Andrei Rodin
Places and Times of Events (draft)
Introduction
Events occur sometime and somewhere. My birth for example happened February 28, 1965 in Moscow. Such a location usually is good enough for practical reasons. To make the location more precise I could ask my mother about a room where I was born and a part of the day when it happened. It may be practically difficult to discover but conceptually it is clear how it may be done. A conceptual difficulty arises if someone ask about the exact location of event in space and time. It is unclear when exactly my birth started and finished and it is even more unclear what space region my birth exactly occupied since it is unclear what states of affairs mark exact temporal and spacious boundaries of my birth i.e. which physiological process should be included as proper parts of my birth and which should be excluded as foregoing and following; which parts of my mother’s body and environment were involved into my birth and which were not. Such questions seem to be futile since any possible answer is obviously conventional. It seems that having a detailed report about every state of affair within space-time region that designedly includes my birth we know everything we need about where, when and how it occurred. In a sense it is certainly true. Establishing spatial and temporal boundaries of event this way or another we get no new information about what actually happens.
Nevertheless a conceptual puzzle remains: we speak about places and times of events, moreover we may know where and when an event happens with more or less precision but we have no idea about what is (or could be) a place and time of event exactly. Compare events with such «regular» objects as balls: besides more or less precise approximate locations of ball in space and time we have the idea of what is ball’s exact location that is a set of points in space-time occupied with material points which are ball’s parts in a moment of ball’s existence. No matter that the idea is abstract and no statement as «ball A has such-and-such exact location in space and time» may be true about any real ball. Having the idea of what is exact location we may understand what is approximate one. In the case of events we can only say that one event’s location is more precise than another location of the same event as when I say that I was born in Russia and then specify that it happened in Moscow. The same is true about such «irregular» objects as mountains: we may easily specify the part of the Earth’s surface where a mountain in question is located in until we come to the question about its exact location. Then again it is a conventional matter of how to define the mountain’s exact spacious and temporal boundaries.
It may be argued that until an explicit convention about temporal and spacious boundaries of event or irregular object is made any reference to its location is vague but such a convention resolves all the problem. The problem however remains when we consider how space and time frames are established. It is obvious that we have no way to measure time but counting consequent events of some sort, particularly astronomical events or ticks of artificial clocks. To count them one need to determine when one event ends and the next begins (or at least that one event ends and the next begins). In the relativistic case it is also necessary to distinguish local clocks and hence to locate ticks in space. It is not so obvious that establishing space frames we deal with events as well but it is also the case. Any length gage should remain unchanged in respect of its length to be valid. No gage can be valid infinitely long. Once happens an event that destroys it. Then to establish temporal limits of gage’s validity it is necessary to know when a destroying event happens. If this problems are solved with a reference to another frame the same problems about this another frame arise and so on. To avoid a regress ad infinitum we should explain what are spacious and temporal relations between events independently of any pre-established frame.
The rest of the paper consists of three parts. In the first part I give a reason why «regular» objects (theoretically) have exact locations while events and some «irregular» objects do not. The idea is that the use of the concept of point presupposed with the notion of exactness is limited with some conditions which are not completed in the cases of events and irregular objects. In the second part of the paper I elaborate the construction of «pointless continuum» which models spacio-temporal relations between events. Finally in the third part I use the model to reconstruct concepts of place, time and construct that of place-time purely in terms of events.
This-Reference to Objects and Events
I use the term «this-reference» for generalization of the notion of ostensive reference to include cases when a direct reference by the word «this» is not accompanied with a move of forefinger or any other gesture. The reason for it is the fact that events may be directly referred by «this» as well as objects but without a pointing gesture. Let me start with usual ostensive reference to objects.
Referring to an object ostensively one gives it what may be called «hic et nunc location»: the object is spaciously «here» and temporally «now». That a thing is here and now is not a good description of its spatio-temporal location for everybody’s use but it may be argued that it is the best description of the thing’s spatio-temporal location for one who refers the thing ostensively herself. Referring to thing as «this» one believes that she herself «knows» where the thing is and when it does exist: namely it is here and exists now. Thus the problem of spatio-temporal location is reasserted as follows: how to explain others where is a thing and when it does exists to give them hic et nunc locations of their own? When hearers share hic et nunc location with a speaker, i.e. they are «immediate hearers» ostensive reference seems to do the job itself: pointing to this a speaker can make it clear for hearers what she does refer to and it gives hearers the same hic et nunc location of the referent that has the speaker: all of them agree that this is here and it exists now. So ostensive reference presupposes «direct contact» with a referent as well as «immediate communication» with hearers (which allows hearers to establish «direct contact» with a referent themselves).
Actually the ostensive reference is often vague for two reasons: 1) there are no clear criteria of what is «direct contact» with a referent; 2)neither there are clear criteria of what communication (particularly speech) is immediate, i.e. of what does it mean for speakers and hearers «to share their hic and nunc location». When someone moves her forefinger toward a table and says «this» you may be confused about what she does refer to: it may be the table itself, the table’s desk or an inscription on the table’s desk. (That is why in ordinary speech the ostensive reference has usually not a form of just «this» but that of «this A», where A is a kind, for example «this table». See [7].) When you communicate people via Internet the difference between on-line and off-line regimes is generally speaking not so sharp to claim that you do share now-location in the former case and do not in the latter. It is even less clear what does it mean to have «the same here-location» since «here» may refer to chair, room, building, city, country and many other things. If you are stranger and you are asked at a party «How long have you been here?» you are totally confused about what exactly you are asked: the room, the city or the country. The idea of «immediate speech» as a speech free from technical support can hardly be helpful either: consider for example a situation when a speaker uses a mike and one group of hearers are in the same room as a speaker while another group of hearers listen her speech in another room. If we add to dated translation equipment a few new monitors which translate into another room a speaker’s image together with her speech and if we also set up such monitors in the room where she speaks to let hearers see her better, we certainly lose any distinction between immediate and mediated speech. With the idea of immediate speech we lose also hic et nunc location defined as a place and time (here and now) of immediate communication.
