Weirdest properties of Banach spaces?
This is soft question, but I've recently have become interested in the odd properties of Banach spaces after studying Banach spaces which do not admit a pre-dual. I also almost work exclusively with...
View ArticleAre all mathematical laws tautologies?
I'm reading a book by Tarski and he says:"Every scientific theory is a system of sentences which are accepted as true and which may be called laws"then after some pages he gives somme laws:for any p...
View ArticleWhat exactly is calculus?
I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:What exactly is calculus? I know who invented it, the...
View ArticleMathematics without coordinate systems
Suppose that I understand mathematics to the level of e.g. Rudin's "Principles of mathematical analysis" I want to completely expel "coordinate systems" and the resulting "analytic geometry" from my...
View ArticleNonlinear mappings between finite dimensional vector spaces
When one studies Linear Algebra, I think it is natural to wonder what happens if we retain the linear structure of the vector spaces (addition and scalar multiplication) and let the mappings between...
View ArticleWas Russell's purpose in Principia Mathematica to formalize all the possible...
In writing Principia Mathematica, was one Russell's purposes to formally describe all the possible variations of logical concepts and reasoning that can be used in mathematics?For example, suppose that...
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An approximation to $\ln(1+e^x)$ and how to use it for splitting the...
Trying to split the logarithm of the sum of two exponential functions (this question), I found the following approximation for the Softplus function$f(x)=\ln(1+e^x)$:$$\ln(1+e^x) \approx \begin{cases}...
View ArticleWhat did Whitehead and Russell's "Principia Mathematica" achieve?
In philosophical contexts, the Principia Mathematica is sometimes held in high regard as a demonstration of a logical system.But what did Whitehead and Russell's Principia Mathematica achieve for...
View ArticleWhat other tricks and techniques can I use in integration?
So far, I know and can use a reasonable number of 'tricks' or techniques when I solve integrals. Below are the tricks/techniques that I know for indefinite and definite integrals separately.Indefinite...
View ArticleWords for describing the quirks of the way that the "average()" operation...
Is there some general way of expressing the fact that the operation of taking an average is sensitive not only to the set of numbers on which it is operating but also the number of instances of those...
View ArticleMemory tricks for maths proofs (soft question) [closed]
Sorry the question is not exactly a maths question. But I want to know how to improve my ability to memorise maths proofs quickly. May I ask:1. What are the common tricks for quickly memorising maths...
View ArticleIf proofs can be checked by computers, will we ever mistakenly believe a...
As I understand, math proofs can be formalized and checked by a computer. Does this mean the math community will never have to deal with believing incorrect proofs?If not, is the issue with adoption in...
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Contest Math--at random vs uniformly
I recently worked on a math contest at a high school level (roughly AMC 12 caliber). In one of my questions, I used the phrase,"...with the third vertex $P$ being chosen at random inside the square."I...
View ArticleUnexpected examples of natural logarithm
Quite often, mathematics students become surprised by the fact that for a mathematician, the term “logarithm” and the expression $\log$ nearly always mean natural logarithm instead of the common...
View ArticleGeneral question about conjectures [closed]
Mathematic conjectures are always fascinat mathematicians to find their secrets and solutions,so my question is who is the authority wich validate the proof of a conjecture and have the ability to say...
View ArticleIntuition: Why is quadratic variation finite for martingales
(Disclaimer: I'm not well-read in this topic, so might be getting some details wrong. Hopefully not wrong enough to make my question for intuition moot)For any martingale $(X_t)_{t \geq 0}$ with...
View ArticleResource for examples and counterexamples [closed]
As an undergrad with finals coming up, I was wondering if anyone had any comprehensive resources with the "to-know" examples and counterexamples. Specifically, I'm looking for this forGalois Theory and...
View ArticleAre every 4 point on a sphere concyclic no matter what?
Are every 4 points on a sphere concyclic no matter what? I need to do this to solve a larger problem. So if we have a sphere and get 4 random points, can we draw a circle where all of them ly? Thank...
View ArticleCan third order derivative (jerk) be use for optimization?
In optimization, ussually people just use second order derivative (or Hessian matrix) to classified local extrema. Suppose that my function is 3 times differentiable, what can I use this information...
View ArticleApplications of the de Bruijn-Newman constant outside of the Riemann Hypothesis
According to Wikipedia,The de Bruijn–Newman constant, denoted by Λ and named afterNicolaas Govert de Bruijn and Charles Michael Newman, is amathematical constant defined via the zeros of a certain...
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