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13861
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Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C]
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Full Idea:
In the Fregean view number theory is a science, aimed at those truths furnished by the essential properties of zero and its successors. The two broad question are then the nature of the objects, and the epistemology of those facts.
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From:
Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro)
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A reaction:
[compressed] I pounce on the word 'essence' here (my thing). My first question is about the extent to which the natural numbers all have one generic essence, and the extent to which they are individuals (bless their little cotton socks).
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13892
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One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C]
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Full Idea:
Someone could be clear about number identities, and distinguish numbers from other things, without conceiving them as ordered in a progression at all. The point of them would be to make comparisons between sizes of groups.
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From:
Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xv)
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A reaction:
Hm. Could you grasp size if you couldn't grasp which of two groups was the bigger? What's the point of noting that I have ten pounds and you only have five, if you don't realise that I have more than you? You could have called them Caesar and Brutus.
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17441
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Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are [Wright,C, by Heck]
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Full Idea:
Wright is claiming that HP is a special sort of truth in some way: it is supposed to be the fundamental truth about cardinality; ...in particular, HP is supposed to be more fundamental, in some sense than the Dedekind-Peano axioms.
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From:
report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 1
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A reaction:
Heck notes that although PA can be proved from HP, HP can be proven from PA plus definitions, so direction of proof won't show fundamentality. He adds that Wright thinks HP is 'more illuminating'.
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13862
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There are five Peano axioms, which can be expressed informally [Wright,C]
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Full Idea:
Informally, Peano's axioms are: 0 is a number, numbers have a successor, different numbers have different successors, 0 isn't a successor, properties of 0 which carry over to successors are properties of all numbers.
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From:
Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro)
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A reaction:
Each statement of the famous axioms is slightly different from the others, and I have reworded Wright to fit him in. Since the last one (the 'induction axiom') is about properties, it invites formalization in second-order logic.
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10164
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Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
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Full Idea:
A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
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A reaction:
This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'!
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10140
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We derive Hume's Law from Law V, then discard the latter in deriving arithmetic [Wright,C, by Fine,K]
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Full Idea:
Wright says the Fregean arithmetic can be broken down into two steps: first, Hume's Law may be derived from Law V; and then, arithmetic may be derived from Hume's Law without any help from Law V.
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From:
report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Kit Fine - The Limits of Abstraction I.4
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A reaction:
This sounds odd if Law V is false, but presumably Hume's Law ends up as free-standing. It seems doubtful whether the resulting theory would count as logic.
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8692
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Frege has a good system if his 'number principle' replaces his basic law V [Wright,C, by Friend]
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Full Idea:
Wright proposed removing Frege's basic law V (which led to paradox), replacing it with Frege's 'number principle' (identity of numbers is one-to-one correspondence). The new system is formally consistent, and the Peano axioms can be derived from it.
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From:
report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.7
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A reaction:
The 'number principle' is also called 'Hume's principle'. This idea of Wright's resurrected the project of logicism. The jury is ought again... Frege himself questioned whether the number principle was a part of logic, which would be bad for 'logicism'.
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17440
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Wright says Hume's Principle is analytic of cardinal numbers, like a definition [Wright,C, by Heck]
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Full Idea:
Wright intends the claim that Hume's Principle (HP) embodies an explanation of the concept of number to imply that it is analytic of the concept of cardinal number - so it is an analytic or conceptual truth, much as a definition would be.
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From:
report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 1
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A reaction:
Boolos is quoted as disagreeing. Wright is claiming a fundamental truth. Boolos says something can fix the character of something (as yellow fixes bananas), but that doesn't make it 'fundamental'. I want to defend 'fundamental'.
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10167
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Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
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Full Idea:
Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
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A reaction:
In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent.
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10169
|
Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
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Full Idea:
Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)
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A reaction:
The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight?
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10179
|
There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
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Full Idea:
The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6)
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A reaction:
This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start.
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10182
|
There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
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Full Idea:
There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9)
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A reaction:
I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind?
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10168
|
Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
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Full Idea:
Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3)
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A reaction:
[very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations.
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10178
|
Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
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Full Idea:
It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
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A reaction:
[compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent.
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10177
|
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
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Full Idea:
Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out.
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From:
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
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A reaction:
I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra.
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13870
|
We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism [Wright,C]
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Full Idea:
We may not be able to settle whether some general form of empiricism is correct independently of natural numbers. It might be precisely our grasp of the abstract sortal, natural number, which shows the hypothesis of empiricism to be wrong.
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From:
Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i)
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A reaction:
A nice turning of the tables. In the end only coherence decides these things. You may accept numbers and reject empiricism, and then find you have opened the floodgates for abstracta. Excessive floodgates, or blockages of healthy streams?
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13899
|
The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals [Wright,C]
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Full Idea:
The Peano Axioms are logical consequences of a statement constituting the core of an explanation of the notion of cardinal number. The infinity of cardinal numbers emerges as a consequence of the way cardinal number is explained.
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From:
Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xix)
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A reaction:
This, along with Idea 13896, nicely summarises the neo-logicist project. I tend to favour a strategy which starts from ordering, rather than identities (1-1), but an attraction is that this approach is closer to counting objects in its basics.
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13895
|
The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems [Wright,C]
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Full Idea:
The general view is that Russell's Paradox put paid to Frege's logicist attempt, and Russell's own attempt is vitiated by the non-logical character of his axioms (esp. Infinity), and by the incompleteness theorems of Gödel. But these are bad reasons.
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From:
Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xvi)
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A reaction:
Wright's work is the famous modern attempt to reestablish logicism, in the face of these objections.
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