What is mathematics?

[ajb]

A quick "google" with give you several definitions of mathematics. To me they tend to be necessary, but far from sufficent.

 

As example; Mathematics is the abstract study of numbers, shape, structure and change.

 

 

Numbers are part of mathematics, so are simple geometric shape, by structures we mean patterens and relations between them, and differential calculus ("rates of change") are all parts of mathematics.

 

This definition seems okay, but a little vague and it may not be very clear what we mean by a structure. Also, as mathematics evolves, solutions to problems often come from outside the area the problem was initially defined, any defintion must be able to include any future mathematics.

 

It maybe better not to define mathematics by what it studies by how it studies them. A first attempt at this would be to reduce all mathematics to logic, but this fails.

 

Mathematics as a science is also a tricky issue. There are many parallels between how a mathematician works and the philosophy of the scientififc method.

 

Anyway...

 

My question to you is what is mathematics?

[Myuncle]

I am an amateur ajb, but just by googling for the word "math", the definition looks always poor and vague, and when a definition is vague, it means that it can be manipulated and used as a weapon to boost your ego. There are so many things to say. Math is an agreement, it can be easily accepted by everyone, as long as we agree that in theory a unity is identical to another unity, 1=1 (all theory, but in practice it might not even exist, since an orange is not exactly identical to an other orange, an atom or quark might not be exactly identical to an other atom or quark). Everything is a consequence of that agreement. That's a very useful agreement between humans, very convenient, that allows us to measure and quantify everything, and to agree on measurements and quantities. Mammals and birds, particularly parrots, can understand various numerical concepts.

Everything can be calculated very slowly with additions and subtractions, and everything else is a shortcut of additions and subtractions. The origin of multiplication was the need to create a shortcut for long tedious additions, and the origin of division was to create a shortcut for long tedious subtractions, and so on. These shortcuts are very useful, they make our life more comfortable.

My new question is: when should we say "stop" to these shortcuts, are they really useful? Do they make our life simpler and more comfortable? How many of these shortcuts are really necessary? How many of them are only created to win a science award or to sell books? How many of them are used, and how many of them are never used? I hope math prodigies and experts on this website can share a honest answer.

Math was born to make our life simpler (just like burocracy and banks...yes...), but if not used porperly it will just make our schools more complicated than needed, discouraging students, and demotivating them.

[studiot]

Well I know that the fattest knight at King Arthur's round table was called Sir Cumference, becaue he ate too much pie.

[John Cuthber]

You may take some consolation from the fact that other professions have similar difficulty with definitions.

http://pubs.acs.org/doi/abs/10.1021/ac00085a709?journalCode=ancham

[ajb]

Everything is a consequence of that agreement.

Stripped to its bare bones, mathematics can be viewed in that light. One proves theorems given some starting axioms that we agree on. But that, to me, seems to undersell what mathematics really is.

 

Everything can be calculated very slowly with additions and subtractions, and everything else is a shortcut of additions and subtractions. The origin of multiplication was the need to create a shortcut for long tedious additions, and the origin of division was to create a shortcut for long tedious subtractions, and so on. These shortcuts are very useful, they make our life more comfortable.

Okay, mathematics grew out of the need of commerce to keep track of things. Indeed multipliction most likely has its origins in tedious additions and so on.

 

But there is more to mathematics than just basic operations on numbers. It is true that the real numbers, or collections of real numbers can be useful in representing more abstract ideas, mathematics in no way is constrained to just numbers.

 

Simple plane geometry is a good example here. The Greeks developed this mathematical theory synthetically. That is "without reference to anything else using logical deductions". Then Decartes applied numbers to geometry by defining the notion of local coordinates and so analytical geometry was born. But even in anaytical geometry, points and lines "exist" independent to the "numbers" used to represent them.

 

As mathematics became more and more abstract a simple connection with numbers was generally lost.

 

My new question is: when should we say "stop" to these shortcuts, are they really useful? Do they make our life simpler and more comfortable? How many of these shortcuts are really necessary? How many of them are only created to win a science award or to sell books? How many of them are used, and how many of them are never used? I hope math prodigies and experts on this website can share a honest answer.

This is more a question of the usefulness of mathematics. This is a seperate question, and one we could go into in another thread.

 

That said, I doubt many people here will need convincing of the usefulness and applications of mathematics.

 

Math was born to make our life simpler (just like burocracy and banks...yes...), but if not used porperly it will just make our schools more complicated than needed, discouraging students, and demotivating them.

Right I take your point. But this is now a question of mathematics education, which is an awkward thing. Not everyone will become pure mathematicians and this needs relfecting in any syllabus. My own experinces are in post compulsary education, so 6th form and university undergrads. Here one still needs to take care of the wide range of motivations each student has for studying mathematics.

 

But again, this is really another question for another thread.

Well I know that the fattest knight at King Arthur's round table was called Sir Cumference, becaue he ate too much pie.

Old joke, but still funny!

