Let's discuss Mathematics

[peter grimes]

Question for the maths people:

Do you consider numbers things that can be said to exist in their own right, or are they simply products or results of us thinking about them?

In other words, If we re-ran the history of Earth, and a different sort of mathematically intelligent being arose, would the axioms and principles current number theory be redeveloped?

 

[ParadigmShifter]

I think someone once said natural numbers exist and everything else is human invention. Maybe he mentioned God in there as well

 

[IdiotsOpposite]

"God made integers; all else is the work of man." - Leopold Kronecker

 

[sanabas]

Question for the maths people:

Do you consider numbers things that can be said to exist in their own right, or are they simply products or results of us thinking about them?

In other words, If we re-ran the history of Earth, and a different sort of mathematically intelligent being arose, would the axioms and principles current number theory be redeveloped?

Yeah, I think the same stuff would be developed. Doesn't mean you can say i or pi or 3 actually exist in their own right.

 

[Atticus]

I believe natural numbers would be like they are now. About the real numbers, not necessarily. I think it's sufficient proof that the axioms of them were developed less than 200 years ago, so other species could be satisfied with the understanding we had before that (which, if I've correctly understood, was that there were no axioms, others than the implicit ones in Euklides' Stokheia). On the other hand they could have nonstandard reals.

On the existence of numbers, I personally find that hard question, since it depends on what we mean by existence in general. Or I could equally well think that it's extremely easy question, that they exist if we define the word "exist" so that it covers numbers as well.

 

[Truronian]

I've occasionally wondered if a non-exact mathematical system could have hypothetically developed. Instead of surds, pi and transcendentals you would simply have approximate quotients.

 

[ParadigmShifter]

Continued fractions can do that can't they?

And computers work on approximate values al the time.

 

[Souron]

Question for the maths people:

Do you consider numbers things that can be said to exist in their own right, or are they simply products or results of us thinking about them?

In other words, If we re-ran the history of Earth, and a different sort of mathematically intelligent being arose, would the axioms and principles current number theory be redeveloped?

Numbers taken along with their properties exist "in their own right" in the sense that the conclusions of mathematics are incontestable, because the stem directly from axioms.

Also, numbers are analog for physical properties such as quantity and ratios. Therefore, in so far as people in alternate earth observed and need to do stuff with those same physical properties, so too would they have the same number system. In this way, math developed as a tool that fits the task. But if we imagine someone intelligent other than people in a world rather unlike earth, who perceive nature in a way rather unlike people, then maybe they would use a different number system.

 

[Atticus]

To open up this little more, the word "number" itself isn't that clear at all. It can mean for example:
a natural number, 1,2,3,....
an integer, ....-2,-1,0,1,2,....
a rational number, n/m, where m is a natural number and n integer,
real number, the set of which are defined by the axioms of real numbers,
or a complex number.

This is probably clear to many of people in this thread, Peter Grimes included, but don't take this as underestimation of your knowledge, since it could equally well mean something else too, a hyper real, or a quaternion.

So, what is the criterion of calling something a number? The man on the street would say first that it's something that is used to count, after being pointed out that reals aren't used for that (solely), he'd say that it's something used to measure things, and then the complex numbers would be problem.

Then, on the other hand, why aren't the vectors of Rⁿ called numbers? Or why the functions from R to R aren't? The only distinction seems to be convention.

Some questions to consider: Are there negative numbers? Is there a number whose square is -1? Were there negative numbers in the year 1? Was there a number whose square is -1 in the year 1?

 

[peter grimes]

I started thinking about this after reading a book on the Reimann Hypothesis. It was written for a lay audience (to which set I certainly belong), so it glossed over some things I would have liked to have read more about.

Much of the book talks about the zeta function, and the 'landscape' it traces if you plot solutions along the number line (x axis) and imaginary axis (y).

I had been exposed to imaginary numbers in other books (it comes up in information theory, as do quaternions - which I don't know anything about). But in learning about the imaginaries, I started to wonder about this idea of whether or not numbers as values exist independent of mathematical axioms.

To my way of thinking they do exist outside of us. I find it hard to imagine that sqrt2 ceases to have the value it does if people lose the knowledge of the pythagorean theorem. Likewise with pi and e (I REALLY don't get e - something else I need to read up on!).

I can't draw a line in the sand as Kronecker did - the different classes of numbers seem to all hang together once you start digging deep enough. But I'm nothing close of a student of math(s) so I thought I'd like to hear what some real math people think.

