A proof is some explanation to convincing others that a statement is true (or false in case of a counterexample). As Yuri Manin once wrote: "A proof becomes a proof only after the social act of accepting it as a proof".   Since Mathematics is not comprehensively 'connected' to the nature, I am trying to compare it with another situation: there were many beliefs in history that every body was convinced their truth, it could even be as far as a superstition. But as time passed their validity changed.   My question is: Do we need other type(s) of intelligent creation (like intelligent aliens) for final check whether our proofs are true or not? Or, human mind is enough for checking their truth?  Note - My question is especially focus on non-empirical Mathematics.   EDIT - Also, there have been many mathematical proofs that had very subtle inconsistency detected by smart people; checking Fermat Last Theorem's proof by a colleague of Wiles came up with a subtle mistake is an example; Is it possible that a 'proof' has that type of mistake but not easy enough to 'see' even by nowadays' people for a long long time? any historical story?    PS - migrated from math.SE
I don't understand philosophy very well, and so I am wondering whether you can "prove" anything in philosophy. It always seems you can go a layer down, and find another question, almost endlessly until you get to the question of "why/how/are we here?"  Can you prove, absolutely, and 100% that something is true, in the field of philosophy?
A matrix and its transpose have the same set of eigenvalues/other version: $A$ and $A^T$ have the same spectrum Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma\left(A^T\right)$ where $A^T$ is the transpose matrix of $A$. Unable to lock the administration directory (/var/lib/dpkg/) is another process using it? I get this error when trying to use apt-get:  E: Could not get lock /var/lib/dpkg/lock - open (11 Resource temporarily unavailable) E: Unable to lock the administration directory (/var/lib/dpkg/) is another process using it?     How can I fix this? How to install pygments on windows 7 When I run LaTeX file its gives error as  "! Package minted Error: You must have pygmentize installed to use this package", you must have pygments installed. when I go through minted documentation minted itself required pygments, so Tex friend How do I install this pygment, I don't know the procedure to install. Please guid me or any link pygement.exe file What is the appropriate public key encryption for secure coin flipping? I quote Bobby's question here since I encountered the same one...    Consider the following protocol for two parties to flip a fair coin.   Trusted party $T$ publishes her public key $K_{pub}$ $A$ chooses a random bit $b_A$, encrypts it under $K_{pub}$, and announces the ciphertext $c_A$ to $B$ and $T$ $B$ does the same and announces a ciphertext $c_B$ (where $c_B \neq c_A$) $T$ decrypts $c_A$ and $c_B$, and announces the results. Both parties $XOR$ the results to obtain the random value $b_A\oplus b_B$.   It was shown that  was not appropriate for the encryption scheme.  seems to fail to work also. Since $c_A=m^e \mod N$. Then if $m=1$, $c=1$. If $B$ receives $A$'s cipher he will know the exact value of $m$.   So what should the appropriate encryption scheme be? Why is the $S_{z} =0$ state forbidden for photons? How long does it take for a black hole to form for an external observer? The well-known fable of an astronaut sending signals out to an external observer while falling toward an event horizon states that the time lapse between such signals becomes greater even if in the astronaut sends them out periodically (as judged in his inertial frame). When viewed from earth and weighing time-dilation due to the gravitational field of the collapsing star, how is it possible that a black hole can form in finite time (for any external observer) if it takes an infinite amount of time to "see" events occurring at the event horizon? Firefox 33 doesn't see Java Plugin I am trying to enable Java (not JavaScript) in my Firefox v. 33 so that I can use applets within the browser.  Currently, I just get a dark gray box across the whole screen when I try to load an applet, without any error message or prompt to install plugin.  As per , I tried going into the Firefox plugin page to activate it, however, it wasn't there listed.  I do have Java 7 installed and it is in the path:  &gt;java -version java version "1.7.0_71" Java(TM) SE Runtime Environment (build 1.7.0_71-b14) Java HotSpot(TM) 64-Bit Server VM (build 24.71-b01, mixed mode)   What do I need to do to get Firefox to see my Java installation and recognize it as a plugin so that I can enable it and view applets inside the browser? I'd like to know if there is any alternative to Deep Freeze for Linux that still under development, because I've read some things on the internet and looks like the same company which developed Deep Freeze had a Linux version of it, but the project was discontinued. How to prove that compact subspaces of the Sorgenfrey line are countable? How to prove that every compact subspace of the  is countable? How can I get Firefox to open torrent files with Transmission? Every time I download a torrent file (.torrent), Firefox thinks I should open it with a text editor. How can I get it to open with Transmission? In Edit -> Preferences -> Applications, there's no setting for torrent files, and no apparent way to add a new filetype. Use a shell variable in awk What is the formula for percent of frame filled at a specific distance and focal length? I am thinking about buying a new lens specifically to take photos of distant targets (birds, bears, etc). I currently own the Canon 70-300 IS USM, which is a nice lens, but I find that if I want to take a photo of anything further away than about 10 meters, the target does not fill a large amount of the frame (meaning I have to significantly crop in post production).  I am currently looking at the Canon prime 400mm, however I would like to determine at what distance an object will fill a reasonable proportion of the frame.  Is there a formula or rule of thumb I can apply that will help me in this situation? I realise the size of the target I am shooting will play a role here, so if we need to make an assumption about the size of the target please let me know. Locking a vehicle bicycle rack We are planning a trip with a bike rack attached to our trailer hitch. What is the best way to lock the bikes while the vehicle is unattended? I see some racks have locks on the knobs but they don't seem very strong.   What do you suggest? Can you fly over a Medium enemy creature in a 10ft tall corridor? How do I tackle this combinatorics problem about married couples around a table? How do you disable browser autocomplete on web form field / input tags? By viewing the polynomials as a difference of two squares, factorise the following polynomial: $$x^4+x^2+1.$$ I searched but couldn't find a way to solve this  Edit: By using Hans Lundmark hint, I get:  $$(x^2+1)^2-x^2$$   Is it fully factorized? Looking for succinct way of describing the act of solidifying an idea into reality Looking for a succinct way of describing the act of solidifying an idea into reality. "Building [idea] out" seems too generic but I'm trying to avoid a long winded, step by step explanation of the process.  Any ideas would be greatly appreciated. Thanks! Project Euler Question 14 (Collatz Problem) Does DTIME(O(n)) = REGULAR?