Combining Philosophers

All the ideas for Hesiod, E.J. Lemmon and Richard Dedekind

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81 ideas

2. Reason / D. Definition / 9. Recursive Definition

22289	Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Dedekind, by Potter]
	Full Idea: Dedkind gave a rigorous proof of the principle of definition by recursion, permitting recursive definitions of addition and multiplication, and hence proofs of the familiar arithmetical laws.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 13 'Deriv'

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic

9535	'Contradictory' propositions always differ in truth-value [Lemmon]
	Full Idea: Two propositions are 'contradictory' if they are never both true and never both false either, which means that ¬(A↔B) is a tautology.
	From: E.J. Lemmon (Beginning Logic [1965], 2.3)

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / a. Symbols of PL

9511	We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon]
	Full Idea: We write 'if P then Q' as P→Q. This is called a 'conditional', with P as its 'antecedent', and Q as its 'consequent'.
	From: E.J. Lemmon (Beginning Logic [1965], 1.2)
	A reaction: P→Q can also be written as ¬P∨Q.

9510	That proposition that either P or Q is their 'disjunction', written P∨Q [Lemmon]
	Full Idea: If P and Q are any two propositions, the proposition that either P or Q is called the 'disjunction' of P and Q, and is written P∨Q.
	From: E.J. Lemmon (Beginning Logic [1965], 1.3)
	A reaction: This is inclusive-or (meaning 'P, or Q, or both'), and not exlusive-or (Boolean XOR), which means 'P, or Q, but not both'. The ∨ sign is sometimes called 'vel' (Latin).

9509	That proposition that both P and Q is their 'conjunction', written P∧Q [Lemmon]
	Full Idea: If P and Q are any two propositions, the proposition that both P and Q is called the 'conjunction' of P and Q, and is written P∧Q.
	From: E.J. Lemmon (Beginning Logic [1965], 1.3)
	A reaction: [I use the more fashionable inverted-v '∧', rather than Lemmon's '&', which no longer seems to be used] P∧Q can also be defined as ¬(¬P∨¬Q)

9512	We write the 'negation' of P (not-P) as ¬ [Lemmon]
	Full Idea: We write 'not-P' as ¬P. This is called the 'negation' of P. The 'double negation' of P (not not-P) would be written as ¬¬P.
	From: E.J. Lemmon (Beginning Logic [1965], 1.2)
	A reaction: Lemmons use of -P is no longer in use for 'not'. A tilde sign (squiggle) is also used for 'not', but some interpreters give that a subtly different meaning (involving vagueness). The sign ¬ is sometimes called 'hook' or 'corner'.

9513	We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P) [Lemmon]
	Full Idea: We write 'P if and only if Q' as P↔Q. It is called the 'biconditional', often abbreviate in writing as 'iff'. It also says that P is both sufficient and necessary for Q, and may be written out in full as (P→Q)∧(Q→P).
	From: E.J. Lemmon (Beginning Logic [1965], 1.4)
	A reaction: If this symbol is found in a sequence, the first move in a proof is to expand it to the full version.

9514	If A and B are 'interderivable' from one another we may write A -||- B [Lemmon]
	Full Idea: If we say that A and B are 'interderivable' from one another (that is, A |- B and B |- A), then we may write A -||- B.
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9508	The sign |- may be read as 'therefore' [Lemmon]
	Full Idea: I introduce the sign |- to mean 'we may validly conclude'. To call it the 'assertion sign' is misleading. It may conveniently be read as 'therefore'.
	From: E.J. Lemmon (Beginning Logic [1965], 1.2)
	A reaction: [Actually no gap between the vertical and horizontal strokes of the sign] As well as meaning 'assertion', it may also mean 'it is a theorem that' (with no proof shown).

