The end of time (defining time and space experimentally)

Andreka, H. and Nemeti, I.

Talk given on the Celebration event in honour of Johan van Benthem, 26-27 September 2014, Amsterdam, Holland.

http://www.illc.uva.nl/J65/

Abstract: The talk is about an interaction between mathematical logic and relativity theory. In our story there is a world without time and space, hence no motion no change, and then time and space emerge as a result of a cognitive process. We define time and space (over timeless objects) as derived notions. Technically, we define two first-order logic theories, SpecRel and SigTh, both representing special relativity. SpecRel has a rich language for talking about time, space, motion, speed, acceleration etc, while SigTh is very much experiment-oriented, the only primitive concept of its language is signaling via photons. Then we show that the two theories are definitionally equivalent in the sense of mathematical logic. This gives an operational semantics for special relativity, and also shows that special relativistic spacetime can be built up without mentioning time or space.

Below comes the full text of the talk, associated to the slides of the presentation.

1. We will talk about an interaction between definability theory and relativity theory. In our story there will be a world without time and space, hence no motion no change, and then time and space will emerge as a result of a cognitive process. We will define time and space (over timeless objects) as derived notions.

2. We work in the framework of mathematical logic . We study a pattern consisting of two theories of First-order Logic (FOL) and our key ingredient will be an interpretation between them. One theory is rich, while the second one is poor, economical, has meager resources. The interpretation will show us how the rich theory emerges from the poor one. An interpretation is a logical homomorphism, in other words a meaning preserving translation function from the rich language to the poor one. Technically, this means a bunch of explicit definitions, one for each primitive notion of the rich language in terms of the poor one. Such an explicit definition can be considered to be an experiment to be made in the poor world to establish whether a notion of the rich theory holds in a particular situation or not. This way we give experimental, operational semantics for the rich theory. In this talk we illustrate our approach with special relativity, but the same is being done for general relativity, too. In our case the poor theory will be a theory of signals (or photons), a theory of communication, while the rich theory will be a spacetime theory, special relativity . The interpretation we are about to show you provides an experimental, operational semantics for special relativity . In the other direction, this interpretation shows us how a society living in the economic word with signaling as the only tool, can discover time and space as useful notions for survival. This society makes experiments, devises and discards concepts, sets up inductive hypotheses, revises its beliefs, and so on. In short: it conducts a cognitive process of knowledge acquisition. Let’s get to work.

3. The timeless theory has a language where we have two sorts of entities, experimenters (or observers) and signals (or photons), and we have two binary relations: an experimenter can send out a signal and an experimenter can receive a signal. So this theory is based on two binary relations only, between experimenters and signals. We took the idea from a paper of James Ax.

4. That was the language. The axioms of Signaling Theory will be all the formulas valid in its intended model. The intended model is the following. We are in four-dimensional Euclidean space R to the four, R is the set of real numbers. Experimenters are represented as straight lines of slope less than 1 (i.e., they are more vertical than horizontal), and signals are represented as finite nonzero segments of lines with slope 1. An experimenter sends out a signal if the starting point of the signal lies on the line representing the experimenter, and an experimenter receives a signal if the end point of the signal lies on the line representing the experimenter. This is the intended model, and the axioms of SigTh are the formulas valid in this model (in the FO language consisting of Sends and Receives). A few words about Signaling Theory, since this will be the main character of our talk today. A line in four-dimensional space represents motion in three-dimensional space. Experimenters represented with vertical lines are motionless and experimenters represented with slanted lines are in motion. Our experimenters, however, have no senses for motion, or time, or space. All what they can perceive of their world is sending out and receiving signals. In this model, from the point of view of sending and receiving signals, all the experimenters are alike: any one of them can be taken to any other by an automorphism. Hence, no matter how clever our experimenters are, they cannot distinguish the motionless among them. This is why they will discover special relativity as a spacetime for their world, and not Newtonian absolute time. By the way, all the signals are like each other, too in this model. In this respect our Signaling Theory resembles Incidence Geometry, where all the lines and points are alike. However, in SigTh there is no duality between experimenters and signals, as in geometry, signals behave differently from experimenters.

5. A visual representation of this intended model. Here are some motionless experimenters, they can send out and receive signals.

6. We have running experimenters, too, they also can send out and receive signals.

7. However, our experimenters have no clocks and no meter rods. Yet, eventually they will discover time and motion. Remember the title: “The End of time”.

8. Julian Barbour published a great book with the title of our talk.

9. Next we recall the language of special relativity theory conceived as a theory of FOL. This language has two universes or sorts, just as Signaling Theory does. These are the sort of bodies and the sort of numbers. Photons and observers are considered as special bodies. These two sorts are connected by a 6-place relation called worldview relation, this represents coordinatization. The worldview relation tells us which observer sees which bodies at what coordinates. One can view the language as consisting of a set of coordinate systems, a coordinate system for each observer. This language is quite convenient for talking about motion, time, space, velocity, acceleration, etc.

