Hi Graham,

This is a *pictorial* depiction of "stucts", which a structural form of *object representation*. Here is the warning from my last paper applicable also to this one:

"Warning: The main difficulty for a scientifically mature reader is not to fall into the trap of the powerful habit of automatically interpreting the information presented (of necessity) in the pictorial form in a “familiar” way, independent of the main text."

So I think you are missing something quite basic: focus on the concept of structured event and its role in object representation. These suggest an absolutely new way to see and represent objects via formative events.

Graham,

I forgot to ask you: What do you mean by "causal time"?

Hi Lev

There are a number of ways to define which event precedes another.

In many cases using a variety of definitions of time wind up with an agreement on which events precede another but it is possible to imagine scenarios where future events seem to cause aspects of the past.

E.g. If somone were able to travel to the past to create a new timeline then in their memory events that seem to be future events are part of remembered history. If knowledge of the previously experienced future events caused you to change the new future then in a sence the old future is in a sense prior to the new future.

In the case of parallel timelines that never influence each other there may be no way to decide which events in other timelines are before or after one on another disconnected timline.

Also as seen in your diagram if an event a causes b and c, we have no way to tell whether b is prior to c or b happened after c. In causal definitions of before and after, it is only when there is a serial connections that we can tell order of events. This of course assumes you have a definition or way to tell whether a caused b or b caused a. In sopme frameworks we assume if we are able to make a free-will choice to make a happen and if b then logically follows we assume a caused b.

In some models decisions we make now decides what states aspects of situations allocated to past clock times. I.e. aspects of the past seem to be written into history after they supposedly happened in clock time. If something we do now can influence a past event is it really in the past?

Where there are clashes between clock time, remembered time, and cause and effect chains we may need to have different types of time to describe such differences.

Graham,

May I suggest that such concepts as "clock time, remembered time, and cause and effect chains " are not adequate, and the whole point of ETS formalism is to introduce a general formalism with which one should be able to address *all* temporal issues?

Please keep in mind that, as has always been he case in science, without some reasonably adequate *formal* language, it is practically impossible to deal with the issues you are concerned with.

I believe that to address temporal side of reality we need a *radically* new formalism, and herein lies all the difficulties. How many people can or are even capable of accepting that after several millennia we have to start almost anew?

By the way, all the ideas about "causal time" one should be able to get from the proposed ETS struct. But let's wait what others have to say (unfortunately, I'm not going to hold my breath).

Hi Lev

It may take a while for me to seen where your ETS struct idea fits in with my general understanding of how things work. It is possible we are both trying to say something very similar but you have a more detailed specific formalism of a similar concept.

One formalized approach by Bars and Weinstein is to have two (or more) time dimensions rather than different types of time.

I have been looking at a number of competing approaches to build a collage of ideas to form an overall picture. One specific analogy I am pursuing is the idea that not all events in the past are set and some aspects of the universe are set after the events supposedly happened.

I use a diagram something similar to your struct diagrams but most likely my ideas were not as detailed as yours.

From what I so far have absorbed from your ideas they seem to fit with one approach I have been exploring.

I suppose it is only when looking at specific examples that one might discover whether our approaches are similar or very different.

One analogy I sometimes use is one where there is a 5D grid of points that are similar to events. This grid can be projected on to lower dimensions through different spacetime frames and possible world lines to create viewpoints or moments in time and space with set aspects of the past and possible futures for each observer, but the poiints on the grid themselves can be projectsed in many ways resulting in completely different realities for different observers.

Graham,

"It may take a while for me to seen where your ETS struct idea fits in with my general understanding of how things work."

May I be frank with you? Don't follow that path: unfortunately, it will not lead you to ETS.

Hi Lev

I could be completely off track.

I look at a lot of ideas but not very closely.

I tend to find a lot of ideas tend to overlap or come up with similar overall structures or elements. When I get time I will have to try to understand your ideas better. From the diagrams I have seen and my imprecise reading of parts of the paper your ideas seem to make sense to me. I am generally of the opinion that most if not all ideas tend to merge eventually if taken far enough. I certainly agree with the idea that non interacting events may not have a absolute truth about which event happened first.

