A Math Art Moment #12

In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it’s the exact opposite. —Paul Dirac

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The Lotto

Here is a polyaesthetic piece of mathematical visual poetry based on the similar triangles poem titled “LOTTO” The photo was shot in Las Vegas. The inspiration for the piece came while being part of a shared "lotto pot" in an office setting. Watching all of the people fantasizing about winning was fascinating.

The poem can be read multiple ways including the following:

The lotto is to financial fantasy as ogling pornograpy is to sexual fantasy. 

-or-

The lotto is to ogling pornograpy as financial fantasy is to sexual fantasy.

Math Test Results (Math Jokes)

Kevin Watters sent me a few math jokes the other day and I thought it may be nice to share them. ------------------------------------------------------------------------------


This is pretty close to being visual mathematical poetry. The student is probably a fine art major and got lost in his/her own thoughts --- why answer the question when you can be creative?
:)

You would think the student would get some extra credit for this ... ok maybe not.

This student deserves no extra credit.



I know this feeling.

Freshness

Here is a Polyaesthetic piece with a 'Similar Triangles Poem' titled "Freshness"

The Question of De Chrico

I have noticed many small mathematical poems on the internet that are in the form of A+B=C . This form is a perfectly legitimate form however, there are times when I think the author intended you to understand it in terms of A*B=C . There is some confusion as to what is the difference between addition and multiplication within the realm of mathematical poetry. I would love to hear anyone try to explain the difference between the two. In the mean time I have posted a piece above that asks that very question.

The piece is titled: “THE QUESTION OF DE CHIRICO” and it poses the question: “Is the image on the right side of the piece equal to the ideas of Baltimore times De Chirico or is the image equal to the ideas of Baltimore plus De Chirico?

I invite anyone to choose between the two statements and explain why and I will post the answers.

The King's Crown

Here is another Similar Triangles Poem entitled "The Kings Crown"

Natural Selection

This entry is a polyaesthetic piece titled "Natural Selection" the structure of the poem inside is a similar triangle poem.

Read Me First

Read me first

In this section of the side bar there are four articles.

The first article is a paper that was published in the journal of mathematics and the arts titled “Polyaesthetics and Mathematical Poetry”. This paper is a good introduction to Mathematical Poetry for it shows some of the main ideas as well as some techniques used to create mathematical poetry. One of the more important ideas it addresses is that of mathematical metaphor. The paper addresses basic theory as well as providing examples.

The second article is a paper published in the 2006 Bridges Proceedings titled “Verbogeometry, The confluence of words and analytic geometry” This paper explains the mechanics of how mathematical poetry can use Cartesian space as a medium for words. It provides examples of analytic geometry as well as the mathematical poetic counterpart.

The third article is an interview published online at word for/word a journal of new writing. The interview was conducted by poet/theoretician Gregory Vincent Thomasino and is formulated in three groups of questions. The first group of questions is about the influences of Kaz Maslanka and the second and third address mathematical poetic theory.

The forth article is a list of terminology that is related to the area where the arts and mathematics meet.

Substitution in Mathematical Poetry

Substitution in Mathematical Poetry

If you have no understanding of similar triangles poems then please read about it at the following link: “Similar Triangles Poem”

This Blog entry will show an example of substitution in mathematical poetry. Substitution can occur when we have two equations that have a common term. For example let’s look at the two equations which have the same form as two similar triangles poems: A = BD/E and A = HJ/U since both equations have the term ‘A’ in common and consequentially they both happen to be solved for ‘A’ then we can set both equations equal to each other as such:

BD/E = HJ/U

We know that we can solve for any of the variables in our new equation and get a new equation in terms of one variable. Let do so and solve for ‘J’ so we now have: J=UBD/EH

So now let’s apply what we have just witnessed to two similar triangles poems.

First of all we must look at the following two poems.

We know from our earlier example that we can solve a mathematical equation for any term in it. If we take the first poem and solve it for “my memories” we then can present the poem as:

Notice (below) that we have the two poems solved for the same term (my memories).

