Is this a fair description of mathematics (in one sentence)?

[tgt]

Mathematics is a vast subject with a very long and rich history divided into many branches that usually begins by making precise and defining new intuitive rational concepts (often inspired by the perceived world or from existing mathematical or scientific theories) and develops by employing logically sound and consistent rules and methods, producing results that are often interesting, interconnected, surprising and useful.

How does that sound (for one sentence) to this big topic?

 

[jedishrfu]

I think your sentence is very hard to read.

To me, mathematics is beautiful. Mathematics is about studying patterns in all forms. It is about making provable assertions that can lead to new mathematical discoveries.

Even my sentence doesn't do justice to what mathematics is.

What are you writing about?

 

[Windadct]

There are so many descriptions of Mathematics - where did you come up with this? -- I prefer to think of mathematics as a language, it is just a tool we use to understand the world (universe) around us - as well as many man made systems, like finance, etc. It seems you are trying to use a lot of vague wooring to describe something that is by nature - not vague.

 

[micromass]

Mathematics is like porn, hard to define but you recognize it immediately when you see it.

 

[tgt]

What are you writing about?

Many people have asked me the question 'what is mathematics'? Even I asked this question seriously in first year uni and read many books on the subject. I've now studied maths for over a decade at uni and now have a much better idea of what mathematics is about. I just want to give a non biased, and a working rather than a philosophical description of what maths is about. I also want to make it very brief in one sentence but as informative as possible.

 

[tgt]

There are so many descriptions of Mathematics - where did you come up with this? -- I prefer to think of mathematics as a language, it is just a tool we use to understand the world (universe) around us - as well as many man made systems, like finance, etc. It seems you are trying to use a lot of vague wooring to describe something that is by nature - not vague.

I've studied mathematical logic before and many of the ideas described came from there. You seem to come from an applied maths background?

 

[jedishrfu]

This is why I like the use of patterns in describing mathematics as it covers geometry, number theory and pretty much whatever discipline you throw at it.

However, Micro's post really hits the point but you'd have to be careful where you'd use it.

 

[Windadct]

Well - yes I am an engineer, but what does that have to do with the reply. You did not ask us to critique a statement like "this is what mathematics is to me" - or a personal narrative as your reply implies. Your topic statement is pretty neutral, as applying to everyone - and in that case the answer, IMO - should be more fundamental. I can say modern art is "a vast subject with a very long and rich history divided into many branches that usually begins by making precise and defining new intuitive rational concepts" - but Modern art is subjective and interpretive - two words that are NOT what mathematics is about. Using the Def of Language - I can not see how my statement is off base. We use English to communicate - yet through poetry for example it can be used to "produce(ing) results that are often interesting, interconnected, surprising and useful." - So this language can be precise AND it can be beautiful.

I do not, at all, disagree with your description, but it would not be the first, or single sentience description. So perhaps your query is out of context - who is your audience ?

 

[Mark44]

Mathematics is a vast subject with a very long and rich history divided into many branches that usually begins by making precise and defining new intuitive rational concepts (often inspired by the perceived world or from existing mathematical or scientific theories) and develops by employing logically sound and consistent rules and methods, producing results that are often interesting, interconnected, surprising and useful.

How does that sound (for one sentence) to this big topic?

Content aside, an editor would have a field day with this. For one thing, the sentence is waaaaaay too long with a number of ideas jumbled together that don't make sense, as written.

• "very long and rich history divided into many branches..." -- the history is divided into many branches?
• "divided into many branches that usually begins ..." -- the clause starting with "that usually begins" desperately needs a comma. This clause modifies "history," but without the comma, seems to modify "branches," thereby disagreeing in number between "branches" (plural noun) and "that ... begins" (singular verb).
• "making precise and defining new intuitive rational concepts" -- making precise what? This would sound better and flow better in the opposite order. You have to define things first before you can make them precise.

Indeed, mathematics is a vast subject, so what's the point of trying to define it in one sentence?