It may be argued that although an ostensive reference is generally speaking equivocal and hence hic et nunc location cannot be shared, such a location is valid for speaker who «knows» that a thing is here and now and who tries to explain the thing’s location to anybody to let her «find» the thing and «meet» it here and now of her own. It sounds doubtful since «here» and «now» are words that should be understood by others. To save the idea of hic et nunc location we should rather say that although the words «here» and «now» may often be misinterpreted in other cases and with certain conditions they may serve to locate things properly. For otherwise we should say that the words «here» and «now» have no meanings at all which is obviously wrong.
To define exact conditions of proper use of hic et nunc locations seems to be interesting and important problem but it may not be considered here. What is of my interest here is this. When we locate things with points in space and time (or space-time) with chosen frames, we take the idea of hic et nunc location for granted. It is what above I called reassertion of location problem as the problem of reduction of any possible location to hic et nunc location. For «point of space» means «here» of its possible occupant, «moment of time» means «now» of possible one presented at the moment and «point of space-time» in Relativity means again «here-and-now» (which may not be separated in this case) for its possible occupant. Without the principle of hic et nunc location we cannot speak about points in space and moments of time (neither about points in space-time) and then consider in the set-theory way spacious and temporal (spatio-temporal) regions as consisting of points.
Although a possibility of ostensive reference to events is doubtful since they cannot be pointed out by forefinger [8] events can be referred directly as this, i.e. referred by this-reference as well as objects: consider a situation when during a party one makes a speech and refers to «this party». This-reference to event also determines its hic et nunc location shared by all the participants of the party. However such a location of event unlike hic et nunc location of object discussed above is not a point but some region in space and time: in our example it is a room where the party is held on and the period of time which it takes. It may be reasonably argued that every object also occupies a region but not a single point in space and exists some period of time but not a single moment. Then referring ostesively to object and locating it «here and now» we also mark rather space-time region than point. What differs the two cases however is this.
In the case of object we presuppose that an object consists of smaller spacious parts which are also objects and can be also referred ostensively and located in «heres». Similarly we presuppose in this case that the period of object’s existence (which may be called the object’s life) consists of smaller periods of time (which are parts of the object’s life) which may be located in «nows». To put it in other words we presuppose that ostensive reference to object may always be specified for its smaller spatio-temporal parts. It may be argued against this that a forefinger is big enough to point small objects and small enough to point big ones. The problem is solved by scaling and mapping: with such technical devices as theodolite and microscope they make images of big and small objects which have a size about that of human’s body and thus can be referred ostensively. I believe that possibility of such an «mediated» ostensive reference to big and small things is essential to consider these things as objects. We could hardly take the globe, planets and molecules together with apples and tables into the same category of objects without possibility to point them out ostensively in images the same way we may point out apples and tables in reality. Being idealized the procedure of sequential ostensive reference to object’s parts and parts of the parts via scaling and mapping is supposed to be such as if an initial object could be arbitrary big and its parts arbitrary small. Notice that such a procedure is the same time the procedure of specification of ostensive reference: when one point out a table she «really» points out only a table’s part; then a reference to table as a whole is taken to be «by reference to referent’s part» that is equivocal for it refers to object and a series of object’s parts simultaneously. Ambiguity of ostensive reference may thus be considered as being due of the fact that one and the same object is a part of different bigger objects. If to accept the idea that within every ostensive reference there is some «univocal» referent that involves no ambiguity then such a referent should has no parts. That makes the idea of point as indivisible spacious part of object.
fig.1
If space is supposed to exist independently of objects as the common place of all the actual and possible objects similar view may by applied to space regions, namely space regions may be considered as consisting of «points in space». In this case it is necessary to accept the idea that not only actual objects and their parts but also places of possible objects and their parts can be referred ostensively as «heres» (although it is not so clear how to use scaling and mapping technique in this case). If again to suppose that such a reference has an univocal referent one gets the idea of point in space. Thus a point may be considered both as minimal, i.e. indivisible space region and minimal, i.e. indivisible part of object (material point).
An existence of indivisible time periods, i.e. moments of time is postulated for an object in a similar way. Such expressions as «present state of object A» are vague being dependent on how fast A’s state changes: «present» of continents is of hundreds millennia, that of human beings is of years, weeks, hours or minutes dependently on context, while present position of racer is measured by hundredth of second. Nevertheless it is assumed that ostensive reference to an object has also an univocal temporal referent that is moment of «now» determined as a moment of act of reference itself. However an ostensive reference takes time. This time is too small to determine nows of geological processes and too large to determine nows of the process of explosion and of many other physical processes. Hence there should exists something as scaling and mapping of time. The only procedure of scaling and mapping of time in a proper sense of the words that I know is making films which present processes as going faster or slower than they go in reality; however it is new and I think it is not yet essentially influenced the ordinary and scientific concepts of time. More traditional way to solve the problem of scaling and mapping of time is to reduce it to that of space («spatialization of time»): that is what we do designing time with a line and moments of time with points on the line. Notice that the idea of moment of time as an univocal temporal referent of ostensive reference is not apparently presented in films: one can scale any process with film just making «now» of the process equal to time interval which takes an ostensive reference without any claim about moments of time (particularly without presupposition that the interval of time consists of moments of time). On the other hand considering that films are prepared with a sequence of separate cadres we go to the idea of moment of time back again.
What was said above about objects and ostensive reference may be summarized as follows.
