What is Abstract math: Definition and 20 Discussions

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.
While pure mathematics has existed as an activity since at least Ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.
Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections, geometrical curves that had been studied in antiquity by Apollonius. Another example is the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure internet communications.It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference than a rigid subdivision of mathematics. In particular, it is not uncommon that some members of a department of applied mathematics describe themselves as pure mathematicians.

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  1. H

    I Is the Order of an Automorphism in a Field with Characteristic p Equal to p?

    Please, I have a question about automorphism: Let ##\mathbb{K}## be a field, if ##\operatorname{char}(\mathbb{K})=p ##, then the order of automorphism ##\phi## is ##p##, i.e. ##\phi^p=\operatorname{id}##, where ##i d## is identity map. Is that right? please, if yes, how we can prove it, and...
  2. AaronQ

    B Predicting outputs of f(x)=(1+i)^x

    I got bored a while back and deiced to create a table of the integer inputs of f(x)=(1+i)^x and I noticed quiet a few patterns which I am trying to catalog here, although most of my work so far deal with Natural inputs, all patterns continue into the negative, see here, I was wondering if anyone...
  3. K

    Schools University Mathematics Abstraction

    I'm currently a first year MathPhys student, and next year I have to decide my stream. I can pick a combination (pure) Mathematics, Applied & Computationtal. Mathematics, Statistics, MathSci, Physics, Theoretical Physics or Physics with Astronomy & Space. Naturally there are restrictions, and I...
  4. TyroneTheDino

    Universal and Existential Qualifiers

    Homework Statement Express the following statement using only quantifiers. (You may only use the set of Real and Natural Numbers) 1. There is no largest irrational number. Homework Equations ##\forall=## for all ##\exists##=there exists The Attempt at a Solution I express the existence of...
  5. P

    Solutions to Hungerford's "Abstract Algebra" 3rd Ed.

    I'm taking an abstract algebra course that uses Hungerford's "An Introduction to Abstract Algebra" 3rd Ed. And while I feel like I'm following the material sufficiently and can do most of the proofs it's hard to learn and practice the material without a solutions guide. How am I supposed to know...
  6. TyroneTheDino

    Intro to abstract math—basic notation

    Homework Statement Simplify the following statement as much as you can: (b). ##(3<4) \wedge (3<6)## Homework Equations ##\wedge= and## The Attempt at a Solution I figured that I could just write this as ##3<4<6##, but then I considered what if I didn't know that ##4<6## If it was just...
  7. D

    Overcoming Struggles with Abstract Math: My Personal Experience

    I'm good at math like stats calc and others that r more process based. But I suck at things like abstract math, pure math , proofs etc. Am I an idiot? I'm OK with being stupid, I know it does not define a persons worth..
  8. A

    Bridge to abstract math: what is wrong with following proof

    See attached picture. The question asks to prove that the statement which I have written on the first line is true. But I somehow proceeded to proving it is false. Basically what I did was simplify the given expression into the form (P or Q) => R and said this is equivalent to (P=>R) ^...
  9. B

    Define set from given function and a subset. Abstract math

    Homework Statement Let f: Z to ZxZ be the function defined f(t)=(3t, 3t+1) . Let B denote the subset of ZxZ defined by B={ (5m, 5m+1) : m is an element of Z}. Determine f^-1(B). This means that you should define set S with a property of S=f^-1(B). In addition, your definition of S should make...
  10. B

    Abstract math, sets and logic proof

    Homework Statement If A is a set that contains a finite number of elements, we say A is a finite set. If A is a finite set, we write |A| to denote the number of elements in the set A. We also write |B| < ∞ to indicate that B is a finite set. Denote the sets X and Y by X = {T : T is a proper...
  11. B

    Discover the Solution to p(x)=xx+x+4 for Set S in Z: Abstract Math Help Needed

    Homework Statement Let p(x)=xx+x+4 Determine a set S such that S ={p(n), n is an element of Z). Your de fition of S should not refer to the polynomial p(x). Homework Equations S={n is an element of Z: p(n)=nn+n+4) The Attempt at a Solution I know that we can't take roots of p(x)...
  12. D

    Is Dummit & Foote's Abstract Algebra the Ultimate Guide for Mathematicians?

    A good book on Abstract Algebra that covers major undergraduate and graduate topics? Something rigorous professional and for mathematicians. Not Hungerford please, or any n x $100 book.
  13. B

    Abstract math prove involwing sets

    Homework Statement Let Ts denote the set of points in the x; y plane lying on the square whose vertices are (-s; s), (s; s), (s;-s), (-s;-s), but not interior to the square. For example, T1 consists of the vertices (-1; 1), (1; 1), (1;-1), (-1;-1) and the four line segments joining them...
  14. B

    Intro to Abstract Math Question about divison of integers.

    (1)Assume a, b and n are nonzero integers. Prove that n is divisible by ab if and only if n is divisible by a and n is divisible by b.I'm wrong and can't remember why. I spoke to the professor about it for ~ 1 minute so it seems to have slipped my mind, it was because in one case it's true and...
  15. J

    Is It is not Monday and it is not Tuesday Equivalent to ~(P v Q)?

    Homework Statement Find a statement for , S, equibalten to ~(P v Q) and show that it is logically equivalent by construction the truth table for "S if and only if ~(P v Q)" and showing that this statement form is a tautology. Homework Equations The Attempt at a Solution My...
  16. H

    Can Abstract Math Prove a 15-Pair Relation Non-Transitive?

    Man I've become desperate. I just signed up needing help on this homework. Can anyone help me with these two problems? Let A be the set {1,2,3,4}. Prove that a relation R on A with 15 ordered pairs is not transitive. I've got no clue on that one. And this second one, which I know...
  17. F

    Sequence Limit (abstract math)

    Homework Statement If x_n\rightarrow L and y_n \rightarrow M prove that x_n - y_n \rightarrow L-M Homework Equations Definition of Limit The Attempt at a Solution I followed and stayed within the definition of limit of a sequence, but I got 0 for x_n - y_n.
  18. T

    Abstract Math: Beyond Category Theory

    Category theory is considered extremely abstract. What are some other branches of mathematics which are considered as abstract or even more abstract then category theory?
  19. F

    Proving 1 + 1 = 3: Abstract Math & Visualization

    I had some colleagues in College who took a degree in Math on their first two years. After finishing their second year, they shifted to a different course. They already finished from Algebra to Calculus and they were in an even higher math. They were asked to prove that 1 + 1 = 3 This is...
  20. D

    Help Needed: Concrete Analogies for Abstract Math Concepts

    I'm giving a talk entitiled "concrete analogies of abstract concepts", where I give examples of concepts in mathematics that might have arisen in things found in every day life. I already have "chess" for isomorphism and "acronyms" for homomorphisms, but I'm running a bit dry. Anyone got any ideas?