Hi Graham,

Have you ever considered a non-Archimedean (e.g., a p-adic) time as a candidate for what you are calling the *causal time*? p-adic numbers can not be ordered, but nevertheless a causality can naturally be modeled in p-adics: for instance, every transducer (an automaton whose input/output alphabets consist of p symbols) is, on the one hand, a model of a particular causality law (since an output at i-th step, the event, depends only on input symbols at steps 0,1,...,i, the causes, and does NOT depend on input symbols at steps i+1, i+2, ...) and on the other hand, the transducer performs a continuous (w.r.t. the p-adic metric) map of the space of p-adic integers (an the latter space is a compact continuum). In the p-adic ergodic theory continuos p-adic time arises quite naturally, see Section 4.8.1 of Applied Algebraic Dynamics (vol. 49 of Degruyter Expositions in Math., Degruyter, Berlin-NY, 2009) by Vladimir Anashin (me) and Andrei Khrennikov. The concept of p-adic time has been discussing in mathematical physics since the beginning of 2000 (you may find some references in the above mentioned monograph). Note that the p-adic time is continuous and 1-dimensional (as our 'conventional' time is), but is NOT ordered: it is impossible to say whether one moment of the time is earlier or later that the other one. But on the other hand, a sequence of time moments can be ordered if needed: for instance, a sequence 1,2,3,... of natural numbers is a dense subset of the p-adic time (which actually is a whole space of p-adic integers), so one may consider a sequence of events w.r.t. the ordering of natural numbers, but this ordering leads to NO 'global' time ordering for the p-adic time. I guess that all these things may sound somewhat odd for a non-mathematician, but mathematically they are quite clear (and are proved rigorously).

Hope this may help you in your studies,

Vladimir

Peter, maybe later I will follow your idea and start a new tread on p-adics, but at the moment I am a little bit too busy. Now I only comment some points from your message.

1. Yes, any set can be totally ordered, but when one speaks about ordering of an algebraic structure this usually means that the order must _agree_ with the structure. To be more exact, for p-adic numbers this means that it is impossible to define such an ordering > that a+c>b+c once a>b. An ordering of this kind exists for real numbers but does NOT exist for complex numbers as well as for p-adic numbers. This is what I meant by saying that p-adic numbers cannot be ordered.

2. Yes, you are right, the p-adic distance between two (rational) integers a and b is equal to p^{-n} where p^n is the greatest power of p which divides a-b. Then the set of all (rational) integers can be expanded to a complete metric space w.r.t. the p-adic distance. This complete metric space is the space of p-adic integers. It is a compact space, and algebraically it is a ring. And yes, the set 0,1,2,3,... is a dense subset in this metric space. This leads to some unusual (at a first glance) things: e.g., any point inside a circle is its center, all triangles are isosceles, appending a segment to itself may result in a shorter segment than the original one (therefore Archimedean Axiom is violated in p-adic metric spaces), a monotonically increasing sequence of natural numbers may converge to a negative (rational) integer. For instance, the sequence 1, 3, 7,...,2^n-1,... converges to (-1) w.r.t. the 2-adic metric (i.e., for p=2).

3. Moreover, our standard metric is in a some meaning _exceptional_: according to Ostrovsky's theorem, there are infinitely many (pairwise non-equivalent) metrics on rational numbers, but only one of these metrics is our *standard* (Archimedean) one, the rest are p-adic metrics, for all prime p. Therefore there exists not a single Calculus, but infinitely many ones which are p-adic analysis', for every prime p. There are many textbooks and monographs on p-adic analysis (Schikhof, Mahler, Gouvea, Koblitz, Katok,...)

4. p-adic numbers were discovered by Kurt Hensel at the end of XIX century and until 1980-th the p-adic theory has been developing as a purely mathematical discipline. But then there were discovered deep relations of the theory with physics, computer science, cryptography, molecular biology, genetics, etc. You may find some references of this kind in my (with Khrennikov) monograph I mentioned in my earlier post. BTW, standard computer instructions, both numerical and bitwise logical ones, can be considered as continuous functions w.r.t. the 2-adic metric. Hence a computer program (at least, a straight line program) combined of these instructions is also continuos and thus its behavior can be studied by means of p-adic analysis (for p=2). You can read about all these things (as well as of 2-adic analysis) in lecture notes of my course at a summer school, see arXiv:cs/0612038

Do p-adic numbers have a digital or quantized or discrete aspect?

Some of my friends are looking at Cliffor Algebras or Clifford Space as a possible vehicle to unite quantum and spacetime.

The continuum aspects of p-adic numbers certainly seems a good example of a non-causal time.

In spacetime relativity and most continuum situations events on parallel causal chains have no set comparitive order.

The answer depends on what you are meaning by the *digital or quantized or discrete aspect*.

As for me, the answer is "yes" just because p-adic integers (which constitute a compact subset in the space of all p-adic numbers) behave themselves somewhat like "usual" (i.e., rational) integers and even may be considered as a (speaking loosely) non-negative rational integers whose base-p expansions are infinitely long. In other words, every p-adic integer has a unique representation of the form (called a canonical representation) a_0+a_1p+a_2p^2+..., where a_0, a_1, a_2,... are p-adic digits, that is, symbols from 0,1,..., p-1. If there are infinitely many 0 coefficients in the canonical form, the corresponding number is just a natural number; if there are infinitely many coefficients equal to p-1, then the number is an "ordinary" (rational) negative integer, if the sequence a_0,a_1,a_2,... is eventually periodic, then the corresponding number is a rational number which can be represented by a ordinary (reduced) fraction whose denominator is not divisible by p. For instance, -1/3=1+4+16+64+256+...+2^{2i}+... in the space of 2-adic integers.

