Ken,
The guidelines state: The goals of the Foundational Questions Institute's Essay Contest (the "Contest") are to: •Encourage and support rigorous, innovative, and influential thinking connected with foundational questions;
Because Reality does not have any abstract "relationships", there are no abstract issues to resolve. Discussion over.
Warm regards,
Joe Fisher
Dear Ken,
I find your axiomatic approach interesting (it reminded me a bit of the approach presented in Richard Shoup's paper).
I did not follow your entire derivation but there are two issues I would like to point out in the spirit of constructive criticism. First, it seems that you would like to derive the introduction of additional coordinate degrees from your original axioms, but I'm not sure that is possible. For example, you may wish to check with a mathematician, but it seems to me that the step
"The only way to satisfy the axioms is to expand the
coordinate system to include a second degree of
freedom.(p3)"
Amounts to an additional unstated axiom (namely, that there is a second degree of freedom). The problem, as I see it, is that without it all you have is an inconsistent system. If this is right, it should still not be a big problem to fix by adding it to your initial set of axioms because, as assumptions go, that is a rather harmless one (this also applies to the Z-dimension).
Second, it is not apparent to me how you arrived at equation 4, which applies to a continuous manifold, given that you apparently started with a discrete geometry. In fact, I got lost right at the first sentence of section 3.4. Perhaps a picture might help, but fundamentally I didn't understand how curvature suddenly entered into the discussion. Even if I take it as a given that curvature did enter the picture, it is still not clear to me how you get to equation 4. One can certainly define curvature for discrete surfaces, but as I understand it, then integrals become sums and differentials become differences. My understanding is that the Ricci tensor and Ricci scalar are defined (at least in their standard form) only for continuous geometries. So it seems at least a possibility that somewhere an extra assumption has crept in by which the discrete geometry has turned into a continuous one.
If that is true, then I would deem it a much more serious hidden assumption because it changes the entire character of your approach.
In any event, as I mentioned your approach is interesting, but I would recommend that you check with a mathematician on the validity of some of your steps. I hope you found my criticism useful.
Best wishes,
Armin
Armin,
Thanks so much for your constructive criticism. You bring up a raft of issues, all of which need to be explored in more detail. First of all, you appear correct in that there are similarities between my paper and the one submitted by Richard Shoup. Thanks for the pointer.
Your primary issue lies in the ability to resolve an apparent inconsistency by simply creating an additional degree of freedom. It seems suspiciously like slight-of-hand. This line of inquiry leads interesting logical and philosophical considerations. A secondary issue is that of continuous versus discrete mathematics.
Expanding a coordinate system, or more generally a geometrical manifold, to support an idea that on the surface, appears to identify a logical inconsistency is nothing new. The prime example of this it expanding the reals to the complex plane to accommodate the notion of 'i' or the SQRT(-1). A much more complex example, and one familiar to most physicists, of expanding a manifold to avoid irritating inconsistencies may be seen in the development of Calabi-Yau manifolds used to create an approximation of a flat space from a curved space. In this example a minimum of 6 extra degrees of freedom are required to perform the necessary manipulations. Finally, the field of string theory requires an expansion of the underlying geometry from 4 to 10 or 11 dimensions (or even sometimes 27 degrees of freedom) to allow for logical consistency.
Your second point serves to illuminate a key point. That is the underlying geometry of the general system is continuous, not discrete. Attached to each continuous point in the manifold is a probability, a veritable field of probabilities where each probability describes the probability of the system being in that particular state. The result is a sea of 0 probabilities sprinkled through with an odd non-zero probability popping up here or there. For example, the probability that the state of the system is in (.5, y, z) is zero since a priori the state of the system (the X coordinate must be an integer). The paper tries to show that the mathematics engendered by the axioms identify which points can have non-zero probabilities. Although outside the scope of the paper the concatenation of all of these points can be describe by a wave equation.
I agree with you that simply creating equation 4 out of thin air is a bit much. It is a leap of faith far too far, one necessitated by the constraint of keeping the paper to 9 pages.
