Taking a page from Hume and Spinoza

[thinkandmull]

**THE PARADOX: To get from point A to point B, you first have to go the first half, than the next half, ect to infinity because if you get to a point that has no half to it, then it has no length and is not part of the line. A million zeros is ZERO. You can’t say its potentially infinite, because all the halves are there, whether you are thinking about them are not. If you cross a bridge with 7 planks, the seven planks are still there and you cross over them even though they are together as one bridge. Suppose there is a smallest unit of space. Put two of them together, draw a line, thus creating a triangle. The Hypotenuse is greater than the unit, but divide the hypotenuse in half, and you have a unit smaller than the smallest unit! In this right triangle all the points on the side opposite the right angle can be drawn to all the points on either side, while this cannot be done when the two lines are put parallel. Or can they? The points on the hypotenuse must be bunched up in a way DIFFERNT than on the other sides, because if you add a segment to a segment, you are adding points. Again let’s recap: if we say that we get to the point, from A to B, where there is only one unit of space between you and the point towards which you are going to, then this intermediate space would have no magnitude, and you would be actually at the point. It doesn’t matter if motion is instantaneous because we are discussing how the line lays there in the first place. Arithmetic doesn’t help me with this, because it very different when it comes to space.

Now, if the diameter of a marble can be crossed, why cannot we reach the prime matter of half of the marble, for Aquinas says in his article on “whether there can be an infinite magnitude” that by division we approach prime matter. The trick is that it is three dimensional and it “full up” in a way that goes behind space.

Again, if something is infinitely divisible, it is infinitely large. Even though the parts get smaller and smaller, the infinity of parts remain.

Therefore there is only conclusion for me. Imagine space going infinitely in each direction from where you are sitting. THAT infinity is what everything posseses. There is no such thing as a “limit”, there are only greater infinities (in a way that is similar to spacial reality, all the odd numbers are not equal to all the odd plus even numbers).

What do you think? **

 

[JuanFlorencio]

**
Again, if something is infinitely divisible, it is infinitely large. Even though the parts get smaller and smaller, the infinity of parts remain.

What do you think? **

Dear thinkandmull:

It just have to be demonstrated that the sum of the terms in the infinite series

S= {1/2, 1/4, 1/8, 1/16, 1/32,…}

is not infinite, but finite.

The demonstration is available in the Internet, and the result of the sum

Sigma= 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +… = 1

The conclusion is that something infinitely divisible is not necessarily infinite in magnitude.

Regards
JuanFlorencio

 

[Bahman]

**
Again, if something is infinitely divisible, it is infinitely large. Even though the parts get smaller and smaller, the infinity of parts remain.

What do you think? **

Yes, that means only that it has infinite parts.

 

[lmelahn]

**THE PARADOX: To get from point A to point B, you first have to go the first half, than the next half, ect to infinity because if you get to a point that has no half to it, then it has no length and is not part of the line. A million zeros is ZERO. You can’t say its potentially infinite, because all the halves are there, whether you are thinking about them are not. If you cross a bridge with 7 planks, the seven planks are still there and you cross over them even though they are together as one bridge. Suppose there is a smallest unit of space. Put two of them together, draw a line, thus creating a triangle. The Hypotenuse is greater than the unit, but divide the hypotenuse in half, and you have a unit smaller than the smallest unit! In this right triangle all the points on the side opposite the right angle can be drawn to all the points on either side, while this cannot be done when the two lines are put parallel. Or can they? The points on the hypotenuse must be bunched up in a way DIFFERNT than on the other sides, because if you add a segment to a segment, you are adding points. Again let’s recap: if we say that we get to the point, from A to B, where there is only one unit of space between you and the point towards which you are going to, then this intermediate space would have no magnitude, and you would be actually at the point. It doesn’t matter if motion is instantaneous because we are discussing how the line lays there in the first place. Arithmetic doesn’t help me with this, because it very different when it comes to space.