The rest of p-adic integers do not lie neither among real nor complex numbers. Addition and multiplication of p-adic integers represented in canonical forms can be implemented by the same school-textbook algorithms as for addition and multiplication of natural numbers represented by base-p expansions. Also, one may reduce p-adic integers modulo p^k for every natural k just in the same way as he takes residues mod p^k of usual integers.

But on the other hand, the set of all p-adic integers is continual, in a contrast to the set of rational integers which is countable. Actually in p-adic analysis the set of all p-adic integers plays a role resembling the one of a closed unit segment in real analysis: both are compact sets, each w.r.t. the corresponding metric. So p-adic integers somehow combine in themselves both 'discrete' and "continuous' features.

A combination of discrete and continuous features is exactly what I am looking for to help solve the problem of uniting quantum and spacetime physics.

I think Clifford algebras (in some situations) can act somewhat like the hands on a clock that ticks from one nunber to another but is never in between.

OK, it would be nice if p-adics helps you in your studies. Please keep me informed about your research in this direction. But note that some work about the p-adic time is already done. Also note that p-adic mathematical physics is a rapidly developing area. In 2014 it will be the 5th (or 6-th?) international conference on p-adic mathematical physics. I also learned from your profile that you are dealing with cognitive science. In this connection maybe it is of interest for you that my co-author Andrei Khrennkov has a number of works where he applies p-adic theory to modeling cognitive processes (and there is a corresponding chapter in our monograph I mentioned in my earlier posting here).

Peter, the p-adic integers contain usual (=rational) integers and some (not all) non-integer rationals but the set of all p-adic integers is much wider. Yes, they constitute a normed ring, and yes, every p-adic integer can be represented as p^n*q where q is a p-adic integer with norm 1 (that is, not divisible by p); so this q is a p-adic unity (=has a multiplicative inverse in Z_p, the ring of p-adic integers). The ring Z_p is an integral domain, and the quotient field of it is Q_p, the field of p-adic numbers which are ordinary fractions whose nominator and denominator are p-adic integers. Also one may extend Q_p to an algebraically closed and (metrically) complete field which is a field C_p, the field of complex p-adic numbers, and the latter field is a very huge one.

The most usual way to represent p-adic integers is a canonical form I mentioned in my posting before; but there are other ways, e.g., via Teichmuller representatives, etc.

Your representation as equi-normed concentric rings (better to say, balls or discs) around zero is OK: indeed, Z_p is a ball of radius 1, pZ_p is a ball of radius p^{-1}, etc.

BTW a p-adic sphere of radius 1 is then Z_p\setminus pZ_p= a disjoint union of p balls z+pZ_p where z=0,1,2,...,p-1 (and a 2-adic sphere of radius 1 is then a 2-adic ball of radius 1/2; a little bit weird, yes). And every ball v+p^nZ_p is a coset w.r.t. the ideal p^nZ_p of the ring Z_p. And this is why every point of the ball is a center of the ball: u+p^nZ_p=v+p^nZ_p once u-v is in p^nZ_p, i.e., once u=v(mod p^n).

And also there is a natural probability measure on Z_p: its elementary measurable subset are balls v+p^nZ_p and the measure of this ball is p^{-n} by definition. So everything is OK with taking integrals on Z_p. And in p-adics a student's dream is getting true: a series u_0+u_1+u_2+ ... of p-adic numbers converge _if_and_only_if the u_n tends p-adically to 0 as n tends to infinity :)

I have been experimenting with the idea that travel in the quantum world , since it doesn't literally have travel through space, is based on time which has a quantitized value of 9. This means that quantum travel is in the form of a particle or wave motion which appears and disappears . An example of this phenomena is the jumping of electrons from orbit to orbit as soon as sufficient energy is present rather than traveling through physical space to get from here to there . The confusing part is that it all appears to us from our three dimensional world vantage point that it is still physical travel through space .Since all states exist in the quantum world you have the past, present and future all at once . This means that you may have non-casual time from the quantum future in the form of a particle or its' wave affecting the quantum past and present of another particle. In other words you have the effect preceding the cause . When we eyeball the quantum world from our real three dimensional world we cause a conversion of the travel time based quantum world to our ( x, y, z ) dimensions at time ( t ) which is a purely arbitrary time . Hence Schrodinger's cat in all its' glory .

Hi Graham

The reply button for your note doesn't appear to work or at least I can't get it to work . Anyway here's my answer .

You're probably right . Time has to come from somewhere . Maybe energy is just a holographic projection that appears in various forms . Time seems to be the only thing that appears everywhere . In other words, if something exists even as a projection time is involved since it is a measure of existence . Perhaps we live on or everything exists on a sine curve with its' ups and downs with the downs being entropy .

Maybe I just said this but if the universe runs on energy whether or not it is a projection then perhaps energy also appears in several guises including time . If so going back to the roots of time which is energy and energy is casual then time itself is casual .