Thanks,
Ken
Dear Ken,
Thank you for your nice answer (not everybody responds nicely to constructive criticism). Let me just comment on the following:
1. Regarding the expansion of coordinate degrees of freedom, my understanding is that if the real line is extended to the complex space, then this has to be accompanied by the definitions of operations involving complex numbers, and I don't know how one can arrive at the fact that there exist entities that satisfy such definitions (namely, complex numbers), without stipulating as an assumption that they do. It may be that I am being a bit of a mathematical stickler here; I am myself by no means a fan excessive rigor, but requiring the clear identification of what it is that one assumes in building a framework does seem reasonable to me. I really don't know anything about the inclusion of CY manifolds in string theory, but if their incorporation is not stipulated at the outset, then I suspect this may have something to do with the fact that sometimes string theory is regarded as non-rigorous mathematics.
2. You said:"Attached to each continuous point in the manifold is a probability". If this is really what you meant, then I think this is bad news. In a continuous probability space, finite probabilities are associated only with continuous intervals. Individual points, or collections of individuals points, are associated with a zero probability measure, so that what you seem to be describing is the type of pathological space which probability theory unfortunately does admit, namely one in which the probability for every event is zero, but the probability of the entire space is one. I strongly urge you to reconsider how you are incorporating probability theory into your theory, or at least consult with a mathematician.
3. If the underlying space is continuous and not discrete, then it seems to me that there is some more to be said about how you arrive at the intervals to which the step sizes at the beginning correspond
4.I still don't understand where curvature came from in your theory. Can you please explain it to me in simple terms?
Thank you and all the best,
Armin
Dear Ken,
I think your system is very interesting and it SEEM to have some flavor similar to my system( and ironically to Armin's, but that is another story).
Now, I have not really delved into the details, but from some reading and the comments I can see that the main issue again surround discrete vs continuous. My system sheds a strong light on the subject, because I use both integer and the reals. However, the real number that I am using is based on a computer system and also I throw a UNIFORM random numbers. I get very nearly the same results, However, the simulation for spin seem to work for only integers. But we also know that quantum mechanics is based on the continuum and we get spin also, hence the strange connection.
I think we can discuss more once you have gotten maybe just a bit familiar with my system.
It is unfortunate that your idea and mine has not been discussed more because it seems that most of the essays have concentrated more on wordy philosophy which are easier reads.
Thanks and good luck.
Armin,
Let me focus on issues 2,3, and 4.
You bring up an excellent point regarding assigning a probability to a point in a continuous system. The key issue revolves around the notion of discrete versus continuous. The underlying geometry is a Riemann manifold, an example of a metric space. General relativity describes the geometry of space-time as a Minkowski space, another example of a metric space. All metric spaces, are by definition, continuous. Now, let's examine a very simple metric space, namely the reals. And on top of the reals let define a traditional random walk, where the metric (the distance measure) is described by stating the distance between any two states is a constant, in this case 1. Now, let the random walk operate over a set of N state transitions. A probability distribution can now be assigned to the set of reals. In this example, each of the possible N+1 states (real numbers 1,2,3, ...,N) is assigned a positive probability while all non-integer states are assigned a probability of one. There is no inconsistency here. In this example a discrete system is overlaid on a continuous geometry. It is the same in the system I described. The system is discrete, while the underlying geometry is continuous.
I hope the above description creates a bit more clarity for your issues 2 and 3. There are a number of subtleties involved. Please get back to me if more explanation is needed.
Your question 4 is deserves a much more complete description than provided so far. The simplest explanation I can come up with to try to imagine a half circle with a radius of n. Lay a string around the circumference of the half circle. The length of the string must be n*(pi). The half circle is made up of n segments of equal length, meaning the length of each of the arc-segments much all be of length (pi). But the system is also defined by a discrete set of velocities, where each velocity is associated with a unique angle. It turns out the the delta between any two of the unique set of discrete angle is always less than (pi)/n. This means the arc-length of each pair of angle must be less than and not equal to (pi)/n. This is an apparent contradiction. However, this apparent contradiction can be resolved if a displacement in a new dimension is assumed. As an example think of a chord connecting to points on the surface of the earth. It is a provable statement that if the earth is not flat, then the length of the geodesic connecting the two point on the earth must be greater than and not equal to the length of the chord. My paper spend a lot of time discussing the relative lengths of both the chord as well as the geodesic. The difference in length is a function of the curvature of the geometry.
I don't know if this helps. If not, feel free to contact me.
Thanks again for the feedback. It was very useful.
