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If there is a category "Empty Set" it has to have the property "emptiness". It must have this property that separates it from every other set. Thus it is not propertyless - contradiction?

I don't see a contradiction here any more than I did back at Question 26649, which is nearly identical. Yes, the empty set has the property of being empty and is the only set having that property. But the emptiness of the empty set doesn't imply that the empty set has no properties. On the contrary, it has the property of being empty, being a set, being an abstract object, being distinct from Mars, being referred to in this answer, etc. Why would anyone think that the empty set must lack all properties?

Doesn't trying to demonstrate how we know anything beg the question?

It needn't. Like Descartes, you might try to demonstrate a priori that you possess perceptual (i.e., external-world) knowledge. Your demonstration needn't presume perceptual knowledge in the course of demonstrating that you possess perceptual knowledge. Therefore, your demonstration needn't beg the question of whether you have perceptual knowledge in the first place. Most philosophers, I think, regard all such demonstrations (including Descartes's) as failures, but I don't see any reason to think that all such demonstrations must fail because they beg the question.

Consider a more interesting case. Suppose I analyze knowledge as true belief produced by a reliable mechanism, i.e., a mechanism that yields far more true than false beliefs in the conditions in which it's typically used. A skeptic then challenges me to show that some perceptual belief I regard as knowledge, such as my belief that I have hands, was in fact produced by a reliable mechanism. In response, I offer empirical evidence in favor of my belief: others verify that I have hands; I clap my hands; I shake the skeptic's hand; I cite other outputs of my perceptual belief-forming mechanism, etc. The skeptic then protests that I'm begging the question because my method of verification simply assumes that I have perceptual knowledge: I simply assume that the evidence I offer was reliably obtained.

I think the skeptic's protest is unfair. It's unfair to ask me to show that my perceptual belief-forming mechanism is reliable while disallowing me the very means I would need to show it, namely, data that I obtain by perception. Alternatively, the skeptic could try to show that my belief-forming mechanisms are not reliable, but skeptics seldom if ever try to show that. Or the skeptic could reject my analysis of the concept of knowledge, but then the skeptic would have to offer grounds for rejecting it, which skeptics seldom if ever do.

Consider the mathematical number Pi. It is a number that extends numerically into infinity, it has no end and has no repeating pattern to its digits. Currently we have computers that can calculate Pi out to many thousands of digits but at a certain point we reach a limit. Beyond that limit those numbers are unknown and essentially do not exist until they are observed. With that in mind, my question is this, if we could create a more powerful computer that could continue to calculate Pi beyond the current limit, and we started at exactly the same time to compute Pi out beyond the current limit on two identical computers, would we observe the computers generating the same numbers in sequence. If this is the case would that not infer that reality is deterministic in that unobserved and unknown numbers only become “real” upon being observed and that if identical numbers are generated those numbers have been, somehow, predetermined. Alternatively, if our reality was non-deterministic would that not mean that the two computers would generate potentially different numbers at each iteration as it moved forward into unobserved infinity inferring that unobserved reality is not set and therefore we live in a reality defined by free will?

You're no doubt right that any computers we happen to have available will only compute π to a finite number of digits, though as far as I know, there's nothing to stop a properly-designed computer from keeping up the calculation indefinitely (or until it wears out.) But you add this:

"Beyond that limit those numbers are unknown and essentially do not exist until they are observed"

Why is that? Let's suppose, for argument's sake, that we'll never build a computer that gets past the quadrillionth entry in the list of digits in π. Why would than mean that there's no fact of the matter about what the quadrillion-and-first digit is? What does a computer's having calculated it or (at least as puzzling) somebody having actually seen the answer have anything to do with whether there's a fact of the matter?

To be a bit more concrete: the quadrillion-and-first digit in the decimal expansion of π is either 7 or it isn't. If it's 7, it's 7 whether anyone ever verifies that or not. If it's not 7, then it's something else whether or not anyone every figures out what. It would take a lot of arguing to give a reason why we should think otherwise.

This is related to another of your questions. Yes: if two properly-programmed, properly-functioning computers kept spitting out the digits in π, they would print out the same digit when they got to the quadrillion-and-first entry. π is the ratio of the radius of a circle to its circumference. This number is the same for all circles not as a matter of incidental empirical fact but because of what it is to be a circle. That ratio is a specific number, and if two computers disagree about one of the digits, at least one computer is wrong.