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL

9516	A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔ [Lemmon]
	Full Idea: A 'well-formed formula' of the propositional calculus is a sequence of symbols which follows the rules for variables, ¬, →, ∧, ∨, and ↔.
	From: E.J. Lemmon (Beginning Logic [1965], 2.1)

9517	The 'scope' of a connective is the connective, the linked formulae, and the brackets [Lemmon]
	Full Idea: The 'scope' of a connective in a certain formula is the formulae linked by the connective, together with the connective itself and the (theoretically) encircling brackets
	From: E.J. Lemmon (Beginning Logic [1965], 2.1)

9519	A 'substitution-instance' is a wff formed by consistent replacing variables with wffs [Lemmon]
	Full Idea: A 'substitution-instance' is a wff which results by replacing one or more variables throughout with the same wffs (the same wff replacing each variable).
	From: E.J. Lemmon (Beginning Logic [1965], 2.2)

9529	A wff is 'inconsistent' if all assignments to variables result in the value F [Lemmon]
	Full Idea: If a well-formed formula of propositional calculus takes the value F for all possible assignments of truth-values to its variables, it is said to be 'inconsistent'.
	From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9531	'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology [Lemmon]
	Full Idea: If A and B are expressible in propositional calculus notation, they are 'contrary' if they are never both true, which may be tested by the truth-table for ¬(A∧B), which is a tautology if they are contrary.
	From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9534	Two propositions are 'equivalent' if they mirror one another's truth-value [Lemmon]
	Full Idea: Two propositions are 'equivalent' if whenever A is true B is true, and whenever B is true A is true, in which case A↔B is a tautology.
	From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9530	A wff is 'contingent' if produces at least one T and at least one F [Lemmon]
	Full Idea: If a well-formed formula of propositional calculus takes at least one T and at least one F for all the assignments of truth-values to its variables, it is said to be 'contingent'.
	From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9532	'Subcontrary' propositions are never both false, so that A∨B is a tautology [Lemmon]
	Full Idea: If A and B are expressible in propositional calculus notation, they are 'subcontrary' if they are never both false, which may be tested by the truth-table for A∨B, which is a tautology if they are subcontrary.
	From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9533	A 'implies' B if B is true whenever A is true (so that A→B is tautologous) [Lemmon]
	Full Idea: One proposition A 'implies' a proposition B if whenever A is true B is true (but not necessarily conversely), which is only the case if A→B is tautologous. Hence B 'is implied' by A.
	From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9528	A wff is a 'tautology' if all assignments to variables result in the value T [Lemmon]
	Full Idea: If a well-formed formula of propositional calculus takes the value T for all possible assignments of truth-values to its variables, it is said to be a 'tautology'.
	From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9518	A 'theorem' is the conclusion of a provable sequent with zero assumptions [Lemmon]
	Full Idea: A 'theorem' of logic is the conclusion of a provable sequent in which the number of assumptions is zero.
	From: E.J. Lemmon (Beginning Logic [1965], 2.2)
	A reaction: This is what Quine and others call a 'logical truth'.

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL

9398	∧I: Given A and B, we may derive A∧B [Lemmon]
	Full Idea: And-Introduction (&I): Given A and B, we may derive A∧B as conclusion. This depends on their previous assumptions.
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9397	CP: Given a proof of B from A as assumption, we may derive A→B [Lemmon]
	Full Idea: Conditional Proof (CP): Given a proof of B from A as assumption, we may derive A→B as conclusion, on the remaining assumptions (if any).
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9394	MPP: Given A and A→B, we may derive B [Lemmon]
	Full Idea: Modus Ponendo Ponens (MPP): Given A and A→B, we may derive B as a conclusion. B will rest on any assumptions that have been made.
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9401	∨E: Derive C from A∨B, if C can be derived both from A and from B [Lemmon]
	Full Idea: Or-Elimination (∨E): Given A∨B, we may derive C if it is proved from A as assumption and from B as assumption. This will also depend on prior assumptions.
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9396	DN: Given A, we may derive ¬¬A [Lemmon]
	Full Idea: Double Negation (DN): Given A, we may derive ¬¬A as a conclusion, and vice versa. The conclusion depends on the assumptions of the premiss.
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9393	A: we may assume any proposition at any stage [Lemmon]
	Full Idea: Assumptions (A): any proposition may be introduced at any stage of a proof.
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9399	∧E: Given A∧B, we may derive either A or B separately [Lemmon]
	Full Idea: And-Elimination (∧E): Given A∧B, we may derive either A or B separately. The conclusions will depend on the assumptions of the premiss.
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9402	RAA: If assuming A will prove B∧¬B, then derive ¬A [Lemmon]
	Full Idea: Reduction ad Absurdum (RAA): Given a proof of B∧¬B from A as assumption, we may derive ¬A as conclusion, depending on the remaining assumptions (if any).
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9395	MTT: Given ¬B and A→B, we derive ¬A [Lemmon]
	Full Idea: Modus Tollendo Tollens (MTT): Given ¬B and A→B, we derive ¬A as a conclusion. ¬A depends on any assumptions that have been made
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9400	∨I: Given either A or B separately, we may derive A∨B [Lemmon]
	Full Idea: Or-Introduction (∨I): Given either A or B separately, we may derive A∨B as conclusion. This depends on the assumption of the premisses.
	From: E.J. Lemmon (Beginning Logic [1965], 1.5)