Now we can set each poem equal to each other because they both have identical terms. (see below)

We also know that we can solve this poetic equation for any of the terms in it. So let us solve this poem in terms of “Delaware River”

Now we can see that the later poem was derived from the two similar triangles poems shown at the top. What is interesting is that all of the logical processes used to create the first two poems are contained in our resultant poem including the subtle differences in the contexts of each initial poem.
Substitution can also be used in poems created by different poets as long as they have a common term. Follow this link to collaborative substitution poems.

The following polyaesthetic piece uses the image of a shipping beacon located at Cedar Swamp on the Delaware side of the Delaware River. The full Delaware River Poem from our example is nestled in the lower left hand corner of the image. The physical size of the digital image is 67” x 31”

Delaware River Correction

I actually made a mistake on my last blog entry. I meant to post the two similar triangles poems (above). If you were on your toes you would have noticed that the last blog entry was actually the same equation (poem) solved for different terms. Today’s entry is two different poems that also share a common term. What is interesting is what we will do with these two poems on the next blog entry. Can you guess what I will do?

Delaware River Memories

Here are a couple of similar triangles poems inspired by a romantic encounter around ‘Cedar Swamp’ on the Delaware River. Notice that they both have one term in common.

My Response To a Critic

I would like to address a comment made in reference to the piece “Peano’s String; A History of Spiritual Stories”(displayed above) … the following (text in green) is a copy of a comment from my blog entry “New Work Accepted At The Bridges Show In Leeuwarden Netherlands Aug 2008”:

This is a strange place. Im all for maths, dont get me wrong. Anyone who's any good at maths needs to make it part of themself but democrats? Abraham? maths is made a cliche with these comparisons. Everything can be expressed in maths but some things shouldnt. Just make a billboard with euler's formula

My response:

I appreciate you giving me some feedback to my blog and I would love to engage you in discourse on any concerns that you may have. I am certainly not going to imply that I am always correct in my assumptions of anything. Furthermore I consider myself a student.

I want to note that I may not defend mathematical poems made by others so if you wish to criticize the axiomatic poem concerning Barack Obama and the democrats you may wish to address your concerns to its author. I also wish to make this same disclaimer concerning any mathematical poetry posted on this blog that is not authored by me. However, I will be happy to address any concerns or criticism involving my work. My Job at this blog is to promote interest in mathematical poetry not criticize it. Yet, I may someday express criticism of someones work if I feel “the discipline” of mathematical poetry is being subverted.

To get to your concerns let’s look at the term cliché and what Wikipedia has to say about it:

A cliché (from French, pronounced [klɪ'ʃe]) is a phrase, expression, or idea that has been overused to the point of losing its intended force or novelty, especially when at some time it was considered distinctively forceful or novel. The term is most likely to be used in a negative context.

It seems that you have applied this term ‘cliché’ to my axiomatic poem titled, “Peano’s String; A History of Spiritual Stories”. So I can only assume that there is something about this mathematical poem that you would consider overused. It is hard to imagine that you may be referring to mathematical poetry in general since there is so little of it. What is it that is overused here? Is your concern related to my references to biblical history? Are you feeling that I have taken biblical references out of context in jest? I can only say that while I can see how one may find this mathematical poem humorous, the root of it can be taken very serious. Maybe, what you may really be trying to say, is that mathematical poetry is aesthetically trivial. This may be is a little more difficult for me to defend due to my belief that just because I find something beautiful I can never assume that anyone else would find it such. However, I do find mathematical poetry extremely beautiful especially in its use of dual aesthetics. My fear is that you, or anyone else for that matter, will discard this entire proposition and never really answer the following questions.

1. From a cognitive scientific point of view what is a metaphor, what are the parts within the structure of a metaphor and what are their relationship to mathematics in general and mathematical equations in particular?

2. What is the difference between connotation and denotation and how do they apply to the language of mathematics?

3. When looking at the structure of a mathematical equation how does that structure relate to other phenomena that can be described with that same mathematical structure?