But this doesn't help us with the question of determinism. Let's suppose that the world is indeterministic. (For all any of us knows, it is.) This doesn't mean that every physical process is also indeterministic. It just means (roughly) that from the laws and the total state of the world at one time, the total state of the world at other times doesn't follow. But the word "total" matters here. Even if quantum processes are indeterministic at some level, the output of a computer running certain sorts of programs isn't. If a computer fits the requirements for being a Turing machine, its output is deterministic. Of course, any physical machine can break down, and it could be that what makes some particular machine crap out has a chance element. But that doesn't mean that all particular physical processes are indeterministic and in any case, doesn't make mathematics mushy.

Finally, on free will. The connection between free will and physical determinism is actually not as simple as it seems. If you're interested in thinking about that, there is a recent book by Jenann Ismael called How Physics Makes Us Free. It's accessible, engaging and rigorously argued. I recommend it highly.

Hi...I'd like to begin reading Hume. Should I begin with the Treatise or the Enquiry?

Well, there are two Enquiries, corresponding to the first and the third books of the Treatise. And I'm sure everyone will have her own strategy for reading Hume. My own opinion is that you can't really appreciate the Enquiries until you see how much is behind them; they're too smooth and polished. So I would recommend starting with the Treatise, but not reading it straight through from beginning to end, and not getting too bogged down in the minutiae. Very roughly, I would recommend reading Book 1 of the Treatise relatively quickly to get an overview of the argument, without attempting to be too precise about it. Then I would skip to Book 3 and do the same, though this one is a bit harder to grasp without attending to the details. It is fashionable these days to claim that the long-neglected Book 2 is just as important etc. as Books 1 and 3, but as a way in to Hume I think you'll find Books 1 and/or 3 more accessible. Also, depending on which you are more interested in (Book 1 if you're more into metaphysics and epistemology, Book 3 if you're more into social and political philosophy), there are the corresponding Enquiries to give you a nice overview from a somewhat different viewpoint, just as the Prolegomena gives you a nice overview of Kant's 1st Critique from a somewhat unexpected point of view.

Also there is some secondary literature that can really help to inform your reading. Among the things I have found helpful are Mossner's old biography, which still has merits even though James Harris's recent intellectual biography has in some respects superseded it (and is also very much worth reading). And depending on whether you're more into the epistemological and metaphilosophical aspects or the social and political aspects, you should look at things like Graciela de Pierris on the former and Russell Hardin on the latter. Also, if you're really focused and committed, you might consider looking at Duncan Forbes on Hume's philosophical politics (a challenging book even then). And there are plenty of new things that consider particular aspects. Look in the journals for new articles about Hume, for instance, and just start reading whatever sounds interesting to you from the title.

Only after that should you then go back and start reading the Treatise from the beginning, line by line, carefully. Again, depending on your interests, focus on Book 1 or Book 3 (or, by then, if you insist, on Book 2). Best of luck, it's worth sticking with, as you'll soon figure out once you get over the initial hurdles!

Are there any true contradictions?

None that I can think of, including none of the candidates that I've seen offered by "dialetheists" (i.e., philosophers who say that some contradictions are true). If you have any promising candidates, please let us know!

If there is a category "Empty Set" it has to have the property "nothingness". Thus it is not propertyless - contradiction?

As far as I can see, the definitive property of the empty set is not nothingness but instead emptiness: It's the one and only set having (containing, possessing) no members at all. The empty set can be empty, in that sense, without itself being nothing. So I see no threat of contradiction here. Indeed, the empty set can belong to a non-empty set, such as the set { { } } , which couldn't happen if the empty set were nothing.

Long time follower, first time asker I deeply identify with the second part Nietzsche's aphorism: 'He who fights with monsters should look to it that he himself does not become a monster. And if you gaze long into an abyss, the abyss also gazes into you.' (Wer mit Ungeheuern kämpft, mag zusehn, dass er nicht dabei zum Ungeheuer wird. Und wenn du lange in einen Abgrund blickst, blickt der Abgrund auch in dich hinein.) As it relates to a thirst for knowledge that takes you deeper into the rabbit hole (one -- potentially wrong -- interpretation). I found that sometimes unanswerable questions have obsessed me past the point of healthy living, and that to get out of this mindset I had to... just stop gazing However I do not think this was a novel idea. Are there any examples of this idea in ancient philosophy? Citations & references appreciated Best,

A little more pedantically than you, I would say, regarding that sentence about the abyss, that an adventurous philosopher looks long and hard even where no explanation seems to lie. That's the abyss. Gazing into an abyss feels like a neutral or innocuous desire to know, until you imagine yourself being looked at in the age of looking. The abyss gazes into you, meaning that it spots you looking for something in it. It sees you actively searching for an explanation.

