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There's no proof in science at all, either. You can never say "prove" in science and be correct because of the problem of induction. You can use inference to support a point, but not to prove it.
However, there is proof/knowledge available through direct perception, and so those who say they had a "spiritual experience" of god or something similar, assuming they are not lying, actually had as much proof as you could spend all of eternity looking to find.
Can you clarify what you mean? How is science without proof? How is math and geometry without proof? How is physics without proof? Also from my limited understanding the problem of induction refers to predictions of things doesn't it? And predictions are just that predictions which one seeks to prove or disprove. If I'm doing geometry I can prove it through repeatable calculations that something is true or false. How can one repeat a "spiritual experience"? If one saw God in a trance can he/she go back and see him again under the same conditions? Are you saying that I can't "prove" that the sun is a fiery ball of mostly hydrogen and I might as well believe it to be the sungod RA and if I get high enough to see his face after looking directly into the sun and that proves it through direct observation of a spiritual experience? Granted there's plenty of theoretical stuff out on the leading edge of science but it's just that "theoretical".
Science is like a courtroom - proof in science is equatable to "proof beyond a reasonable doubt," and so that's why you read in published studies things like "The findings show that a statistically significant correlation exists between x and y, p < .05" or something similar. The problem of induction arises whenever you try to infer predictions from a set of observations. If you see me eat breakfast at McDonald's for 30 straight mornings, this is not proof that I will eat breakfast at McDonald's on the 31st morning, nor is it even an indicator of likeliness that I will visit McDonald's on the 31st day. With respect to things that we assume are laws such as the laws of gravity or entropy, the same thing applies. And while some might say, "Well it's just simply obvious that gravity is a law, try jumping off a building if you aren't so sure of it." Well, that's fine and dandy and I might not jump off a bridge, but to try to apply validity to inductive reasoning using your experiences as a means for that application only means that you're using inductive reasoning to give inductive reasoning validity!
Math and geometry are abstract, self-contained languages. Math proofs and geometry proofs are called proofs because of this self-containment - in and of themselves, they are proof of the solutions they express. However, when you take a math or geometry proof and try to pin it to something tangible in reality, then the problem of induction creeps up again. You might consider some mathematical expression to constitute proof of some real-life occurrence (E.g. d = m/v), and this might seem true because your observations have aligned with that particular expression, but it's still inductive reasoning.
Can you "prove" the sun is a fiery ball? Well, I have a close example that exemplifies what I'm saying...
Suppose you go outside and feel the warmth of the sun on your face. You know it's warm, you can feel it. You can touch your fingers to your face, feel the heat on it, and you say, "Wow, that's warm!" And I say, "Warm, you say? Prove it to me."So, you get out your spot thermometer and take a temperature reading of your face and you say, "Look, 102 degrees. It's warm." What you don't know is that I just got back from a stream room at the gym where the temperature was 140 degrees. I say, "I don't know, seems pretty cool to me." So, which is it? Is it warm or hot? We have some arbitrary piece of evidence (i.e. 102 degrees) and yet there is no possible way for you to prove to me that 102 is warm unless I had the same direct perception of that warmth as you did while feeling the warmth on your face.
Now, you might say, "But that's a relative example! Warm and cool are relationally opposed - I proved to you that it was 102 degrees! Well, I say that 102 degrees is relational! 102 degrees is relational to something that we, for example, defined to be 0 degrees, and if we had defined that other thing to be something other than 0 degrees, then the object you just measured at 102 degrees would be measured at some other figure. Geometry is relational to math, and math and physics are relational to philosophy.
Inductive reasoning has been called "the glory of science and the downfall of philosophy." Because philosophy is a self-contained system that is inherently entwined with logical expression, it is possible to form logical tautologies that are totally unbreakable. But when philosophy branches into mathematics (abstract) and physics (physical) and you attempt to link the two, you automatically begin formulating assumptions that aren't present with a strict philosophical model, namely that mathematical models can always be applied to their physical analogues.
Edit: Don't forget, ratio is the root word of rationale.