I'm sure the mathematical anomaly that .999 repeating equals 1 has been brought up, but I was wondering what you think of it. Why is this possible?
Subtract one x from the 10x
and you get
divide both sides by 9
I was wondering if you could explain why this happens. Does it show a flaw in our math system? Or is it just a strange occurrence that should be overlooked? Or is it true?
As a teacher of high school mathematics and a former student of philosophy, I try to merge the two to engage my students in meaningful conversations about the significance of some mathematical properties. Recently, however, I could not adequately defend the statement "a=a" as being necessary for our study of geometry when one student challenged "When is a never NOT equal to a?" What would you tell them?
(One student did offer the defense that "Well, if we said a=2 and a=5 then a=a would be false, causing problems.")
How do we resolve the fact that our finite brains can conceive of mental spaces far more vast than the known physical universe and more numerous than all of the atoms?
For example, the total possible state-space of a game of chess is well defined, finite, but much larger than the number of atoms in the universe (http://en.wikipedia.org/wiki/Shannon_number). Obviously, all of these states "exist" in some nebulous sense insofar as the rules of chess describe the boundaries of the possible space, and any particular instance within that space we conceive of is instantly manifest as soon as we think of it. But what is the nature of this existence, since it is equally obvious that the entire state-space can never actually be manifest simultaneously in our universe, as even the idea of a board position requires more than one atom to manifest that mental event? Yet through abstraction, we can casually refer to many such hyper-huge spaces. We can talk of infinite number ranges like the integers, and "bigger"...
This is more like a comment to the question in Mathematics that starts with:
"If you have a line, and it goes on forever, and you choose a random point on that line, is that point the center of that line? And if you ..."
The answer provided by the panelist, as well as the initial question, assume that one can distinguish between points at infinity. As far as Math goes however, one cannot do that, and this is the reason the limit for cos(phi) does not exist, as phi goes to infinity. Revisiting the argumentation provided by the panelist, the error starts with the 'definition' of the distance between a fixed point and infinity - this distance cannot be defined, and therefore it cannot be compared (at least, as math goes).
A somewhat similar problem can be stated, without the pitfalls of the infinity concept, for a point on a circle, or any closed curve.
If you have a line, and it goes on forever, and you choose a random point on that line, is that point the center of that line? And if you picked a new point, would that become the center of the line (since to either side of the point is infinity, and infinity is congruent to infinity)? Also if the universe has no middle and no end, am I, and everyone, at the center of the universe? (Of course the middle of the universe thing only works if you believe the universe has no middle and no end.)
I recently heard about mathematical paradoxes and I have a perhaps strange question: It seems to me that the goal is to figure out what the fundamental problem is, i.e. what gives rise to the paradox, so we can perhaps rewrite the axioms so that the problems disappear. But why not just say: "Well, paradoxes arise when you talk about sets that contain every set, so let's avoid talk about sets that contain empty sets." (Kind of like saying that bad things happen when you divide something with zero, so don't do it!)