Before Everett's Many Worlds interpretation of QM, it appeared that reality may have been indeterministic, although people like Einstein famously argued that this was not possible. When you think about it, I don't see how indeterminism could even be possible. Logically, the universe would have to behave logically and mathematically, and there is no algorithm composed of logic and math which can produce TRULY random numbers. The best you can do is have those pseudo random number algorithms which produce random looking numbers but in fact if given enough data you would be able to reverse engineer the algorithm. So, this means that an indeterministic universe is logically impossible, correct?
You presume we have reached the pinnacle of our understanding of logic, mathematics, probability and the universe at large. Why do you think that, because we know of no algorithm, the universe is obliged to behave according to the ones we know?
Well, I assumed that it is mathematically impossible to have an algorithm which produces truly random numbers. I just assumed that this has been mathematically proven, but in my 2 minutes of googling I couldn't find a proof. But, I'm sure someone could come up with a proof that true randomness is impossible in math. For example: an algorithm is composed of functions, and functions map one value to another. By definition, with a given input, a function will always output the same value, so then, it is impossible for an algorithim which is composed of functions to produce different results given the same input. Therefore, if the universe dictated by math, then given a set of initial conditions, the result will always be the same, which means there is no element of true "randomness" possible.
So, what you're saying is that the Newtonian mechanical universe is inconsistent with quantum indetermincy? So, what? What's the problem?
All you're saying is that we don't have a mathematical model of it. Mathematics is a very active field. Don't close the coffin on it yet.
Without getting into the mess of trying to define exactly what a random sequence of bits is, a mathematical proof of their existence would have to be either constructive or non-constructive. If it were constructive it would give a computational algorithm to produce a random sequence; by any definition it would be predictable and hence not random. If the proof were non-constructive it would be essentially useless. Take the set of all infinite sequences of 0's and 1's; it is uncountable. The set of all computational algorithms is countable. Therefore there are an uncountable number of infinite sequences of bits which cannot be produced by any computational algorithm i.e. they are random. So in the world of mathematics, random sequences exist but we are incapable of displaying them --- useless. The crux of your argument is whether it is possible to encapsulate the objective nature and evolution of the universe in a set of mathematical formulae or, if physics uses mathematics can only describe our interactions with the universe with various models. While the belief in ultimate mathematical laws of nature is widespread it is not universal: "Law without law" James Archibald Wheeler in "Quantum Theory and Measurement" ed J. A. Wheeler & W. H. Zurek 1983 also available online: http://what-buddha-said.net/library/pdfs/wheeler_law_without_law.pdf "Are there dynamical laws?" by J. Anandan, Found.Phys. 29 (1999) 1647-1672 also available online at Arxiv:quant-ph/9808045v4. Skippy
Interesting. Im not sure if this was your point, but since there are uncountable random sequences which cannot be described by math, then you could have a universe in which the quantum events are dictated by a finite or infinite string of random numbers. And then if you assume Max Tegmark's view that the universe is a mathematical object, and all mathematical objects exist, then that would mean that if there was one universe which had events dictated by random strings of numbers, then there would be infinite universes like that, each with a different random string of digits dictating events in that universe, and so these events would seem genuinely random to observers inside the universe, no matter how much data they collected, because there is no algorithm dictating the events, they are dictated by the random string of digits. However, this would not be indeterministic, because the whole future of the universe would be spelled out in the string of random digits. Edit: I saw your link to the article titled "Law without Law". Logically, you can't have something and not have it. This is an example of what I mean by logically impossible. Basic logic says that A and ~A cannot both be true, yet this is the type of thinking that I believe is required for true indeterminism. i.e. true indeterminism is logically impossible, which is why there is so much confusion about quantum mechanics, because the fundamental idea (that the quantum events are truly random) goes against basic logic. Despite their fundamental assumptions being wrong, their math (the Schroedinger wave function) was correct, and was experimentally verified, so people began to believe their fundamental assumption too. So this is why you had people like Feynman talking about how no one understands quantum mechanics, because the idea that events can be truly random is impossible. But, if you look at it from the Everett Many Worlds approach (which I admit I don't understand very well, but I know the overview and the implications of it) then these logical impossibilities that seemed to be real, suddenly disappear. So what I am saying is that, the universe MUST obey logic. The early quantum mechanics pioneers confused themselves by taking the assumption that it is possible for events to be truly random, which I believe fundamentally violates logic. People should have figured this out sooner, they should have realized right away that if something does not obey the rules of logic, it must be wrong, but the excellent predictions of the Schroedinger wave equation, and the lack of a good alternative interpretation, made people believe it.
That the outcomes of a series of measurements of, say, the the spin of an electron, is a random sequence of bits is a genuine falsifiable physical theory. If you can find an algorithm which matches all the bits in the sequence AND an arbitrary number of future bits then you have proved the sequence to be non-random; the theory is false; the world is deterministic. What sort of experiments would show that the assumption of randomness fundamentally violates logic? You must make the assumption of universal mathematical laws which describe the objective nature and evolution of everything. You must basically assume determinism to logically disprove indeterminism. A little circular. Can you show any fundamental violation of logic which does not presuppose determinism? Skippy PS If you liked "Law without Law" by Wheeler then you must look into his beautiful (but, alas failed) theories of "charge without charge" and "mass without mass".
