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. A

    What are some recommended books for improving mathematical physics skills?

    So I really want to get good at mathematical physics. Below is a list of books I could get from various libraries, but I don't know which one's I should start with and which to do next and so on. I currently understand intro CM, EM, and QM. My math skills are very basic (ODES as of now). I thank...
  2. J

    Leibniz formula using mathematical induction

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  3. M

    Exploring the Mathematical Conversion between ∆ and dX

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  4. F

    Splitting light into colors, mathematical expression (fourier transforms)

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  5. E

    MATLAB Writing mathematical equations in MATLAB

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  6. B

    Good programs for mathematical physics in canada

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  7. H

    Math Structures: Proving Existence w/ Accurate Solutions?

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  8. B

    Developing mathematical maturity?

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  9. W

    Mathematical treatment of signals and systems

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  10. S

    Show the mathematical relationship between Relative Risk and Difference in

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  11. C

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  12. T

    Curvatures in space-time: actual reality or mathematical concept?

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  13. W

    Help with a Mathematical Induction Problem

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  14. M

    Which Mathematical Physics Text Should I Choose?

    I found myself having a hard time choosing what mathematical physics text should I stick with. I'd like to think of myself as mathematically inclined, and I would really prefer a mathematical physics book that emphasizes 'the maths' and the proofs and not just the methods while maintaining its...
  15. M

    Proficiency in Mathematical Proofs

    Where does one attain the skill/ability (skill vs. ability is a good question in itself) to do proofs? By "do proofs" I mean prove existing theorems. For instance I am really no good with ε-δ proofs. The concept is so easy, but when I have to prove something using ε-δ I either have no idea what...
  16. K

    Otto cycle mathematical model - Heat added by combustion?

    I am attempting to improve a spreadsheet I created back in uni some 5-6 years ago, which models all the velocities and accelerations of the pistons and rods, as well calculating pressure, temperature and resulting torque. It is only a simple model so far, originally it started simply by...
  17. L

    A mathematical explanation of hysteresis?

    I wish to know why and how does hysteresis occur from a purely mathematical perspective, using analytical functions and multivariable calculus. Thank you!
  18. V

    Pairwise disjoint mathematical proof

    Set 1 {{1, 2}, {2, 3}, {1, 2, 3}, {2, 3, 4}, {2, 3, 5}, . . . , {2, 3, n}}; Set 2 {{3, 5}, {5, 4}, {3, 5, 4}, {4, 5, 1}, {4, 5, 2},{3, 5, 6}, {3, 5, 7}, . . . , {3, 5, n}}; Set 3 {{3, 1}, {1, 4}, {3, 1, 4}, {1, 4, 2}, {1, 3, 5}, {1, 3, 6}, . . . , {1, 3, n}}; Set 4 {{1, 5}, {5, 2}, {1...
  19. K

    Using induction technique in mathematical logic

    Homework Statement Prove the following LEMMA: For every proposition A[P_{1}, \dots, P_{n}] and any two interpretations v and v', if v(P_{i})=v'(P_{i}) for all i=1, \dots,n, then v^{*}(A)=v'^{*}(A). Homework Equations The Attempt at a Solution Sure this is obviously an...
  20. P

    Mathematical description of Refractive index in crystal/glass

    Hi, I have read on a website that the refractive index of glass is related to the number of electrons it has per unit volume. I don't recall which website it was, but is it true? & if so what is the quantitative description behind it? or is it a lot more complicated :) Many thanks for...
  21. C

    Can anyone solve this mathematical induction problem?

    Homework Statement Prove that for all n>1, P(n) :1 + 1/2 + 1/3 +...+1/n = k/m where k is an odd number an m is an even number.Homework Equations The Attempt at a Solution 1)Base Case n =2 P(2) = k/m 3/2 = k/m which is true. 2) Inductive Step Assume P(n) is true for some arbitrary n. 3)...
  22. Crazymechanic

    How can I calculate gas pressure in a cylinder using mathematical formulas?

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  23. C

    Mathematical logic & its branches

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  24. M

    Filling holes or gaps in mathematical knowledge.

