what is math?

[cameron marical]

i know this is a reall general and broad question{and stupid/new to this world sounding}, im just wondering what kind of answers ill get. i love math, but when i ask myself this question, i seem to lack the answer.

[samtheflash82]

mathematics, in a general sense is the study of digits. maybe?

[D H]

Math is not the study of digits. Math is not science. Math is about constructing theorems from a set of axioms. Science is about constructing theories from a set of hypotheses. The key difference: Science is tested against reality by means of experimentation while math is tested against logic. Scientific theories can never be proven true. Mathematical theorems can be.

[samtheflash82]

I see, i have wondered about this topic myself.

[cameron marical]

Math is not the study of digits. Math is not science. Math is about constructing theorems from a set of axioms. Science is about constructing theories from a set of hypotheses. The key difference: Science is tested against reality by means of experimentation while math is tested against logic. Scientific theories can never be proven true. Mathematical theorems can be.

 

but we scientists use math for experimentation. how does that connect?

[Kaeroll]

but we scientists use math for experimentation. how does that connect?

We use it to quantify data in order to make precise predictions, which are then investigated to test the hypothesis. In my area at least, quantification and prediction improve syntheses.

 

Kaeroll

[ajb]

A (not hugely satisfying answer) is mathematics is what mathematicians do.

 

By this I mean largely it is defined by its practitioners.

 

To me, mathematics is the science* of abstraction.

 

My field is probably one of the oldest; mathematical physics. It is about seeing through all the "distractions" of real life systems and getting to the basic, fundamentally important aspects.

 

Pure mathematics (if there really is such a thing) comes from taking this abstraction one stage further. There is massive debate on mathematical philosophy and pure mathematics. I try not to get to involved in this, but wiki will give you some ideas.

 

Even more abstract again in category theory, which is the mathematics of mathematics.

 

*Many people are undecided if mathematics is a science or not. Some mathematicians think of themselves as scientists (I do for example, but then my work is very much rooted in physics) and others do not.

[D H]

but we scientists use math for experimentation. how does that connect?

Just because scientists write things on paper does not make them lumberjacks.

 

Recording a bunch of experimental measurements in the form of numbers is merely accountancy. Accounting is no longer a branch of mathematics. Scientists also use calculus and statistics. The key word here is "use". Mathematicians have created some extremely useful concepts that the rest of us use, sometimes to the chagrin of mathematicians.

 

From G.H.Hardy, A Mathematician's Apology, Cambridge University Press (1940), full text at http://www.math.ualberta.ca/~mss/misc/A%20Mathematician's%20Apology.pdf,

It will probably be plain by now to what conclusions I am coming; so I will state them at once dogmatically and then elaborate them a little. It is undeniable that a good deal of elementary mathematics—and I use the word ‘elementary’ in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus—has considerable practical utility. These parts of mathematics are, on the whole, rather dull; they are just the parts which have the least aesthetic value. The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly ‘useless’ (and this is as true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work.

[the tree]

Of course, it should be noted that the quote is ridiculously out of date. All of those mathematicians created tools that in fact, are pretty useful in the real world.

[ajb]

Of course, it should be noted that the quote is ridiculously out of date. All of those mathematicians created tools that in fact, are pretty useful in the real world.

 

It is a huge point of fascination that just about all areas of "pure mathematics" can be useful if not essential in describing the real world.

[antimatter]

*Many people are undecided if mathematics is a science or not. Some mathematicians think of themselves as scientists (I do for example, but then my work is very much rooted in physics) and others do not.

 

It is a huge point of fascination that just about all areas of "pure mathematics" can be useful if not essential in describing the real world.

 

 

 

I think of math as the language of science. Quite similar to what you said in the second quote. It's what we use to explain and back up our observations, and to make theoretical predictions.

Maybe I'm wrong to think that math 'belongs' to science, and isn't really its own separate field.

[ajb]

I think of math as the language of science.

 

It is more than just a language.

 

For example the dictionary contains all the words, but it is not poetry.

[antimatter]

It is more than just a language.

 

For example the dictionary contains all the words, but it is not poetry.

 

How is it more than just a language? Maybe I don't have enough experience with it, but to me it just always seemed like a tool in understanding science.

[Patrick Henry]

Math is the study of patterns.

[antimatter]

Math is the study of patterns.

 

I think that you're over simplifying it. Sure, math frequently involves patterns, but that is not what it is meant for specifically.

[Patrick Henry]

I think that you're over simplifying it. Sure, math frequently involves patterns, but that is not what it is meant for specifically.

 

Then what is it meant for, specifically?

[antimatter]

Then what is it meant for, specifically?

 

To be totally honest, I'm not entirely sure myself. I stated above that I believe it is a tool to be used as the language of science, but according to others it is much more. I feel that describing it as just the study of patterns is honing in too specifically on one aspect of the broad field.

[coke]

Interestingly almost any other subject/science is remarkably easier to explain its use...i.e. chemistry, physics, english, etc.

[BigMoosie]

I agree with D H.

 

In my view, math is simply: The persuit of any logic that is flawless under its assumptions.

