Are Real Numbers Really Real?

[Esote]

Indeterminism in Physics, Classical Chaos and Bohmian Mechanics, from Nicolas Gisin (Dated: March 20, 2018) :

https://arxiv.org/pdf/1803.06824.pdf

Physics is often presented as the example of a deterministic explanation of our world. Furthermore, it is often claimed that all good explanations must follow that structure.
This is usually illustrated by classical physics, a theory whose explanatory power is truly impressive, despite that (or because?) its limits are well understood.
Indeed, the domain of validity of classical mechanics is limited by relativity and quantum theory whose predictions are more accurate when speed and size (or action) get close to critical values determined by the universal constants c and ̄h, respectively.
Classical mechanics is a set of dynamical equations, with initial conditions - typically position and momentum of point particles - given by real numbers. Except for particular cases, these dynamical equations together with the initial conditions determine completely and uniquely the solutions at all future and past times.
Hence, the conclusion that classical physics is deterministic.
This has huge consequences. First, as said, this is often taken as the goal of all good scientific explanations.

For example, many philosophers and physicists try to formulate quantum physics in such a way as to recover something like classical determinism, despite quantum randomness; in sections VII and VIII I discuss Bohmian mechanics in this context. Second, if scientific determinism would be the only good scientific explanation, then it would be highly tempting to conclude that everything covered - at least in principle - by science happens by necessity, i.e. is determined since the big-bang, including all physiological processes.
In my opinion - but this paper is independent of this opinion - this has dreadful consequences: time and free- will would be mere illusions, our world would be like a movie in a closed box without any spectator.
Even life would be just an accident due to peculiar initial (or final?) conditions of the world.
In the first part of this paper I argue that there is another theory, similar but different from classical mechanics, with precisely the same set of predictions, though this alternative theory is indeterministic.
In a nutshell, this alternative theory keeps the same dynamical equations as classical mechanics, but all parameters, including the initial conditions are given by numbers containing only a finite amount of information.

In sections III-V I argue that this alternative classical mechanics is more natural because it doesn’t assume the existence of inaccessible information. One way to argue in favour of limiting physics to numbers with finite information is that any finite volume of space can contain only a finite amount of information. Consequently, the huge empirical evidence for classical mechanics equally applies to the alternative indeterministic theory. The alternative theory has the same (enormous) explanatory power, section VI.
It is thus not correct to claim that the empirical evidence and the explanatory power of classical mechanics supports a deterministic world view, as the same could be said about an empirically equivalent but indeterministic alternative classical mechanics theory.
In the second part of this paper I argue that every indeterministic theory can be supplemented by additional variables in such a way to render it deterministic. In brief, it suffices to assume that all the indeterminism that is required at some point in time when, according to the indeterministic theory, God plays dice, i.e. when potentialities becoming actual, could be hidden as supplementary variables in the initial condition of the equivalent deterministic theory, i.e. God played all dice at the big-bang.
This closes the circle: deterministic theories are equivalent to an indeterministic alternative theory in which real numbers are replaced by finite-information numbers, and indeterministic theories can be supplemented by additional hidden variables in such a way that the supplemented theory is deterministic. Moreover, it seems that applying our rule twice one may recover the initial theory.

In sections VII and VIII the above rule to supplement indeterministic theories is illustrated on the alternative classical mechanics theory and on standard quantum theory, leading to standard classical mechanics and to Bohmian mechanics, respectively.
Admittedly, in these two examples, the supplemented deterministic theories have, in addition to determinism, some elegance which speaks in their favour. However, I conclude that determinism is too high a price to pay to accept these supplementary hidden variables. Indeed, indeterminism explains nicely, among other things, why probabilistic tools are so powerful in statistical mechanics.
Moreover, indeterminism opens the future, makes potentialities a real mode of existence and describes the passage of time when potentialities become actual...

 

[Approaching Infinity]

Indeterminism in Physics, Classical Chaos and Bohmian Mechanics, from Nicolas Gisin (Dated: March 20, 2018) :

https://arxiv.org/pdf/1803.06824.pdf

Not sure if your title reflects the main point of this article, at least from the description posted. (That said, I'm no mathematician!) I agree with this thought:

In my opinion - but this paper is independent of this opinion - this has dreadful consequences: time and free- will would be mere illusions, our world would be like a movie in a closed box without any spectator.