Now, if the diameter of a marble can be crossed, why cannot we reach the prime matter of half of the marble, for Aquinas says in his article on “whether there can be an infinite magnitude” that by division we approach prime matter. The trick is that it is three dimensional and it “full up” in a way that goes behind space.

Again, if something is infinitely divisible, it is infinitely large. Even though the parts get smaller and smaller, the infinity of parts remain.

Therefore there is only conclusion for me. Imagine space going infinitely in each direction from where you are sitting. THAT infinity is what everything posseses. There is no such thing as a “limit”, there are only greater infinities (in a way that is similar to spacial reality, all the odd numbers are not equal to all the odd plus even numbers).

What do you think? **

This is basically one of Zeno’s paradoxes. I think Aristotle gave the best answer to Zeno. (Have a look at VI, 9, 269b5-240a4Physics. It’s kind of a difficult read, but very interesting.)

It is important, I think, not to substantify mathematical or purely geometrical objects. Mathematical objects are, in the proper sense of the term, figments of our imagination: squares and circles and line segments (as we imagine them in Euclidean geometry–or any kind of geometry; it doesn’t matter) don’t really exist in the world. Sure, sheets of paper appear to be perfect rectangles on first inspection, but when we examine them, we notice that they have a thickness and volume, the ability to be bent and folded, and numerous small imperfections.

The same can be said for the imaginary half-distances between A and B. They don’t really exist. What exists is the real distance between the real object A and the real object B. (We might have to add the real observer O who is travelling between the two.)

So, it is true that our observer O has the option to occupy any of the points (a point is also an imaginary mathematical abstraction, by the way) between A and B, the fact is, O can only occupy one of those points at a time.

If O does stand between A and B, then he is effectively “dividing” the distance between A and B. But the imaginary points between O and A and between O and B are imaginary; they don’t exist.

If O makes the trajectory from A to B, then the whole time there are only three objects in question: A, B, and O (which is moving; Aristotle would say that O is partly “in potency” to being at point B and partly “in act,” because he is in transition). The rest of the things (points, lines, etc.) are merely mathematical constructs that help us to measure the distances and velocities.

 

[JuanFlorencio]

This is basically one of Zeno’s paradoxes. I think Aristotle gave the best answer to Zeno. (Have a look at Physics VI, 9, 269b5-240a4. It’s kind of a difficult read, but very interesting.)

It is important, I think, not to substantify mathematical or purely geometrical objects. Mathematical objects are, in the proper sense of the term, figments of our imagination: squares and circles and line segments (as we imagine them in Euclidean geometry–or any kind of geometry; it doesn’t matter) don’t really exist in the world. Sure, sheets of paper appear to be perfect rectangles on first inspection, but when we examine them, we notice that they have a thickness and volume, the ability to be bent and folded, and numerous small imperfections.

The same can be said for the imaginary half-distances between A and B. They don’t really exist. What exists is the real distance between the real object A and the real object B. (We might have to add the real observer O who is travelling between the two.)

So, it is true that our observer O has the option to occupy any of the points (a point is also an imaginary mathematical abstraction, by the way) between A and B, the fact is, O can only occupy one of those points at a time.

If O does stand between A and B, then he is effectively “dividing” the distance between A and B. But the imaginary points between O and A and between O and B are imaginary; they don’t exist.

If O makes the trajectory from A to B, then the whole time there are only three objects in question: A, B, and O (which is moving; Aristotle would say that O is partly “in potency” to being at point B and partly “in act,” because he is in transition). The rest of the things (points, lines, etc.) are merely mathematical constructs that help us to measure the distances and velocities.

Dear lmelahn:

I agree that Aristotle responded effectively to Zeno’s paradoxes. Zeno proposed a problem to demonstrate rationally that movement is impossible. Aristotle responded accordingly, in a rational fashion. You can’t make any reference to “reality” to solve this kind of problems, because they are proposed to define what “reality” is, or how “reality” is. It is like trying to “prove” an idealist that the world is “real” applying the method of kicking him in the shins.

