tq-System

The tq-system is a formal system that consists of the following components:

1. Axioms: $xqxt-$
2. Inference Rule: If $xqytz$ is a theorem, then $xyqytz-$ is a theorem.

It’s easy to see that the theorems of the tq-system are of the form $xqytz$, and must meet $x=y\times z$.

Describe Composite and Prime Numbers

It’s easy for us to describe composite numbers. We could utilize the tq-system to define a new set of theoroms of the form $Cx$.

Inference Rule: If $xqy-tz-$ is a theorem, then $Cx$ is a theorem.

This system works because $x=(y+1)(z+1)$, which means $x$ could be multiplied by two numbers greater than 1. Then the $x$ of all the $Cx$ theoroms are all the composite numbers.

However, when it comes to prime numbers, things get a little bit tricky. We could not define a set of theoroms of the form $Px$ to describe prime numbers. Because prime numbers are indivisible, we could not find two numbers that could multiply to get a prime number.

We may create a new set of theoroms $Px$ with the rules “if $Cx$ is not a theorem, then $Px$ is a theorem”. The fatal flaw here is that checking whether $Cx$ is not theorem is not an explicitly typographical operation. To know whether a theorem is not of a system, we have to go outside of the system.

Figure and Ground

The relation between prime numbers and composite numbers makes people recall the famous artistic distinction between figure and ground. This concept is not only applicable to the field of painting, but also to the field of music, literature, and even mathematics.

This picture shows a theorems tree. The tree provides a considerable visual representation of the relationship between the theorems and the negations of theorems. More importantly, it reveals the Gödel’s Incompleteness Theorems in a more intuitive way. There exists theorems that are neither provable nor disprovable within the system which are unreachable truths.

Recursively Enumerable Sets vs. Recursive Sets

It is intuitive to think that a figure and its ground carry exactly the same amount of information. However, in the field of mathematics, this is not the case. It is fact that, there exist formal systems whose negative space is not the positive set of any formal system. In a more thechnical term, there exist recursively enumerable sets that are not recursive. We could conclude that typographical decision procedures do not exist for all system.

Hofstadter left a puzzle for us to think about:

Can you characterize the following set of integers

$$1\quad 3\quad 7\quad 12\quad 18\quad 26\quad 35\quad 45\quad 56 \cdots$$

The “ground” of this set is

$$2\quad 4\quad 5\quad 6\quad 8\quad 9\quad 10\quad 11\quad 13\cdots$$

Ground here has two different meanings: 1. The diff between any two consecutive numbers in the set; 2. All the numbers that are not in the set.

This puzzle is a good example of the concept of figure and ground.

Primes as Figure Rather than Ground

We skip right over multiplication, and go straight to the concept of nondivisibility. Here an axiom schema and rule for producing theorems which represent the notion that one number does not divide(DND) another number exactly.

Axioms: $xyDNDx$
Inference Rule: If $xDNDy$ is a theorem, then $xBZCxy$ is a theorem.

So the solution of describing prime numbers is as follows:

Axioms: $P--$. ($2$ is a prime number.)
Inference Rules:

1. If $--DNDz$ is a theorem, then $zDF--$ is a theorem. ($z$ is not divisible by $2$, then $z$ is divisor free up to $2$.)
2. If $zDFx$ is a theorem and also $x-DNDz$ is a theorem, then $zDFx-$ is a theorem. ($z$ is divisor free up to $x$ and $x+1$ is not divisible by $z$, then $z$ is divisor free up to $x+1$.)
3. If $z-DFz$ is a theorem, then $Pz-$ is a theorem. ($z+1$ is divisor free up to $z$, then $z$ is a prime number.)

$DF$ here means the number is divisor free up to x.

And there we have it. The principle of representing primality formally is that there is a test for divisibility which can be done without any backtracking.

Contracrostipunctus

The contracrostipunctus is a piece of music that is a counterpoint to itself when played backwards. Hofstadter wrote a poem which is not noly a contracrostipunctus, but also a contracrostipunctus in meaning. The poem is a palindrome, and the meaning of the poem is also a palindrome. More interestingly, the poem is a acrostic, which means the first letter of each line spells out a word.

HOFSTADTER’S CONTRACROSTIPUNCTUS ACROSTICALLY BACKWARDS SPELLS J.S. BACH

赫赫有名的德国作曲家给了我侯世达灵感 在此我借用他的对位技巧写下一个对话并嵌进他的名字以表示我对他卓越才能由衷的景仰 大家也许都记得他曾把他的名字写进一首赋格曲的尾巴