4. Are the commonalities between identical mathematical structures purely linguistic? Or are they physical?... Or maybe spiritual? Could there be something such as archetypical equations?

5. What are the differences between the aesthetics of mathematics and the aesthetics of poetry or art? How can those differences be delineated when analyzing a mathematical poem?

6. How does mathematical poetry relate to the history of art, poetry and applied mathematics? Can mathematical poetry be considered a legitimate field of applied mathematics?


And now let’s address this mathematical poem in particular:


7. What is the relationship of Natural numbers to linear historical events?

8. What do the descendents of Abraham have to do with current cultural events especially ones that concern the military of the United States of America? Who are the children of Abraham and what is the historical and spiritual relationship that they share.

9. How are cultural stories passed from generation to generation?

10. How are mytho-spiritual (religious) stories created? How does deities and deification come to be? What is the source of the ‘so called’ divine inspirations that create works of poetry and art? And what is their relationship to this piece of art in particular.

11. What is the relationship of cats in mytho-spiritual literature? What is the meaning of cat when applied to a human being? What is the meaning of a cat when applied to a God?

12. When looking at the proofs using these axioms what can be said poetically from the proofs.

13. What are the proofs that can be created from Peano’s axioms?

14. How do questions 7 through 13 relate to questions 1 through 6?

I am not going to discount that you may provide an argument to the idea that my work is cliché and trivial but I would hope you address the latter questions within your argument.

Thanks!

Kaz

The Gift of San Shin 산신 (Polyaesthetic)

Here is the Polyaesthetic version of "The Gift of San Shin" which utilizes a Similar Triangles Poem.

In the vernacular this mathematical poem can be spoken four ways but the two most important ways are: 1.) Wisdom is to Adversity as the Wind is to a Cage  2.) Wisdom is to the Wind as Adversity is to a Cage.  It can also be put into the syntax of an orthogonal space poem.   I like to think of the denominator of  orthogonal space poem as some kind of valve that controls the value on the other side of the equal sign. For example I like to look at the limit of "The Cage" as it approaches zero thereby making "Wisdom" near infinite. 

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General Music

Here is another “Similar Triangle Poem” Titled “General Music” Inspired by the differences in their philosophy of battle execution.

공의 옉 설 The Empty Paradox

Here is the Korean version of “The Empty Paradox” "공의 옉 설"

The Empty Paradox

Here is a new piece titled "The Empty Paradox"
C= Compassion and W= Wisdom
The Chinese character is 'Buddha's mind'
So we have C multiplied times W equals the limit of (1/x) as 'x' approaches Buddha's mind.

The equation is the familiar function of x equal to 1/x which yields a hyperbolic curve when graphed and results an asymptote when x = 0. Compassion multiplied by Wisdom is equal to 1 over X as the limit of X approaches Buddha’s mind. Buddhist philosophy tells us that Buddha’s mind is emptiness yet the philosophy also tells us that emptiness is different than nothingness or zero. In fact it is quite paradoxical for we are told that emptiness is very much something. This piece also uses visual imagery for poetic expression with Buddhist symbolism of flexibility and eternity represented by bamboo and pine trees respectively.

산신 의 선물

Here is the Korean version of the Similar Triangle Poem titled “The Gift of San Shin / 산신 의 선물

” shown in the previous post.

The Gift of San Shin 산신

Here is another Similar Triangles Poem inspired by the Korean Mountain spirit San Shin 산신

Gregory Vincent St Thomasino Interviews Kaz Maslanka

Gregory Vincent St Thomasino Interviews Kaz Maslanka

I am happy to announce that my interview with The Poet/Philosopher Gregory Vincent St Thomasino has now been published at Word For/Word. I was fortunate enough to have met Gregory last summer in his home town of NYC and really appreciate the effort he made for this interview. I also want to thank Jonathan Minton at Word For/ Word for being kind enough to publish it.

If you are interested in who I am and what drives me then this interview will answer most of your questions. It also explains much of the theory behind mathematical poetry. Check it out here

New Work Accepted At The Bridges Show In Leeuwarden Netherlands Aug 2008

"Peano’s String; A History of Spiritual Stories" has been accepted into the Bridges show in Leeuwarden Netherlands Aug 2008.