Best regards!
JuanFlorencio

 

[thinkandmull]

I’ve already refuted that it is only potentially divided. We are talking about PARTS, not separation of parts, which is completely irrelevant. And again {1/2, 1/4, 1/8, 1/16, 1/32,…} equally one is arithmetic. Its totally different when you are considering parts of a line, because we are talking about infinite space in what appears to common sense to be finite

 

[JuanFlorencio]

I’ve already refuted that it is only potentially divided. We are talking about PARTS, not separation of parts, which is completely irrelevant. And again {1/2, 1/4, 1/8, 1/16, 1/32,…} equally one is arithmetic. Its totally different when you are considering parts of a line, because we are talking about infinite space in what appears to common sense to be finite

Dear Thinkandmull:

We use arithmetic to represent relationships between any kind of magnitudes: temperature, mass, density, voltage, speed, …, and length. This way, if 1 represents the whole length of your segment, 1/2 represents half of it, 1/4 represents a quarter of it, and so on. The union of those infinite parts is represented by the infinite sum that I mentioned. The series is convergent, and it can be demonstrated that for any infinite convergent series, the sum of its elements is finite.

Kind regards
JuanFlorencio

 

[thinkandmull]

When we are dealing with length, to say that the sum if finite is just to name the set. The infinite pieces of length would still go on, infinitely, even if if the mind says “there is a limit”. To divide one infinitely is simply mental, because out in the world one always represents something with magnitude

 

[JuanFlorencio]

When we are dealing with length, to say that the sum if finite is just to name the set. The infinite pieces of length would still go on, infinitely, even if if the mind says “there is a limit”. To divide one infinitely is simply mental, because out in the world one always represents something with magnitude

Yes, but the set has infinite elements, as you said. It is only that when you sum them, the result is finite.

It is not our mind who says “there is a limit”, but our experience.

In practice you cannot divide indefinitely; but you cannot stop thinking that divisibility has no end. Even with that, the total length -as the sum of the infinite partial lengths- is finite.

Have a good night!
JuanFlorencio

 

[jbp]

When we are dealing with length, to say that the sum if finite is just to name the set. The infinite pieces of length would still go on, infinitely, even if if the mind says “there is a limit”. To divide one infinitely is simply mental, because out in the world one always represents something with magnitude

Take calculus for crying out loud.

 

[thinkandmull]

Calculus doesn’t answer this. It using infinities as if they are finite, and some infinities larger than others.

But if there are infinite lengths, there is no finite length as the sum. That’s logically contradictory. Saying they add up to One is saying they add to a complete set, like an infinite line is a complete line. An infinite line going in each direction is the same as any other line, except it has more points. The hypotenuse has more points than the side opposite it, so there is no one to one correspondence between them

 

[jbp]

Calculus doesn’t answer this. It using infinities as if they are finite, and some infinities larger than others.

But if there are infinite lengths, there is no finite length as the sum. That’s logically contradictory. Saying they add up to One is saying they add to a complete set, like an infinite line is a complete line. An infinite line going in each direction is the same as any other line, except it has more points. The hypotenuse has more points than the side opposite it, so there is no one to one correspondence between them

I think you would have a greater appreciation of the simplicity of your problem if you have a background in higher mathematics. It is embodied in the concept of the limit. Or maybe convergent and divergent series and sums with a limit.

Or perhaps, finite area bounded by an infinite length (yes, they do exist. lots of them). Try watching khanacademy.org lecture videos on math and calculus because they are so good and short and easy to understand.

 

[thinkandmull]

A surface has no depth, so how could it be greater than what it contains?

Benjamin Allen “It is mathematically possible for a faster thing to pursue a slower thing forever and still never catch it, so long as both the faster thing and the slower thing both keep slowing down in the right way.” He says something about convergence vs divergences, but I highly doubt he’s right

Aquinas said that an infinite set cannot be numbered when speaking of the alleged past infinite ages, but are you not saying your can call an infinite finite and thus number it?

 

[Bahman]

A surface has no depth, so how could it be greater than what it contains?

