Uncertainty about uncertainty

  1. Philosophy Forum
  2. » Metaphysics
  3. » Uncertainty about uncertainty

Get Email Updates Email this Topic Print this Page

Reply Wed 28 Jan, 2009 05:53 am
I think that I understand how it is that we can know an electron's position but not its velocity and vice versa. I also understand that Einstein did not think that this affected determinism but that nowadays, most physicists do believe that uncertainty would cast doubt on a deterministic interpretation of the universe.

What I don't understand is how uncertainty would practically affect causality. If cause and effect does nort apply everywhere, what can be there in its place? Is the uncertainty in my brain electrons analogous to my actually exercising free will? Surely no-one really believes this, do they? What then are the implications of the uncertainty principle for free will and consciousness?
 
Khethil
 
Reply Wed 28 Jan, 2009 07:44 am
@Lord Lucan,
Lord Lucan wrote:
I think that I understand how it is that we can know an electron's position but not its velocity and vice versa. I also understand that Einstein did not think that this affected determinism but that nowadays, most physicists do believe that uncertainty would cast doubt on a deterministic interpretation of the universe.

What I don't understand is how uncertainty would practically affect causality.


You're right - I don't think it would.

Because we can't know what's going to happen with 'X' because of its uncertainty, is not to say that whatever happens doesn't have an effect. This only means that we can't, with 100% precision, predict it.

I believe that were all factors known, that events could be predicted. Which, I believe, brings me to your core question. If things could be known or predicted, how does this affect free will?

Lord Lucan wrote:
If cause and effect does nort apply everywhere, what can be there in its place? Is the uncertainty in my brain electrons analogous to my actually exercising free will? Surely no-one really believes this, do they? What then are the implications of the uncertainty principle for free will and consciousness?


I'm no expert - to be sure - but what I'd say is this: I do have free will -and- you could predict my decisions beforehand were you to know all factors, all variables and their appropriate weights, my past, my feelings, emotions, physical state and more. To say that the cause and effect principle enables the possibility to predict future events doesn't preclude the concept of free will one iota.[INDENT]The possible choices I have on my menu before me still exist and I will still choose of my own accord, I'm simply not aware of the near-infinite variables involved that will lead me to that decision. This doesn't mean I don't have a choice; Even if I could know all the variables that'll ultimately lead me to that decision, I'd still have a choice. Despite how it 'feels', these aren't mutually-exclusive.
[/INDENT]Here's my qualifier and what I think is the Key: We can't know all the variables involved in events, actions or decisions; there's simply too much to take into account - too much to weigh. As you mentioned with your electron-predictability point, there are some aspects of physics we simply can't predict. This does not invalidate the cause-and-effect principle at all, it simply says we haven't the means by which to exploit it to its logical permutation.

Does that make sense?
 
Bones-O
 
Reply Wed 28 Jan, 2009 07:55 am
@Lord Lucan,
Uncertainty does not disrupt causality at all, you'll be pleased to know. The 'uncertainty' in a particle's position and velocity is a feature of a deterministically time-evolved wavefunction. I actually think its a bit of a misnomer, but it's a matter of interpretation. Some people interpret the wavefunction (or its square) at a given position as nothing more than the probability of finding the particle in each of those positions. I read it as any normal wave: that is, the particle is spread across all those positions at once. Thus the 'uncertainty' in position of a wave is nothing more than its spatial spread. Likewise the uncertainty in its momentum is nothing more than the dynamics of the wave components.

The breakdown in determinism lies in measurement, not propagation, since there is no known mechanism for time-evolving an 'uncertain' particle into a certain state (position, velocity, spin, energy, etc) that would explain observed statistical variation and no known time-interval over which this would happen if there were, though people are working on it. This is referred to as the 'instantaneous collapse of the wavefunction'.

As for free will and consciousness, this probabilistic collapse is unlikely to have any implications since the units of the body and brain, from atoms to cells to organs, are so large compared to their wavelengths that one observes no probabilistic behaviour (though it has been measured for a fairly large molecule, it is a very small effect that is manifest in very specific circumstances measured by very sensitive apparatus).
 
