# What is Mathematics: Definition and 980 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. ### B What is a closed form solution?

Hi friends, I was wondering if you could give the definition of 'closed form', with examples of closed form solutions and open? form solutions. Foe example, is this a closed form solution? $$\sum_{k=1}^\infty \frac{1}{2^k}$$ Or this? $$\sum_{k=1}^5 \frac{1}{2^k}$$ Thanks.
2. ### Partial Differentiation of this Equation in x and y

Hi; please see below I am trying to understand how to get to the 2 final functions. They should be the same but 6 for the first one and 2 for the second? (I hope my writing is more clear than previously) There is an additional question below. thanks martyn I can't find a standard derivative...
3. ### A Concrete examples of randomness in math vs. probability theory

A recent question about interpretations of probability nicely clarified the role of the Kolmogorov axioms: [... some excursions into QM, negative probabilities, and quasiprobability distributions ...] Conclusion: the Kolmogorov axioms formalize the concept of probability. They achieve this by...
4. ### I Determining the dimension of a given PDE

Now in my understanding from text ...just to clarify with you guys; the pde is of dimension 2 as ##t## and ##x## are the indepedent variables or it may also be considered to be of dimension 1, that is if there is a clear distinction between time and space variables. Your insight on this is...

6. ### B Maze Proof and Statistical Mechanics

https://www.quantamagazine.org/maze-proof-establishes-a-backbone-for-statistical-mechanics-20240207/
7. ### Find the time taken by the truck to move ##1## metre.

Unless i am missing something; there is an error with the textbook It ought to be , ## t= \sqrt {\dfrac{1}{0.04}}= \sqrt 25 = 5## seconds.
8. ### Work out an estimate for the total number of ponies in the forest

My question; Which branch of maths is this? Also, can you give me a clue as to where to start regards solving this. Just a hint please, not a full explanation. I'm struggling to even guess at this one. I did think, '60 ponies, 5 of which are tagged, so, 5/60 tagged, which is 1/12 1/12 of the 60...
9. ### Number of ways of arranging 7 characters in 7 spaces

The rightmost position has 3 possibilities: ##x,y,z## The remaining two letters are to be arranged in 6 spaces: ##\frac{6!}{4!}## Now the 3 can be placed in ##\frac{4!}{3!}## Total no of ways =$$3×\frac{6!}{3!}=12×30$$ $$OR$$ Since ##x,y,z## are three different boxes/variables, we can use the...
10. ### A SDE valuation equation (stochastic calculus)

I read from a text: "suppose a stock with price ##S## and variance ##v## satisfies the SDE $$dS_t = u_tS_tdt+\sqrt{v_t}S_tdZ_1$$$$dv_t = \alpha dt+\eta\beta\sqrt{v_t}dZ_2$$ with $$\langle dZ_1 dZ_2\rangle = \rho dt$$ where ##\mu_t## is the drift of stock price returns, ##\eta## the volatility of...
11. ### Given value of vectors a,b, b.c and a+(b×c), Find (c.a)

I thought this was too easy $$a+(b\times c)=0\implies a=-(b\times c)=(c\times b)$$ Then $$3(c.a)=3(c.(c\times b))=0$$ Since cross product of vectors is perpendicular to both vectors and dot product of perpendicular vectors is zero. Now here's the problem, correct answer given is 10. But how do...
12. ### DeepMind AlphaGeometry: Is AGI just around the corner?

DeepMind's new geometry problem-solver AlphaGeometry can solve Euclidean plane geometry questions from the International Mathematics Olympiad (IMO) almost as well as a human gold medallist. https://deepmind.google/discover/blog/alphageometry-an-olympiad-level-ai-system-for-geometry/...
13. ### Find an irrational number that satisfies the given inequality

I just came across this question and the ms indicates, Would ##31.5## be correct? ...i think it is rational as it can be expressed as ##31.5 = \dfrac{63}{2}##.

39. ### Why are Physicists so informal with mathematics?

I realize of course that this will probably not apply to all physicists, but at least every physicist in my university's math department is very unrigorous when it comes to mathematics. This is frustrating because some of the physics material seems genuinely interesting, but the lack of an...
40. ### Difference between -3² and (-3)² ?

What is the difference between these to. -3² and (-3)² ? I know - x - = + I am told -3²= -9 not +9 -3² means -3 x -3 = 9 I am told (-3)² = 9
41. ### A Summing over continuum and uncountable numerocities

Here I want to address of the question if it is possible to make a sum over an uncontable set and discuss integration rules involving uncountably infinite constants. I will provide introduction in very condensed form to get quicker to the essense. Conservative part First of all, let us...
42. ### Partial Fraction Decomposition

##\frac {1} {x^2 -c^2}## with ##c \neq {0}## So the first thing I do is split the ##x^2 -c^2## into the difference of squares so ##x +c## and ##x - c## I then do ##\frac {A} {x + c}## ##+## ##\frac {B} {x-c}##, and then let ##x=c## to zero out the expression. And that is where I am getting...
43. ### Programs What subject is better for an aspiring experimental physicist?

I'm in my last 2 years of high school, and I have to pick a speciality to study before becoming an undergraduate and studying in college. In the future, I'm hoping to become an experimental physicist. My high school offers 3 specialities that are relevant to physics to pick from, all of them...
44. ### New book out by Routledge called "Mathematical Conundrums" with many interesting problems

TL;DR Summary: new book with interesting problems There's a new book out by Routledge called Mathematical Conundrums with many interesting problems in algebra, arithmetic, route-drawing, and logic. Good for schools as algebra is no higher than high school. Challenging though.

46. ### I Where can I find explanations about physics applications in real life?

I am a curious physics student who wants to learn how to use its knowledge to create things, to understand phenomenons and so on. I am looking for detailed explanations that use physics and maths. (books, websites, videos, etc.)
47. ### Style of teaching/learning mathematics: by proofs of theorems only?

I remember there was a method of learning/teaching mathematics where all they do in class is to force students to prove the theorems themselves. What was this method again? It was named after someone.... @fresh_42 ?
48. ### I Order of an element in ##\mathbb{Z}_n##

Doing some self study here; my understanding of order of an element in a group is as follows: Order of ##3## in ##\mathbb{z_4}## can be arrived by having, ##3+3+3+3=12≡0## likewise, the order of ##12## in ##\mathbb{z_{20}}## can be arrived by ##12+12=24 ≡4≠0## ##12+12+12=36≡16≠0##...
49. ### Relativity Looking for a good introductory Tensor Analysis Textbook

Hello all, I've taken math through differential equations and linear algebra, am in my senior year of physics curricula while conducting McNair research regarding General Relativity. I found a NASA document outlining Einstein's field equations, which suggests only preparative familiarity with...
50. ### Prospective ArXivLabs project

In short, I'm interested in working on a web-app to make landmark papers in theoretical physics and mathematics more broadly accessible, especially to undergraduate and graduate students who are looking to catch up to modern topics (without sacrificing rigor or exactness of understanding), and...