Just take as accepted, anything not counted as physical is not counted as empirical, and anything not counted as empirical in some way is counted as a priori, and anything not counted as empirical in any way whatsoever is counted as pure a priori. It follows that whatever is there that makes changes in one’s subjective condition merely possible, is pure a priori. But it must be something, and thus is established and justified, a precursory condition. — Mww
The sound a lead ball makes is different than the sound a rubber ball makes, and the sound a ball makes is different than the sound a trash compactor makes. That all these make a sound is determined by the the matter of each, but the matter of these, while affecting the senses with sound, do not carry the information of what form the matter has. It is impossible for us to get “ball” out of the sound an object makes when it hits something solid. Without antecedent experience, you cannot get “telephone” out of some arbitrary ringing/clanking/buzzing sound. — Mww
You just observe evidence with no inference? — Metaphysician Undercover
Deductive inferences if valid are certain, so they do constitute proof. — Janus
You meant “don‘t constitute evidence” right? — Srap Tasmaner
What I have been claiming about the number of coins in a jar is simply that we can know a priori that if they can be counted then there is already a specific number of coins in the jar; we can only know a posteriori what that number is.
I do not think I have ever had occasion to make a claim to knowledge that so clearly fits the definition of a priori. Whaddya know. — Srap Tasmaner
If there is a "platonic world" M of mathematical facts, what does M contain precisely? I observe that if M is too large, it is uninteresting, because the value is in the selection, not in the totality; if it is smaller and interesting, it is not independent from us. Both alternatives challenge mathematical platonism. I suggest that the universality of our mathematics may be a prejudice hiding its contingency, and illustrate contingent aspects of classical geometry, arithmetic and linear algebra. — Michelangelo's Stone: an Argument against Platonism in Mathematics - Carlo Rovelli
What type of knowledge do you assume that a "hypothetical" gives someone? — Metaphysician Undercover
The ancient Greek deiknymi (δείκνυμι), or thought experiment, "was the most ancient pattern of mathematical proof", and existed before Euclidean mathematics,[6] where the emphasis was on the conceptual, rather than on the experimental part of a thought-experiment. — Thought experiment - Wikipedia
When you assume hypothetically that it is raining, this does not mean that you have knowledge that it is raining. — Metaphysician Undercover
No. But the hypothetical shows the consequences that follow when it is raining. Namely that Alice knows that it is raining when, in addition to it raining, she has a justified belief that it is raining. — Andrew M
The issue was, if it must be raining in order for Alice to know that it is raining (i.e. true in your sense), then knowledge is infallible. How does this example show that knowledge is fallible? — Metaphysician Undercover
Perhaps this also says something about how the word "count" is used. — Andrew M
all we need is this:
(Card) If and only if there is a one-to-one correspondence between the coins in a jar and the set of natural numbers less than or equal to k, for some natural number k, then the number of coins in the jar is k and there is a definite number of coins in the jar.
That’s just the definition of cardinality for finite sets plus existential generalization. We don’t need counterfactuals for that, and we don’t need them for this:
(Count) If and only if a jar contains k coins, then counting the coins in the jar yields the value k.
This definition of cardinality for finite sets might as well be a description of counting; there’s almost nothing else to say. — Srap Tasmaner
I see how your coins and boxes are analogous to photons and interferometers, but I’m still not getting the point here. — Srap Tasmaner
But! I think I have thought of the perfect example, because it also involves making calculations based on values that you should not be using: the two envelopes problem.
Refresher: The only right way to do this is to treat the envelopes as X and 2X; you don’t know which one you got, so you stand to gain X or to lose X by switching, and the expected value of switching is 0. But if instead, you call whatever you got Y, and then reason that if it’s the bigger the other is Y/2, and if it’s the smaller then the other is 2Y, then the expected value of switching is Y/4.
It could be that exactly what’s wrong with this analysis is that it relies on counterfactual definiteness. (Oddly, like the black boxes and the interferometers, there are points in the defense of this analysis that rely on the principle of indifference giving equal chances to events, and then relying on those chances as if they were real values. Among many many other issues.) — Srap Tasmaner
Talk of switching in either the X or the Y analysis is counterfactual. Why does one of them work and the other not? — Srap Tasmaner
The example shows that human fallibility doesn't preclude Alice from knowing that it is raining. — Andrew M
You and I know up front because I created the hypotheticals that way. The question is not about what you and I know, which is a given, but about what Alice knows. — Andrew M
Regarding the rejection of the idea of intellectual intuition, would you say that is on account of the impossibility of inter-subjective and cross-sensory corroboration? — Janus
Empirical evidence in itself does not justify a belief, what is required is empirical evidence plus logic. — Metaphysician Undercover
Assuming the coin always has a definite heads or tails state, even when not measured, what definite state could it have had when it was between the two black boxes? It seems that the coin couldn't have had a definite state, contrary to assumption. — Andrew M
Easy part first....cross-sensory collaboration is a physiological impossibility, and inter-subjective collaboration is impossible within the reference frame of its occurrence. We do inter-subjectively collaborate, which is at that point merely a euphemism for post hoc relative agreement. — Mww
Something like that? — Janus
I was talking about inter-subjective and cross sensory corroboration, not collaboration. — Janus
The example shows that human fallibility doesn't preclude Alice from knowing that it is raining.