Bones-O
 
Reply Wed 28 Jan, 2009 08:09 am
@Khethil,
Khethil wrote:

[INDENT]The possible choices I have on my menu before me still exist and I will still choose of my own accord, I'm simply not aware of the near-infinite variables involved that will lead me to that decision. This doesn't mean I don't have a choice; Even if I could know all the variables that'll ultimately lead me to that decision, I'd still have a choice. Despite how it 'feels', these aren't mutually-exclusive.
[/INDENT]Here's my qualifier and what I think is the Key: We can't know all the variables involved in events, actions or decisions; there's simply too much to take into account - too much to weigh. As you mentioned with your electron-predictability point, there are some aspects of physics we simply can't predict. This does not invalidate the cause-and-effect principle at all, it simply says we haven't the means by which to exploit it to its logical permutation.

Does that make sense?

Yes, and I agree with this sentiment personally. However, if one states the question in the context of quantum mechanics, as per the OP, one cannot really pick and choose its laws. Standard QM describes deterministically propagated but probabilistically measured systems. Thus it is not a question of ignorance:- one can posit, for instance, hidden variables as per Bohm's theory; however this is then a different description of the Universe, not a quantum mechanical one (even if it predicts the same outcomes).

On the other hand, I don't think QM is the last refuge of traditional ideas of free will either: because collapse of the wavefunction is spontaneous and probabilistic it removes intent. (On the other hand, it is consistent with the illusion of free will, since once collapse occurs, past causal chains are set in stone).

The physicist and author Roger Penrose wrote a book called 'Shadows of the Mind' that tried to link QM with mental processes and phenomena. I have no idea if it was along these lines (I haven't read it) but I don't think it was taken very seriously at all.
 
kennethamy
 
Reply Wed 28 Jan, 2009 08:09 am
@Bones-O,
Bones-O! wrote:
Uncertainty does not disrupt causality at all, you'll be pleased to know. The 'uncertainty' in a particle's position and velocity is a feature of a deterministically time-evolved wavefunction. I actually think its a bit of a misnomer, but it's a matter of interpretation. Some people interpret the wavefunction (or its square) at a given position as nothing more than the probability of finding the particle in each of those positions. I read it as any normal wave: that is, the particle is spread across all those positions at once. Thus the 'uncertainty' in position of a wave is nothing more than its spatial spread. Likewise the uncertainty in its momentum is nothing more than the dynamics of the wave components.

The breakdown in determinism lies in measurement, not propagation, since there is no known mechanism for time-evolving an 'uncertain' particle into a certain state (position, velocity, spin, energy, etc) that would explain observed statistical variation and no known time-interval over which this would happen if there were, though people are working on it. This is referred to as the 'instantaneous collapse of the wavefunction'.

As for free will and consciousness, this probabilistic collapse is unlikely to have any implications since the units of the body and brain, from atoms to cells to organs, are so large compared to their wavelengths that one observes no probabilistic behaviour (though it has been measured for a fairly large molecule, it is a very small effect that is manifest in very specific circumstances measured by very sensitive apparatus).


I agree with all you say. Most experts in QM are now willing to say that causality breaks down on the micro-level. The notion that the problem is merely epistemic is a view of more than a half-century ago.

I want only to add that the supposed problem about free will and micro-physics pertains only to what philosophers and theologians think of as the problem of free will having to do with determinism. But nothing whatsoever to do with what we ordinarily think of as free will which is just the absence of compulsion; not the absence of causation.
 
Bones-O
 
Reply Wed 28 Jan, 2009 09:02 am
@kennethamy,
kennethamy wrote:
I agree with all you say. Most experts in QM are now willing to say that causality breaks down on the micro-level. The notion that the problem is merely epistemic is a view of more than a half-century ago.

Cheers.

kennethamy wrote:

I want only to add that the supposed problem about free will and micro-physics pertains only to what philosophers and theologians think of as the problem of free will having to do with determinism. But nothing whatsoever to do with what we ordinarily think of as free will which is just the absence of compulsion; not the absence of causation.

True, true.

As an aside, many argue it lets God, gods, soul, etc. in through the back door. It would seem a valid theological conjecture, though it seems odd that with all His commandments, angels and son on Earth, or equivalent human-god interactions in other religions, there was no mention of wavefunction collapse. I mean... usually when someone introduces themselves to you, the first thing you ask is what they do for a living. And it must be His main job: there's a lot more collapsing wavefunctions than imminent floods or plagues of locusts.

