Unity+

I was originally thinking that the book was going into simplification(combining like terms) because it was weird how a textbook would present the equation as such. Good eye, I must say.

The equations that I worked with mainly work with the y direction. You added on to the idea, which I appreciate greatly. Yes, I have difficulty presenting ideas. It has always been my weakest points. Thank you for taking the time to allow other members to understand it. I will try to improve my skills in presentation. EDIT: I need to make a correction in my generalized equation: [math]\eta(x,F)=\frac{\kappa x-x}{\kappa }-\sum_{n=1}^{\infty }\frac{x}{F^{2n}\kappa}[/math]

Here is a more generalized equation that both takes into account the fraction of the line the midpoint is and the size of each line after each new line. [math]\eta (x,F)=\frac{\kappa x-x}{\kappa }-\sum_{n=1}^{\infty }\frac{1}{F^{2n}\kappa}[/math] Where F is the new fraction size after each time while κ is the positioning of the newline based on what particular fraction the new point will be on.

I also got 11/9. (x+11)/3=2(x+14x)/9 (x+11)/3=2(15x)/9 (11/9+11)/3=2(15(11/9))/9 (11/9+99/9)/3=2(55/3)/9 110/9/3 = 110/3/9 110/27 = 110/27

Recent work I have done has relations with this kind of problem, as I think there seems to be a difference between axioms founded upon nature and axioms founded upon the un-natural(for lack of a better word). At the beginning of Mathematics, many mathematicians developed axioms of logic that were related to the reality of nature, not upon the construction of the human mind. Thus, mathematics came to be founded upon the axioms of nature. However, soon enough we begin developing axioms that relate to other systems. Therefore, humans have the capability of discovering the axioms of nature, but they also have the capability of developing axioms of other types of systems. For example, there are many abstract constructs which are used very much in physics, such as many geometries that are abstract and numbers(complex numbers) which are used in quantum mechanics. There are also abstract constructs that do not fit within our CURRENT Universal landscape. In regards to parallel lines, I compare the change to the change between Newtonian mechanics and Einstein's theories of relativity and the forces that exist, such as gravity. Einstein didn't change the particular ideas surrounding the existence of the force of gravity, but completed Newton's ideas as to add a mechanism for the gravitational forces that exist. Therefore, Newton was not wrong but incomplete with his theories. Ideas develop over time. This does not mean that the original idea is wrong, but needs modifications to fit more and more closely to the natural world. Discovery is not one particular event, but a group of events that leads us to understand more fully the mechanisms of the Universe.

Well, this is something that isn't really useful, but more fun. Well, here is the idea. The idea is there is a line of size x which then another line is placed at the other line's midpoint, or any given point for generalization, where one end of the new line is positioned at this particular coordinate. This is to repeat onto infinity. The idea is to find the distance between the end of the first line to the end of the last line that is placed in the particular fashion told about above. The distance of this is equal to the following function, given x as the length of the first line and F being the fraction at which to place the new line at on the original line. [math]\eta (x,F)=\frac{Fx - x-\sum_{n=1}^{\infty }\frac{Fx}{F^{2n+1}}}{F}[/math] The way this was found was first to add up all the lengths of the lines that added distance to the line that is being analyzed. Since the length of the lines are being halved their parents, the way to find the distance is to simply subtract the sum of these lines from the length x of the original line. Also, the beginning must be taken into account when subtracting the sum because the original line's height must also be accounted for to find the distance. [math]\eta (x,F)=x-\frac{x}{F}-\sum_{n=1}^{\infty }\frac{x}{F^{2n+1}}[/math] [math]\eta (x,F)=\frac{Fx - x}{F}-\sum_{n=1}^{\infty }\frac{x}{F^{2n+1}}[/math] [math]\eta (x,F)=\frac{Fx - x-F\sum_{n=1}^{\infty }\frac{x}{F^{2n+1}}}{F}[/math] [math]\eta (x,F)=\frac{Fx - x-\sum_{n=1}^{\infty }\frac{Fx}{F^{2n+1}}}{F}[/math] I'm going to be adding more to this later on, but that is what I have so far. There are some other interesting patterns I am finding when I modify the rules. Tell me if I have some incorrect math here if I do.

