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1answer
108 views

Soft Question:Is the following a Paradox?

Can the statement: "I swear by God that I will never swear" be regarded as a variant of the Paradox of Self-Reference like the one "I am a liar"?
4
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1answer
88 views

Is there a specific notion as to what 'forgetfulness' is?

Concrete categories $A$ carry a forgetful functor $U:A\rightarrow Set$, whose left adjoint if it exists is the free functor. There are other forgetful functors such as $U':Ass\rightarrow Vect$ whose ...
1
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0answers
169 views

Proofs involving Disjunctions [Velleman, Chapter 3.5]

$\Large{{1.}}$ Are proofs using strategies $P136, P143$ always easier than those using $P140$? In the former two, only one statement (either $P$ or $Q$) must be proven. In the latter, both $P$ and $Q$ ...
2
votes
2answers
110 views

Greedy algorithm for reading a single paper in Dutch

I want to read the paper of Freudenthal and van der Waerden that proves there are only 8 convex deltahedra. (“Over een bewering van Euclides” Simon Stevin 25 (1946–7), pp. 115–121.) I have a copy of ...
1
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3answers
260 views

Introductory Algebra Book Suggestions

General Requirements The algebra book must be no more than 400-500 pages in length and should contain end-of-lesson/chapter exercises. Required Topics linear equations linear inequalities ...
4
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2answers
46 views

Property With Specific Properties

I am thinking of making a game (for the mathematicians that study numbers) in which players try to construct a set with certain properties. Which properties satisfy the following: Formally, every ...
7
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1answer
98 views

How to look at a polynomial ring based on a ring that is not commutative?

When I first met polynomial rings $R[X]$ I wondered: 'where do they come from?' Later the idea that - if $R$ is commutative - they could be interpreted as $R$-algebras free over a singleton brought ...
5
votes
2answers
537 views

Knowledge of proofs and proof-writing before studying spivak's or apostol's Calculus

I want to start reading spivak or apostol calculus. But I believe a certain level of proof knowledge is essential to solve the problems. Do these books teach you how to prove results or do we learn ...
6
votes
2answers
99 views

How should we think of maps to the intial object?

A final object in a category is one that has a unique map from any other object. An intuitive way of thinking about the final object is as the 'point'. Then we think of maps from the 'point' to ...
3
votes
1answer
529 views

Lang as a first algebra book

I think I am ready to learn algebra from Lang, but wanted some perspective. I have been exposed to: Linear algebra: All of Axler From my other, legendary honors course: -Order theory (lattices, ...
0
votes
0answers
421 views

Spivak, courant and Apostol for beginners

Are the above books : spivak, courant and apostol good even for beginners in calculus? I mean do they build up your basics before plunging into hardcore calculus. And also about writing proofs : ...
3
votes
2answers
73 views

Soft question: $(a_n) \in A$ or $(a_n) \subseteq A$ f0r sequences?

I have always used, in place of the full, unambiguous (but clumsy?) statement namely "Let $(a_n)_{n\geq 1}$ be a sequence where $a_n \in A$ for $n\geq 1$." the short version "Let $(a_n)_{n\geq 1} \...
2
votes
1answer
180 views

Derivative existence theorem

Has anyone here heard of the Derivative existence theorem? Derivative existence theorem: For $f$ defined on some interval including $a$, $f$ is differentiable at $a$ if and only if there ...
2
votes
1answer
81 views

Non 0-1 integer programming

Many interesting combinatorial problems - graph coloring, maximal matching, set cover, and knapsack among others - can be reformulated as integer linear programs. One thing that all of these ...
10
votes
5answers
923 views

Necessity of learning programming for math major.

How necessary it is for a person planning to do research in pure mathematics to learn programming as an undergrad? And if it is, then which languages would you recommend to her or him? And why? I am ...
0
votes
1answer
64 views

Detailed Visual Introduction to Complex Numbers with Problems and Solutions

I'm aware of http://tutorial.math.lamar.edu/Extras/ComplexPrimer/ComplexNumbers.aspx. It's very detailed and helpful but I'm looking for something with more pictures. It also doesn't have enough ...
2
votes
2answers
112 views

Information on crucial results concealed as exercises or neglected in a textbook

First, where can students find lists, information, or resources on the crucial results, inequalities, theorems, etc... which a textbook might not explictly feature or even bring up at all? Second, ...
20
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3answers
1k views

A problem V.I. Arnold solved as a primary school student

According to a 1995 interview that Vladimir I. Arnold gave to the Notices of the AMS, his primary school teacher I.V. Morozkin gave in 1949 (when Arnold was 12 years old) to a Soviet classroom, most ...
2
votes
1answer
135 views

Does every major genre of mathematics have applications?

I know that it used to be said, in praise by some and as criticism by others, that Number Theory had no applications. Now it is used in cryptography and Quantum Theory. Since the mathematics that ...
6
votes
1answer
84 views

Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation?

Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows. $$(f+' g)(x) = f(x)+g(x)$$ Question. The ...
24
votes
6answers
1k views

Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we ...
12
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5answers
268 views

Scholarly work on the beauty of math

When reading mathematical books written for a general audience, or even searching questions on this site, the adjective beautiful is often used to describe mathematics. My question is whether there ...
2
votes
3answers
134 views

Esoteric knowledge regarding statistic tests like $F$-test, $t$-test and $X^2$ (Chi-Square) etc.