    I graduated high school nine years ago. The highest math I took then was Algebra II, as I had little interest then. After a seven year career, I was able to go back to school and have made it to Calculus I this semester. I struggled some in college algebra, finding out that there were...
  25. B

    My Mathematical Methods in Physics is written to weed out students, help me

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  26. L

    Proving n3 + n < 3n for n >= 4 using Mathematical Induction

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  27. H

    Best place to study mathematical logic uk

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  28. M

    Visualising mathematical concepts in QM - comments needed

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  29. T

    Standard Model Mathematical Representation

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  30. M

    Mathematical Induction - Algebra Manipulation

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  31. G

    Exploring the Pauli Vector: Mathematical Definition

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  32. D

    Use mathematical induction to prove the following statements are true

    Homework Statement Use mathematical induction to prove the following statements are true for n≥1 a) 1^2+3^2+5^2+...+(2n-1)^2= [n(4n-1)]/3 Homework Equations The Attempt at a Solution Attempt at showing for n+1 is true: n[4(n+1-1)]/3+ 2[k+1-1)^2
  33. A

    Step-by-step mathematical physics from classical to quantum physics

    Dear folks, Under the aalaniz blog, I just posted step-by-step notes starting for ODEs and PDEs both linear and nonlinear, of the type you will see in grad physics programs. The notes connect Lagrangian, Hamiltonian and Poisson Bracket methods in classical physics to the various approaches...
  34. M

    Can I Visualize Mathematical Terms with Wolfram Alpha?

    hello! I think it would help a lot to visualize mathematical terms I would like to see the graphs of various f(x), then their f'(x), their f''(x), in order to understand what derivatives mean example: http://en.wikipedia.org/wiki/File:Graph_of_sliding_derivative_line.gif same with...
  35. R

    Using second Principle of Mathematical Induction Proof

    Homework Statement Let F: N->N be defined by f(1)=1 and f(2)=2 and f(n+2)=1/2(f(n+1)+f(n)) Use Theorem 1.3 to prove that 1<=f(n)<=2 for all n in N Homework Equations For each n in N let P(n) be a statement about the positive integer n. If a) P(1) is true, and b) for k>1, P(k) is...
  36. M

    Mathematical Physics vs. Honours Physics vs. Physics and Astronomy

    Hi, I'm a Grade 12 student and I'm very interested in physics. I would like to apply to Waterloo University becasue of their renowned Co-op program, but they have the three types on Physics Undergraduate Degrees; Mathematical Physics...
  37. H

    Momentum, impulse and force (mathematical)

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  38. A

    Understanding Mathematical Induction: Why Do We Use n Instead of k?

    what is mathematical induction? explain with an example.
  39. M

    Complete list of mathematical notations

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  40. G

    What Are the Best Resources for Learning Mathematical Finance?

    Can anybody provide some resources (books, websites etc. - preferably websites or course notes) for a (somewhat rigorous) introduction to Mathematical Finance? Basically, I want to have the background to understand stuff like this http://www3.stat.sinica.edu.tw/library/c_tec_rep/c-2002-10.pdf...
  41. E

    Explaining the Mathematical Reasoning of Finding Image in Convex Mirrors

    I am having a problem understanding the mathematical reasoning of finding the description of image in convex mirror... We know the relation, 1/v + 1/u =1/f is true for all mirrors... we can prove this for both concave and convex mirrors... In case of convex mirrors we use the conventional...
  42. TrickyDicky

    Some doubts concerning the mathematical bases of GR

    I'm a little confused about certain assumptions usually made in GR as to how rigorous they are mathematically speaking. For instance the assumption generally presented without proof that GR spacetime manifold is a Hausdorff space seems not to be warranted given the fact that pseudometric spaces...
  43. I

    Mathematical Quantum Statistics: Why is A*rho of trace class?

    Hi, as we know a density operator \rho is defined to be a non-negative definite operator of trace class (with trace 1). We also know that for a given observable A, which is a (possibly unbounded) self-adjoint operator, the expectation value can be calculated as \operatorname{tr}(A \cdot...
  44. E

    Anyone know of good Mathematical Modeling classes/books?

    I've recently gotten into Mathematics more and more, and I wanted to do some mathematical modeling just to see where I stand on it and maybe try to get better at it. I've looked around as well as I could, and I didn't really find much, so I was wondering if anyone here knew of any good...
  45. C

    Which mathematical subjects must I learn to understand basic QFT?

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  46. S

    Mathematical tools for quantum mechanics

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  47. S

    What are some resources for understanding mathematical moments?

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  48. T

    Mathematical induction - inequality

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  49. S

    Is f(n) = 4^n + 6n - 1 Divisible by 9 for All Positive Integers n?

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  50. P

    Suppose traffic laws allowed mathematical reasoning

    Some time ago I had a thought regarding turning rules in the US, back when right turn on (a) red (light - unless otherwise marked) was becoming universal. If U-turn was legal on the cross street, then by smoothing out a sequence of legal maneuvers: - left turn on red should be legal (right...
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