# Ancient Philosophy and Contemporary Problems

## Friday, March 31, 2006

i think i missed the email we're supposed to send our paper to in class. could someone help me out? the paper is due at midnight tonight too, right?

## Wednesday, March 29, 2006

I have two quick questions about the paper:

Is there a length requirement/limit?

Do we have to define view that we wish to discuss (second point on the paper explaining essay 2)?

I'm only asking this because I'm discussing motion and it seems self explanatory after I give a definition of this. There is not much of a specific theory of motion that I'll be discussing.

Is there a length requirement/limit?

Do we have to define view that we wish to discuss (second point on the paper explaining essay 2)?

I'm only asking this because I'm discussing motion and it seems self explanatory after I give a definition of this. There is not much of a specific theory of motion that I'll be discussing.

## Saturday, March 18, 2006

I have an interesting proposal to make concerning the nature of space-time, granted that I have no clue at all whether its right or not.

A lot of our discussion on the nature of space-time is concerned with the attempt to break space-time down into infinitesimals; we have argued on and off the blog about the nature of limits, infinity, motion, and other related issues. However, I would like to propose that it is impossible to break space-time down into infinitesimals; not because space-time isn’t made of infinitesimal quantities, but because of the following (this is not a formal argument, but rather a listing of facts that I will use to show that space-time cannot be broken down into infinitesimals):

1.) Any finite amount subtracted from infinity will produce infinity. Stated in a more rigorous (i.e. mathematical) way, whenever any finite number of members of an infinite set is removed from an infinite set, the cardinality of the set does not change.

2.) Any finite amount added to any other finite amount will always produce a finite amount. Once more, it can be stated in set theory as follows: in the union of any two finite sets, a finite set will always be produced.

3.) An infinitely small amount added to any other infinitely small amount will always produce another infinitely small amount.

4.) Infinity subtracted from infinity is undefined. This is fairly easy to show. Imagine that we have the sum of all positive integers (which is equal to infinity):

S=1+2+3+4+5+6+7+8…

And the sum of all negative even integers:

N=-2-4-6-8…

Note that the sum of all negative even integers is negative infinity. Therefore, if we add S to N, we will get this: S+N=1+3+5+7… That is, we will be left with the sum of all positive odd numbers. Of course, that is also equal to infinity. But suppose that we had the sum of all negative integers:

A=-1-2-3-4-5-6-7-8…

And next suppose that we added S to A. Obviously, that would produce zero – or A+S=0. But also notice that A=N (because they are both equal to negative infinity.) Therefore, this leads us to conclude that S+N=S+A, or zero equals infinity. Obviously, that’s not true. Therefore, it must be the case that infinity minus infinity is undefined (actually, this is very similar to how any number divided by zero is undefined.) For mathematical reasons that I will not include here, none of this is contradictory – if it is the case that infinity minus infinity is undefined or indeterminate (because a contradiction would only result if infinity minus infinity could only have one value. Saying that it is indeterminate or undefined means that it either has no value or that its value cannot be determined. It's similiar to the end state of one of Black's infinity machines.)

Let us assume that space-time is continuous - or, at any rate, that it consists of infinitesmals (for technical reasons, there can exist a difference between a continuum and a space consisting of infinitesmals, although there doesn't have to. That is, a continuum is a subset of the set of infinitesmal-consisting spaces.) Due to (1), there is no operation by which space-time could be broken down into infinitesimals unless infinite amounts were taken away. But according to (4), there does not exist the operation of taking away infinity because it gives indeterminate results. Also note that due to (2) there is no way in which any finite number of objects can be placed together to produce an infinite number. Thus, no matter how many infinitesimals are added together, you could never get continuous space (or – any finite set united with any other finite set yields a finite set.) Also note that this is further confirmed by (3). Therefore, I propose that space cannot be broken down into infinitesimals and that there is no operation by which space could be built back up by infinitesimals. In fact, it is enough to just say that one of these processes cannot actually be performed. Imagine that space could be broken down into infinitesimals. Now imagine that we have a movie of space being broken down into infinitesimals. If we simply ran the movie backwards, we would see space being built by infinitesimals. Therefore, if we can eliminate one process, we also have eliminated the other. Furthermore, it is easy to see how this could be extended to sets in general (after all, space-time is nothing more than a metrizable set - this arises out of set topology.) That is, the properties of the proper subsets of an infinite set do not reflect the properties of the set in general - i.e. the properties of an infinite set appear to be emergent properties which are not easily related to its members.

That being said, I am not convinced that a continuous view of space-time is incorrect (or at least that the arguments given here are efficient reasons to believe that that is the case. See the latest special issue of Scientific American - specifically the article on "Atoms of Space And Time" for an introduction to loop quantum gravity, which apparently requires that space-time is quantized.) I just think that it may be the case that there is a distinction that must be drawn between the infinitesimals that make up space and space itself. Perhaps this distinction, if it in fact exists, gives the infinitesimal units of space properties which macro-space does not have. What I am proposing is that all of our difficulties concerning space-time are simply due to the fact that this distinction exists. That the impossibility of breaking motion down into infinitely small parts, for instance, does not constitute an argument against motion because a continuum has the property of being unable to be broken down into its parts. If continuous motion were possible, it would have the property that we could not break it down into its parts through any finite operation. I do not see this as an actual problem – only a conceptual one. While it may be difficult to imagine that there could exist such a huge distinction between an object and its parts, I hold that this difficulty is due only to the finite size of our brains and has nothing to do with any actual logical impossibility.

Also, I would like to point out that what we are discussing is not whether or not the space-time we live in is actually quantized or whether motion is impossible or what have you, only whether or not motion in continuous space is logically impossible. Of course, if it is logically impossible, then it could not be case that we live in a continuous space-time. But if it is logically possible to have motion in a continous space-time, then it could still be the case that the space-time we live in is actually quantized. A logical possibility does not necessarily gaurantee the truth of the matter (i.e. while something may be logically possible, it might be physically impossible or simply untrue.) In other words, if it is the case that either possibility is logically possible, then the matter cannot be concluded apriori, but must be concluded through experimentation. As an example, examine the hypothesis that grass is green. If you had never seen grass, you would have no reason to suspect that this statement was true or even that it was false based upon logic alone. Obviously, the statement is sound, but if one had never seen grass, the statement may appear just as sound as the statement that grass is blue. The fact of the matter is that until the color of grass was observed, its color-state could not be determined (and, therefore, the soundness of the proposition was on equally shaky grounds.) Likewise, if it is the case that there is, in fact, no logical contradiction that occurs from assuming that space-time is continuous or that it is quantized, then the soundness of such a proposition cannot be determined in terms of logic alone.

A lot of our discussion on the nature of space-time is concerned with the attempt to break space-time down into infinitesimals; we have argued on and off the blog about the nature of limits, infinity, motion, and other related issues. However, I would like to propose that it is impossible to break space-time down into infinitesimals; not because space-time isn’t made of infinitesimal quantities, but because of the following (this is not a formal argument, but rather a listing of facts that I will use to show that space-time cannot be broken down into infinitesimals):

1.) Any finite amount subtracted from infinity will produce infinity. Stated in a more rigorous (i.e. mathematical) way, whenever any finite number of members of an infinite set is removed from an infinite set, the cardinality of the set does not change.

2.) Any finite amount added to any other finite amount will always produce a finite amount. Once more, it can be stated in set theory as follows: in the union of any two finite sets, a finite set will always be produced.

3.) An infinitely small amount added to any other infinitely small amount will always produce another infinitely small amount.

4.) Infinity subtracted from infinity is undefined. This is fairly easy to show. Imagine that we have the sum of all positive integers (which is equal to infinity):

S=1+2+3+4+5+6+7+8…

And the sum of all negative even integers:

N=-2-4-6-8…

Note that the sum of all negative even integers is negative infinity. Therefore, if we add S to N, we will get this: S+N=1+3+5+7… That is, we will be left with the sum of all positive odd numbers. Of course, that is also equal to infinity. But suppose that we had the sum of all negative integers:

A=-1-2-3-4-5-6-7-8…

And next suppose that we added S to A. Obviously, that would produce zero – or A+S=0. But also notice that A=N (because they are both equal to negative infinity.) Therefore, this leads us to conclude that S+N=S+A, or zero equals infinity. Obviously, that’s not true. Therefore, it must be the case that infinity minus infinity is undefined (actually, this is very similar to how any number divided by zero is undefined.) For mathematical reasons that I will not include here, none of this is contradictory – if it is the case that infinity minus infinity is undefined or indeterminate (because a contradiction would only result if infinity minus infinity could only have one value. Saying that it is indeterminate or undefined means that it either has no value or that its value cannot be determined. It's similiar to the end state of one of Black's infinity machines.)