I think most mathematicians would identify themselves, like me, as a platonist. It seems, without a doubt, that nature has constructs of mathematics based on mathematics. Of course, this could be caused by the limitation of the human mind.

I meant all of this. When I was watching a film about the two philosophies, they referred to the idea of numbers existing in reality as "platonism" while they referred to realism as being numbers only existing in the mind of humans. EDIT: The poll has been reworded. I apologize for the confusion.

I just wanted to take a little poll of everyone's view of the world. Are you a Platonist of mathematics, or someone that thinks that Mathematical structures exist in nature and are abstract, but real forms or do you believe that Mathematics simply is a human construct that allows us to interpret the nature of the world?

I think you must realize that all mammals contain traits related to killing their own race. A mammal's mistakes shouldn't automatically declare it inferior to another.

Just to remind you, the system I am talking about is different than Dynamic Collatz-Matrix equations that is logic functions. It is simply something that I found that I thought would be interesting. Also, the system doesn't contain any axioms besides the first axiom of contradiction. The problem is encountered when you do add more axioms. So, the only axiom that would exist within the system would be that particular axiom. What I was saying when referring to formality is that the axioms that are used in Dynamic Collatz-Matrix equations are not the common system of axioms used in mathematics. They are simply systems from the logic equations. That is what I mean, if you understand what I am saying. The problem with that example, as you stated, is there are similar axioms within both systems, which means they, of course, will have similar axioms that arrive from both systems. I apologize if I am being confusing.

The theorem isn't an actual theorem I made, persay, but it is something that I looked into when developing an example Fundamental Theorem that contains such an axiom to be used for Dynamic Collatz-Matrix equations that deal with logic functions I described in the earlier post. The idea is being able to take an axiom of a set of axioms to be able to develop an a branch of other theorems or axioms. The notation is simply saying that there is the Fundamental Theorem which contains axiom a which is now having to comply with axiom b, which axiom b must not comply with axiom N. It is also implying that it must not comply with axiom a. And actually, I should have done this earlier, but the notation should have been this to show that these axioms must not comply with axiom a. [math]\delta_{F}\rightarrow\left\{A_a\star A_{b}\right \}\left( A_{a}: A_{b}\star_{!}A_{N}\right)[/math] [math]A_{N}\ni A_{a}[/math] NOTE: Because of this, axiom a, which is an element of axiom N, or the axioms of the system, axiom a must also contradict axiom a. This doesn't make sense because the rule of the axiom is all axioms contradict each other. Therefore, axiom a contradicting itself because axiom is compliant with axiom a since they are both the same and have the same rules. I see now where you get the relation with Russell's paradox. To avoid an inconsistent contradiction with the statements, [math]A_{b}=\varnothing[/math] EDIT: Realized this was correct. Wouldn't it be that if the axioms have to all contradict that no new theorem could be introduced because even if you were trying to find one that contradicted axiom a, the axiom of contradiction, then it would still contradict and therefore the theorem could not be proved true? I can see, though a theorem that could develop that would say "axiom a contradicts axiom N, an axiom that represents axioms that contradict axiom a because they contradict each other for they all comply with axiom a." The theorem wouldn't hold because then it doesn't contradict axiom N, the axioms that exist within the Fundamental Theorem of such a system. Therefore, not theorem could be proven with such a system. The whole idea isn't meant to be formal. The idea is developing systems using axioms that are not perceived in the natural sense of Mathematics. Dynamic Collatz-Matrix equations use a completely new set of Fundamental axioms or a new Fundamental Theorem consisting of a set of axioms. The idea would be that such an inconsistent system could exist within the spectrum of axioms that produce systems. It goes off the concept of taking forms of reasoning to look at new ways of thinking of systems of mathematics. Instead of the formal axioms that exist to produce more theorems, this takes a whole new set of axioms that are forms of reason for a completely different system, which produce a completely new system. However, this means now systems now must be determined to be consistent with each other now. Here is a question: If you have two completely different systems that have a different set of axioms, is it possible that each could produce similar theorems? I would think not because if the logic of both systems are not similar then they would produce completely different results. EDIT: I took a further look at the statements presented in the Principle of Explosion, and here is the statement I found: "If one claims something is both true () and not true (), one can logically derive any conclusion ()." The paradox doesn't, or at least I don't see it does, address statements that are both true and false. It is merely stating that axiom A states that all what comes after must contradict itself, meaning nothing most comply with its logic. Therefore, if an axiom were to comply with its rules of logic then it already has defined itself as complying with its logic therefore it hasn't complied with its logic, making it not liable as an axiom.

