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 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!

 

[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 philosphy) 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?

 

[JoeDawg]

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

You britons are a backwards lot.

 

[D H]

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

Its one of those divergent words. American English uses math. The Queen's English, being ever so proper, wouldn't deign to use such uncouth shorthand, be it maths or math. Brits who wouldn't deign to speak the Queen's English use maths.

What I want to know is how "universal" maths is, considering the only animals that apply it are us and chimps.

Ancient lore and many modern studies hold that crows and their kin do count.

 

[dst]

Its one of those divergent words. American English uses math. The Queen's English, being ever so proper, wouldn't deign to use such uncouth shorthand, be it maths or math. Brits who wouldn't deign to speak the Queen's English use maths.


Ancient lore and many modern studies hold that crows and their kin do count.

There's a difference between physical counting, and abstract maths. I mean, our brains HAVE to count to do everything in life, so that's a given, but can other animals think abstractly?

 

[JoeDawg]

There's a difference between physical counting, and abstract maths. I mean, our brains HAVE to count to do everything in life, so that's a given, but can other animals think abstractly?

Rudimentary abstract thinking is certainly observable. Dolphins seem to recognize 'their name', chimps can use simple sign language. Many animals can recognize themselves in mirrors.

Its not algebra, but math is really just another language, with descriptors, its own syntax, logic... etc..

 

[j0nis0n]

if aliens from an advanced civilization came to Earth and tried to communicate with us using math, do you think we will be able to understand? for example: being shown on the history channel, an astronaut brought back footage of an object in space catching up with the space shuttle, stopping, then moving the other way. that action just broke the laws of physics. so will we learn how to do the same if the aliens tried to teach us the math and physics behind it? if we do understand and learn all of this from the aliens, i think it would be safe to say that math is a universal, absolute language, not created (instead discovered) by humans.
and to the previous post: the reason why chimps can use sign language and dolphins can remember their names are because humans have embedded "human" language in their heads. we made them follow our way of things.

 

[JoeDawg]

i think it would be safe to say that math is a universal, absolute language, not created (instead discovered) by humans.

I don't think its safe to say that at all. Math describes concepts that aliens would probably recognize, but so does Russian, so does Java. Math is simply a specialized language used for a specific purpose. Its the underlying meaning that is important however, not how it is described.

We... those of us not wearing tinfoil hats... have no clue what aliens would be like.

 

[al-mahed]

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.

Wittgenstein is hard to read... I read "philosophycal investigations", if you don't understand a point the following arguments become obscure, you need to back and back again in order to follow the ideas.

I think this particular sentence that you posted means that to Wittgenstein mathematics is in the mathematicians head, like an invention that could be different if another been endowed with intelligence thought on the subject.

I disagree of that because if it went true we would never have theorems that definitively are true necessarily implicating in other. We would have degrees of different truths instead of different forms of expressing the same idea. In the mathematics I never saw two conflicting ideas being valid for the same conclusion.

 

[pace]

I maybe find it weird to talk about mathematics. To me it's as if mathematics is there to tell us what is alike, and language that we and things are different and that we should enjoy it. It's as if when you do mathematics it takes control of language, and when you do language it takes control of mathematics.

Yeah, I also found Wittgenstein hard to read, so I started on Plato Nah, well, I've read Spinoza's Ethics also. I started on his Tractatus myself, but as we all know that's not easy reading. I find it fun how reading such twistingly stuff can be so gripping. But I think that it would help a lot by knowing all the propositions by head before you start to have an oppinion about his work(like with Spinoza). And oddly to me that doesn't seem so unwanting to do. Logic freak I guess.
On the critical side I find him a little depressive. Like if he knew he was depressive and were a little embarresed that it showed in his early writings. Ie. he comes at early age at large to very subjective conclusions based on these. Not that I get things and enjoyment out from reading him.

------------------------------------------

I wonder, when it's discovered that 'only humans chimps and dolphins 'apply' math', how you are going to ponder why it is so. And when only 'humans, chimps, dolphins and gorillas do so' , and when.. Study magpies for some time. Just look at them and see all the crazy things they'll do and you're going to ponder when these guys are getting arms popping out of their sides. I think they'll could come up with some theorems, let alone a whole lot of fun stories. I've seen them play around a box at a kindergarden, play what looked to be catch. Picking up a lock on a gate. It's fricking scary.

 

[al-mahed]

Pace, I'm a little "slow" in english, this finishes phrase :

Study magpies for some time. Just look at them and see all the crazy things they'll do and you're going to ponder when these guys are getting arms popping out of their sides. I think they'll could come up with some theorems, let alone a whole lot of fun stories. I've seen them play around a box at a kindergarden, play what looked to be catch. Picking up a lock on a gate. It's fricking scary.

