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Both idealism and materialism have convincing arguments for me, yet within each

Both idealism and materialism have convincing arguments for me, yet within each argument are refutations of the other. How can I reconcile which to believe when they both seem equally as likely? I've thought that perhaps idealism explains our own subjective worlds, and materialism explains the objective external world, but can both be true when they contain refutations of the other?

I'm not sure what arguments you have in mind for these two views, but I understand, I think, the main feature that raises your question: each implies that the other is false. And yet, as you say, you find that each is equally likely. If each implies that the other is false, then they can't both be true, to answer your last question. One of them, at least, must be false. Perhaps they are both false, if they both rest on a false presupposition, for example. This seems to be the situation with the two views, (a) that the present king of France is bald and (b) that the present kind of France is not bald. They contradict each other, so one, at least, must be false. In fact, it seems they are both false, since there is no present kind of France at all--though I should note that, at one time, more philosophers thought the two views lacked a truth value at all (that is, they were neither true nor false, so on such a view we could say that the two theories you are considering might lack a truth value at all).
So much for that last question, whether they can both be true. They can't. However, there's another interesting question in what you wrote. If you find both theories equally likely, and assuming that there isn't some third theory that is also a competitor, then why not just be equally confident in one as you are in the other? For example, if you think that Idealism and Materialism are the only options, and they are equally likely, then you should think there is a 50% chance, or probability, that each of them is true. This is what you would do when it comes to a coin flip. So why not split your confidence in this way with these theories? Not every contradiction that is unsettled by our evidence is a paradox. Sometimes it makes sense to simply be as confident as our evidence calls on us to be, and no more, in each option. This doesn't mean stop thinking about it. You may, for example, soon discover that the case for Idealism is not as good as you are thinking it is now, and that the case for Materialism is actually better. So you'll get more confident in Materialism.

We know for now, at least, it's impossible to go back in time scientifically.

We know for now, at least, it's impossible to go back in time scientifically. But what if you really needed to, say if you had done something really bad and had ever desperation to go back in time and correct what you did, so you don't suffer the consequences you are suffering in the present. Provided you would not cause a disaster by going back in time, and that you would only change the bad things you did, it is an interesting concept. With this context, if you could be given a drug, that would leave you asleep for the rest of your life (coma), would you do it? Read on, there's more. In this sleep, you will have a dream, which is set from just before your mistake. So essentially, it causes you to simulate the past and the rest of your life in your head. It seems real, but it isn't. My question is, would this be the same as going back in time and changing things in reality? Does reality matter more, or our interpretation of it?

First a terminological quibble. By "scientifically impossible," I take it you mean "technological infeasible," i.e., impossible given the limits of current technology. As I see it, what's scientifically possible or impossible depends only on the laws of nature, which are standardly taken to be unchanging over time (or at least over any time that humans will experience). I think the jury's still out on whether backward time-travel is scientifically impossible in this latter sense.

To your question: I think there's something self-contradictory in the idea of "correcting what you did" if that means "bringing it about that you never did what you in fact did." Either (1) you did it, or (~ 1) you never did it. I can't see how any consistent story features both (1) and (~ 1).

In that sense, then, there's no such thing as (2) "going back in time and changing things in reality" and therefore nothing that's the "same as" (2). See section 1.2 of the SEP article on time-travel.

If technology advances to the point of recreating the world almost perfectly in

If technology advances to the point of recreating the world almost perfectly in a virtual reality (i.e. The Matrix), would it be morally acceptable to "move" into that world indefinitely? Let us assume there is a moral disparity between someone with/without family, friends, attachments moving into this virtual reality. Let us also assume there is no cost to sustain anyone's well-being in this distant future, either in this virtual world or the real world, such as rent or food. Perhaps in this virtual world there are new fun things to do, like flying freely, that in the real world one could not do. There is seemingly no catch to this, but is there a moral obligation to remain in the "real world" and do "real things?"