Let us assume that space-time is continuous - or, at any rate, that it consists of infinitesmals (for technical reasons, there can exist a difference between a continuum and a space consisting of infinitesmals, although there doesn't have to. That is, a continuum is a subset of the set of infinitesmal-consisting spaces.) Due to (1), there is no operation by which space-time could be broken down into infinitesimals unless infinite amounts were taken away. But according to (4), there does not exist the operation of taking away infinity because it gives indeterminate results. Also note that due to (2) there is no way in which any finite number of objects can be placed together to produce an infinite number. Thus, no matter how many infinitesimals are added together, you could never get continuous space (or – any finite set united with any other finite set yields a finite set.) Also note that this is further confirmed by (3). Therefore, I propose that space cannot be broken down into infinitesimals and that there is no operation by which space could be built back up by infinitesimals. In fact, it is enough to just say that one of these processes cannot actually be performed. Imagine that space could be broken down into infinitesimals. Now imagine that we have a movie of space being broken down into infinitesimals. If we simply ran the movie backwards, we would see space being built by infinitesimals. Therefore, if we can eliminate one process, we also have eliminated the other. Furthermore, it is easy to see how this could be extended to sets in general (after all, space-time is nothing more than a metrizable set - this arises out of set topology.) That is, the properties of the proper subsets of an infinite set do not reflect the properties of the set in general - i.e. the properties of an infinite set appear to be emergent properties which are not easily related to its members.

That being said, I am not convinced that a continuous view of space-time is incorrect (or at least that the arguments given here are efficient reasons to believe that that is the case. See the latest special issue of Scientific American - specifically the article on "Atoms of Space And Time" for an introduction to loop quantum gravity, which apparently requires that space-time is quantized.) I just think that it may be the case that there is a distinction that must be drawn between the infinitesimals that make up space and space itself. Perhaps this distinction, if it in fact exists, gives the infinitesimal units of space properties which macro-space does not have. What I am proposing is that all of our difficulties concerning space-time are simply due to the fact that this distinction exists. That the impossibility of breaking motion down into infinitely small parts, for instance, does not constitute an argument against motion because a continuum has the property of being unable to be broken down into its parts. If continuous motion were possible, it would have the property that we could not break it down into its parts through any finite operation. I do not see this as an actual problem – only a conceptual one. While it may be difficult to imagine that there could exist such a huge distinction between an object and its parts, I hold that this difficulty is due only to the finite size of our brains and has nothing to do with any actual logical impossibility.

Also, I would like to point out that what we are discussing is not whether or not the space-time we live in is actually quantized or whether motion is impossible or what have you, only whether or not motion in continuous space is logically impossible. Of course, if it is logically impossible, then it could not be case that we live in a continuous space-time. But if it is logically possible to have motion in a continous space-time, then it could still be the case that the space-time we live in is actually quantized. A logical possibility does not necessarily gaurantee the truth of the matter (i.e. while something may be logically possible, it might be physically impossible or simply untrue.) In other words, if it is the case that either possibility is logically possible, then the matter cannot be concluded apriori, but must be concluded through experimentation. As an example, examine the hypothesis that grass is green. If you had never seen grass, you would have no reason to suspect that this statement was true or even that it was false based upon logic alone. Obviously, the statement is sound, but if one had never seen grass, the statement may appear just as sound as the statement that grass is blue. The fact of the matter is that until the color of grass was observed, its color-state could not be determined (and, therefore, the soundness of the proposition was on equally shaky grounds.) Likewise, if it is the case that there is, in fact, no logical contradiction that occurs from assuming that space-time is continuous or that it is quantized, then the soundness of such a proposition cannot be determined in terms of logic alone.

## Monday, March 13, 2006

I strongly disagree, namely for the reasons given in my response to his post.

Additionally, here are some other reasons that velocity at an instant must be defined to have some numerical value in some manner:

1. The derivative of the velocity is very well defined (namely, it is the accelleration.) If it were truly the case that the velocity was not defined at each slice of time, but instead had only a numerical value over a broad set of times, then the velocity curve could not be said to be continous. If the velocity curve were not continuous, then one could not find a value for its derivative at all times. This is easier to see if you imagine plotting the velocity versus the time. For any point along one axis, there is a corresponding point along the other. This is the definition of a function. In other words, since velocity can be expressed as a function of time, the cardinality of the set of velocities is identical to the cardinality of the set of times (they are both continuum); furthermore, for every member of the set of velocities, there is a corresponding member of the set of times. If this were not the case, velocity would not be a function; if velocity were not a function, then you could not take its derivative. Thus, if velocity were not defined at every time t then accelleration would not exist.

2. There is one object whose velocity is the same irrespective of time. That object is light, whose velocity, c, is independent of time (this arises out of Maxwell's equations and is used as a postulate in Einstein's Special Relativity.) If velocity could only be defined by using a large number of space-time slices, then it would be the case that the velocity of light wasn't defined at any particular point in time (but rather had to be taken over a series of times.) However, this is not the case - the velocity of light is defined at a time precisely because the equations for it are independent of time.

3. Could it not be that a limiting operation gives the behavior of a function at a point? Why are you resistent to think that it might be the case that the limit of a function at a point is identical to the function's behavior at that point? Although it is true that not all functions behave at a given point as their limits do, there do exist functions which do behave exactly as their limits do. For these functions, I believe it is safe to assume that the behavior of the point is identical to the behavior of the limits precisely because we already know, a priori, that this is so. For instance, the limit of the function y=5x at x=3 is 15, but the value of y at that point is also 15. Therefore, for that particular function, the value of the limit is identical to the value of the function - something that we all have to agree upon precisely because 15=15. There is no reason to suppose, as far as I can see, that the limit involved in the definition of the derivative would be any different for many functions.

By the way, are you supposing that if every individual member of a given set does not have some property or feature F then the whole set might? Please note that the argument against motion given in class depends entirely upon the opposite (i.e. that if every individual member of a given set does not have F then the whole set will not either.) If some property of a set (such as the set of times) can only be defined over several members of the set (such as your idea of motion) then this must be the case. However, I believe that there exist several difficulties in supposing that this is, in fact, the case. For instance, if for all times t, some statement A is not true, then it cannot be the case that collectively, over the times t, A is true. Under your ideas of motion, you would be forced to say something like, "Motion is not defined for all times t, but it is defined for the set collectively", which would seem to contradict the statement I just made. That is, if a logical statement is false at one time, then it must be the case that it is false for all times, simply because such statements are independent of time.

Here's some sites you might find useful:

Velocity versus time graphs (high school physics and kinematics): http://www.glenbrook.k12.il.us/gbssci/phys/Class/1DKin/U1L4a.html

Definition of a function (including how it relates to set theory): http://mathworld.wolfram.com/Function.html

Definition of the derivative: http://mathworld.wolfram.com/Derivative.html

Maxwell's Equations (including the definition of the speed of light arising therefrom): http://en.wikipedia.org/wiki/Maxwell%27s_Equations

Additionally, here are some other reasons that velocity at an instant must be defined to have some numerical value in some manner:

1. The derivative of the velocity is very well defined (namely, it is the accelleration.) If it were truly the case that the velocity was not defined at each slice of time, but instead had only a numerical value over a broad set of times, then the velocity curve could not be said to be continous. If the velocity curve were not continuous, then one could not find a value for its derivative at all times. This is easier to see if you imagine plotting the velocity versus the time. For any point along one axis, there is a corresponding point along the other. This is the definition of a function. In other words, since velocity can be expressed as a function of time, the cardinality of the set of velocities is identical to the cardinality of the set of times (they are both continuum); furthermore, for every member of the set of velocities, there is a corresponding member of the set of times. If this were not the case, velocity would not be a function; if velocity were not a function, then you could not take its derivative. Thus, if velocity were not defined at every time t then accelleration would not exist.