For the theory on this piece please check out "Axiomatic Poems"

Ego Pride

Here is a new similar triangles poem titled “Ego Pride”

Heisenberg and Pablo Kagioglu

Pablo Kagioglu sent me a few slides showing some Mathematical Paradigm Poems. I am extremely limited in my understanding of Quantum Mechanics so I am sure there will be much reflexive didactic that I will miss however, I do find it interesting that he has substituted the idea of "quanta" for “self” in our human identity. I am interested in pondering these further I hope you do as well.

Boundaries? Thierry Brunet

Here is an orthogonal space Poem submitted to us by Thierry Brunet via France.

a+b+c Does Not Equal c+b+a

In Delancyplace's excerpt for 4/16/08 --as discussed by political advisor Frank Luntz, the sequential arrangement of information often creates the very meaning of that information:

"[In film, when] two unrelated images are presented, one after the other, the audience infers a causal or substantive link between them. A shot of a masked killer raising a butcher knife, followed by a shot of a woman opening her mouth, tells us that the woman is scared. But if that same image of a woman opening her mouth is preceded by a shot of a clock showing that it's 3 a.m., the woman may seem not to be screaming, but yawning. The mind takes the information it receives and synthesizes it to create a third idea, a new whole. ...

"The essential importance of the order in which information is presented first hit home for me early in my career when I was working for Ross Perot during the 1992 presidential campaign. I had three videos to test: a) a Perot biography, b) testimonials of various people praising Perot, and c) Perot himself delivering a speech. Without giving it much thought, I'd been showing the videos to various focus groups of independent voters in that order--until, at the beginning of one session, I realized to my horror that I'd failed to rewind the first two videotapes. So I was forced to begin the focus group with the tape of Perot himself talking.

"The results were stunning.

"In every previous focus group, the participants had fallen in love with Perot by the time they'd seen all three tapes in their particular order. No matter what the negative information I threw at them, they could not be moved off their support. But now, when people were seeing the tapes in the opposite order, they were immediately skeptical of Perot's capabilities and claims, and abandoned him at the first negative information they heard. ... I repeated this experiment several times, reversing the order, and watched as the same phenomenon took place. Demographically identical focus groups in the same cities had radically different reactions--all based on whether or not they saw Perot's biographical video and the third-party testimonials (and were therefore predisposed and conditioned to like him) before or after the candidate spoke for himself.

"The language lesson: A+B+C does not necessarily equal C+B+A. The order of presentation determines the reaction."

Dr. Frank Luntz, Words that Work, Hyperion, Copyright 2007 by Dr. Frank Luntz, pp. 40-41

A Math Art Moment #11

Delineation #11

Mathematics, among other things, is a language.
Art, among other things, uses language.

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On The Dangers of Spiritual Art

Karl Kempton sent me the piece (shown above). It, as well as other works of his spawned the following essay.

I feel that one of the most dangerous areas of contemporary art comes when the artist makes him/herself a target by embracing spiritual concerns. Our society enjoys pointing fingers at the inadequacies of institutionalized religion (there are many) and ignoring the archetypical ideas of the spirit that have brought us the wonderful icons of the past. These spiritual metaphors have manifested themselves throughout history in many forms always relating to the culture of the artist. Many of the ideas of religions are obsolete and don’t function well in societies as diverse and ever-changing as ours. The artistic challenge of spirit is an extremely difficult task especially when trying to use historically loaded iconography of current dominant religions. I think many of the artistic phobias associated with the spirit are due to our experience of so many ‘so-called’ spiritual artists, who have created cliché kitsch or dogmatic concepts that accent the hypocritical ideas of the church or yet have forged an audaciously different direction aligning themselves with likes of aliens from other planets. Also to note, there seems to be a direct conflict between science and the spirit, which is anxiously evident when scientific minds address spiritual matters. I believe the problem is based in the illusiveness of Truth in both arenas. There are many that think that Truth is defined by science using the language of mathematics. Others believe Truth is beyond logic and only evoked through the metaphoric language of a spiritual ritual. Then there is my personally distasteful category of those who think that Truth is defined by their particular religion or should I say defined by their particular church. Focusing on the later idea we see that human nature tends to have many conflicts and from a historical perspective, one of the most destructive is the religious “us versus them conflict.” I see churches tending to promote this kind of behavior due to its doctrine being fed through so many egos. On the same line of thinking, the testosterone of the self righteous seems to have made its way into religion and spiritual matters to set up so many of the conflicts that we humans engage in. Many have died and continue to die in spiritual wars created by the religious intolerant.

