The Conundrum - Free Will and Choice

Mathematical Conundrum Series

This post is part of a series of posts that should be read in order:

Mathematical Conundrum Series

• Introduction
• Foundations - The Counting Numbers
• Foundations - The Fundamental Theorem of Arithmetic
• The Fundamental Theorem of Arithmetic - Part 2
• The Inverse Naturals and The Euler Product
• The Conundrum
• The Conundrum - Free Will and Choice

In my previous post, I presented 'The Conundrum'. In this post, I will look deeper at 'The Conundrum' and ask the reader, is it real or is it not?

In my previous post, I looked at Euler's infinite product over the primes derived from the harmonic series (the inverse natural numbers). Euler's infinite product over the primes is in fact an infinite product of inverse prime geometric power series. I consider this mathematical object the generating object of all inverse natural numbers. It kind represents the reverse of factorization. Instead of taking the inverse natural numbers and determining what prime factors a specific inverse natural number is composed of, we are instead generating/creating all inverse natural numbers from their generating object that supplies all of those prime factors, and this is basically just simple algebra, multiplicative distribution of the infinite product of inverse prime geometric power series.

But then I closed my prior post with this.

The Conundrum

What is this?

The above infinite paths, which can also be represented in the infinite multiplicative sequences model, never terminate to 1,1,1,1,1,... in their rightward progression. There is always another term in the rightward progression that is not a selected value of 1. The question is, do they really exist?

Remember that all of this information comes from a math paper that I was asked to write, and it took about six months to complete the paper, and I posted the paper in the forum and thread of the original requester asking me to write it. I received very few responses (actually 2 members responded, neither of which was the original requester). One of the responses from an individual that seemed to kind of grasp what I was trying to point out was this:

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.

It has been 30 months since I received that response. I never replied back, as I was baffled. I couldn't understand how the infinitely factored (multiplicatively distributive) paths could not exist, how anyone could not see from both the visual trace path model and the infinite multiplicative sequences model that these infinitely factored inverse numbers existed. I also did not think the part of the response about infinities being divergent and difficult was even relevant to the question. The question is whether the infinitely factored numbers exist or not, not that they may be difficult to deal with if they do exist. That difficulty has nothing to do with whether they are extant or not. I had hoped someone would actually look at the logic and methods, and comment on the validity of the logic and methods. If the logic and methods are wrong, then why are they wrong?

I have come to the conclusion that not being able to see the infinitely factored numbers may come down to what I started this series of posts about to begin with, and that is the shortcuts and assumptions built into our mathematical education whose foundation is the counting numbers 1, 2, 3, 4, 5,...

More Than the Sum of Its Parts?

Euler's infinite product over the primes was derived from the harmonic series, the infinite summation series of all inverse natural numbers (all inverse counting numbers). How could the derived infinite product result have more numbers in it than Euler initially started with? It is counterintuitive. But that is indeed the assumption drilled into everyone from our mathematical education based on these counting numbers. The counting numbers are the foundation of everything we have today in mathematics and probably in society as a whole, all science, engineering, physics, etc. This foundation is drilled into us from when we are three years old and dominates our world view, with all of its assumptions and shortcuts. The counting numbers are basically the first principles in the hierarchy of mathematical education from day one. The question is, should they be the first principles that everything else is built on?

To me, the answer to my previous question is No. I think the infinite product of prime geometric power series should be the base, the foundation, the fundamental object, the first principles, and what arises from it becomes the second step.

If we examine the definition of the counting numbers or inverse counting numbers (naturals or inverse naturals), every element of each set has a finite value, meaning a finite set of prime factors. That is their definition. But if we take the infinite product of prime geometric power series as the foundation, it is plain as daylight that there are infinite multiplicative distributive paths that never have that infinite progression to the right where all values progress with an infinite signature of all ones, 1,1,1,1,1,... The only way to get rid of the infinite paths that always have another term to the right that is not a selected value of 1 is to impose a rule, an artificial constraint from the outside on the infinite product of prime geometric power series. We would have to create a rule that said, "No distinct/unique infinite multiplicative distributive path can have a signature where the infinite progression on the righthand side never becomes 1,1,1,1,1,... Now we can create that rule and inject it into our mathematical logic, but to me that is nonsense.

