In my previous two posts, I outlined the Platonist view of mathematics, and the empiricist alternative. There are also two other alternatives:
The truths of mathematics appear to be different in nature from the truths of physics. Formalism accepts this, but suggests that the nature of mathematics is inherently cultural. Different branches of mathematics are essentially just games with symbols and arbitrary rules – games that don’t have any particular meaning. Mathematicians simply work within the chosen rules. However, apart from the problem of the “unreasonable effectiveness of mathematics,” the idea that these rules are chosen arbitrarily runs counter to the experience of most mathematicians. When the concept of “number” was extended to include the imaginary numbers, for example, consistency with the existing rules meant that there was very little choice about how imaginary numbers behaved. In the words of mathematician Jacques Hadamard: “We speak of invention: it would be more correct to speak of discovery… Although the truth is not yet known to us, it pre-exists and inescapably imposes on us the path we must follow under penalty of going astray” (from the introduction to The Psychology of Invention in the Mathematical Field).
Many officially formalist mathematicians are Platonists at heart. Jean Dieudonné once wrote with refreshing honesty: “On foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hide behind formalism and say: ‘Mathematics is just a combination of meaningless symbols,’ and then we bring out Chapters 1 and 2 on set theory. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working on something real. This sensation is probably an illusion, but is very convenient.” (from “The Work of Nicholas Bourbaki,” American Mathematical Monthly, 77(2), Feb 1970, p. 134–145).
Most mathematicians feel that the truths of mathematics are indeed in a different category from the truths of physics – that the truths of mathematics in a sense come first. Logicism is a way of rescuing this aspect of Platonism while avoiding the more mystical aspects. The basis for logicism is that logic also comes before physics – all sciences assume logical thought as a starting point. Logical truths “exist” in some sense, and logicists assume that there are no philosophical difficulties about this kind of existence. In other words, a mystical Platonic world is not needed to explain logic. Consequently, if we can provide a foundation for mathematics in terms of pure logic, we can retain all the benefits of Platonism without any of the problems.
In logicism, numbers are defined as being particular kinds of sets. Logicism began with Gottlob Frege, who published two volumes of his Die Grundgesetze der Arithmetik in 1893 and 1903. Sadly for Frege, his fellow mathematician and philosopher Bertrand Russell found a major flaw – now known as “Russell’s paradox” – in the work, just before the second volume was published. With Alfred North Whitehead, Russell was able to repair the flaw, in a three-volume work called Principia Mathematica (published in 1910, 1912, and 1913).
The logicist programme, however, is not free of problems. First, it is extremely complex. It took Whitehead and Russell hundreds of pages of complicated logic to prove that 1 + 1 = 2. Normally, we try to explain complex things in terms of simple ones. It seems a little perverse to give such a complicated explanation of numerical facts that we understood in kindergarten. And, by including set theory as part of the basis, it isn’t really “pure logic” any more.
Second, there is more than one way of defining numbers as sets, and none of them is obviously “right.” This has led to the suggestion that sets are “what numbers could not be” (the title of an article by Paul Benacerraf in The Philosophical Review, 74, Jan 1965, pp. 47–73), and that numbers must be fundamentally different in nature from sets – if not Platonic objects satisfying certain axioms, then something else which exists in a non-contingent way.
Third, it is unclear whether logicism has actually gained anything. The starting assumption was that logic was simple and obvious, raising no philosophical problems. But if all of mathematics is hidden deep inside the structure of logic, then perhaps logic is not as simple as it first seemed. Mathematics and logic may in fact be different aspects of the same thing, but this may not make the fundamental questions about mathematical existence go away.
Personally, I still see Platonism as the best answer. How about you?