2. There is one object whose velocity is the same irrespective of time. That object is light, whose velocity, c, is independent of time (this arises out of Maxwell's equations and is used as a postulate in Einstein's Special Relativity.) If velocity could only be defined by using a large number of space-time slices, then it would be the case that the velocity of light wasn't defined at any particular point in time (but rather had to be taken over a series of times.) However, this is not the case - the velocity of light is defined at a time precisely because the equations for it are independent of time.

3. Could it not be that a limiting operation gives the behavior of a function at a point? Why are you resistent to think that it might be the case that the limit of a function at a point is identical to the function's behavior at that point? Although it is true that not all functions behave at a given point as their limits do, there do exist functions which do behave exactly as their limits do. For these functions, I believe it is safe to assume that the behavior of the point is identical to the behavior of the limits precisely because we already know, a priori, that this is so. For instance, the limit of the function y=5x at x=3 is 15, but the value of y at that point is also 15. Therefore, for that particular function, the value of the limit is identical to the value of the function - something that we all have to agree upon precisely because 15=15. There is no reason to suppose, as far as I can see, that the limit involved in the definition of the derivative would be any different for many functions.

By the way, are you supposing that if every individual member of a given set does not have some property or feature F then the whole set might? Please note that the argument against motion given in class depends entirely upon the opposite (i.e. that if every individual member of a given set does not have F then the whole set will not either.) If some property of a set (such as the set of times) can only be defined over several members of the set (such as your idea of motion) then this must be the case. However, I believe that there exist several difficulties in supposing that this is, in fact, the case. For instance, if for all times t, some statement A is not true, then it cannot be the case that collectively, over the times t, A is true. Under your ideas of motion, you would be forced to say something like, "Motion is not defined for all times t, but it is defined for the set collectively", which would seem to contradict the statement I just made. That is, if a logical statement is false at one time, then it must be the case that it is false for all times, simply because such statements are independent of time.

Here's some sites you might find useful:

Velocity versus time graphs (high school physics and kinematics): http://www.glenbrook.k12.il.us/gbssci/phys/Class/1DKin/U1L4a.html

Definition of a function (including how it relates to set theory): http://mathworld.wolfram.com/Function.html

Definition of the derivative: http://mathworld.wolfram.com/Derivative.html

Maxwell's Equations (including the definition of the speed of light arising therefrom): http://en.wikipedia.org/wiki/Maxwell%27s_Equations

## Saturday, March 11, 2006

I agree, it seems that the concept of an instantaneous slice of time wouldn't make a lot of sense. A frame of time, to be defined, would have to have some measureable dimensions- such as length. If the length of the time frame is zero, wouldn't the time frame not exist?

Also, usually the only time we talk about an instantaneous time frame is in a context such as physics with instantaneous velocity - velocity is displacement / time so if time is zero, this clearly becomes undefined. What is really happening here, is we take the limit as x -> 0 of displacement / x so x is extremely small, but not really equal to zero.

SO, when we take an instant of time of length 0, the velocity of an object becomes undefined, and therefore the object can not be determined to be moving. But this is not correct - an object in motion has velocity of greater than zero, but by this definition, we need more than one zero-length time frame to determine the velocity.

I think based on this, you can't look at any single instant of time and find motion, motion needs to be defined in terms of multiple consecutive frames of time.

Also, usually the only time we talk about an instantaneous time frame is in a context such as physics with instantaneous velocity - velocity is displacement / time so if time is zero, this clearly becomes undefined. What is really happening here, is we take the limit as x -> 0 of displacement / x so x is extremely small, but not really equal to zero.

SO, when we take an instant of time of length 0, the velocity of an object becomes undefined, and therefore the object can not be determined to be moving. But this is not correct - an object in motion has velocity of greater than zero, but by this definition, we need more than one zero-length time frame to determine the velocity.

I think based on this, you can't look at any single instant of time and find motion, motion needs to be defined in terms of multiple consecutive frames of time.

## Thursday, March 09, 2006

The Fourth Dimension

Before the twentieth century many people believed that every person experienced only three dimensions. This included most physicists who described the world within mathematics as mainly three dimensional and that time was absolute. In the beginning of the twentieth century came the reformulation of the basic principle of physics. This was the theory of special relativity.

The great mind of Albert Einstein has introduced to the world that time is no different than space, they are interchangeable. For example, what if you had an appointment in the heart of Manhattan? You are given instructions to meet someone as follows. Go to the corner of Fourth Street and Ninetieth Street. Then proceed in the building up to the tenth floor. Here you were given the three spatial directions to meet someone. However, you might be standing there for eternity because you never had the indication of when you were suppose to meet someone. Therefore, time is just as important as all three spatial dimensions.

In class discussion we talk about the arrow not moving at an instant. Now that we concluded that space and time are one in the same we have concluded that just as there no possible way to move an infinite amount of positions in a finite amount. Similarly, with time there is no way to capture an instant of a moment in a time equal to zero. This would violate the laws of special relativity. We can perceive this idea in our mind of a slice of the reality loaf and assume that it is an instant of time, but this is impossible. To capture a picture at an instant you would have to open and close you eyelids in an instant which would be physically impossible.

With the argument of Zeno I have concluded that it impossible to view an instant of a motion. It is easy to presume that at an instant something is not moving when flash the image in our mind, but with our human senses this impossible. If you looked at object moving and took a snap shot it would appear to be not moving in the photograph. The truth is that the object was moving during the time when the shutter opened and then closed. Furthermore, it is impossible to slice the reality loaf in an infinitely thin slice.

(Note: This is Chad's post.)

Before the twentieth century many people believed that every person experienced only three dimensions. This included most physicists who described the world within mathematics as mainly three dimensional and that time was absolute. In the beginning of the twentieth century came the reformulation of the basic principle of physics. This was the theory of special relativity.

The great mind of Albert Einstein has introduced to the world that time is no different than space, they are interchangeable. For example, what if you had an appointment in the heart of Manhattan? You are given instructions to meet someone as follows. Go to the corner of Fourth Street and Ninetieth Street. Then proceed in the building up to the tenth floor. Here you were given the three spatial directions to meet someone. However, you might be standing there for eternity because you never had the indication of when you were suppose to meet someone. Therefore, time is just as important as all three spatial dimensions.

In class discussion we talk about the arrow not moving at an instant. Now that we concluded that space and time are one in the same we have concluded that just as there no possible way to move an infinite amount of positions in a finite amount. Similarly, with time there is no way to capture an instant of a moment in a time equal to zero. This would violate the laws of special relativity. We can perceive this idea in our mind of a slice of the reality loaf and assume that it is an instant of time, but this is impossible. To capture a picture at an instant you would have to open and close you eyelids in an instant which would be physically impossible.

With the argument of Zeno I have concluded that it impossible to view an instant of a motion. It is easy to presume that at an instant something is not moving when flash the image in our mind, but with our human senses this impossible. If you looked at object moving and took a snap shot it would appear to be not moving in the photograph. The truth is that the object was moving during the time when the shutter opened and then closed. Furthermore, it is impossible to slice the reality loaf in an infinitely thin slice.

(Note: This is Chad's post.)

## Wednesday, March 08, 2006

Sets, Infinity, Limits, and Functions

I hope I have substantiated my claim that one does not need facility with modern mathematics in order to grasp the philosophical problems posed by Zeno's paradoxes. Regardless, some of you have asked about the mathematics employed in addressing the mathematical problems posed by Zeno's paradozes. I have intentionally not focused on the mathematics in class. In case you are curious, I have posted two discussions by Wesley Salmon that deal with the mathematical concepts involved. The first discusses the concept of a limit, the sum of an infinite series, and mathematical functions. The second discusses sets and infinity. Both expositions assume no background in modern mathematics. If you read either and have questions about them, please comment to this post.

