What is Mathematical: Definition and 1000 Discussions

Mathematics (from Greek: μάθημα, máthēma, 'knowledge, study, learning') includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition.Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof. When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.

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  1. R

    Reducing a mathematical expression

    Hi guys i need your help! How can i reduce this equation: (Nd-Na)*[B*sqrt((Nd-Na)^2) + n^2) - sqrt((Na-Nd)^2) + n^2)] into this: (B-1)*(Nd-Na) + (B+1)*sqrt(Nd-Na)^2) + n^2)? I'm totally depressed, I'm always stocked in sqrt part of the equation: if there's a techniques in dealing this kind...
  2. ZetaOfThree

    V.I. Arnol'd's Mathematical Trivium

    Check out this collection of mathematics problems, published in 1991, by V.I. Arnol'd called "A Mathematical Trivium". Here's the link: http://www.math.upenn.edu/Arnold/Arnold-Trivium-1991.pdf Apparently, these problems are meant to be solvable by the end of your undergraduate (math) education...
  3. Strangelet

    Theoretical Physics or Earth Physics?

    Good Evening to everyone! I'm currently a student of theoretical physics. I have a Bachelor Degree in Physics and I'm from Italy. When you finish your B.D. you can choose to go on with your studies in order to get a Master Degree (other 2 years of study) and here you can choose your curricula...
  4. epenguin

    My first mathematical discovery. What was yours?

    This is occasioned by the thread https://www.physicsforums.com/threads/proof-multiplication-is-commutative.782057/ but being somewhat personal I thought the sometimes rather sobersided mods could find it out of place there. I was first interested by the proposition that (n + m) = (m + n) (1) -...
  5. WineRedPsy

    Our Mathematical Universe: My Quest for the Ultimate Nature?

    Is Our Mathematical Universe by Max Tegmark any good, as a work of popsci? Apparantly it's related to some TOE thing Tegmark's got, is that one actually something legitimate? (Edit: Sorry if this isn't appropriate for this forum!)
  6. Greg Bernhardt

    Interactive Introduction to Mathematical Analysis by Lewin

    Author: Jonathan Lewin Title: An Interactive Introduction to Mathematical Analysis Amazon Link: https://www.amazon.com/dp/1107694043/?tag=pfamazon01-20
  7. S

    Mathematical discoveries caused by misprints?

    There is at least one story about mathematical discoveries caused my a misunderstood homework assignment, http://en.wikipedia.org/wiki/George_Dantzig . Are there any stories about mathematical discoveries caused by misprints?
  8. ShayanJ

    Why is the term space commonly used in geometry but not in algebra?

    There are different things in mathematics that are called space: Topological space, vector space, etc. Wikipedia defines space as below: And in the page related to the word "structure", algebraic structures are listed too. So why people never use the word "space" when talking about groups...
  9. aditya ver.2.0

    Where should I ask about mathematical problems with Riemann curvature tensor

    I have come about few mathematical problems related to Riemann Tensor analysis while learning General Relativity. Should I ask these questions in this section or in the homework section. They are pretty hard!
  10. S

    Help Make Math Connect in General Relativity

    As some may know, I have been studying the Morris-Thorne wormhole metric for quite some time now. ds2= -c2dt2 + dl2 + (b2 + l2)(dθ2 + sin2(θ)d∅2) Now, from this space-time interval, it is easy to see how I would deduce the following metric tensor: g00= -1 g11 = 1 g22= (b2 + l2) g33= (b2 +...
  11. S

    Mathematical Reformulation of Polarization Equation

    Hello, I hope I got the section right ;). I orginally posted this in the physics section, but as the problem is more mathematical. It would be nice if someone knows the right direction. I've stumbled upon a math problem while going through some physics and got stuck with some mathematical...
  12. A

    Math Applying Einstein's Theory of General Relativity

    Just a quick question. I've recently thought about the various equations I'm learning about calculating velocity or vertical motion and have wondered that sense these equations are using Gravity as a constant force (9.8m/s^2) Is it not true that Einsteins' theory would denounce this idea and...
  13. V

    Mathematical Considerations of Spacetime in Classical Mechanics

    When studying the motion of particles in space, what are the mathematical considerations that have to made of spacetime? Could I say there exists a bijection between spacetime and ##\mathbb{R}^4##? Is the topology under consideration the usual product topology of ##\mathbb{R}^4##? Are there any...
  14. D

    Flow through Interconnected Tubes

    Hello all, this is my first post here, so please excuse me if there are any mistakes or important information is left omitted. Now, my problem is quite intricate and specific so I will be thankful for any information I can get. Using Excel, I am looking to model the volume flow of air through...
  15. _N3WTON_

    Mathematical Truths: Discovered or Invented?