I am your Lord God. I collapse all your wavefunctions. No-one else shall collapse your wavefunctions, just Me.
 
click here
 
Reply Fri 30 Jan, 2009 02:50 am
@Khethil,
Khethil wrote:
You're right - I don't think it would.

I'm no expert - to be sure - but what I'd say is this: I do have free will -and- you could predict my decisions beforehand were you to know all factors, all variables and their appropriate weights, my past, my feelings, emotions, physical state and more. To say that the cause and effect principle enables the possibility to predict future events doesn't preclude the concept of free will one iota.[INDENT]The possible choices I have on my menu before me still exist and I will still choose of my own accord, I'm simply not aware of the near-infinite variables involved that will lead me to that decision. This doesn't mean I don't have a choice; Even if I could know all the variables that'll ultimately lead me to that decision, I'd still have a choice. Despite how it 'feels', these aren't mutually-exclusive.
[/INDENT]Here's my qualifier and what I think is the Key: We can't know all the variables involved in events, actions or decisions; there's simply too much to take into account - too much to weigh. As you mentioned with your electron-predictability point, there are some aspects of physics we simply can't predict. This does not invalidate the cause-and-effect principle at all, it simply says we haven't the means by which to exploit it to its logical permutation.

Does that make sense?


Khethil wrote:

Even if I could know all the variables that'll ultimately lead me to that decision, I'd still have a choice.


Ummm no? If the variables lead you to the choice you have no choice or else the variables didn't lead you to that choice. If you still have a choice you can not have calculated the variables correctly.

Sure if you wanna say we don't have the ability to calculate your choice thats fine. But if we theoretically could then we would find out that you don't have a choice or we did the math wrong.
 
Lord Lucan
 
Reply Fri 30 Jan, 2009 05:23 am
@Bones-O,
Thanks for all these very helpful replies.

Bones-O! wrote:
The breakdown in determinism lies in measurement, not propagation, since there is no known mechanism for time-evolving an 'uncertain' particle into a certain state (position, velocity, spin, energy, etc) that would explain observed statistical variation and no known time-interval over which this would happen if there were, though people are working on it. This is referred to as the 'instantaneous collapse of the wavefunction'.


Could you possibly enlarge on this? I don't really understand how the breakdown of determinism is affected in this way. I probably lack the background so any pointers would be very gratefully accepted.

Quote:

Ummm no? If the variables lead you to the choice you have no choice or else the variables didn't lead you to that choice. If you still have a choice you can not have calculated the variables correctly.



Determinism implies that it is in principle possible to predict a given choice but surely that does not affect our free will. We do still have the choice. At worst, perhaps determinism means that we are not quite as capricious as we might otherwise be.
 
click here
 
Reply Fri 30 Jan, 2009 05:44 am
@Lord Lucan,
Lord Lucan wrote:

Determinism implies that it is in principle possible to predict a given choice but surely that does not affect our free will. We do still have the choice. At worst, perhaps determinism means that we are not quite as capricious as we might otherwise be.


Oh I would say that it must. How are you defining free will? Do you define it as something outside of possession of a soul? There is no such thing as free will if you do not believe in a soul. Free will is then a concept or an illusion.

It is not that you are predicting. You are calculating. If EVERYTHING is put into the massive equation there is only ONE final out come. You can't add 2+2 and get 7. You will always get 4. When you calculate everything you don't end up with 2 choices at the end of your calculation or you seriously messed up. Or are you saying that you calculate everything to one response and then choose something else? Well then again you messed up your calculations.

I would like to know how you calculate everything and then some how get this ability to still change the answer to the calculation. That would seem too be a soul.

Lord Lucan wrote:

We do still have the choice.

Would you please tell me what part of your brain allows to to make a choice outside of the calculations? You see if your 'making a choice' that is part of your calculations.

I don't see any room for questioning:

A human being without a soul has no free will. It has as much control of its self as a pinball in a pinball machine. (actually slightly less)
 
Oh phil
 
Reply Fri 30 Jan, 2009 07:53 am
@kennethamy,
kennethamy wrote:
I agree with all you say. Most experts in QM are now willing to say that causality breaks down on the micro-level. The notion that the problem is merely epistemic is a view of more than a half-century ago.

I want only to add that the supposed problem about free will and micro-physics pertains only to what philosophers and theologians think of as the problem of free will having to do with determinism. But nothing whatsoever to do with what we ordinarily think of as free will which is just the absence of compulsion; not the absence of causation.


