Jack Meiland and Richard Taylor, Spatio-Temporal Analogy
Rick Garlikov

Richard Taylor[1]  has argued that time and space (or perhaps more accurately, temporal relations and spatial relations) are far more alike or analogous than one might think.  Jack Meiland[2] in answer to Taylor has argued that time and space or temporal and spatial relations are not so analogous as Taylor might think.  Meiland has pointed out what he considers to be a neglect or confusion in Taylor's argument.  In this paper I wish (1) to point out what that confusion is  (2) to show that with the proper distinctions Taylor’s arguments escape Meiland's criticisms  (3) to show that Meiland himself falls victim to a very similar confusion[3] and  (4) to present an aspect in which space and time or spatial and temporal relations do appear to be disanalogous.

I will construct my analogies in the same way both Taylor and Meiland do, by exchanging every spatial term in a statement with the corresponding temporal term and vice versa.  First, however, I should like to make a few distinctions.

Let us assume that all objects consist of and are the sum total of (except for arrangement) units we shall call spatio-temporal parts -- parts which have some size and last for some time.  These units are to be arbitrarily small both in size and duration; just as ordinarily we conceive of spatial parts as being able to be arbitrarily small. And, although I will not argue for it, I would think that these spatio-temporal parts would be either limited or unlimited as to brevity of size and duration in the same way ordinarily spatial parts are limited to size—i.e., if the smallest conceivable spatial part must be some finite size and if the smallest conceivable temporal parts must be some finite duration, then the smallest conceivable spatio-temporal units would probably also have to be of some finite spatial and temporal length.  But if no logical or practical difficulties follow from considering either the smallest temporal or spatial parts (here again relying on the common notion of a spatial part—e.g., gears are spatial parts of a watch, cogs are spatial parts of a gear, etc.)as being infinitely small (difficulties such as 'but then no matter how many you had together, they would still be invisible because they would still take up no space or time, etc.'), then I should imagine none would follow from considering the smallest spatio-temporal parts as infinitely small in either or both size and duration.

Now at different times an object will consist of different spatio-temporal parts.  For instance an old man consists of different spatio-temporal parts from those of himself as a baby; this is by virtue of the 'flow of time' since he still has the same basic 'spatial parts' (ordinary sense of spatial parts as above), fingers, eyes, feet, etc. And at any given time an object will have different spatio-temporal parts by virtue of its having different spatial parts (again, ordinary sense of spatial parts), e.g., at any given time a man has different spatio-temporal parts because he has different spatial parts-fingers, eyes, feet, etc.

Taking then the idea of a spatio-temporal part as primitive I wish to define (technically, though I think closely allied to common usage)

(1)  a temporal part, t, of an object 0 as all the spatio-temporal parts of 0 at a given time T.

(2) a spatial part, p, of an object 0 as all the spatio-temporal parts of 0 at any given place (or slice') P.

For example, the temporal part of me at my birth is or was all 7 Ibs. 15 oz. of me then; spatial parts of me are my fingers, lungs, etc. Now there can be spatial part of temporal parts and temporal parts of spatial parts; and in fact 'the temporal part t_x of spatial part p_x  of an object' is or refers to the same spatio-temporal part(s) as 'the spatial part p_x of temporal part t_x of an object.'  For example, a spatial part of me at my birth --a temporal part of me-- is or was my right index finger then, just as my right index finger at birth is or was a temporal part of my right index finger --a spatial part of me.

Of course, also temporal parts may have temporal parts just as spatial parts may have spatial parts.  E.g., my right index finger has three bones, skin, etc.; the 'me' of February 1967 was made of the "me's" of February 1, February 2, February 3, etc. of 1967.

Now on page 4 of his paper Meiland claims:

"But even if we do say that temporal parts have spatial_parts_of this sort, there is still a crucial similarity between spatial parts of physical objects_and temporal parts of physical objects.   [Meiland, of course, is not necessarily using my 'technical' notion of "temporal parts" or "spatial parts," but the 'common sense' notion of such, which may be confused or which may differ in some other way from my definition.]  For spatial parts of physical objects are composed wholly of temporal parts, but temporal parts of physical objects are not composed wholly of spatial aspects.  If a Taylorian analogy existed between spatial and temporal parts of physical objects, then both of the following statements wouId be true:

"(S₁):  A spatial part of a physical object can be regarded as a set of temporal parts.

"(S₂): A temporal part of a physical object can be regarded as

a set of spatial parts (of the sort known as 'spatial aspects').