Let us go back to events. What was said above about equivocality of ostensive reference to objects seems to be applicable to direct this-reference to events, although an ordinary language intuition hardly gives good evidences for it. In ordinary speech an expression «this thing» is usually more ambiguous than an expression «this event» (notice that in both cases only a direct reference is meant). One may easily be confused about what part of a whole is referred as «this thing» (see the example with a table above) but I can hardly imagine the situation when participating some event and being said something about «this event» one could be similarly confused about parts and wholes. Nevertheless we can easily consider such a confusion theoretically. Consider for example a football championship that consists of number of games. Being directly referred as «this» a game may by in principle confused with a championship in a whole. We can say that events have parts (which are also events) as well as objects (whose parts are objects) and we can suppose that this-reference confuses parts with wholes similarly in both cases, i.e. that in both cases this-reference is equivocal.
There is however a difference between the two cases that I believe is crucial: a (speech) act of direct this-reference to event is an entity of the same kind as a referent (namely of that of events) while an act of ostensive reference to object is an entity of another kind than a referent. Moreover in the example of direct this-reference to event the act of reference is a part of event to be referred (i.e. a party) itself. Is it a peculiarity of the example or a feature of direct this-reference to events in general? I believe that the latter option is true. Counter-example may be this. Someone in a room hears unusual noise from outside and asks another person «What is this?». In this case a speaker may be said is not involved into event she refers as «this». Such cases however may be excluded as linguistically loose if to presuppose that to give hic et nunc location is an essential property of this-reference. For in such cases an event referred as «this» is located «there», outside. Consider the example that apparently is similar. Light in a room is unexpectedly turned off and someone in the room asks again «What is this?» In this case an event referred as «this» is located here, in the room and it may be argued that people in the room are involved into this event. On the other hand it may be said that here are two different events - physical event of turning light off and linguistic event of asking «What is this?» - such as the former causes the latter. (We could also add an intermediate mental event of speaker’s surprise.) This claim is not inconsistent with the claim that a speaker is involved into «turning light off» event since both physical and linguistic events mentioned above as a cause and effect may be considered as two part of one and the same event which is picked up by his physical part.
The problem is however that the same is applicable to previous example of noise in a street. Then we lose a desired meaning of «involvement» into event. Actually we should define this important notion which is analog of the notion of «direct contact» with an object within ostensive reference mentioned above. To keep the analogy closer we demand that a speech act S of this-reference to event A which locates A here and now presupposes a direct involvement of speaker into A that is S is a part of A. Both notions of «direct contact» with object and «direct involvement» into event are based on the idea of some kind of «direct knowledge» of hic et nunc location of this-reference’s referent. To locate an object in «here and now» you should have it at hand. To locate an event in «here and now» you should participate it. I think it basically corresponds our intuition. It does not mean of course that there are no event without a speaker involved. A physical event of turning light off (caused by short-circuit which in its turn is caused by lightning) is a counter-example. What I claim here is that event may not be directly this-referred unless a speaker is not directly involved into it.
Going back to the problem of alleged equvocality of direct this-reference to events (which is supposed to be similar to equivocality of ostensive reference to objects) and considering the fact that a speech act of direct this-reference to event is a part of event so referred we conclude that among possible referents of direct this-reference to some event is the (speech) event of this reference itself. Thus a direct this-reference to event which is supposed to be equivocal the same way as ostensive reference to object is, involves autoreference.
It is actually striking how «here» and «now» are used to locate events in ordinary speech. Consider again a party where a speaker refers to party’s temporal part locating it in «now» as follows: «now John will present his new poem» or «now you heard Mary’s last opus». In these examples «now» locates events not in presence but in immediate future and past which allows to call such a use of «now» metaphorical. Consider now the example of «literal» use of «now» when event temporally located in «now» includes the act of speaking «now» as its part: an announcements about the beginning of show that may be reasonably considered as a part of the show. In this example a larger temporal location that is a period of time taken with the show is determined by its part that is a period of time taken with the announcement. It may be argued that event that is temporally located with the announcement is not the show itself but show’s beginning. Then the situation is reversed. Beginnings are instantaneous or at least very short events. In our example the beginning of show is certainly shorter than the announcement about it because considering the announcement as the show’s part we presume that the announcement ends later than the show begins. Thus in this case narrower temporal location of the beginning of show is determined by a larger location of the announcement about it which includes the former as its part. Both cases seem to be similar with cases of temporal location of objects whose observable changes take much longer or much shorter periods of time than takes an act of ostensive reference. (See above).
Similar things may be noticed about spacious location of events in «here». Consider the situation when one comes to group of people gathered at lawn and notice something unusual in people’s behavior. Then she asks «What happened here?» and is answered that «John broke his leg». «Here» in the question refers to some region of space around the speaker’s body however it remains ambiguous what is the region exactly. The event mentioned in the answer and correspondingly located in «here» of the first speaker may be argued to be located in some small region of space around John’s leg at the time of its braking (see [6] about event’s «minimal location»). Again the situation seems to be similar to that of objects’ spacious location in «here» with a forefinger. Since there is no a pointing gesture of forefinger in the case of events we may suppose that in this case the role of forefinger is played by all speaker’s body. Then it might be said that as well as ostensive reference to big and small objects locates them equivocally «here», similarly speech act of saying «here» about event may locate it in large and small places, also equivocally.