Hi Kenneth,

This is something I have been thinking about recently, so it's great to find this discussion.

QM is just the latest theory, isn't it? Won't it be overturned completely in due course by a more powerful theory? And QM itself hasn't got to the bottom of things yet, has it? The original poster made reference to particles, Bones-O has shown us that there is still disagreement about wave/particle duality.

My own current thinking is that there are no particles, because the universe is fundamentally an indivisible whole. There are no individual objects, there are also no individual events, and therefore ultimately no causation, all these things are extremely useful fictions, partial descriptions of aspects of the universe as we experience them. If our experience was removed from the picture, the universe would be a very different thing, and we find it difficult to appreciate how different.

I'd be interested in your views, as well as those of Bones-O and the other contributors.
 
kennethamy
 
Reply Fri 30 Jan, 2009 09:08 am
@Oh phil,
Oh! wrote:
Hi Kenneth,

This is something I have been thinking about recently, so it's great to find this discussion.

QM is just the latest theory, isn't it? Won't it be overturned completely in due course by a more powerful theory? And QM itself hasn't got to the bottom of things yet, has it? The original poster made reference to particles, Bones-O has shown us that there is still disagreement about wave/particle duality.

My own current thinking is that there are no particles, because the universe is fundamentally an indivisible whole. There are no individual objects, there are also no individual events, and therefore ultimately no causation, all these things are extremely useful fictions, partial descriptions of aspects of the universe as we experience them. If our experience was removed from the picture, the universe would be a very different thing, and we find it difficult to appreciate how different.

I'd be interested in your views, as well as those of Bones-O and the other contributors.


Yes, it is the latest theory. But I don't know that it is only the latest theory. It may be the correct theory. That other theories have been "overturned" does not show that QM will be overturned. We may not be certain it is true, but that doesn't mean that it isn't true, does it? In any case, the point is that indeterminacy is not epistemic, as far as we know. It is metaphysical. But there is no good reason to think it affect human actions and choices.

Well your view that there are no particles is not one held by most scientists, so I will have to go along with them. It is hard to see why the postulation of particles would be so useful unless it was true.
 
Bones-O
 
Reply Fri 30 Jan, 2009 09:26 am
@Oh phil,
Oh! wrote:

QM is just the latest theory, isn't it? Won't it be overturned completely in due course by a more powerful theory? And QM itself hasn't got to the bottom of things yet, has it? The original poster made reference to particles, Bones-O has shown us that there is still disagreement about wave/particle duality.

Overturned completely? How? Will we suddenly discover that microchips and CD players don't actually work? There is still a lot to figure out about QM, and there may be a better theory turning up that explains QM from a more fundamental level.

Oh! wrote:

My own current thinking is that there are no particles, because the universe is fundamentally an indivisible whole. There are no individual objects, there are also no individual events, and therefore ultimately no causation, all these things are extremely useful fictions, partial descriptions of aspects of the universe as we experience them. If our experience was removed from the picture, the universe would be a very different thing, and we find it difficult to appreciate how different.

I think anyone who accepts QM has already accepted that the universe is very different to our experience of it. I'm not sure, though, how those experiences would arise out of your indivisible universe theory. My experiences are not your experiences, and yet both exist in the universe. If the universe is indivisible, why do our experiences seem utterly separated?
 
Oh phil
 
Reply Fri 30 Jan, 2009 02:42 pm
@kennethamy,
kennethamy wrote:
Yes, it is the latest theory. But I don't know that it is only the latest theory. It may be the correct theory. That other theories have been "overturned" does not show that QM will be overturned. We may not be certain it is true, but that doesn't mean that it isn't true, does it? In any case, the point is that indeterminacy is not epistemic, as far as we know. It is metaphysical. But there is no good reason to think it affects human actions and choices.

Well your view that there are no particles is not one held by most scientists, so I will have to go along with them. It is hard to see why the postulation of particles would be so useful unless it was true.


By chance I read something today about Thomas Kuhn, who I didn't previously know much about, except that he was responsible for the popularisation of the phrase "paradigm shift":

Quote:

The history of science, Kuhn argued, is punctuated by violent intellectual revolutions that overturn long periods of conservative puzzle-solving. Periods of so-called "normal" science are characterised less by independent and objective research than by adherence to agreed assumptions and expected outcomes. During periods of normal science, anomalous or unexpected findings get brushed aside as either irrelevant or problems to be solved another time. Original research that questions the current assumptions of accepted theory are often debunked as wild and useless speculation. This gives rise to Kuhn's notion of a paradigm. The current paradigm is a a web of interwoven assumptions and beliefs shared by a particular community which underlies and sets the agenda for current research.

