What is Pure mathematics: Definition and 49 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.
TL;DR Summary: The university pure maths courses feel rough and uninsightful, yet exclude (temporally) the possibility of studying good texts. What and how does one study in order to attain such understanding as to be profound, but allowing, by structure, still to follow the curriculum...
TL;DR Summary: education, history
I need to devise a module for next academic year which is an introduction to pure mathematics. They need to use this module as a stepping stone module such as number theory, group theory, combinatorics, real analysis.
What should I cover to make this an...
Hey y'all, I'm starting college as a Physics major this fall and I started working through Steve Warner's Pure Mathematics for Beginners in my downtime because I love the math courses I've had, wanted to learn more about proofs and how those things came to be, and most importantly had nothing...
I am recreationally self studying physics and math with the ultimate goal of understanding theoretical/mathematical framework of classical mechanics, general relativity, and quantum mechanics (in addition to the actual physics). The question I have is how deep in the realm of pure math does one...
I want to produce a resource that has a narrative and includes the following topics:
Sets, logic and proofs, numbers (irrational, integers, rational, …), binomial theorem, geometric series, inequalities, define things like identity, polynomial, symmetry, sigma and product notation.
It is in aid...
This is perhaps the single most important mathematical physics papers I have ever read; I think everyone - especially (theoretical) physicists - interested in theoretical physics should read it. In fact, read it now before reading the rest of the thread:
Klainerman 2010, PDE as a Unified Subject...
Hi,
So I am currently a mechanical engineer working in aerospace designing and testing different interior assemblies, video arms, meal tables, etc. For a while now I have been thinking that this field is just not up my alley. I probably use Creo for about 80% of my time while the other 20% I am...
Hi fellow thinkers, my main area of interest is in QM. However I am always interested in new discoveries and experimental proofs of outstanding theories. Looking forward to joining the discussion with others who, like me, enjoy absorbing new information...
S.P.W...
have a nice day. I was torn between mathematics and physics. I do not know which one I wanted. How can I decide? for me; theoretical physics, applied mathematics - topology (dif.topology, alg.topology, geom.topology), differential geometry easy than pure, proof based mathemics. I'm better...
The Rearrangement Inequality states that for two sequences ##{a_i}## and ##{b_i}##, the sum ##S_n = \sum_{i=1}^n a_ib_i## is maximized if ##a_i## and ##b_i## are similarly arranged. That is, big numbers are paired with big numbers and small numbers are paired with small numbers.
The question...
Introduction
If Quantum Mechanics is more fundamental than General Relativity as most Physicists believe, and Quantum Mechanics is described using Hilbert Spaces wouldn't finding a compatible version of General Relativity that operates within the confines of a Hilbert Space be of utmost...
I've asked many... what does it mean to major in math (pure). I am a first year sophomore and still undecided about my major.I used to get full marks on math tests n high school, but how can I know if I'm able to be something in math. Anyone can learn how to do math, but few can actually make...
I am looking for a book concerning the foundations of mathematics. I am not talking about algebra or calculus but the foundation of the mathematical language( Axioms, sentential logic...). I hope you have understood the prior rubish. Thanks for any help.
Hello everyone! :smile:
I am almost finishing a bachelors degree in computer science and I would like to have some insight on the possible academic backgrounds and career opportunities for someone who wants to be working with the creation and development of a physics engine.
Considering that...
I use to hate math during high school. However some eight years after graduating and being in the military help with my attitude towards math. I am reviewing a GRE math review book and I haven't been able to put it down. I just learned quadriatic equations and am pursuing into intermediate...
I am planning to study the following pure mathematics areas (on my own) and wanted to know if this is the best sequence:
1- Formal Logic
2 -Philosophical Logic
3- Sentential Logic
4- Predicate Logic
5- Symbolic Logic
6 -Set Theory
7 -Pure Mathematics (Intro, Pure Math I and II and Hardy) -...
I was just wondering what people thought about this program. I know a couple of kids who are graduating with their M.S. in pure math from DePaul and they are going to university of Oregon and Toronto for the phd's.
Hello. I am in desperate need for advice. I have searched for advice everywhere and each time failed to obtain it. This is my last resort. Please help me and bare with me while I explain the problem.
I am 28 years old guy. I have literally whole day free to pursue whatever goals I see fit...
Hello,
I've just been accepted and will begin as an undergraduate in physics this fall. I'm planning on taking classes, in addition to the required courses, concentrated in particle physics, QM, field theory, etc. Also, I took courses on Set Theory and Topology, and fell in love with what I...
I need some advice,
I've just finished my 2nd year at UCL studying the Bsc in mathematics. I've so far really enjoyed my course and I'm hoping to go on to do a phd somewhere. Now, like everyone else, I want to go to the best place possible.
So far, my grades are fairly good, every exam...
I'm in my last year of school and starting a mathematics degree at a British university this Autumn. I've always liked mathematics, especially 'pure' side of mathematics. I love poofs and always enjoyed solving puzzles and brainteasers for fun. Some of my favourite topics of maths are geometry...
I have only taken mathematics up to trig and I was curious if I would be able to start reading books on more advanced topics like Abstract Alegebra and topology??
Applied mathematics vs. "pure" mathematics in high school?
I've just started working through "Basic Mathematics" by Serge Lang. It immediately struck me, when I skimmed through the pages, that there is a large emphasize on proving things and manipulating expressions, and very little exercises...
Getting into a Phd-good or bad??
Hi,this might sound familiar.I quit my masters in automotive engineering from a very reputed university in Germany.I had finished all my course-work in 2 years,had 1 year of thesis and internship left.But,I couldn't take the emperical dominated engineering...
