Let's go.........RANDOM!

[Jackdapantyrip]

If you want a re-match, we look forward to ruling you all over again. Just don't bring those damned French back into it this time, OK?

Talking about rematches I thought Edinburgh was in Scotland..


Hey guys if you leave your red coats behind I'm cooking some ribs and going to blow some stuff up if you want to come party.

[shinygerbil]

July the Fourth be with you?

[Jackdapantyrip]

July the Fourth be with you?

I just watched a Russian American almost blow his family up. Put a flat basketball in an aluminum cylinder with lots of gunpowder pumped the basketball up and lit the fuse, with a beer in his hand. Ha close call buddy :p

The basketball did not launch out by the way..

This is what I love about the 4th.. Dangerous yet so fun..

lets make fireworks illegal /sarcasm

[jelco]

Jelco

[Mas Tnega]

If you want a re-match, we look forward to ruling you all over again. Just don't bring those damned French back into it this time, OK?

Talking about rematches I thought Edinburgh was in Scotland..


Hey guys if you leave your red coats behind I'm cooking some ribs and going to blow some stuff up if you want to come party.

That's 24th June.

[xander]

Stupid blowy-uppy holiday. Why won't you fuckers let me sleep!?

xander

[Feud]

Because you won't let us play on your lawn.

[xander]

Because you won't let us play on your lawn.

We moved last year. The new place has a xeriscaped yard. So now I don't let the kids play on my sharp, pointy rocks and decomposed granite! Ha!

xander

[Jackdapantyrip]

[NeatNit]

fail

[NeatNit]

I don't understand how he could possibly claim that this is actually a paradox (at 6:12-ish), and a baffling one at that. Welcome to the real world, where motion is continuous and doesn't work in "frames" nor "steps". On every graph we draw, there is an infinite number of points. The graph clearly passes through all of them. This is not a paradox, it is the definition of continuous.

How is that baffling?

[xander]

I don't understand how he could possibly claim that this is actually a paradox (at 6:12-ish), and a baffling one at that.

How much mathematics have you studied, and how deep down the rabbit hold do you want to go? Without rigorous mathematical definitions, the paradox is that an infinite process (i.e. one with no end) can be "completed."

Welcome to the real world, where motion is continuous and doesn't work in "frames" nor "steps".

Sometimes mathematics can be used to model the real world, but mathematics is not the real world. What does the real world not moving in frames or steps have to do with the mathematical abstraction of an infinite series? Even if that abstraction was based on an observation in the real world? Moreover, I am pretty sure that many modern quantum physicists would argue that space and time are discrete (Planck length and Planck time being the smallest possible units). In that case, the whole problem is irrelevant, as these sums seem to model both space and time as infinitely divisible (i.e. continuous) objects.

On every graph we draw, there is an infinite number of points. The graph clearly passes through all of them. This is not a paradox, it is the definition of continuous.

Actually, no it is not. One definition of continuous is

A function f is continuous at a point a if for every epsilon > 0 there exists a delta > 0 such that |f(x)-f(a)| < epsilon whenever |x-a| < delta. f is continuous on an interval (a,b) is f is continuous at c for every c in (a,b).

There are other more general definitions (the most general probably coming from topology: a function is continuous if the inverse image of an open set is open), but when considering functions that take the reals to the reals, they are all equivalent to this one.

Moreover, your definition fails in a couple of ways. For instance, I can find functions whose graphs pass through an infinite number of points, but which are continuous nowhere (consider the function that is defined to be 1 on the rationals and 0 on the irrationals). Maybe you meant to say that the function can be graphed without lifting the writing utensil. This definition is the one normally given to students until they take a rigorous class in calculus or analysis, and is implied by the formal definition, but is not, in and of itself, a complete definition (what does it even mean? how do you use it to prove anything?).

How is that baffling?

Have you ever seen the movie Good Will Hunting? I am not claiming that it is a great movie, but there is a particular scene that I recall, which I think is quite appropriate. The protagonist hears some freshman philosophy undergrad waxing on about whatever theory had been taught that week, then proceeds to explain exactly what new theories this person is going to attach to over the next several years of their undergraduate career, thus demonstrating how shallow that freshman's knowledge really is. The freshman knew enough to think that he was really clever, until the protagonist put him in his place.

