What is the difference between Zeno's paradox and the theory of limits?

[mark_d]

The difference between the number 0 and 1 is 1. So is the difference between 3 and 4.

Proposition:
The difference between 0 and 1 is much larger than the difference between 3 and 4.

That is because regardless of how many decimal places you go to in the calculations, there is a qualitative difference in the first statement. Nothing contrasted with something. In the second place there are two "somethings" being compared. A fundamental category shift from "nothingness" to "being" is huge.

Thoughts?

 

[Stephen Tashi]

My thought: you should post subjective questions like this the Lounge section of the forum, not in the math sections.

 

[SteveL27]

The difference between the number 0 and 1 is 1. So is the difference between 3 and 4.

Proposition:
The difference between 0 and 1 is much larger than the difference between 3 and 4.

That is because regardless of how many decimal places you go to in the calculations, there is a qualitative difference in the first statement. Nothing contrasted with something. In the second place there are two "somethings" being compared. A fundamental category shift from "nothingness" to "being" is huge.

Thoughts?

Sure, you can look at it in percentages. When you age from 10 to 20, that's an additional 100% of your age. When you go from 80 to 90, it's only a 12.5% increase. So it's much less significant.

And there is a very great difference between 0 and 1. Because if you think of some event, it can happen zero times -- it never happens. Or it can happen once, or it can be very commonplace and happen a million times.

But the difference between 0 and 1 in this context is the difference between something that never happens, and something that happens. So it's a huge difference.

 

[Mark44]

The difference between the number 0 and 1 is 1. So is the difference between 3 and 4.

Proposition:
The difference between 0 and 1 is much larger than the difference between 3 and 4.

No, it isn't. As you said above, the two differences are equal. 1 - 0 = 4 - 3 = 1.

Now, if you were talking about relative differences, that would be a different matter. The relative change from 0 to 1 is very much larger than that from 3 to 4, but if you're only talking subtraction (hence differences), then as I said, they're the same.

That is because regardless of how many decimal places you go to in the calculations, there is a qualitative difference in the first statement.

Both subtractions can be done without resorting to decimal places.

Nothing contrasted with something.

Zero is not "nothing". It is a number, just like any other real number.

In the second place there are two "somethings" being compared. A fundamental category shift from "nothingness" to "being" is huge.

Thoughts?

 

[Thetes]

Mark44,

You are in good company! Qualitative and quantitative changes are a vast subject of debate among reductionists and scientists. Qualitative changes will lead you into a better understanding of geometry and the complex domain.

Just a little from Leibniz from here:

Leibniz's ingenious attack on this Cartesian model of qualitative variety proceeds in two steps. The first step charges that motion alone is unable to account for qualitative variety at an instant: since all qualitative variety in the Cartesian system depends on motion, and there is no motion in an instant, it follows that in a Cartesian world there could be no qualitative variety at an instant.[12] The second step of Leibniz's argument charges that if the world is qualitatively homogenous at every instant, then it must be qualitatively homogenous over time as well. For if the world is qualitatively undifferentiated at each instant, then every instant will be qualitatively identical, and so the world as a whole will not undergo any qualitative change as it passes from one instant to the next. To use an anachronistic analogy, the two steps taken together imply that a Cartesian world would be like a filmstrip whose every frame was blank, and thus whose projection would not only be homogenous at each instant, but through time as well.​

 

[Mark44]

Mark44,

You are in good company! Qualitative and quantitative changes are a vast subject of debate among reductionists and scientists. Qualitative changes will lead you into a better understanding of geometry and the complex domain.

Just a little from Leibniz from here:

Leibniz's ingenious attack on this Cartesian model of qualitative variety proceeds in two steps. The first step charges that motion alone is unable to account for qualitative variety at an instant: since all qualitative variety in the Cartesian system depends on motion, and there is no motion in an instant, it follows that in a Cartesian world there could be no qualitative variety at an instant.[12] The second step of Leibniz's argument charges that if the world is qualitatively homogenous at every instant, then it must be qualitatively homogenous over time as well. For if the world is qualitatively undifferentiated at each instant, then every instant will be qualitatively identical, and so the world as a whole will not undergo any qualitative change as it passes from one instant to the next. To use an anachronistic analogy, the two steps taken together imply that a Cartesian world would be like a filmstrip whose every frame was blank, and thus whose projection would not only be homogenous at each instant, but through time as well.​

This makes me think of one of Zeno's paradoxes, in which the object under consideration is an arrow in flight. The argument goes like this: At each instant in time, the arrow is motionless at a precise point in space. The arrow is not moving toward that point, nor away from it, so the arrow must be motionless at all times.

This might pose a conundrum for philosophers, but mathematicians and physicists have this figured out.

 

[nucl34rgg]

0 is not "nothing." This is where your problem arises.
0 is an integer. The magnitude of the difference between any two consecutive integers is 1.
Here you are looking at the difference between two "somethings." (integers)

 

[SteveL27]

This makes me think of one of Zeno's paradoxes, in which the object under consideration is an arrow in flight. The argument goes like this: At each instant in time, the arrow is motionless at a precise point in space. The arrow is not moving toward that point, nor away from it, so the arrow must be motionless at all times.

This might pose a conundrum for philosophers, but mathematicians and physicists have this figured out.

I do not agree. Mathematicians have resolved the paradox with the theory of limits; a theory that is only possible to carry out in a continuum.

Physical space may or may not be a continuum. We have no evidence either way; and some theorists suggest space may consist of discrete particles or points.

The real numbers are a mathematical model. It is unknown and (IMO) doubtful that the actual, physical space is like the real numbers. And if it is -- then you have all the problems of set theory suddenly becoming problems for experimental physics. Is the Continuum Hypothesis true in physical space? This question must have a definitive answer in the physical universe; even though it's independent of the usual axioms of set theory.

So you see that imagining that the physical universe is a continuum is fraught with problems. But if the universe isn't a continuum, then the mathematical solution to Zeno's paradoxes does not apply.

Therefore Zeno's paradoxes of motion are not resolved in physics; only in mathematics. In my opinion anyway.

 

[Mark44]

This makes me think of one of Zeno's paradoxes, in which the object under consideration is an arrow in flight. The argument goes like this: At each instant in time, the arrow is motionless at a precise point in space. The arrow is not moving toward that point, nor away from it, so the arrow must be motionless at all times.

This might pose a conundrum for philosophers, but mathematicians and physicists have this figured out.

I do not agree. Mathematicians have resolved the paradox with the theory of limits; a theory that is only possible to carry out in a continuum.

I lumped physicists in with mathematicians because physicists are generally aware of mathematics concepts. That's all I meant by my remark.