It is normally expected that the laws of physics are the general end-product of the scientific process. In this paper, consistently with said expectation, I produce a model of science using mathematics, then I use it to derive the laws of physics by applying the (formalized) scientific method to the model. Specifically, the laws of physics are derived as the probability measure that maximizes the quantity of information produced by the scientific method as the observer traces a path in the space of all possible experiments. In this space said probability measure describes a general linear ensemble of programs which is a foundation sufficient to express all known physics. Since the definitions are purely mathematical and contain no physical baggage of any kind, yet are nonetheless able to derive the laws of physics from first principles, then it follows that the present derivation of said laws, as it is ultimately the product of the (formalized) scientific method, is the minimal mathematical foundation of physics as well as its philosophical less controversial formulation. We end with applications of the model to open problems of physics and produce testable predictions.
Since the definitions are purely mathematical and contain no physical baggage of any kind, yet are nonetheless able to derive the laws of physics from first principles,
In (2) of 3.1.1, are you defining the expression |p| with a bar over the top on the right side? — jgill
Without physical axioms or principles, you could not use math alone to determine physical truths. — fishfry
How can you use math by itself, without any physical principles, to know anything about the world? — fishfry
Both Euclidean and non-Euclidean geometry are logically consistent, and mathematics can't say which is true about the universe. — fishfry
your attitude guarantees your work, in terms of your claims, is nonsense. — tim wood
It's time for you to lay out just how a UTM works, because I do not see how one stops on an undecidable proposition. — tim wood
All facts are historical. What I suspect you mean by "fact" is "true." But that presents a whole other set of problems, true being hard to define except in abstract terms, and that at best a partial listing of the conditions under which something may be called true. — tim wood
Just for the heck of it, I can almost make sense of the above sentence. Care to try for English? — tim wood
Without physical axioms or principles, you could not use math alone to determine physical truths.
— fishfry
What is your definition of physical truth? — Alexandre Harvey-Tremblay
Instead of defining a theory via a finitely axiomatic system — Alexandre Harvey-Tremblay
I didn't use the phrase, and I included my own words that you quoted in my quote, so that you can see that. — fishfry
I am sorry. I am not sure I follow your point - you are saying you did not write the words in your quote? (I must be misunderstanding). — Alexandre Harvey-Tremblay
You have been misinformed. The standard axioms of set theory are infinite in number. That's because the axiom schema of specification and the axiom schema of replacement are templates that instantiate a new axiom for each predicate. What is true, if it helps your ideas, is that the axioms are recursively enumerable. — fishfry
There's no physics in math. You can use math to model real physics, as in the physics of the universe; or fake physics, as in the physics governing a video game. Math doesn't distinguish. How do you determine what's true about the world we live in, without any physical principles or empirical evidence? — fishfry
I meant finite in the sense that the number of bits required to specify the axioms is finite, thus schemas are fine. The correct therm I should have used is formal instead of finite... thus I will correct this to formal axiomatic system to avoid future confusion. Sorry for the trouble and thanks for pointing it out. — Alexandre Harvey-Tremblay
Do you believe one can define a computer (such as Alan Turing did) purely mathematically? I assume you say yes. — Alexandre Harvey-Tremblay
Do you believe computers exists in nature? I assume you say yes. — Alexandre Harvey-Tremblay
Thus, you have one example of a theory that is purely mathematically derived, which nonetheless corresponds to the rules of computations which are found in nature. — Alexandre Harvey-Tremblay
I have therefore produced a counter example to your claim that "There's no physics in math.".
Would you like to respond or revise your claim? — Alexandre Harvey-Tremblay
Why do you think my paper is purely about Turing machines and computation? — Alexandre Harvey-Tremblay
But the set of all natural numbers {1, 2, 3, 4, ...} is only an abstract conceptual fiction appearing in pure mathematics in the form of the axiom of infinity. There aren't infinitely many of anything in nature. — fishfry
Haha you didn't think you'd sneak that one by me, did you? Of course computers exist in nature, I'm typing on one as we speak. But we are now using the word computer in two distinct ways: A TM, which is a purely conceptual object with an unbounded tape; and my laptop, with a fixed, finite amount of memory. That's two distinct and inconsistent usages of the word computer. — fishfry
Allowing repetitions within the manifest is necessary because it allows one to build evidence, and thus increase one's confidence, that nature allows for the construction of a reliable universal Turing machine. Thus, a valuable use for repetitions is as a quality check on the UTM. Indeed, if one randomly or pseudo-randomly repeats many different experiments across the logical spectrum, and are indeed correctly reproduced, then one knows with a high degree of confidence that one uses a reliable computational system. Whereas for a manifest theory as a set, which includes no repetitions, the existence of a reliable UTM is merely assumed and in fact conditional to the framework, and as such cannot be inferred or proven. To prove with absolute certainty that the world allows for a universal Turing machine, one has to repeat every pair infinitely many times. Consequently, in any practical case one only infers, to a finite degree of certainty, that one has access to such a machine. Knowing that nature allows for the construction of a UTM, or if it does not then what are the specific limits to its construction --- is essential knowledge about reality. Using a tuple of experiments grants us this knowledge, whereas using a set of universal facts does not. This difference defines a classification between a mathematical and a scientific theories --- and any enumeration strategy which involves maximizing the entropy will differ in the scientific case from the mathematical case as a result.