Here is a paradox I found while working on Dynamic Collatz-Matrix equations. If someone solves the paradox I will give reputation points. The Contradicting-Axiom Paradox Here is the mathematical statement: [math]\delta_{F}\rightarrow\left\{A_a\star A_{b}\right \}\left( A_{a}: A_{b}\star_{!}A_{N}\right)[/math] This paradox is a statement about the nature of axioms. The paradox is that there could be a Fundamental Theorem, consisting of a set of axioms, that has the main axiom state "All other axioms and/or theorems must contradict each other." The paradox here is if all axioms must contradict each other, even the main axiom, then these axioms have complied with such an axiom and therefore do not contradict all axioms, even if the axioms or theorems afterword contradict each other. Therefore, either the axioms and theorems comply with the main axiom or do not comply with it.

Adding on to the earlier post: Because of the way theorems work, being branched off the Fundamental Theorem, or set of axioms, the equation must be a Dynamic Collatz-Matrix equation. This means that using a particular logic constant the parameters change per the change in distance of the initial point of the matrix solution. The logic constant is not a numerical value, but a set of logical statements that influence the change in the parameters per movement in the matrix solution. For example, the initial parameter could be that set A is equal to set B. Then, after such a statement is applied to the initial statement in the matrix solution, the statement could become set B's elements are equal to set C's elements. The following states that after two movements of the initial element in a matrix solution, the parameter of addition would become an operation of multiplication. [math]\Delta \Lambda _{a}\Rightarrow O_{a}\star O_{b},\left \{ \star : +\to\times \right \}[/math]

Well I imagine a tree branch-type system where theorems branch off from the axioms that are given. If we are to define simplification, there must be a limit to the definition of a theorem. For example, maybe a theorem that comes from the Fundamental Theorem, or set of axioms, must only contain 2 to 3 of the axioms that exist within the Fundamental Theorem.

A friend I know has recently got an interest into the theoretical of Black holes and I wanted to recommend some literature on Black Holes. Since I mainly work in the field of mathematics and computer science, I wanted someone else's recommendations. So, can anyone give a list of recommendations of literature for new physicists about Black Holes?

I think there needs to be consideration of what disorder and order are defined by in the idea of natural cleverness. Cleverness is simply being able to understand and apply an idea quickly to achieve a result that is desired. Therefore, both order or disorder could be the goal and entropy has a different meaning in such a context. There would have to be a different approach similar to entropy.

