What is the Study of Mathematics?

[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?

 

[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.

 

[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

 

[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.

 

[arildno]

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

 

[HallsofIvy]

numbers.

So you would say that geometry is not mathematics?

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

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.

 

[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.

 

[arildno]

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

 

[HallsofIvy]

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 philosophy of mathematics- it is a subset of the "Platonist" philosophy.

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!

 

[Hurkyl]

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

 

[granpa]

So you would say that geometry is not mathematics?
.

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.

 

[HallsofIvy]

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.

That is simple nonsense!

 

[trueuniverse]

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

My definition is that math is the language of measurements.

 

[rook_b]

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

 

[arildno]

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

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

 

[D H]

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

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.

My definition is that math is the language of measurements.

What does knot theory or category theory (to name but two) have to do with measurements?

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

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

 

[coberst]

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?

Math is the study of pattern.

 

[n1mrod]

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

 

[viet_jon]

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.

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

i guess it should be no?

 

[bfitts]

Mathematics is the study of the human race. I am dead serious about this.

 

[Dragonfall]

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

Actually it's supposed to go:

Biologists think they are biochemists,
Biochemists think they are Physical Chemists,
Physical Chemists think they are Physicists,
Physicists think they are Gods,
And God thinks he is a Mathematician.

 

[nicky nichols]

the religion of hypothesis

 

[Crosson]

Mathematics is the study of our intuitions of time and space.

-Immanuel Kant

 

[JoeDawg]

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

And in English, Math is both singular and plural.

 

[Kummer]

I will begin by saying how much I dislike it when non-mathematical philosophers think about math. There are many threads in the mathematics forum like this and they belong in the philosophy forum, so one good thing about this thread is that it at least is in the philosophy subforum. I am not going to say what the answer to that question because it has been already mentioned by all knowlegable members in math so their is no point. I will however mention which posters you should pay much attention to because they certainly know what they are talking about: arildno, HallsofIvy, Hurkyl, and mathwonk (if he posts). <edit - MIH>.

My last comment is about using quotations in an argument. Just because you quote a famous charachter from history (such a philosophy) does not suddenly make your argument correct. Thus, quoting Immanuel Kant is garbage, for one thing he was not a mathematician, why would you ever listen to him then?

 

[D H]

Thus, quoting Immanuel Kant is garbage, for one thing he was not a mathematician, why would you ever listen to him then?

Not only that, he was a real pissant who was very rarely stable.

 

[nicky nichols]

yellow, as is the cover of springer verlag lecture notes in mathematics...

 

[pace]

Mathematicians are not explorers, but inventors. - Wittgenstein.

That's a bit funky line, but I don't see how he could get to that conclusion.

 

[nicky nichols]

somewhere there is research on computational irreducability of mathematical statements, which sort of tends to suggest that maths is just replacement for language, a sort of common denominator of description, hmm...

 

[dst]

And in English, Math is both singular and plural.

Just so you know, in English, "Maths" is the correct word, as an abbreviation of "Mathematics". :)

What I want to know is how "universal" maths is, considering the only animals that apply it are us and chimps. I remember someone somewhere proving that e and pi were "universal" constructs, does anyone know more about that?