As usual, the answer will depend on your ethical theory. For instance, some forms of utilitarianism might require that you go into the Matrix if doing so would maximize happiness (e.g., because you'd be much happier, outweighing any unhappiness you might cause to people in the 'real world' by being hooked up to the machine). Indeed, Robert Nozick used his Experience Machine thought experiment (a prequel to The Matrix) to argue that there must be something wrong with utilitarianism precisely because he thought we would not (and should not) hook up to the machine, in which our happiness would not be based on real actions and accomplishments. (There's some interesting experimental work on whether and why people say they would or would not be hooked up.)

For various reasons (not just utilitarian), I think everything depends on what you would be leaving behind and what you would be doing in the Matrix. I'm not sure what you meant when you wrote that we should "assume there is a moral disparity between someone with/without family, friends, attachments." But I take it that there would be nothing wrong with entering the Matrix for a person who would not thereby betray her obligations to others or herself (e.g., she had no family or friends or was on the verge of suicide or could do no jobs that would help society, etc.). But there would be something wrong for the person who would be betraying such obligations to others (perhaps also including obligations to develop her own abilities, create a meaningful life, etc.)

But note that there may be Matrix setups that allow people to develop their abilities (creating art or learning new skills--like flying!), to help other people (who are interacting with them in the Matrix), to fulfill obligations, etc. It might be that people in online gaming communities create real friendships and really help other people, even in a world that is, in many ways unreal.

When the word" exist "occurs like "numbers exist "does it mean what it means in

When the word" exist "occurs like "numbers exist "does it mean what it means in sentences like "Dogs exist"?

I think it does, or at least I think the burden of proof is on anyone who says that "exist" is systematically ambiguous, meaning one thing when applied to numbers and another thing when applied elsewhere.

It's widely held that abstract objects such as numbers, if indeed they exist, don't exist in spacetime, whereas concrete objects such dogs clearly do exist in spacetime. But that doesn't affect the meaning of "exist" itself. In particular, it doesn't imply that "exist" means "exist in spacetime." Otherwise, the expression "exist in spacetime" would be redundant and the expression "exist but not in spacetime" would be self-contradictory, neither of which is the case.

Analogy: It's a fact that some things exist aerobically and some things exist anaerobically, but that fact doesn't tempt anyone to say that one or the other kind of thing doesn't really exist, or to say that "exist" just means "exist aerobically." So I see no reason not to say that numbers, if they exist, exist nonspatiotemporally, whereas dogs exist spatiotemporally: the adverbs differ in meaning, but not the verbs.

I am wondering if there is an alternative between nominalism and realism. Both

I am wondering if there is an alternative between nominalism and realism. Both of these theories agree that particulars exist but they disagree about the relationship between the particulars and their properties. However, what if there are no individual particular entities that are independent with their own self-nature? In our modern times, we often discuss how entities are interconnected and, to some extent, dependent on other entities. In the field of Social Work, for example, the micro, mezzo, and macro environments affect each other and the line between the three are constantly blurred and uncertain. Believing in a "mechanical" world that is made up of separate individual parts is becoming increasingly difficult to believe.

I can't do better than to recommend that you consult the SEP entry on monism written by Jonathan Schaffer. It's not clear to me from your question whether you favor what Schaffer calls "existence monism" or instead what he calls "priority monism," but you'll find both views expertly discussed.

We all know beyond the universe, there's nothing. How come is that possible?

We all know beyond the universe, there's nothing. How come is that possible? Theory says the big bang happened, and that theory has been accepted since it was "released". But where was the energy that caused it? And how did it existed, if there was nothing? Is there anything in the "nothing"? And, if we talk about religion, how did god exist? Who made it? How was he created if there was nothing? Some people say it created himself, but how the heck that happened if there was nothing?

I realise this may not be satisfactory, but many philosophers and scientists think that we cannot know the answers to your questions.

