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I wasn't clear about something; A+B=B+A, in abstract, is no more an axiom than anything else. There are some systems where it is an axiom (especially as absolutely anything can be an axiom, so one could always create a system if one desired). However, it is a bad example of an axiom, especially without reference to which system we are talking about, since it is frequently not one. Furthermore, in cases where it is an axiom, it cannot "be shown to be true" without either assuming it, or without assuming another axiom which then makes the original A+B=B+A case true regardless of being an axiom, thus making it useless as an axiom.
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In other words, we can't really ever know anything, because you can always argue that we don't really know one of the foundations upon which we are basing some other knowledge.
No point in discussing anything, since we can't even be sure what words mean. In English this sentence has meaning, but I can come up with examples where it's meaningless, so I've just proven myself wrong.
I think I just won the Internet!
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quote:Originally posted by fugu13: If you take a look at MC's link, that's one way to define addition via axioms that include a commutativity axiom. You'll see that derivation if you take a good real analysis class.
If you aren't willing to agree that an axiom is actually an axiom, I don't think you're ever going to be satisfied in this discussion.
A statement which is an axiom in one system is not necessarily an axiom of another system; it can be a theorem, it can be false, it can be unprovable, and its negation can be an axiom. You have apparently found a system in which 'addition commutes' is, indeed, an axiom; this says nothing about other systems. In Peano arithmetic, for example, I believe it's a theorem.
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quote:Originally posted by fugu13: I wasn't clear about something; A+B=B+A, in abstract, is no more an axiom than anything else. There are some systems where it is an axiom (especially as absolutely anything can be an axiom, so one could always create a system if one desired). However, it is a bad example of an axiom, especially without reference to which system we are talking about, since it is frequently not one. Furthermore, in cases where it is an axiom, it cannot "be shown to be true" without either assuming it, or without assuming another axiom which then makes the original A+B=B+A case true regardless of being an axiom, thus making it useless as an axiom.
In the example given it is basically a convenient short cut. To actually do the work of extending the integers to rationals and rationals to reals and also extending the concretely defined notion of addition on the integers is a huge amount of work with little pay-off: everyone already basically knows how to add and (more or less) what real numbers are. If your first analysis course was anything like mine about 3 mins will be used mentioning Dedekind cuts and then you will be told to read the appendix to chapter one if interested. This axiomatization works because not only has someone proved that the familiar notion of addition is consistent with these properties, but that it is also uniquely defined by these properties. Notice that these properties don't provide a concrete way of actually adding two numbers.
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I, at least, was arguing to correct your understanding, apparently faulty, of what an axiom means in mathematics.
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quote:None of those are axioms, they are all judgements based on experience, which rests on the axiom of "my senses are reasonably accurate." Which is far more agreed upon than "God exists."
That only works if you are being fuzzy with the reasoning. One can just as easily say that "God exists" is a judgment based on experience, and thus that it also is derived from the axiom that "my senses are reasonably accurate". Similarly, you can say that moral claims are also judgments based on experience. You could count anything as something you "sensed" to be true, and thus use that axiom to support it.
However, if by senses you really only mean what you see, hear, smell, touch, and taste, then I think you are going to be very very hard pressed to derive predictions of the stock market solely from the axiom that "your senses are reasonably accurate" without any additional axioms. You can't directly observe a stock after all - even in the present, much less the future. You have to rely on some authority to tell you current and past prices, etc. Then you'd have to assume the future is going to be similar to past trends. And so on.
Most "factual" beliefs are based on many different unprovable assumptions in a similar fashion.
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When you read a newspaper or a website telling you current and past stock prices, you are basing that off of experience that such things are usually accurate. You know this because you may have seen them corroborated in the past, or because a large number of people have reported to you that they were accurate for them. You assume the future will be similar to the past because you've watched the future be similar to the past over and over again throughout your entire life.
And sometimes, this all still turns out to wrong. Sometimes it turns out you were reading the Weekly World News and the bat baby wasn't really born last week. Sometimes the website is out of date. And when that happens, people generally go "oh" and change their minds. The things I consider myself "certain of" to the degree that I can be certain about anything are things I have studied extensively and seen firsthand, or read about from a wide variety of sources that I have other reasons to believe are trustworthy.