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL

9521	'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q [Lemmon]
	Full Idea: 'Modus tollendo ponens' (MTP) says that if a disjunction holds and also the negation of one of its disjuncts, then the other disjunct holds. Thus ¬P, P ∨ Q |- Q may be introduced as a theorem.
	From: E.J. Lemmon (Beginning Logic [1965], 2.2)
	A reaction: Unlike Modus Ponens and Modus Tollens, this is a derived rule.

9522	'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q [Lemmon]
	Full Idea: 'Modus ponendo tollens' (MPT) says that if the negation of a conjunction holds and also one of its conjuncts, then the negation of the other conjunct holds. Thus P, ¬(P ∧ Q) |- ¬Q may be introduced as a theorem.
	From: E.J. Lemmon (Beginning Logic [1965], 2.2)
	A reaction: Unlike Modus Ponens and Modus Tollens, this is a derived rule.

9525	We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q) [Lemmon]
	Full Idea: The proof that P→Q -||- ¬(P ∧ ¬Q) is useful for enabling us to change conditionals into negated conjunctions
	From: E.J. Lemmon (Beginning Logic [1965], 2.2)

9524	We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q [Lemmon]
	Full Idea: The proof that P→Q -||- ¬P ∨ Q is useful for enabling us to change conditionals into disjunctions.
	From: E.J. Lemmon (Beginning Logic [1965], 2.2)

9523	De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions [Lemmon]
	Full Idea: The forms of De Morgan's Laws [P∨Q -||- ¬(¬P ∧ ¬Q); ¬(P∨Q) -||- ¬P ∧ ¬Q; ¬(P∧Q) -||- ¬P ∨ ¬Q); P∧Q -||- ¬(¬P∨¬Q)] transform negated conjunctions and disjunctions into non-negated disjunctions and conjunctions respectively.
	From: E.J. Lemmon (Beginning Logic [1965], 2.2)

9527	The Distributive Laws can rearrange a pair of conjunctions or disjunctions [Lemmon]
	Full Idea: The Distributive Laws say that P ∧ (Q∨R) -||- (P∧Q) ∨ (P∧R), and that P ∨ (Q∨R) -||- (P∨Q) ∧ (P∨R)
	From: E.J. Lemmon (Beginning Logic [1965], 2.2)

9526	We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q) [Lemmon]
	Full Idea: The proof that P∧Q -||- ¬(P → ¬Q) is useful for enabling us to change conjunctions into negated conditionals.
	From: E.J. Lemmon (Beginning Logic [1965], 2.2)

4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables

9537	Truth-tables are good for showing invalidity [Lemmon]
	Full Idea: The truth-table approach enables us to show the invalidity of argument-patterns, as well as their validity.
	From: E.J. Lemmon (Beginning Logic [1965], 2.4)

9538	A truth-table test is entirely mechanical, but this won't work for more complex logic [Lemmon]
	Full Idea: A truth-table test is entirely mechanical, ..and in propositional logic we can even generate proofs mechanically for tautological sequences, ..but this mechanical approach breaks down with predicate calculus, and proof-discovery is an imaginative process.
	From: E.J. Lemmon (Beginning Logic [1965], 2.5)

4. Formal Logic / B. Propositional Logic PL / 4. Soundness of PL

9536	If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology [Lemmon]
	Full Idea: If any application of the nine derivation rules of propositional logic is made on tautologous sequents, we have demonstrated that the result is always a tautologous sequent. Thus the system is consistent.
	From: E.J. Lemmon (Beginning Logic [1965], 2.4)
	A reaction: The term 'sound' tends to be used now, rather than 'consistent'. See Lemmon for the proofs of each of the nine rules.

