Axiomatic Poems

This is a page devoted to collect information on axiomatic poems.

Introduction to axiomatic poems -- Peano’s string; a history of spiritual stories.

Axiomatic Poems part two -- More structure added to Peano’s string; a history of spiritual stories.

The addition of another stanza and creating a metamorphic poem.

Proof that no cat is the God of itself (Peano’s proof by Professor Ray Balbes)

Proof That No Cat Is The God Of Itself

The Mathematician, Professor Ray Balbes will prove to you that “No Cat Is The God Of Itself”.

At the end of my blog entry for Axiomatic poems dated January 29, 2008 I asked the question, “Can these axioms create interesting theorems?” And the answer is definitely yes! Professor Balbes has shown us four theorems that are proven from the Peano’s axioms. Due to the poetic nature of the new axioms, not only are the four theorems poetic but even the proof of theorem #1 is poetic. Furthermore, the choices Professor Balbes made for the terminology in his proof shows his perceptions of how the poetic nature of the axioms should extend. Therefore, there can be uncountable variations of poetic form in the proofs one could make for the theorems. I find this very exciting.

Kaz

The Professor Ray Balbes wrote the following text:

For reference, here is what we have so far.

The Peano Axioms

1. One is a number
2. If x is a number, the successor of x is also a number.
3. One is not the successor of any number.
4. If two numbers have equal successors, they are equal.
5. If a set of numbers contains the number one and it contains all the successors of its members then the set contains all the numbers.

Let us replace “number” with “cat” and let us also replace “successor” with “God”. Lastly, I am going to replace “One” with “Abraham”.

The Poetic Peano Axioms

1. Abraham is a cat
2. If x is a cat, the God of x is also a cat.
3. Abraham is not the God of any cat.
4. If two cats have equal Gods, they are equal.
5. If a set of cats contains the cat Abraham and it contains all the Gods of its members then the set contains all the cats.

OK, now to make the theorems more succinct, lets set up some conventions. With regard to the Peano Axioms, let us call the set of all numbers N and let us denote by n’, the successor of n

Also let:

1₁ = 1’

1₂ = 1’’

1₃ = 1’’’

etc.

We will refer to 1’ by the name of 2, 2’ will be called 3, etc.

Axiom 3 says that there is no n such that n’=1.

Axiom 4 says that if m’ = n’ then m=n.

Axiom 5 says that if S is a non-empty subset of N with these 2 properties:

i) 1 is in S

ii) If n is in S then n' is in S.

Then S = N.

Here are three theorems that lead up to the Well Ordering Principle. First, I will state them in terms of the Peano Axioms, next in terms of the Poetic Peano Axioms and finally I will prove something.

Theorem 1. For every n in N, n’≠n.

Theorem 2. If n ≠ 1 then n=m’ for some m.

We will say that m ≤ n provided that m = n or m_p = n, for some p

Theorem 3. For every n in N, 1 ≤ n

Theorem 4 (The Well Ordering Principle) If S is any non empty subset of N then there is a number m in S such that m ≤ n for all n in S.

Here are the theorems in terms of the Poetic Peano Axioms. We will say that m is the source of n provided that m ≤ n. In other words, a finite number of Gods of m, yields n.

Theorem 1 No cat is the God of itself.

Theorem 2. Every cat, other than Abraham is the God of some other cat.

Theorem 3. Every cat has Abraham as a source.

Theorem 4 (The Well Ordering Principle) In any (non-empty) set of cats, there is one that is the source of all the others.

Here is the proof of Theorem 1 in terms of the Peano Axioms

Let S = {n| n’ ≠ n}. We will show that S satisfies the conditions i) and ii) of Axiom 5. By Axiom 3, 1 is in S so i) is true. To prove ii), suppose that n is in S then n’≠n. But if n’’=n’ then, by Axiom 4, we would have n’=n, a contradiction, so n’’<>n’. Hence n’ is in S. This means that S satisfies the conditions of Axion 5 and therefore S= N. So that n’<>n for all n in N.

Here’s the proof of Theorem 1 in terms of the Poetic Peano Axioms. Note that in the proof, I am referring to the Poetic Peano Axioms, not the Peano Axioms.

Consider the set S of all cats that are not Gods of themselves. We will show that S satisfies the conditions i) and ii) of Axiom 5. By axiom 3, Abraham is a member of S so i) is true. To prove ii), suppose that Isaac is a cat in S then Isaac is not the God of Isaac. Suppose the God of Isaac is Moishe. Now if the God of Moishe is Moishe then by Axiom 4, Moishe would be Isaac; that is the God of Isaac would be Isaac, a contradiction. Hence Moishe is in S. Since Moishe is the God of Isaac, we have shown that the God of Isaac is in S; in other words, the condition ii) of Axiom 5 is satisfied and thus S is the set of all cats. This means that all cats satisfy the property that they are not Gods of themselves.

The proofs of the other theorems are similar to this.