If we look at the infinite product of inverse prime geometric power series and use the infinite multiplicative sequences model as a proxy for the multiplicative distribution and we add that artificial new rule we just created, what would that mean?

Let's look at that.

In the infinite multiplicative sequences model, each term corresponds to a term in the infinite product of inverse prime geometric power series. There are infinite terms in the infinite product of inverse prime geometric power series, so we must have infinite terms in our infinite multiplicative sequences model. The values that are available to choose for each term of a multiplicative sequence are the corresponding values available from the corresponding term of the infinite product of inverse prime geometric power series; those values are either 1 (one) or an integer power of the corresponding prime number.

But that new rule we have just created limits us. It tells us that every distinct/unique combination of sequences we create must, at some point in its infinite rightward progression, somewhere there in that progression, we no longer have a choice. For a number to be finitely factored, every distinct/unique sequence you come up with, somewhere in that sequence, the choice of selecting either the number 1 (one) or a corresponding prime power for the next term, there is no choice, from that point rightward, you can only select the value 1 (one) for all of the rest of the infinite terms to the right. In effect, at some point you have no choice, your choice has already been determined for you, choice is no longer possible.

To me, this almost comes down to a philosophical argument. Do we have free will? Do we have choice? That is why I included 'Free Will and Choice' in the title of this post. Does Free Will and Choice exist?

To me, the answer is Yes. To me, I am unable to impose that artificial constraint on the infinite product of prime geometric power series. This is also why I think it is the 'infinite product of prime geometric power series' mathematical object that should be foundational, fundamental. If we make the 'infinite product of prime geometric power series' mathematical object foundational and we do not put any external artificial constraints on it, then what we see emerge from it are two sets of numbers. One where every member of the set has finite factors, and a second set where every member of the set has infinite factors.

Further observations of what I call this foundational mathematical object lead me to think that of all the numbers created by this foundational mathematical object, 1/2 are finite and 1/2 are infinite, and you cannot have one of the sets (finitely valued members) without the other set (infinitely valued members). In fact, both emerge bilaterally, dually, and simultaneously. The mathematical object responsible for creating members of both sets is a prime geometric power series term in the infinite product. If you remove, for instance, the inverse two power series from the infinite product, not only do all finitely factored members of the set of inverse natural numbers disappear from the result (all finitely factored numbers with a factor of 2 in them), but also all members of the parallel infinitely factored set also disappear (infinitely factored numbers with a factor of 2 in them). The two sets are bonded together, one cannot exist without the other existing.

So if these infinitely factored numbers do exist and I cannot see how they do not, then where are they in the literature? Well, basically, they are not in the literature. The closest thing in the literature is the formulation of the surreal numbers with the dyadics, birthdays, and day w, or the recent formulation of non-standard analysis and its notion of infinitesimals. But as I have mentioned before, this information (the parallel sets of infinitely factored numbers) could easily be derived 2300 years ago simply from Euclid's Elements or the Sieve of Eratosthenes, which were both available somewhere around the third century BCE.

I will not go into further information in this post on expanding this observation (the parallel infinitely factored sets arising from the infinite product of prime geometric power series), as proceeding further requires the acceptance that these infinitely factored numbers arise naturally just as the finitely factored numbers arise naturally from the infinite product of prime geometric power series, and until that Conundrum is decided upon, going further would be futile because all further expansion of the concept requires the acceptance of these infinitely factored numbers.

I will, however, briefly mention where my math paper goes that all this information comes from.

Accepting that the infinitely factored numbers exist, my paper continues and expands the same observations to the rational numbers and then to the real numbers via the introduction of a real valued variable into the exponents of every prime power series, similar to Euler's zeta function, then it continues to expand to two additional levels. In all, my paper proposes that there are six levels of numbers that are created by the infinite product of prime geometric power series.

1. The inverse natural numbers
2. The natural numbers
3. The rational numbers
4. The real numbers via an Euler like zeta function
5. A first level where only infinitely factored numbers reside
6. A second, even greater level where only infinitely factored numbers reside

In the first four levels, all resultant numbers created by the infinite product of prime geometric power series produce 1/2 of the numbers as being finite and 1/2 of the numbers as being infinite.