I now am applying primitive logic functions to Collatz-Matrix equations. [math]C(\delta _{F})_{k\times d}\begin{Bmatrix} \Lambda _{a} &\Lambda _{b} \\ \Lambda _{u}& \Lambda _{v} \end{Bmatrix},s(k_{p},d_{p})[/math] Basically, given a Fundamental theorem, or set of axioms, [math]\delta_{F}[/math] there is a set of theorems that are derived from this fundamental theorem, where the parameters [math]\Lambda _{a}[/math], [math]\Lambda _{b}[/math], [math]\Lambda _{u}[/math], and [math]\Lambda _{v}[/math] are not equations but logical statements that are added to the fundamental theorem or set of axioms. Here is an example concept: [math]C(\delta _{F})_{2\times 2}\begin{Bmatrix} \Lambda _{a} &\Lambda _{b} \\ \Lambda _{u}& \Lambda _{v} \end{Bmatrix},s(1,1)=\begin{bmatrix} \delta _{F} & \delta _{F}+ \Lambda _{u}\\ \delta _{F}+ \Lambda _{u} + \Lambda _{v}+\Lambda _{a}& \delta _{F}+ \Lambda _{u} + \Lambda _{v} \end{bmatrix}[/math] In this case, the order of the logic equations matters. There are also other rules about how these work. All logic functions must comply with each other's rules. The rules of logic functions must not contradict. That is just the start. More is coming to the concept. Here is an application of the concept: [math]C(A\cap B)_{2\times 2}\begin{Bmatrix} O_{a}=Q_{b}& O_{a}\in O_{b} \\ O_{a}\ni O_{b}&O_{a}\neq Q_{b} \end{Bmatrix},s(1,1)=\begin{bmatrix} A\cap B &\left (A\cap B \right )\rightarrow A\ni B_{e} \\ 0& 0 \end{bmatrix}[/math] Therefore, the theorem arrived from this is that the elements of A and B will intersect when either some or all elements of B are within set A. The reason why order within this is important is because if it were the other way around then it would say that either some or all elements of B are within A if A and B intersect. Though, in this case, both cases are true for other instances there would be a difference.

Another thing that would be interesting to see is the limitations of proofs for a given system of axioms. For example, would it be provable to determine the exact amount of simplified theorems that would result from a set of axioms?

That depends on the meaning of 'correct.' Usually, there is a guideline for the use of standard notation and it isn't as if a person can create their own notation and get away with it easily. If there wasn't a standard notation then it would make it difficult to keep learning more and more notations of a particular problem or solution.

In mind, I would be referring to the Fundamental theorem of algebra, for example.

That depends on what you mean by an automated theorem prover. While they used computer algorithms to be proved, there was a specific task for such a proof given to the algorithm. What I am referring to is a generalized form of an algorithm. While I agree that the point of proofs are to give a human understanding of mathematics, it is important to realize that computers would give much help to providing an understanding if this algorithm were produced. In fact, if such an algorithm were developed it would provide a way for mathematicians to use similar patterns of proofs to understand theorems and conjectures. I can see the issue here. Machines are limited to how they can interpret truth because truth is ultimately defined by their own rules as well. However, doesn't this also apply to Mathematics in general? We base proofs off of theorems that are based off of theorems, which are eventually based off axioms of Mathematics. As the definition of an axiom states, an axiom is a starting point of reasoning(http://en.wikipedia.org/wiki/Axiom). Therefore, the same principle could apply to a machine as axioms would be defined for a machine from which it would know truth from in the same way we hold truth to our axioms of Mathematics. Therefore, wouldn't truth be defined by the axioms presented to such a machine and provide an end point of a proof just like how we use axioms to provide an end point to our proofs? This is a very interesting topic, I must say. Though there isn't really a fundamental theorem of mathematics, it seems there must be because of the existence of many other fundamental theorems that exist that are potentially semi-fundamental theorems of mathematics. That is what I was approaching.

I am currently working(though with little success on a concrete algorithm) on an algorithm that takes the Fundamental Theorem of Mathematics and tries to "tree-branch" theorems using a fundamental and semi-fundamental theorem of mathematics. It involves a "theorem constant", which is not a numeral value but a logic-based value that is detected through out theorems, which would help determine new theorems. Though, again, it is conceptual and is merely at the drawing board still. If I get any progress, I will post it and see if anyone things it has potential.