You start by saying that we know that beyond the universe there is nothing. But this may be one of the things that we cannot know. If there was something, something physical, before the universe then this would be the origin of the energy that gave rise to the Big Bang. One possible explanation, then, is that there are or have been many other universes (the multiverse theory). Of course, there is considerable difficulty (impossibility?) in collecting any empirical evidence for this theory, since such evidence would need to come from beyond the limits of spacetime itself.

But perhaps something - some universe or other - has always existed; there was never nothing. Many people find this claim more puzzling than the thought that once there was nothing, and then there was something. But why? The philosopher David Hume asked us to consider the limits of our knowledge about matters like this. It seems conceivable that something has always existed, and each thing has in turn caused the next. You may object that this just pushes the problem back. Your questions apply to any other universe as well. If this universe was caused by a previous (or another) universe, and so on, infinitely, that doesn't help. For instance, science tells us that time came into existence with the universe. Time itself ‘began’ with the beginning of the universe just under 14 billion years ago. That means that whatever caused the universe (if it has a cause) cannot exist ‘before’ the universe – there is no ‘before’ the universe! Instead, the cause of the universe must exist outside time. We think incorrectly then if we think that another universe, one that existed before this universe, caused this universe. If there is an infinite series of causes, this cannot be how it takes place. Hume might respond that we simply can’t know the answer here. So we should draw no conclusions. Bertrand Russell once said that the universe is ‘just there, and that’s all’.

Perhaps there was nothing and then something. What should puzzle us here? First, must everything have a cause? Hume argues that it is not self-contradictory to deny it. The same is true of ‘Something cannot come out of nothing’. That means that these claims are not certain. Our experience clearly supports these claims, but experience cannot establish that a claim holds universally. And we have no experience of such things as the beginnings of the universe. Second, the beginning of the universe is not an event like events that happen within the universe. For instance, it doesn’t take place in space or time, since both come into existence with the universe. We cannot apply principles we have developed for events within the universe, such as ‘everything has a cause’, to the universe as a whole.

When we turn to the question of God, there is the possibility of a distinct kind of reply. To think of God as self-causing is not usually to think that there was a moment in which God caused God's own existence. A better way of putting the thought, which some religious philosophers have done, is to claim that God's existence is necessary. God is the kind of being that must exist, that could not not exist. So God never came into existence. The philosopher Frederick Copplestone thought that your questions lead us to conclude that a God of this kind exists:

1. Things in the universe exist contingently (they may or may not exist, i.e. they can come into existence and go out of existence).
2. Something that exists contingently has (and needs) an explanation of why it exists; after all, its existence is not inevitable.
3. This explanation may be provided by the existence of some other contingent being. But then we must explain these other contingent beings.
4. To repeat this ad infinitum is no explanation of why anything exists at all.
5. Therefore, what explains why contingent beings exist at all can only be a non-contingent being.
6. A non-contingent being is one that exists necessarily, and doesn’t need some further explanation for why it exists.
7. This necessary being is God.

There are at least two problems with this argument. First, as Hume and Russell would argue, perhaps it is not true that every contingent thing requires an explanation for its existence. Second, it is unclear whether the concept of God as a being that must exist is coherent.

I haven’t answered your questions, but I hope I have given you a sense that they are widely shared!

The Kalam Cosmological Argument has as its first premise "Everything that begins

The Kalam Cosmological Argument has as its first premise "Everything that begins to exist has a cause" (at least in the form I've heard it). Often examples of "chairs" or "people" are given for things that began to exist. But this seems to be a category error - the Universe did not begin to exist in the same way that a chair does. Indeed a chair doesn't "begin to exist" in that it was created from other things. So to me it sounds like the argument overstates its case with "everything that begins to exist" since the only thing that has begun to exist is, well, everything. One could restate this premise as "The universe began to exist" could it not? Is I missing something or is this what is meant by this argument? If so it seems to be more of an assumption than the inductive reasoning I hear it being used as (e.g. "you've never seen a chair 'pop into existence' have you?").

I think you've put your finger on a dubious feature of the KCA. While I would say that a chair does begin to exist when it's created from pre-existing materials, I agree with you that if the universe began to exist, the universe didn't begin to exist in the same sense in which a chair does.