And this:

...indeterminism opens the future, makes potentialities a real mode of existence and describes the passage of time when potentialities become actual...

If I remember correctly, Ark wrote something similar: the future is quantum, the past classical.

As for your title, "are real numbers really real?", that's a deep question, too. I'd ask the mathematicians: is this any different from saying "are numbers really real?" Because that's a big question in philosophy of mathematics that materialists have a tough time answering. While mathematicians can't help but presume that numbers and other mathematical objects are "real" while doing math, their reality can't be adequately defended in a materialistic philosophy. That led one philosopher to say something to the effect that mathematicians are materialists in theory, but Platonists in practice. The problems are: how/where can numbers exist? How can they have any causal effect on the minds of people doing mathematics? And how can humans "perceive" them, if they are not physical objects and if humans only perceive via the senses?

Those problems led Whitehead to argue that mathematical objects are "eternal objects" in the cosmic mind that are "perceived" non-sensorily and given causal power by the agency of said cosmic mind.

 

[Esote]

Thanks Approaching infinity for your clever insight.

Hence the question could be : Is reality really real ?..

 

[Maat]

As for your title, "are real numbers really real?", that's a deep question, too. I'd ask the mathematicians: is this any different from saying "are numbers really real?" Because that's a big question in philosophy of mathematics that materialists have a tough time answering. While mathematicians can't help but presume that numbers and other mathematical objects are "real" while doing math, their reality can't be adequately defended in a materialistic philosophy. That led one philosopher to say something to the effect that mathematicians are materialists in theory, but Platonists in practice. The problems are: how/where can numbers exist? How can they have any causal effect on the minds of people doing mathematics? And how can humans "perceive" them, if they are not physical objects and if humans only perceive via the senses?

Yes it's a big question which Marie-Louise von Franz tried to address in her book Number and Time. Not an easy reading. Here's are some notes about it from someone. And here are some quotes in it from wiki :

• To sum up: numbers appear to represent both an attribute of matter and the unconscious foundation of our mental process. For this reason, number forms, according to Jung, that particular element that unites the realms of matter and psyche. It is “real” in a double sense, as an archetypal image and as a qualitative manifestation in the realm of outer-world experience.
  • p. .52

• It [number] preconsciously orders both psychic thought processes and the manifestations of material reality.
  • p. 53

• Nevertheless, this individual aspect [just-so-ness] of number appears to contain the mysterious factor that enables it to organize psyche and matter jointly.
  • p. p60-61

 

[Approaching Infinity]

Yes it's a big question which Marie-Louise von Franz tried to address in her book Number and Time. Not an easy reading. Here's are some notes about it from someone. And here are some quotes in it from wiki :

Those are great quotes, Maat. Thanks for posting. Probably one of the coolest things about math to me is the fact that some theoretical mathematicians will come up with really cool ideas - and only years later it is discovered that those ideas are manifest in some previously obscure physical process. Math really is the language of God.

As for Esote's question, I'd say it only makes sense to say yes. We have no frame of reference except our experience. And any "philosophizing" that isn't grounded in experience is a flight of fancy. If reality isn't real then "real" means nothing. It's just having fun with words.

 

[Esote]

Actually it's not reality (or numbers) which wouldn't be real, but merely our interpretation of said reality...

 

[Approaching Infinity]

Actually it's not reality (or numbers) which wouldn't be real, but merely our interpretation of said reality...

So reality is more expansive than our current understanding of it? I can get behind that. We can never know the truth completely, there's always more to learn, but some interpretations are more valid than others.

 

[kenlee]

Actually it's not reality (or numbers) which wouldn't be real, but merely our interpretation of said reality...

Perhaps one can look at reality from 2 perspectives, from which interpretations can be made. From one perspective reality can be viewed from the perspective of pure form with little to no essential content and from the other perspective from pure essential content with no form. For example, you could have a conceptual essential rose (the idea of a rose) with all its qualities (that has yet to take form) which starts out as an idea first, in a deeper reality, and then it eventually becomes an individual actual rose in a "lower" reality (with more mechanical laws) that has mostly form with less essential content. From one perspective there is the essential rose with all its essential content, packed with information, and from the other there is the existential (actual) rose with its multiplicity of external forms that has less essential content when it exists as an individual form, although it still possess the quality of "rose-ness" at even the lowest level of reality. So at one end of the 'spectrum' there is the pure essence of the rose, the 'great rose' (so to speak), with all its possibilities and qualities. At the other end of the spectrum there is the actual multiplicity of forms it can take.