 

[Svein]

Mathematics is a study of

1. What is true and can be proved true
2. What is false and can be proved false

Unfortunately, since Kurt Gödel proved his famous theorem, we have to deal with two inconvenient categories

• What is true, but cannot be proved true
• What is false, but cannot be proved false

And so we end up with Shakespeare:

There are more things in heaven and earth, Horatio,
Than are dreamt of in your philosophy.
- Hamlet (1.5.167-8), Hamlet to Horatio

 

[micromass]

Mathematics is a study of

1. What is true and can be proved true
2. What is false and can be proved false

Unfortunately, since Kurt Gödel proved his famous theorem, we have to deal with two inconvenient categories

• What is true, but cannot be proved true
• What is false, but cannot be proved false

I'm sorry. I see your point but I do not accept your definition. Consider the following neat deductive system from Underwood's "Mathematical Cranks"

Primitive Concepts
Arthroid
Order
Junction
Exterior
Bounding
Occupy 1. To each arthroid is assigned a positive integer which is called the order of the arthroid.
2. An arthroid may be said to be the junction of some set of arthroids.
3. An arthroid or set of arthroids may or may not be exterior.
4. If an arthroid or set of arthroids is not exterior it may be fully bounded: if it is exterior it is either partially unbounded or fully unbounded.

Defined Concepts
5. An arthroid of order 1 is called a plere. An arthroid of order two is called a bine. An arthroid of order three is called a trine. An arthroid of order 4 is called a quane.
6. If an arthroid of order A is the junction of a set S of arthroids, then S is said to join to form A. (Symbolically, S >- A or A -<S.) Any subset S' of S is said to participate in joining to form, A, and A is said to be the co-junction of Sf. Symbolically, S' r- A or A-\S'.
7. A set of arthroids is said to fully bound an arthroid A if and only if Ar- every arthroid in S and the arthroids in S are of order one more than the order of A. If A is not exterior and S contains some but not all of the arthroids, of order one more than the order of A, which A participates in joining to form, then A is said to be partially bounded by S. If A is not exterior and if S contains all of the arthroids, of order one more than the order of A, which A/-, then A is said to be fully bounded by S. If A does not s- any arthroid it is said to be unbounded.

Axioms
In the following axioms S and S' will always denote sets of arthroids, while A, A\, A2 etc. will denote single arthroids.
I. If S>- A, then the arthroids in S must all be of the same order, and this must be less than the order of A.
II. Given any two positive integers m and n, with m < n, every arthroid of order n (n-arthroid) is the junction of precisely one collection of m-arthroids. This collection contains (^) arthroids. (See explanation.)
III. If A2r-A\ and S>-A\ where the order of A2 is greater than the order of the arthroids in S, then there is a subset Sf of S such that S'>-A2.
IV. Suppose S >- A where S is a collection of m-arthroids and A is an n- arthroid. Suppose p is an integer such that m < p < n. Then every subset of (£) m-arthroids in S joins to form a p-arthroid which participates in joining to form A.

THEOREM A
If a plere participates in forming a trine, it participates in forming at least two bines.
Proof:
Let Pi be the plere and let T be the trine. By II, T is the junction of precisely 3 pleres. By hypothesis one of these is Pi, and we denote the others,P2, P3. By IV, (Pi,P2)>- a bine which ^-T; call this Bx. By IV, (Pi,P3)>- a bine which r-T\ call this B2. Therefore Pi^-Pi and P\^-B2. (Note: Pi ^ B2 since by II a bine is the junction of precisely one pair of pleres.) The next four theorems are:
B: If two pleres do not form a mutual bine they do not form a mutual trine.
C: A plere that participates in joining to form a bounded bine also participates in joining to form some other bine.
D: If two pleres form more than one mutual bine, then no two of those bines are adjacent.
E: If three bines join to form a trine, each of those bines is the junction of two of the same three pleres that join to form the trine.

Here we have defined a very rich deductive system and proved theorems to be rigorously true. Is this mathematics? You might stick to your guns and say that yes, this is mathematics. I say no, this isn't mathematics. Mathematics is supposed to be about something. It's not just meaningless deductions like these. There is supposed to be a goal, interrelations, connections with some reality. None of this is to be found here. No mathematician would want to study his system.

 

[tgt]

Mathematics is like porn, hard to define but you recognize it immediately when you see it.

This is why I like the use of patterns in describing mathematics as it covers geometry, number theory and pretty much whatever discipline you throw at it.

However, Micro's post really hits the point but you'd have to be careful where you'd use it.

Mathematics is like porn, hard to define but you recognize it immediately when you see it.

Or the old mathematics is like love, the idea is simple but it can get complicated.