Ray

Axiomatic Poems Part Two

I have been having some wonderful conversations with the mathematician Ray Balbes. Ray has been asking some very important questions concerning the axiomatic poem. Ray has also helped me by correcting mathematical errors in my nomenclature.

Ray also has had concerns with the idea of God being a viable substitute for successor within the Peano axioms. For God in this sense must be comparable to a mathematical function. I personally have no problem with this idea for my understanding of the word God is metaphorical anyway. Therefore, I can see this metaphoric structure of “God IS mathematical function” as being nested e.g. metaphors within metaphors. The question then would be is God a mathematical function? Alternatively, can we say God functions mathematically? Historically God is described beyond language so I would not try to convince anyone otherwise. I personally do not see God functioning mathematically as a mathematical Platonist would however, I do see the accessibility of ideas mathematically expressed as phenomena attributed to a deity. I believe if you denote phenomena with words, you can do the same with math. Furthermore, I would go on to say that if you can be inspired to connote it with words you can do the same with math for those type of inspirations fuel mathematical poetry.

Therefore, the poem addresses the dichotomy of God being created by men or men being created by God.

To help anyone see how the logic in Peano’s axioms is functioning correctly in the Blog entry of January 29^th, I created another axiomatic poem to show some more structure. The disadvantage to creating another ‘equal’ poem is that the new poem focuses the semantics in such a way that limits the metaphorical content. The advantage is that it gives more semantic structure, which enables one to see the Peano logic with ease. So in essence, we now have an axiomatic poem, which has metamorphic qualities. We see that the Peano axioms function as the underlying paradigm for the poem however, it could be viewed as the source domain with the other two ‘axiomatic stanzas’ as the target domains for the ‘overall metaphor’. In this case, we have three structures separated by two equal signs.

The Peano Axioms

1. One is a number
2. If x is a number, the successor of x is also a number.
3. One is not the successor of any number.
4. If two numbers have equal successors, they are equal.
5. If a set of numbers contains the number one and it contains all the successors of its members then the set contains all the numbers

Poem #1 -- Peano’s string; a history of spiritual stories

1. Abraham is a story
2. If x is a story, the unique inspiration of x is also a story.
3. Abraham is not the unique inspiration of any story
4. If two stories have equal unique inspiration, they are equal.
5. If a set of stories contains the story Abraham and it contains all the unique inspirations of its members then the set contains all the stories.

Poem #2 -- Peano’s string; a history of spiritual stories

1. Abraham is a cat
2. If x is a cat, the God of x is also a cat.
3. Abraham is not the God of any cat.
4. If two cats have equal Gods, they are equal.
5. If a set of cats contains the cat Abraham and it contains all the Gods of its members then the set contains all the cats.

Poem #1 = Poem#2

Axiomatic Poems

Peano’s string; a history of spiritual stories (Image above)


Axiomatic Poems

I would like to introduce a new mathematical structure to be used with mathematical poetry.

I understand that for two thousand years Euclid’s axioms stood alone as a meaningful axiomatic system. However, in 1889 Italian mathematician Giuseppe Peano created a new axiomatic system based on two primitive notions and the five following statements:

1. One is a number
2. If x is a number, the successor of x is also a number.
3. One is not the successor of any number.
4. If two numbers have equal successors, they are equal.
5. If a set of numbers contains the number one and it contains all the successors of its members then the set contains all the numbers.

What is interesting is that this system does not have to be limited to number. Calvin C. Clawson in his book “Mathematical Sorcery: Revealing the Secrets of Numbers” gives us the same five statements in the following form:

1. Heinsforth is a gelb
2. If x is a gelb, the ranker of x is also a gelb.
3. Heinsforth is not the ranker of any gelb.
4. If two gelbs have equal rankers, they are equal.
5. If a set of gelbs contains the gelb Heinsforth and it contains all the rankers of its members then the set contains all the gelbs.

Clawson has substituted the number “one” with Heinsforth, the term “number” with “gelb” and used “ranker” in place of successor. The point that Clawson is trying to make is that we need not be concerned with the primitive notions per se. What we need to be concerned with is the relationship of these notions within the axiomatic structure. From what I understand there could be incalculable different ways to describe the primitive notions however, only one way to logically relate them to each other. After reading Clawson’s axioms, I became aware of the ability of this structure to create metaphor. The source domain of the metaphor is the Peano axioms. The target domain is the same set of axioms with poetic substitutions placed inside the axioms. Therefore, I have created the axiomatic poem shown below:

Let us replace “number” with “cat” let us also replace “successor” with “God”. Lastly, I am going to replace “One” with “Abraham”.

1. Abraham is a cat
2. If x is a cat, the God of x is also a cat.
3. Abraham is not the God of any cat.
4. If two cats have equal Gods, they are equal.
5. If a set of cats contains the cat Abraham and it contains all the Gods of its members then the set contains all the cats.

Now the next interesting idea is:

Can these axioms create interesting theorems?