4. Formal Logic / B. Propositional Logic PL / 5. Completeness of PL

9539	Propositional logic is complete, since all of its tautologous sequents are derivable [Lemmon]
	Full Idea: A logical system is complete is all expressions of a specified kind are derivable in it. If we specify tautologous sequent-expressions, then propositional logic is complete, because we can show that all tautologous sequents are derivable.
	From: E.J. Lemmon (Beginning Logic [1965], 2.5)
	A reaction: [See Lemmon 2.5 for details of the proofs]

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / a. Symbols of PC

13909	Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....' [Lemmon]
	Full Idea: Just as '(∀x)(...)' is to mean 'take any x: then....', so we write '(∃x)(...)' to mean 'there is an x such that....'
	From: E.J. Lemmon (Beginning Logic [1965], 3.1)
	A reaction: [Actually Lemmon gives the universal quantifier symbol as '(x)', but the inverted A ('∀') seems to have replaced it these days]

13902	'Gm' says m has property G, and 'Pmn' says m has relation P to n [Lemmon]
	Full Idea: A predicate letter followed by one name expresses a property ('Gm'), and a predicate-letter followed by two names expresses a relation ('Pmn'). We could write 'Pmno' for a complex relation like betweenness.
	From: E.J. Lemmon (Beginning Logic [1965], 3.1)

13911	The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E [Lemmon]
	Full Idea: I define a 'symbol' (of the predicate calculus) as either a bracket or a logical connective or a term or an individual variable or a predicate-letter or reverse-E (∃).
	From: E.J. Lemmon (Beginning Logic [1965], 4.1)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / b. Terminology of PC

13910	Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers' [Lemmon]
	Full Idea: Quantifier-notation might be thus: first, render into sentences about 'properties', and use 'predicate-letters' for them; second, introduce 'variables'; third, introduce propositional logic 'connectives' and 'quantifiers'. Plus letters for 'proper names'.
	From: E.J. Lemmon (Beginning Logic [1965], 3.1)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / c. Derivations rules of PC

13904	Universal Elimination (UE) lets us infer that an object has F, from all things having F [Lemmon]
	Full Idea: Our rule of universal quantifier elimination (UE) lets us infer that any particular object has F from the premiss that all things have F. It is a natural extension of &E (and-elimination), as universal propositions generally affirm a complex conjunction.
	From: E.J. Lemmon (Beginning Logic [1965], 3.2)

13906	With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro [Lemmon]
	Full Idea: If there are just three objects and each has F, then by an extension of &I we are sure everything has F. This is of no avail, however, if our universe is infinitely large or if not all objects have names. We need a new device, Universal Introduction, UI.
	From: E.J. Lemmon (Beginning Logic [1965], 3.2)

13908	UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one [Lemmon]
	Full Idea: Univ Elim UE - if everything is F, then something is F; Univ Intro UI - if an arbitrary thing is F, everything is F; Exist Intro EI - if an arbitrary thing is F, something is F; Exist Elim EE - if a proof needed an object, there is one.
	From: E.J. Lemmon (Beginning Logic [1965], 3.3)
	A reaction: [My summary of Lemmon's four main rules for predicate calculus] This is the natural deduction approach, of trying to present the logic entirely in terms of introduction and elimination rules. See Bostock on that.

13901	Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules [Lemmon]
	Full Idea: In predicate calculus we take over the propositional connectives and propositional variables - but we need additional rules for handling quantifiers: four rules, an introduction and elimination rule for the universal and existential quantifiers.
	From: E.J. Lemmon (Beginning Logic [1965])
	A reaction: This is Lemmon's natural deduction approach (invented by Gentzen), which is largely built on introduction and elimination rules.

13903	Universal elimination if you start with the universal, introduction if you want to end with it [Lemmon]
	Full Idea: The elimination rule for the universal quantifier concerns the use of a universal proposition as a premiss to establish some conclusion, whilst the introduction rule concerns what is required by way of a premiss for a universal proposition as conclusion.
	From: E.J. Lemmon (Beginning Logic [1965], 3.2)
	A reaction: So if you start with the universal, you need to eliminate it, and if you start without it you need to introduce it.