 

[tgt]

Well - yes I am an engineer, but what does that have to do with the reply. You did not ask us to critique a statement like "this is what mathematics is to me" - or a personal narrative as your reply implies. Your topic statement is pretty neutral, as applying to everyone - and in that case the answer, IMO - should be more fundamental. I can say modern art is "a vast subject with a very long and rich history divided into many branches that usually begins by making precise and defining new intuitive rational concepts" - but Modern art is subjective and interpretive - two words that are NOT what mathematics is about. Using the Def of Language - I can not see how my statement is off base. We use English to communicate - yet through poetry for example it can be used to "produce(ing) results that are often interesting, interconnected, surprising and useful." - So this language can be precise AND it can be beautiful.

I do not, at all, disagree with your description, but it would not be the first, or single sentience description. So perhaps your query is out of context - who is your audience ?

My audience are beginners and math undergraduates at most.

 

[tgt]

Mathematics is a study of

1. What is true and can be proved true
2. What is false and can be proved false

Unfortunately, since Kurt Gödel proved his famous theorem, we have to deal with two inconvenient categories

• What is true, but cannot be proved true
• What is false, but cannot be proved false

But Godel constructed very 'unnatural' scenarios. I'd be much more impressed if there was a 'genuine' maths problem that falls into Godel's category.

 

[micromass]

But Godel constructed very 'unnatural' scenarios. I'd be much more impressed if there was a 'genuine' maths problem that falls into Godel's category.

The axiom of choice and the continuum hypothesis are not unnatural at all!

 

[tgt]

Content aside, an editor would have a field day with this. For one thing, the sentence is waaaaaay too long with a number of ideas jumbled together that don't make sense, as written.

• "very long and rich history divided into many branches..." -- the history is divided into many branches?
• "divided into many branches that usually begins ..." -- the clause starting with "that usually begins" desperately needs a comma. This clause modifies "history," but without the comma, seems to modify "branches," thereby disagreeing in number between "branches" (plural noun) and "that ... begins" (singular verb).
• "making precise and defining new intuitive rational concepts" -- making precise what? This would sound better and flow better in the opposite order. You have to define things first before you can make them precise.

Indeed, mathematics is a vast subject, so what's the point of trying to define it in one sentence?

I only wish someone had written something like this when I was an undergraduate. I loved maths but didn't know what I was studying.

 

[mathwonk]

i like the discussion by euler on page 1 of his elements of algebra, as to what mathematics is. it may not capture it all but it is useful anyway.

https://archive.org/stream/elementsofalgebr00eule#page/n5/mode/2up

 

[pwsnafu]

But Godel constructed very 'unnatural' scenarios. I'd be much more impressed if there was a 'genuine' maths problem that falls into Godel's category.

See Goodstein's[/PLAIN] theorem and Paris-Harrington theorem.

Mathematics is like porn, hard to define but you recognize it immediately when you see it.

Can't forget this thread:

She told me that "back-in-the-day", non-applied mathamaticians looked down upon applied mathamaticians as the working class. She told me that she was about halfway inbetween the 2 opinions, although the sections in the book she is choosing to skip tell a different story. She referred to math as "mental msturb...ion"

 

[Ssnow]

I believe that your description is complex to read. I don't know an exact definition but Mathematics, as the poetry, uses the essentials to say much and I think also the definition must respect this feature ...

 

[Svein]

Here we have defined a very rich deductive system and proved theorems to be rigorously true. Is this mathematics? You might stick to your guns and say that yes, this is mathematics. I say no, this isn't mathematics. Mathematics is supposed to be about something. It's not just meaningless deductions like these. There is supposed to be a goal, interrelations, connections with some reality. None of this is to be found here. No mathematician would want to study his system.

Have you ever read the axiomatic definitions of a first-order system?

 

[micromass]

Have you ever read the axiomatic definitions of a first-order system?

Yes. And I definitely do not think that mathematics is in any way synonomous with "the study of first order systems" or anything similar.

 

[Svein]

Yes. And I definitely do not think that mathematics is in any way synonomous with "the study of first order systems" or anything similar.

I did not say that, but the axiomatic definitions of a first-order logic system (https://en.wikipedia.org/wiki/First-order_logic) is at least as convoluted as your example.

 

[micromass]

I did not say that, but the axiomatic definitions of a first-order logic system (https://en.wikipedia.org/wiki/First-order_logic) is at least as convoluted as your example.

I find it very easy to understand this definition and to get a feel for it. I do not all feel the same way with my example. I don't think my example counts as mathematics at all.