosing [Shapiro]
10253 Either logic determines objects, or objects determine logic, or they are separate [Shapiro]
10254 Can the ideal constructor also destroy objects? [Shapiro]
10255 Presumably nothing can block a possible dynamic operation? [Shapiro]
10256 For intuitionists, proof is inherently informal [Shapiro]
10257 Intuitionism only sanctions modus ponens if all three components are proved [Shapiro]
10258 Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro]
10262 Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro]
10259 The two standard explanations of consequence are semantic (in models) and deductive [Shapiro]
10268 Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro]
10266 Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro]
10270 The main versions of structuralism are all definitionally equivalent [Shapiro]
10272 The notion of 'object' is at least partially structural and mathematical [Shapiro]
10274 Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
10273 Some structures are exemplified by both abstract and concrete [Shapiro]
10276 Mathematical structures are defined by axioms, or in set theory [Shapiro]
10275 A blurry border is still a border [Shapiro]
10277 Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro]
10280 A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro]
9554 We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro]
9626 A structure is an abstraction, focussing on relationships, and ignoring other features [Shapiro]
8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
8730 'Impredicative' definitions refer to the thing being described [Shapiro]
8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
8731 Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
8744 Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
8749 Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
8752 Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
8753 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
8760 Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
8763 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
8764 Categories are the best foundation for mathematics [Shapiro]
18249 Cauchy gave a formal definition of a converging sequence. [Shapiro]