Ken
Adel,
Thanks for you comments. It turns out there are number of essays that roughly lay out a similar idea, namely that an abstract, axiomatic foundation of physics is indeed possible to formulate. As you note, the notion of discrete versus continuous is a common thread connecting your essay as well as mine. But I think an even deeper thread is the notion of deterministic versus stochastic. If I understand your essay correctly you have developed a probabilistic model where the state of the system is essentially the expectation value generated by your simulations.
The idea of a fundamentally stochastic/probabilistic model of physics is a radical concept, one that most physicists would reject out of hand (although I strongly adhere to a stochastic approach). I would be interested in your ideas regarding a non-deterministic model of physics. Although quantum mechanics operates on probabilities, the math itself is decidedly and unambiguously deterministic.
Ken
Dear Ken,
A few comments:
"All metric spaces, are by definition, continuous."
No, a space with a discrete metric is a counterexample. In fact, originally, I thought that that something like that was what you had in mind in your initial steps of building your theory.
"And on top of the reals let define a traditional random walk, where the metric (the distance measure) is described by stating the distance between any two states is a constant, in this case 1"
Yes, after you pointed out to me that the underlying space is continuous, I understood that this is what you meant, and my question 3 still stands: What is it that gets you to a particular step size and not some other in your random walk? If you don't provide any argument for that, then I have to take that to be another hidden assumption.
"However, this apparent contradiction can be resolved if a displacement in a new dimension is assumed. As an example think of a chord connecting to points on the surface of the earth. It is a provable statement that if the earth is not flat, then the length of the geodesic connecting the two point on the earth must be greater than and not equal to the length of the chord."
Ah, I think I am beginning to get a better idea of where the curvature is coming from. Apart from the same issue as before regarding to whether this amounts to a hidden assumption, I am not clear how it connects to the explanation of curvature in GR. Note that in GR, the curvature can manifests itself in modifications of both time and space coordinates (relative to those associated with a flat space), whereas your explanation (as best as I can tell) of curvature only gives potentially an explanation for modifications of spatial coordinates relative to those of flat space.
Again, hope you find my critique useful.
Best,
Armin
Dear Ken Matusow
Your approach seem promising, at least for pedagogical visualization of quantum mechanics, for instance something as Bohr's model of an atom. But, alhough you wish to show, that your formulae are independent from real space and time, you can draw them with figures in real space. They can be easier to comprehend.
I do not understand some your expressions, for instance[math]\beta(n), i\beta(n)[/math] because this means always the same angle, 45° in complex space. Is this correctly written?
Best regards,
Janko Kokosar
Janko,
I'm not sure I understand your question. The order paired (B(n),iB(n)) represent displacements in both the X and Y dimensions. The displacements represent an angle on the unit circle and may be explained as follows:
A priori we know the distance to the ordered pair from the origin must be n*c(bar). This distance is imposed directly by the metric axiom. From some of the early theorems developed we know that the distance in the X dimension is proportional to B(n). Given these two pieces of information we can now use the Pythagorian equation to compute the displacement in the Y dimension, namely:
d(y)=SQRT((n*c(bar))**2 + (B(n)*c(bar))**2)
Again, a priori, the order pair (B(n),iB(n)) is a point of a circle with a radius of n*c(bar) that has a displacement in the X dimension of c*B(n) and a displacement in the T-dimension of SQRT((n*c(bar))**2 + (B(n)*c(bar))**2). This displacement angle from the Y axis to the point (B(n),iB(n)), corresponds to a velocity since it represents a spatial displacement divided by a temporal displacement. In this scenario a value of 45° represents a velocity of exactly c/SQRT(2).
Admittedly, these ideas are quite difficult to get across with lots of diagrams and pictures, but I hope this helps.
Ken
Dear Ken,
AS I understand, your (B(n),iB(n)) really means 45° on the unit circle in (x,iy)? Thus, the first and the second B(n) are the same?
But, once you wrote above a formula for d(y) = ..., and another time you said ''displacement in the T-dimension of'' and you gave the same formula. So, d(y)=''displacement in the T-dimension of'' ??
I suggest that once in a future you draw your derivation, it will be easier to understand.
Regards
Janko Kokosar
Dear Ken,
I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.
All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.
Joe Fisher