You may take some consolation from the fact that other professions have similar difficulty with definitions.

http://pubs.acs.org/doi/abs/10.1021/ac00085a709?journalCode=ancham

Thank you for the link.

Edited August 19, 2013 by ajb

[HalfWit]

Well I know that the fattest knight at King Arthur's round table was called Sir Cumference, becaue he ate too much pie.

A friend of Sir Cumflex, who always wore a little pointed hat.

Edited August 19, 2013 by HalfWit

[studiot]

 

Old joke, but still funny!

 

 

Do I look old?

I don't feel old.

 

I don't feel anything at all until noon

and then it's time for my nap

ajb, my apologies for the humerous diversions.

 

I consulted my trusty shorter Oxford, volume 1 and it had this to say

 

mathematics probably after French "mathematiques".

 

Originally treated as plural, frequently with the.

 

The sciences or disciplines of the quadrivium, collectively.

 

Later these and optics, architecture navigation etc.

 

Now treated as singular

 

The abstract deductive science of space, number, quantity, and arrangement

 

Colloquial abbreviation maths (N America, Math)

 

I think that basically covers everything discussed so far, and shows why the OED is so reliable.

[ajb]

ajb, my apologies for the humerous diversions.

Not at all, any humor is welcomed.

 

The abstract deductive science of space, number, quantity, and arrangement

I think just about all dictionary definitions of mathematics are equivalent to this. What is meant by arrangment? I guess this is the same as "structure".

[studiot]

I doubt anyone's list is exhaustive.

 

Personally I think of structure as being more in the way of the formal axioms eg a space with one binary operation or two. So structure can be induced or inherited.

 

Whereas I think of arrangements in terms of combs and perms or left handed v right handed or a star v convex region and so on.

But arrangements just are. So the fruit is hanging on the 5th or the 15th branch of the tree or whatever. But that is no reason to expect every tree to have fruit on those branches.

 

I think mathematics is about analysis and self consistency.

[seriously disabled]

Mathematics is made of numbers and symbols.

 

But the numbers and symbols are just pixels on the computer monitor or ink on a paper, numbers and symbols are completely human creations and as such I don't think that they can exist in the wild nature.

Edited August 24, 2013 by seriously disabled

[uncool]

The way I'd put it is everything that can be proven. Specifically, this includes statements of the form "If we make these assumptions, this is what we can prove".

[arc]

Mathematics is made of numbers and symbols.

 

But the numbers and symbols are just pixels on the computer monitor or ink on a paper, numbers and symbols are completely human creations and as such I don't think that they can exist in the wild nature.

 

Arthur C Clark hosted; Fractals - The Colors of Infinity. It forever changed my perception of the universe and mans perception of it through maths. There seems to be a simple underlying form to reality.

http://www.youtube.com/watch?v=qB8m85p7GsU

 

It poses in my mind the question; Is math ours or the universes.

Edited August 25, 2013 by arc

[ajb]

Mathematics is made of numbers and symbols.

 

But the numbers and symbols are just pixels on the computer monitor or ink on a paper, numbers and symbols are completely human creations and as such I don't think that they can exist in the wild nature.

I guess we have to be a little careful with the distinction between the objects themselves and the notation used to represent them. Though one school of thought, "formalism", would not make too much of an issue of this. Here mathematics is a formal game using symbols and have nothing to do with nature at all. It is all a game of human invention.

 

The other extreme is "realism"; mathematical entities exist independently of the human mind, thus mathematics is discovered and not invented.

 

Personally I have no idea which end of the spectrum here is closer to being right or wrong. I don't think we can ever actually decide this. But, given how mathematicians work I am tending towards realism.

 

Any thoughts?

It poses in my mind the question; Is math ours or the universes.

This is a great question. See my comments above.

The way I'd put it is everything that can be proven. Specifically, this includes statements of the form "If we make these assumptions, this is what we can prove".

At the fundamental level that is how mathematicians work. You take some starting statements and use whatever tools you have, maybe even inventing some, to derive new statement.

 

My opinion is that this is a too simple point of view that undersells the beauty and power of mathematics.

[Myuncle]

 

 

It poses in my mind the question; Is math ours or the universes.

 

Since we are part of the universe, is both. Math allows us to have an approximation of reality, without math we wouldn't have that approximation either.

[uncool]

At the fundamental level that is how mathematicians work. You take some starting statements and use whatever tools you have, maybe even inventing some, to derive new statement.

 

My opinion is that this is a too simple point of view that undersells the beauty and power of mathematics.

Just because this definition doesn't show the beauty or power doesn't make it a bad definition, though. A definition doesn't have to describe every aspect of something.

[ajb]

Since we are part of the universe, is both. Math allows us to have an approximation of reality, without math we wouldn't have that approximation either.

But is it tied to the human mind in a fundamental way? Or really any mind as I assume any intelligent aliens would have mathematics and further more we would understand and agree on this mathematics. (Mod complications of notation and any time scales to actually absorb this information etc.)