I hope I have substantiated my claim that one does not need facility with modern mathematics in order to grasp the philosophical problems posed by Zeno's paradoxes. Regardless, some of you have asked about the mathematics employed in addressing the mathematical problems posed by Zeno's paradozes. I have intentionally not focused on the mathematics in class. In case you are curious, I have posted two discussions by Wesley Salmon that deal with the mathematical concepts involved. The first discusses the concept of a limit, the sum of an infinite series, and mathematical functions. The second discusses sets and infinity. Both expositions assume no background in modern mathematics. If you read either and have questions about them, please comment to this post.

## Monday, March 06, 2006

I came across why Black's argument is fallacious on the Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/entries/spacetime-supertasks/)....

Black's argument, in case you don't remember it or didn't read, says basically that you've got a machine that can take a ball and move it from point A to point B over some time t. It keeps moving it back and forth, until at some final time it's oscillating back and forth infinitely fast, so that in the time t the machine performs a super task. We ask whether this is logically possible in order to deduce whether a supertask can be performed... Of course, Black derives a contradiction from this, saying that if a supertask was performed, then the state of the ball (whether its at A or B) cannot be determined (by one argument it's at A; by another, it's at B; it can't be at both places at the same time, so there's a contradiction.) What the Stanford Encyclopedia of Philosophy says is that Black's entire reasoning is false for the following reason: the state of the ball (whether it's at A or B) is not determined by what has occured to the ball before that time. In their own words:

"His argument is fallacious because he seeks to reach a logical conclusion regarding instant t* = 1 P.M. from information relative to times previous to that instant.. In the standard argument against Zeno's dichotomy one could similarly specify Achilles's position at t* = 1 P.M. saying, for instance, that he is at B (x = 1), but there is no way that this is going to get us a valid argument out of a fallacious one, which seeks to deduce logically where Achilles will be at t* = 1 P.M. from information previous to that instant of time."

What do you guys think about this?

Black's argument, in case you don't remember it or didn't read, says basically that you've got a machine that can take a ball and move it from point A to point B over some time t. It keeps moving it back and forth, until at some final time it's oscillating back and forth infinitely fast, so that in the time t the machine performs a super task. We ask whether this is logically possible in order to deduce whether a supertask can be performed... Of course, Black derives a contradiction from this, saying that if a supertask was performed, then the state of the ball (whether its at A or B) cannot be determined (by one argument it's at A; by another, it's at B; it can't be at both places at the same time, so there's a contradiction.) What the Stanford Encyclopedia of Philosophy says is that Black's entire reasoning is false for the following reason: the state of the ball (whether it's at A or B) is not determined by what has occured to the ball before that time. In their own words:

"His argument is fallacious because he seeks to reach a logical conclusion regarding instant t* = 1 P.M. from information relative to times previous to that instant.. In the standard argument against Zeno's dichotomy one could similarly specify Achilles's position at t* = 1 P.M. saying, for instance, that he is at B (x = 1), but there is no way that this is going to get us a valid argument out of a fallacious one, which seeks to deduce logically where Achilles will be at t* = 1 P.M. from information previous to that instant of time."

What do you guys think about this?

I went through the geometry of the whole Weyl Tile thingy and I've come up with some interesting results regarding the percent deviation between Euclidean (Pythagorean) right triangles (specifically, 45-45-90 triangles) and the triangles produced in the Weyl tiling (those weird stair case psuedo triangles we discussed in class.)

Suppose that we have a 45-45-90 right triangle whose legs both measure n units in length. Then, by the Pythagorean Theorem, n^2+n^2=r^2, where r=the length of the hypotenuse. So:

n^2+n^2=r^2

2n^2=r^2

Thus, r=sqrt(2n^2).

Therefore, r=n*sqrt(2).

So, for any 45-45-90 right triangle on the Euclidean plane, the Pythagorean Theorem tells us that the length of the hypotenuse is given by the length of one of the legs multiplied by the square root of 2. For the Weyl Tiling, the psuedo triangle's hypotenuse has a length of n. Therefore, the perecent deviation between the length of a psuedo triangle's hypotenuse and a Euclidean right triangle's hypotenuse is found through the following:

%Deviation = ([(n-n*sqrt(2)])/(n*sqrt(2)))*100 where we take the value given by the Pythagorean theorem as the reference value. Note that the "[" and "]" are absolute value symbols. For those unfamiliar with %dev calculations, the equation is simply given by the absolute value of the difference of the value you found to the reference value divided by the reference value and then multiplied by 100 (to turn it into a percentage.) It gives the deviation between any two values as a percentage of the reference value.

Now here comes the important part, which most people have already noticed already - the n's cancel out. So, we get:

%Deviation = ([(n-n*sqrt(2)])/(n*sqrt(2)))*100 = ([1-sqrt(2)]/sqrt(2))*100 for any 45-45-90 triangle whatsoever, no matter its size! So what is that percent deviation? Well, using my trusty TI-83, I get a value of approx 29.29%. Once again, that's true for any 45-45-90 triangle (on the Euclidean plane, of course!)

Now let's calculate how certain physical measurements would change if we lived in a universe in which space was quantized in a manner identical to the Weyl tiling. According to http://www.newscientist.com/channel/fundamentals/quantum-world/mg18424752.300, the smallest distance that any one could ever measure even in principle is about 10^-33 meters (the Planck length), other wise the measuring device would collapse into a black hole of its own making. So, let's suppose that space quanta are as small as they possibly could be, while still being measurable (if it was any smaller, it wouldn't even matter because no one could ever even hope to measure it.) Therefore, based on that assumption, space quanta must be on the scale of 10^-33 meters. Now, let's also suppose that we have a 45-45-90 triangle whose base is one meter. Then, it would measure 10^33 space quanta units on its two bases. Therefore, we have (10^33)^2+(10^33)^2=2*10^66 units^2 - giving a hypotenuse of sqrt(2)*10^33 units (or the sqrt(2) meters.) But the hypotenuse of the corresponding psuedo triangle measures 10^33 units, or 1 meter. That's a difference of approx 0.414 meters, or approx 1.36 feet. Since it's entirely reasonable to measure 1.36 feet, I conclude that if our universe does have a quantized space-time, then it is not quantized in the form of a Weyl tiling (assuming that you actually believe our experimental results from over 4000 years of measuring things - the ancient Babylonians and Chinese had figured out some of the useful properties of 3-4-5 right triangles and so forth - which are derived from the Pythagorean theorem - more than 4000 years ago.)

Next, I will show that there is a very large set of plane tilings which have this property. I will make use of the fact that for any plane tiling there is a corresponding planar network. Suppose you have a line of dots running horizontally across a page. Then, vertically down the page, you have an equal number of dots. For each dot running horizontally, there is a corresponding dot in the vertical direction. Draw a line down from each horizontal dot, and another outwards from each vertical dot. End the line where the two intersect. You should now have two lines, at 90 degrees to each other, one coming down from a horizontal dot and the other outwards from a vertical dot. Repeat this procedure for each pair of vertical/horizontal dots. The nodes formed by all those intersections now becomes a diagonal running at 45 degrees between the horizontal and the vertical. This pattern is a network which can be mapped onto any uniform tiling to form the right triangle given by that particular tiling. For instance, if we take this network and map it onto the Weyl tiling, we produce that same old stair case pattern. If we take a tiling of circles on the plane, a similiar thing happens. In fact, for any uniform periodic tiling of the Euclidean plane, this turns out to be true. Notice that each node corresponds to a tile on the tiling, and that there is a one to one correspondence between the set of nodes in one of the legs of the triangle and the number of nodes in the hypotenuse. Therefore, for any uniform periodic tiling of the Euclidean plane the number of polygons in the legs correspond to the number of polygons in the hypotenuse, meaning that the deviations from the Pythagorean theorem I derived above hold in general for the entire set.

