# 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. ### Double integration - switching limits

I get that the bottom answer isn't a constant - but does this physically represent anything? When I set the two answers equal to each other, I get x = +- 1/sqrt(2) and I am wondering if this represents anything significant. I don't think (mathematically) there is anything wrong with the bottom...
2. ### Format for Mathematical Expressions

TL;DR Summary: Why has format for mathematical expressions changed? The formula I am using as an example here is from classical physics so that is why I have chosen the classical physics forum. I have been running into a new format very different from what I have been used to for decades...
3. ### 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.
4. ### Neural networks as mathematical models of intelligence

Why do almost all people not think that neural networks are the mathematical model of intelligence? I briefly explain what I understand: -A neuron is a mathematical object that takes numerical inputs from other nearby neurons, applies a nonlinear function (combining the input with numbers...
5. ### Should I skip labs/experiments for Mathematical Physics program?

I am attending University of Waterloo and my school will allow me to graduate as a Mathematical Physics major without taking any labs/experiment courses (in my school lab is not integrated to physics courses, they are separate courses with separate credits). This could be great because : -...

7. ### Mathematical Methods Book That Uses SI Units

I'm looking for an undergraduate-level 'mathematical methods' or 'engineering mathematics' book that uses SI units for the purpose of self-study. I've had my eyes on Zill's Advanced Engineering Mathematics, but it seems to use US customary units. So ideally I'm looking for a book that covers...
8. ### Exploring "The Mathematical Theory of Black Holes" by S. Chandrasekhar

In page 67 of book "The mathematical theory of black holes" by S. Chandrasekhar in chapter 2 "Space-Time of sufficient generality" there is a theorem that metric of a 2-dimensional space $$ds^2 = g_{11} (dx^1)^2 + 2g_{12} dx^1 dx^2 + g_{22} (dx^2)^2$$ can be brought to a diagonal form. I would...
9. ### [L'Hospital's Rule] Can I Use Mathematical Induction to Prove This?

The Student's Manual simply applies l'Hospital's Rule n-times to arrive at ##\frac{e^x}{n!}\to\infty## as ##x\to\infty##. However, I'm wondering if I could use Mathematical Induction to prove this. Is the following correct and sufficiently rigorous (at least for an undergraduate-level Calculus...
10. ### Mathematical representation of two-entangled q-bits?

Dear all,I have four questions. Hopefully, someone can answer. Thank you :) 1. A qubit is described as a two-orthogonal basis state. How about two entangled qubits? 2. What is the actual reason for a qubit cannot be cloned/copied? Is it because without knowing the value of the complex...
11. ### Open problems and suggestions of great mathematical journals

Hi! Suppose that someone had solved an old but open problem in the great area of mathematics and physics, for instance, dynamical systtems, algebraic geometry and differential equations. Based on your broad experience, what are the best scientific journals to submit such a discovery? In...
12. ### Mathematical equations to ask questions about physics- Who teaches this skill?

I'm a retired Engineer. I want to learn methods to formulate equations to ask questions of physics phenomenon. What school teaches this and by whom? Can anyone point me there?
13. ### Understanding Spatial Description: A Mathematical Perspective

I understand the mathematical difference, that the independent variables are the "reference position vector+time" for material description, and "current position vector+time" for the spatial description. But I can't seem to wrap my head around the concept of the spatial description.
14. ### What are the basic mathematical objects in QFT?

I found a copy of David McMahon's "Quantum Field Theory Demystified" and I'm already confused on page 4 where he says, " . . in order to be truly compatible with special relativity, we need to discard the notion that \phi and \psi in the Klein-Gordon and Dirac equations respectively describe...
15. ### Prove the 2nd axiom of mathematical logic using the Deduction Theorem

prove: The 2nd axiom of mathematical logic 2) $((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))$ By using only the deduction theorem
16. ### Use mathematical logic to prove this proposition

Given the following axioms: 1) ##P\implies(Q\implies P)## 2) ##((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))## Where ##P,Q,R## are any formulas 3)##(\neg P\implies\neg Q)\implies (Q\implies P)## then prove: ##\{A\implies B,B\implies C\}|- A\implies C## Without using the...
17. ### I have a question about the mathematical model of a reflective curve

Hello : Have a question regarding the mathematical model of reflective curve where could i find information on it ? (pdf , webpages , ebooks ,...etc ) Other than Wikipedia Best Regards HB
18. ### Vectors as geometric objects and vectors as any mathematical objects