— Andrew M
The example cannot serve this purpose, because it premises that we can know up front, infallibly whether or not it is raining. — Metaphysician Undercover
Before continuing with this, I want to point out that truth is very much the issue at stake in all of these apparent detours.
... — Srap Tasmaner
Is this the claim? If each coin left box 1 with a definite state, then it would enter box 2 with a definite state, and if all of the coins entered box 2 with a definite state, then we should see some coins not in their initial orientation? — Srap Tasmaner
Since we don't, it must not be true that coins leave box 1 and enter box 2 with a definite state. — Srap Tasmaner
What I don't get is that the behavior of the boxes is defined only for coins entering with a definite state, and as emitting coins only in a definite state. — Srap Tasmaner
What are the boxes doing if not that? Isn't this a way of saying that the behavior of the boxes is not entirely definite? — Srap Tasmaner
No, not infallibly. One can possibly be mistaken about what the premises of the hypotheticals are. But since they are clearly stated, there's no good reason why anyone should be mistaken. — Andrew M
As a result of looking out the window, Alice justifiably believes that it is raining outside. For Alice to know that it is raining outside, her justifiable belief also has to be true. Those are the conditions for knowledge. Let's look at two different scenarios:
(1) If it is raining outside, then Alice knows that it is raining outside. She knows that even though she didn't exclude the possibility that it was not raining and that Bob was hosing the window. She knows it is raining because her belief is both justifiable and true. Alice has satisfied the conditions for knowledge.
(2) If it is not raining outside (say, Bob was hosing the window which Alice mistakenly thought was rain), then Alice's belief is false. Thus she doesn't know that it is raining, she only thinks that it is. Alice has not satisfied the conditions for knowledge. — Andrew M
The boxes may operate in a well-defined (definite) way, but are instead able to input and output coins in an indefinite state. But that can't be directly confirmed since a coin is always measured to be in a definite state. — Andrew M
A hypothetical doesn't provide us with the required knowledge. — Metaphysician Undercover
The point was that we cannot say whether or not "Alice has knowledge" under your description of "knowledge", unless we infallibly know whether or not it is raining. — Metaphysician Undercover
So the reliance on counterfactual definiteness is here? That perhaps a coin was emitted in an indefinite state but we can’t observe indefinite states, only definite ones. This is like your grid-world example with the direction of the unobserved arrow. — Srap Tasmaner
So the issue is that in some cases there might be no fact of the matter, no definite state, but if we take a measurement, we’ll always find that there is. — Srap Tasmaner
And then counterfactual definiteness is specifically the claim that since our measurements always show definite states, then what we measure — or, more specifically, what we intend to measure or consider or imagine measuring, must always be in a definite state because indeed that’s what measuring it would show. — Srap Tasmaner
In this review we shall adhere to the view that [state] p is only a mathematical expression which encodes information about the potential results of our experimental interventions. The latter are commonly called "measurements" - an unfortunate terminology, which gives the impression that there exists in the real world some unknown property that we are measuring. Even the very existence of particles depends on the context of our experiments.
...
The essential difference between the classical and quantum functions which change instantaneously as the result of measurements is that the classical Liouville function is attached to objective properties that are only imperfectly known. On the other hand, in the quantum case, the probabilities are attached to potential outcomes of mutually incompatible experiments, and these outcomes do not exist “out there” without the actual interventions. Unperformed experiments have no results. — Quantum Information and Relativity Theory - Peres, Terno
When I point out that a premise of the hypothetical is that it is raining, I'm not claiming that it's actually raining outside, here in the real world. — Andrew M
Then your hypothetical does squat, as Srap says, toward justifying your claim. We still cannot ever correctly judge that what Alice has is "knowledge", in the real world, because any such judgements could always turn out to be incorrect. Your example only applies to a hypothetical world, in which it actually is raining. What good is it, if it doesn't apply to the real world? — Metaphysician Undercover
Sometimes, in the real world, it is actually raining, and sometimes, in the real world, it actually isn't raining, irrespective of our certainty and judgements and justifications. — Michael
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