"Statement S₁ is true.  But what could statement S₂ mean?  We might try to give S₂ a meaning in this way.  Suppose that the temporal part of X in question is (T₃ – T₇).  This temporal part has its own temporal parts, namely (T₃-T₄), (T₄-T₅, (T₅-T₆), and (T₆-T₇).  And each of the latter has a spatial aspect.  So it might be thought that the temporal part (T₃-T₇) could be regarded as the set of these spatial aspects, that is, as the set of spatial aspects of its own temporal parts.  This would provide a meaning for statement S₂ but it would also render S₂ false.  For even with the set of these spatial aspects, there is still something missing from temporal part (T₃ – T₇), namely its temporal aspect.  Hence the temporal part (T₃ – T₇) cannot be regarded as composed wholly of spatial elements.  This temporal part is not the set of certain spatial aspects."

Elsewhere after extended argument (pp. 3 and 4) Meiland correctly concludes what is present in the temporal part of an object is not the spatial parts themselves, but rather temporal parts of those spatial parts.

The non-underlined portions of the above offer a very astute observation, but the underlined statements are simply false.  S₁  is false or meaningless as is S₂.  A Taylorian analogy does not say both S₁ and S₂ must be true, but that they must both have the same truth value (both can be false) or both be meaningless.  Meiland gives S₂ a meaning, a meaning which he correctly argues renders it false. But as it stands, S₁, though Meiland thinks it to be true, is also either meaningless or false, for contrary to Meiland's statement, spatial parts of physical objects are not composed wholly of temporal parts, just as (as Meiland correctly claims) temporal parts of physical objects are not composed wholly of spatial aspects.

I will accept Meiland's argument and (particularly) his conclusion that a temporal part of a physical object must have both spatial and temporal aspects -- and consider his paper as offering support for my definition of temporal part.  Now I wish to support a bit more my definition of spatial part and argue that Meiland’s S₁ is false if given a meaning in the same way that he gave S₂ a meaning.

Paraphrasing Meiland in the last above quoted paragraph:

Suppose that the spatial part [in Meiland's or Taylor's or common-sense notion of spatial part, disregarding my definition whether it characterizes that notion or not] of 0 in question is (P₃ - P₇) (e. g., say my right index finger).  This spatial part has its own spatial parts, namely (P₃-P₄), (P₄-P₅), (P₅-P₆), and (P₆-P₇) (say, bones, muscle, skin, nail).  And each of the latter has a temporal aspect.  So it might be thought that the spatial part (P₃-P₇) could be regarded as the set of these temporal aspects, that is as the set of temporal aspects of its own spatial parts.  This would provide a meaning for statement S₁ but it would also render S₁ false.  For even with the set of these temporal aspects, there is still something missing from spatial part (P₃–P₇), namely its spatial aspect.  Hence the spatial part (P₃–P₇) cannot be regarded as composed wholly of temporal elements.  This spatial part is not the set of certain temporal aspects.-- BUT OF TEMPORAL AND SPATIAL ASPECTS; OF SPATIO-TEMPORAL PARTS.

If at T₁ I cut off my finger and look at the cross section of bone, muscle, skin, etc. I do not see things that only endure through time; I see things which also take up space-bone, muscle, skin, etc. Likewise, as Meiland might argue, if I take a temporal part of me, say me now,  I do not just have a chunk of material which takes up space, but a part which also takes up time, which endures -- for as long as 'now' 'endures.'

With this as background I should now like (1) to review the analogies of Taylor's which Meiland reviewed  (2) to try to get the correct or precise statement of each analogue according to definitions 1 and 2 and the notion of spatio-temporal parts and  (3) to try to show that Taylor's original statements using my definitions of temporal parts and spatial parts gives the analogies he sought.  I shall exchange temporal terms for spatial terms and vice versa, but will consider there to be no analogue to the term spatio-temporal, considering this term to be equally characteristic of both spatial and temporal relations, of both space and time, being neither more a space term than a time term nor vice versa.  I shall not go into detail as to what phenomena or objects can actually represent the objects portrayed in these analogues; Taylor does, and I shall simply refer to his examples, presupposing that the reader is familiar with those examples.

I.  A:  Entities can move in space.

Taylor A;  At T₁, 0₁ is north of O₂. At T₂, 0₁ is south of 0₂.

Meiland A: At T₁, temporal part N of 0₁ is north of temporal part G of 0₂. At T₂, temporal part S of 0₁ is south of temporal part H of 0₂.

Garlikov A:  At T₁, spatio-temporal part N of 0₁ is north of spatio-temporal part G of 0₂.  At T₂, spatio-temporal part S of 0₁ is south of spatio-temporal part H of 0₂.