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀

13905	If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers [Lemmon]
	Full Idea: If all objects in a given universe had names which we knew and there were only finitely many of them, then we could always replace a universal proposition about that universe by a complex conjunction.
	From: E.J. Lemmon (Beginning Logic [1965], 3.2)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃

13900	'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional → [Lemmon]
	Full Idea: It is a common mistake to render 'some Frenchmen are generous' by (∃x)(Fx→Gx) rather than the correct (∃x)(Fx&Gx). 'All Frenchmen are generous' is properly rendered by a conditional, and true if there are no Frenchmen.
	From: E.J. Lemmon (Beginning Logic [1965], 3.1)
	A reaction: The existential quantifier implies the existence of an x, but the universal quantifier does not.

4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets

10183	An infinite set maps into its own proper subset [Dedekind, by Reck/Price]
	Full Idea: A set is 'Dedekind-infinite' iff there exists a one-to-one function that maps a set into a proper subset of itself.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], §64) by E Reck / M Price - Structures and Structuralism in Phil of Maths n 7
	A reaction: Sounds as if it is only infinite if it is contradictory, or doesn't know how big it is!

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V

22288	We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter]
	Full Idea: Dedekind had an interesting proof of the Axiom of Infinity. He held that I have an a priori grasp of the idea of my self, and that every idea I can form the idea of that idea. Hence there are infinitely many objects available to me a priori.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], no. 66) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 12 'Numb'
	A reaction: Who said that Descartes' Cogito was of no use? Frege endorsed this, as long as the ideas are objective and not subjective.

4. Formal Logic / G. Formal Mereology / 1. Mereology

10706	Dedekind originally thought more in terms of mereology than of sets [Dedekind, by Potter]
	Full Idea: Dedekind plainly had fusions, not collections, in mind when he avoided the empty set and used the same symbol for membership and inclusion - two tell-tale signs of a mereological conception.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], 2-3) by Michael Potter - Set Theory and Its Philosophy 02.1
	A reaction: Potter suggests that mathematicians were torn between mereology and sets, and eventually opted whole-heartedly for sets. Maybe this is only because set theory was axiomatised by Zermelo some years before Lezniewski got to mereology.

5. Theory of Logic / B. Logical Consequence / 8. Material Implication

9520	The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q [Lemmon]
	Full Idea: The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q. That is, since Napoleon was French, then if the moon is blue then Napoleon was French; and since Napoleon was not Chinese, then if Napoleon was Chinese, the moon is blue.
	From: E.J. Lemmon (Beginning Logic [1965], 2.2)
	A reaction: This is why the symbol → does not really mean the 'if...then' of ordinary English. Russell named it 'material implication' to show that it was a distinctively logical operator.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers

9823	Numbers are free creations of the human mind, to understand differences [Dedekind]
	Full Idea: Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things.
	From: Richard Dedekind (Nature and Meaning of Numbers [1888], Pref)
	A reaction: Does this fit real numbers and complex numbers, as well as natural numbers? Frege was concerned by the lack of objectivity in this sort of view. What sort of arithmetic might the Martians have created? Numbers register sameness too.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers

10090	Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
	Full Idea: It was primarily Dedekind's accomplishment to define the integers, rationals and reals, taking only the system of natural numbers for granted.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by A.George / D.J.Velleman - Philosophies of Mathematics Intro

17452	Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
	Full Idea: Dedekind and Cantor said the cardinals may be defined in terms of the ordinals: The cardinal number of a set S is the least ordinal onto whose predecessors the members of S can be mapped one-one.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 5

7524	Order, not quantity, is central to defining numbers [Dedekind, by Monk]
	Full Idea: Dedekind said that the notion of order, rather than that of quantity, is the central notion in the definition of number.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4
	A reaction: Compare Aristotle's nice question in Idea 646. My intuition is that quantity comes first, because I'm not sure HOW you could count, if you didn't think you were changing the quantity each time. Why does counting go in THAT particular order? Cf. Idea 8661.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers

14131	Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
	Full Idea: Dedekind's ordinals are not essentially either ordinals or cardinals, but the members of any progression whatever.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - The Principles of Mathematics §243
	A reaction: This is part of Russell's objection to Dedekind's structuralism. The question is always why these beautiful structures should actually be considered as numbers. I say, unlike Russell, that the connection to counting is crucial.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers

17611	We want the essence of continuity, by showing its origin in arithmetic [Dedekind]
	Full Idea: It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity.
	From: Richard Dedekind (Continuity and Irrational Numbers [1872], Intro)
	A reaction: [He seeks the origin of the theorem that differential calculus deals with continuous magnitude, and he wants an arithmetical rather than geometrical demonstration; the result is his famous 'cut'].