And even then, I keep in the back of my mind that I could be wrong. I was careful to say "my senses are reasonably accurate," not that they are perfect. You have to constantly be on the lookout for competing evidence that shows you when your perceptions of the world are off. (And yes, I'm including things like the sensation of emotion.)
Religious beliefs often ARE based on experience, either through strong, sudden emotional changes or seeing lots of "meaningful patterns" in the world. The problem is that these things are only good evidence if you're ignoring vast other swaths of data, such as the fact that plenty of people go through the same powerful religious experiences for reasons completely unrelated to religion or related to religions that are diametrically opposed to your own. Or the fact that statistically, the "mysterious patterns" are almost guaranteed to appear to anyone who looks for them.
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quote:Originally posted by Raymond Arnold: The problem is that these things are only good evidence if you're ignoring vast other swaths of data
Nope.
But please continue to tell me what I'm doing. It's a large part of what has made Hatrack into the vibrant, happy, thriving community it is today!
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quote:Originally posted by rivka: But please continue to tell me what I'm doing. It's a large part of what has made Hatrack into the vibrant, happy, thriving community it is today!
quote:Factual beliefs tend to rely on axioms such as "my senses are reasonably accurate," which, even if false, would make the navigating the world completely impossible if we didn't assume.
That still means that factual beliefs are based on unprovable axioms.
It should also be noted that many factual beliefs are complicated enough that they require far more questionable axioms than "my senses are reasonably accurate". For instance, you can't simply observe with your senses that George Washington was our first president - that factual belief requires axioms that allow you trust things you have heard from other people and/or speculate based on historical artifacts. And then there's far more complicated factual beliefs like "Investing in stocks will bring me more money in the long run than investing in bonds", or "My friend is lying to me when he says he likes his mother-in-law", or "Health care is inefficient in America"...
What axioms are required to believe that George Washington is the first president? It seems to me that once you have enough axioms for logical reasoning then you're set.
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quote:It should also be noted that many factual beliefs are complicated enough that they require far more questionable axioms than "my senses are reasonably accurate". For instance, you can't simply observe with your senses that George Washington was our first president - that factual belief requires axioms that allow you trust things you have heard from other people and/or speculate based on historical artifacts. And then there's far more complicated factual beliefs like "Investing in stocks will bring me more money in the long run than investing in bonds", or "My friend is lying to me when he says he likes his mother-in-law", or "Health care is inefficient in America"...
None of those are axioms, they are all judgements based on experience, which rests on the axiom of "my senses are reasonably accurate." Which is far more agreed upon than "God exists."
I don't think it is agreed upon at all. If you take it as an axiom that "[your] senses are reasonably accurate" then that means that there is no evidence that could convince you otherwise.
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quote:Originally posted by Raymond Arnold: The problem is that these things are only good evidence if you're ignoring vast other swaths of data
Nope.
But please continue to tell me what I'm doing. It's a large part of what has made Hatrack into the vibrant, happy, thriving community it is today!
Are your experiences different than the two that I had just described? If they are, what are they? (Most of the religious experience I've had described to me fell into one of those two categories), and if they are, how exactly do you distinguish them from perfectly mundane human experiences.
I've had an experience that I'd describe as revelatory regarding Zoroastrianism. I've had moments when I suddenly found a part of my life to become very important, helping me through a difficult time, and soon afterwards I started finding patterns of messages everywhere I looked. (The part of my life in question? Gaming). I often talk to myself using imaginary versions of my friends as sounding boards, and eventually, as I got to know my friends better, I reached a point where I'd imagine them responding realistically. (Early on, if I was mentally going through an argument with a friend, I'd always pretend I won the argument, later on if I imagined saying one thing, I'd immediately think of the sort of logic my friend would respond with). If I had substituted God for real friends and had people telling me from birth that the person I was talking to was real, I have no doubt that I could have developed an imaginary relationship with him, regardless of whether or not he really existed.