There is the important difference between the cases of objects and events however that can be clearly seen with mentioned analogy. In the case of objects the problem of equivocality of ostensive reference is transferred from theoretical to practical sphere, namely it is assumed that in principle every ostensive reference can be cleared up to be univocal with a single (material) point as a referent which is localized correspondingly in a single point of space and in a single moment of time (or in a single point in space-time) although the univocality can never be practically reached but can be only approximated. The problem of approximation is treated then as practical or even purely technical one. The distinction between theoretical and practical, i.e. between what can be done in principle and what can be done really (or what was actually done) is of importance here. The distinction stresses the fact that a material point is not an object as any other but it is idealized object; correspondingly point in space is idealized place and moment of time is idealized period of time. As I mentioned above such an idealization is based on scaling and mapping procedures that make an image of a referent of ostensive reference which has a size suitable to be pointed out ostensively in an intermediate way. Supposing that arbitrary small objects may be scaled and mapped to be ostensively referred, one may speak about a point as a limit case. Similarly the idea of moment of time is obtained.
Such a way to avoid (theoretically) equivocality of this-reference and hence ambiguity of hic et nunc location is inapplicable to events. No hic et nunc location of event univocally refers to point in space and time (or in space-time) even in principle. The claim that no event occurs in a single point may not be the reason for it because the similar is true about objects: no object consists of single point either. The reason is this. No scaling specifying this-reference to event is accessible because a direct this-reference to event involves autoreference. Scaling events we scale equally a referent and a mean of reference that is a speech act of speaking «this». It is similar to imaginary situation when scaling objects we would be equally scaled ourselves. Obviously in this case scaling would be useless. What we need to have access to events which are too quick or too slow temporally as well as too small or too large spaciously is to change ourselves to adopt our acts of reference to events in question. To participate Lilliputians’ and Brobdingnagian parties one should obtain Gulliver’s abilities. It is not matter of mere fiction however. The same instruments which we use for scaling and mapping of objects may be used and understood another way to scale ourselves. It is partly conceptual and partly technical question. Looking through a microscope at a sample we may say that microscope gives us a picture of sample’s microstructure that is adopted to human natural scales; making a micro-photo we can point out ostensively elements which are too small to be pointed out ostensively in reality. But on the other hand we can take a microscope as a tool that improves our vision and adopts it to witness micro-events. It may be argued that to witness an event is not enough to be involved in it, then I may take a micro-surgery as an example of real involvement. We have not yet instruments that make our speech acts faster and smaller spaciously but we have tools to make it slower (writing technique in general) and broader (radio, TV and Internet as well as traditional writing again which extends speech not only in time but also in space). Without these instruments such broadly located events as Lady Di’s campaign would be impossible.
Note that never during such Gulliver Travels one reaches Final Point practically or at least theoretically. Neither it may be supposed that such a point exists. Remember what was the reason to introduce points in the case of objects. The reason was to make a limit of infinite progress of «embedded» equivocal ostensive references by supposing a point as a referent of univocal ostensive reference which is embedded into all the equivocal ones (see fig.1). Since we may suspect a this-reference to event to be also equivocal the same may be desirable for events. However it is impossible for the reason which is topological-like. Since an object is in front of my eyes I may suppose that it is (constricted into) a point. Since I am involved into event and thus an event is around me it may not be (constricted into) a point because a point may not be around anything. It applicable for space, time or space-time with minimal corrections.
fig.2
(Note that the same reason is valid for such «irregular» objects as mountains which may not be constricted into a point either. Mountain for example unlike stone (no matter how big) is not divisible from its basis and that prevents from the constriction.)
If it is true it means that events cannot be located within frames of the same type as (regular) objects can. We locate objects within point continuum. It not necessary that we consider continuum as consisting of points, i.e. as a set of points of certain kind, but anyway to distinguish some region R of continuum means to be able to answer whether a point A belongs to R or not. To locate events we need pointless continuum any segment of which would be infinitely divisible in a stronger sense than it is the case for the standard point continuum.
Going back to the question of supposed equivocality of direct this-reference to events we may say this. Since events are located in pointless continuum the idea of a point as an univocal referent of direct this-reference is inapplicable to events. If equivocality of direct this-reference to event is a case as in mentioned example where a football championship was supposed to be confused with a single game it may be easily avoided other way than by specification of its hic et nunc location. To avoid equivocality it is enough to point a kind of event which is referred as «this»; in considered example it is enough to distinguish between «this game» and «this championship». Practically the equivocality of ostensive reference to objects is avoided the same way: in the example where a table was supposed to be confused with an inscription of table’s desk it is enough to distinguish between «this table» and «this inscription» and it is exactly what is done in ordinary language. Nevertheless the idea of point as univocal referent of ostensive reference is of great theoretical importance since it allows to locate tables, inscriptions and any other objects in space and time as accurately as accessible keeping in mind the ideal of «absolutely exact» location. It could be also desirable for events but it is impossible for the reasons mentioned above.
Since events unlike objects have no exact locations no location of event may be called approximate either. Nevertheless we may distinguish between more and less detailed locations as in the case when I say that I was born in Russia and then add that it happened in Moscow. Actually in the case of location of events comparatively to that of objects we lose nothing but the ideal.
Pointless Continuum of Events
1) Informal discussion
The idea is this. Take an infinite straight line AB. In this informal discussion I feel free to put a question naively: what does the line consists of? I know three options to answer the question. The first is to say that line consists of points and all the line’s parts are some sets of points. To identify a line it is enough to take any two of its points, for example A and B.
Another option is dualistic: a line AB consists of its ends A and B which are points and continuous stuff between A and B which is potentiality of points. If you pick up, i.e. actualize another point C anywhere between A and B then you get AC and CB as parts of AB (fig.3). Roughly speaking the second option says that a line consists of elements of two types: points and continuous segments. Both options have large mathematical traditions based upon them: the first is relatively new (contemporary set-theory based mathematics), the other is old (all ancient and modern mathematics).
fig.3
There is however the third option that was never developed mathematically. That is: a line consists of its continuous segments and nothing else. Immediately a question arise: how different line’s segments may be distinguished if not by points? What about A, B and C which distinguish segments AC and CB at fig.3? To switch from this intuition let us consider a duality between segments and points of the same line. Changing segments for points and otherwise at fig.3 we have this:
fig.4
(To make the duality complete suppose that all the line at fig.3 is divided into the infinite number of intervals such a way as the line is the union of the intervals.)