 

The next two levels I have no name for, levels 5 and 6, have no finite numbers at all. All members of these two levels are infinite.

Thus was born 'The Conundrum'.

Will this effort of mine lead anywhere or will it collapse? 

Stay tuned...

The Conundrum

 

Mathematical Conundrum Series

This post is part of a series of posts that should be read in order:

Mathematical Conundrum Series

• Introduction
• Foundations - The Counting Numbers
• Foundations - The Fundamental Theorem of Arithmetic
• The Fundamental Theorem of Arithmetic - Part 2
• The Inverse Naturals and The Euler Product
• The Conundrum
• The Conundrum - Free Will and Choice

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.

1. Introduction 
2. Foundations - The Counting Numbers
3. Foundations - The Fundamental Theorem of Arithmetic
4. Foundations - The Fundamental Theorem of Arithmetic - Part 2
5. 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.

$$\left(\begin{array}{c}\\ 1\\ +\\ 2^{-1}\\ +\\ 2^{-2}\\ +\\ 2^{-3}\\ +\\ 2^{-4}\\ +\\ 2^{-5}\\ ⋮\\ +\\ 2^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ 3^{-1}\\ +\\ 3^{-2}\\ +\\ 3^{-3}\\ +\\ 3^{-4}\\ +\\ 3^{-5}\\ ⋮\\ +\\ 3^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ 5^{-1}\\ +\\ 5^{-2}\\ +\\ 5^{-3}\\ +\\ 5^{-4}\\ +\\ 5^{-5}\\ ⋮\\ +\\ 5^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ 7^{-1}\\ +\\ 7^{-2}\\ +\\ 7^{-3}\\ +\\ 7^{-4}\\ +\\ 7^{-5}\\ ⋮\\ +\\ 7^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ {11}^{-1}\\ +\\ {11}^{-2}\\ +\\ {11}^{-3}\\ +\\ {11}^{-4}\\ +\\ {11}^{-5}\\ ⋮\\ +\\ {11}^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ {13}^{-1}\\ +\\ {13}^{-2}\\ +\\ {13}^{-3}\\ +\\ {13}^{-4}\\ +\\ {13}^{-5}\\ ⋮\\ +\\ {13}^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ {17}^{-1}\\ +\\ {17}^{-2}\\ +\\ {17}^{-3}\\ +\\ {17}^{-4}\\ +\\ {17}^{-5}\\ ⋮\\ +\\ {17}^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ {19}^{-1}\\ +\\ {19}^{-2}\\ +\\ {19}^{-3}\\ +\\ {19}^{-4}\\ +\\ {19}^{-5}\\ ⋮\\ +\\ {19}^{-n}\\ \end{array}\right)\times \cdots$$

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,...

$$\begin{array}{cl}\frac{1}{n}& \text{sequence}\\ 1& 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{2}& 2^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{3}& 1,3^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{4}& 2^{-2},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{5}& 1,1,5^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{6}& 2^{-1},3^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{7}& 1,1,1,7^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{8}& 2^{-3},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{9}& 1,3^{-2},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{10}& 2^{-1},1,5^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{11}& 1,1,1,1,{11}^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{12}& 2^{-2},3^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{13}& 1,1,1,1,1,{13}^{-1},1,1,1,1,1,1,1,1,1,1,1,1,...\end{array}$$

The Conundrum

What is this?

Continue to - The Conundrum - Free Will and Choice

The Inverse Naturals and The Euler Product

 

Mathematical Conundrum Series

This post is part of a series of posts that should be read in order:

Mathematical Conundrum Series

• Introduction
• Foundations - The Counting Numbers
• Foundations - The Fundamental Theorem of Arithmetic
• The Fundamental Theorem of Arithmetic - Part 2
• The Inverse Naturals and The Euler Product
• The Conundrum
• The Conundrum - Free Will and Choice

In this post I am going to look at the inverse natural numbers, and the easiest way to do that is to look at 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.