Next, I will show that this mapping is even true for a large number of non-Euclidean geometries. For this I will use the topological property of a surface known as homeomorphism. Two geometries are said to be homeomorphic (i.e. topologically identical) if their topological properties remain invariant as one is continuously transformed into the other. In other words, two shapes are homeomorphic if a continous transformation can change one into the other without tearing or connecting the surface in a way that it wasn't connected before. In this way, a coffee cup is homeomorphic to a donut. Since the surfaces would remain internally connected in the same way as it was before under a topological transformation (that is, topological invariants would be, well, invariant) the mapping I outlined above wouldn't really change - it would just get stretched or bent or what not. Imagine that you have a picture laid down onto a piece of silly putty. When you stretch or bend the silly putty, you are performing a topological transformation. Under that transformation, the network I constructed above would also remain topologically invariant. In other words, a given network when laid down upon a surface remains topologically invariant under a given transformation of the surface so long as the resultant surface is homeomorphic to the original surface. So, the resultant network/tiling would be homeomorphic to the original network/tiling. Of course, under such transformation, the relationship between the lengths of the legs of a right triangle and its height is not invariant - that is, the Pythagorean Theorem does not apply to surfaces other than the Euclidean plane. So, if the Weyl tiling's psuedo triangle was consistant with the non-Euclidean surface’s analogue of the Pythagorean Theorem, then the whole thing would be consistent and space could be quantized (because there would be either little or no deviation between the result given by the Pythagorean theorem and the result obtained by constructing the surface’s tiling.)

So, now we must show that the Euclidean version of the Pythagorean Theorem truly does hold in our universe – in other words, that the Pythagorean Theorem isn’t simply just an approximation. We could suppose that space-time was bent in such a way that both the pseudo triangle’s hypotenuse and the Euclidean triangle had corresponding hypotenuses – as outlined above. Therefore, we could ask what is it that is known to bend space-time. From General Relativity, it is known that the only thing that affects the curvature of space-time is the mass density of whatever occupies a given spatial region. So, for instance, space-time is curved far more around the Sun than it is around the Earth. However, space-time curvature around the Earth, for instance, is small enough that we can assume it to be negligible locally – that is, at sufficiently small scales, it appears as though space-time isn’t curved at all, just as a circle’s edge appears flat upon being sufficiently magnified. Therefore, I will conclude that the Pythagorean Theorem does hold in our universe, if only on the small scale. Since space-time quanta are presumably extraordinarily small (beyond our current ability to measure) I do not believe that they are affected in a noticeable way by typical space-time curvature (except, perhaps, at singularities or discontinuities in space-time – as in a black hole.)

From all of this, I conclude that our observations of the universe in which we live is inconsistent with the idea that space-time is quantized. If our observations guarantee us a correct answer, then space-time is not quantized.

One possible response to my argument is that perhaps the quanta of space-time are not uniform or aperiodic. There are two ways in which this could occur – there are special space-time quanta that appear any time a right triangle is formed or measured. These quanta lie along the hypotenuse and trick us into measuring a different value than we would have measured otherwise. But I think the idea that space-time would “know” that we’re measuring it is ridiculous.

Another way in which space-time could be non-uniform or aperiodic is if it were tiled in an aperiodic way. For example, space-time quanta could take on the form of the shapes in the Penrose tiling, which is aperioidic, but has no special points whatsoever. This is plausible, although I see no reason for space-time to be aperiodic – why should one region of space differ from any other region of space in any appreciable way? Invoking Occam’s Razor, I deduce that this is an unreasonable assumption (i.e. space-time cannot be tiled aperiodically without a motivation for doing so. There is no motivation for doing so. Therefore, space-time is not tiled aperiodically.)

Suppose that we have a 45-45-90 right triangle whose legs both measure n units in length. Then, by the Pythagorean Theorem, n^2+n^2=r^2, where r=the length of the hypotenuse. So:

n^2+n^2=r^2

2n^2=r^2

Thus, r=sqrt(2n^2).

Therefore, r=n*sqrt(2).

So, for any 45-45-90 right triangle on the Euclidean plane, the Pythagorean Theorem tells us that the length of the hypotenuse is given by the length of one of the legs multiplied by the square root of 2. For the Weyl Tiling, the psuedo triangle's hypotenuse has a length of n. Therefore, the perecent deviation between the length of a psuedo triangle's hypotenuse and a Euclidean right triangle's hypotenuse is found through the following:

%Deviation = ([(n-n*sqrt(2)])/(n*sqrt(2)))*100 where we take the value given by the Pythagorean theorem as the reference value. Note that the "[" and "]" are absolute value symbols. For those unfamiliar with %dev calculations, the equation is simply given by the absolute value of the difference of the value you found to the reference value divided by the reference value and then multiplied by 100 (to turn it into a percentage.) It gives the deviation between any two values as a percentage of the reference value.

Now here comes the important part, which most people have already noticed already - the n's cancel out. So, we get:

%Deviation = ([(n-n*sqrt(2)])/(n*sqrt(2)))*100 = ([1-sqrt(2)]/sqrt(2))*100 for any 45-45-90 triangle whatsoever, no matter its size! So what is that percent deviation? Well, using my trusty TI-83, I get a value of approx 29.29%. Once again, that's true for any 45-45-90 triangle (on the Euclidean plane, of course!)

Now let's calculate how certain physical measurements would change if we lived in a universe in which space was quantized in a manner identical to the Weyl tiling. According to http://www.newscientist.com/channel/fundamentals/quantum-world/mg18424752.300, the smallest distance that any one could ever measure even in principle is about 10^-33 meters (the Planck length), other wise the measuring device would collapse into a black hole of its own making. So, let's suppose that space quanta are as small as they possibly could be, while still being measurable (if it was any smaller, it wouldn't even matter because no one could ever even hope to measure it.) Therefore, based on that assumption, space quanta must be on the scale of 10^-33 meters. Now, let's also suppose that we have a 45-45-90 triangle whose base is one meter. Then, it would measure 10^33 space quanta units on its two bases. Therefore, we have (10^33)^2+(10^33)^2=2*10^66 units^2 - giving a hypotenuse of sqrt(2)*10^33 units (or the sqrt(2) meters.) But the hypotenuse of the corresponding psuedo triangle measures 10^33 units, or 1 meter. That's a difference of approx 0.414 meters, or approx 1.36 feet. Since it's entirely reasonable to measure 1.36 feet, I conclude that if our universe does have a quantized space-time, then it is not quantized in the form of a Weyl tiling (assuming that you actually believe our experimental results from over 4000 years of measuring things - the ancient Babylonians and Chinese had figured out some of the useful properties of 3-4-5 right triangles and so forth - which are derived from the Pythagorean theorem - more than 4000 years ago.)

Next, I will show that there is a very large set of plane tilings which have this property. I will make use of the fact that for any plane tiling there is a corresponding planar network. Suppose you have a line of dots running horizontally across a page. Then, vertically down the page, you have an equal number of dots. For each dot running horizontally, there is a corresponding dot in the vertical direction. Draw a line down from each horizontal dot, and another outwards from each vertical dot. End the line where the two intersect. You should now have two lines, at 90 degrees to each other, one coming down from a horizontal dot and the other outwards from a vertical dot. Repeat this procedure for each pair of vertical/horizontal dots. The nodes formed by all those intersections now becomes a diagonal running at 45 degrees between the horizontal and the vertical. This pattern is a network which can be mapped onto any uniform tiling to form the right triangle given by that particular tiling. For instance, if we take this network and map it onto the Weyl tiling, we produce that same old stair case pattern. If we take a tiling of circles on the plane, a similiar thing happens. In fact, for any uniform periodic tiling of the Euclidean plane, this turns out to be true. Notice that each node corresponds to a tile on the tiling, and that there is a one to one correspondence between the set of nodes in one of the legs of the triangle and the number of nodes in the hypotenuse. Therefore, for any uniform periodic tiling of the Euclidean plane the number of polygons in the legs correspond to the number of polygons in the hypotenuse, meaning that the deviations from the Pythagorean theorem I derived above hold in general for the entire set.