I don't want to "put you in your place," but I think that you have a similar problem. You have studied enough to believe that you know what you are talking about, and in a few years, when you have looked into it a bit more, I suspect that you will feel differently. Or perhaps you are like my engineer students, who don't really care about the philosophy of mathematics, but only want to know how to use mathematics to make computations. If that is the case, I would suggest that you not comment on the philosophy of mathematics, because you are way out of your depth (I, also, am way out of my depth on that topic, but not quite as much as you are :P).

xander

[tllotpfkamvpe]

If Paradoxes are possible how can mathematicians use proof by contradiction.

[NeatNit]

Not in the slightest disagreeing that I'm way out of my depth, and the terminology I picked may not have been the best. I also tried reading that definition of continuous you posted and have no clue what you mean. But I haven't been in the best mental or physical health lately so I'm not going to let that stop me.

But hear me out for a minute.

Suppose a marble at t=0 starts to move from y=0 to y=1m at a constant rate of 1m/s. In order to do so it must pass through y=0.5m. And in order to do that it has to pass through y=0.25. Yada yada yada, you get the point. So STARTING a motion is just as "problematic" as ending it. But for any y between 0 to 1m inclusive you can find a t between 0 to 1s in which the marble was at said point y. You can zoom in as far as you possibly want. How is that supposed to "stop" the marble from moving? The marble doesn't need to think "okay, now I'm going to reach 0.125. Okay, now I'm going to reach 0.12500000005". It just DOES it. it passes through those points at infinite density of positions, just as there is an infinite density of time as well to correspond to every position. That's what I meant by 'continuous'.

The fact that you could make a series that contains an infinite number of points that the marble must pass through, does not cause any sort of paradox. It simply confirms the obvious and the definition: a line is a collection of infinite points.




Also, five stars:

If Paradoxes are possible how can mathematicians use proof by contradiction.

[xander]

If Paradoxes are possible how can mathematicians use proof by contradiction.

Simple. Paradox isn't a synonym for contradiction. In mathematics, a paradox is generally an unexpected result, or a result that confounds expectation and/or common sense. A paradox is a surprising or counter-intuitive result.

Suppose a marble at t=0 starts to move from y=0 to y=1m at a constant rate of 1m/s. In order to do so it must pass through y=0.5m. And in order to do that it has to pass through y=0.25. Yada yada yada, you get the point. So STARTING a motion is just as "problematic" as ending it.

That is one of Zeno's paradoxes, yes.

But for any y between 0 to 1m inclusive you can find a t between 0 to 1s in which the marble was at said point y. You can zoom in as far as you possibly want. How is that supposed to "stop" the marble from moving?

It isn't.

The marble doesn't need to think "okay, now I'm going to reach 0.125. Okay, now I'm going to reach 0.12500000005". It just DOES it. it passes through those points at infinite density of positions, just as there is an infinite density of time as well to correspond to every position. That's what I meant by 'continuous'.

You are very close to asking the kinds of questions that lead to good definitions of limits, and maybe continuity.

The biggest problem that you seem to be having is the assumption that there is a correspondence between a mathematical abstraction and the real world. Obviously, marbles move, Achilles catches the tortoise, and the arrow can leave the bow and eventually arrive at a target. But let's abstract the last example for a moment:

Suppose that you have a bow and an arrow. You knock the arrow, pull back the bowstring, aim at a target, and loose the arrow. It travels half way to the target, then half of the remaining distance, then half of the remaining distance, and so on. Intuitively, the distance that the arrow must travel is an infinite sum: 1+(1/2)+(1/4)+(1/8)+... . An infinite sum has an infinite number of terms. You can never stop adding new terms and arrive at a number. If you use this sum to model the motion of the arrow, then it appears that the process never stops (since an infinite sum never stops), and the arrow never reaches its target. Since you KNOW that the arrow eventually gets to the target (this is what you keep arguing), there are only two possible conclusions to draw: your mathematical model is wrong, or there is some way of terminating an infinite process.

Either conclusion is not intuitive. Either result is unexpected. Hence, it is a paradox.

xander