How does a collection of points have dimensions?

[elementbrdr]

Recently, I learned that, in a probability density function, the probability of the occurrence of any specific x-value is in fact zero, for the relevant interval on the function is a point, which has zero width and therefore has zero area associated with it under the probability curve. This made me realize that, although I understand how Riemann sums work for intervals of >0 width, I cannot make sense of the concept of an integral once the constituent intervals become point-sized (i.e., an interval of 0 width multiplied by any finite y-value will always produce a partition of 0 area). More generally, this means I don't understand how a collection of adjacent points can form a line having nonzero dimensions. Presumably there is a relatively simple explanation of how to conceptualize this, but I can't find anything. Any guidance would be appreciated. Thank you.

 

[jbriggs444]

The concept of "measure" may be what you are after. http://en.wikipedia.org/wiki/Measure_(mathematics )

 

[PeroK]

Recently, I learned that, in a probability density function, the probability of the occurrence of any specific x-value is in fact zero, for the relevant interval on the function is a point, which has zero width and therefore has zero area associated with it under the probability curve. This made me realize that, although I understand how Riemann sums work for intervals of >0 width, I cannot make sense of the concept of an integral once the constituent intervals become point-sized (i.e., an interval of 0 width multiplied by any finite y-value will always produce a partition of 0 area).

The Riemann Integral is a limit of Riemann sums. A vital part of the concept of a limit is that you never reach the limit. In this case, you are never adding areas of zero width. Instead, the limit is defined using the properties of areas of finite width.

If you want to do rigorous mathematics, this is something you need to understand fully.

More generally, this means I don't understand how a collection of adjacent points can form a line having nonzero dimensions. Presumably there is a relatively simple explanation of how to conceptualize this, but I can't find anything. Any guidance would be appreciated. Thank you.

This is one of the apparent paradoxes of the real numbers. It's never lost its magic or mystery for me. I always think: how can there be no next real number after 0?

Also, the rationals are countable and the reals are not. But, between every two real numbers, there is a rational (a countable infinity of rationals, in fact). How can that be?

Perhaps this is not helping you conceptualize it. The way I look at it, you have to believe where rigorous mathematical logic leads you; even if it seems counter-intuitive. And, perhaps, embrace these apparent paradoxes rather than be troubled by them.

 

[gopher_p]

The real line is more than just a collection of points. It's a collection of points with additional structure.

1. It's totally ordered, meaning that for every pair $a,b$ of real numbers, either $a<b$ or $b<a$ or $a=b$.

2. This order is dense, meaning that for all $a,b$ with $a\neq b$ there is $c$ satisfying $a<c<b$. (note that this rules out the possibility of "adjacent" points)

3. There are no endpoints according to the order; i.e. there is no largest or smallest real number.

4. It is complete, which roughly means that there are no gaps.

5. It is connected, which roughly means it's one piece.

Our geometric intuition of what a "one-dimensional line" should be satisfies all of these properties. In fact, the modern definition of "one-dimensional" is basically something along the lines of "looks like $\mathbb{R}$ if you zoom in close enough".

Also you need to be very careful when applying finite logic/reason to that which is not finite.

 

[mathman]

4. It is complete, which roughly means that there are no gaps.

This is not quite accurate. For example, the rational numbers have no gaps, if by a gap you mean an non-zero length interval outside the set.
Complete is usually defined to mean that the limit of any convergent sequence is in the set. To be precise, topolgy needs to be introduced.

 

[gopher_p]

This is not quite accurate.

Hence "roughly".

For example, the rational numbers have no gaps,

There is a gap where $\sqrt{2}$ should be.

if by a gap you mean an non-zero length interval outside the set.

I don't. By gap I mean "missing part".

Complete is usually defined to mean that the limit of any convergent sequence is in the set. To be precise, topolgy needs to be introduced.

As long as we're being pedantic, you absolutely do not need any notion of topology or a metric in order to very precisely define what completeness means for the real line; http://en.wikipedia.org/wiki/Dedekind_completeness.

 

[micromass]

As long as we're being pedantic, you absolutely do not need any notion of topology or a metric in order to very precisely define what completeness means for the real line; http://en.wikipedia.org/wiki/Dedekind_completeness.

Indeed. In fact, completeness is not a topological property at all. You need an extra structure.

 

[mathman]

Indeed. In fact, completeness is not a topological property at all. You need an extra structure.

I stand corrected. I was thinking about the notion of completeness as defined for toplological spaces. As others have noted, Dedekind cuts or something equivalent is the concept needed for the real line.