The fact that the conflicts exist, illustrate how illusive Truth is. It seems to me that science is no better when it comes to Truth. The eminent scientist David Boehm points out that science does not find Truth, its purpose is to correlate experience. Also in this vein, we can see that there are those who provide great arguments against the platonic nature of mathematics pointing out numerous problems with using mathematics as a true model for reality. I see the bottom line being that the terra firma of veracity is constantly shifting; therefore, we must accept this fact and move on. The eastern mystics use the metaphor, “form is emptiness and emptiness is form”.

I believe it is the function of special artists to assimilate as much information as possible from the diverse cross-planet cultural ideas not limited to including the concepts of science so that they can re-contextualize, synthesize and synergize their metaphors to be acute and pertinent to the global culture today. They must fully embody the ideas of love and tolerance as if the ideas were new so as to debride the cliché skins attached to them. As impossible this task seems, it is the challenge of those artists to reconnect the loose strands of past archetypical works and re-contextualize them to breathe new life in today’s world. Their job is not to run from the spiritual confusion that permeates the ever-changing cultures on this globe by hiding in some self-conceived scientific illusion of truth without spirit. That is not to say that science cannot be the new religion … it can. However, the spiritual scientist must connect the magical and irrational mind to scientific metaphors so that our spiritual understanding can be flexible as science metamorphoses. The past mytho-spiritual ideas were always based in the science of the times. It takes courage to navigate through the mental minefield of past ‘truths’ finding new veracity that resonates in ones psyche as they express it and expose themselves to the ridicule of being an irrational kook.

I believe the special artist/poets should focus their efforts to make metaphors current to our historical and sociological condition. The purpose of a metaphor is to bridge the infinite to the concrete. Many people feel that past mytho-spiritual/religious metaphors are absolute in the notion that they permanently point to the infinite. Personally speaking, I see the veracity of metaphors being temporal with their cultural relevance having different half-lives. What can confuse matters is that the half-life in some metaphors have existed for such a long time that they seem absolute. There is an argument that the Bastian elemental ideas and Jungian archetypes are absolute. Even if this is true, the metaphors employing those elemental ideas always need recontextualizing to be relevant to the current cultural thought. The frustrating aspect for the artist is having so little control over the fertility of the inspiration process. I wish I could say that artists had full control over the source and production of their metaphors. However, it seems to me that their strength, viability and temporality are a function of graciousness, imparted from the muses. I believe it is though the struggle and success with life that these special artists acquire the molecular building blocks of a vocabulary that becomes the means of their expressions. These ideas logically coagulate around an infinite idea provided to them by the unknown.

A Math Art Moment #10

Delineation #10

Mathematical creations are not unique in the sense that they could be discovered by anyone.

Artistic creations are uniquely invented by individuals.

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Tag Math Art Moment

The Muses

Here is another similar triangles poem

Bravery

Here is the orthogonal space poem "Bravery" realized as a polyaesthetic work.

Is Pure Mathematics Poetic?

I receive a very important comment the other day from Jonathan who uses the JD2718 to identify himself on his blog. His comment was in reference to axiomatic mathematical poetry. However, I think his question is much broader.

Jonathan expressed the following:

Abraham, cats, Gods.

One, numbers, successors.

Which is really more poetic?