Hello guys. Some of you might remember my posts regarding the talent required to become a pure mathematician. I have done much introspection as of late, and I keep coming to the conclusion that I am not enjoying "Pure" mathematics as much as I thought I would. I originally became interested in...
Should i choose pure mathematics or statistics as a major for my degree? I intend to do my masters n PhD to become a lecturer. which is d better path if i want to be an academician? i also heard dat pure maths esp math proof is vry rigorous but I'm nt sure if i'll like stats.. any advice?
I'm sure everyone gets tired of this question, but it's important, so I'll ask anyway. Can anyone make suggestions for good grad schools for pure mathematics? Especially number theory or algebra. I already have a list of about ten of the top tier schools, but it would be especially helpful to...
Hi,
Just wanted to know if any of you guys had any useful advice with regards to careers from a maths degree. I am in my second year of a integrated masters (in the UK) and would be open to further study. I am doing well academically so far (top 15% of my uni, which is prob 2nd best in UK)...
I have been wanting a book that would contain all mathematical theorems and proofs, such as G.S. Carr's A Synopsis of Pure Mathematics, which I could use for reference. I have been able to get his book digitally, but I want a hard copy, and one that is up to date with the mathematics. Does...
This year, I will apply for University of Waterloo, and I am quite certain that I will attend this university. But I have been having trouble of deciding my major. Of course, I don't want you to determine my major on in stead of my self! I am asking for advice.
I really liked playing with...
Hi, I am applying in college this year, and I consider to major in mathematics. My desire is to understand the actual beauty of math, as well as its application. Therefore, I consider between Pure math and applied mathematics. Is it good to pursue in both branches? and How long it would take me...
I am a Mathematics and Physics double major, currently in my second year. I really enjoy both subjects, but my interests are progressing towards Theoretical physics/mathematical physics. My academic goal is to improve my understanding of how the universe works and thus I would like to pursue a...
I have been thinking much on the nature of pure mathematics. I believe this forum would make the best place to post over say the philosophy section, as i am more interested in the opinions of working mathematicians and physicists than philosophers.
In my opinion pure mathematics is the core...
1. Homework Statement [/b]
3. if X m and n are positive rational numbers, X(m^2 ~ n^2) , 2Xmn , and X(m^2+n^2) are positive rational numbers. Hence show how to determine any number of right-angle triangles the lengths of all whose sides are rational
Homework Equations
The Attempt...
Well, I'm 19, I love pure mathematics and I'm determined to follow my heart in my life. but recently this idea has been in my head that there is no job for a pure mathematician except teaching math. I love teaching math to other people but don't want it to be my future career. Do you think that...
General Relativity Spacetime is pure Mathematics. But we live in a real world. How do we get coupled to Spacetime? We are real. Spacetime is math. Math just describe reality and doesn't affect it. So Spacetime only describe reality as metaphors. But how come physicists believe Spacetime is real...
Hi, I am in need of some expert advice please.
I am enrolling for my honours degree in mathematics next year. I am trying to mix it up in order to have a strong foundation in pure mathematics with specialisation in applied mathematics. My interests are financial mathematics, probability...
For a while now I have been looking for a series of texts that I can read in order to learn some of the basics of abstract/pure mathematics. I am currently taking Calculus II with Stewart's book as my text, and this summer I plan on going through Spivak's Calculus.
I have finished my...
"Pure" Mathematics vs. Applied Math vs. Discrete Math
I'm approaching the point where I'm going to have to decide which four-year university I'm going to finish my Bachelor's degree at. I'm pretty much restricted to colleges in Georgia, and I am primarily looking at Georgia State and Georgia...
Homework Statement
Prove that 10^n + 1 cannot be expressed in the form a*a*c, where n,a,c are positive integers.
Homework Equations
By considering the reminder of 10^n + 1 when it is devided by 3, I arrived at the conclusion that:
a*a = 3z +1;
c = 3v + 2
for some positive integers...
Square Root Of 2 (from Hardy "Course Of Pure Mathematics")
I was surprised to find it in my local bookstore amidst math "cheat" books in the one-shelf math section (and the fact that it was the last copy left... ??).
Is he asking to simply carry out the calculation? Or is he asking to...
Hi.
I have to write a paper (about 20-25 pages) and I'm likely going to choose the topic 'The Unreasonable Effectiveness of Pure Mathematics' as was suggested by one my prof's (of course, I'm familiar with Eugene Wigner's article). I was curious if anyone could point me towards more modern...
Can you give me a nonselfish reason to study pure mathematics? How does one justify spending an entire life pushing around symbols and definitions which may have no connection to the physical world? I enjoy doing math, but I also enjoy playing chess and minesweeper, and even if someone were to...
i am now studying pure maths in uni. is it possible for someone to tell me wht kind of jobs would pure maths lead to? i believe it would be some jobs tht requires analytical and theorectical skill. bt is tht any exact examples pleasE?
I am trying to get into an ivy league school and I'm trying to decide what is the best approach. I eventually would like to receive a phd in Mathematics and teach at the University level. My questions are:
1. What would increase my chances more of getting into an ivy league phd math program...
What does pure mathematics describe exactly? What are its laws based on?
Recall the psi beta function. It was used for purely mathematical persuits and it was believed it did not pertain to our physical reality. In 1969, Venetziano observed a relationship between the behavior of particles (or...
I've always been of the opinion that mathematicians, especially pure mathematicians, are the most intelligent segment of our society, and also that pure mathematicians are born rather than made. Anyone can improve their mathematical skills and knowledge, but blood, sweat and tears are not enough...