I did not understand very well...

 

[pace]

Oh I'm sorry, that part wasn't directed at you but to others at here. :)

 

[mubashirmansoor]

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!

This has been exactly what I have thouht all through my life, but what about Kurt Godel's incompletness theorem? ...

This theory contradicts with our thouhts...

 

[knine143]

This is a response to the original post.

I agree that math is a study of pattern, but there is a very specific reason for this.

When we look at everything around us, it is obvious that there are patterns. For example, when there is one proton in an atom, its hydrogen, two protons becomes helium and so on. That is the pattern of atoms. The pattern of all trees is that they branch continuously all the way up to their leaves. The pattern of humans is that we live, and reproduce to create more of ourselves.

These are all specific patterns that apply to specific things. However, the pattern of everything that exists is math. Nothing that exists breaks any laws of mathematics. For example, everything that has a size of (t^3) where t is measured in seconds changes size at a rate of 3(t^2). That is simply the way things are.

So, in conclusion, math is the one pattern common to all things. It is the universal pattern. Math is the pattern of existence. Everything that exists follows the pattern of mathematics.

 

[ripcurl1016]

Math is make believe to make sense of something so it works. The pieces to a puzzle...sure it makes a picture but is it the right picture or the right pieces.

 

[knine143]

What are you talking about ripcurl?

 

[JoeDawg]

So, in conclusion, math is the one pattern common to all things. It is the universal pattern. Math is the pattern of existence. Everything that exists follows the pattern of mathematics.

No, math is a language, we can easily describe things with math which do not exist.

2 + 2 = 5

The fact that it can be represented with math, doesn't mean its true.

Its the underlying concepts that either agree with reality or not.

If I have one apple and I get another apple, I now have two apples.

That is true, it involves no math, simply english and observation.

 

[knine143]

There is a difference between math as a concept and math as syntax (or language).

The symbol for the number 2 is part of the syntax that we use, but we're really talking about the concept of two things.

The concept of math as a whole is not a language, but a universal pattern.

You use the example of having one apple, obtaining another, and then having two. This is described by math. 1+1=2.

Therefore, your example fits the pattern of math, just like everything does. There is nothing that has the property that you can have one of it, obtain another, and the not have two. EVERYTHING follows the pattern of math.

The language you are talking about is simply the syntax we use, such as the symbol "+" or "3". That is not what math really is, but rather how we are able to express it to each other.

 

[JoeDawg]

There is a difference between math as a concept and math as syntax (or language).

The symbol for the number 2 is part of the syntax that we use, but we're really talking about the concept of two things.

The concept of math as a whole is not a language, but a universal pattern.

You use the example of having one apple, obtaining another, and then having two. This is described by math. 1+1=2.

Math is an abstraction from reality.

Therefore, your example fits the pattern of math, just like everything does.

No.

1+1=2 fits the pattern that is reality. We keep the concepts that describe reality and discard the ones that do not. We use mathematics to describe these concepts.

There is nothing that has the property that you can have one of it, obtain another, and the not have two. EVERYTHING follows the pattern of math.

Really? Show me this 'everything' you speak so confidently about. Because it sounds like you are making this up as you go.

I observe certain patterns in nature, from those patterns I develop abstractions; concepts and rules. When I want to communicate or develop these rules/concepts further I use math to organize my thoughts. Its language, pure and simple. Then I check my results against reality. 1+1=1 exists in math, but it doesn't correspond to reality, so its not something we consider true.

Reality is the yardstick, math is just formalized generalizations derived from observing reality.

 

[ripcurl1016]

EVERYTHING follows the pattern of math.
QUOTE]

Everything follows the pattern of math because we created the pattern in the first place to follow everything.

 

[jostpuur]

We have experimentally found, that coming up with definitions, and proving theorems related to them, in certain rigor style, has worked well. It is difficult to give precise definition to mathematics for the same reason why it is also difficult to give precise definition to some direction in fashion or in culture. It is a matter of style.

(This was from the point of view of comparing mathematics to other sciences. Not really related to the few previous posts, which attempt to be more depthful.)

 

[jostpuur]

So, in conclusion, math is the one pattern common to all things. It is the universal pattern. Math is the pattern of existence. Everything that exists follows the pattern of mathematics.

It could be this is true, but I don't think it helps in the problem of some fundamental definition of the mathematics (or more precisely, the definition of the thing that mathematics is study of). Now we have to wonder what precisely is the concept of "pattern", and I think it is equally difficult/mysterious as the concept of mathematics (similar correction here) itself.

No, math is a language, we can easily describe things with math which do not exist.

knine143's claim doesn't have so clear meaning that it could be proven wrong. Only way to prove it wrong is to first guess what its meaning was more precisely, and guess it so that the more precise meaning is wrong.