According to Kuhn, only results which tend to strengthen the normal paradigm get accepted during periods of normal science. The paradigm itself is never questioned or criticised. However, from time to time paradigms are overthrown by intellectual revolutions, when the paradigm fails to provide adequate models for observed phenomena, or a new, more powerful model has a greater explanatory force but requires a "paradigm-shift", a revolution takes place.


My suggestion is that particles are a very useful fiction. The universe is actually indivisible, but we have the ability to make countless artificial distinctions. I think this is initially a passive ability, I think it comes to us as a consequence of consciousness, but higher animals can also actively exploit it. All objects are fictional in this sense, but the fictions are so useful and so densely woven into our existence that we are inclined to assume they are "true".

I find it interesting that scientists and others want to defend the existence of particles, even when the entities in their theories don't seem to have the qualities of particles as we think of them in everyday life. They are wave/particles. Bones-O, if I remember rightly, said he thinks they are more like waves than some other physicists think (I speak very loosely). Some theories talk about wavicles, or strings, and then there are ideas about distant events being linked at this particulate level. Do scientists really believe in particles at all?

Some of the leading scientists in the current paradigm are searching for the fundamental particle: what if there isn't one? If the universe is fundamentally just one thing, we could find smaller and smaller effects at a smaller and smaller scale, ad infinitum.

Is that impossible? Does it not fit in rather well with what we observe?
 
Oh phil
 
Reply Fri 30 Jan, 2009 03:36 pm
@Bones-O,
Bones-O! wrote:
Overturned completely? How? Will we suddenly discover that microchips and CD players don't actually work? There is still a lot to figure out about QM, and there may be a better theory turning up that explains QM from a more fundamental level.


Clocks didn't stop working when Einstein overturned Newton.

One of the implications of my theory that the universe is all one thing is that there is no fundamental level. Levels, like everything else we talk about, are just partial and therefore faulty descriptions, and the only true description is the whole universe itself.

Quote:

I think anyone who accepts QM has already accepted that the universe is very different to our experience of it.


Goodness me. Are you sure that is right? I am not a physicist, I have to take a lot of what physicists say on trust. I kind of expect them to have experienced things in some way before they accept them. Like with the CD player, I take it on trust that there is some kind of phenomenon there that is dependent on QM, a phenomenon I could experience, however remotely.

But you seem to be saying that QM says something about parts of the universe that you haven't experienced, maybe that you couldn't experience, however remotely. So that would mean, nothing we know about would stop working if that theory was overturned.

That isn't really what I am paying my scientists to do.

Quote:

I'm not sure, though, how those experiences would arise out of your indivisible universe theory. My experiences are not your experiences, and yet both exist in the universe. If the universe is indivisible, why do our experiences seem utterly separated?


So, the next step in my theory is, experience creates differentiation. The arrival of experience in the universe instantaneously gives rise to entirely new kinds of phenomena and interrelationships. For example, most of what we mean by "time" is related to our experience, or our projection of our experience into distant eras.

One of our experiences is the experience of being an individual persisting over time. You're right, our experiences are separated, nobody else could have my childhood memories, even if I lost them.

Let me know if that hasn't entirely transformed your understanding of the universe.
 
Bones-O
 
Reply Sat 31 Jan, 2009 06:10 am
@Oh phil,
Oh! wrote:
Clocks didn't stop working when Einstein overturned Newton.

Of course they didn't - Newton was right (in the classical, non-relativistic limit).

Oh! wrote:

But you seem to be saying that QM says something about parts of the universe that you haven't experienced, maybe that you couldn't experience, however remotely. So that would mean, nothing we know about would stop working if that theory was overturned.

I'm just saying that a lot of aspects of QM are counter-intuitive, that things we take for granted are ofteh thwarted in QM.

Oh! wrote:

That isn't really what I am paying my scientists to do.

Maybe you should pay them more. Smile

Oh! wrote:

Let me know if that hasn't entirely transformed your understanding of the universe.