Reference: E072en.pdf

Theorem 7 of Euler's work concludes that the harmonic series is equal to the infinite product "where the numerators are all the prime numbers and the denominators are the numerators less one unit". 

$${\displaystyle 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\cdots =\frac{2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdots }{1\cdot 2\cdot 4\cdot 6\cdot 10\cdot 12\cdot 16\cdot 18\cdot 22\cdots }}$$

Euler's infinite product is also, like in the natural numbers, an infinite product of prime geometric power series. How do I know that?

$${\displaystyle \frac{2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdots }{1\cdot 2\cdot 4\cdot 6\cdot 10\cdot 12\cdot 16\cdot 18\cdot 22\cdots }}$$

Each term of the Euler product can be written as a prime geometric power series.

The first term: $$\begin{array}{rl}\frac{2}{1}& =1+1\\ & =1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\dots \end{array}$$
The second term: $$\begin{array}{rl}\frac{3}{2}& =1+\frac{1}{2}\\ & =1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{243}+\dots \end{array}$$
The entire Euler product can be written as a product of prime power series. $$\left(\begin{array}{c}\\ 1\\ +\\ 2^{-1}\\ +\\ 2^{-2}\\ +\\ 2^{-3}\\ +\\ 2^{-4}\\ +\\ 2^{-5}\\ ⋮\\ +\\ 2^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ 3^{-1}\\ +\\ 3^{-2}\\ +\\ 3^{-3}\\ +\\ 3^{-4}\\ +\\ 3^{-5}\\ ⋮\\ +\\ 3^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ 5^{-1}\\ +\\ 5^{-2}\\ +\\ 5^{-3}\\ +\\ 5^{-4}\\ +\\ 5^{-5}\\ ⋮\\ +\\ 5^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ 7^{-1}\\ +\\ 7^{-2}\\ +\\ 7^{-3}\\ +\\ 7^{-4}\\ +\\ 7^{-5}\\ ⋮\\ +\\ 7^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ {11}^{-1}\\ +\\ {11}^{-2}\\ +\\ {11}^{-3}\\ +\\ {11}^{-4}\\ +\\ {11}^{-5}\\ ⋮\\ +\\ {11}^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ {13}^{-1}\\ +\\ {13}^{-2}\\ +\\ {13}^{-3}\\ +\\ {13}^{-4}\\ +\\ {13}^{-5}\\ ⋮\\ +\\ {13}^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ {17}^{-1}\\ +\\ {17}^{-2}\\ +\\ {17}^{-3}\\ +\\ {17}^{-4}\\ +\\ {17}^{-5}\\ ⋮\\ +\\ {17}^{-n}\\ \end{array}\right)\times \left(\begin{array}{c}\\ 1\\ +\\ {19}^{-1}\\ +\\ {19}^{-2}\\ +\\ {19}^{-3}\\ +\\ {19}^{-4}\\ +\\ {19}^{-5}\\ ⋮\\ +\\ {19}^{-n}\\ \end{array}\right)\times \cdots$$

For a modern reference, you can checkout Wofram's Mathworld reference on the Euler product.

Can we model the factorization of the inverse naturals using this infinite product of inverse prime geometric power series?

Yes.

The infinite product of prime geometric power series in the naturals and the infinite product of inverse prime geometric power series are similar. Factorization in the naturals is in the numerator, and in the inverse naturals we have the exact same factorization but in the denominator. Outside of that, the models are identical.

The visual model of tracing distinct/unique multiplicative paths through the infinite product of inverse prime geometric power series.

The infinite multiplicative sequences model:

$$\begin{array}{cl}\frac{1}{n}& \text{sequence}\\ 1& 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{2}& 2^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{3}& 1,3^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{4}& 2^{-2},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{5}& 1,1,5^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{6}& 2^{-1},3^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{7}& 1,1,1,7^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{8}& 2^{-3},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{9}& 1,3^{-2},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{10}& 2^{-1},1,5^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{11}& 1,1,1,1,{11}^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{12}& 2^{-2},3^{-1},1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ \frac{1}{13}& 1,1,1,1,1,{13}^{-1},1,1,1,1,1,1,1,1,1,1,1,1,...\end{array}$$

So, as can be seen here, both the infinite product of prime geometric power series in the naturals and the infinite product of inverse prime geometric power series can model the factorization of their respective sets. 