At fig.3 segments AC and CB contact by the point C. Let us say now that at fig.4 points AC and CB contact each other by the segment C.
Consider a simpler example: point C divides a line into half-lines A and B (fig.5a). The dual representation is a construction of two points A, B and the segment C between them (fig.5b). In both cases I say that C is a boundary between A and B and/or C is bounded by A and B.
Note an argument that Aristotle (Phys. 231a) uses to prove that points cannot contact each other. It is based on two definitions:
D1: Two things contact each other when their boundaries (partly or wholly) coincide.
D2: Point is what has no parts (Cf. Eucl. El. Book1 Def.1)
The argument is independent of how exactly a boundary is defined; it is enough to assume that
P: a boundary of thing is its part.
In both D2 and P a «part» is supposed to be a smaller part, not a thing as a whole (Cf. Eucl. El. Book1 Axiom 4: a part is less than its whole.)
Then let two points contact. Hence by D1 they have a common boundary. It implies that each of contacting points has a boundary which by P is its part. But it is impossible by D2. Hence points cannot contact.
Then to say that points at fig.5b contact we should reject some of D1, D2, P. Let us reject D2. Then by D1 the interval C at fig.5b is a common boundary of points A and B the same way as the point C at fig.5a is a common boundary of intervals A and B. Of course rejecting D2 we have no more reason to call A and B at fig.5b «points» and distinguish them from intervals. We lose the distinction between points and intervals as well as the intuition that segments of a line are necessary separated from each other by objects of other kind, namely by points.
Then we consider a line’s parts separated one from another by other parts which are boundaries. All the parts are of the same «nature», we do not say any more that «proper» parts are segments while boundaries are points:
fig.6
Here parts A and B are separated by boundary C or speaking dually, A and B separate C from the rest of the line.
If the first formulation is accepted then the only case in which a boundary C may be said «smaller» than parts A and B which it separates is the case when A and B are divided into smaller parts and C is not. Note that it is not the case when A and C may be divided into smaller parts (for C may be divided into smaller parts as well) but the case when A and B are actually divided end C is actually not:
fig.7
Then it may be said that C separates the two aggregates A1-A2-A3-A4-A5-A6-A7-C and C-B1-B2-B3-B4-B5 which are identical with A and B correspondingly. Thinking about the line at fig.6 as representing time we may say that there are two processes A and B separated by event C. Considering this interpretation let me call an «event» any part of the constructed continuum. A process I assume to be an event which is actually distinguished into smaller parts. The reader should bear in mind this conventional use of the term «event» in the following paragraphs but later I argue that everything what is said about conventional «events» is true about real ones.
What still remains puzzling about pointless continuum is the question of events’ identity. We may consider none of events A, B and C at fig.6 as a separate individual without taking into consideration two others. Considering C as a part of A we may not to distinguish the event A\C (A without C) such as A is a mereological sum of A\C and C. What we can do is to distinguish events Bg(C), En(C) and Con(C) which are the «beginning», the «end» and the «content» of C correspondingly:
fig.8
Then we may go ad infinitum distinguishing «beginning of the beginning of C» Bg(Bg(C)) and so on but at any step it does not let us separate A from C by some boundary.
The only event which may be considered alone within the constructed continuum without taking into consideration of other ones is an event which is the continuum as a whole. On the other hand we may reasonably distinguish between the «unbounded» continuum and any its «bounded» part such as A or C at fig.6. The «minimal» configuration which consists of more than one event is presented at fig.6. It consists of three events (to distinguish exactly two events appears to be impossible) which are of two types. The first type is of «properly bounded» events which have two boundaries (or to put it dually - are common boundaries of two events); this type is instantiated at fig.6 by C. The second type is of «partly bounded» events which have the only boundary (which are boundaries of the only event); this type is instantiated at fig.6 by A and B. To put a criteria of identity for both properly and partly bounded events we should distinguish between events’ left and right boundaries. At fig.6 C’s left boundary is A, C’s right boundary is B, A has only right boundary C and B has only left boundary C. Then we may distinguish between the two kinds of partly bounded events: those having only right boundary and those having only left boundary. Let us call a partly bounded event of the first kind left-oriented and a partly bounded event of the second kind right-oriented», then at fig.6 A is left-oriented and B is right-oriented (the reason to chose between the terms «right» and «left» this way is clear from fig.6). Let us also call the property of partly bounded events to be right- or left-oriented their orientation. Then the criterion of identity for properly bounded events is this:
I2B: Properly bounded events X and Y are identical iff they have the same left and right boundaries.
Note that this criterion is analogous to Davidson’s criterion of events’ identity that is the assumption that events are identical iff. they have the same causes and effects [4]. As well as Davidson’s criterion I2B is obviously circular since boundaries in the right part of I2B are also events which are demanded to be «the same». (Although as well as in Davidson’s case the circularity may be formally hidden with a suitable notation.)
For partly bounded events the criterion of identity is that:
I1B: Partly bounded events X and Y are identical iff they have the same orientation and the same boundaries.
Notice however that the mentioned classification of events depends on supposed topology of pointless continuum, namely of the fact that its topological dimension is 1. If we change a line for plane or other surface the classification would be different.