This is a sticky question because I want to avoid slipping into the bottomless void of the “What is poetry? What is art?” question However; I can discuss elements of poetry from which my idea of poetics is derived. I also want to add the following statements are not a value judgment on the aesthetics of mathematics. The mathematical aesthetic is one of the most wonderful experiences one may realize.

To answer Jonathans question; I am assuming that his question implies that pure mathematics is poetic. It is my view that pure mathematics is not poetic. Furthermore, the quick and dirty response to this question is that pure mathematics is different from poetics the same as pure mathematics is different from physics. Physics and Mathematical poetry, although vastly different, live in the realm of applied mathematics. Even when we ‘feel’ that pure mathematics is poetic, we are applying mathematics to some preconceived notion of what we believe poetry is without actually applying it. We may choose to argue that mathematics contains elements of poetry such as rhythm and pattern. Yet one may argue that it is not maths that has poetic elements but poetry that has mathematical elements. For the sake of argument, let us say that poetry possesses the mathematical element of pattern. I would like to make the point that it is difficult to get excited about these metric patterns when taken out of the context of poetry and view in only the light of mathematics. I know we are starting to get away from the intention of our question however, the point I want to make is that the aesthetics of mathematics is much different from the aesthetics of poetry and poetics. The ‘polyaesthetic experience’ that we are discussing is a vector sum experience of the aesthetics of art/language poetry and the aesthetic of mathematics. (They are different aesthetics) If we were to separate the mathematical aesthetic from a language poem how beautiful is it? Now let us look at the aesthetics of mathematical pattern by comparing the beauty of the pattern in iambic pentameter (or any other meter for that matter) to the beauty of self-similar patterns in a mathematically generated fractal. Which is more beautiful? Is the ‘isolated’ metric pattern in poetry more beautiful than a fractal? How about asking, “Is the fractal poetic?” If so what are the elements of poetry in the fractal. Is it the concept of rhythm that makes maths poetic? Are all things displaying rhythm poetic? The point I am trying to produce is that mathematical poetry, makes the structure of mathematics poetic only by application of poetics within that structure. Pure mathematics is not poetic by itself.

When addressing the metric beauty in language poetry; the metric beauty is not relevant to the mathematical pattern per se. It is relevant to the aesthetics involved in the relationship of the pattern to the words and the sounds of the words with its synesthetic energy igniting the meaning of the words as they point further to the cultural and historical relationships within the poem. The mathematical aesthetic devoid of the poetic aesthetic plays an extremely limited role in the aesthetics of language poetry. Yes, there is maths in the poetry however, break it out of the poetry, isolate it and I believe it becomes aesthetically trivial.

Let us look at metaphor – Does pure mathematics express metaphor? How could it? for pure mathematics is more about illuminating the logical structure of thinking. The key word that I want to stress is “logical”. Metaphor requires logical tension if not paradox to function as a concept to bridge the infinite to the concrete. However, I must say that mathematics does provide us with the linguistic structure to express metaphor. Again, this is the issue of pure mathematics relative to applied mathematics. To express metaphor you have to have an application of poetic concepts. You need a source domain and a target domain. (see the section on metaphor structure at Wikipedia) Pure mathematics does not have these metaphoric domains until we apply the poetic idea to the structure of maths as we do in mathematical poetry. The essay “Polyaesthetics and mathematical poetry” goes into more detail on this matter as well as an interview conducted by poetic aesthetician Gregory Vincent St. Thomasino. The interview will soon be published at “word for/word” an online journal of new poetry. I hope to announce the interview soon at this blog.

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)

Download Polyaesthetics and Mathematical Poetry

In March of 2007 I announced “Polyaesthetics and Mathematical Poetry.” published by Taylor and Francis in the Journal of Mathematics and the Arts Edited by Professor Gary Greenfield. This paper outlines many of the basic principles of mathematical poetry and polyaesthetics.

The contents of the paper are available for downloaded free at this link.

Journal of Mathematics and the Arts published “Polyaesthetics and Mathematical Poetry” March 2007 Volume 1 Number 1 ISSN 1751-3472

The published paper can be purchased at this link.

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