Note: I keep saying factorization in these posts regarding the infinite prime geometric power series products in the naturals and inverse naturals, but here I am not really working with factorization. Instead of taking the numbers in the respective sets and determining their factors, here I am starting with the actual creators, the mathematical objects that create/generate the members of the respective sets, the naturals and the inverse naturals. I am literally performing a reverse process. I am demonstrating that the prime geometric power series creates the infinite canonical representation of the factors of each member of each set.

Next up, 'The Conundrum'.

The Fundamental Theorem of Arithmetic - Part 2

Mathematical Conundrum Series

This post is part of a series of posts that should be read in order:

Mathematical Conundrum Series

• Introduction
• Foundations - The Counting Numbers
• Foundations - The Fundamental Theorem of Arithmetic
• The Fundamental Theorem of Arithmetic - Part 2
• The Inverse Naturals and The Euler Product
• The Conundrum
• The Conundrum - Free Will and Choice

I left off with the following

The Fundamental Theorem of Arithmetic is an infinite product of prime geometric power series.

$$\left(\begin{array}{c}\\ 2^n\\ +\\ ⋮\\ 2^5\\ +\\ 2^4\\ +\\ 2^3\\ +\\ 2^2\\ +\\ 2^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ 3^n\\ +\\ ⋮\\ 3^5\\ +\\ 3^4\\ +\\ 3^3\\ +\\ 3^2\\ +\\ 3^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ 5^n\\ +\\ ⋮\\ 5^5\\ +\\ 5^4\\ +\\ 5^3\\ +\\ 5^2\\ +\\ 5^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ 7^n\\ +\\ ⋮\\ 7^5\\ +\\ 7^4\\ +\\ 7^3\\ +\\ 7^2\\ +\\ 7^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {11}^n\\ +\\ ⋮\\ {11}^5\\ +\\ {11}^4\\ +\\ {11}^3\\ +\\ {11}^2\\ +\\ {11}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {13}^n\\ +\\ ⋮\\ {13}^5\\ +\\ {13}^4\\ +\\ {13}^3\\ +\\ {13}^2\\ +\\ {13}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {17}^n\\ +\\ ⋮\\ {17}^5\\ +\\ {17}^4\\ +\\ {17}^3\\ +\\ {17}^2\\ +\\ {17}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {19}^n\\ +\\ ⋮\\ {19}^5\\ +\\ {19}^4\\ +\\ {19}^3\\ +\\ {19}^2\\ +\\ {19}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {23}^n\\ +\\ ⋮\\ {23}^5\\ +\\ {23}^4\\ +\\ {23}^3\\ +\\ {23}^2\\ +\\ {23}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {29}^n\\ +\\ ⋮\\ {29}^5\\ +\\ {29}^4\\ +\\ {29}^3\\ +\\ {29}^2\\ +\\ {29}^1\\ +\\ 1\\ \end{array}\right)\times \cdots$$

There are two models we can use to examine factorization or multiplicative distribution of the above infinite product of prime geometric power series.

The first model is just a visual model of tracing distinct/unique multiplicative paths through the above infinite product of prime geometric power series.

The second model or method is to create a countably infinite multiplicative sequence.

Rules for the multiplicative sequences modeling the multiplicative distribution of the infinite product of prime geometric power series.

• A multiplicative sequence must be distinct (unique).
  
  
• A multiplicative sequence must have infinite terms, as there is a one-to-one correlation between a prime power series term and a term in the countably infinite multiplicative sequence model. There are countably infinite prime power series terms in the infinite product. There will be countably infinite terms in a multiplicative sequence.
  
  
• The values that are selectable as a choice for each term are either 1 or an integer power of the respective prime power series.
  
  
• A value must be selected for each and every infinite term.

Listing multiplicative sequences in the order of the natural numbers.

Finitely factored sequences

$$\begin{array}{cl}n& \text{sequence}\\ 1& 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 2& 2^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 3& 1,3^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 4& 2^2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 5& 1,1,5^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 6& 2^1,3^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 7& 1,1,1,7^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 8& 2^3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 9& 1,3^2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 10& 2^1,1,5^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 11& 1,1,1,1,{11}^1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 12& 2^2,3^1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...\\ 13& 1,1,1,1,1,{13}^1,1,1,1,1,1,1,1,1,1,1,1,1,...\end{array}$$

Listing multiplicative sequences can continue this way infinitely.

