The (Possible) Structure(s) of Space
According to the dominant philosophical view concerning the structure of space, the following theses are true:
Points (P): There are spatial points.
(Where:
Necessarily, for all x, x is a spatial point iff (i) x is a spatial region, (ii) x is unextended, and (iii) x has no proper subregions.
Necessarily, for all x and y, x is a proper subregion of y iff (i) x is a subregion of y and (ii) x is distinct from y.)
Density (D): Between any two spatial points there is a third spatial point.
Extended Regions (ER): For any extended spatial region R, there are some points pps such that R is an S-fusion of the pps.
(Where:
Necessarily, for any x and any yys, x is an S-fusion of yys iff (i) for all y such that y is one of yys, y is a subregion of x and (ii) for all z such that z is a subregion of x, z S-overlaps at least one of yys.
Necessarily, for any x and y, x S-overlaps y iff there is a z such that z is a subregion of x and z is a subregion of y.)
Universal S-Fusion (US): For any spatial regions xxs, there is a y such that y is an S-fusion of xxs.
Let the Standard View be the conjunction of (P), (D), (ER), and (US). It is well-known that some philosophers have challenged the Standard View or, at least, its necessity. There are several views distinct from the Standard View that have been discussed. For instance, according to the Gritty View, space is discrete. Proponents of the Gritty View accept (P), (ER), and (US) but deny (D). Whereas (D) entails that between any two points there are at least as many points as there are rational numbers, the Gritty View denies this. According to the Gritty View, the following thesis is true:
Grittiness (G): For any points x and y, either (i) there is a point p such that p is between x and y and there is no point p' such that p' is between p and x or (ii) there is no point p between x and y.
The Gritty View, then, is the conjunction of (P), (G), (ER), and (US).
Another view that has been discussed is the Extended Simple Regions View. This view denies (P). However, like the Standard View and the Gritty View, the Extended Simple Regions View is committed to the existence of spatial regions. But its proponents think that all spatial regions, including those without proper subregions, are extended. Thus, proponents of the Extended Simple Regions View accept (US) (and (D), but only because they hold it to be vacuously true) as well as:
Extended Simple Regions (ESR): There are extended simple regions.
(Where necessarily, for all x, x is an extended simple region iff (i) x is a spatial region, (ii) x is extended, and (iii) x has no proper subregions.)
No Points (NP): There are no spatial points.
Extended Non-Simple Regions (ENR): For any spatial region R, if R is not an extended simple region, then there are some extended simple regions xxs such that R is an S-fusion of xxs.
Abutment(A): For any extended simple regions x and y, either (i) there is an extended simple region z such that z is between x and y and the distance between x and z is zero or (ii) the distance between x and y is zero.
A final view that has been discussed in opposition to the Standard View is the Gunky View. The Gunky View agrees with the Extended Simple Regions View that (P) and (ER) are false. However, it denies that there are extended simple regions. That is because, according to the Gunky View, every spatial region has proper subregions. This is the first thesis of the Gunky View (and it entails the falsity of both (P) and of (ESR)):
Gunkiness (Gu): Every spatial region has proper subregions.
The Gunky View also holds that (US) is true and that every spatial region is extended:
Only Extended Regions (OER): Every spatial region is extended.
[Note that (OER) is also endorsed by the proponent of the Extended Simple Regions View. I do not include it among the theses endorsed by the Extended Simple Regions View since it seems to follow from (ESR) and (ENR) along with other plausible theses (for instance, that if a spatial region is an S-fusion of some extended spatial regions, then it too is an extended spatial region).]
The Standard View, the Gritty View, the Extended Simple Regions View, and the Gunky View are, then, the main proposals concerning the structure of space. It is important to note, though, that one may accept one of these views as the correct account of the actual structure of space while holding that space could have the structure described by one of the other views (or while remaining agnostic about this). It is also important to note that the Standard View, the Gritty View, the Extended Simple Regions View, and the Gunky View do not exhaust the views that one may have concerning the actual or possible structure(s) of space. In a future post I hope to explain what other views concerning the actual or possible structure of space one might have.
