Wikipedia:Reference desk/Archives/Mathematics/2012 June 30

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June 30

Components of Mathematics?

Is all of math a form of dividing, adding, subtracting, and multiplying? I have completed trigonometry, algebra I and II, geometry, precalculus, and calculus. Although I found the subject matter interesting and beautiful, I found later subjects complicated an intimidating.

I honestly have no intention taking advanced math courses such as calculus 3, differential equations, and linear algebra.

My question is, since mathematics is sometimes the only language that is universal, should I feel insecure not willing to take higher level math courses as it does not quite fit in with my daily activities? — Preceding unsigned comment added by 184.88.243.39 (talk) 02:05, 30 June 2012 (UTC)

If you need it, you should learn it. If you like it, you should learn it. If you don't need it and don't like it, I don't see why you need to learn it. Regarding the first question, there is a strong argument that all mathematics is ultimately set theory -- it is certainly not true that all mathematics is dividing, adding, subtracting, and multiplying. That is only arithmetic, hardly mathematics at all. Looie496 (talk) 02:24, 30 June 2012 (UTC)

Since you've had the courses you mentioned, you probably already know that there's more to math than +,-,*,/ (geometry and calculus come to mind). The courses you've mentioned should leave you fairly well prepared for life, mathematically speaking. A bit of statistics would be useful for almost anyone. Aside from that, what Looie496 said: If you need it or like it, learn it; if not, don't.--Wikimedes (talk) 02:36, 30 June 2012 (UTC)

If you don't need math in your job, the most complex math that the average person really needs to know is how to deal with interest rates and compounding. People who don't understand that stand to lose lots of money. StuRat (talk) 02:56, 30 June 2012 (UTC)

If you "found the subject matter interesting and beautiful" you should not be discouraged by the later subjects being complicated and intimidating. Even the later subjects are interesting and beautiful if you give it a chance. The capability of mathematics is truly impressive. The joy of understanding and the fun of solving problems makes mathematics worthwhile. Bo Jacoby (talk) 16:57, 2 July 2012 (UTC).

There are definitely some forms of mathematics that don't involve numbers and the arithmetical operations that appear on calculators. The study of computer science is a refuge for people who like formalism and proofs, but not numbers. The lambda calculus, for example, is a mathematical system that (in its simplest form) manipulates functions. (The functions are naturally functions from functions to functions... it is turtles all the way down.) There are also a multitude of different logics, each capturing rules for proving statements of a particular character. There's type theory, which is the application of logic to programs. And then there's denotational semantics, which (from what I've heard) uses concepts from topology to explain the meaning of programs. And so on...

I was a math-oriented student in high school, but in college, I drifted toward computer science, and I'm happy with my choice. Now I study names, which are like numbers, except that the only thing you are allowed to do to them is compare them for equality and create new ones. It's a ridiculously simple concept, until you put those names into programs, and then doing the Right Thing is a major challenge. I'm tempted to blather on, but I probably shouldn't... Paul (Stansifer) 19:41, 3 July 2012 (UTC)

Is Mathematics discovered or invented?

I looked online at a variety of discussions, and just can't seem satisfied with a particle answer.

Obviously everything in nature is set, it is only when we use numbers we are actually analyzing the elements in the realm of the universe.

So did we just come up with things or are we using what already exists in nature. — Preceding unsigned comment added by 184.88.243.39 (talk) 02:09, 30 June 2012 (UTC)

Most of mathematics exists in nature. For example, the ratio of the perimeter of a circle to it's diameter is 3.1415926..., everywhere, even on an alien planet.