As we see from the above, the infinite product of prime geometric power series is basically 'The Fundamental Theorem of Arithmetic'.

In the next post, I will look at the inverse naturals and the harmonic series, thanks to a great mathematician, Leonhard Euler.

Continue to - The Inverse Naturals and The Euler Product

Foundations - The Fundamental Theorem of Arithmetic

 

Mathematical Conundrum Series

This post is part of a series of posts that should be read in order:

Mathematical Conundrum Series

• Introduction
• Foundations - The Counting Numbers
• Foundations - The Fundamental Theorem of Arithmetic
• The Fundamental Theorem of Arithmetic - Part 2
• The Inverse Naturals and The Euler Product
• The Conundrum
• The Conundrum - Free Will and Choice

I am continuing my previous post, 'Foundations - The Counting Numbers', where I spoke of some of the shortcuts we are taught kind of unknowingly in mathematics. This post is about another one of those shortcuts that at first glance seems very trivial or even non-sensical. It is about factorization.

We know that the atoms per se of all natural numbers, inverse natural numbers, and rational numbers are the prime numbers, and this is all wrapped up in the Fundamental Theorem of Arithmetic, the Unique Factorization Theorem. 

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. This means that every natural number is either a prime number or a composite number made of primes or powers of primes.

$$137,200=2^4\cdot 5^2\cdot 7^3$$
You will notice that in the above factorization there is no representation for the prime number 3. It may seem silly to write the factorization as this:
$$137,200=2^4\cdot 3^0\cdot 5^2\cdot 7^3$$ but it is entirely acceptable. It is one of those shortcuts. Why include $$3^0$$ in this factorization? $$3^0$$ is just 1 and 1 times a number is just the number.

This actually becomes important in later posts I am going to make. For now, I will provide a reference to the Canonical representation of factorization here: Canonical representation of a positive integer, which includes the following: 

In fact, any positive integer can be uniquely represented as an infinite product taken over all the positive prime numbers 

The quoted text basically means this:
$$137,200=2^4\cdot 3^0\cdot 5^2\cdot 7^3\cdot {11}^0\cdot {13}^0\cdot {17}^0\cdot {19}^0\cdot {23}^0\cdot {29}^0\cdot {31}^0\cdot \dots P_n^0$$

There is one caveat to this. We are working in the natural numbers, the positive integers. The Fundamental Theorem of Arithmetic is defined for the natural numbers, the positive integers. All natural numbers, positive integers, are finite. There are countably infinite elements in the set of natural numbers, but every single element is a finite number. This means writing any natural number as its factorization with infinite terms requires that there are only finitely many terms with prime factors with powers that are non-zero. No matter how large the number, writing its infinitely termed factorization will have a point in its progression to the right where every further prime power term is the respective prime raised to the zero power.

So every natural number, positive integer, will have a specific form or signature in its infinitely termed factorization, where everything in the rightward progression eventually becomes the respective prime to the zero power, or 1, 1, 1, 1, 1, 1, 1,... This is for every natural number, positive integer, no matter how large.

The Fundamental Theorem of Arithmetic directly implies the following:

The Fundamental Theorem of Arithmetic is an infinite product of prime geometric power series.

