Let's discuss Mathematics

[ParadigmShifter]

Much thread derailment has gone on in the OT atheism thread about the nature of scientific and mathematical proof and it was suggested a thread should be started in this section.

So fire away!

My maths skills are a bit rusty but I like discussion about the subject.

 

[Perfection]

It's math not maths.

 

[Aramazd]

The OP is too vague to have a really good discussion. However, I think maybe a quick overview of statistics would be good considering how useful it is for experiments.

 

[ParadigmShifter]

It's math not maths.

Wrong! :smugger:

 

[Perfection]

As a filed of study, it's a singular noun so there's no need for the disgusting "ths" sound.

 

[ParadigmShifter]

Hmm a quick overview of statistics... basically it applies probability theory to trials i.e. experimental data.

You get some data. You look how likely that would be to occur at random using probability, often assuming a model for your data source (most usually the Normal distribution, which is what all repeated trials will converge to if repeated infinitely, by the central limit theorem). That gives you a confidence level of how likely/unlikely the observed situation would be to occur by chance.

That's the basics of it...

 

[Aramazd]

Hmm a quick overview of statistics... basically it applies probability theory to trials i.e. experimental data.

You get some data. You look how likely that would be to occur at random using probability, often assuming a model for your data source (most usually the Normal distribution, which is what all repeated trials will converge to if repeated infinitely, by the central limit theorem). That gives you a confidence level of how likely/unlikely the observed situation would be to occur by chance.

That's the basics of it...

A meant a bit longer than that. I suppose I could write some more detailed stuff. Or just link to some wiki articles, either way would be good.

Wrong! :smugger:

Maths sounds terrible.

 

[ParadigmShifter]

Well I was hoping for more philosophical discussion but links would be good as well.

Null hypothesis testing is probably the best starting place, link here

http://en.wikipedia.org/wiki/Null_hypothesis

 

[Aramazd]

Statisitical Significance
Correlation
and Sample Size are all good articles as well for a beginner.
The Stats Portal on wiki is good as well

As to a philosophical discussion on math, I'd be interested in discussing it. Anything you had in mind?

 

[ParadigmShifter]

Well in the atheism thread we were talking about undecidability and the limits of proof via axioms when reasonable axioms can produce counter-intuitive results.

But any mathematical discussion is fine by me as long as we keep those pesky physicists with their "real world" applications out of it Just kidding of course although I suck at applied maths.

My main strengths are linear algebra, combinatorics, analysis, a bit of number theory, statistics (only because of lack of pure maths courses in my 2nd year ), chaos/fractals, and 3D stuff. I liked topology but it was WAY HARD.

EDIT: A bit of geometry too but only from linear algebra really.

 

[Aramazd]

Sorry about leaving you waiting for so long. You posted right before I went to dinner.

Well in the atheism thread we were talking about undecidability and the limits of proof via axioms when reasonable axioms can produce counter-intuitive results.

That is something I've never put a lot of thought into. My gut reaction would be to reject the axiom as incorrect, but if it works in other instances, I guess it would be valid.

But any mathematical discussion is fine by me as long as we keep those pesky physicists with their "real world" applications out of it Just kidding of course although I suck at applied maths.

My main strengths are linear algebra, combinatorics, analysis, a bit of number theory, statistics (only because of lack of pure maths courses in my 2nd year ), chaos/fractals, and 3D stuff. I liked topology but it was WAY HARD.

EDIT: A bit of geometry too but only from linear algebra really.

I'm an ecology major, just Calculus (up to multivariable), diff eqs and stats. Although I am kinda interested in math.

 

[ParadigmShifter]

Well the classic example of an axiom which can't be proven and can be formulated in several ways is the parallel postulate giving rise to Euclidean and non-Euclidean geometry.

Parallel lines never meet = Euclidean geometry
Otherwise: Non-Euclidean geometry.

EDIT: I'm wrong however. Euclidean geometry is where any point not on a line in 2D has a unique line that passes through it which never meets the other one.

 

[ParadigmShifter]

Ha ha, I've been reading wikipedia about Euler and following links led me to this gem which shows how a square wheel can roll smoothly if the ground consists of evenly shaped inverted catenaries of the right size and curvature, with animation.

http://en.wikipedia.org/wiki/Square_wheel

 

[Aramazd]

That is pretty cool.

 

[Mise]

Ha ha, I've been reading wikipedia about Euler and following links led me to this gem which shows how a square wheel can roll smoothly if the ground consists of evenly shaped inverted catenaries of the right size and curvature, with animation.

http://en.wikipedia.org/wiki/Square_wheel

Lets say I were standing in the centre of that square, and experienced no rotation myself; that is, I am completely unaware that the square I'm standing on is moving from left to right, or that I am rotating with it. Rather, all I see is those catenaries rolling around the outside of the square. Am I right in assuming that it would look to me as though a circle of radius equal to half the diagonal width of the square were rolling around the square?

 

[peter grimes]

No, I don't think that's what it would look like. There is a definite perigee and apogee (not sure if those are the correct terms), so, I think it would seem as if the catenaries weren't curved at all; rather it would appear as if the edge of the square were receding and approaching - like the tide.

 

[Belisar]

as long as we keep those pesky physicists with their "real world" applications out of it Just kidding of course although I suck at applied maths.

So you throw us peskys out, toghether with our real world
Personally, I always found it more fascinating when a mathematical model can be applied to real world (approximately), but I guess others find it fascinating working with immaculate equations.

 

[Mise]

I've thought about it, and, no, it wouldn't look like a circle, for the obvious reason that, if you tracked a single point, it would get closer and closer, then further and further away, to infinity, whereas if it were a circle it would disappear after 90 degree rotation of the circle.

Anyway, it would only make sense to a person if there was either a really, really long line of catenaries that are revolving around the square, or the person would realise that they are in fact rolling on a line of catenaries. I guess that's how people realised they were revolving around the sun, and the sun was revolving around the centre of the galaxy, that other stars were also revolving around -- it was the only way to make sense of it all.

 

[ParadigmShifter]

Another interesting thing about rolling is the question:

If you roll a coin around the circumference of another coin, how many complete rotations has it done when it arrives back at the start point?

I also like this question:

Say you have a rope which fits snugly around the equator of the Earth. How much longer would it have to be if it were 1 metre above the ground instead?

 

[Mise]

To the first one, 1?

To the 2nd one, 2*PI meters bigger, so 6.28m?