Every piece of the "God experience" that's been described to me, I've experienced at some point in my life, just not necessarily related to God. If each of those experiences had been religious in nature and had happened in the right order, the evidence in favor of religion would have appeared very compelling, but in the end would be no more real than the mental simulacra of my friends that I talk to.
So unless you either have additional experiences that cannot possibly have been driven by anything other than supernatural forces, or can explain how to distinguishing between your experiences and what mine would have been if they were more religiously oriented (I grew up in a largely secular environment so I didn't have as much opportunity such spiritual ideas to sink into my subconcious), I don't see a reason to take your beliefs seriously. And while I don't see a reason you should change them if you're happy and not hurting anyone (I was on Chris' side in the King of Men/Chris' Wife debate) I do think you are choosing happiness over truth. Not inherently bad, but worth pondering.
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If you think I am going to discuss personal experiences -- particularly religious ones -- on the current incarnation of Hatrack, you are deluded.
quote:I've experienced at some point in my life, just not necessarily related to God.
And you know that how?
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quote:If you think I am going to discuss personal experiences -- particularly religious ones -- on the current incarnation of Hatrack, you are deluded.
In all seriousness, why would you mind? It's not like you care what, say, KoM thinks of you.
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quote:Originally posted by rivka: If you think I am going to discuss personal experiences -- particularly religious ones -- on the current incarnation of Hatrack, you are deluded.
I definitely do not blame you there. I didn't really mean to get into an argument with you - everything I've seen you post has been well thought out and respectful. I assume that you've put enough thought into your religious beliefs that they are satisfying for you, but if you are going to be participating in a religious debate thread, well, obviously your beliefs are going to come under scrutiny. I don't think there was anything particularly unreasonable about my post that set off this particular sub-debate. I stated how most of human beliefs boil down to a particular axiom and why many of the religious mindsets that have been described to me do not follow well from that axiom. I have no idea what your mindset actually is and I apologize if you felt unfairly judged
quote:
quote:I've experienced at some point in my life, just not necessarily related to God.
And you know that how? [/qb]
I think I explained that pretty clearly in my previous post. Everything that I've heard other people describe about their religious experiences matched up perfectly with how I would have described experiences I myself have went through. I know that my experiences are heavily influenced by my own life, which revolves around storytelling and games, and I think it's fair to assume that if my life revolved heavily around religion, the experiences would have as well. And if they had been constantly reinforced by a community telling me they were real, I certainly would not have been able to tell the difference between a "false" religious experience and a true one.
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quote:If you think I am going to discuss personal experiences -- particularly religious ones -- on the current incarnation of Hatrack, you are deluded.
In all seriousness, why would you mind? It's not like you care what, say, KoM thinks of you.
Tom: I believe we had a pretty extensive discussion as to why deep meaningful religious experiences often cannot be discussed with people with whom the teller does not have a relationship of trust with.
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Except that rivka, as far as I know, does not share the Mormon belief that something spiritual can be cheapened by sharing it with mockers.
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quote:I definitely do not blame you there. I didn't really mean to get into an argument with you - everything I've seen you post has been well thought out and respectful.
Thank you.
quote:I think I explained that pretty clearly in my previous post.
Perhaps I was unclear. You did not initially say religion; you said God. Leaving aside for the moment what the relationship between those two is or ought to be, what if those experiences were actually interactions with God that you have explained in other ways? How can you know? (This is a serious question. I realize it may seem like a leading one.)
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quote:Originally posted by TomDavidson: Except that rivka, as far as I know, does not share the Mormon belief that something spiritual can be cheapened by sharing it with mockers.
In my case, it's more like: I have finite time and energy. I spend way too much of it online anyway. Why would I invest more when I know precisely how useless it would be?
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quote:Originally posted by TomDavidson: Except that rivka, as far as I know, does not share the Mormon belief that something spiritual can be cheapened by sharing it with mockers.
Oh I don't think that principle is germane to mormonism Tom. I'm fairly certain we both agreed that is was a possibility that when a person shares deeply personal convictions to an audience of uncharitable skeptics that the result can be to the hearers detriment.
Not attempting to be disparaging to skepticism per se.