$$\left(\begin{array}{c}\\ 2^n\\ +\\ ⋮\\ 2^5\\ +\\ 2^4\\ +\\ 2^3\\ +\\ 2^2\\ +\\ 2^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ 3^n\\ +\\ ⋮\\ 3^5\\ +\\ 3^4\\ +\\ 3^3\\ +\\ 3^2\\ +\\ 3^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ 5^n\\ +\\ ⋮\\ 5^5\\ +\\ 5^4\\ +\\ 5^3\\ +\\ 5^2\\ +\\ 5^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ 7^n\\ +\\ ⋮\\ 7^5\\ +\\ 7^4\\ +\\ 7^3\\ +\\ 7^2\\ +\\ 7^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {11}^n\\ +\\ ⋮\\ {11}^5\\ +\\ {11}^4\\ +\\ {11}^3\\ +\\ {11}^2\\ +\\ {11}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {13}^n\\ +\\ ⋮\\ {13}^5\\ +\\ {13}^4\\ +\\ {13}^3\\ +\\ {13}^2\\ +\\ {13}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {17}^n\\ +\\ ⋮\\ {17}^5\\ +\\ {17}^4\\ +\\ {17}^3\\ +\\ {17}^2\\ +\\ {17}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {19}^n\\ +\\ ⋮\\ {19}^5\\ +\\ {19}^4\\ +\\ {19}^3\\ +\\ {19}^2\\ +\\ {19}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {23}^n\\ +\\ ⋮\\ {23}^5\\ +\\ {23}^4\\ +\\ {23}^3\\ +\\ {23}^2\\ +\\ {23}^1\\ +\\ 1\\ \end{array}\right)\times \left(\begin{array}{c}\\ {29}^n\\ +\\ ⋮\\ {29}^5\\ +\\ {29}^4\\ +\\ {29}^3\\ +\\ {29}^2\\ +\\ {29}^1\\ +\\ 1\\ \end{array}\right)\times \cdots$$

This is the Fundamental Theorem of Arithmetic.

Now several people will say, "Prove that the above infinite product of prime geometric power series is the Fundamental Theorem of Arithmetic. If that is requested, I will provide it. It will be simple but long. I did it 30 years ago by pencil and paper and 20 years ago by Excel spreadsheet. It goes back to 2300 years ago and the sieve of Eratosthenes and its removal sets / Smallest Common Factor (SCF) sets. Analyzing the SCF sets shows that they are the above construction of an infinite product of prime geometric power series.

Continue to - The Fundamental Theorem of Arithmetic - Part 2

Foundations - The Counting Numbers

Mathematical Conundrum Series

This post is part of a series of posts that should be read in order:

Mathematical Conundrum Series

• Introduction
• Foundations - The Counting Numbers
• Foundations - The Fundamental Theorem of Arithmetic
• The Fundamental Theorem of Arithmetic - Part 2
• The Inverse Naturals and The Euler Product
• The Conundrum
• The Conundrum - Free Will and Choice

Discussing my educational foundation in mathematics, I think, is probably very similar to many average people. But first, I think I have to provide a little background to begin this first post—my own background. What is my educational background in mathematics?

I really have no advanced mathematical education. I have no degree. I have no CV (curriculum vitae). My mathematical education is simply from the American educational system, grades 1–12, from the 1960s and 1970s. So my blog posts here will be limited to that, mostly basic stuff, basic algebra. I have forgotten most of the geometry, trigonometry, probabilities and statistics, and pre-calculus. I was a good student, but I did not take it any further than grades 1–12. 

That being said, how did I learn about numbers?

I think like most people, I started learning about numbers either in pre-school or first grade by learning to count: 1 2 3 4 5 6... What are the counting numbers? They are the natural numbers, the positive integers, the positive integers greater than 0, etc. 

In those early formative years, whether it was the first grade or earlier, we learn about these numbers. We learn how to count. Then, over time, and as we move on to the next grade level in school, we learn how to add them, how to subtract them, and then multiplication and division. Eventually we learn the properties: commutative property, associative property, distributive property, and identity property. Moving on, we learn about fractions (ratios) and how to add, subtract, multiply, and divide them. We learn about exponentiation (powers).

Where is this all going? Well, I will try to speed it up a bit. Eventually I want to point out some shortcomings in this educational process—the shortcuts and assumptions I spoke of in my 'Introduction' post. So let's just make a list here of things I learned through grades 1–12. A brief list, not necessarily complete or in perfect order, but an approximation, and then I will speak of those shortcomings, the shortcuts and built-in assumptions that, in later posts to follow, will become very important.

• Simple high school algebra
• Solving equations with one variable
• Solving equations with two variables
• Logarithms
• More exponentiation
• Set theory - union, intersection,...
• Types of sets - naturals, inverse naturals, rationals, irrationals, reals, cardinality
• Prime numbers, factorization, LCM, GCD,...
• Symbolic logic
• Graphing curves and slopes of various equations
• Geometry - angles, polygons, areas, volumes, pi
• Trigonometry - sine, cosine, tangent, cotangent, secant, cosecant,...
• Intro to probabilities and statistics
• Pre-calculus

There, brief and done.

