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diff --git a/mindmap/Maxwell's Equations.org b/mindmap/Maxwell's Equations.org index 187f7a5..c092100 100644 --- a/mindmap/Maxwell's Equations.org +++ b/mindmap/Maxwell's Equations.org @@ -56,17 +56,17 @@ By definition of electromagnetic induction (and to make Ampere's law consistent \vec{\nabla} \times \vec{E} = - \frac{\partial\vec{B}}{\partial t} \end{align*} Instead of assuming induction as an axiom, it is possible to fix Ampere's equation with the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] first, and then -assume Lorentz symmetry. This explanation is a work in progress. +assume Lorentz Covariance. This explanation is a work in progress. * Implications Maxwell's Equations can be used to calculate all electromagnetic phenomena on the macro scale all the way down to the atom. In practice, solving Maxwell's Equations can be analytically impossible, so several simplifying assumptions are often made. To recap, these are the four equations: -\begin{align*} +\begin{align} \vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_{0}} \\ \vec{\nabla} \times \vec{E} = -\frac{\partial\vec{B}}{\partial t} \\ \vec{\nabla} \cdot \vec{B} = 0 \\ \vec{\nabla} \times \vec{B} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t} -\end{align*} +\end{align} * Speed of Light Maxwell's Equations can be shown to reproduce the speed of light in a vacuum, where: \begin{align*} @@ -76,3 +76,31 @@ Maxwell's Equations can be shown to reproduce the speed of light in a vacuum, wh It is known that Maxwell's Equations are consistent with [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]] and can be expressed in terms of curved spacetime. In fact, if relativity is taken as an axiom, it can be proven that the electric and magnetic fields are descriptions of the same phenomena; this can be taken as a specific example of a [[id:1b1a8cff-1d20-4689-8466-ea88411007d7][duality]]. +The result is a Lorentz Invariant theory of Electromagnetism. +** Relativistic Electrodynamics +The [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] has a Lorentz Invariant counterpart. We unify the charge density and the current density +under a four-vector quantity ~j~: +\begin{align*} +j = (\rho , J) +\end{align*} +Where ~J~ is the current density, and $$ \rho $$ is the charge density. The continuity equation can be reformulated +like so: +\begin{align*} +\partial_{\alpha} j^{\alpha} = 0 +\end{align*} +Likewise, the electric scalar potential and the magnetic vector potential create an invariant four-vector: +\begin{align*} +a = (V, A) +\end{align*} +We notice that Gauss' Law: +\begin{align*} +\nabla^{2} V = \frac{\rho}{\epsilon_{0}} +\end{align*} +needs a more general formulation, as the potential and the charge density need to be replaced with Lorentz Invariant +quantities. We then use Ampere's law: +\begin{align} +\vec{\nabla} \times \vec{\nabla} \times \vec{A} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial(\nabla V)}{\partial t} \\ +\vec{\nabla} \cdot \vec{\nabla} \times \vec{\nabla} \times \vec{A} = \nabla \cdot \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial(\nabla^{2} V)}{\partial t} \\ +\vec{\nabla} \cdot \vec{\nabla} \times \vec{\nabla} \times \vec{A} = \mu_{0} \nabla\cdot\vec{J} + \mu_{0}\frac{\partial\rho}{\partial t} \\ +\vec{\nabla} \cdot \vec{\nabla} \times \vec{\nabla} \times \vec{A} = \mu_{0} (\nabla\cdot\vec{J} + \frac{\partial\rho}{\partial t}) \\ +\end{align} diff --git a/mindmap/egoism.org b/mindmap/egoism.org new file mode 100644 index 0000000..fb16307 --- /dev/null +++ b/mindmap/egoism.org @@ -0,0 +1,28 @@ +:PROPERTIES: +:ID: 326eb3f8-680a-432c-bf69-42ba4d366116 +:END: +#+title: egoism +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t +* Introduction +Egoism, broadly defined, is the philosophical view that one should maximize for their own utility function, or well-being, +or self-satisfaction, etc..., whatever you want to call it. The general statement is that there is some value one would +want to maximize for themselves. This philosophical view is a metaethical view in general but can be applied descriptively. +* Descriptively +In economics, homo-econimus is a kind of idealized economic agent that maximizes for their own economic value. Descriptive +egoism is in a sense the statement that people tend to maximize for some utility function. In a sense, there are many games +that people can play, and economics is just one of them; there's homo-politicus, homo-vitamus, and many other games that +agents try to play. Descriptive egoism is the descriptive formulation of a utility function that people have in-built, +and a statement that people tend to play according to said utility functions with generalized rational expectations. +* Perscriptively +The perscriptive egoism is a meta-ethical framework that simply asserts that one should act in their own self interest. +Self interest is a loosely defined term that is subjective to the individual. +* Critisisms +Often many will point out the fact that it is a circular system, that it has no descriptive power because one can formulate +anything in terms of an egoist outlook. Therefore, while one can accept or deny the validity of egoism, the [[id:6d8c8bcc-58b0-4267-8035-81b3bf753505][framework]] of +egoism has weak axioms and it is therefore possible to formulate many sub-philosophies that are compatible with egoism. +(Meta-comment: I would describe myself as someone that formulates sub-philosophies inscribed within the egoist framework, +and I would myself describe my own views in an egoist manner). diff --git a/mindmap/everything.org b/mindmap/everything.org index 8c92bd9..57321af 100644 --- a/mindmap/everything.org +++ b/mindmap/everything.org @@ -30,6 +30,8 @@ and you have to omit details. However, I think this is a pretty reasonable attempt at doing so and you will soon notice that the frameworks that I am