It looks like you're using an Ad Blocker.
Please white-list or disable AboveTopSecret.com in your ad-blocking tool.
Thank you.
Some features of ATS will be disabled while you continue to use an ad-blocker.
Let us pretend you have three apples--go ahead, picture it in your mind. These apples may be oddly shaped, but more importantly we are going to consider them to be normal apples. What does normal mean? Well, statistically it means that all three of your apples fall within 3 standard deviations (or 99.7%) of all the apple data you can think of. Their shape, weight, nutritional value, ripeness, age, color, etc. all fall into the category of normal. Normal from what? In statistics we can relate everything to a, more or less, perfect apple. This apple we relate data to is the average-in-every-way apple. It has the exact mean, or average, of every apple that ever existed, and in every category assumed with apples. Now, this specific apple we relate things to may have a mean shape, weight, nutritional value, ripeness, age, color, etc, but to say that an apple with "perfect" everything (or average everything) at one point existed may not necessarily be true. There is a very likely chance that that apple never existed, and may never exist in time (all speculative of course). There may have been apples very close to being perfect, but there may have always been one conflicting variable in which it was slightly above or below average. And of course, as a mathematician, I can say that there may have even been a perfect apple at one point in time, but likely not many more than one, if any.
What is, just 1? 1 is a number and aside from the units and imprecision in the "real-world", 1 can be dealt with graphically. Graphs, mathematically, are perfect. If you graph "1" in a Cartesian plane without defining parameters, it exists as nothing but a concept. If you define it, say by saying that y=1, then you have a horizontal line. For every x point that you can discover then has a set value of y=1. For that matter, regardless of x, you know that y=1 (for the only y that exists here is 1).
1. The normal distribution is not a distribution of apples, it is a distribution of some measurable property of apples, such as their weight or volume.
2. Of course when we speak of the population of apples, we don't really know if any of the measured properties of apples is actually normal. For example, if we are measuring weight, we may have a skewed distribution with all those Macintosh apples pulling our average weight down counterbalancing he larger but fewer delicious apples.
But then of course what is 0? There is a set theoretic approach that starts with the idea that a number can be defined in terms of sets. We start with the empty set "e", then take the set consisting of just the empty set [e], then the set of consisting of the two sets we have just defined [e,[e]]. Now we define a unique property to each set so constructed called its cardinality. Then 0 is defined as card(e), 1 is defined as card([e]) and 2 is defined as card([e,[e]]).
No need to resort to graphing or to make any reference to any real world properties or ideas about what a number "is, and you don't need to resort to any geometrical intuition (graphs are perfect? huh?)
Now, before we continue on graph-theory, I do wish you to keep some things in mind. We are made up of atoms. Each atom is made up of subatomic particles, and perhaps each subatomic particle is made up of other things. Now, atoms make up molecules, molecules make up substances, substances make up objects, objects make up systems of objects, systems of objects make up the world, worlds make up a solar system, a solar system makes up a universe, and blah blah blah. It goes on forever as far as we know—in both directions. This means, that for everything you can think of small scale, there could be some macro version of it somewhere, sometime. So to say this does not apply graphically, would be inaccurate. Still have that circle in mind, with radius 1? That circle has a radius of 1 and therefore a hypothetical area value of π. So an area of π perhaps, yet there seems to be one interesting aspect to this circle. Inside the circle lie an infinite number of points! Each data point has a value of zero, because as you zoom in further and further on a graph you can never actually see a data point, and you never will. There are an infinite number of points, each with an area of zero… yet the addition of all the points brings the circle to an area of π? So 0*∞= π? No, that does not seem to be right…
The real numbers are much much bigger than the rational numbers and include the transcendental numbers e (natural logarithm base) and pi.
Now in terms of area, your argument only applies if we consider ONLY some countable set of numbers since limits only work over countable sets, we have to use different sorts of things (called filters and nets) to take limits over uncountable sets.
"That just isn't right."
Originally posted by PhysicsAdept
reply to post by metamagic
1. The normal distribution is not a distribution of apples, it is a distribution of some measurable property of apples, such as their weight or volume.
Correct, no dispute there. A theoretical graph plotting a multi-variate chart of the apples. You take one property, compare it to the rest of the sample (in the case every apple ever, or to be statistically more professional about 10% of the population id fine) and you receive a z-score for each property. Then you would do some other test comparing all the collected variance away from zero, or residuals, of each trait and give that a z-score of sorts. By doing this, would you not then conceive some notion of how "perfectly normal" some apple may score
2. Of course when we speak of the population of apples, we don't really know if any of the measured properties of apples is actually normal. For example, if we are measuring weight, we may have a skewed distribution with all those Macintosh apples pulling our average weight down counterbalancing he larger but fewer delicious apples.