Benjamin Allen “It is mathematically possible for a faster thing to pursue a slower thing forever and still never catch it, so long as both the faster thing and the slower thing both keep slowing down in the right way.” He says something about convergence vs divergences, but I highly doubt he’s right

Aquinas said that an infinite set cannot be numbered when speaking of the alleged past infinite ages, but are you not saying your can call an infinite finite and thus number it?

Aquinas didn’t know much about math. There is no problem with infinite regress. In simple word, you can have a set of infinite beings with finite magnitude.

 

[thinkandmull]

So the length is infinite but it ends? You don’t believe in the Law of non-Contradiction?

 

[Bahman]

So the length is infinite but it ends?

Yes.

You don’t believe in the Law of non-Contradiction?

I do only believe in I to be honest. The rest is just construct of mind, endless.

 

[JuanFlorencio]

So the length is infinite but it ends? You don’t believe in the Law of non-Contradiction?

Dear Thinkandmull:

The statement “an infinite length ends” is contradictory, of course. But the statement “a finite length is infinitely divisible” is not; or if you prefer “a finite length has infinite parts”, is non contradictory either. I will suggest you something, which is not a demonstration, but might make it easier for you to move on: take an Excel sheet and write 1/2 in cell A2. Then write this formula in cell A3: “=A2/2”. The result displayed in A3 will be 1/4, which is half of 1/2. Copy cell A3 to cells A4…A100. In each one of the subsequent cells you will see a value which is half of the previous one.

Then, write in cell A1 the formula "=sum(A2:A100). This is the sum of all the values in the cells from A2 to A100. The result will be a number smaller than 1.

Extend now the formula in cell A2 till A1000.

Modify the formula in cell A1 to “=sum(A2:A1000)”. I guess you would expect to see a value greater that 1, but it will not happen. You will see a value closer to 1, but still less than it.

Extend now the formula in cell A2 till A10000.

Modify the formula in cell A1 to “=sum(A2:A10000)”. You will see a value even closer to 1, but still less than it.

You can continue indefinitely, but the sum in A1 will never be greater than 1.

I hope this will help you. The principle of non contradiction holds.

Best regards
JuanFlorencio

 

[Bahman]

Dear Thinkandmull:

The statement “an infinite length ends” is contradictory, of course. But the statement “a finite length is infinitely divisible” is not; or if you prefer “a finite length has infinite parts”, is non contradictory either. I will suggest you something, which is not a demonstration, but might make it easier for you to move on: take an Excel sheet and write 1/2 in cell A2. Then write this formula in cell A3: “=A2/2”. The result displayed in A3 will be 1/4, which is half of 1/2. Copy cell A3 to cells A4…A100. In each one of the subsequent cells you will see a value which is half of the previous one.

Then, write in cell A1 the formula "=sum(A2:A100). This is the sum of all the values in the cells from A2 to A100. The result will be a number smaller than 1.

Extend now the formula in cell A2 till A1000.

Modify the formula in cell A1 to “=sum(A2:A1000)”. I guess you would expect to see a value greater that 1, but it will not happen. You will see a value closer to 1, but still less than it.

Extend now the formula in cell A2 till A10000.

Modify the formula in cell A1 to “=sum(A2:A10000)”. You will see a value even closer to 1, but still less than it.

You can continue indefinitely, but the sum in A1 will never be greater than 1.

I hope this will help you. The principle of non contradiction holds.

Best regards
JuanFlorencio

So knowing the fact that you know all this math shows that you know that infinite regress is possible?

 

[JuanFlorencio]

So knowing the fact that you know all this math shows that you know that infinite regress is possible?

Hi Bahman!,

I did not understand you question. Could you please re-phrase it?

Best regards
JuanFlorencio

 

[Bahman]

Hi Bahman!,

I did not understand you question. Could you please re-phrase it?

Best regards
JuanFlorencio

I can give you a math function Y=Y(X) (read it X cause Y). I can define a set produced by this function S={S0,S1…} where S1=Y(S0) etc. Now I define an amplitude a=A (for example the sum of a geometric series) which exists and it is finite. Hence there is no problem with infinite regress if the function and the norm are defined properly.