Thursday, 24 July 2014
Frege's prime purpose for writing his well-known and important Foundations of Arithmetic was to show us that mathematics is really analytic; as well as to disprove Kant’s view that it is synthetic a priori. In that, Frege was at one with Hume. This analyticity of mathematics, according to Frege, could only be proved and shown by reducing mathematics to the elementary laws of logic – hence ‘logicism’.
Frege took these logical laws to be more basic than any truths and laws in mathematics because they "must be accepted if there is to be reasoning at all" (385). It can be said, therefore, that Frege’s position on the logical laws is not unlike Aristotle’s on his ‘laws of thought’ which are required in all reasoning, even reasonings which deny their truth or dispute their fundamental nature.
Interestingly enough, Leibniz was basically a proto-logicist. He provides these proofs that arithmetical statements can be expressed logically:
2 = df. 1+1
4 = df. 1+1+1+1
2+2 = df. 1+1+1+1 = 4
As can be seen, however, Leibniz still uses numbers in his ‘logical’ reductions of numbers and arithmetical statements. In a sense, every reductionist logical definition only uses the number 1, along with the equality sign and other operators.
However, surely 4 + 2 = 6 is more illuminating than 1+1+1+1+1+1 = 6 because that too could become 1+1+1+1+1+1 = 1+1+1+1+1+1 and so on.
In one sense Leibniz is also committed to a proto-extensionalist logic in which numerals can be substituted within an arithmetical statement if the substitutions have the same extension – the same number – as its extension.
However, Leibniz’s reduction does not function as correct classical logic because it misses out the brackets needed in his:
2+2 = df. 1+1+1+1
It should be this:
2+2 = df. (1+1) + (1+1)
In other words, without brackets we don't recognise the logical scope of the original arithmetical operators in their statements. That is, 2 = df. (1+1), not 2 = df. 1+1. Similarly, 2 + 2 = df. (1+1) + (1+1), not the initial 2+2 = df. 1+1+1+1.
S puts it this way: "What entitles us to drop the brackets and convert (1+1) + (1+1) into (1+1+1+1)?" (386). The operation + enables us to add 2 + 2, so 1+1+1+1 isn't allowable because it is 2 that is added to 2, not 1 + 1 that is added to 1 + 1. Not only that: these brackets show us the scope of the arithmetical ‘2’ in terms of the operator of addition. So if Leibniz reduces it to 1 + 1+ 1 + 1 only by illicitly or tacitly using a mathematical operator in his ‘reduction’. And if he has done that, then he has not reduced arithmetic or mathematics to logic at all (just as a Tarskian meta-language cannot use terms from the object-language).
Dedekind, at the end of the 19th century, reduced the basic notions of arithmetic (rational, real and complex numbers) to the theory of natural numbers, if not to logic itself. Of course we need to know what natural numbers are, and how they differ from rational, real and complex numbers.
Peano too reduced arithmetic to a set of axioms. Peano’s ‘postulates’, of course, are far better known than anything offered by Frege or Dedekind, for example. What are Peano’s postulates or axioms? These:
i) 0 is a number.
ii) Every number has at least one and at most one successor which is a number.
iii) 0 is not the successor of any number.
iv) No two numbers have the same successor.
v) Whatever is true of 0, and is also true of the successor of any number when it is true of that number, is true of all numbers.
It can be seen that Peano’s postulates are intuitively acceptable and also very simple in nature. Presumably he said that ‘0 is a number’ because other mathematicians and philosophers didn't actually think this.
In terms of postulate number ii). If every number has one successor, then this by definition seems to create or accept an infinite class of numbers. In addition, if the number 0 is not itself a successor of a number, then Peano must have rejected negative numbers like -1 and -44 and so on. They must have come later, as it were.
However, postulate v) seems to be incorrect. It says that whatever "is true of 0… must be true of all numbers". But postulate iii) has already claimed that "0 is not the successor of any number". Not being a successor, then, is a property of 0, so how can ‘all numbers’ have the same properties as 0? More correctly, how can the statement "whatever is true of 0… is true of all numbers" be true? If not true, then correct according to its other fellow axioms, specifically axiom three.
The last axiom just stated, interestingly enough, is the ‘ell-known axiom of mathematical induction. In other words, we have a strange juxtaposition of induction and a mathematical axiom. This is especially interesting because many philosophers and logicians say that the so-called ‘logical law of induction’ is not a genuine logical law at all, primarily because it deals with probabilities and not necessities and also, for example, because induction is a psychological phenomenon; at least according to Wittgenstein.
Anyway, the fifth axiom is inductive in nature because it "enables us to prove theorems about all the numbers by considering only three of them" (386). In other words, what is said to be the case in three of Peano’s axioms can be used inductively to show why the other two axioms are true, and also true about the nature of all numbers. So the axioms themselves, when taken separately, are themselves a micro-deductive or inductive system in that it has two ‘meta’-axioms, from which two lower axioms, as it were, can be deductively derived. And when we have all Peano’s postulates together, then we can again derive things; though this time theorems, not more axioms. Indeed not only can the axioms engender theorems and also two more axioms, but what is said or stated in them about numbers provides the basis of a mathematical system in which pure numerical axioms can be used to derive more numerical theorems from them (just as logical premises engender conclusions).
According to Peano’s postulates, all of arithmetic can be derived from them.
Are his postulates logical in nature?
In terms of logicism, the logicist seeks to define the three primitive terms – ‘number’, ‘successor’ and ‘0’ – and show that the postulates can be derived by logic from the definitions. So, in that sense, Peano carried on the programme begun by Frege a few decades earlier.
How were the primitive terms shown by Peano and the logicists to be explainable in terms of logic and logical terms?