 

I hear you cry "But there's lots of controversy in math, how can it be flawless?", well controversy arrives in the bridging between that logic and the real world, for instance:

 

* which axioms are chosen to align the logic with reality

* controversy surrounding the introduction of imaginary numbers was based on the fact that it "makes no sense in reality", not that it isn't self consistent (and if people claimed it wasn't self consistant then that's all part of the "persuit")

* controversy over machine proofs: under the assumption that proven machine instructions that provide results are valid, then whatever the results happen to be are self contained under that assumption, therefore it is maths. The problem occurs in the real world argument over whether that assumption is suitable for reality.

 

Under my view, some philisophical arguments can be considered math which I agree with, if you don't agree those sorts of arguments are math then you will need to expand upon my definition or reject it (I'd be interested in arguments to either).

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Interestingly almost any other subject/science is remarkably easier to explain its use...i.e. chemistry, physics, english, etc.

 

I don't think so. For one, we are not discussing its use, it's clear what it is used for (whatever "it" is). And also there are many people (not just religious people) who reject definitions of branches of science, including the scientific method, evolution as fact, the importance of double blind experiments etc.

Edited April 2, 2009 by BigMoosie

[ajb]

How is it more than just a language? Maybe I don't have enough experience with it, but to me it just always seemed like a tool in understanding science.

 

Two points to make here.

 

First, mathematics is more than a language in the sense that you cannot simply do whatever you like. There are rules and unarguable truths.

 

Secondly, mathematics does indeed give us the tools to describe what we see around us. But what is more fascinating is that it can guide us in what to look for next. Mathematics and physics are great partners.

 

The classical example here is the work of Dirac on antimatter. Dirac in trying to unite special relativity and quantum mechanics discovered that his equations gave "two kinds of matter". The stuff he knew was around us and a kind of "mirror matter" known as antimatter. At the time there was no experimental evidence for antimatter, yet Dirac made a firm prediction in his paper.

 

A few years later antimatter was discovered by Anderson.

 

The mathematics guided Dirac to this prediction.

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Math is the study of patterns.

 

I would agree with that to a large extent.

 

A large part of mathematics is looking for structures and objects that have "the same properties". The idea is that you can cut out all the unnecessary complications due to specific examples and describe many things in a unified way. The apex of this philosophy is category theory (and its generalisations which are being worked out at the present time.)

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* which axioms are chosen to align the logic with reality

 

I think this is the big point to much of what I have been saying. You don't need to initially set things up to agree with nature, that seems to arise naturally after. It is a not very well posed question, but "is all mathematics like this?"

 

Well, it is clear that some mathematics was always going to be useful in physics. Many of these areas were developed with physics in mind. For example calculus. Rates of change are a very physical idea.

 

What is much more interesting is the idea that maths originally devoid of "physical reality" (whatever that is exactly) turns out to be extremely important or even fundamental in theoretical physics. Complex numbers are a good example of this. They are simply needed in quantum mechanics.

 

Another is the work of Riemann on number theory. Patterns very similar to the zeros of the Riemann zeta function can be seen in nature!

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Let me give you a quote;

 

 

“One sometimes hears expressed the view that some sort of uncertainty principle

operates in the interaction between mathematics and physics: the greater the

mathematical care used to formulate a concept, the less physical insight to be

gained from the formulation. It is not difficult to imagine how such a viewpoint

could become popular. It is often the case that the essential physical ideas of a

discussion are smothered by mathematics through excessive definitions, concern

over irrelevant generality, etc. Nonetheless, one can make a case that mathematics

as mathematics, if used thoughtfully, is almost always useful–and occasionally

essential–to progress in theoretical physics.”

 

Robert Geroch, Mathematical Physics, Chicago Lectures in Physics, (1985) 351p

[Kaeroll]

Slighty tangential anecdote - apparently JW Gibbs (of Gibbs free energy fame, amongst other things), a quiet man who rarely spoke at faculty meetings of his university, once witnessed a debate. It concerned whether the teaching of languages or mathematics is more important, and he simply interjected with: "Mathematics is a language."

 

Thought I'd throw it in there for historical interest.

[Mr Skeptic]

Mathematics is the "study of numberlike things" or "philosophy of numbers"

[ajb]

Mathematics is the "study of numberlike things" or "philosophy of numbers"

 

You will need to be more specific with the notion of "number-like".

 

Many things in mathematics are noncommutative, [math]a*b \neq b*a[/math] and even nonassociative [math]a*(b*c) \neq (a*b)*c[/math] which is not very number-like.

 

But if you are referring to algebraic structures more generally then things are in some sense "number-like".

[antimatter]

Mathematics is the "study of numberlike things" or "philosophy of numbers"

 

I'm curious as to what you mean by 'philosophy of numbers'.

The way that they function? Philosophy is defined as the investigation of truths and principles, so is math a way of describing the truths and principles of the world through numbers?

[coke]

I don't think so. For one, we are not discussing its use, it's clear what it is used for (whatever "it" is). And also there are many people (not just religious people) who reject definitions of branches of science, including the scientific method, evolution as fact, the importance of double blind experiments etc.

 

Well, no I meant like if you say

What is chemistry?

Study of chemical interactions, etc. etc.

 

What is physics?

Study of forces, matter, etc. etc. etc.

 

What is english?

Study of the language english.

 

Math?

Study of something- patterns? Who knows?