In geometry, a vector ##\vec{X}## in n-dimensions is something like this $$\vec{X} = \left( x_1, x_2, \cdots, x_n\right)$$ And it follows its own laws of arithmetic. In Linear Analysis, a polynomial ##p(x) = \sum_{I=1}^{n}a_n x^n ##, is a vector, along with all other mathematical objects of...
19. ### Prove this problem that involves Mathematical induction

Ok, would i be correct to approach it this way, Let ##n=1##. If ##n=1##, then ##5^1+3## is divisible by ##4##, the statement is true for ##n=1##. Assume its true for ##n=k## ∀ ##kε\mathbb{z}^{+}.## Then ##5^k+3## is divisible by ##4.## i.e ##5^k+3=4m## ∀ ##m ε\mathbb{z}^{+}## Let ##n=k+1.##...
20. ### Can one do mathematical physics and theoretical physics at the same time?

I finished my 1st-year physics, took analysis, linear algebra, mathematical logic, classical mechanics, quantum mechanics(I was exempted from intro phy and took some 3rd-year physics courses) I internal transferred to pure maths. The reason is that the curriculum of the physics programme in our...
21. ### Mathematical prerequisites for general relativity

What mathematical topics do I need to know to start studying general relativity? From which textbooks can I learn them? I don't currently know anything about differential geometry. I know calculus, linear algebra, mathematical methods of physics (the necessary topics for quantum mechanics) and...
22. ### I am revisiting the mathematical formulation of quantum mechanics with the dimensional (MLT) perspective

Hi Fellas! My first post after a long hiatus from forums. Feeling nostalgia (this is the place where it all began, my fuel for quantum fascination so to speak). I am revisiting the mathematical formulation of quantum mechanics with the dimensional (MLT) perspective. I want to understand what...
23. ### Space/Spacetime Metrics: Comparing Routes in PHY 101

[Moderator's note: Spin off from previous thread due to more advanced level of discussion.] The above is true even on an ordinary position-vs-time graph in PHY 101, when "distance" is appropriately defined. OOPS (Thanks @PeterDonis ) This turns out to be true only in the Galilean/Newtonian...
24. ### Impedance Matching: The mathematical Conditions for Matching Network

Assume we have a network consisting of a source with impedance ##Z_S## and load with impedance ##Z_L## and we want to perform impedance matching on them in order to obtain the maximum power transfer: Note that in practice there may occur sitations where causes more harm than profit (see eg...
25. ### How Does Mathematical Theory Explain Multiple Wave Reflections?

I know for a wave moving from left to right, ##\psi_i = Ae^{i(\omega t - k_1x)}## The first reflection where ##Z_1## is ## R_{12}Ae^{i(\omega t - k_1x)}## The second reflection. The wave moves from 2 to the limit between 2 and 3 then reflect... Thus, ##T_{12}R_{23}T_{21} Ae^{i(\omega t - k_1 x...
26. ### Rigorous mathematical definition of sampling

Please could you help me find a rigorous mathematical definition of sampling as it is used in mathematical statistics? Let ##X:\Omega\rightarrow\mathbb{R}## be a random variable and ##X_{1},...,X_{n}## is a statistical sample. What is its mathematical meaning in probability theory? Are we...
27. ### Non Mathematical Quantum Field Theory Books?

Are there any QFT books that use little to no math? If there is a little math that is okay. I don't know much about math. I am looking for good explanations on how it works without math. Any help would be great!
28. ### Distances in Knn: Are they metric functions in the Mathematical sense?

Hi, Just curious as to whether distances 'd' , used in Knn ; K nearest neighbors, in Machine Learning, are required to be metrics in the Mathematical Sense, i.e., if they are required to satisfy, in a space A: ##d: A \times A \rightarrow \mathbb R^{+} \cup \{0\} ; d(a,a)=0 ; d(a,b)=d(b,a) ...
29. ### Mathematical equation for the deviation from ideality of real gases

Hi guys, We have this very common graph where pV deviates from ideality. May I know the equation for such a curve? Secondly, if the x-axis were changed to V, what would the graph look like?
30. ### Math Treatment of Approaching Event Horizon

[This thread can be considered the A-level footnote to https://www.physicsforums.com/threads/is-there-an-inside-to-a-black-hole.1007588/] For a static [admits a hypersurface orthogonal timelike Killing field ##k##], spherically symmetric spacetime, a time coordinate ##t## can be chosen as the...
31. ### Edward Witten and the Mathematical Multiverse?