Well, ok fine we take a more specific example to explain it, that is beside the point...
1. Nothing is exact, it is impossible to determine the actual quantity of a real object, as a quantity is only expressed as a unit which is a generalization.
2. I ask you to broaden your mind, and think about how the related rates of the world and the interaction between all the numbers you can think of what they really mean. Consider the fact that between each integer lies an infinite possible amount of numbers, and that each decimal has some decimal slightly larger than it as well as some decimal slightly less than it.
...
Still have that circle in mind, with radius 1? That circle has a radius of 1 and therefore a hypothetical area value of π. So an area of π perhaps, yet there seems to be one interesting aspect to this circle. Inside the circle lie an infinite number of points! Each data point has a value of zero, because as you zoom in further and further on a graph you can never actually see a data point, and you never will. There are an infinite number of points, each with an area of zero… yet the addition of all the points brings the circle to an area of π? So 0*∞= π? No, that does not seem to be right…
...
Ok, so now where did we leave off… ah yes, an infinite amount of zero can actually equal some number. Yet, different infinite amounts of zero will lead us to different numbers each time.
3. Infinity itself is a quantization. It is an assumption that we make, and the definition can be treated similarly to the rest of the numerical symbols that have been created, and thusly makes some infinities larger than others.
Originally posted by PhysicsAdept
reply to post by metamagic
But then of course what is 0? There is a set theoretic approach that starts with the idea that a number can be defined in terms of sets. We start with the empty set "e", then take the set consisting of just the empty set [e], then the set of consisting of the two sets we have just defined [e,[e]]. Now we define a unique property to each set so constructed called its cardinality. Then 0 is defined as card(e), 1 is defined as card([e]) and 2 is defined as card([e,[e]]).
Exactly, what is zero? Zero is some example of nothingness, or emptyness. However, zero is somewhat scalar. I examined s piece of vacuum in space, is that zero? It would be hard to detect, but who is to say that there aren't neutrinos bumbling about in there? Or, what about some mass of energy that very finitely makes up space? Zero does not exist, but in concept.
No need to resort to graphing or to make any reference to any real world properties or ideas about what a number "is, and you don't need to resort to any geometrical intuition (graphs are perfect? huh?)
How are graphs not perfect? They show the exact conceptual idea, and mold it to a scale that you can perceive. The closest you will ever get to a circle that equals pi, is through graphing. Otherwise, it is approximate, but this is aside from the point I was making in the thread. It is not about the numerical premise of a graph, or a unit, of of objects, but the conceptual impossibility that something can be as exact as a number can tell us.
Originally posted by PhysicsAdept
reply to post by metamagic
The real numbers are much much bigger than the rational numbers and include the transcendental numbers e (natural logarithm base) and pi.
Riiiight, again failing to see where this argument leads to describing how I am incorrect... you referencing Euler's formula has nothing, at least that I see, to do with anything as of now...
Now in terms of area, your argument only applies if we consider ONLY some countable set of numbers since limits only work over countable sets, we have to use different sorts of things (called filters and nets) to take limits over uncountable sets.
The number itself is a limit, which is pretty much what I am pointing out throughout the post...
Originally posted by PhysicsAdept
I post this due to recent contemplations I have been considering. This is a long thread that I hope you read, but if you really don’t want to then skip to the bottom, the last line, before you decide not to give it a chance. Honestly, I really think there needs to be a new "math-topics" forum, but anyway...
We shall examine a few simple (real) numbers and what they really represent in this thread.
There are 3 main points in this thread (which I will try very hard to relate at the end, despite my inability to communicate my ideas fluidly):
1. Nothing is exact, it is impossible to determine the actual quantity of a real object, as a quantity is only expressed as a unit which is a generalization.
2. I ask you to broaden your mind, and think about how the related rates of the world and the interaction between all the numbers you can think of what they really mean. Consider the fact that between each integer lies an infinite possible amount of numbers, and that each decimal has some decimal slightly larger than it as well as some decimal slightly less than it.
3. Infinity itself is a quantization. It is an assumption that we make, and the definition can be treated similarly to the rest of the numerical symbols that have been created, and thusly makes some infinities larger than others.
edit on 16-3-2012 by PhysicsAdept because: (no reason given)