Mathematical Conundrum Series
This post is part of a series of posts that should be read in order:
Mathematical Conundrum Series
I will provide a little bit of background information here once more. The main point of these posts so far is that my own personal experience in mathematical education has left me thinking that I was misled (to some degree) with assumptions and shortcuts that were built into that education. This post will delve into more of the specifics of that feeling I have.
About 5-7 years ago, I came upon information that did not make sense to me regarding the foundations of arithmetic. I knew for years that the fundamental theorem of arithmetic directly implied the infinite product of prime geometric power series in the natural numbers, but I wanted to also see if the same structure existed in the inverse natural numbers, and I found that information in Leonhard Euler's 'Theorem 7', also known as Euler’s Product Formula, from his 1744 published work 'Variae observationes circa series infinitas' Several Remarks on Infinite Series. I already covered this information in the prior post, 'The Inverse Naturals and The Euler Product'. However, I got the idea to do something further with the Euler Product. I wanted to reverse the process that Euler used, to perform something like a validation of Euler's Theorem 7. Instead of starting with the harmonic series and deriving Euler's infinite product over the primes, I wanted to reverse that process by starting with Euler's result, the infinite product, and recover the harmonic series elements.
This gave birth to The Conundrum.
I gathered a lot of this information and compiled it into a pdf document and posted it to a forum I participate in back in 2019. I received a few responses from participants, but most, I think, did not understand what I was talking about. Then in the fall of 2021, I made another attempt to mention this information that I had gathered in a more scientific and mathematical oriented thread where I thought the audience had more mathematical knowledge. Most of the participants' mathematical knowledge far exceeded my own, including several whose careers are in advanced mathematics and research. Here is the response I received.
The problem is with your presentation of the results. In mathematics we have definitions and theorems and proofs. Can you condense your results in a theorem, clearly stated, and a proof, clearly written? Then perhaps it will be readable.
The responder was correct. Taking a good look at what I had compiled into a document was very disjointed and would be hard for anyone to discern, especially to someone actually in the field of mathematics. So I responded.
Thank you. I will try.
I have never written a math paper before. I understand and can read math papers (if they are not too advanced). I had written proofs before in that old American 1-12 education decades ago. But the information I had gathered was not very amenable to formal theorems and proofs. Still, I thought I could make the information more discernible and logical by using a math paper structure with logic and observations. I have used some of that information in these blog posts already and I will continue to present more of that information and logic.
It took me six months to find a proper math paper template and I created a very lengthy html file to which I added a math JavaScript library and a few custom JavaScript scripts for custom linking and citations. I posted the completed document to the forum as a response to the actual post where I was asked to write a math paper. The next morning someone responded very enthusiastically, which was encouraging, but the responder went kind of overboard about their own interests, talking about loop time in my posted document. The responder wanted more time to read my paper in depth before responding further.
I pre-read what you wrote. You've put a lot of work into it. I want to be more specific on this, but I need a little more time for that. I think it will be possible next week.However, I sincerely congratulate you on some of your insights. This is something that very few human beings will understand. I understand you, but I need to read it a few more times.Nevertheless, you see very well what I just said a few days ago (loop logic, linear versus loop time, and maybe even a time-light relationship? It can be seen here!), that nobody sees. It is beautiful. But bear in mind that I am a bit crazy. I am in love with time etc.I will look at what you wrote and I promise to analyze it in depth before speaking. It may take a while, but I will keep it in mind.
Three and one half weeks later, nothing.
So I tried to mention my post and paper in another thread with a much greater audience, a thread where the original first responder who requested the math paper was also commenting. I had some success in this second thread. The original requester of the paper did not respond, but another forum member with an advanced mathematics background did. This was their response.
It would be difficult to connect your work with things like the different types of particles and forces. People will tend to want to produce the "loops" with math that fits better with the math already being used for particles/forces. Sometimes there can actually even be different correct ways to do something but some ways may fit better with what you want to do next.
What is it with these responses about particles, forces, and loops? My paper was about basic simple high school algebra, with observations and logic about something being amiss with our foundations of arithmetic. So I responded to this forum poster with what I was trying to get at in my paper, which I will reveal here shortly in 'The Conundrum', but that seemed to elicit a response from the responder that was a little dismissive, which I understand, as it is a little difficult to look at simple things you may have learned as a child, but that also has to do with the point of my posts here and the assumptions and shortcuts ingrained in all of us by our early mathematical education. Here is the response.
Things like the harmonic series are certainly fun. I would tend to think your infinitely factored set is related to the harmonic series set having infinite members (even though no member itself is infinite). Infinities are headache inducing and divergent infinite series means you can't even just check for approaching a limit but it actually is fun reading your work.
So I will now attempt to explain 'The Conundrum' and what this is all about.
The Conundrum, relates back to all my earlier posts. If you haven't already read the previous posts, this will not make a lot of sense. Please read the prior posts, of which there are five. Each takes at most five minutes.
- Introduction
- Foundations - The Counting Numbers
- Foundations - The Fundamental Theorem of Arithmetic
- Foundations - The Fundamental Theorem of Arithmetic - Part 2
- The Inverse Naturals and The Euler Product
In 'The Conundrum', the assumptions and shortcuts I have mentioned become overwhelmingly apparent.
In the previous posts, I have shown that the natural numbers have a generating object, the infinite product of prime geometric power series. I have also shown that this very same structure exists in the inverse natural numbers via Leonhard Euler's 'Theorem 7', also known as Euler’s Product Formula, where Euler produces via simple algebra the inverse infinite product of prime geometric power series that produce all inverse natural numbers.
I will use Leonhard Euler's inverse infinite product of prime geometric power series that produce all inverse natural numbers for my demonstration of 'The Conundrum'.
Here it is:
In the 'inverse infinite product of prime geometric power series,' we can easily see that the infinite product produces all members of the inverse natural numbers. Using either the visual trace model or the infinite multiplicative sequences model we can see that the 'inverse infinite product of prime geometric power series' produces every single countably infinite inverse natural number.
In the visual trace model, we can see that every inverse natural number generated has a specific form or signature. That signature is such that every distinct infinite trace path always, eventually becomes a flat line where every value selected to the infinite right progression of the trace path all becomes ones, 1,1,1,1,1,1,....
In the infinite multiplicative sequences model, we see the same thing, and we should see the exact same thing, as each term of the infinite multiplicative sequences model corresponds to a value selected from the corresponding prime power series term of the infinite product. Just as in the trace model, in the infinite multiplicative sequences model, we see that for every natural number, eventually the progression to the right always becomes 1,1,1,1,1,1,...
This is the very definition of an inverse natural number. The set of all inverse natural numbers is countably infinite (meaning there are countably infinite members), but each individual member is finite. If any member of the inverse natural numbers had countably infinite factors, then it would never assume the signature in the rightward progression of 1,1,1,1,1,... If an inverse natural number never acquired that rightward progression signature of 1,1,1,1,1,..., in other words, there was always another term in the rightward progression that was not a 1, then it would be an infinite number, not a finite number, and by definition it would not be an inverse natural number.
Here is a review of Euler's infinite product, which he derived from the harmonic series (infinite sum of all inverse natural numbers).
The infinite product of inverse prime geometric power series derived from the harmonic series.
Here is the visual trace path model, demonstrating that all inverse natural numbers have an infinite progression to the right (eventually) of 1,1,1,1,1,...
Here is the infinite multiplicative sequences model, also demonstrating that all inverse natural numbers have an infinite progression to the right (eventually) of 1,1,1,1,1,...
The Conundrum
What is this?
Continue to - The Conundrum - Free Will and Choice
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