The former has all the essential content of the idea of the rose (the essential rose). The latter has a multiplicity of forms but when perceived thru the senses as an individual form in time and space it lacks much of the essential content of its essential nature. But the higher and lower nature of the rose are still connected. So it's really one reality in terms of a mixture of quality and quantity, but from our perspective we can see it from one or the other extremes or from a mixture of both and within that limited context we come up with an "interpretation" of what a rose really is.

 

[Jeffrey of Troy]

So reality is more expansive than our current understanding of it? I can get behind that. We can never know the truth completely, there's always more to learn, but some interpretations are more valid than others.

"All models are wrong, but some are useful." (wikipedia link)

In the 3D / 4D etc. model, taking 7th D's perception of "what is" as "what is", even when we graduate from 3D to 4D, we will only improve from being able to perceive a maximum of 3/7th's of "what is"to a max of 4/7th's of "what is".

Perfect is the enemy of Good. As long as we are moving in a direction away from error, we are "walking the path." (youtube)

 

[Esote]

Perhaps one can look at reality from 2 perspectives, from which interpretations can be made. ...So it's really one reality in terms of a mixture of quality and quantity, but from our perspective we can see it from one or the other extremes or from a mixture of both and within that limited context we come up with an "interpretation" of what a rose really is.

An interpretation of our conditioned projections in a more or less limited context.
Which doesn't mean it isn't real, only that it's a distorted, partial view of what really is...

 

[mrtn]

I think I don't really understand that text, especially the math and science behind it, but anyway I had some thoughts :

Since a finite volume of space can’t contain more than a finite amount of information ...

That's seems to be the main base assumption in the paper, and its a big assumption. Why would you assume that actually? From there - which is at the 3. sentence of the PDF - it's not even about numbers anymore. If you assume that the nature of some real numbers (for example infinity) doesn't exist in the first place, then there is no point in showing that there is no reality to the numbers representing this nature.

Today, it is quite natural to assume that no finite volume of space can hold more but a finite amount of (Shannon) information, as measured by bits.

Firstly, it's becoming more and more natural these days to assume the opposite, be it true or not. Secondly, 'measured by bits' means to apply a limited system. So the [amount of] information you will find is limited already by the measurement. It's like saying you won't find an unlimited amount of centimeters in a meter.

Consequently, assuming that information has always to be encoded in some physical stuff, a finite volume of space can not contain more than a finite amount of information. At least, this is a very reasonable assumption.

Is it? The whole thing looks as if he has some kind of god particle in mind, or a somewhat smallest 'thing' the universe will ever do (I doubt that).

This illustrates the absurdly unlimited amount of information that real numbers contain. Real numbers are monsters!

It's not absurd. And we don't know that our reality isn't a monster, I rather guess it really is, as seen from half of all the possible perspectives.

after the first bits, the next bits of almost all real numbers are random, they don’t follow any structure.

I don't understand that random thing he is talking about at different passages. The best I can imagine what he means is that each subsequent digit or bit is only one further step of approaching the target value, kind of like bouncing around it.

... First, because computable numbers like, e.g., π have all their bits fully determined by a finite program.

Thats another thing that confuses me. Maybe its my ignorance about the terminology. A program might be finite in its conceptual representation, like a sourcecode of limited size. But it will most likely run within a limited framework, like computer hardware, and so it will need infinite amount of time. Infinity doesn't go away just because you put it oustide or elsewhere. So if its always there somehwere anyway, why not just letting it be in the numbers. Thought there are probably practical reasons for that.

But today that we know that “real numbers” contain an infinite amount of information and that they would be better called “random numbers”, we should realize that such numbers can’t be the basis for determinism.

Yeah, but maybe determinism isn't a basis for reality.

<edit>

In brief, any storage of a bit of information requires some energy and large enough energy densities trigger black holes.

Well we don't know if we live inside a black hole, or if our universe is perveived as such from outside.

Given that I don't have the background to understand it properly, I feel a bit snotty to comment on it in such a way, but that's what came to my mind.