Consider another question stressed by Davidson: are events particulars. Let us talk again for now about our conventional events i.e. parts of pointless continuum. Are they particulars? They are not kinds for sure. But neither it sounds acceptable to call them particulars. For it seems to be the essential feature of any particular that it can be pointed out independently of its relations to other particulars. It allows to find out later how particulars relate to each other. With events we have a different situation. What for example the name «C» at fig.5 refers to? We may say that it is the event which is the common boundary of or which is bounded by events A and B. What is problematic in these individuating descriptions it is the fact that the descriptions refer to other things presumably of the same kind as the referent. It causes or infinite regress or circularity as in the considered case. Thus the worry about circularity of I1B-I2B (and I believe that equally about the circularity of the Davidson’s criterion) is something more than a worry about the logical structure of theory. It may be argued that any philosophical theory is circular if it is general enough. But here is another case. The worry is about a referent which is referred as «event C». We may refer to point C as a point which lay between points A and B (and has some additional relational property that makes the description individuating). But we may also consider the point C itself and ask about its relational properties. On the other hand no event may be considered itself. The worry is that in this case apparently there is no «it» to make a description of. What seems to make sense however it is to consider a triple of events (A,C,B) such as C is the common boundary of and is bounded by A and B. Then we may refer to A, B and C separately as to elements of the triple (however when A or B is properly bounded it is not individuated within this triple). But hardly then A, B and C may be called particulars even in such a special sense in which we may consider mathematical points as particulars.
However the situation does not seem unusual if to accept the axiomatic approach to design a theory. Within Gilbert’s axiomatic of geometry [5] for example there is no need to consider points and straight lines as particulars each of those may be considered in any sense itself. Only relations between points and lines defined by axioms matter for theory. Although Gilbert himself apparently keeps the idea that points and lines are «things» which may be straightwardly denoted as A, B etc., I believe that it is in vein of axiomatic approach to leave this together with the idea that such «things» have essential properties which matter for theory.
2)Formal Theory
a) linear model
Consider a set T of elements which I shell call events. Suppose T to be partially ordered; the partial order « » is axiomatized in the usual way as a reflexive, antisymmetric and transitive relation. Define the relation of «involvement» between events; actually «A involves B» means simply «B follows immediately after A», I accept the terminology for the reason of interpretation:
Def.: (A,B) (read: A involves B) iff. A < B & A< B & " C (C< A or B < C);
Then demand that every event involves another and is involved by another.
AV: " A $ B$ C ((A,B) & (C, A))
AV implies that T is infinite.
Def.: Event set is a set T such as AV is satisfied if to quantify over T.
Now I want to find a way to combine events into aggregates which would form another event sets. The other words I want to find a suitable mapping from T to the set of T’s subsets M(T). Firstly I demand that the mapping keeps the partial order. Secondly as it was explained above a minimal event aggregate that «makes sense» itself and the same time «gives a sense» to its constituents consists of three elements. To combine three adjacent events into one means to miss a boundary between two of them (fig.6). Denote for simplicity
Def.: (E1, ... Ek) (read: a set of events {E1, ... Ek} makes an interval) iff. (E1,E2) & ... & (Ek-1,Ek).
When it obviously does not cause any confusion I use the same notation (E1, ... Ek) for the predicate defined above and for the set {E1, ... Ek} which satisfies the predicate.
Def.: [A,B,C] (read: A,B and C are together) iff. (A,B,C) or (A,C,B) or (B,A,C)or (B,C,A) or (C,A,B) or (C,B,A)
Consider a mapping EInv (read: elementary involution) from T to M(T) such as
AI1: If A e T and Pe M(T) then Ae P or Einv(A) = P
AI2: If EInv(A) = EInv(B) then $ !C (Ce T & [A,B,C] & EInv(C) = EInv(A))
Then every A’= EInv(A) has or one prototype A in T or exactly three prototypes in T A,B C such as [A,B,C]. Let T’ be a set of all the images of Einv: T’= {A’ I A’= Einv(A), Ae T}. Consider an ordering <’ on T’ such as
AI3: Einv(A)<’Einv(B) iff A< B
It is straightward to see that there is only one such ordering. Suppose that EInv induces an ordering on the set of its images accordingly with AI2 every time it is applied to some event set.
Consider a composition of (a finite) k elementary involutions.
Def. Inv (read: involution) = EInv k ° ... ° EInv2 ° EInv1
Lemma 1: If A’= Inv(A) then or A’ has the only prototype A from T or the set of prototypes of A’ from T is an interval which consists of odd number of events.
Lemma 2: Any event from T is mapped by some involution into some event of M(T) which has more than one prototype.
Proof: Due to AV for any event A from T there exist B, C from T such as [A,B,C]. Then take Inv: T® M(T) which maps A, B and C into {A,B,C} and maps all other events into themselves.
Lemma 3: Any event A from T is an image of more than one event from some T’ within some involution.
Proof: Take for T’ all the elements of T but A replace for a triple {A’,B’,C’}. Then consider the involution which maps all the elements of T’ except {A’,B’,C’} into themselves and maps {A’,B’,C’} into A.
Lemma 4: Consider some involution Inv from T . Let T’ be a set of all the images of events of T with Inv. Then T’ is an event set.
Proof. It is sufficient to show that T’ satisfies AV. In the case of elementary involution it is checked immediately. Since the number of elementary involutions which an involution is superposition of supposed to be finite, the general claim follows by induction by the number of elementary involutions in one involution.
Lemma 5: Composition of two involutions is involution.
Lemma 6: Inv3 ° (Inv2 ° Inv1) = (Inv3 ° Inv2) ° Inv1 (transitivity of involutions’ composition)
Lemmas 2 and 4 allow to compose involutions unlimitedly. It is essential that due to AV an event set is infinite for otherwise it could be mapped by involution into a set of two or one events to which no further involution would be applicable. Lemma 3 says that every event «infinitely divisible». Lemmas 5 and 6 allow to define the linear continuum of events:
Def. (Linear) event continuum is a category [2] Ev = Cat (T, Inv) where objects are event sets, arrows are involutions, composition is that of involutions and identity of object T is the involution that maps T into itself.
