About the definition and the branches of mathematics , and some quotes .

The material for this post is taken or inspired from two answers I wrote at quora.com about mathematics.

First I’ll start with a general definition of mathematics taken from the Encyclopedia Britannica (click here to view the online source ) :

“Mathematics, the science of structure, order, and relation that has evolved from elemental practices
of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and
quantitative calculation, and its development has involved an increasing degree of idealization and
abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable
adjunct to the physical sciences and technology, and in more recent times it has assumed a similar
role in the quantitative aspects of the life sciences.”

There are numerous (a word related to number , and hence to mathematics) branches and areas in mathematics , and they are all related and interconnected in one way or another.
Branches can also be divided into sub-branches .

The image below is a combination of (mathematical) images , it was made with Photoshop.

math images

Here are a few examples of mathematical branches or fields of study :
Some would say that the main branches of mathematics are algebra , geometry , and arithmetic (the science of numbers was an important branch of mathematics until the Middle Ages) , others would add analysis , logic , etc.

Algebra generally comprises abstract algebra , the study of rings and fields , linear algebra , and vector spaces. Arithmetic and number theory are viewed as sub-fields of algebra , as well as group theory , representation theory (the study of the elements of algebraic structures as transformations of vector spaces) , homological algebra , commutative algebra …

Analysis involves infinitesimal quantities and the passage to a limit . It comprises (integral and differential ) calculus , vector analysis , complex analysis , real analysis , Fourier analysis , the study of differential equations …

Geometry , the study of space , shapes , and sizes , includes differential geometry , algebraic geometry , analytic geometry , affine geometry , projective geometry , Euclidian and non-Euclidian geometry , noncommutative geometry , topology , etc.

The theory of the foundations of mathematics includes the study of mathematical and symbolic logic , set theory , model theory , proof theory ,…

Pure mathematics is sometimes distinguished from applied mathematics , which comprises disciplines such as probability theory and statistics , numerical analysis , computer algebra , …

Other fields of mathematics include combinatorics , graph theory , ergodic theory , etc.

All the branches of mathematics are used in various ways by the exact sciences  (physics , astronomy , …) and by the social sciences. There is by the way an area of mathematics called ‘mathematical physics’ .

A certain field of mathematics is sometimes interconnected with a number of other fields of mathematics. For example , tensor analysis is closely related to differential geometry , multilinear algebra , mathematical physics , algebraic topology , …

Below are some useful links about the branches of mathematics :

Branches of Mathematics

Areas of mathematics

Glossary of areas of mathematics

Next I give some revealing quotes , opinions and definitions  about mathematics:

  • “Mathematics is the most beautiful and most powerful creation of the human spirit.”
    Stefan Banach.
  • “Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.”
    Galileo Galilei .
  • “Neglect of mathematics work injury to all knowledge, since he who is ignorant of it cannot know the other sciences or things of this world. And what is worst, those who are thus ignorant are unable to perceive their own ignorance, and so do not seek a remedy.”
    Roger Bacon.
  • “Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.”
    David Hilbert .
  • “Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.”
    Deepak Chopra .
  • “Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley.”
    Yōko Ogawa
  • “[When asked why are numbers beautiful?] It’s like asking why is Ludwig van Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.”
    Paul Erdős .
  • “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
    G.H.Hardy .
  • “Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.”
    Felix Klein .
  • “But in my opinion, all things in nature occur mathematically.”
    René Descartes .
  • “The study of mathematics is apt to commence in disappointment… We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet’s father, this great science eludes the efforts of our mental weapons to grasp it.”
    Alfred North Whitehead .
  • “Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.”
    E.T.Bell .
  • “Mathematics takes us into the region of absolute necessity, to which not only the actual word, but every possible word, must conform.”
    Bertrand Russell .
  • “The object of pure physics is the unfolding of the laws of the intelligible world; the object of pure mathematics that of unfolding the laws of human intelligence.”
    James Joseph Sylvester .
  • “Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion—not because it makes no sense to you, but because you gaveit sense and you still don’t understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.”
    Paul Lockhart .
  • “[Replying to G. H. Hardy’s suggestion that the number of a taxi (1729) was ‘dull’, showing off his spontaneous mathematical genius] No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 13 + 123 and 93 + 103.”
    Srinivasa Ramanujan .
  • “There cannot be a language more universal and more simple, more free from errors and obscurities…more worthy to express the invariable relations of all natural things [than mathematics]. [It interprets] all phenomena by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes”
    Joseph Fourier .
  • “Humans are like Variables in mathematics, some Dependent, some Independent. Variables are in relationship but remain Variable. Of course, there are some Constants too both in mathematics and humans. Constants help define precisely the relationship between variables. Maybe, that is why humans keep adding (to problems), subtracting (from happiness), multiplying (what else, we are all over earth) and dividing (the earth among themselves).”
    R.N.Prasher .

These are eloquent quotes , not easy  to add to them.
The most important language one could know is not English or French or Spanish (these are of course necessary and useful) , it’s math.

Combined coherently with the rules of the scientific method  , mathematics can be used efficiently to explain and explore  the Natural and Physical World.
Applied adequately to the social sciences , mathematics can inseminate them and bring the human sciences closer and closer to the exact sciences.

One should never underestimate the power of mathematics , reasoning , and exact science.

Related online sources :

Quotes about mathematics from goodreads 

Mathematics Quotes at BrainyQuote