In other words, I don't exactly take the sentence as you do, but we're reading it similarly. And I am grateful for your response to it: giving up on gazing when you see you're being gazed at. You can understand Kant's advice to metaphysicians as similar to what you're describing. Stop trying to answer these traditional questions as if they were real questions; learn to diagnose the questions in their unanswerability, down to the human desire to exceed empirical human knowledge.

But you wanted to know specifically about ancient philosophers who might have said as you do: “just stop gazing.” As far as I know, the systematic approach to philosophical questions as unanswerable, hence as questions we respond to with the refusal to answer, dates back to Hellenistic philosophy. The Hellenistic Skeptics sound like Kant or Hume sometimes in the way they urge curious philosophers to give it up and stop making proclamations about the way nature reveals itself (or fails to).

For the Skeptics, most metaphysical inquiries lead to error and dismay, all the more unacceptable outcomes given the Skeptics’ recommendation of ataraxia “tranquility, an untroubled condition.” Worrying about the truth or falsity of a metaphysical assertion only gave you troubles you were better off without; so the Skeptics proposed that you assert your indifference to the question with a simple ou mallon “not more” or “not particularly”: no more true than false. Is change real? – it no more is than it isn’t. The skeptic Pyrrho is credited with having originated this response, promising that talking back to metaphysical utterances in this way will ease your mind.

Pyrrho wrote nothing, so we know of his pioneering skepticism through the writings of others. Of the Skeptics who adopted and elaborated his views, Sextus Empiricus is the best known and his books the best preserved. But rather than spell out a reading list I would just recommend the Stanford Encyclopedia of Philosophy entry “Ancient Skepticism,” an excellent and extensive discussion by Katja Vogt.

As far as Nietzsche’s relationship to such ideas goes, you might be interested in Jessica Berry’s book “Nietzsche and the Ancient Skeptical Tradition” (Oxford University Press, 2010). Berry finds a deep resonance between Nietzsche’s account of the inquirer and the Skeptics’ cautions. I don’t always agree with her analogies, but her argument is informed and reasonable; and your own thoughts about the abyss remark tell me that you’re inclined to agree with her.

I want to ask about the truth, universal truth. There is any standard about universal truth?I mean the truth which every one agree about it. what is the real truth? Why people have their own argument about their own truth? is it possible all of people agree about one truth?

Do you really believe that there is any truth that you are going to get 7,511,772,360 (the number on the rolling world census at the time of writing) people to agree to? That includes new born infants, who are people. So perhaps we should cut off your question at some later age, say 18. But how will this age be decided? I think probably people have their own views because they have their own ideas, and they have their own ideas because their experiences are different and they are very different people anyway. Still, most people, though by no means all, accept elementary mathematics and the elementary ethics of everyday life. There are standards of evidence and argument that apply to both, to ethics in practical settings and to mathematics in theoretical ones as well. It seems very improbable that short of the coming of the Kingdom of God everyone will agree on everything. But that doesn't mean that what they should agree on, because it is true, isn't true.

Philosophers like Wittgenstein and Plato are known for their distinctive, and challenging, writing styles. Perhaps unsurprisingly, commentators generally don't write like Wittgenstein and Plato in writing about them. Does this show that works like the Tractatus and the Republic could have been written just as well in ordinary prose? My underlying presumption here is that when people write about philosophers, this largely amounts to restating the claims of those philosophers. So if a Wittgenstein scholar insists that Wittgenstein's oracular style is essential to his philosophy, and yet argues as much in an article written in straightforward, conventional prose, she is actually contradicting herself in a way.

Not all of Plato's writings are challenging or difficult. But some are, the Parmenides for example. And Wittgenstein writes perfectly ordinary sentences in the Philosophical Investigations , though people complain that the writing is hard to follow. I can't agree with this, as I think it's a view that insinuates itself when you don't know the arguments very well. The Tractatus is of course different. But both the Parmenides and the Tractatus could have been written in ordinary prose, apart from the symbolic bits, for example in connection with the symbol omega and the theory of numbers, say, or the theory of truth-functions and the general form of the proposition. But your presumption is wrong. When people write about philosophers' work, very little restates the claims of those philosophers. The only point of that would be educational. Much more of the writing about works of philosophy is about what the claims mean and whether they are true. So your contradiction disappears. It is possible to try to unravel the meaning of sentences in the Tractatus and then to argue that they are true or that they are false or that we can't decide, without just repeating what the book says, and so, by your premise, descending in oracularity.