In my studies of Mathematics and (mostly) Theoretical Computer Science I've encountered what is known as  which purports to demonstrate nothing can be actually be proved because all proofs are one of the following:   The circular argument, in which the proof of some proposition is supported only by that proposition The regressive argument, in which each proof requires a further proof, ad infinitum The axiomatic argument, which rests on accepted precepts which are merely asserted rather than defended   My response: What's wrong with circular reasoning in the first place? I simply disagree fundamentally with the notion that we can't have &quot;closed systems&quot; of knowledge. Why is there this idea that we need some kickstarter axioms to prevent going around in circles?  It's not that I don't appreciate the need for axioms from a convenience perspective. It's that I don't think that there's really no merit to going around in circles in the first place! &quot;I am God therefore I am God&quot; is valid. Things don't need to make sense. After all sense is not an inherent property of a statement. It requires interpretation to judge that. Interpretation from another language - in the form of symbolic manipulation. It's all inherently meaningless - so why bother with making sense when you can just be valid. This is my argument for why circular reasoning is a blessing. Instead of hunting down possibly ad infinitum the &quot;source&quot;, the most &quot;fundamental&quot; axiom, the &quot;point&quot;, we can prevent the existence of the Hydra altogether. It's either circular reasoning, or the same thing in disguise but with more stress on top of it - the difference is that in the latter the circle never runs out. Why choose stress?
Suppose the following dialogue:  ... &quot;I accept only one notion of land property. Namely, 'I am doing my stuff here, therefore I am here&quot;. &quot;But this means,&quot; he responded, &quot;you can break into any place and stay there, using this rule.&quot; &quot;There is another rule, though. Right to act in its most natural sense, 'I am here, therefore I am doing my stuff here.&quot; &quot;But do not these rules put together become circular reasoning?&quot; &quot;Of course, they do,&quot; I put my head closer to him, &quot;and this is exactly that rare case when circular reasoning is not only sufficient, but is necessary.&quot; &quot;But why is that? Circular reasoning never was a correct argument.&quot; &quot;There are many circular things happening in nature. Life forms, for example, exist only in order to produce new life forms or to prolongate their own lifetime. Is there any reason to say this can't be the case for reason itself? Why must reasoning not be circular?&quot; ...  Can circular reasoning ever be justified? For example, we know all the words in our languages can't be defined using other words without the use of circularity. We know logic can't prove itself without circularity. Are these arguments rational? Are there any other arguments where circular reasoning is still rational?
Riddle (simple arithmetic problem/illusion) I'm not sure how well known this "riddle" is but here it goes.  3 people go to a restaurant, each buy food worth 10.00.  When they're done, they give 30.00 to the waitress.  She gives the money to the manager, manager says they paid too much, gives 5.00 back to the waitress.  She comes back to the table, gives a dollar back to each person and then puts the remaining 2.00 in her pocket for tip.  So now, each person has paid 9.00.  9.00 x 3 = 27.00.  Plus the 2.00 in the waitresses pocket is 29.00.  What happened to the 30th dollar?  So what's the issue with this way of calculating (since if you backward it works fine), that doesn't give us the correct result? How to scale and shift voltage signal? I want to scale and shift my voltage signal by below relation:  $$ V_{out} = \frac {10} {12} \cdot V_{in} - 10 $$  Indeed my primary voltage is from 0v to 24v and I want to put it from -10v to +10v.  I try to use the below circuit but its gain is always greater than 1.    Image from SLOA097 TI.  Does anyone know how I can achieve my goal? I have ±12v and 5v power lines.  EDIT: According to the innovative solution by scorpdaddy, I thought it would be valuable to edit my question:   High input impedance is required to decrease load effects. How can it be achieved? How can I achieve the more and more accuracy? Primary ideals in Noetherian rings For an $R$-module $M$ I have the following definition for a submodule $N\subset M$ to be $\mathfrak{p}$-primary: this is the case when $\text{Ass}(M/N) = \{\mathfrak{p}\}$, that is, $M/N$ is coprimary or $(0)$ is a primary submodule.  Now for an ideal $\mathfrak{q} \subset R$ this definition gives that $\mathfrak{q}$ is $\mathfrak{p}$-primary iff $\text{Ass}(R/\mathfrak{q}) = \{\mathfrak{p}\}$. It can be shown that this is equivalent with $\mathfrak{p}^n \subset \mathfrak{q}$ for some $n\in \mathbb{N}$ and the condition \begin{align*} \forall r,s \in R: rs \in \mathfrak{q}, r\not\in \mathfrak{p}\Rightarrow s \in \mathfrak{q}. \end{align*}     Now I suspect that if $R$ is Noetherian, and if $\mathfrak{q}$ is $\mathfrak{p}$-primary, then we have $\sqrt{\mathfrak{q}} = \mathfrak{p}$. I have shown this for PID's, but does this generaly hold?    And if so, is every ideal $\mathfrak{q}\subset R$ then $\sqrt{\mathfrak{q}}$-primary?   Many thanks. Finding minimum-area-rectangle for given points? As you see in the figure,  the question is:    How to find the minimum-area-rectangle (MAR) fitted on the given points?    and a supporting question is:    Is there any analytical solution for the problem?    (A development of the question will be to fit a box (3D) to a cluster of points in a 3D point cloud.)  As a first stage I propose to find the convex-hull for the points which reforms the problem (by removing those points are not involved in the solution) to: fitting a MAR to a polygon.  The required method will provide X (center of rectangle), D (two dimensions) and A (angle).      