Two sides of the same coin

This reminds me of a bit from Stranger in a Strange Land. One of the characters, Anne, is a Fair Witness, someone trained to witness and reliably recount events, sort of a notary on steroids.

When explaining this, another character asks Anne, "What color is the house." And she says, "This side of the house is white." He goes on, "And that's all you'll get her to say, unless she walks up there to see for herself, and even then, she'll only testify to that it was white while she was looking at it."

need some important assumptions here. Assuming the world in which the coin was flipped follows the laws of physics, such that the coin won't be magically replaced with a 2-headed coin once it's flipped; and assuming I've already looked at the can before flipping it and verified that it, indeed, has a tails side and a heads side, I can be 100% certain the other side is tails.

In fact, this is the notion of quantum entanglement. It can be said that the 2 faces of a coin are entangled such that when you observe one of the faces, you can be certain about the state of the other face. This is exactly how when we observe an election with SPIN UP, we can be certain the other electron in the same orbital has SPIN DOWN.

I think not knowing the state of a coin flip is a little different from Schrödinger's paradox. I'll try to explain it.

The concept of a coin flip serves as an excellent analogy to illustrate the differences between classical and quantum mechanics. In the macroscopic world, a coin's state is determined upon flipping, regardless of observation. However, in the quantum realm, observation itself plays a crucial role in determining the state of a particle.

In classical physics, exemplified by a traditional coin toss, the outcome is determined as soon as the coin lands, even if no one observes it. The coin will always have a heads side and a tails side, and its state is fixed upon landing, irrespective of whether someone looks at it or not. Our lack of knowledge about the outcome does not affect the coin's actual state.

Quantum mechanics, however, introduces a fundamentally different perspective. In the quantum world, particles can exist in a superposition of states until they are observed or measured. A hypothetical "quantum coin" would not have a definite state until interaction occurs. This is not merely a lack of information but a fundamental property of quantum systems.

The famous Schrödinger's cat thought experiment illustrates this concept on a macroscopic scale. In this scenario, a cat is placed in a sealed box with a radioactive source and a poison that will be released when the source decays. According to quantum mechanics, until the box is opened, the cat is theoretically in a superposition of states - both alive and dead simultaneously.

However, this apparent paradox can be resolved through the principles of quantum decoherence and the Copenhagen interpretation. Decoherence explains how quantum superpositions break down when a quantum system interacts with its environment. In the case of Schrödinger's cat, the cat, being a macroscopic object, constantly interacts with its environment inside the box. These interactions cause rapid decoherence, collapsing the quantum superposition into a definite state (either alive or dead) long before the box is opened.

This resolution bridges the gap between quantum mechanics and our classical understanding of reality. It demonstrates that while quantum effects are prevalent at the microscopic level, they typically don't persist in large-scale systems due to constant environmental interactions. The act of observation doesn't determine the cat's fate; rather, it reveals the already-determined state resulting from decoherence.

Without picking the coin up to confirm the side that is down is tails. Could you ever know that it is tails ?

No, and in fact by Bayes Theorem you should be slightly more suspicious that the coin is a double-headed coin. On the other hand, you should now be confident that the coin is not a double-tailed coin, and if your initial suspicion was symmetric that's where the increased skepticism would come from.

Suppose you start by thinking that there's a 1-in-a-million chance that a randomly-tossed coin is double-headed, the same chance that it's double-tailed, and the residual (999,998-in-a-million) that it's a fair coin.

After the toss, you observe that one side of the coin is heads. If the coin was double-headed, the probability of it being double-headed was 100%, if it was fair then 50%, and if it was double-tailed then 0%. To write this in conditional probability notation, where P(A|B) means "probability of A if B is true/observed:"

• P(heads | 2 head coin) = 100%
• P(heads | fair coin) = 50%
• P(heads | 2 tail coin) = 0%

We also know that absent any information about the coin, we'd expect P(heads) = 50%.

Bayes' theorem states P(A|B) = P(B|A)*P(A)/P(B), essentially allowing us to reverse the way conditioning works. Rather than "probability of observation given an assumption about the underlying truth", we end up with "probability of the underlying truth given an assumption" (and our previously-existing belief!).