But answer me this. What principle of mathematics says that bowling balls fall down, instead of up? After all if I implement Newtonian physics in a computer simulation but I reverse the sign of gravitational acceleration, bowling balls will fall up. The math doesn't care. — fishfry
The laws of physics are simply not what you think they are. — Alexandre Harvey-Tremblay
If your theory can't prove that bowling balls fall down, it's not a good theory. — fishfry
Of course it does - I told you how in the previous post. In my framework, gravity, and anything else for that matter, are simply the consequence of invariant transformations associated with the general linear probability amplitude. — Alexandre Harvey-Tremblay
Right, but if the tape is not infinite, then it is just an automata and these were also described by Alan Turing purely mathematically along with universal Turing machines (perhaps even earlier than that). — Alexandre Harvey-Tremblay
Thank good I am not using Peano's axioms then - if I did I would be in real trouble because indeed as you say there are more facts in PA than there are in nature. — Alexandre Harvey-Tremblay
To prove with absolute certainty that the world allows for a universal Turing machine, one has to repeat every pair infinitely many times.
Because you could certainly simulate Newtonian physics by reversing the sign of G and thereby make it a repulsive force. Math does not distinguish these two conditions. — fishfry
You have to understand that facts such 'the apple I am looking at falls down', and 'the fridge in my home has milk in it' are facts that may be found in a given manifest, but the laws of physics are statements about transformations of states between manifests. — Alexandre Harvey-Tremblay
To know whether bowling balls fall down, you have to drop a bowling ball. That's the only point I'm making. — fishfry
As hard to believe as that may be, I guaranteed you it is not. — Alexandre Harvey-Tremblay
Bar on top, by convention indicates the average of a value. Will add a note under the equation to specify. — Alexandre Harvey-Tremblay
My sense is that a professor of modern physics could not tell me whether bowling balls fall down either. — fishfry
If he is any good he will tell you that wether the balls falls down, up or side ways, is dependant on the reference frame of the observer. — Alexandre Harvey-Tremblay
If your theory can't tell if gravity is attractive or repulsive, then it's a bad theory. I can't state for sure whether the modern notion of a physical theory passes my test. — fishfry
I personally have not investigated the question. — Alexandre Harvey-Tremblay
Well you should go out and get a bowling ball. Let me know when you've got one and I'll tell you the next step! LOL. — fishfry
So is there any fact about the world that I I can go out and verify with my own eyes, that your theory predicts? — fishfry
I was also wondering about your focus on TMs, After all, as I noted, a TM can not solve the Halting problem, but for all we know, the universe can. Or maybe it can't. The question is at least open. — fishfry
I get that, but it doesn't mean you account for all possible experimental scenarios by looking out the window and playing with one ball. If one uses induction one gets an empirical theory and such theories can be falsified (fundamental theorem of science). — Alexandre Harvey-Tremblay
Glad you asked. If you look at the last section of the paper you can see the experiment I propose. It involves the observation of geometric interference exceeding that which is possible with complex interference of QM. I think it is very interesting and ought to be fairly easy to test (no need to build a bigger particle accelerator for instance). — Alexandre Harvey-Tremblay
Well that would be one way to falsify my claims. However, such a thing would violate a lot of things... the fundamental assumption science (a definition in my paper) would be violated, and would further make it impossible to formulate a scientific theory around those parts with exceed Turing computation. My probability measure 'travels' from manifest to manifest using a cumulation of computing steps. It would be unable to 'travel' to regions which cannot be reached by computation. — Alexandre Harvey-Tremblay
Hmmm. By that logic Newton shouldn't have based his theory of gravity based on studying just one solar system. — fishfry
I'm just giving you a hard time, as I've said, I haven't followed your paper in detail and I'm not in a position to offer any more substantive points than I already have. — fishfry
It was beneficial no doubt not only for his career and reputation but also for the history of science as a whole, but Newton got an empirically derived theory which was eventually falsified, because he looked at a subset of experimental space. Whereas, my framework allows one to look at all of experimental space at once, thus allowing for an "absolute" definition of the laws of physics. — Alexandre Harvey-Tremblay
I appreciate your feedback greatly, and I would not be engaging if I didn't. — Alexandre Harvey-Tremblay
There's no physics in math. — fishfry
A point that Metaphysician Undercover and I agree on, even though he blames it on math. But it's not math's fault. A hammer can build a house or be used by a vandal to break your car window. The tool isn't responsible for the user's bad behavior. — fishfry
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