Logic, too, is an active field of research. It is simply too early in our knowledge to claim we can mathematically or logically rule out possibilities.
I think the error in the argument of the OP lies in the fact that it is impossible to describe the whole universe in logical way. From GĂ¶del's theorems we know that no theory can be both complete and consistent at the same time. Thus, if you construct a theory to be logically consistent, then it is impossible for that theory to describe the whole universe (and so there can be randomness in the parts we have not been able to describe). On the other hand, if you construct a theory that describes the whole universe, then it must be inconsistent, which may be viewed as a form of randomness.
Indeterminacy is just another property we observe like the mass of an object. People don't often go around insisting that mass is impossible or illogical simply because it is considered absurd to do so in most worldviews, but give them some other property like "time" and they feel free to argue the point with their last breath. We observe what we observe and to argue that what we observe is illogical or impossible without a concrete explanation is either mysticism or just plain foolishness. Likewise, the idea that because we don't have the math to describe something it must be impossible or illogical is again either mysticism or just plain foolishness.
Why? I don't see this is logically necessary at all. What does it mean to say that something behaves "mathematically" - mathematics is a tool we use to describe how something behaves, not the other way around. Everything behaves "mathematically" - even indeterministic systems. Define Rand(1) as a function which produces a "truly" random number between 0 and 1. The function Rand(1) then produces a "truly" random number. If quantum events are truly indeterministic then all we need do is use a quantum event as our random number generator.
Indeterminism may be against mathematics, but not against logic (rand(1) function definition I saw here seems to me a tautology). Maths defines things. Logic does not. We probably cannot achieve a complete mathematical description of the Universe (see https://www.physicsforums.com/showthread.php?t=411747 ) but that does not exclude a complete logical framework. There are things that are out of mathematics, but in the realm of logic, for example Philosophy. The sentence "I think therefore I am" is pure logic, but it is not maths.
This is not a function however, you have not produced a rule to determine the outcome. Furthermore, it doesn't really make sense to produce a random outcome mathematically. ---------- The OP has a point however in that the world must behave logically. But that is not the same as behaving mathematically, which makes no sense. The reason for that the world must behave logically is simple. We describe the world through language in which logic is built, and hence our descriptions, as far as they refer to natural objects and phenomena, must behave logically. We cannot describe paradoxes, and if we did we would rather conclude that our descriptions were wrong than to conclude that we have found a "paradox of nature". More strongly still, it is apparent that even our sense of perception must behave accordingly. We cannot observe paradoxes or inconsistencies, as our mind has a certain way of structuring external stimuli in a logical and causal fashion. This must not be confused with the models we use-which may or may not be in accordance with strict causality-but rather only the raw output of immediate perception. The statement that the world behaves mathematically is absurd however. It is not saying that mathematics refers to physical objects, which it doesn't, but it is actually more strongly saying that mathematical notions actually applies innately to physical phenomena. Mathematics (or I should say:doing mathematics) is in its simplest sense applying predefined abstract rules in some way or another, and has therefore no logical connection to physics-for which we at best can have descriptions-mathematical in form or not. The connection is only through mathematical models which serve as no more than descriptions. I believe this interpretation as close as we can get to a meaningful sense of the statement "the world behaves logically". That logic is innate to nature is as absurd as if it were mathematics, and hence I propose that innate determinism is absurd as well. Determinism as a term can only meaningfully be applied to sense representations, abstract or perceptive, not to nature itself. And I actually believe it to be so.
Anything that can be described logically can be described mathematically (though that's not to say a new branch of mathematics might have to be invented). The mathematics is nothing more than quantifying the logical entities and relationships.
So for example, do you mean that "I walked into the room. Therefore I was in the room." as a logical argument can be described mathematically? If it could, I wouldn't see the use of it, and it would certainly not add anything to what we have already said. In any case I did not intend to deny what you are saying, but I fail to see the point. One has to be careful about the notion of describing something mathematically. It can mean various things. At its weakest sense it's merely the use of mathematics to aid a representational description of reality. Like using arithmetic to count people in order to group them in a certain way etc.. In a stronger sense it can mean to capture a physical system into a mathematical theory, in which the full causal structure one wishes to describe has its mathematical counterpart. For example Newtonian mechanics and, say, planetary movement. In any case there is no way of denying that the mathematical part purely consists of the calculations involved, and that any interpretation of it as a model will require language. While the translation from theory to representation (through language) is commonly understood (like from elementary calculations in Newtonian physics to the physical counterpart), the fact that it is there must be affirmed. The logic involved is still by means of language (unless of course one has abstracted from this and formalized the logic as well, but in this case the formalization is just a concatenation of the underlying logic involved).
Certainly. It is the point of the thread. Whether a logical universe could be described mathematically. Logic is defining entities and the rules they follow using words. Mathematics is simply assigning values to the entities and equations to the rules.
Of course it can. Lay it out logically - the entities involved and the logical rules they follow, and then simply* assign values to the entities and equations to the rules. *it's not really simple. The issue is to determine what entities are required to describe the system, and then what equations describe changes to the system.