Just because this definition doesn't show the beauty or power doesn't make it a bad definition, though.

Sure.

 

A definition doesn't have to describe every aspect of something.

I think the definition in terms of "axioms -> theorems" describes how mathematicians work, rather than what is mathematics. You now have to define carefully what a mathematician is!

 

This can be a problem with any definition. You want to avoid shifting the question to an equally as hard one.

[Unity+]

 

Since we are part of the universe, is both. Math allows us to have an approximation of reality, without math we wouldn't have that approximation either.

I think the question was pertaining to whether it is a human construct or a Universal language.

 

Many mathematicians hold the platonic view of Mathematics, where nature actually has it embedded within itself and uses it as a way of developing itself to form structures such as we observe within science.

 

The other view is that though mathematics has a build up that is similar to nature, it is only a prediction to what it is observed(I hold the platonic view).

 

Popular culture within science mostly holds the platonic view because it just seems more philosophically fit that there is an underlying language behind the structure of the Universe. For one, it shows there is an underlying unity between everything whether observably connected or not. The other factor is if everything can be represented with mathematics then we know we can explain everything of the Universe in a mathematical point of view(though, this would only apply to the observable).

Edited August 25, 2013 by Unity+

[Myuncle]

But is it tied to the human mind in a fundamental way? Or really any mind as I assume any intelligent aliens would have mathematics and further more we would understand and agree on this mathematics. (Mod complications of notation and any time scales to actually absorb this information etc.)

It seems any intelligent animal or alien, would agree on math, it's always a win-win situation, any primate or parrot would agree with us on the same math. If in a distant galaxy there are aliens with only three fingers per hand, I bet they would have a base 6 numeral system, if we met each other we would agree on math, but we would have to compromise, change even the symbols, change our base 10 system, or learn their 6 base system, or compromise, agree, and create something in the middle, like a base 8 system...

[Unity+]

Just because this definition doesn't show the beauty or power doesn't make it a bad definition, though. A definition doesn't have to describe every aspect of something.

I don't think the beauty of it is the problem. The problem lies in its vagueness.

 

 

Mathematics is the abstract study of numbers, shape, structure and change.

In my opinion, the definition needs more specific explanations. The fact that it only includes numbers, shapes, structure and change just shows that it lacks an understanding of predictable additions to mathematics.

It seems any intelligent animal or alien, would agree on math, it's always a win-win situation, any primate or parrot would agree with us on the same math. If in a distant galaxy there are aliens with only three fingers per hand, I bet they would have a base 6 numeral system, if we met each other we would agree on math, but we would have to compromise, change even the symbols, change our base 10 system, or learn their 6 base system, or compromise, agree, and create something in the middle, like a base 8 system...

The problem with this assertion is we assume that everything will have humanly concepts or thoughts. For example, in our culture of how things will look on a different planet, we always seem to give a structure to the anatomy of the organisms going to be found on other planets. The problem is we can only make accurate predictions on what we have observed, and then build onto those predictions.

 

Aliens may even have a better way of handling the measurements of systems than mathematics. However, this could lead to a possibility that mathematics is simply a branch of a higher unknown study.

[ajb]

... but we would have to compromise, change even the symbols, change our base 10 system, or learn their 6 base system, or compromise, agree, and create something in the middle, like a base 8 system...

Changing base should not be a big problem.

 

But anyway, there are problems with notation across branches of mathematics here and now!

Aliens may even have a better way of handling the measurements of systems than mathematics.

Or would it just be mathematics that we have not discovered/invented yet?

[EdEarl]

One can read about, talk about, and think about a thing without ever truly understanding it; then, when one experiences something they understand what it is. For example, what is Chartreuse?

 

1. an aromatic liqueur, usually yellow or green, made by the Carthusian monks at Grenoble, France, and, at one time, at Tarragona, Spain.

[ajb]

One can read about, talk about, and think about a thing without ever truly understanding it; then, when one experiences something they understand what it is.

That sounds a lot like mathematics. Maybe we cannot define mathematics, but one knows when one is using mathematics!

[arc]

Changing base should not be a big problem.

 

But anyway, there are problems with notation across branches of mathematics here and now!

 

Or would it just be mathematics that we have not discovered/invented yet?

 

If it belongs to or is of at least this universe, what is it's totality. Can it with enough time be revealed entirely. Is our utilization of it its "true purpose" or are we just using a microscope to break rocks into smaller pieces.

[ajb]

Can it with enough time be revealed entirely.

 

Great question you pose; is there only a finite amount of mathematics and are we indeed going to be privy to all of it?

 

 

[Yevgeny Karasik]

Mathematics is the science about equal dualities. In other words, axioms of any mathematical theory can be replaced with another set of axioms, each of which asserts equivalence (or equality) of dual expressions. This fact has nothing to do with duality principles in various branches of mathematics. 

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Edited October 25, 2021 by Phi for All
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