 

[Petek]

I think that the concept of a "number" is so fundamental that it should be treated as an undefined term in arithmetic, similar to the concepts of "points" and "lines" in geometry. For an interesting alternate point of view, see Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. The authors, who are cognitive scientists, treat various mathematical concepts as arising from metaphors. Their metaphors for numbers include:

* Numbers are object collections.
* Numbers are physical segments.
* Numbers are points on a line.
* Numbers are sets.
* Numbers are things in the world.

I found the book to be tough going, but it has lots of discussion about these matters.

 

[ParadigmShifter]

I started thinking about this after reading a book on the Reimann Hypothesis. It was written for a lay audience (to which set I certainly belong), so it glossed over some things I would have liked to have read more about.

Much of the book talks about the zeta function, and the 'landscape' it traces if you plot solutions along the number line (x axis) and imaginary axis (y).

I had been exposed to imaginary numbers in other books (it comes up in information theory, as do quaternions - which I don't know anything about). But in learning about the imaginaries, I started to wonder about this idea of whether or not numbers as values exist independent of mathematical axioms.

To my way of thinking they do exist outside of us. I find it hard to imagine that sqrt2 ceases to have the value it does if people lose the knowledge of the pythagorean theorem. Likewise with pi and e (I REALLY don't get e - something else I need to read up on!).

I can't draw a line in the sand as Kronecker did - the different classes of numbers seem to all hang together once you start digging deep enough. But I'm nothing close of a student of math(s) so I thought I'd like to hear what some real math people think.

e^x is the function which is equal to the value of its derivative at every point.

EDIT: Ok, f(x) = 0 is as well But that's not very interesting.

 

[IdiotsOpposite]

e^x is the function which is equal to the value of its derivative at every point.

EDIT: Ok, f(x) = 0 is as well But that's not very interesting.

Technically f(x) = a e^x, for any a, is the SET of functions which are equal to the value of their derivatives at every point. This includes f(x) = 0.

 

[IdiotsOpposite]

Thomae's Function is Riemann integrable over [0,1], according to Wikipedia. So I wanna know... is there any estimate of what it integrates to over that interval?

 

[Atticus]

Had to wiki it, but it intergrates to 0, since it's nonzero only in a countable set.

Lebesgue integral is the easier way to approach these kind of things: If a function is both Lebesgue- and Riemann integrable, then the integrals are the same. The news of that statement is in the integrability, it reminds a lot the function which is 1 in rational points and 0 elsewhere, which is Lebesgue integrable, but discontinuous at every point and thus not Riemann integrable.

This function has so many values near the 0 that it's continuous almost everywhere. Compare to function [-1,1]-> R : f(x)= x when x is rational and -x when x is irrational, and which is somewhat counter intuitively continuous at 0.

 

[ThinkTank]

Question for the maths people:

Do you consider numbers things that can be said to exist in their own right, or are they simply products or results of us thinking about them?

This is one of the basic problems in the philosohy of mathematics, which has been dicussed since Plato, and to my mind, no definitive answer has been given. You have basically given two common answers (realism, conceptualism), but there are other answers (see http://en.wikipedia.org/wiki/Philosophy_of_mathematics).

I'm inclined to answer "Do numbers exist?" with "yes", but then the next qestion is "Where are they?", and then you easily get into metaphysical answers/positions that are often quite questionable, like the alternate abstract reality of Platonism - I mean what evidence do we have of that?
But somehow answering "Do numbers exist?" with "no" doesn't seem to be an option - if so, then what are we doing in methematics?

I find Fictionalism an interesting view (see the section in the wikipedia link above), which sees mathematics as interesting and useful fiction, but the entities in that fiction are no more real than the characters in a novel.
But almost nobody believes that ...

 

[Spoonwood]

But somehow answering "Do numbers exist?" with "no" doesn't seem to be an option - if so, then what are we doing in methematics?

Studying the properties of the meth maybe?

 

[Petek]

Not much going on in this thread lately. Try this:

Factor (a + b + c)³ - a³ - b³ - c³

 

[nc-1701]

(b+c)((a+b+c)^2+a(a+b+c)+a^2-b^2+bc-c^2)=3(b+c)(a^2+ab+ac+bc)

 

[Petek]

(b+c)((a+b+c)^2+a(a+b+c)+a^2-b^2+bc-c^2)=3(b+c)(a^2+ab+ac+bc)

Technically correct, but you can do better. Hint: The original expression is symmetric in a, b and c. If it's divisible by b + c, then it must also be divisible by ... .