Tuesday, August 11, 2020

2 + 2 = 5

"One must be able to say at all times - instead of points, straight lines, and planes -- tables, chairs, and beer mugs." -- David Hilbert (as quoted by Constance Reid, Hilbert, Springer-Verlag, 1970: p. 57)

Consider the Fibonacci sequence: 1, 2, 3, 5, 8, 13, ... The first two terms in this sequence are 1 and 2. After this, each term is the arithmetic sum of the previous two terms. Let A be the set of elements in this series. Let s be the function mapping an element in A onto another element of A, where s(n) is the term in the sequence following n. I want to read s as "successor". (I do not claim originality for the following use of the Fibonacci series.)

Notice that the set A and the successor function satisfy the following properties:

  1. 1 is in the set A.
  2. For any n in the set A, its successor s(n) is in the set A
  3. There does not exist an elment n of A such that its successor s(n) = 1.
  4. For all n and m in A, if s(n) = s(m), then n = m.
  5. Suppose a set B contains 1. And further suppose that for any n in B, the successor s(n) is also in B. Then the set A is a subset of B.

The above five properties are the Peano Axioms. The last property is known as the principle of induction.

Some, such as the formalist David Hilbert, would say that mathematical objects are defined by axioms. And the above axioms define the set of natural numbers. Taking the succesor of a number is defined as adding one. And you can define taking the succesor of the successor of a number as adding 2. So, by these definitions, one has:

1 + 1 = s(1) = 2
2 + 2 = s(s(2)) = 5

(By the way, Hilbert was what the kids these days call "woke":

"David Hilbert sought to bring [Emmy Noether] into the mathematics department at the University of G├Âttingen in 1915, but other faculty objected. 'What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?' one professor complained. Hilbert was indignant. 'I do not see that the sex of the candidate is an argument against her admission,' he retorted. 'We are a university, not a bath house.' (See here.))

Bertrand Russell objected to formalism. He thought that whatever mathematicians meant by numbers, these meaning should allow us to assert that most people have one nose, two eyes, and five fingers on each hand. I find his Introduction to Mathematical Philosophy the most approachable statement of his logicist position. Russell found a way to define numbers based on set theory. He required the universe to contain a countable infinity of things, in some sense. His theory of types was a weakness of his approach. Frank Ramsey comes into the story here, with a theory I do not understand. Anyways, various ways can be found to define number in Zermelo-Fraenkel set theory, which I gather avoid the problems with the theory of types.

But this story is incredible. Did people not know how to count or what they were doing until the twentieth century? Maybe a logical foundation for mathematics is not needed. Rather, one might try to provide an anthropological description of what people do when they are counting, adding, multiplying, manipulating infinite series, and so on. Does this capture the "must" in following a rule, though? When a judge consults a law book, he does not treat it as a work in anthropolgy. And who is in doubt on how to proceed when they come to a sum they have never calculated before? Here I am gesturing towards the work of Ludwig Wittgenstein.

I am near my intellectual limits, but I know I have hardly exhausted positions that have been taken over the last century on the philosophy of mathematics. By the way, if I take Jean Piaget seriously, the most advanced mathematics some tried to teach me is crucial to understanding how children think and important for anybody that wants to attempt to design curricula for mathematics.

This post should have a bibliography.


Anonymous said...


Blissex said...

Dear blogger, it so happens that I have done in the past extensive readings in these topics, and they are quite subtle, and very often misdescribed or even misunderstood by mathematicians; because the practice of mathematics is sort of a rather experimental practice, especially as all the easy or simple results seem to have been obtained, and a lot of current mathematical research is mostly on very difficult, very obscure, or very complicated issues. Also most mathematical research is not very rigorous, and is almost "experimental" in nature.

The issues you describe arise from the "paradox"/"self contradiction" crisis of the 19th century, which then resulted in metamathematical investigation of what "to prove" means, and how provability and truth are different concepts. and that involves really very subtle issue that most mathematicians disregard as most mathematical research is not very rigorous (lots of mathematical work has "bugs"), and is almost "experimental" in nature, especially when it comes to computational methods. Hilbert hoped to avoid most of them by taking shortcuts (to avoid issues of semantics), but it turns out that is not possible.

Some interesting perspectives can be gained by looking at so called "weak logics" (where it is accepted that false theorems can be proven), modal/intensional logic and their Tarski/Kripke models.

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