    Ok, so I don't doubt that this discussion has been had on these forums hundreds of times, but I want to take part in it rather than just read old threads, so here is the (admittedly trite) question: do you think that mathematical truths are discovered or are they merely invented?
  16. J

    VI Arnold: Mathematical Understanding of Nature

    One of the advantages of not all speaking the same language is that a great, dead but legendary mathematician might still author a book. It's only recently translated, apparently: https://www.amazon.com/dp/1470417014/?tag=pfamazon01-20
  17. learnings_sweet

    Mathematical background concerning classical physics/mechanics.

    Hello, I was wondering what mathematical skills I would need to have in order to grasp classical mechanics as a whole; Newtonian, Lagrangian and Hamiltonian. I already have a small understanding of some classical concepts; such as motion in a two dimensional space (circular/arc motion...
  18. M

    How can I improve my ability to visualize mathematics?

    Hello everyone! I am having difficulties interpreting mathematical phenomenons visually speaking. I am trying to create an image in my head to explain how the mathematics works, for example, I have tried explaining the term brightness with close results but still flaws. So how do you become...
  19. S

    Integral problem / mathematical induction

    Ok so i got step (a) and found that $\int_{0}^{\infty} \,d (x^0)*e^(-ax)dx=1/a$ But i do not get how i should go about starting the next steps using the info from the first step(have not done a similar problem before so i need to get a grasp on the method)
  20. C

    Mathematical Game Theory (Drawing gametree for Tic-Tac-Toe

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  21. G

    Guide to learning mathematical physics

    Currently, I am a graduate student specializing in algebraic geometry. On the other hand, I have also become extremely interested in the mathematical physics. However, I am not sure what steps I should take to get to the modern frontiers of mathematical physics research. My mathematical...
  22. Quarlep

    What is required to write a mathematical theory?

    I want to write to theory about math but I don't have enough information about writing process.What can I do to write correct theory or article. I use theory and article in same meanings. Example a physicist makes experiments to proof his theory.Whats a mathematician do to...
  23. H

    Truth of Mathematical Statements: Which is Correct?

    Which one of these statements is true? $$ \exists y >0 : \forall x > 0, y < x $$ or $$ \forall x > 0 \exists y > 0 : y < x $$I think the second statement is correct, since for all x greater than 0, there exists at least one value of y > 0 such that y <x. The first statement doesn't really...
  24. H

    Mathematical model for pearlite interlamellar spacing

    It is seen that interlamellar spacing in pearlite becomes finer by increasing rate of cooling. theoretically this is because the rapid cooling rate produces more nucleation sites for lamella percipitation. so i tried to create a mathematical model using gibbs free energy change during...
  25. S

    Linear Component of Polarization - Mathematical transformation

    Hello, I'm currently going through Agrawal's book 'Nonlinear Fiber Optics' and got stuck with some mathematical cosmetics (pp. 40). It is the substition of: \vec{P_L}(\vec{r},t) = \frac{1}{2} \hat{x} \left(P_L \exp{(-i \omega_0 t)} + c.c.\right) into \vec{P_L}(\vec{r},t) = \epsilon_0...
  26. moriheru

    Mathematical description of Hilbertspace

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  27. P

    Mathematical help in understanding this integral (Dirac formulation)

    I was studying Dirac delta function, when I ran across this statement: The Dirac delta function is defined by: $$\delta(x) = lim _{K \to \infty } d_K(x), $$ where $$ d_K(x) = \int_{-K/2}^{K/2} \frac{dk}{2\pi} e^{ikx}$$ I was thinking that as $$K\to \infty$$ the integral won't be defined...
  28. moriheru

    Introdutions to mathematical astronomy?

    Does anybody know of a good Introduction to the mathematics of astronomy or astrophysics?
  29. M

    Symmetric Tensor Product in Pic. 1 - Explained

    we all know that a tensor has a symmetric part and anti-symmetric part and the anti-symmetric product (of the anti-symmetric part) called [wedge product] in pic.(2). then what is the name of the product the represents the symmetric part of a tensor in pic.(1) ?
  30. P

    Math Symbols in PF 4.0 - Where Did It Go?

    In the previous version of PF, there was some sort of list from which one can select mathematical symbols (for the LaTex layman). However, I can't seem to find this option in PF 4.0. Can someone please tell me where it got moved to (if not deleted)? This feature was extremely helpful and...
  31. N

    Can Newton's Second Law be Derived from the Work Energy Theorem?

    i was just thinking over line that my teacher told - "Newton's second law is ultimate truth" i feel why one can say 'F=ma' is more prestigious th 'work energy method' Though i know they are one and same thing, i was planning to derive (using principle of reverse engineering) second law of...
  32. X

    Mathematical Analysis on Cold Dark Matter

    Hi Guys, Can anyone provide links, books which explain the mathematical background of the field theory that explains the current accepted ^CDM model of the universe? Thanks Adarsh
  33. P

    Infinity and mathematical phyisics

    If the bulk universe is infinite (IF it is), would mathematical physics have trouble describing it because infinity is not a number?
  34. D