Running this through with our numbers gives:

• P(2 head coin | heads) = P(heads | 2 head coin) * P(2 head coin) / P(heads) = 100% * 1e-6 / 50% = 2e-6 (2 in a million)
• P(fair coin | heads) = P(heads | fair coin) * P(fair coin) / P(heads) = 50% * (1-2e-6) / 50% = (1-2e-6) (999,998 in a million, no change)
• P(2 tail coin | heads) = P(heads | 2 tail coin) * P(2 tail coin) / P(heads) = 0 * 1e-6 / 50% = 0 (no chance)

Now, if the same coin is flipped (without you seeing both sides) and lands on heads again, we start to become slightly more skeptical of the 'fair coin' hypothesis. P(2 heads | fair coin) is 25%. Interestingly, P(2 heads) is not 25% thanks to the very small influence of the unfair coin: it's 25% * P(fair coin) + 100% * P(2 heads) = (25%*(1-1e-6) + 1e-6) = (25% + 0.75e-6)[†].

• P(2 head coin | 2 heads) = P(2 heads | 2 head coin) * P(2 head coin) / P(2 heads) = 100% * (1e-6) / (25% + 0.75e-6) ≈ 4e-6 (4 in a million)
• P(fair coin | 2 heads) = P(2 heads | fair coin) * P(fair coin) / P(2 heads) = 25% * (1-1e-6) / (25% + 0.75e-6) ≈ (1 - 4e-6) (999,996 in a million)

[†] — This seems weird, but the unfair coin contributes disproportionately to this result. Consider P(1 billion heads): this will essentially only happen if the coin is a double-headed coin, so if we know nothing about the coin ahead of time then we must assume that the chance of 1 billion heads is equal to the chance that the coin is a double-headed coin.

It seems related to the “problem of induction”. Paraphrasing, we expect past events to be predictive of future events because, in the past, they have been, but that is only satisfactory as evidence if you’ve already assumed the hypothesis. In this case, we expect the flip side to be tails because every time in the past we’ve picked up the coin it’s been true.

From the perspective of a zero-knowledge proof, you could flip 50 coins, check the other side of 49 of them, and, assuming that your choice of 49 was made freely, that would give you progressively increased confidence that the last one is tails. Approaching, but never equal to 100% probability.

In this case, you are saying that you have a normal coin, which you observe to have a head side and a tail side, and you toss it and it shows heads; can you be sure that the other side is tails?

I think this is an Epistemological question. This is really asking "what can we know about the universe?" There are a few schools of though on the matter. A Skeptic would say that we can't know. A Fallibilist would say that we don't know for certain but we have a pretty good guess. An Empiricist or Rationalist would say that we know what is on the other side of the coin. Or one might subscribe to a Reliabilist point of view, and say that given a proposition - "the other side of the coin is tails" - we would believe the other side to be tails if we have observed it to be tails and we know that coins do not typically switch faces.

This is a reductive answer, but basically, the answer to the question depends on what school of thought you subscribe to. I don't know what the answer is, but I think that Rationalism has a lot going for it; we can know that the coin isn't going to change it's face, because coins don't do that. But by the very nature of the question you have posited, it's unsolvable; one could say that the coin has a different face only so long as the other face is not observed, and that is not something that can be tested for, because all tests require observation.

This is distinct from Schrödinger’s Cat, which is a thought experiment to explain quantum superposition, notably put forth by Schrödinger to explain how ridiculous quantum superposition is; something cannot be both alive and dead. I guess in a way, this is related to the coin question - coins cannot change their faces, just like cats cannot be in both states.

(IANAP: I am not a physicist)

I’d argue that Schrödinger’s cat is a framing of quantum superposition in a macroscopic context in order to underline its unintuitiveness as a principle, whereas the coin flip argument is that you can’t see something which is hidden.

Notably, the intended takeaway of Schrödinger’s cat as a thought experiment is that there is no hidden state: a superposition of states is 100% probabilistic until the state “collapses” by being “observed” (quotes around words with differing meanings from their common connotations). It’s also a thought experiment in that it’s impossible to perform: the box would need to have literally no connection to the outside world. You can’t be able to run it through an x-ray machine, shake a box of treats in front of it, or bark repeatedly to provoke a reaction.

All of this said: glass table and a mirror?

Sorry for suggesting Reddit but that's a good question for either /r/askphilosophy or /r/logic.