presenting can in fact be used to explain the article itself, albeit in a somewhat contradictory way. +** Isn't this just another Framework? +Yes. In fact, I have a page about [[id:6d8c8bcc-58b0-4267-8035-81b3bf753505][frameworks]]! * Something I linked them above, but if you want to read about something that is not everything, go here: diff --git a/mindmap/framework.org b/mindmap/framework.org new file mode 100644 index 0000000..114913f --- /dev/null +++ b/mindmap/framework.org @@ -0,0 +1,38 @@ +:PROPERTIES: +:ID: 6d8c8bcc-58b0-4267-8035-81b3bf753505 +:END: +#+title: framework +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t +* Introduction +Frameworks are an essential thing to understand are fundamental to how we work, and this statement will +never be contradicted. The main [[id:1b1a8cff-1d20-4689-8466-ea88411007d7][two]] hallmarks of a framework are as follows: +1. the framework has a set of axioms i.e. statements that are taken as a given. +2. the framework can use some form of pseudo-logical deduction in order to reach conclusions. +Note that this sounds a lot like [[id:a6bc601a-7910-44bb-afd5-dffa5bc869b1][mathematics]]. Indeed, mathematics is an instantiation of a framework, but there +are other frameworks as well. Mathematics is a framework detached from reality, and we often develop mathematics +in order to fit those mathematical models to reality. Mathematical frameworks aren't wrong or right, they're contradictory +or logical (i.e. they either degenerate into principle of explosion or they are logically consistent). In other words, +a theory or framework can be independent of reality. + +Applying mathematical models to reality are a statement about that particular system you are modelling i.e. you are +supposedly applying said mathematical model because there is some belief that reality conforms to whatever axioms +the mathematical model has. General mathematical models tend to model more things, but the axioms are weaker, meaning +the predictions of said models are also weaker. Adding additional assumptions to models generally makes those predictions +stronger. + +Of course, there are other ways to model things as well other than with mathematics; one can also use the framework of +word-based logical reasoning in order to make models. This mindmap utilizes this kind of modelling to a great extent, +and this very article is an instantiation of word-based logical reasoning. In this mindmap, the same model will be used +in order to model many things, leading to a fine structure. +** Perscription and Description +Perscriptive models tend to make value judgements about a particular system, meaning they in general view agents of +said system as responsible and non-deterministic agents, whereas descriptive models in general view agents of said system +to be deterministic and not able to be self aware. In other words, perscription is where "thinking" ends and where +"action" begins. Adding more levels of determinism to the model tend to make it more descriptive of reality (this is +an emperical statement, and not a theoretical one). +** Metaframeworks +This article is a metaframework, another instantiation of [[id:8f265f93-e5fd-4150-a845-a60ab7063164][recursion]]. 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a/mindmap/ltximg/org-ltximg_f6df9e1246dbdd06793bf411095452fbf1e4e7d4.png b/mindmap/ltximg/org-ltximg_f6df9e1246dbdd06793bf411095452fbf1e4e7d4.png Binary files differnew file mode 100644 index 0000000..a160412 --- /dev/null +++ b/mindmap/ltximg/org-ltximg_f6df9e1246dbdd06793bf411095452fbf1e4e7d4.png diff --git a/mindmap/philosophy.org b/mindmap/philosophy.org new file mode 100644 index 0000000..b1a26e5 --- /dev/null +++ b/mindmap/philosophy.org @@ -0,0 +1,30 @@ +:PROPERTIES: +:ID: f4d70abf-242c-41b7-b0dd-d7f1813cfb33 +:END: +#+title: philosophy +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t +* Introduction +Philosophy is a hard to describe term, but this mindmap defines this term as the study of living life optimally. We use +this definition to ground meta-ethical frameworks in [[id:326eb3f8-680a-432c-bf69-42ba4d366116][egoism]], which gives meaning to perscriptive ethical statements (which +this mindmap holds there is regularly no meaning to in the colloquial sense). This mindmap defines an ethical statement +such as "it is wrong to do x" as a statement which says, "it maximizes utility for your own life if not x". Of course, +because it is an emperically justified statement to say that most people act the same, this can ground commonly held +moral beliefs. There are several possible refutations to this point of view, but this mindmap maintains that refutations +to this point of view usually appeal to some feeling of wrongness rather than being definitionally inconsistent, which +is again an instantiation of this point of view (to argue moral truths from a point of view of a feeling of wrongness is +an instantiation or a confirmation of this point of view). + +For instance, one possible counterargument that is brought up involves the fact that this theory equates preferences to +moral statements. "I prefer red" and "x is morally right" indeed /feel/ like two separate things. This mindmap maintains +that they are two different things in many senses, but that the fundamental assertion of these two statements is the same. +It is just a different kind of emotion, but there is no underlying fact of the matter that one can point to with regards +to moral theory. + +Note that there are several arguments that facts are treated on a separate footing to moral theory under such a framework. +Indeed it is true that this mindmap will rest on some emperical facts, but this mindmap maintains that doing this is a +perfectly internally consistent and descriptive standpoint. From here on, we will use ethical and moral statements as +a description of people, rather than a description of some real moral fact. |
