So some day ago a friend of mine send me this: http://www.jainmathemagics.com/page/10/default.asp Which because of all those emotional and religious words the site author uses (e.g. divine, purify, sacred etc.), I know it is crackpot However I do read through the whole thing (including going through his "lollypop construction", and found it indeed can contruct a circle of radius 1/2 using 1 and gold ratio alone) and with the help of wikipedia and other sources, sucessfully found out his flaw and confirmed my suspicion on it: That he incorrectly use the mathematical coincidence that \(\frac{4}{\pi} \approx sqrt{\phi}\) and risk squaring the circle illegally (since it was proven impossible, thus all attempts were just approximate squarings) However the analysis of this brought me some thoughts Given that in mathematics, there are many systems out there that does not obey the convenient properties as in real numbers e.g. in general EDIT: Clarification: (Assuming each example does nto involve any addition structures, e.g. matrices when I mention about complex numbers, so if I mention the property in the contex of complex numbers, then a,b, are elements of complex numbers not complex matrices) 1. Matrix multiplication: AB=/=BA 2. Complex numbers: exp(a)exp(b)=/=exp(a+b)(exp(a))^b=/=exp(ab) (EDIT: Mistake correct as noted by Dinosaur and rpenner) 3. Non transitive relations that a implies b b implies c does not mean a implies c (http://en.wikipedia.org/wiki/Transitive_relation) 4. http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorem 5. Discontinuous functions, nowhere differentiable functions, divergent integrals etc. etc. (P.S. is there an umbrella term that describe all these properties?) Is there exist a class of mathematical systems or problems that have properties so "broken" such that the following will happen? If there isn't, then using our current established framework of mathematics, how to proof in general that "Claim" does not exist? Reference: http://en.wikipedia.org/wiki/Proof_by_contradiction http://en.wikipedia.org/wiki/Constant_problem http://en.wikipedia.org/wiki/Undecidable_problem
For any two complex numbers a and b, exp(a)exp(b) = exp(a + b). I think you were thinking of √a√b ≠ √(ab) where the easiest example is a = b = −1, since −1 = i × i = √(−1)√(−1) ≠ √[(−1)(−1)] = √1 = 1 Or possibly you were thinking of matrices again as exp(a)exp(b) ≠ exp(a + b) means a b ≠ b a Example: \( a = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad b = a^{\textrm{T}} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \quad a^2 = b^2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}, \quad ( a + b )^2 = (a + b )(a + b ) = a^2 + a b + b a + b^2 = ab + ba = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \exp (a) \exp(b) = (I + a)(I + b) = I + a + b + ab = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \neq \begin{pmatrix} \cosh 1 & \sinh 1 \\ \sinh 1 & \cosh 1 \end{pmatrix} = I + \sinh \, 1 \, ( a + b ) + (\cosh \, 1 \; - \; 1 ) ( a + b )^2 = I + \sum_{k=1}^{\infty} \frac{1}{k!} (a +b)^k = \exp (a + b) a b = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \neq \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = b a \)
There's a gem in every paragraph. How about this one: The Pin Number for the Universe! I guess it's time to change the combination on my luggage...
Maybe you didn't notice it but his page is full of hidden secrets. For example figure 18a shows indeed that the pope's hat can hide rabbit ears because the pope should be a rabbit
R Penner: In your Post #2 you discuss some equations relating to square root Since every number has two values for the square root, it is essential that there be consistency in choosing one of the values. Without some consistency, you do not need to use complex numbers to get strange results Code: sqrt(4) = 2 sqrt(9) = 3 sqrt(4)*sqrt(9) = 6 sqrt(36) = -6 Ergo: sqrt(36) not equal to sqrt(4)*sqrt(9) In your example you could have made the following choices Code: sqrt(1) = -1 sqrt(-1)*sqrt(-1) = i*i Ergo:sqrt(1) = i*i = -1 No strange result due to choice made for sqrt(1)
Note these replies is about a topic that is tangent to the one in the OP, however it is interesting enough in its own rights as it provide deeper understanding of complex numbers While this topic is not what I have in mind, (and in fact I have made a mistake due to memory failure because I was trying to mention this property http://en.wikipedia.org/wiki/De_Moivre's_formula (exp(a))^b=/=exp(ab), (but it seems there's some extra notes about the generalization and multivalued issues now which is not there back in 2011, which now make this claim strictly speaking, false (at least for complex numbers, without additional structures on top of it like matrices))) Your question also raised an interesting point about convention of square roots. My opinion after reading this post: While conventionally when we take a square root of a positive number, we only get the positive root of it, I think when you start putting in complex numbers, in order for all laws to generalise properly, we should treat the square root as multivalued, like what we did when we solve for x in x^2=a, that way, as you mentioned, the problem will go away, as well preserving the fact that 1^2=1 also Or more generally, if a function is multivalued, it should always give all the values when you act it onto some other mathematical object, (e.g. complex logarithms, trigonometric functions etc.), that way , we won't miss out solutions and cause some of the "laws" to break down because of some convention. That way, if the laws DO break down, it is mainly because of the intrinsic property of the mathematical system or object, and IMO I think it might help understand them better as each of them has their own characteristics (e.g. Minkovski space always has the signature of magnitude 2 (+-2 depending on whether you use +--- or -+++ convention) A nice example, thanks I should have made my examples more specific, whenever I mention about a property in a certain system (e.g. complex numbers), I don't have more advanced objects/additional structures (e.g. matrix) in mind because for those examples, I want to said that it breaks down even for some simple enough elements in said system Tangent ends here The question in the OP is asking, since there are so many different mathematical objects out there that does not obey the rules or identities we took for granted in some familiar systems such as complex numbers, real numbers etc. Could there exist a mathematical object or problem/question that is so broken/pathological that the following will occur? PS I am not really good at mathematical logic..., apologies and guildelines welcome in case I missed something that is obvious...
For a consistent logical axiom system where the law of explosion is valid, then it follows that your example cannot "exist." (That word isn't used in propositional logic.) Specifically, in standard axiomatic logic: \( \vdash ((\varphi \rightarrow \psi) \rightarrow (\varphi \rightarrow \neg \psi) \rightarrow \neg \varphi)) \) can be quickly proven from axioms. And from that, it follows that \((\varphi \rightarrow \psi) \wedge (\varphi \rightarrow \neg \psi)\) is logically equivalent to \(\neg \varphi\). http://us.metamath.org/mpegif/pm2.65.html http://us.metamath.org/mpegif/pm4.82.html That's with the standard definition of implication, which satisfies: \(\vdash (((\varphi \rightarrow \psi) \rightarrow \varphi) \rightarrow \varphi)\) Which you may see is a tautology in classical logic with a truth table: \(\begin{array}{cc|cc|c} \varphi & \psi & (\varphi \rightarrow \psi) & ((\varphi \rightarrow \psi) \rightarrow \varphi) & (((\varphi \rightarrow \psi) \rightarrow \varphi) \rightarrow \varphi) \\ \hline \\ T & T & T & T & T \\ T & F & F & T & T \\ F & T & T & F & T \\ F & F & T & F & T \end{array}\) http://us.metamath.org/mpegif/peirce.html
Nice No wonder the claim that said Proof of contradiction is one of the most powerful weapon in mathematical proof and that a mathematical object that which this does not apply will be too broken to be useful (like a division by zero system that reduces to a trivial ring, as discussed some time ago in a past thread) <Thread solved> PS concluding joke https://xkcd.com/704/ Imagine the disaster that can result if this is possible...