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts

10572	A cut between rational numbers creates and defines an irrational number [Dedekind]
	Full Idea: Whenever we have to do a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut.
	From: Richard Dedekind (Continuity and Irrational Numbers [1872], §4)
	A reaction: Fine quotes this to show that the Dedekind Cut creates the irrational numbers, rather than hitting them. A consequence is that the irrational numbers depend on the rational numbers, and so can never be identical with any of them. See Idea 10573.

14437	Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
	Full Idea: Dedekind set up the axiom that the gap in his 'cut' must always be filled …The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - Introduction to Mathematical Philosophy VII
	A reaction: This remark of Russell's is famous, and much quoted in other contexts, but I have seen the modern comment that it is grossly unfair to Dedekind.

18094	Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock]
	Full Idea: One view, favoured by Dedekind, is that the cut postulates a real number for each cut in the rationals; it does not identify real numbers with cuts. ....A view favoured by later logicists is simply to identify a real number with a cut.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by David Bostock - Philosophy of Mathematics 4.4
	A reaction: Dedekind is the patriarch of structuralism about mathematics, so he has little interest in the existenc of 'objects'.

18244	I say the irrational is not the cut itself, but a new creation which corresponds to the cut [Dedekind]
	Full Idea: Of my theory of irrationals you say that the irrational number is nothing else than the cut itself, whereas I prefer to create something new (different from the cut), which corresponds to the cut. We have the right to claim such a creative power.
	From: Richard Dedekind (Letter to Weber [1888], 1888 Jan), quoted by Stewart Shapiro - Philosophy of Mathematics 5.4
	A reaction: Clearly a cut will not locate a unique irrational number, so something more needs to be done. Shapiro remarks here that for Dedekind numbers are objects.

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure

9824	In counting we see the human ability to relate, correspond and represent [Dedekind]
	Full Idea: If we scrutinize closely what is done in counting an aggregate of things, we see the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, without which no thinking is possible.
	From: Richard Dedekind (Nature and Meaning of Numbers [1888], Pref)
	A reaction: I don't suppose it occurred to Dedekind that he was reasserting Hume's observation about the fundamental psychology of thought. Is the origin of our numerical ability of philosophical interest?

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic

17612	Arithmetic is just the consequence of counting, which is the successor operation [Dedekind]
	Full Idea: I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself is nothing else than the successive creation of the infinite series of positive integers.
	From: Richard Dedekind (Continuity and Irrational Numbers [1872], §1)
	A reaction: Thus counting roots arithmetic in the world, the successor operation is the essence of counting, and the Dedekind-Peano axioms are built around successors, and give the essence of arithmetic. Unfashionable now, but I love it. Intransitive counting?

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite

9826	A system S is said to be infinite when it is similar to a proper part of itself [Dedekind]
	Full Idea: A system S is said to be infinite when it is similar to a proper part of itself.
	From: Richard Dedekind (Nature and Meaning of Numbers [1888], V.64)

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits

18087	If x changes by less and less, it must approach a limit [Dedekind]
	Full Idea: If in the variation of a magnitude x we can for every positive magnitude δ assign a corresponding position from and after which x changes by less than δ then x approaches a limiting value.
	From: Richard Dedekind (Continuity and Irrational Numbers [1872], p.27), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.7
	A reaction: [Kitcher says he 'showed' this, rather than just stating it]

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers

13508	Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD]
	Full Idea: Dedekind's natural numbers: an object is in a set (0 is a number), a function sends the set one-one into itself (numbers have unique successors), the object isn't a value of the function (it isn't a successor), plus induction.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by William D. Hart - The Evolution of Logic 5
	A reaction: Hart notes that since this refers to sets of individuals, it is a second-order account of numbers, what we now call 'Second-Order Peano Arithmetic'.