    Distribution of named and important mathematical constants

    I've noticed that the vast majority of named or important mathematical constants, are what you might call small numbers. Their modulus lies very often in the range [0,5]. Here's two examples of tables: http://en.wikipedia.org/wiki/Mathematical_constant#Table_of_selected_mathematical_constants...
  35. nuuskur

    Proof by mathematical induction

    Hello, I need to prove the following: \sum_{i=0}^n\binom{n}{i} = 2^n by using something called mathematical induction. I understand, somewhat, what it is - we propose a statement and show that is true for n=1, then we assume that the statement is true for all n \in \mathbb{N}, which should also...
  36. M

    Understanding the Metric Tensor to Tensor Products and Basis Vectors

    why the metric tensor is the dot product of two basis (co-variants or contra-variants) if it's a tensor then the metric tensor should be written as the tensor product of two basis (basis of a vector) >>> why ?!
  37. ShayanJ

    Physical reason vs. Mathematical reason

    Sometimes people ask a question and when I answer them in terms of equations, they say they want the physical reason, the real reason.. 1- Do you think there is a distinction between physical and mathematical reasons? 2- If your answer to the last question was yes, explain the distinction! if...
  38. micromass

    Your favorite mathematical theorems

    A theorem in mathematics can be beautiful in a lot of different ways. The result can be surprising or satisfying. Or perhaps the proof is very elegant and beautiful. Or maybe the theorem nice because it can be applied to other mathematics, or physics or engineering, or anything. So give here...
  39. K

    Proof by mathematical induction

    I'm not sure if I'm posting in the right subforum because there is one about proofs, but it requires the question not to be homework-like, but I also need an explanation... Homework Statement Prove by mathematical induction: Let m be the smallest element of A\subseteq\mathbb{N}. If A...
  40. M

    Mathematical Products: Geometric vs. Tensor

    hello every body. may i know what is the difference between geometric product and tensor product ?
  41. mxam

    Is n^n+1 a Valid Formula for the Sum of n^n Series?

    Can you help me with this exercise? 1^{1}+2^{2}+3^{3}+4^{4}+...+n^{n} = n^{n+1} Thanks! PD. I was trying to solve, and i have this: 1^{1}=1^{1+1} = 1 = 1 a) k^{k+1} b) k+1^{k+1} = k+1^{(k+1)+1} a in b) k^{k+1} + k+1^{k+1} = k+1^{(k+1)+1}...
  42. A

    Is transfer Function of a mathematical system unique?

    I have gone through a mathematical modelling of a mass spring dashpot system in which if substitute differently we get different transfer functions. How does this really happen as we say Transfer function of a system is unique.
  43. samgrace

    Math for Physics: Solid State, Thermal, Maxwells, Relativity & Optics

    Hello, it is said that maths is the language of science, specifically physics. I am posting to ask which areas of mathematics are used in sections of physics. In a months time I will be studying: Solid state physics Thermal and statistical physics Maxwells equations, relativity and optics...
  44. H

    Mathematical name of time dilation curve

    In mathematics, what is the name (type) for a curve given by velocity and time dilation? Specifically, I want to find a name for the curve y=1/(1-x^2)^{1/2} This curve is derived from the equation of a unit circle (x-a)^2+(y-b)^2=r^2 where y=(1-x^2)^{1/2}
  45. L

    Is double majoring necessary for a career in mathematical physics?

    Hello, I believe I presented myself before, but correct me if I'm wrong. I'm a math/physics major at a community college. I'm going to transfer to an university for spring. However, I don't know if I should major in astrophysics, Math, or Math/Astrophysics. I want to go to graduate school to...
  46. microsansfil

    Is the Game of Life a metaphor for the search for a Theory of Everything?

    Hello, How is it perceived the "Mathematical universe hypothesis" from Max Tegmark, in the scientific community ? Thank Patrick
  47. gfd43tg

    Mathematical induction ''all horses are the same color''

    Homework Statement Find the error in the following proof that \all horses are the same color" 4. Claim: In any set of h horses, all horses are the same color. Proof: By induction on ##h##: Base Case: For ##h = 1##. In any set containing just one horse, all horses are clearly the same color...
  48. Math Amateur

    The Amplitiuhedron - mathematical device or predicted real particle

    One of the things that pervades my mental life is whether theoretical physicists are implying that the ampituhedron is a (predicted) real particle or just a helpful mathematical device? Can anyone shed light on this question ... ? Peter
  49. P

    EPQ: Mathematical model of tea bag diffusion

    I've done some research, and I've looked at the derivation of the diffusion equation, but I don't want to reproduce even a more simple version of it. I'd like something that can I do almost exclusively myself. The point of this is not to plot some graphs, and extract some line of best fit or...
  50. gfd43tg

    Proving Relationships with Mathematical Induction

    Hello, We had a short unit on mathematical induction, and I know my final exam will probably have one problem that says ''prove this relationship with mathematical induction''. I was wondering, is there some sort of step by step procedure to proving something using induction? Or is it...
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