What is the Study of Mathematics?

[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.

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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.

 

[jostpuur]

The title What is Mathematics?, and the question in the original post

What is Mathematics the study of?

are in fact different questions. It could be clearer if you make it clear which one of these you are trying to answer. My post #49 dealt with the first one, and #50 with the second one.

 

[Holocene]

Mathematics is more than just a language; it describes some general truths.

Take for example the ratio between the circumference of a sphere, and its diameter. That ratio is ALWAYS the same, regarless of the size of the sphere. It doesn't matter if it's an object the size of a glass marble, or an object the size of a star.

The ratio is ALWAYS the same, regardless of the size of the object.

 

[JoeDawg]

Mathematics is more than just a language; it describes some general truths.

Mathematics IS a language that describes general relations.

A circle is a circle, by definition. Its a generalization based on experience.
We say something is not a circle when it contradicts, or expands on, the definition of a circle.
Mathematics allows us to describe a circle very precisely.
A square is not a circle, by definition, these two definitions conflict.
A sphere is not a circle, by defintion, a sphere expands on the definition into 3 dimensions.

The definition of a circle comes about because we live in a certain type of spacetime and can perceive certain dimensions. Its a generalization, the fact we can apply this generalization to many observed things is a function of the fact its a generalization. The definition is always true, because we have defined it precisely that way.

 

[oldman]

Mathematics IS a language ...

While I agree with you as far as this, there seems to be some confusion in the replies to the question raised in the OP, namely "What is Mathematics the study of?"

This may be because other disciplines mentioned in the OP, like mechanics and biology, study aspects of the "real" world (if this indeed exists --- as the evidence of our senses suggests it does) --- whereas mathematics involves the study of properties of an abstract entity, which some assume is a structure that exists in the Platonic world, fully formed, waiting to be discovered by any creatures capable of doing so.

Others, like myself, believe that mathematics is an ephemeral construct like all languages --- whether they be invented by human beings, as with Zulu, Latin and English, or by birds, who use various twitterings for communications that must seem pretty important to them.

There is much scope for confusion here. Perhaps it would be useful to consider how
subjects like mechanics, biology and mathematics arose -- their provenance, as with works of art, so to speak --- and especially what kind of animals use them and why.

 

[yhp266]

Mathematics is what mathematicians do.

this definition is useless and also wrong. It seems to imply mathematics is a human construct.

 

[Hootenanny]

this definition is useless and also wrong. It seems to imply mathematics is a human construct.

If mathematics is not a human construct, a framework/formalism constructed by humans to solve problems, then what is it?

 

[yhp266]

My definition of mathematics is that it is merely an abstraction of ideas

My personal view is that mathematics is permanent outside of our existence.

 

[Noone]

Math is a dictator that tell's logicaly how each line must effect another line, hence every number is a line, 2=ll 1=l and in which ways can those lines effect each one another, in every way shape and form through a fixed logical dictation.

So math is the studie of how line's effect one another in everyway shape and form, bound by a logical dictation of how they must effect each one another in everyway shape and form.

Its like a lang, used like a tool, a true gift to have and use, for that of which you need or seek.

 

[al-mahed]

perhaps one can say that math is a system such that there is no exceptions ( a particular process leads to a particular result any time) and when the "rules" are followed there is no absurd conclusions according to these rules

 

[oldman]

perhaps one can say that math is a system such that there is no exceptions ( a particular process leads to a particular result any time) and when the "rules" are followed there is no absurd conclusions according to these rules

You're confusing maths with "reality" here, I think. Maths is an abstract language --- and it's constructive to give examples, rather than just express an opinion.

The simplest example of this abstract language involves the counting numbers we all use. They are an abstraction and their arithmetic is another. The language of maths is used to quantiatively describe "reality" -- (the "reality" of your "particular process", al-mahed?).

Another quite simple example most of us are familiar with is the description: "exponential increase" , where we talk of the rate of increase of something (population for example) being proportional (another abstract math concept) to the value of that something (yet another abstract math concept).

Or, to be more succint, we write down an abstraction; a differential equation, namely (with N for quantity and t for time) dN/dt proportional to N, and then solve it -- as N proportional to (the mysterious number) e raised to a power proportional to t.

One should appreciate that this sort of abstract gobbledygook, used with physics, proves --- from an evolutionary point of view --- to be the supremely useful lever that has enablied us to infest this planet with billions of individuals. Maths is one heck of an important language!

By the way, an "abstraction" is just another human concept, too. I'll bet dogs find smells both as abstract and as useful as we find, say, the counting numbers --- and that smells stimulate their doggy imaginations as much as abstractions do ours!