For a year or two I've been doing/learning statistics using books written both for engineers and semi-professionals. I know how to apply most of the theory that statisticians use on job, but I'm still ...
0
votes
1answer
66 views

Adding together curves or shapes to approximate something more complex

I'm looking for proper terminology / references for the following sort of problem: Say we have some one-dimensional curve like $y = 10$ defined over the real valued domain $[0,1]$, and we ask, how ...
61
votes
4answers
5k views

Mathematicians don't quit, they fade away [closed]

Edit: This question is now closed for being not related to math, but many people pointed out that becoming an actuary is one of the most viable career path for someone with skills in pure math. Noone ...
2
votes
0answers
164 views

Prime number represented by spiral

There is this image on 9gag, with description: "Comparison between 5,000 and 50,000 prime numbers plotted in polar coordinates" I thought it might have something to do with Ulam spiral, but ...
2
votes
3answers
559 views

Decision Calculus text [duplicate]

I was wondering out of these three which would you take calculus by Spivak calculus by Hughes-Hallet calculus by Morris Kline I'm taking calculus III next semester but I want a better book that's ...
0
votes
1answer
207 views

Why is the number Pi more popular than any other constant? [closed]

What is so special about the number $\pi$? There are many more interesting constants, such as e, $\gamma, \sqrt{2}$ or the catalanian number. $\pi$ has been calculated to more digits than any other ...
1
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0answers
99 views

The inverse matrix |${\delta}$| does it have an application

The jacobian is the determinant of |${\delta}$| this means that |${\delta}$| is invertible. Does this inverse have any use in the real world? Maybe I am not clear in my question: Does the inverse of ...
1
vote
2answers
162 views

Infinite sum with 0 terms: comparison to infinite product

Depend on what text you read, an infinite product with an infinite number of terms that are 0 is either divergent, or diverge to 0. Even though, obviously, the partial product is still a convergence ...
5
votes
1answer
125 views

Why the Little Methuselah form is the “Little Methuselah”s form?

This is my first question on MathStackexchange. Let me know if I am violating rules, or my question is somewhat ugly. I am reading Conway's book "Sensual (Quadratic) Form". He introduces a tenary ...
5
votes
1answer
191 views

What determines when an inner model is “canonical”?

I've read several places that L, the constructible universe, is the least canonical inner model. Grigor Sargsyan explains in his slides that L is canonical due to $\mathbb{R}^L$ being $\Sigma_2^1$ and ...
4
votes
1answer
517 views

Solvable and nilpotent groups, normal series and intuition

I'm reading Hungerford's algebra and I'm on Nilpotent and solvable groups chapter. Hungerford starts with: Consider the following conditions on a finite group G: i) G is the direct product ...
4
votes
2answers
123 views

What is a Kählerian variety?

I know what a Kähler manifold is, and I (roughly) know what a variety is. However, I don't know what a Kählerian variety is. Is it just a variety which is also a Kähler manifold, or is it a separate ...
11
votes
1answer
258 views

“Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended. A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and ...
6
votes
1answer
183 views

Soft Question: Suggestions on mathematics resources for problem solving.

I'm doing my final year of under graduation through distance education and would be appearing for entrance tests for various graduate schools in a few weeks. I am looking for a database of algorithms/...
24
votes
6answers
1k views

How can a high schooler get more involved in mathematics?

For a high school student interested in majoring in math and learning more about math, what kinds of mathematical research can a student in high school get involved in? How can a high school student ...
6
votes
3answers
445 views

How can people understand complex numbers and similar mathematical concepts?

In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural ...
18
votes
3answers
861 views

What good is infinity?

I am becoming increasingly convinced that Wildberger's views are, if a little bizarre, at least not hopelessly inconsistent. When I was reading the comments in the video following (MF17), somebody ...
2
votes
1answer
53 views

Gaussian Discriminant Analysis

A Gaussian discriminant analysis (GDA) is a type of generative learning algorithm. What kind of mathematics background do I need to understand what it is? Are there any books that explain this subject ...
284
votes
21answers
16k views

On “familiarity” (or How to avoid “going down the Math Rabbit Hole”?)

Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole". For example, suppose you come across the novel term vector space, and want to learn more ...
4
votes
4answers
1k views

If we accept a false statement, can we prove anything? [duplicate]

I think that the question is contained in the title. Suppose we begin from something that is false for example $1=0$. Is it possible using only $\Rightarrow$ (and of course $\lnot ,\wedge,\lor$) to ...
22
votes
3answers
955 views

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

I've seen a lot of definitions of notions like boundary points, accumulation points, continuity, etc, and axioms for the set of the real numbers. But I have a hard time accepting these as "true" ...
11
votes
3answers
765 views

Why are conjectures about the primes so hard to prove?

I recently started learning number theory, and I've noticed there are many conjectures about the prime numbers that are unproven. Some examples would be whether there are infinite Mersenne, Sophie-...
6
votes
1answer
380 views

Was Fermat's last theorem proved based on Peano's postulates?

Is the proof of Fermat's last theorem solely based on the Peano's postulates $+$ first order logic? Or it contains other axiomatic systems as well? What does it mean from foundations of math ...
5
votes
4answers
273 views

how many $1$s in the first n digits of $\pi$?

how many $1$s are there in the first n digits of $\pi$? Any good approximation of its distribution? How about the place of the $n$th $1$? Are these two questions related?
4
votes
1answer
342 views

About constructive mathematics and Homotopy type theory

I am a CSer and I am reading the HoTT book and found that doing math with computer is fascinating. I found that constructive math compared with classical math is beautiful because: type theoretic ...
4
votes
1answer
426 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...
5
votes
0answers
151 views

definition of determinant in Artin

In Artin, the discussion on determinant starts from the standard recursive expansion by minors. Artin defines determinant as a function $\delta$ from a square matrix to a real number. Then Artin lists ...
5
votes
1answer
108 views

Is there any way to define differentiablity without any reference to the Euclidean space?

We define metric spaces based on the properties of the real numbers $\Bbb{R}$. In the same spirit we define smooth manifolds. But there is a more general and elegant way to formulate our intuition of ...