So I think you're right to detect a questionable move from "Everything within the universe that begins to exist has a cause" to "Everything, including the universe itself, that begins to exist has a cause." It's not at all clear that the phrase "begins to exist" is being used in the same way both times.

To the question "You've never seen a chair pop into existence, have you?" one can reply as follows: "I've never seen anything arising from pre-existing materials pop into existence, but that isn't relevant to whether something not arising from pre-existing materials can pop into existence."

If something can’t be defined can it exist? and vice versa

If something can’t be defined can it exist? and vice versa

Some things can be defined that cannot exist, such as "A square circle in two dimensional space" or "2+2=1" --and some things can be described that do not exist but could have existed or might come to exist (unicorns). And, I suggest, that there may be indefinitely many things that exist for which we do not have any successful definition. "Consciousness" might be a candidate, insofar as some philosophers are right in thinking we may never have a good or at-least problem-free definition.

As an aside, your question raises the need for a good definition of definitions. I will not attempt such a philosophy of definitions here, but you might check out the Stanford Encyclopedia entries bearing on philosophy of language for further, useful material. Paradoxically, if nothing can exist than cannot be defined, and we have no definition of being defined, we all might be in trouble.

Thinking further: I suspect you may be principally concerned with the problem of affirming that something (X) exists, and whether this affirmation is meaningful if we lack a definition of X. On the face of it, there would be a problem with someone claiming: "Call the reporters. There is something I will refer to as 'N,' but I have absolutely no idea or definition of what 'N' might be. It could be an animal or number or time of day, for I know." Such a claim would be as bizarre as what we find in Alice in Wonderland. Even so, I suggest that we should distinguish claims about meaningful speech and claims about what does or does not exist. Even if we cannot make claims about what does or does not exist without (at least vague) definitions, it is another thing to claim that there only exists things we can make meaningful claims about. Sadly, we can imagine the whole human species perishing from some force which we cannot comprehend (and thus we cannot define) That is such a grim thought to end this reply, let me change the example: we can imagine that cancer and depression might be eradicated by a force that we human beings cannot comprehend or define.

Is there a clear way to distinguish physical and non physical things? I'm not

Is there a clear way to distinguish physical and non physical things? I'm not implying that there are non physical things. I would prefer if you didn't define "physical" as whatever is studied by physicists.

How about this: All physical things occupy spacetime. But not all nonphysical things occupy spacetime, and maybe none do.

The clearest example of allegedly nonphysical things would, I think, be abstract objects such as numbers and sets. Platonists say that there are infinitely many such things. See this SEP entry.

Do rainbows exist? I assume rain drops and sunlight exist, but the rainbow is

Do rainbows exist? I assume rain drops and sunlight exist, but the rainbow is not a collection of rain drops, nor a region of the atmosphere where passing rain drops get some colour, is it? Should we say that rainbows are optical illusions? Or what?

Lots of things should be said to exist, even though they are not material entities (like raindrops) nor energy forms (like sunlight). We're happy to talk about numbers or abstract concepts as existing, for example, and likewise dreams, or things that happened in the past. We might provisionally say that X exists if it were to matter in some way if someone asserted that X did not exist. (This is a pragmatist definition. I'm not endorsing it so much as finding it useful.) If someone said that you DID NOT have the dream last night you say you had, then that would matter, because they would be saying you are lying; if someone says that we have no concept of causation (Hume), then that matters because whole bits of philosophy, and maybe physics too, become valid or invalid. An optical illusion exists because its happening matters to the person experiencing it. (So, even an atheist would have to admit that God existed, because it matters so much to so many that he does; however, the atheist would assert that God exists more in the way of rainbow or dream, than in the way of a rainbow.)

To my mind, the interesting question is not 'what kinds of things can be said to exist?' but rather 'what different types of existence are there, how do they relate together, and how is it we come to understand such a variety of types?'