Introduction and Peano

AssignedQuestionsIntroductionandPeano

ResponsePapersIntroductionandPeano

-- ShaughanLavine - 01 Sep 2006

False or misleading claims in what we read

1. Frege was little read.
2. Paradoxes produced a crisis in mathematics (though they did in "foundations of mathematics").
3. Burali-Forti discovered a paradox. !. Hilbert took mathematics to be about uninterpreted strings of symbols.

I assigned the Kneebone article for two reasons: It gives a rough overview of logicism, formalism, intuitionism. We shall get a more accurate picture as the semester goes on. Kneebone's presentation of his "dialectical logic of mathematics" includes a description of ways in which mathematics changes, evolves, involves the replacement or embedding of theories in successor theories, and the like complete with examples.

We shall often look at simple examples like embedding the natural numbers in the rational numbers. It is important to be aware that allied procedures play an important role in what mathematicians actually do, even if the real examples that are mathematically important are sufficiently récherché that we shall avoid them. Though we shall emphasize deriving results from axioms, note that our main subject is introducing new mathematical objects/structures. That is a critical task within mathematics and often the main innovation involved in making a breakthrough. (The work of Ken Manders argues that in some detail.) Thus, understanding how new mathematical objects/structures get introduced is important not just for rigor or foundational purposes, but is a question of great internal importance to mathematics. It is easy to lose track of that when our main examples are so elementary.

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Peano's axioms are preceded by Peano's explanations, which "no one" mentions when they talk about the axioms. The explanations include, for example, "the sign %$n+1$% means the successor of %$n$%." Thus, when Russell says that the axioms are compatible with taking the successor to be %$n+2$%, while he may be right about that, he is ignoring something about Peano's characterization of the numbers, namely, that that is not compatible with Peano's explanations.

Russell ignores the explanations, but it is clear that they play a real role in what Peano thinks he is doing, at least in this article. Kennedy suggests that Peano may have moved some distance toward Russell's picture in later work. Peano's explanations are a lot like some of Euclid's axioms: "a point has no extension." He, unlike traditional translations of Euclid, distinguishes the explanations from the axioms, but he is doing much the same thing as Euclid is ordinarily described as having done, he is isolating certain basic principles about an antecedently known domain as being sufficient to enable the deduction of everything we know about the domain. His axioms represent a substantial piece of mathematical analysis: what we can take to be the fundamental principle about numbers, and that is what he takes himself to be doing. It is no part of his project, or least not a central part, to "define" 0, number, and successor.

Peano even seems to think that we cannot define 0, number, successor. He is said to have said that Russell's definitions define things that are simple in terms of things that are more complex. From the point of view of the mathematician trying to prove theorems of elementary number theory (who are, I think, Peano's intended audience) there is no question that that is correct.

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So, what is that Russell wants that Peano does not provide? What he says is that he wants a "definition" of 0, successor, number. Definition has to stop somewhere. Russell wants to push it as far back as he can. He wants some basis on which we can introduce all of the rest of mathematics, and he believes he has found it in "logic," his theory of classes.

Is he right that Peano's axioms do not suffice for applications? He says if we start with 100 instead of 0 then we will not have ten fingers. It seems to me that 0 is the number that '0' stands for, 1 is its successor, and so on. If we count in terms of Peano's theory, it will turn out that we have %$0+1+1+1+1+1+1+1+1+1+1$% fingers, in the notation of the theory. It is not at all clear that anything has in fact gone wrong.

Liz asks whether we need to take applications to be fundamental. Can't we just take the axioms and mathematics to be quite independent of applications? Russell, Frege, Hilbert, Dummett, Wright, Hale and many others have taken the application of the natural numbers to counting to be fundamental to what the numbers are. Cantor and Benacerraf agree that it is necessary that the numbers be suitable for application. Dedekind doesn't seem to think that that is true, though he presumably does think that his numbers can be so applied. Frege argues that merely formal systems of numbers (Thomae) are defective in part because they are not suitable for counting.

The argument that I gestured at is in the tradition of Dedekind and Benacerraf, and it leads toward the conclusion that any system of numbers that obeys the Peano axioms is suitable for counting, and so the question doesn't arise in any simple form. When you move beyond the natural numbers to the real numbers, related questions do arise. According to Frege, just as the natural numbers are "for" counting, the real numbers are ratios of quantities. For that reason, he argues that it is a mistake to first introduce the rational number then the real numbers, that is, as he says, to introduce the real numbers "piecemeal," since the rational numbers don't have any separate use. (That story is told in Dummett91.

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What the taught-us said to a-kill-ease. The usual way the distinction Lewis Carroll is pointing at is analyzed today is in terms of the distinction between an axiom and a rule:

From the axiom %$A\land (A \rightarrow B) \rightarrow B$% nothing at all follows. Axioms just sit there on the page. We need a rule:

From %$A$% and %$A \rightarrow B$%, you may infer %$B$%.

Not just Peano, but everyone in his era lacked anything like that distinction until Frege.

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Recursive definition:

Peano says:

A definition, or Def. for short, is a proposition of the form [%$x=a$% or %$x$% if and only if %$a$% or from %$\alpha$% one deduces %$x=a$% or from %$\alpha$% one deduces %$x$% if and only if %$a$%], where %$a$% is an aggregate of signs having a known meaning, %$x$% is a sign or an aggregate of signs, hitherto without meaning, and %$\alpha$% is the condition under which the definition is given.

He then "defines"

From %$a$% and %$b$% are in %$N$% one deduces %$a+(b+1)=(a+b)+1.$%

How can that possibly be a definition of addition, as Peano claims: it has the + sign on both sides, and hence doesn't have only signs with a "known meaning" on the right. Here is his answer:

Note. This definition has to be read as follows: if %$a$% and %$b$% are numbers, and if %$(a+b)+1$% has a meaning (that is, if %$a+b$% is a number) but %$a+(b+1)$% has not yet been defined, then %$a+(b+1)$% means the number that follows %$a+b$%.

What does that note mean? Since we need to define addition from something already defined, it is confusing that Peano takes %$+1$%, which has a + sign in it as already defined: the '+' never disappears. To make matters clearer, I shall use %$S$% to mean successor. Peano's "explanation" of %$+1$% then becomes %$a+1=Sa$% and his definition of addition becomes %$a+Sb=S(a+b).$%

We can now, for example, add 2+4 (that is, %$S1+SSS1$%) in virtue of the following sequence of equations, each of which is either

• an instance of %$a+1=Sa$% or of %$a+Sb=S(a+b)$% (that is, is a definition), or
• results from some prior line %$n$% by substituting the right-hand side of some prior line %$m$% for an occurrence of left-hand side of line %$m$% in line%$n$% (that is, "substituting equals for equals"), and
• in each of which no expressions involving + occur on the right-hand side that have not already been defined on a previous line (that is, that occur on the left-hand side of a prior line):

1. %$S1+1=SS1$% (explanation, %$a=S1$%)
2. %$S1+S1=S(S1+1)$% (definition, %$a=S1$%, %$b=1$%)
3. %$S1+S1=SSS1$% (substitution of line 1 in line 2)
4. %$S1+SS1=S(S1+S1)$% (definition, %$a=S1$%, %$b=S1$%)
5. %$S1+SS1=SSSS1$% (substitution of line 3 in line 4)
6. %$S1+SSS1=S(S1+SS1)$% (definition, %$a=S1$%, %$b=SS1$%)
7. %$S1+SSS1=SSSSS1$% (substitution of line 5 in line 6)

-- ShaughanLavine - 05 Sep 2006