No comment. :whistling:
 
Oh phil
 
Reply Sat 31 Jan, 2009 06:16 am
@Bones-O,
Bones-O! wrote:
I think anyone who accepts QM has already accepted that the universe is very different to our experience of it.


Bones-O! wrote:

I'm just saying that a lot of aspects of QM are counter-intuitive, that things we take for granted are often thwarted in QM.


Those seem to be two very different statements. How are things you take for granted thwarted by QM? How does the thwarting take place?
 
Bones-O
 
Reply Sat 31 Jan, 2009 07:30 am
@Oh phil,
Oh! wrote:
Those seem to be two very different statements. How are things you take for granted thwarted by QM? How does the thwarting take place?

For examples: that empty space isn't empty (Casmir effect); or that two simultaneous events separated by great distances can be causally linked (EPR paradox); or that objects can be two things at once (Schrodinger's cat); or that processes can occur backwards in time (CPT symmetry); or that an object can get somewhere by more than one path (double slit experiment, path integral formalism); or that something can have spin but not be spinning; or that matter can be created or destroyed (pair creation/annihilation); or that energy isn't quite conserved (uncertainty principle).
 
Oh phil
 
Reply Sun 1 Feb, 2009 06:14 am
@Bones-O,
Bones-O! wrote:
For examples: that empty space isn't empty (Casmir effect); or that two simultaneous events separated by great distances can be causally linked (EPR paradox); or that objects can be two things at once (Schrodinger's cat); or that processes can occur backwards in time (CPT symmetry); or that an object can get somewhere by more than one path (double slit experiment, path integral formalism); or that something can have spin but not be spinning; or that matter can be created or destroyed (pair creation/annihilation); or that energy isn't quite conserved (uncertainty principle).


Ok, these are supposed to show that the universe is different to our experience of it, that things we take for granted are thwarted by QM.

So, what experience have we had of empty space, what do we take for granted about empty space?

What experience have we had of causal connection? What do we take for granted about it?

What experience have we had of objects being two two things at once?

And so on.

Maybe we could just take one of your examples: would it be ok to take the first, and maybe you could tell me what you think is our experience of empty space, and how the Casimir effect shows that the universe is different to that experience?
 
Bracewell
 
Reply Thu 5 Feb, 2009 07:35 pm
@Oh phil,
Why does the behaviour of particles generate so many -----isms? These things are so tiny, many times smaller than an atom, with no dimensions and behaviours that do not owe allegiance to time and yet, somehow, science has managed to fire the imagination that humans can relate to them. True, particles move thru the brain but feelings and actions are decided by atoms, which are hugely more weighty, complex and interactive.
Was it not Feynman who said that anyone who claimed to understand QM almost certainly did not, which sounds like good advice.
However, if you can accept there must be a point where the electro/magnetic starts a march to solidness then the particle (which is not a particle of anything) is a good starting point but there is definitely a very, very long way to go.
The discussions are good even if I am scrambling for a dictionary too often but good lord yes, our actions and feelings are too easily influenced by unwanted interference - read any star sign.
 
Bones-O
 
Reply Fri 6 Feb, 2009 11:11 am
@Oh phil,
Oh! wrote:

Maybe we could just take one of your examples: would it be ok to take the first, and maybe you could tell me what you think is our experience of empty space, and how the Casimir effect shows that the universe is different to that experience?

I think you're misunderstanding my post. I gather you're interpreting me as saying that what we experience is thwarted by QM. This is not the case. What I'm saying is that the universe contains phenomena where ordinary experience doesn't apply. Things that you take for granted, like something having an exact position or speed, or causal links requiring time to evolve, no longer hold when studying systems on the quantum scale. Now, the universe isn't just planet earth and your understanding of it, it contains all of the phenomena modelled by QM also, hence: the universe is different to our experience. It naturally includes phenomena that we experience because we are in it, but it also includes phenomena that are contrary to our everyday understanding.

By the way, it was a just flippant comment that you're taking waaaaay too seriously. But I can defend it if you want me to, if it's really, really important to you personally. :a-ok:
 
 

 
  1. Philosophy Forum
  2. » Metaphysics
  3. » Uncertainty about uncertainty
Copyright © 2024 MadLab, LLC :: Terms of Service :: Privacy Policy :: Page generated in 0.03 seconds on 04/23/2024 at 04:20:17