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To the extent that having your beliefs demolished is unpleasant, yes, that can be detrimental; but since they are not in fact true, you should welcome that unpleasantness. It's better to rip off the mistakes in one big OOPS than to hang on to them for years and only gradually lose faith. Beliefs which are actually true do not need to be shielded from either criticism or mockery; they can stand both the light of day and the laughter of the ignorant. When you hide the reasons for your inmost beliefs, that's a danger signal; if your brain were evolved to do actual rationality as opposed to running away from tigers, you would be doing the equivalent of moving your head back and forth, breathing deeply through your nose. "Hmm," you would say to yourself, "is there a whiff of tiger in the air? Perhaps I'd better get my spear." Unfortunately we are descended from the ones who could smell tigers, not from the ones who could spot a defensive reaction in themselves.
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quote:Perhaps I was unclear. You did not initially say religion; you said God. Leaving aside for the moment what the relationship between those two is or ought to be, what if those experiences were actually interactions with God that you have explained in other ways? How can you know? (This is a serious question. I realize it may seem like a leading one.)
It's possible that my Zoroastrian experience was a message from God. I did seriously consider the possibility at the time, since Zoroastrianism is a far more plausible religion to me than most other monotheistic variants. I gave Ormazed a few days to clarify anything. I got nothing. Given that even if I HAD gotten something, I still wouldn't be able to distinguish that from similar experiences people have had from mutually exclusive religions, I decided that was the last I was going to worry about it. If God had ever wanted to communicate with me, that was the last chance he was going to have to do so.
As for my other experiences, well, I doubt God is going to try and communicate with me via card games and imaginary friends. And if he is, it's not my fault that I have a hard time noticing his presence.
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quote:What axioms are required to believe that George Washington is the first president? It seems to me that once you have enough axioms for logical reasoning then you're set.
The minimum would be an axiom stating that some source X is trustworthy, a second axiom stating that some sense Y is trustworthy, and a third observation that sense Y told you that source X says George Washington was the first President. Y might be something like "my eyes" and X might be something like "a history book". But if you wanted to actually do research to verify the things said in the history book, that would require further axioms allowing you to establish that historical pieces of evidence exist and imply the things we assume they imply. You could use the "trust your senses" axiom to establish the existence of such evidence, but that would not allow you to deduce what the evidence implies - for that you'd need more assumptions.
"Factual" beliefs rely on assumptions and axioms no less than moral beliefs. That's because moral beliefs are a type of factual belief, rather than a separate sort of thing.
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Those aren't minimal axioms at all. Trustworthiness is almost always an earned condition, and in such cases by definition not axiomatic (being based on evidence).
edit: and the trustworthiness could be earned from consistency of experience. The entire construction can thus rest only on the reliability of the senses, which is a more minimal set.
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Senses can't observe the past (or the future). You can't use senses to logically establish the trustworthiness of a source about events that took place well before you were born - not without other assumptions or axioms from which to draw that conclusion.
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If trustworthiness is enough as an axiom, then trustworthiness based on consistency of sensory experience is enough as well. You didn't have any axioms about the past in what you incorrectly described as your minimal set.
If you like, we can add an axiom about reality being generally consistent. Since one's senses can certainly tell one what one currently perceives as having happened in one's own past, that's definitely enough to build trustworthiness on.
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I didn't have any axioms about the past in the minimal set because I skipped over proving the trustworthiness of source X. I simply made the trustworthiness of source X an unproven axiom. I had to do that to cut it to so few axioms.
I suppose if we wanted to be really absurd, we could prove it with one axiom, if that axiom was "George Washington was the first president". P therefore P. That would be one assumption only. Simply asserting the conclusion as an axiom is always a quick way to "prove" the conclusion.
As for the axiom "reality is generally consistent"... If our premises are: 1) Senses are generally trustworthy 2) Reality is generally consistent I think it would be many many steps before we could go from there to "George Washington was the first President" and would require other axioms along the way.
