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.

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