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic

18096	Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock]
	Full Idea: Dedekind's idea is that the set of natural numbers has zero as a member, and also has as a member the successor of each of its members, and it is the smallest set satisfying this condition. It is the intersection of all sets satisfying the condition.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by David Bostock - Philosophy of Mathematics 4.4

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order

18841	Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind]
	Full Idea: It is Dedekind's categoricity result that convinces most of us that he has articulated our implicit conception of the natural numbers, since it entitles us to speak of 'the' domain (in the singular, up to isomorphism) of natural numbers.
	From: comment on Richard Dedekind (Nature and Meaning of Numbers [1888]) by Ian Rumfitt - The Boundary Stones of Thought 9.1
	A reaction: The main rival is set theory, but that has an endlessly expanding domain. He points out that Dedekind needs second-order logic to achieve categoricity. Rumfitt says one could also add to the 1st-order version that successor is an ancestral relation.

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction

14130	Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell]
	Full Idea: Dedekind proves mathematical induction, while Peano regards it as an axiom, ...and Peano's method has the advantage of simplicity, and a clearer separation between the particular and the general propositions of arithmetic.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - The Principles of Mathematics §241

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

8924	Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride]
	Full Idea: Dedekind is the philosopher-mathematician with whom the structuralist conception originates.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888], §3 n13) by Fraser MacBride - Structuralism Reconsidered
	A reaction: Hellman says the idea grew naturally out of modern mathematics, and cites Hilbert's belief that furniture would do as mathematical objects.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism

9153	Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K]
	Full Idea: Dedekindian abstraction says mathematical objects are 'positions' in a model, while Cantorian abstraction says they are the result of abstracting on structurally similar objects.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §6
	A reaction: The key debate among structuralists seems to be whether or not they are committed to 'objects'. Fine rejects the 'austere' version, which says that objects have no properties. Either version of structuralism can have abstraction as its basis.

9. Objects / A. Existence of Objects / 3. Objects in Thought

9825	A thing is completely determined by all that can be thought concerning it [Dedekind]
	Full Idea: A thing (an object of our thought) is completely determined by all that can be affirmed or thought concerning it.
	From: Richard Dedekind (Nature and Meaning of Numbers [1888], I.1)
	A reaction: How could you justify this as an observation? Why can't there be unthinkable things (even by God)? Presumably Dedekind is offering a stipulative definition, but we may then be confusing epistemology with ontology.

18. Thought / E. Abstraction / 3. Abstracta by Ignoring

9189	Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dedekind, by Dummett]
	Full Idea: By applying the operation of abstraction to a system of objects isomorphic to the natural numbers, Dedekind believed that we obtained the abstract system of natural numbers, each member having only properties consequent upon its position.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Michael Dummett - The Philosophy of Mathematics
	A reaction: Dummett is scornful of the abstractionism. He cites Benacerraf as a modern non-abstractionist follower of Dedekind's view. There seems to be a suspicion of circularity in it. How many objects will you abstract from to get seven?

9827	We derive the natural numbers, by neglecting everything of a system except distinctness and order [Dedekind]
	Full Idea: If in an infinite system, set in order, we neglect the special character of the elements, simply retaining their distinguishability and their order-relations to one another, then the elements are the natural numbers, created by the human mind.
	From: Richard Dedekind (Nature and Meaning of Numbers [1888], VI.73)
	A reaction: [compressed] This is the classic abstractionist view of the origin of number, but with the added feature that the order is first imposed, so that ordinals remain after the abstraction. This, of course, sounds a bit circular, as well as subjective.

18. Thought / E. Abstraction / 8. Abstractionism Critique

9979	Dedekind has a conception of abstraction which is not psychologistic [Dedekind, by Tait]
	Full Idea: Dedekind's conception is psychologistic only if that is the only way to understand the abstraction that is involved, which it is not.
	From: report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by William W. Tait - Frege versus Cantor and Dedekind IV
	A reaction: This is a very important suggestion, implying that we can retain some notion of abstractionism, while jettisoning the hated subjective character of private psychologism, which seems to undermine truth and logic.

23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues

20239	Unlike us, the early Greeks thought envy was a good thing, and hope a bad thing [Hesiod, by Nietzsche]
	Full Idea: Hesiod reckons envy among the effects of the good and benevolent Eris, and there was nothing offensive in according envy to the gods. ...Likewise the Greeks were different from us in their evaluation of hope: one felt it to be blind and malicious.
	From: report of Hesiod (works [c.700 BCE]) by Friedrich Nietzsche - Dawn (Daybreak) 038
	A reaction: Presumably this would be understandable envy, and unreasonable hope. Ridiculous envy can't possibly be good, and modest and sensible hope can't possibly be bad. I suspect he wants to exaggerate the relativism.