Denote also for further needs:
Def.: Event A’ of T’ is a process of involution Inv: T® T’ iff A’ has more than one
prototype in T.
Def.: Event A from T is a left boundary of process A’ from T’ of involution Inv: T® T’
iff (Inv(A) = A’) & O ($ B((B,A) & Inv(B) = A’).
Def.: Event A from T is a right boundary of process A’ from T’ of involution Inv: T® T’
iff (Inv(A) = A’) & O ($ B((A,B) & Inv(B) = A’).
Def.: Event A from T is a boundary of process A’ from T’ of involution Inv: T® T’ iff
it is its left or right boundary.
b) generalized model
The presented model is a model of linear event continuum. It may help to locate events in time but not in space nor in space-time. To go beyond this limitation the model may be generalized this way.
Consider a graph G with an infinite number of nodes which I shell call events. Denote (A,B) a relation of incidence between graph’s nodes (which is by definition a reflexive and symmetric relation). I shell also call the relation «involvement» and the graph «event graph». Then an event set considered above is a particular event graph, namely a chain:
fig.9
Def.: A subgraph S of graph G is a star iff. $ A0"Ai (i = 1, 2, ...) " B (A0, Ai, B eS &(A0, Ai) & ((B, Ai) ® B = A0)).
Let us call A0 a center of star S. Denote for further needs Star(S) a predicate which says that S is a star.
A triple of events which make an interval is a particular case of event star. To make a star one should take some event A and all the events which it involves. An event star is an event taken together with its neighbors. The simplest non-trivial star consists of two events (fig.10a; unlike linear case here A0 is properly bounded by only one event Ai); event triple considered above is a particular case of event star (fig.10b); fig.10c shows a case of star which consists of four elements:
fig.10
The idea is that a star in a whole may be taken as a new event the same way we do it with triples in the linear case. Let T to be a set of G’s nodes. Then involution is to be redefined as follows. Consider a mapping EInv from T to the set of T’s subsets M(T) such as
AI1: If Ae T and Pe M(T) then Ae P or Einv(A) = P;
AI2’: If EInv(A) = EInv(B) = P then $ S(Star(S) & (CI S « Einv(C) = P))
Then every A’= EInv(A) has or one prototype A in T or its prototypes are nodes of the same star in T. Let T’ be a set of all the images of Einv: T’= {A’I A’= Einv(A), Ae T}. Consider a graph G’ such as its nodes are elements of T’ and
AI3’: (Einv(A),Einv(B)) iff (A,B)
Consider again a composition of (a finite) k elementary involutions.
Def. Inv = EInv k ° ... ° EInv2 ° EInv1
Lemma 1’: If A’= Inv(A) then prototypes of A’ are elements of connected subgraph of G. (A graph is connected iff for any two its nodes A,B there exists a «path» of its nodes C1, C2,...,Cn such as (A, C1) & (C1, C2)& ...&( Cn, B).)
Lemma 1 is a particular case of Lemma 1’. Lemmas 2, 3, 5 and 6 are true with the generalized involution. Lemma 4 has no more sense because in general case no condition as AV is necessary. (For the same reason AI3 is formulated in general case in a more natural way.)
Def. Event continuum is a category Ev = Cat (G, Inv) where objects are event graphs, arrows are involutions, composition is that of involutions and identity of object G is the involution that maps G into itself.
For the reason that there is no better term I keep a definition of process in general case.
Left and right boundaries of process in the general case make no sense but boundary in general is defined as follows:
Def.: Bd(A,A’) (read: A is a boundary of A’) iff. Inv(A) = A’ & $ !B(Inv(B) = A’ & (A,B));
i.e. an event of process is its boundary iff. within the process it involves only one other event. In the case of elementary involution when a process is made of one star all the elements of star but its center are boundaries. As well as in linear case «boundary» here means «partial boundary»; all the partial boundaries of the same process make what may be called its whole boundary.
How to make times and places with events
The are two senses in which they speak about «location»: one presupposes an universal frame within which things are located, the other does not. They presuppose an universal frame locating something «in the moment of time t at the point x». A presupposed frame is universal in the sense that anything may be located within this frame if not practically then theoretically. This type of frames are commonly used in science. Even in General Relativity where only local frames are possible they do not introduce another type of frames but simply say that such and such frame (which is of the same type as any universal one) actually is applicable only locally within some neighborhood of the point of its origin. However outside science they use frames which are bona fide local. The location «Broadway and 116-th street» presupposes a frame that is not applicable anywhere outside New York City. Speaking that something «happened in Europe between the World Wars» they presuppose the frame where the Moon and events that caused the present appearance of Moon’s surface may not be located. From the point of view of classical mechanic the universal character of frames which it uses makes a difference between science and ordinary discourse, however considering the General Relativity we may reasonably doubt that it is the case. It is sufficient reason to take local frames seriously.
To stress the difference between universal and local frames we may also say this. While universal frames come together with the concepts of space and time, local frames come with places and times. Certainly NYC and Europe are places but it is questionable how they relate to the idea of space in general. «Times» I use primary in the sense they speak about «ancient times» or «good times» although the idea of times as countable occurrences of similar things is also relevant here. Again it is a problem how times relates to time.
