For any statement, you can always ask "how do you know?". Even if you give an answer, how do you know that answer is true? Let me give an example. Take the statement 'I think, therefore I am'. How do you know you think? Because you're thinking right now? But how do you know you're thinking right now? etc. etc. Basically, for any statement, it produces an endless cycle of "how do you know?" and you can never truly know anything... even as simple as "I am conscious right now". How do philosophers get around this? It seems like they must have a way, but I can't seem to find it anywhere.
The issue you describe is one which is thousands of years old, and still unresolved. It's known either as the Aggripan Trilemma or the Münchhausen Trilemma, depending on who you want to credit it to.
If we ask of any knowledge: "How do I know that it's true?", we may provide proof; yet that same question can be asked of the proof, and any subsequent proof. The Münchhausen trilemma is that we have only three options when providing proof in this situation:
- The circular argument, in which theory and proof support each other (i.e. we repeat ourselves at some point)
- The regressive argument, in which each proof requires a further proof, ad infinitum (i.e. we just keep giving proofs, presumably forever)
- The axiomatic argument, which rests on accepted precepts (i.e. we reach some bedrock assumption or certainty)
If anything, it is a demonstration of a limitation of knowledge. Philosophers wishing to deal with it typically either choose one of the three options, or seek to refine their philosophy in a way that makes it less dependent on a need to "know" things. I find the most popular to be declaring a set of basic axioms, such as "I think therefore I am," but its certainly not the only way to go.
Swami Vivekananda (1863 – 1902) says on this (Complete Works, V7, p 54-55; also here under the heading Inspired Talks, sub-heading Wednesday July 17 - http://cwsv.belurmath.org/volume_7/vol_7_frame.htm):
Shankara further asks, can you see existence separate from everything else? Where is the differentiation between two objects? Not in sense-perception, else all would be one in it. We have to perceive in sequence. In getting knowledge of what a thing is, we get also something which it is not. The differentiae are in the memory and are got by comparison with what is stored there. Difference is not in the nature of a thing, it is in the brain. Homogeneous one is outside, differentiae are inside (in the mind); so the idea of "many" is the creation of the mind.
Differentiae become qualities when they are separate but joined in one object. We cannot say positively what differentiation is. All that we see and feel about things is pure and simple existence, "isness". All else is in us. Being is the only positive proof we have of anything…
Shankara says again, perception is the last proof of existence. It is self-effulgent and self-conscious, because to go beyond the senses we should still need perception. Perception is independent of the senses, of all instruments, unconditioned. There can be no perception without consciousness; perception has self-luminosity, which in a lesser degree is called consciousness. Not one act of perception can be unconscious; in fact, consciousness is the nature of perception. Existence and perception are one thing, not two things joined together. That which needs no cause is infinite; so, as perception is the last proof of itself, it is eternal. It is always subjective; perception itself is its own perceiver. Perception is not in the mind, but perception brings mind. It is absolute, the only knower, so perception is really the Atman [innermost soul of man]. Perception itself perceives, but the Atman cannot be a knower, because a "knower" becomes such by the action of knowledge; but, Shankara says, "This Atman is not I", because the consciousness "I am" (Aham) is not in the Atman. We are but the reflections of that Atman; and Atman and Brahman are one [innermost soul of man and the universal Soul].
You perceive that you think. Perception is the ultimate proof.
I would like to add to the other answers that conventionally the response to this problem is pragmatism. In the context of a philosophical debate, the degree of pragmatism is usually significantly lower than in everyday conversations, which is why skeptical hypotheses are entertained for longer.
In everyday conversations, you accept an answer based on your personal standard for evidence and arguments. Of course, I would argue that the way to get over this problem is, again, to pragmatically look for common ground with your partner in conversation and start with arguments and evidence from there.
If I'm asked how I know that 1+1=2, I will ask the person if they are familiar with a given set of mathematical axioms. If they say no, I explain them to them and say 1+1=2 follows directly from these axioms and the definition of addition. If they ask "How do you know?" again, I say I use deductive reasoning and show my reasoning to them. I could still have made mistakes in my deductive reasoning, of course, but again pragmatism with regards to topics like these is eventually unavoidable.
Use your senses. Knowledge is empirical verification of what is - else how do you know what is?? What is is that which is empirically verified. Use your words. Truth is merely a condition of propositions satisfied when utterance corresponds with what is. Don't take my word for it, just consider that you did not invent language. Consider that certainty is only a mood. Consider that despite epistemic limits - that knowledge is partial, imperfect, not "absolute" - we build skyscrapers, bridges, replace human hearts, send men to the moon and back, etc. with imperfect and fallible knowledge (read: empirical verification of that which is empirically verified).
Yes, a child can do as much as ask "why?" after the answer to every question and as little or as much may be understood as their capacity to keep their audience engaged.