Physicist Max Tegmark proposed that every mathematically possible universe exists [1]. According to Tegmark himself, when he published such ideas, he received mostly positive feedback from Edward Witten and David Vogan [2]. Due to this, I was wondering whether Witten has made reference to...
32. ### Help with Mathematical Description & Calculations of Space Charge Density

Guys I have Problems with this task The arrangement consists of a point charge Q at a distance (x0, y0,0) from the origin and two perfectly conductive surfaces in the (x, z) and (y, z) plane a) Mathematical description of the space charge density p of the original and mirror charge using the...
33. ### Best layman non mathematical interesting book on Ray and wave optics

A book on optics which is less mathematical maybe a similar one to physics for poets or gamow gravity classics
34. ### How to conclude a mathematical expression with the provided background

Hi PF In my textbook, the Spanish 6th edition of "Calculus", by Robert A. Adams, at Chapter 3.2, Theorem 1 states: If ##x>0##. $$\dfrac{d}{dx}\ln{x}=\dfrac{1}{x}$$ PROOF For ##x>0## and ##h>0##, ##\ln{(x+h)}-\ln{(x)}## is the area of the shaded part (...) Regard the shaded area at Figure...
35. ### I need to write a paper on mathematical chemistry -- Any suggestions?

I can choose any topic. The paper doesn't have to be original. Please suggest a simple paper to write which involves math and chemistry. I am in second year university.
36. ### Which CAS use multiple representations for mathematical objects?

How do various computer algebra systems (CAS) compare with respect to keeping multiple representations of the same mathematical object? For example, a polynomial could be represented by a list of coefficients, or a list of roots, or a list of factors with some factors non-linear. One design...
37. ### Construct a mathematical formula

Hello, I have been thinking of it for a long time, and I would appreciate suggestions from math experts. I am working on a simulation of human agents. I want to set up a formula that defines the consumption probability (0,1), which consists of X, a value between 0 and 1, and two positive and...
38. ### What is the geometric approach to mathematical research?

I read this article History of James Clerk Maxwell and it talks about Maxwell and Dirac also at some point. It is said that Maxwell thought geometrically, and also Dirac said he thought of de Sitter Space geometrically. They say their approach to mathematics is geometric. I see this mentioned...
39. ### Importance of "A Mathematical Theory of Communication"

I am not sure this is the right section to ask this question, but here it goes. So, I was studying Stat. Physics and I came across this paper, A Mathematical Theory of Communication. What it's so important about this paper?

41. ### Physics equations with the mathematical constant Phi?

I know Phi appears often when modelling exponential growth and, probably because of that, also in Biology/Ecology. But does it appear spontaneously in the mathematical description of some fundamental physics phenomenon at all? (As does Pi, the ubiquitous irrational number) Hope I'm posting on...
42. ### Can you use Taylor Series with mathematical objects other than points?

I was recently studying the pressure gradient force, and I found it interesting (though this may be incorrect) that you can use a Taylor expansion to pretend that the value of the internal pressure of the fluid does not matter at all, because the internal pressure forces that are a part of the...
43. ### Drop out of PhD in fluid mechanics and switch to Mres in mathematical physics?

A post doc in an area that differs from my PhD? I am currently doing a PhD in fluid mechanics but want to do mathematical physics tbh. In another thread I got an answer about a user who had done a PhD in accelerator physics and went to do a post-doc in condensed matter, vice versa even, but in...
44. E

### Neuroscience with mathematical emphasis

What is a nice resource to learn about the mathematics of neuroscience? I read a little bit about the Hodgkin-Huxley model for propagation of action potentials and also stuff like synaptic junctions, but I like to learn some more about modelling networks of more than one neuron, stochastic...
45. ### Recommended mathematical courses for a Physics major

Hi, you all, I am an undegrad Physics student and I'm choosing my optional courses for my third year. I'm looking for advice (or opinions, if you prefer) since I'm not sure what of the following mathematical, optional courses would be more "beneficial" (I know this term is abstract) to my...

47. ### Is the concept of "potential" just a mathematical model?

is the concept of potential just a mathematical model
48. ### Mathematical term for physicist's "Lattice Theory"?

Lattices are studied in mathematics. What physicists call a "lattice theory" uses the mathematical object that is a lattice, but it involves other things, such as associating elements of a group with the links between nodes of a lattice. Is there a mathematical term for lattices with this...
49. ### A problem of mathematical induction

I have gone through the principle of mathematical induction. I cannot understand why do we need to prove every statement for n=1. I mean why is it necessary? Why can't we start directly from n=k then n=k+1. For example see the below image. Thanks!
50. ### Error in Sadri Hassani's Mathematical Physics

I am reading his excellent book "Mathematical Physics Part 1, Second Edition", which has benefited me a lot in many ways. However, I have a doubt about the correctness of the theorem 2.3.23, which states that for any subspace U of V, the map T ' : V/U -> T(V) defined by T ' ([a]) = T|a> is...