Next, I will show that this mapping is even true for a large number of non-Euclidean geometries. For this I will use the topological property of a surface known as homeomorphism. Two geometries are said to be homeomorphic (i.e. topologically identical) if their topological properties remain invariant as one is continuously transformed into the other. In other words, two shapes are homeomorphic if a continous transformation can change one into the other without tearing or connecting the surface in a way that it wasn't connected before. In this way, a coffee cup is homeomorphic to a donut. Since the surfaces would remain internally connected in the same way as it was before under a topological transformation (that is, topological invariants would be, well, invariant) the mapping I outlined above wouldn't really change - it would just get stretched or bent or what not. Imagine that you have a picture laid down onto a piece of silly putty. When you stretch or bend the silly putty, you are performing a topological transformation. Under that transformation, the network I constructed above would also remain topologically invariant. In other words, a given network when laid down upon a surface remains topologically invariant under a given transformation of the surface so long as the resultant surface is homeomorphic to the original surface. So, the resultant network/tiling would be homeomorphic to the original network/tiling. Of course, under such transformation, the relationship between the lengths of the legs of a right triangle and its height is not invariant - that is, the Pythagorean Theorem does not apply to surfaces other than the Euclidean plane. So, if the Weyl tiling's psuedo triangle was consistant with the non-Euclidean surface’s analogue of the Pythagorean Theorem, then the whole thing would be consistent and space could be quantized (because there would be either little or no deviation between the result given by the Pythagorean theorem and the result obtained by constructing the surface’s tiling.)

So, now we must show that the Euclidean version of the Pythagorean Theorem truly does hold in our universe – in other words, that the Pythagorean Theorem isn’t simply just an approximation. We could suppose that space-time was bent in such a way that both the pseudo triangle’s hypotenuse and the Euclidean triangle had corresponding hypotenuses – as outlined above. Therefore, we could ask what is it that is known to bend space-time. From General Relativity, it is known that the only thing that affects the curvature of space-time is the mass density of whatever occupies a given spatial region. So, for instance, space-time is curved far more around the Sun than it is around the Earth. However, space-time curvature around the Earth, for instance, is small enough that we can assume it to be negligible locally – that is, at sufficiently small scales, it appears as though space-time isn’t curved at all, just as a circle’s edge appears flat upon being sufficiently magnified. Therefore, I will conclude that the Pythagorean Theorem does hold in our universe, if only on the small scale. Since space-time quanta are presumably extraordinarily small (beyond our current ability to measure) I do not believe that they are affected in a noticeable way by typical space-time curvature (except, perhaps, at singularities or discontinuities in space-time – as in a black hole.)

From all of this, I conclude that our observations of the universe in which we live is inconsistent with the idea that space-time is quantized. If our observations guarantee us a correct answer, then space-time is not quantized.

One possible response to my argument is that perhaps the quanta of space-time are not uniform or aperiodic. There are two ways in which this could occur – there are special space-time quanta that appear any time a right triangle is formed or measured. These quanta lie along the hypotenuse and trick us into measuring a different value than we would have measured otherwise. But I think the idea that space-time would “know” that we’re measuring it is ridiculous.

Another way in which space-time could be non-uniform or aperiodic is if it were tiled in an aperiodic way. For example, space-time quanta could take on the form of the shapes in the Penrose tiling, which is aperioidic, but has no special points whatsoever. This is plausible, although I see no reason for space-time to be aperiodic – why should one region of space differ from any other region of space in any appreciable way? Invoking Occam’s Razor, I deduce that this is an unreasonable assumption (i.e. space-time cannot be tiled aperiodically without a motivation for doing so. There is no motivation for doing so. Therefore, space-time is not tiled aperiodically.)

A while back someone asked where Elea was, and whether it was actually part of Greece...I did a little bit of research...and found that Elea was a Greek coastal city founded in 540 BCE in the south of Italy, by some Greeks who were fleeing from a Persian attack.

So, Elea was part of Greece, even though it is not somewhere that we generally think of as being Greek territory.

~Lisa~

So, Elea was part of Greece, even though it is not somewhere that we generally think of as being Greek territory.

~Lisa~

## Wednesday, March 01, 2006

The problem with finding the state of the light switch after it has been switched an infinite number of times is that the sequence of switching the light on and off is an ocillatory process.

To simplify matters, suppose that we take the simplest oscillatory curve in existence - the sine wave. y=sine(x) is constantly oscillating throughout its graph, back and forth between 1 and -1. If you ask what value y is at when x=infinity, then the answer is that its indeterminate. There is no problem with having an indeterminate answer either, since we could never have an oscillation, in physical reality that occurs infinitely fast. An oscillation that occurs infinitely fast would involve infinite amounts of energy (breaking the law of conservation of energy) and it would involve moving your hand to flip the switch faster than the speed of light (breaking special relativity.) Since the process isn't even physically possible to begin with, I find that its silly to wonder why rediculous answers come about as a result. In fact, from the point of view of a physics major, I find that this example gives a perfect thought experiment to try to figure out why oscillatory motion cannot occur infinitely fast. Silly results come about when you ask questions about things which are physically impossible.

However, the fact that the limit of an oscillating sequence is indeterminate has no bearing whatsoever on the limit of a non-oscillatory sequence. Since ordinary motion - that is, motion from one point to another - does not involve any oscillatory terms whatsoever, we do not have this problem. In fact, no law of physics is broken in ordinary motion either - so the light switch flipping thingy is a really poor argument.

If we are going to define passing by a point as being one particular example of an infinitesmal task, then I find no problem in likewise thinking that passing by an infinitesmal instant of time is also a task. To pass over even one femtosecond, we need to perform an infinite number of tasks - that is, we need to move through an infinite number of infinitesmal times. Doesn't any one have a problem with that definition of a task? Please note that, as Einstein showed, space and time are basically interchangeable (through d=sqrt(x^2+y^2+z^2-(ct)^2)), so if we assume that space is discrete, then time must be discrete as well. Likewise, any other problems that we find with space should necessarily apply to time as well.

To simplify matters, suppose that we take the simplest oscillatory curve in existence - the sine wave. y=sine(x) is constantly oscillating throughout its graph, back and forth between 1 and -1. If you ask what value y is at when x=infinity, then the answer is that its indeterminate. There is no problem with having an indeterminate answer either, since we could never have an oscillation, in physical reality that occurs infinitely fast. An oscillation that occurs infinitely fast would involve infinite amounts of energy (breaking the law of conservation of energy) and it would involve moving your hand to flip the switch faster than the speed of light (breaking special relativity.) Since the process isn't even physically possible to begin with, I find that its silly to wonder why rediculous answers come about as a result. In fact, from the point of view of a physics major, I find that this example gives a perfect thought experiment to try to figure out why oscillatory motion cannot occur infinitely fast. Silly results come about when you ask questions about things which are physically impossible.

However, the fact that the limit of an oscillating sequence is indeterminate has no bearing whatsoever on the limit of a non-oscillatory sequence. Since ordinary motion - that is, motion from one point to another - does not involve any oscillatory terms whatsoever, we do not have this problem. In fact, no law of physics is broken in ordinary motion either - so the light switch flipping thingy is a really poor argument.

If we are going to define passing by a point as being one particular example of an infinitesmal task, then I find no problem in likewise thinking that passing by an infinitesmal instant of time is also a task. To pass over even one femtosecond, we need to perform an infinite number of tasks - that is, we need to move through an infinite number of infinitesmal times. Doesn't any one have a problem with that definition of a task? Please note that, as Einstein showed, space and time are basically interchangeable (through d=sqrt(x^2+y^2+z^2-(ct)^2)), so if we assume that space is discrete, then time must be discrete as well. Likewise, any other problems that we find with space should necessarily apply to time as well.

An Infinite Series With A Precise, Well Defined Sum

Suppose that we have the infinite series 1, 1/16, -1/512, 1/8192… and we wish to take its sum (that is, we wish to add all the numbers in the sequence together, up to whatever number happens to occupy the infinite position in the sequence. Those numbers might look random, but, as I will show later, they are produced by a definite mathematical rule such that any term whatsoever in the sequence can be produced.

I can guarantee you that the sum of the terms of the sequence I just gave is 3/sqrt(8) – that is, 3 divided by the square root of 8. How can I do that? The sequence of numbers that I just gave happens to be the binomial expansion for (1+1/8)^(1/2) – in other words, the binomial expansion of the square root of 1+1/8. Of course, if you multiply that square root by the square root of 8, you get the square root of 1+8, which is the square root of 9. As we all already know, the square root of 9 is 3. Thus, if you really believe that the square root of 9 is 3, then you must also accept that the sum of the sequence I just gave is 3/sqrt(8).