My proposal for solution:     Find the centroid of the polygon (see )   [S] Fit a simple fitted rectangle i.e., parallel to the axes X and Y   you may use minmax function for X and Y of the given points (e.g., polygon's vertices)  Store the area of the fitted rectangle Rotate the polygon about the centroid by e.g., 1 degree Repeat from [S] until a full rotation done Report the angle of the minimum area as the result   It looks to me promising, however the following problems exist:   choose of a good resolution for the angle change could be challenging, the computation cost is high, the solution is not analytical but experimental. I am currently learning electromagnetism and I am getting really annoyed using a physical hand to accomplish the right hand rule. I get it, i've seen diagrams, and i've solved problems with it but still somehow I mess up because I bend a finger or twist my wrist a strange way. I am tired of using my hand.  So my question is can I find an entirely mathematical basis for this?  Lets say for any given diagram I define the positive $x$ axis as going to the right, the positive $y$ axis as going up and the positive $z$ axis as going into the page. Then I just did unit vectors for everything to determine the missing component?  As I understand it $\vec{F} = q (\vec{V}\times\vec{B})$ and with that definition I am confident I could find out the direction of the electric field given the velocity of a particle of charge and the magnetic field but how can I re-arrange this so I can solve for other variables? For example what does $\vec{B}$ equal in terms of a cross product involving $\vec{V}$ and $\vec{F}$? I need to find my external IP address from a shell script. At the moment I use this function:   myip () {      lwp-request -o text checkip.dyndns.org | awk '{ print $NF }' }   But it depends on perl-libwww, perl-html-format, perl-html-tree installed. What other ways can I get my external IP? Buffering GeoJSON file using Python? Finding all solutions for a system of equations with constraint on the sum of absolute values I am looking for a fast algorithm to find all integer solutions for the following system of linear equations ($c_1,\ldots,c_n \in \mathbb{Z}$ and $r\in \mathbb{N}$ ):  $x_1+x_2=c_1$  $x_3+x_4=c_2$  ...  $x_{2n-1}+x_{2n}=c_n$  $|x_1|+|x_2|+\ldots+|x_{2n}|=r$  $x_1,\ldots,x_{2n} \in \mathbb{Z}$   Also can we find the number of solutions before solving the system? Is every invertible matrix a change of basis matrix? var.replace is not a function Can a single molecule have a temperature? Who is the Community user? I see the  on the user page.  It seems to edit posts sometimes.  What is it? And why has it downvoted over 1800 times? See also: I am running Ubuntu Gnome 16.10 (64bit). Gnome-shell version 3.20.4.   When I start my computer, I eventually see GDM3 offering a login prompt, as expected.   After I login, my monitors flash a little (like they're unplugged and then replugged in), and then I am brought to the gnome 3 desktop.   However, if I hit  CTRL + ALT + F1, I am brought back to GDM3, offering a login prompt.   If I then hit CTRL + ALT + F2,  I am brought back to my Gnome 3 desktop.  So it seems like GDM3 continues to run in TTY1 while the desktop runs in TTY2.   Questions   Is this how things are supposed to work? I thought the desktop was supposed to use the same TTY (or whatever you call it) as the display manager.  Am I using more graphical resources by having things setup this way? Is there a way to have GDM3 pass off the display to Gnome Desktop after I login? Is there any advantage do doing #3? I need to clip a vector layer to the extent of a raster. It has to be very precise that's why I would not want to draw a polygon myself (and clip the two polygons). There seem to be some ways to clip a raster to a polygon but I have difficulties to find a solution to clip a polygon to a raster. I already tried to create a polygon out of my raster several times in ArcGIS and R, so I could clip the result. But every time it crashes because the raster is to big. There must be a simple way to solve this, and somehow I can just not see it at the moment.  How can I do this using GDAL? There are a plethora of words for user accounts, like logon, login, signon, and also the action of logging in (or logging on) or signing in.  Are there any usage guidelines here? Change line spacing inside a node in TikZ Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian. it seems like it... should be? In that I can't think of any counterexamples off the top of my head. I was looking up these  and saw the quaternion group, but the subgroups of the quaternion group aren't abelian... Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc. Exists a dedicated non-O(n²) Algorithm for avoiding to have overlapping elements? I am generating positions in a random fashion.  In the game world, instantiated elements have a radius, and shouldn't overlap.  The problem is that they will overlap because I use a usual random number generator combined with other fanciness. I can't use other random number generation algorithms where I can guarantee that the elements will never overlap.  A way to solve this is to use a spatial data structure and search for the nearest neighbors in that tree. When I insert an element into that tree, I update it accordingly.  This solution would do the work and it would be fast but my question is:  Is there a dedicated algorithm without that Tree implementation complexity which is faster than O(n²)? Temperature in space Temperature is a measure of kinetic energy transferred to particles, henceforth,  being vacuum, temperature cannot be measured. But then, there is cosmic background radiation. It is the leftover heat from the Big Bang, but then heat in space doesn't make sense, how does this radiation, heat up space time when it technically cannot be heated up? Moreover why is it referred to as heat when it is approximately just $2.5$ Kelvin, which is just above absolute zero?