The reason why local frames, places and times are of my interest here is the fact that they may be defined exclusively in terms of events. I cannot investigate here the big problem of how local and universal frames relate to each other, but I want to notice one obvious thing. Any universal frame presupposes a repeatable unite of space and time. It may be a period of clock’s ticking and some gage of length. A repetition of unit allows to extend a frame as far as possible: we may easily represent the age of the Earth in years, hours or seconds as well as the distance from Earth to Sun in inches. Thus the idea of repeatable unite is essential for universal frame. However there is a puzzling thing about units and their gages: one cannot check a gage but comparing it with another one and hence one should believe that some gages are good a priory. How can one prove that clocks are good, i.e. time periods between their ticks are perfectly equal? To measure the periods with another clocks. But what about quality of this other clocks? How one can prove that a gage of length is good, i.e. its length does not change. To measure it with another one. The reasons to believe that some processes are perfectly periodical and some objects (or speaking generally some physical configurations) perfectly keep distances may be rather theoretical: if we do not presuppose this we can not built a consistence scientific theory about things in question. If to presuppose that the periods between sun’s rises and downs are equal during a year then it seems to be impossible to explain why a pendulum oscillates in summer faster than in winter. If I suppose that my body is of the same size during all my life then I have no idea what is going on with everybody else’s body particularly with that of my mother and that of my daughter. Actually in science there is no need to presuppose that some processes are absolutely periodical and some distances are absolutely unchangeable. It is enough to point processes and distances such as if the processes are supposed to be absolutely periodical and distances absolutely unchangeable then they get a good theory about real processes and distances. If what is supposed to be unchangeable is itself a subject of slow change the theory may be improved but not entirely given up. Notice that what was said is true in relativistic case as well as in non-relativistic one, because the problem is the same for local clocks and scales.
The fact that local frames do not presuppose any puzzling idea about a repetition of equal space and time units seems to be an important advantage. The local space frame that is a system of 5 continent on the Earth says nothing about their equality (although it seems to be essential that the continents are «comparable» in size). Nor blocks in New York should be equal to find the crossroad of Broadway with 116-th street. Let us see now how local frames work with events.
What does it mean «John kissed Mary at Tom’s party» (let «Tom’s party» to be univocal)? It means that the kissing was an episode of the party. We should be accurate to say that the kissing was a part of the party since as I argued above the objects’ mereology is not applicable to events. However now we have enough machinary to make the statement strict. Let the kissing be A and the party be P. Denote «A happened at P» as L (A, P). Suppose two event graphs G and G’ such as AI G and PI G’. If L(A, P) is true then there exists an involution from G to G’ such as P is an image of A. The latter condition is not sufficient however because the following restrictions are reasonable:
Finally I define
Def.: L (A,P) iff $ (Inv: G® G’) (Ae G & Pe G’ & Inv(A)=P & $ B(Be G & B is not A & Inv(B)=P) & not Bd(A,P)).
Theorem: If L(A,B) and L(B,C) then L(A,C) (transitivity of location).
Now I define for events an analog of «exact location» of object. Following Lombard I call it «minimal location» (denote ML):
Def.: ML(A,P) iff $ (EInv: G® G’) (Ae G & Pe G’ & Inv(A)=P & $ B(Be G & B is not A & Inv(B)=P) & not Bd(A,P)).
It means that if P is a minimal location of A the only prototypes of P are A and P’s boundaries; all the prototypes make a star with a center A. As an example consider: «At the last Friday’s seminar students discussed ontology of events». Suppose that «the last Friday’s seminar» is univocal and that the discussion took the whole seminar so that nothing occurred at the seminar but this discussion. Then «the last Friday’s seminar» is a minimal location of the discussion. It may be argued that in this case the discussion and the seminar are identical. I believe however that it is not the case. This example obviously differs from the case of rotating and heating ball. It is nonsense to say that the seminar occurred at the discussion. The reason is that the seminar made a frame for the discussion determining its beginning and end as well as its spacious location. Notice that events of the discussion’s beginning and end are not located at the seminar.
Events’ location prima facie does not make any difference between times and places while location is a relation between events and every event takes place and time. Consider again «John kissed Mary at Tom’s party». Obviously this location is both temporal and spacious. Every event location prima facie complies with the principle accepted in Relativity according to which any location is that in space-time. Opposing places to space and times to time I also introduce the term place-time for the similar opposition to space-time. Thus an event’s location is prima facie a place-time.
From this point of view one can hardly take the idea of Relativity that space and time make the unity of space-time as revolutionary. The idea was revolutionary only in the context of classical mechanics. Moreover within the present theory of events’ location the idea of some sort of independence of space and time (or more accurately - of places and times) seems to be non-trivial.
Roughly speaking the idea is this: there are locations of two special types of «places» and «times» such as events may occur at the same place in different times and otherwise at different places in the same time. For example «Rome» is a place where in different times Romul killed Rem, Brutus killed Caesar, Michelangelo painted the ceiling of the Sistine Chapel and my former student recently made her Ph.D.. One the other hand Modernity is a time which extends over many places simultaneously. Formally the idea may be elaborated as follows.
Suppose two different sets of processes P0,...,Pk and T0,...,Tk. Then consider events Eij such as:
APT: L(Eij, Pi) & L(Eij, Tj)
Then it is said that Pi are places and Tj are times of Eij.
Consider a simple example. Let G be an infinite square grid. Take four events E11, E12, E21 and E22 as shown at fig.11.
fig.11
Then processes P1, P2 (fig.11a) and T1,T2 (fig.11b) may be constructed with (different) two-step involutions (check it) such a way that P1 ,P2 may be called places of events E11, E12 and E21, E22 correspondingly while T1, T2. may be called times of events E11, E21 and E21, E22 correspondingly. Each of P1, P2, T1 and T2 is also a location of third event which is a temporal (for places) or spacious (for times) boundary between events Eij of each pair. Notice that relation between places and times so defined is symmetric; since usually they presume a linear structure of times and lack of such a structure of places the presented theory of times and places should be specified to meet this condition. I cannot however do it here. Notice finally that places and times so defined remain local and do not make universal space and time. Times have spacious limits as well as places have temporal limits. Particularly Modernity hardly ever took place in Antarctica and Rome is Eternal City only metaphorically.
Literature: