# Does everything have a infinitesimal amount of uncertainty?

Suppose a soldier has to fight unarmed against 200 opponents. Most people would say that the soldier is certainly dead. However, quantum mechanics dictates that there is always some degree of uncertainty. This means that there is always some factor that the statistician has not accounted for. Maybe one enemy breaks his arm. Maybe another one trips and falls. On and on until all enemies are gone. What I am trying to ask is if there is always a 0.00000000000000000000000000000000000000...1% error on any probability calculation. A historical example and not a hypothetical one would be the existence of the universe. The universe might have come from the big bang. But what caused the big bang if there was nothing? If there is nothing, then by human calculations the probability of something just forming is 0%. But if there is always an infinitesimal degree of freedom in all of the possible scenarios, then that would explain how a spark just appeared and since the universe exists, the universe might prove that there is always a
0.00000000000000000000000000000000000000000000000...1% chance of an "impossible" event happening. Is my logic correct?

• Maybe... But you can easily translate the probability above (that the single soldier will survive) into a frequency: how many fights must occur to test it? Commented Sep 2, 2021 at 14:39
• @Mauro ALLEGRANZA I am assuming that time is infinite. Commented Sep 2, 2021 at 14:46
• I don't know if your specific logic is right in terms of quantum mechanics as such, but I can really recommend 3Blue1Brown's youtube series on probability distributions when it comes to talk about "infinitesimal probabilities". There is a sense in which this a 0% probability is still a possible outcome! youtube.com/watch?v=8idr1WZ1A7Q Commented Sep 2, 2021 at 18:26
• If you wish to consider probability as a degree of credence, then a probability of zero indicates not an impossible event but something of which we are absolutely certain that it is false. It is commonplace in Bayesian reasoning to avoid assigning zero (or one) to any proposition, because it would imply that no amount of evidence could move you to consider its probability to be anything but zero. If your prior is zero, your posterior is zero under any evidence. This is called Cromwell's rule. Commented Sep 2, 2021 at 20:34
• While you mention quantum mechanics here, it really seems like more of a red herring here. What probability interpretation are you using here? Commented Sep 2, 2021 at 22:31

Yes, everything does have an infinitesimal amount of uncertainty. See this post for some discussion on it. As you say, there is always the possibility for events to randomly play out in an unexpected way.

Even if, for a particular event, the laws of physics demand the event must play out in a certain way - we can never be certain of the laws of physics, nor can we ever be fully certain that our calculations using those laws were correct.

Because our brains are flawed and stochastic machines, to be rational we must be slightly uncertain of everything - including even the degree of uncertainty.

• Right... even our uncertainty is uncertain, so there is always the possibility that there is some truth statement we might be able to be 100% certain about. Commented Sep 2, 2021 at 23:11

There are still some QM "ideas" and interpretations that do not necessarily guarantee a statistician wouldn't be able to provide an exact prediction in principle as far as the quantum mechanical portion of an experiment is concerned. I have in mind superdeterminism and interpretations that treat the wavefunction as epistemic with an underlying physical causality (a la Rob Spekkens).

There are other areas of physics like general relativity that do prevent arbitrarily precise experiments from being done, so even in superdeterminism and psi-epistemic interpretations, one can't measure certain physical states perfectly because these aspects need to be factored in too. Without a perfect initial measurement, chaos will lead to uncertainty in predictions as the system evolves. Not the same kind uncertainty as in textbook QM, but still an insurmountable obstacle for the experimenter. One example is general relativity preventing volumes beneath the Planck scale from being probed as the energy requirements increase and you just get bigger and bigger black holes. Another physical limitation outside of textbook quantum randomness is to get the "true" likelihood of any physical outcome you'd need to sample it infinite times (frequentism), meaning you'd never get to run the actual experiment you want because you'd need an infinite amount of time before you run it.

So under GR and QM, certainly a human statistician can never achieve perfect certainty. And under many interpretations, not even a god-like being could achieve perfect predictions. As for the universe coming from "nothing", there still has to be a physical process that would allow that. Uncertainty doesn't guarantee any possibility can occur, only those physically possible. We don't know if such a thing is physically possible, and in principle we might rule out nothing->something as our genesis. Just like I can rule out an infinite amount of energy appearing before me in the next second. Within GR that is not possible. We still can rule out some things even with the above uncertainties. Similarly, we might one day rule out nothing->something.

No, the number of grey cups that I perceive on my desk at this time is exactly 1.

• The question seems to be about possible futures, not the present state of the world. In a quantum world, I'm not convinced that even assertions about the present state are ever unambiguously true. Commented Sep 2, 2021 at 22:21
• Ah, so you say that when you assess what you perceive, you cannot be wrong? Think of your mind as a computer, and you will see that there are always uncertainties. If a computer reports it has a file in a particular place - is that necessarily so? No! The kernel could be corrupted, so it reports a file where nothing is there on the hard drive at all. Reporting that you perceive a grey cup is akin to reporting on the contents of a certain "perception" file. Maybe what you report is there, in the perception file - maybe it is not. Commented Sep 2, 2021 at 22:53
• It could be an optical or other illusion, this tome even though not other times. You can talk about your history of interactions, but you can't be ontologically certain. You could be in a simulation made by your descendents. Commented Sep 2, 2021 at 23:05