In the case that you don’t believe me that the binomial expansion can yield a definite answer, or that the binomial expansion is actually correct, I suggest you check out this web site (which contains a small proof): http://www.rism.com/Trig/binomial.htm

The same arguments which I just presented can be made, in a similar form, for many other series and sequences.

Also, please note that it is possible find the exact value of a definite integral. For those of you who know anything at all about calculus, it should be readily apparent why that’s important in this context.

Also, note that for any system of logic which may be constructed, there is a corresponding system of mathematics. Therefore, we are left with the inescapable conclusion that if our logic is sound, then there is a sound mathematical theorem to go along with it.

Suppose that we have the infinite series 1, 1/16, -1/512, 1/8192… and we wish to take its sum (that is, we wish to add all the numbers in the sequence together, up to whatever number happens to occupy the infinite position in the sequence. Those numbers might look random, but, as I will show later, they are produced by a definite mathematical rule such that any term whatsoever in the sequence can be produced.

I can guarantee you that the sum of the terms of the sequence I just gave is 3/sqrt(8) – that is, 3 divided by the square root of 8. How can I do that? The sequence of numbers that I just gave happens to be the binomial expansion for (1+1/8)^(1/2) – in other words, the binomial expansion of the square root of 1+1/8. Of course, if you multiply that square root by the square root of 8, you get the square root of 1+8, which is the square root of 9. As we all already know, the square root of 9 is 3. Thus, if you really believe that the square root of 9 is 3, then you must also accept that the sum of the sequence I just gave is 3/sqrt(8).

In the case that you don’t believe me that the binomial expansion can yield a definite answer, or that the binomial expansion is actually correct, I suggest you check out this web site (which contains a small proof): http://www.rism.com/Trig/binomial.htm

The same arguments which I just presented can be made, in a similar form, for many other series and sequences.

Also, please note that it is possible find the exact value of a definite integral. For those of you who know anything at all about calculus, it should be readily apparent why that’s important in this context.

Also, note that for any system of logic which may be constructed, there is a corresponding system of mathematics. Therefore, we are left with the inescapable conclusion that if our logic is sound, then there is a sound mathematical theorem to go along with it.

Zeno and Black

Below are notes on Zeno and Black from 03/01. The Dichotomy and Black's argument are presented in numbered premise-conclusion form, as is the argument that Black's conclusion has vast implications for mathematical physics. Comments about which premise or premises are false and why are warmly encouraged.

The Dichotomy

The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics, 239b11)

Call the starting point 'Z' and the endpoint 'Z*'. We may present the argument as follows:

1. If A travels from Z to Z* and space is continuous, then A can perform an infinite number of tasks.

2. A cannot perform an infinite number of tasks.

3. Therefore, it's not the case that A travels from Z to Z*.

4. If (3), then motion in continuous space is impossible.

5. Therefore, motion in continuous space is impossible.

Note that a parallel argument would apply to continuous times and a similar idea is employed by the paradox of Achilles and the Tortoise. We might think of Achilles and the Tortoise as arguing that if space is continuous, Achilles can never reach his destination. We might think of the Dichotomy as arguing that if space is continuous, Achilles can never get started. Either way, assuming nothing is special about Achilles, we may conclude that motion in continuous space is impossible.

Zeno's paradoxes pose mathematical problems. But do they pose problems that extend beyond mathematics? Descartes, Pierce, Whitehead, and high school calculus teachers all across the land say 'no'. Black says 'yes'. Why does Black think that?

Here's a quote from Black:

"The series of distances traveresed by Achilles is convergent. This means that if Achilles takes enough steps whose sizes are given by the series 100 yards, 10 yards, 1 yard, 1/10 yard, etc., the distance still to go to the meeting point eventually becomes, and stays, as small as we please . . . But the distance, however much reduced, still remains to be covered; and after each step there are infinitely many steps still to be taken. The logical difficulty is that Achilles seems to be called upon to perform an infinite series of tasks, and it does not help to be told that the tasks become easier and easier, or need progressively less time in the doing. Achilles may get nearer to the place and time of his rendezvous, but his task remains just as hard, for he still has to perform what seems to be logically impossible. It is just as hard to draw a very small square circle as it is to draw an enormous one: we might say both tasks are infinitely hard . . . I think Zeno had enough mathematical knowledge to understand that if Achilles could run 111 1/9 yards -- that is to say, keep going for 11 1/9 seconds -- he would indeed have caught the tortoise. The difficulty is to understand how Achilles could arrive anywhere at all without first having performed an infinite number of tasks." (Black 1951: 122-123.)

"I am going to argue that the expression, 'infinite series of acts', is self-contradictory, and that failure to see this arises from confusing a series of acts with a series of numbers generated by some mathematical law." (Black 1951: 123.)

Black supposes, for the sake of argument, that we can construct an infinity machine. Here is a simple one (due to Thompson 1954-5): Suppose we have a push-button lamp. Suppose someone pushes it an infinite number of times by taking 1/2 min for the first push, 1/4 for the second, 1/8 for the third, and so on. Now Black argues (roughly) as follows:

1. If it is logically possible to perform an infinite series of acts, then the lamp was switched off each time it was on and on each time it was off.

2. If the lamp was switched off each time it was on and on each time it was off, then in its final state the lamp is neither on nor off.

3. Therefore, if it is logically possible to perform an infinite series of acts, then in its final state the lamp is neither on nor off.

4. It's not the case that in its final state the lamp is neither on nor off.

5. Therefore, it is logically possible to perform an infinite series of acts.

This is, in short, Black's argument. It follows from this that, on the assumption that the race can be run, the physical situation does not correspond to the mathematical operation of summing an infinite series. Here is the argument:

6. If it is logically impossible to perform an infinite series of acts, then the running of a race cannot be correctly described as completing an infinite series of tasks.

7. If the running of a race cannot be correctly described as completing an infinite series of tasks, then mathematical physics needs a radically different mathematical foundation in order to adequately describe physical reality.

The consequent of (7) follows from HS on (6,7) and MP on the resulting premise and (5).

Further Reading:

Benaceraff, Paul. 1962. "Tasks, Supertasks, and the Modern Eleatics." Journal of Philosophy 59:24, 765-84.

SEP entry on Supertasks (Warning: Much of this is difficult reading. However, it contains a discussion of Black and Benaceraff's reply which are not too hard to follow.)

Parsons and Prosser papers (especially Parsons)

Below are notes on Zeno and Black from 03/01. The Dichotomy and Black's argument are presented in numbered premise-conclusion form, as is the argument that Black's conclusion has vast implications for mathematical physics. Comments about which premise or premises are false and why are warmly encouraged.

The Dichotomy

The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics, 239b11)

Call the starting point 'Z' and the endpoint 'Z*'. We may present the argument as follows:

1. If A travels from Z to Z* and space is continuous, then A can perform an infinite number of tasks.

2. A cannot perform an infinite number of tasks.

3. Therefore, it's not the case that A travels from Z to Z*.

4. If (3), then motion in continuous space is impossible.

5. Therefore, motion in continuous space is impossible.

Note that a parallel argument would apply to continuous times and a similar idea is employed by the paradox of Achilles and the Tortoise. We might think of Achilles and the Tortoise as arguing that if space is continuous, Achilles can never reach his destination. We might think of the Dichotomy as arguing that if space is continuous, Achilles can never get started. Either way, assuming nothing is special about Achilles, we may conclude that motion in continuous space is impossible.

Zeno's paradoxes pose mathematical problems. But do they pose problems that extend beyond mathematics? Descartes, Pierce, Whitehead, and high school calculus teachers all across the land say 'no'. Black says 'yes'. Why does Black think that?

Here's a quote from Black:

"The series of distances traveresed by Achilles is convergent. This means that if Achilles takes enough steps whose sizes are given by the series 100 yards, 10 yards, 1 yard, 1/10 yard, etc., the distance still to go to the meeting point eventually becomes, and stays, as small as we please . . . But the distance, however much reduced, still remains to be covered; and after each step there are infinitely many steps still to be taken. The logical difficulty is that Achilles seems to be called upon to perform an infinite series of tasks, and it does not help to be told that the tasks become easier and easier, or need progressively less time in the doing. Achilles may get nearer to the place and time of his rendezvous, but his task remains just as hard, for he still has to perform what seems to be logically impossible. It is just as hard to draw a very small square circle as it is to draw an enormous one: we might say both tasks are infinitely hard . . . I think Zeno had enough mathematical knowledge to understand that if Achilles could run 111 1/9 yards -- that is to say, keep going for 11 1/9 seconds -- he would indeed have caught the tortoise. The difficulty is to understand how Achilles could arrive anywhere at all without first having performed an infinite number of tasks." (Black 1951: 122-123.)