B (the analogue):  Entities can move in time.

-- e.g., an earthquake’s occurring before the stroke of a clock in one town and after that stroke of the clock in another town.

Taylor B;  At P₁, 0₁ is future to 0₂. At P₂, 0₁ is past to 0₂.

Meiland B:  At P₁ temporal part F of 0₁ is future to 0₂. At P₂ temporal part Q of 0₁ is past to 0₂, [which is not a real analogue to Meiland A as Meiland correctly points out].

However, Meiland does not present statement or situation B correctly (disregarding his representation of A) because F and P are not only different temporal parts of the earthquake, but also different spatial parts (hence, different spatio-temporal parts) given that earthquakes are the types of things that have spatio-temporal parts.  And it would seem that if an earthquake can be said to have different temporal parts, parts which occur at different times, then it can be said to have different spatial parts -- parts which occur at different places.  Hence, it could be said:

Garlikov B:  At P₁ spatio-temporal part F of 0₁ is future to spatio-temporal part R of 0₂.  At L₂ spatio-temporal part P of 0₁ is past to spatio-temporal M of 0₂.

This is analogous to Garlikov A since “spatio-temporal” is not one of the terms needed to be exchanged in the statement (or perhaps some might wish to say it changes to tempero-spatial, which would be the same thing as spatio-temporal).

Notice further that if we substitute '0₁' for 'spatio-temporal part F (a part of 0₁ occurring at some time and place, place P₁) of 0₁'   and if we substitute '0₁' also for 'spatio-temporal part P of 0₁' in Garlikov B, and if in Garlikov B we also substitute '0₂' for both 's-t part R of 0₂' and 's-t part M of 0₂' we get Taylor B.

Making a similar substitution in Garlikov A we get Taylor A.[4]  Further this is the type of substitution we make all the time in everyday usage.  We do call both the earthquake in town P₁ at T₁ and the earthquake in town P₂ at T₂ 'the' earthquake or "the same' earthquake, though to be accurate or precise we should perhaps refer to them as different spatio-temporal parts of 'the' earthquake, assuming, of course, that we still wish to speak about 'the' earthquake.  Likewise we call the face or whole body of Rick Garlikov now when we see it “Rick Garlikov” just as we called the facial or all 7 Ibs. 15 oz. spatio-temporal part of Rick Garlikov some 62 years ago “Rick Garlikov” when we looked toward the blanket and saw it.[5]  We often refer to the spatio-temporal part of a thing as the whole thing.

Hence, here, Taylor A and Taylor B are analogous when interpreted as Garlikov A and Garlikov B respectively, these latter two being analogous.  And this interpretation does in fact reflect ordinary usage, though perhaps it might be better, ordinary language might be more clear (though more cumbersome) if it were of the expanded Garlikov form, rather than in the Taylor form as it is now.

II.  A:  An entity can be at two times in one place.

Taylor A:  0 is at P₁ at T₁ and T₂.

Meiland A:  At  P₁, a temporal part of 0 is at T₁ and another temporal

part of 0 is at T₂.

But, since temporal parts (on Meiland's notion of temporal parts, not just my technical notion) cannot exist without spatial aspects, what must be the case is:

Garlikov A:     At P₁ a spatio-temporal part of 0 is at T₁ and another spatio-temporal part of 0 is at T₂.

B:  An entity can be at two places at the same time.

Taylor B:  0 is at T₁ at P₁ and P₂.

Meiland B:  At T₁ a temporal part of 0 is at P₁ and P₂.

Garlikov B:  (for the same reasons as above)  At T₁ a spatio-temporal part of 0 is at P₁ and another spatio-temporal part of 0 is at P₂.

Again here in II, as in I, with a normal substitution (i.e., referringto a spatio-temporal part of a thing as the whole thing) we get Taylor A and Taylor B from Garlikov A and Garlikov B respectively.  Hence, Taylor A and Taylor B are or can be analogous as Taylor claimed, since Garlikov A and Garlikov B are analogous.

III.A: Things can move closer together or farther apart in space.

Taylor A:  At T₁, temporal part 1 of object 0₁ and temporal part 1

of object 0₂ are separated by a spatial interval x.

At T₂, temporal part 2 of object 0₁ and temporal part 2

of object 0₂ are separated by a spatial interval y, y

being either smaller or larger than x.

Meiland A:  Same as Taylor A, I believe.

Garlikov A:  At T₁, spatio-temporal part 1 of object 0₁ and spatio-temporal part 1 of object 0₂ are separated by a spatial interval x. At T₂, spatio-temporal part 2 of object 0₁ and spatio-temporal part 2 of object 0₂ are separated by a spatial interval y, y being either smaller or larger than x.

