# What is Mathematics?

1. Sep 15, 2007

### Skhandelwal

Physics is the study of Mechanics(understand trajectories, forces, etc.), relativity, etc.

Biology is the study of understanding life, evolution, etc.

What is Mathematics the study of?

2. Sep 15, 2007

### coberst

I would say that mathematics is the science of pattern. Math studies pattern.

Mathematics is a way of seeing. Mathematics is the science of pattern. Imagine a very elaborate Persian rug. Imagine that you have only a small fragment of that rug. Mathematics offers a means whereby you might be able to construct the rest of that rug to look exactly like the original. Math can perhaps create a formula for the pattern in the rug such that you can, by following that math formula, exactly duplicate the pattern from which that rug was created.

Understanding is a stage of comprehension whereby a person can interject them self into the pattern through imagination. ‘Understanding is math’ because it helps the individual to ‘walk in the shoes’ of some other entity.

3. Sep 15, 2007

### granpa

numbers.

all you need to form math is the number one and the function '+1'.
2 is defined as (1)+1
3 is defined as (2)+1=1+1+1
and so on

2+3=(1+1)+(1+1+1)=1+1+1+1+1=5
multiplication is defined in terms of addition
division is defined in terms of multiplication

4. Sep 15, 2007

### out of whack

I see it as a language first before a science. But unlike natural languages, math is systematically defined to be unambiguous and internally consistent. This way, anyone anywhere can know precisely what a mathematical expression represents, and this is what makes it useful. Math fill the need for clarity of expression in science and technology.

As a science, I see it as a form of artificial linguistic, the study of its own form and meaning. This is where experts can spend a lifetime working out what inescapable conclusions can be reached from initial premises when the language is rigorously applied to them.

5. Sep 15, 2007

### arildno

Mathematics is what mathematicians do.
Mathematicians decide who are mathematicians.

6. Sep 15, 2007

### HallsofIvy

Staff Emeritus
So you would say that geometry is not mathematics?

No, multiplication is not, except in the very simple situation of the integers, "defined in terms of additon". And, to most mathematicians, division is not an operation at all- it isn't "defined" at all.

I think it is a very bad idea to try to define "mathematics" on the basis of elementary and secondary mathematics.

7. Sep 15, 2007

### Coto

In regards to all these posts:

Mathematics may have had its beginnings in numbers and patterns, however current day mathematics is far broader than simply that. A course in real or complex analysis, tensor analysis, or set theory, all help to explain why mathematics is more than just the study of numbers and patterns (i.e. number theory, geometry, applied mathematics in general).

Mathematics has a part which explains numbers and patterns, but it is also a logical playground for humans. It allows us to explore the outer limits of logic, or in other words to find something that very well 'exists' without human consciousness -- something that is universally true regardless even of species (or so philosophically I tend to believe.) This logic is continually generalized to any object that exists in the human mind.

Last edited: Sep 15, 2007
8. Sep 15, 2007

### arildno

To be slightly more serious, I'd say maths is one branch of applied logic.

9. Sep 15, 2007

### HallsofIvy

Staff Emeritus
I would be inclined to define mathematics as the study of "relationships" rather than "patterns" but they are obviously closely related(!). There is a field of mathematics called "category theory" that is just about as abstract as you can get (the textbook, in the preface, said category theory is often called "abstract nonsense" with no sense of that being derogatory at all). A category has "objects" and "relations". The collection of all sets is a category with sets as objects and functions between them as "relations". The collection of topological spaces is a category with the topological spaces being the objects and continuous functions from one topological space to another being the relations.

One basic theorem of category theory is that a category is completely defined by its relations- you don't have to mention the objects at all!

In fact "relationism" is a recognized philosphy of mathematics- it is a subset of the "Platonist" philosphy.

Here's another point, related(!) to that: Mathematical "structures", consist of: axioms, definitions, undefined terms, theorems etc. Back when I was in high school geometry, they explained the "undefined terms" by saying that a "definition" is an explanation in words- of course, you need to know the definitions of the words in that definition in order for it to make sense. Hopefully the words in a definition are simpler and more basic that then word they define. Eventually, you get back to the simplest possible concepts which cannot be "defined" because there are no simpler words.

That's perfectly good but it is only recently that I realized how very fundamental to mathematics "undefined terms" are. Mathematical structures are "templates" and the undefined words are the "blanks" that have to be filled to apply the template to a specific purpose.

Why is it that Calculus, originally developed to solve problems in physics (specifically the orbits of planets) can be used so effectively for problems in economics, biology, etc.?
All of calculus, like any mathematics, is based on theorems proved from axioms and definitions, both of those containing undefined terms. To apply it to any field, you give meaning to those "undefined terms" using terms of your application. If, then, you can show that the axioms are "true" in terms of your application, then you know that all theorems, and all methods of solving problems based on those theorems, still work!

10. Sep 15, 2007

### Hurkyl

Staff Emeritus
One of the characters of mathematics is precision of thought; one of its important applications is the process of taking some fuzzy, intuitive idea and transforming it into a precise, explicit mathematical idea.

Among the benefits of this process is:
. A precise, explicit idea is easier to convey to others
. A precise, explicit idea can be systematically analyzed to discover its limitations
. A precise, explicit idea can be expaned to much greater generality than our intuition could have imagined

11. Sep 15, 2007

### granpa

geometry is multi-dimensional math. it is math with an extra axiom defining the 'hypotenuse'. in our universe a^2+b^2=c^2 but it can be, within certain limits, anything.

12. Sep 16, 2007

### HallsofIvy

Staff Emeritus
That is simple nonsense!

13. Oct 3, 2007

### trueuniverse

My science dictionary says the following.
Mathematics: science of relationships between spaces.

My definition is that math is the language of measurements.

14. Oct 4, 2007

### rook_b

I'm curious HallsofIvy, if multiplication is a fundamental operation how is its use defined for irrational numbers?

15. Oct 4, 2007

### arildno

Well, you could go back and read Dedekind's work on that in his construction of the reals in terms of cuts.

16. Oct 4, 2007

### D H

Staff Emeritus
Mathematicians use theorems and proofs (mathematical rigor) while scientists use theories and experiments (the scientific method). Only a small part of mathematics follows the scientific method.
What does knot theory or category theory (to name but two) have to do with measurements?

To be slightly less serious, I'll add that mathematics is the one branch of logic that involves the use of a wastebasket.

17. Oct 5, 2007

### coberst

Math is the study of pattern.

18. Oct 21, 2007

### n1mrod

In college, you learn that:
Biology is applied Chemistry
Chemistry is applied Physics
Physics is applied Maths
and Maths, it's something else..

19. Nov 14, 2007

### viet_jon

that made me think....I also think of math as a universal language....but is it really?

i guess it should be no?

20. Nov 15, 2007