10. We presented the language, next come the axioms. We present SpecRel by listing its 4 axioms. 1:All observers see photons move with the same speed in all directions. (Hence there is such a notion as the speed of light.) 2:All observers see the same events (an event is meeting of two or more photons). 3:The owner of the coordinate system sits tight at the origin of the space. 4:The numbers are the usual. By this we presented SpecRel. We note that all the characteristic predictions of special relativity are provable from these four axioms. E.g., moving clocks slow down. Having introduced our two theories, now we have to show an interpretation between them.

11. Theorem: SpecRel can be interpreted in Signaling Theory. Moreover, the interpretation is not only a logical homomorphism, it is a logical isomorphism. So, from the point of view of logic, these two theories are the same theory, just told with different words. Proofidea (the full proof can be found in Johan’s Outstanding Logicians volume): We have to give explicit definitions for the primitive notions of SpecRel in terms of SigTh. The key notion of SpecRel is that of a coordinate system. So, given an experimenter, we have to give an explicit definition for a coordinate system in terms of just signaling.

12. Space. Our experimenter has to define his space. The points of space will be the things that do not move relative to him. Hence he has to give an explicit definition for when an experimenter is motionless relative to him.

13. Whether two experimenters are motionless relative to each other is definable in the intended model. Here is the usual geometric definition for when two straight lines are parallel: e and e’ are parallel if they do not meet and there is a pair of intersecting lines connecting them. How do we express that e and e’ do not meet? Remember: all they can conceive is signaling. Def: two experimenters meet if and only if there is a signal that both of them send out. Next, how do we define whether two signals intersect? We will use signal-triangles for this purpose, so we have to define when two signals are sent out at the same time (remember, time is not in our language). Def: two signals s and s’ are sent out at the same time if and only if there are two experimenters who both sent out each of them. These are the main steps in the definition of relative motionlessness. The explicit definition we get this way will contain a universal quantifier (two experimenters do not meet if there is no signal …). Thus they have to wait for an infinity of time for learning that they are indeed motionless wrt each other. Existential formulas are more experiment-friendly.

14. The good news is that there is an existential explicit definition, too, the experimenters do not have to wait for an infinity of time. We can invoke Desargue’s theorem from geometry and design an experiment in finite time. The definition is represented geometrically on the left. Experimenters e and e’ want to decide whether they are motionless wrt each other. They ask a friend to send three balls towards them as shown in the figure. The three balls in the figure are drawn in three different colors. First e checks whether the speeds of the three balls satisfy the pattern shown in the lower left part in the figure, and then e’ checks for the same pattern but now on his lifeline. If he finds that the triple meeting takes place, then they are motionless, otherwise not. This is an existential definition for two experimenters being motionless wrt each other. Lower left part of spacetime diagram is animated on the left.

15. Time. To give an explicit definition for the time-part of the coordinate systems, the experimenters define a structure on the events happening on their lifelines. An event is sending out or receiving a signal.

16. They can define temporal order on these events by this pattern, they can define equiduration of time-lapses by this pattern, and using equiduration they can define addition. They can define multiplication on the events on their lifelines by using this pattern (from Hilbert’s coordinatization method). After this, they find out that the so defined structure is a real closed field. We are almost done. To finish the explicit definition of their coordinate systems, the experimenters need a notion of abstract number that is the same for all of them. And here we hit a wall.

17. Usual definability theory of FOL does not allow us to define new entities, it only allows us to define new relations on already available entities. What do we do about this? We elaborated a new branch of definability theory. (Thus logic benefits from such applications.) The idea of defining a new universe of abstract numbers that the experimenters all can use is to abstract from all the “private” number-fields of the experimenters that they have on the events happening with them. These little fields are all isomorphic to each other, and the important thing is that there is a uniform definable isomorphism between them. The “name” (i.e., the explicit definition) of this isomorphism makes it possible to convert the little fields into a linguistic entity that will then become the new sort of abstract numbers in the language. This is how to create an abstract notion of real numbers. This rounds up the proofidea for showing that Special Relativity and Signaling Theory are definitionally equivalent, i.e., they are indeed the same theory despite all appearances, told with different words. They are two wrappings for the same content. Do we want to have different wrappings for the same content? Would not it be better to have just one wrapping for one content?

18. Feedback to definability theory: defining new universes and new entities. Logic and relativity (in the light of definability theory). Madarasz, J., PhD Dissertation, ELTE Budapest, 2002.

19. Famous physicist Richard Feynman said in his Nobel Prize lecture that it is extremely important to have many different equivalent theories for the same known portion of physical reality, because these different theories will suggest different ideas when moving towards the unknown.

20. The same works for arbitrary fields: finite axiom system for signaling theory. Undirected signals connect two experimenters, but we do not know who sent and who received, there is no order in the sending and receiving acts. They both sent and received them. “Mapping” Schwarzshild static black hole spacetime goes similarly using the suspended as well as the free-falling observers together with the light-like geodesics as signals. These applications can contribute to the trend of pushing the limits of today’s notion of computation (beyond-Turing).

21. The end. Of time. As a primitive notion.