B.  Objects can move closer together or father apart in time.

Taylor B:  At P₁, spatial part 1 of object 0₁ and spatial part 1 of object 0₂ are separated by a temporal interval X. At T₂, spatial part 2 of object 0₁ and spatial part 2 of object 0₂ are separated by a temporal interval Y, Y being either smaller or larger than X.

Meiland B:  At P₁ temporal part 1 of 0₁ and temporal part 1 of 0₂ are separated by a temporal interval X. At P₂ temporal part 2 of 0₁ and temporal part 2 of 0₂ are separated by a temporal interval Y, Y being either smaller or larger than X.

Garlikov B:  (again since strictly speaking there can be no temporal parts on Meiland's or Taylor's notion without spatial aspects, just as there can be no spatial parts on Meiland's or Taylor's or the common sense notion of spatial parts without temporal aspects) At P₁, spatio-temporal part 1 of 0₁ and spatio-temporal part 1 of 0₂ are separated by a temporal interval X. At P₂, spatio-temporal part 2 of 0₁ and spatio-temporal part 2 of 0₂ are separated by a temporal interval Y, Y being either larger er smaller than X.

Garlikov B is analogous to Garlikov A, and although Meiland B is not analogous to Meiland A, that is because Meiland has again not seen that spatial parts have temporal aspects, although he has noticed that temporal parts have spatial aspects.  What is heard when one hears the two rolls of thunder in different places at different times (Taylor's example here) is both different spatial and different temporal parts of 'the' thunder.

And, if in Taylor's account, one understands 'spatial parts' and 'temporal parts' as I have above technically defined them -- as each having both temporal and spatial aspects by virtue of being about certain groups of spatio-temporal parts, groups either in the same time 'plane' or in the same space 'plane,' -- then Taylor A and Taylor B are correct and analogous.  (That is, for example, in Taylor B substitute 'spatio-temporal part 1 ["where spatio-temporal part 1 is composed of smaller spatio-temporal parts 0, 0.1, 0.2,..., 1] of 0-1' for 'spatial part 1 of 0-1' in order to get the corresponding part of Garlikov B.)  We would do this normally conceptually anyway, though we may not, of course, be aware that we are doing so.  For example, it is tempting: to speak of me now as a temporal part of the object Rick Garlikov  (or as noted before, as the whole object Rick Garlikov) when actually this temporal part of me now is the collection of spatio-temporal parts of Rick Garlikov at the present or now.

In short, Meiland has helped show that Taylor’s use of the terms temporal parts and spatial parts needed further explicating.  But I do not think Meiland has successfully pointed up any disanalogy between time and space.

I now, however, would like to present a rather simple example of how time and space or temporal and spatial relations seem disanalogous. I say seem, because perhaps they really are analogous, but that we have not yet discovered that time really is or can be correctly represented as three-dimensional.  For that is the difference-that space seems to be three-dimensional (although it perhaps can be represented one- or two-dimensionally) whereas time seems to be only one-dimensional.  To put this into Taylor's type of statement, consider the following as one (perhaps the simplest) example of stating three-dimensionality:

At any time, T_x, there can be four objects (or four spatio-temporal parts of (an) object(s) )  0₁, 0₂, 0_3, 0₄ such that the spatial distance from 0₁ to 0₂ equals the spatial distance from 0₂ to 0_3, which equals the spatial distance from 0₃ to 0₄ which equals the spatial distance from 0₁ to 0₃ which equals the spatial distance from O₂ to 0₄ which equals the spatial distance from O₁ to 0₄ = x (where x is greater than zero).

Such a description fits the  vertices of a regular tetrahedron.  The temporal analogue would have to be:

At any place, P_x there can be four objects (or four spatio-temporal parts of (an) object(s) )  0₁, 0₂, 0_3, 0₄ such that the temporal distance from 0₁ to 0₂ equals the temporal distance from 0₂ to 0_3, which equals the temporal distance from 0₃ to 0₄ which equals the temporal  distance from 0₁ to 0₃ which equals the temporal distance from O₂ to 0₄ which equals the temporal distance from O₁ to 0₄ = X (where X is greater than zero).

But what objects or parts of (an) object(s) could this represent?  It is easy to see how one object or spatio-temporal part can be separated equally from two others by the same amount of time -- e.g., me now from me last year and me next year;  but it is not easy to see how four objects or spatio-temporal parts can all be equally separated from each other in time.  Hence, in this regard, it seems time and space or temporal and spatial relations are not analogous,

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