03/03/2014, 09:13 AM

Hello.

I am a graduate student working toward my PhD in Physics. Aside from physics, I enjoy programming, camping, and parkour (though I don't have much time for the last 2 activities these days).

I admit that I come bearing an idea that will undoubtedly require guidance in the future. That being said, I am also trying to read as many related posts as I can on this forum so that I have a good idea of what progress has already been made in related areas. The problem with that is that I don't understand much of what I read on the forum. Is the language something that I should expect to pick up over time? Is the type of jargon used on the forum common to mathematicians, or only to tetration enthusiasts? Are there any specific references that you can recommend as starting points?

My idea involves a hyperoperator of which addition, multiplication, exponentiation, are cross-sections. While I am still chiseling at it, I am learning a lot as I go. At first, I was perfectly happy to make generalizations from the top down: Formulating conjectures by observing mathematical patterns and symmetries. But, since then, I have realized that conjectures are not adequate: Some of my conjectures, which seemed obvious from observation, were inconsistent with each other. The problem is that a conjecture can be wrong, and there is no particular reason that a conjecture should be true. Lately, I am been trying to take a more axiom-based approach. An axiom is true by definition, and I have realized that some of the conjectures I was coming up with (from mere observations of patterns) were actually already built in to the axioms I was using. I just hadn't figured out how to prove them yet. One of my current problems is evaluating my expressions: As I am learning, proving a relation between 2 hyperoperators does not necessarily mean that you can evaluate either of them. Around every turn, I find more questions than answers, and as my methods become more rigorous, the road becomes more treacherous. As I evolve, I gain new insights, but this means that my goal-posts are constantly moving. Each insight leads me to realize that I am further from my goal than I previously thought.

So far, this forum is the greatest resource I have found. There are mathematical concepts that, for the most part, are of interest to all members of this forum from what I can tell, and yet cannot be found anywhere else on the internet. There are conversations held here that are not held anywhere else. For people interested in certain concepts, there is nowhere else to go, it seems.

I am currently typing up an overview of my idea in LaTeX, so I will start a thread on that in a few weeks or so, depending on how much time I have to work on it. In my write-up, I am trying to be as clear and as thorough as possible, while also being as concise as possible. I would like to say that I am being as thorough as possible, but I would have to amend that I am only being as thorough as possible for a physicist. I hope that the progression of my document is not unbearable. More so, I hope that my current hyperoperator framework is thoughtful so as to be irrefutable (if that is possible). Another thing I would like to know is whether or not there are (among forum members) any accepted properties that a hyperoperator SHOULD have. For example, I read in more than one thread that

\( a[x]a=a[x+1]2 \).

(and similar claims at this, and this. Why is \( 2[\alpha]2=4 \) a beautiful result?)

Where is this coming from? Is this an axiom or is it derivable?

In the meantime, I will try to read through and understand the threads on related topics. I would like to post a comment or question in each of these threads, if only to confirm that I read/understood it. I think that my biggest barrier to making use of this forum right now is understanding the language, as I mentioned before.

Some threads that seem related:

1) http://math.eretrandre.org/tetrationforu...r+operator

This sounds very similar to what I am aiming for, though I can't say I understand it yet.

2) http://math.eretrandre.org/tetrationforu...hp?tid=546

Ideally, the end product of my formulation is an actual plot of values for intermediate operations, such as the one at the above link.

3) http://math.eretrandre.org/tetrationforu...r+operator

Sorry for the long introduction. I look forward to interesting discussions on this forum in the future, and I look forward to better getting to know the community here.

hixidom

I am a graduate student working toward my PhD in Physics. Aside from physics, I enjoy programming, camping, and parkour (though I don't have much time for the last 2 activities these days).

I admit that I come bearing an idea that will undoubtedly require guidance in the future. That being said, I am also trying to read as many related posts as I can on this forum so that I have a good idea of what progress has already been made in related areas. The problem with that is that I don't understand much of what I read on the forum. Is the language something that I should expect to pick up over time? Is the type of jargon used on the forum common to mathematicians, or only to tetration enthusiasts? Are there any specific references that you can recommend as starting points?

My idea involves a hyperoperator of which addition, multiplication, exponentiation, are cross-sections. While I am still chiseling at it, I am learning a lot as I go. At first, I was perfectly happy to make generalizations from the top down: Formulating conjectures by observing mathematical patterns and symmetries. But, since then, I have realized that conjectures are not adequate: Some of my conjectures, which seemed obvious from observation, were inconsistent with each other. The problem is that a conjecture can be wrong, and there is no particular reason that a conjecture should be true. Lately, I am been trying to take a more axiom-based approach. An axiom is true by definition, and I have realized that some of the conjectures I was coming up with (from mere observations of patterns) were actually already built in to the axioms I was using. I just hadn't figured out how to prove them yet. One of my current problems is evaluating my expressions: As I am learning, proving a relation between 2 hyperoperators does not necessarily mean that you can evaluate either of them. Around every turn, I find more questions than answers, and as my methods become more rigorous, the road becomes more treacherous. As I evolve, I gain new insights, but this means that my goal-posts are constantly moving. Each insight leads me to realize that I am further from my goal than I previously thought.

So far, this forum is the greatest resource I have found. There are mathematical concepts that, for the most part, are of interest to all members of this forum from what I can tell, and yet cannot be found anywhere else on the internet. There are conversations held here that are not held anywhere else. For people interested in certain concepts, there is nowhere else to go, it seems.

I am currently typing up an overview of my idea in LaTeX, so I will start a thread on that in a few weeks or so, depending on how much time I have to work on it. In my write-up, I am trying to be as clear and as thorough as possible, while also being as concise as possible. I would like to say that I am being as thorough as possible, but I would have to amend that I am only being as thorough as possible for a physicist. I hope that the progression of my document is not unbearable. More so, I hope that my current hyperoperator framework is thoughtful so as to be irrefutable (if that is possible). Another thing I would like to know is whether or not there are (among forum members) any accepted properties that a hyperoperator SHOULD have. For example, I read in more than one thread that

\( a[x]a=a[x+1]2 \).

(and similar claims at this, and this. Why is \( 2[\alpha]2=4 \) a beautiful result?)

Where is this coming from? Is this an axiom or is it derivable?

In the meantime, I will try to read through and understand the threads on related topics. I would like to post a comment or question in each of these threads, if only to confirm that I read/understood it. I think that my biggest barrier to making use of this forum right now is understanding the language, as I mentioned before.

Some threads that seem related:

1) http://math.eretrandre.org/tetrationforu...r+operator

This sounds very similar to what I am aiming for, though I can't say I understand it yet.

2) http://math.eretrandre.org/tetrationforu...hp?tid=546

Ideally, the end product of my formulation is an actual plot of values for intermediate operations, such as the one at the above link.

3) http://math.eretrandre.org/tetrationforu...r+operator

Sorry for the long introduction. I look forward to interesting discussions on this forum in the future, and I look forward to better getting to know the community here.

hixidom