However, there are some conventions in mathematics, like our use of base 10, that an alien intelligence might find odd (even though it's clearly even). :-) StuRat (talk) 02:59, 30 June 2012 (UTC)

But one could argue that there exist no perfect circles (nor perfect lines, for that matter) in nature, and thus the value of pi derived from Euclidean circle is a human construct and an aberration. 152.97.171.80 (talk) 05:42, 11 July 2012 (UTC)

This is a question that mathematicians have been asking themselves and each other for a long time, with no clear answer. Mathematical realism is a strong position that mathematical truths are discovered. See below in that article for other viewpoints. There is no real consensus among mathematicians, but it's not something that most of us think about from day to day, and it doesn't really affect how most researchers work. Staecker (talk) 11:19, 30 June 2012 (UTC)

Here's my amateur take on the question. Notation and other conventions are clearly invented. When you go deeper to the conceptual level, I think some of it is invented and some is discovered. For example, number is surely a discovered idea; it's impossible to imagine mathematics without a concept equivalent to that of number, and number is a very salient property of our reality. However, smooth functors are probably invented. In fact, it would be easy to argue that all of category theory is. You could recast the mathematical content of category theory in a completely different form and solve the same problems, and you don't see categories or morphisms in nature. Developments such as rational trigonometry suggest that more mathematics than you'd guess might be invented; and the fact that you can often solve one problem in many ways suggests to me that our invented mathematical description of reality contains redundancies. Ultimately, the mathematics we've constructed is purposed with approximating or modelling the laws underpinning reality. When that construction obviously coincides exactly and somewhat uniquely with these laws – as in the case of arithmetic or number – we can say that piece of mathematics has been discovered; in other cases, it has been invented. —Anonymous Dissident^Talk 13:07, 30 June 2012 (UTC)

It seems that I asked more or less the same question a year ago. --Theurgist (talk) 14:05, 30 June 2012 (UTC)

But did you (re)invent the question or (re)discover it? —Tamfang (talk) 22:07, 1 July 2012 (UTC)

I think that the concepts and definitions are invented, while the theorems and proofs are discovered. Bo Jacoby (talk) 22:18, 30 June 2012 (UTC).

What about $n^{0}=1$ and $0!=1$? Are they defined/invented, or calculated/discovered? --Theurgist (talk) 22:42, 30 June 2012 (UTC)

Surely to a large extent the fuzziness of the boundary between the categories "discoveries" and "inventions" will bedevil any answer to this question? So in a sense, trying to answer it is really exploring the definition of these categories. E.g., are genes patentable? The specific conceptual frameworks in which all the necessary concepts are constructed is largely an accident of evolution of our mental hardware and the building of this framework through experience. It feels as though there is some immutable "logical universe" that we are approximating through mathematics that we could view as a scaffolding around the reality, but along the way we can make certain arbitrary choices (e.g. of definition) along the way that affect the form of specific answers (Is 1 a prime? What is 0⁰?). The very process of trying to put everything into categories is a clumsy discretizing mechanism to help our mental framework cope with our mental modelling of the world. Until we have a firmer grasp on our conceptual framework and its limitations, the answer to the OP's question will surely perforce remain fuzzy. — Quondum^☏ 07:02, 1 July 2012 (UTC)

What about $n^{0}=1$ ? The recursive definition

$n^{1}=n$

$n^{1+k}=n\times n^{k}$

is an invention. It works for k=1,2,3,... The observation that this definition can be extended in a uniquely beautiful way to include the case k=0 like this

$n^{0}=1$

$n^{1+k}=n\times n^{k}$

is a discovery. Bo Jacoby (talk) 13:04, 1 July 2012 (UTC).

Nominalism says maths is all made up, and that mathematical objects merely make up a useful language for discussing things which do exist. This is a valid ontological viewpoint to take, though I would suspect it to be distasteful to the clientèle of the Maths Ref-Desk. That said, it is not an uncontroversial one. For interests sake I once attended a lecture by a physicist who had undertaken the task of writing all known laws of physics without reference to mathematical objects. He had been reasonably successful and at that time had successfully completed descriptions of classical and quantum mechanics, and gauge field theory. Arguably though, whether or not such objects exist is inconsequential to the validity of statements made using mathematics. Hope this helps!