Points (P): There are spatial points.
(Where:
Necessarily, for all x, x is a spatial point iff (i) x is a spatial region, (ii) x is unextended, and (iii) x has no proper subregions.
Necessarily, for all x and y, x is a proper subregion of y iff (i) x is a subregion of y and (ii) x is distinct from y.)
Density (D): Between any two spatial points there is a third spatial point.
Extended Regions (ER): For any extended spatial region R, there are some points pps such that R is an S-fusion of the pps.
(Where:
Necessarily, for any x and any yys, x is an S-fusion of yys iff (i) for all y such that y is one of yys, y is a subregion of x and (ii) for all z such that z is a subregion of x, z S-overlaps at least one of yys.
Necessarily, for any x and y, x S-overlaps y iff there is a z such that z is a subregion of x and z is a subregion of y.)
Universal S-Fusion (US): For any spatial regions xxs, there is a y such that y is an S-fusion of xxs.
Let the Standard View be the conjunction of (P), (D), (ER), and (US). It is well-known that some philosophers have challenged the Standard View or, at least, its necessity. There are several views distinct from the Standard View that have been discussed. For instance, according to the Gritty View, space is discrete. Proponents of the Gritty View accept (P), (ER), and (US) but deny (D). Whereas (D) entails that between any two points there are at least as many points as there are rational numbers, the Gritty View denies this. According to the Gritty View, the following thesis is true:
Grittiness (G): For any points x and y, either (i) there is a point p such that p is between x and y and there is no point p' such that p' is between p and x or (ii) there is no point p between x and y.
The Gritty View, then, is the conjunction of (P), (G), (ER), and (US).
Another view that has been discussed is the Extended Simple Regions View. This view denies (P). However, like the Standard View and the Gritty View, the Extended Simple Regions View is committed to the existence of spatial regions. But its proponents think that all spatial regions, including those without proper subregions, are extended. Thus, proponents of the Extended Simple Regions View accept (US) (and (D), but only because they hold it to be vacuously true) as well as:
Extended Simple Regions (ESR): There are extended simple regions.
(Where necessarily, for all x, x is an extended simple region iff (i) x is a spatial region, (ii) x is extended, and (iii) x has no proper subregions.)
No Points (NP): There are no spatial points.
Extended Non-Simple Regions (ENR): For any spatial region R, if R is not an extended simple region, then there are some extended simple regions xxs such that R is an S-fusion of xxs.
Abutment(A): For any extended simple regions x and y, either (i) there is an extended simple region z such that z is between x and y and the distance between x and z is zero or (ii) the distance between x and y is zero.
A final view that has been discussed in opposition to the Standard View is the Gunky View. The Gunky View agrees with the Extended Simple Regions View that (P) and (ER) are false. However, it denies that there are extended simple regions. That is because, according to the Gunky View, every spatial region has proper subregions. This is the first thesis of the Gunky View (and it entails the falsity of both (P) and of (ESR)):
Gunkiness (Gu): Every spatial region has proper subregions.
The Gunky View also holds that (US) is true and that every spatial region is extended:
Only Extended Regions (OER): Every spatial region is extended.
[Note that (OER) is also endorsed by the proponent of the Extended Simple Regions View. I do not include it among the theses endorsed by the Extended Simple Regions View since it seems to follow from (ESR) and (ENR) along with other plausible theses (for instance, that if a spatial region is an S-fusion of some extended spatial regions, then it too is an extended spatial region).]
The Standard View, the Gritty View, the Extended Simple Regions View, and the Gunky View are, then, the main proposals concerning the structure of space. It is important to note, though, that one may accept one of these views as the correct account of the actual structure of space while holding that space could have the structure described by one of the other views (or while remaining agnostic about this). It is also important to note that the Standard View, the Gritty View, the Extended Simple Regions View, and the Gunky View do not exhaust the views that one may have concerning the actual or possible structure(s) of space. In a future post I hope to explain what other views concerning the actual or possible structure of space one might have.
posted by Greg at 12:59 PM
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