So what is the point?

The point is, all of this is based on counting numbers, learning to count 1, 2, 3, 4, 5,...

If you continue on with your mathematical education, mathematics is so vast and so complicated that, in comparison to the simple education I have talked about above, this simple education is maybe one quadrillionth of one percent of the vast knowledge produced in the field of mathematics. It is probably even way, way less than that. But still, everyone started with counting 1, 2, 3, 4, 5,...

So now I will point out what I feel are some of the built-in assumptions and/or shortcuts in that grade 1–12 education that I think affect everyone in their journey of the math world.

The counting numbers, natural numbers or positive integers. {1,2,3,4,5,6,7,...} n+1. The set of natural numbers has countably infinite members. Each element of the countably infinite set of natural numbers, however, is finite.

The inverse natural numbers. {1,1/2,1/3,1/4,1/5,1/6,1/7,...} 1/n+1. The set of inverse natural numbers has countably infinite members. Each element of the countably infinite set of inverse natural numbers, however, is finite.

Because we begin to learn with counting numbers, we do not learn immediately that they are also a ratio, a fraction, 2 = 2/1, 3 = 3/1, 4 = 4/1, 5 = 5/1,... We do learn this eventually, and everyone knows this, but it seems silly to write them as a fraction/ratio because any number divided by 1 is that number, understandable, but it is a shortcut.

There is no shortcut for writing the inverse naturals. We are forced to write it as a ratio/fraction, there is no other way, we have to write 1 divided by n. This is entirely understandable, as I cannot imagine learning how to count where you are taught from the get-go 1/1, 2/1, 3/1, 4/1, 5/1,... You are imposing the notion of fractions/ratios and division on a child just learning how to count, and the first thing they are going to ask is why all the counting numbers are written so strangely. None the less, these shortcuts exist.

Another example is when we learn exponentiation. Generally we write 7, instead of $$7^1$$, but to write seven to the inverse first power we have to write $$7^{-1}$$.

Once again, not a big deal, but once again a shortcut.

Now this next shortcut I am going to discuss I think is more of a big deal. Maybe even catastrophic. It is not that we do not know they are shortcuts or that we were never taught the long way of writing these expressions, but after years and years of reinforcing the use of these shortcuts over and over, we tend to build reinforcing circuits in our brains and we tend to forget the other ways or the not shortcut ways of expression, and I think in at least one case that is very detrimental.

Factorization of natural numbers and inverse natural numbers

 

At some point in our early mathematical education, we learn factorization. The prime numbers, the 'Fundamental Theorem of Arithmetic', also known as the 'Unique Factorization Theorem'. This basic theorem goes all the way back to the third century BCE and Euclid's Elements. It is foundational. 

Continue to - Foundations - The Fundamental Theorem of Arithmetic

Introduction

Mathematical Conundrum Series

This post is part of a series of posts that should be read in order:

Mathematical Conundrum Series

• Introduction
• Foundations - The Counting Numbers
• Foundations - The Fundamental Theorem of Arithmetic
• The Fundamental Theorem of Arithmetic - Part 2
• The Inverse Naturals and The Euler Product
• The Conundrum
• The Conundrum - Free Will and Choice

I started this blog back in 2005 and really never made use of it. I have decided now, 19 years later, to revive it and hopefully use it. I am not sure if anyone at all will ever read it, but none the less, I would like to document some of the things that I think about and have questions about. 

The first topic I think I will begin talking about are some simple questions I have about mathematics. I will start with my own personal experience in mathematical education and how I think I was misled (to some degree) with assumptions and shortcuts built into that education that to some degree shaped my world view and have robbed me from seeing things in a more open and unlimited way. I am then going to try to build on that first post, build on it slowly with further posts, and refer back to it periodically to try to demonstrate why I think that something is wrong with the foundations of mathematics.

This first topic may take 20-30 posts. I would like to make the posts not too long, and there is a lot of ground to cover.

In the future, I will expand my topics into other areas of interest.

Continue to - Foundations - The Counting Numbers