"I am going to argue that the expression, 'infinite series of acts', is self-contradictory, and that failure to see this arises from confusing a series of acts with a series of numbers generated by some mathematical law." (Black 1951: 123.)

Black supposes, for the sake of argument, that we can construct an infinity machine. Here is a simple one (due to Thompson 1954-5): Suppose we have a push-button lamp. Suppose someone pushes it an infinite number of times by taking 1/2 min for the first push, 1/4 for the second, 1/8 for the third, and so on. Now Black argues (roughly) as follows:

1. If it is logically possible to perform an infinite series of acts, then the lamp was switched off each time it was on and on each time it was off.

2. If the lamp was switched off each time it was on and on each time it was off, then in its final state the lamp is neither on nor off.

3. Therefore, if it is logically possible to perform an infinite series of acts, then in its final state the lamp is neither on nor off.

4. It's not the case that in its final state the lamp is neither on nor off.

5. Therefore, it is logically possible to perform an infinite series of acts.

This is, in short, Black's argument. It follows from this that, on the assumption that the race can be run, the physical situation does not correspond to the mathematical operation of summing an infinite series. Here is the argument:

6. If it is logically impossible to perform an infinite series of acts, then the running of a race cannot be correctly described as completing an infinite series of tasks.

7. If the running of a race cannot be correctly described as completing an infinite series of tasks, then mathematical physics needs a radically different mathematical foundation in order to adequately describe physical reality.

The consequent of (7) follows from HS on (6,7) and MP on the resulting premise and (5).

Further Reading:

Benaceraff, Paul. 1962. "Tasks, Supertasks, and the Modern Eleatics." Journal of Philosophy 59:24, 765-84.

SEP entry on Supertasks (Warning: Much of this is difficult reading. However, it contains a discussion of Black and Benaceraff's reply which are not too hard to follow.)

Parsons and Prosser papers (especially Parsons)

If we look at the formal definition for the sum of a geometric series (which I do not contest, is in mathematics a finite number) you can see that the sum of the series as n-> infinity *converges* on a finite number given by the gemoetric summation formula.

http://mathworld.wolfram.com/GeometricSeries.html

This means that the series approaches an asymptotic bound. It is a limit, it will not actually reach it. In theory, the sum of the series WILL be exactly equal to a finite number, but this will happen if and only if every one of the infinite number of elements is summed. Untill this happens, the sum is only approaching the finite number, it will still fall a bit short of it. By definition, an infinite number of elements is never ending - it is impossible to finish the task of adding all of them.

The finite value of the summation is therefore a theoretical bound, and a limit the function is approaching - it is not an actual value.

Also, lim (x-> infinity) 1/x = 0, does not require 1/infinity to equal zero - it just means that 1/infinity approaches zero and is so close, in fact, that the distance becomes negligable.

Graph 1/x - it will never actually reach 0, because y=0 is an asymptotic lower bound on the function.

Despite the fact that 1/infinity = 0 is used in math, it is still a limit, not an actual equality. There's a better explanation of this here though: http://mathforum.org/library/drmath/view/62486.html

Any thoughts?

http://mathworld.wolfram.com/GeometricSeries.html

This means that the series approaches an asymptotic bound. It is a limit, it will not actually reach it. In theory, the sum of the series WILL be exactly equal to a finite number, but this will happen if and only if every one of the infinite number of elements is summed. Untill this happens, the sum is only approaching the finite number, it will still fall a bit short of it. By definition, an infinite number of elements is never ending - it is impossible to finish the task of adding all of them.

The finite value of the summation is therefore a theoretical bound, and a limit the function is approaching - it is not an actual value.

Also, lim (x-> infinity) 1/x = 0, does not require 1/infinity to equal zero - it just means that 1/infinity approaches zero and is so close, in fact, that the distance becomes negligable.

Graph 1/x - it will never actually reach 0, because y=0 is an asymptotic lower bound on the function.

Despite the fact that 1/infinity = 0 is used in math, it is still a limit, not an actual equality. There's a better explanation of this here though: http://mathforum.org/library/drmath/view/62486.html

Any thoughts?

Chad on Zeno and Black

I agree with Black’s position that mathematics alone cannot break the argument put forth by Zeno. Moreover, his argument examines the point that trying to count an infinite number of steps in a finite amount of time as impossible. In nature, it is true that the faster runner will overtake the slower runner. In my opinion the answer does not lie in trying to prove that it is impossible to count an infinite number of something in finite amount of time. The answer lies in the whole definition of infinity.

To my knowledge no one has ever proved infinity can exists. It is just simply an approximation used to put at an end of what the human mind is unable to conceive of. Human beings have been approximating nature since the beginning of time. As the centuries have passed since the time of Zeno our approximations have become better and better.

A good example is from the time of Isaac Newton. He invented calculus and applied it to celestial and earth bound mechanics. This new formulation explained everything from the distance a baseball could be thrown in a finite amount of time to the motion of planetary objects. Except, one thing was unable to be explained by Newton’s mechanics. The orbit of Mercury did not fit the calculations or any other explainable reason for its peculiar orbit. However, Newton’s approximations were close on nearly everything else in nature, but they were still approximations none the less.

Three centuries later Albert Einstein formulated his own approximations and clearly explained the orbit of Mercury. More in particular he defined that nature does have finite values for what was previously thought to be infinite. For example, Newton believed that the speed of light and gravity are instantaneous or at a speed of infinity. Einstein discovered that the speed of light in not instantaneous. Therefore, light travels a finite distance in a finite amount of time.

Two of the greatest minds in all of history understood that mathematics is invented as just approximations. It is the best tool we have and the approximations are used to explain how we believe nature works. Therefore, their arguments, including Zeno’s, still hold until a better approximation can explain them otherwise.

note: This post is Chad's despite the fact that it is not posted by Chad.

I agree with Black’s position that mathematics alone cannot break the argument put forth by Zeno. Moreover, his argument examines the point that trying to count an infinite number of steps in a finite amount of time as impossible. In nature, it is true that the faster runner will overtake the slower runner. In my opinion the answer does not lie in trying to prove that it is impossible to count an infinite number of something in finite amount of time. The answer lies in the whole definition of infinity.

To my knowledge no one has ever proved infinity can exists. It is just simply an approximation used to put at an end of what the human mind is unable to conceive of. Human beings have been approximating nature since the beginning of time. As the centuries have passed since the time of Zeno our approximations have become better and better.

A good example is from the time of Isaac Newton. He invented calculus and applied it to celestial and earth bound mechanics. This new formulation explained everything from the distance a baseball could be thrown in a finite amount of time to the motion of planetary objects. Except, one thing was unable to be explained by Newton’s mechanics. The orbit of Mercury did not fit the calculations or any other explainable reason for its peculiar orbit. However, Newton’s approximations were close on nearly everything else in nature, but they were still approximations none the less.

Three centuries later Albert Einstein formulated his own approximations and clearly explained the orbit of Mercury. More in particular he defined that nature does have finite values for what was previously thought to be infinite. For example, Newton believed that the speed of light and gravity are instantaneous or at a speed of infinity. Einstein discovered that the speed of light in not instantaneous. Therefore, light travels a finite distance in a finite amount of time.

Two of the greatest minds in all of history understood that mathematics is invented as just approximations. It is the best tool we have and the approximations are used to explain how we believe nature works. Therefore, their arguments, including Zeno’s, still hold until a better approximation can explain them otherwise.

note: This post is Chad's despite the fact that it is not posted by Chad.

Zeno Made Practical

The Eleatic Hangover Cure by Josh Parsons

The Eleatic Non-Stick Frying Pan by Simon Prosser

The Eleatic Hangover Cure by Josh Parsons

The Eleatic Non-Stick Frying Pan by Simon Prosser