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quote:Originally posted by King of Men: To the extent that having your beliefs demolished is unpleasant, yes, that can be detrimental; but since they are not in fact true, you should welcome that unpleasantness. It's better to rip off the mistakes in one big OOPS than to hang on to them for years and only gradually lose faith. Beliefs which are actually true do not need to be shielded from either criticism or mockery; they can stand both the light of day and the laughter of the ignorant. When you hide the reasons for your inmost beliefs, that's a danger signal; if your brain were evolved to do actual rationality as opposed to running away from tigers, you would be doing the equivalent of moving your head back and forth, breathing deeply through your nose. "Hmm," you would say to yourself, "is there a whiff of tiger in the air? Perhaps I'd better get my spear." Unfortunately we are descended from the ones who could smell tigers, not from the ones who could spot a defensive reaction in themselves.
You'll note in my previous post I said "to the hearers" detriment not to the tellers.
quote:I think it would be many many steps before we could go from there to "George Washington was the first President" and would require other axioms along the way.
It's certainly a huge number of steps (trusting teachers based on past experience, trusting newspapers and books based on cross referencing and the fact that having a bunch of books conspiratorily leading you to believing some random guy was president when he really wasn't would be kinda silly). But no, it doesn't require any more axioms. Bearing in mind that when people say "George Washington was the first president" what they're generally really saying is "as far as I know, based on the evidence available, George Washington was most likely the first president." In the case of George Washington the evidence is fairly overwhelming and even if it turned out to somehow be a lie, it would a kinda weird one to be propagating, so it would take an awful lot of counter-evidence to assume otherwise.
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quote:It's certainly a huge number of steps (trusting teachers based on past experience, trusting newspapers and books based on cross referencing and the fact that having a bunch of books conspiratorily leading you to believing some random guy was president when he really wasn't would be kinda silly). But no, it doesn't require any more axioms. Bearing in mind that when people say "George Washington was the first president" what they're generally really saying is "as far as I know, based on the evidence available, George Washington was most likely the first president."
So how is this fundamentally different from how it is for moral beliefs? When I say killing is wrong, what I'm really saying "As far as I know, based on the evidence available, killing is most likely wrong." It might take many steps, including trusting teachers based on past experience, trusting books based on cross referencing, etc., but you could end up justifying that moral claim through the same process.
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The difference is in the quality of the axiom. Have you really never had a discussion of first principles before?
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The same axiom is being used: "Senses are trustworthy."
Raymond is suggesting that we can use our senses to observe that sources are trustworthy, and then based on that conclude those sources are likely telling the truth when they agree George Washington was the first president. We can do the exact same thing if we establish sources are trustworthy using our senses, and then based on that conclude those sources are likely correct when they agree that killing is wrong.
I don't agree that all of that is possible from just the one axiom, but if it IS possible in the way being suggested, it works for moral beliefs in the same way it does for "factual" beliefs.
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Okay, I'm not sure what the original argument actually was. One can (and people have) derived moral frameworks from the "my senses are reasonably accurate and reality is consistent" axiom.
My point was that "my senses are reasonably accurate and reality is consistent" is a far more universally accepted axiom than "my interpretation of God exists and his word is moral law."
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Tres: what other axiom might be required? Can you give a specific step in the process that another axiom would be required at?
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Raymond, My original argument was in response to KoM who said: "I have been quite careful to distinguish moral and factual beliefs. Moral beliefs at some point boil down to primate instinct, or axiom, or something else un-provable." He said "factual" beliefs don't rely on unprovable axioms. I argued that they do, and that moral beliefs are simply a type of factual belief. So the argument is really about whether moral beliefs are fundamentally different in the way they are justified than other "factual" beliefs.
Fugu, Let's say I have a bunch of bills signed by George Washington that could serve as evidence. My senses, if I trust them, can tell me that those bills exist and have a George Washington signature on them. However, I can't infer anything about history from those bills without another axiom (or more) - my senses cannot even say whether or not they all simply popped into existence at this particular moment. In order to take the step from "I have a bunch of bills that have a George Washington signature in front of me" to "George Washington signed these bills in the late 1700s" I'd need a premise my senses can't provide.
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quote:Originally posted by Tresopax: In order to take the step from "I have a bunch of bills that have a George Washington signature in front of me" to "George Washington signed these bills in the late 1700s" I'd need a premise my senses can't provide.
You can account for any uncertainty that you have by assigning a probability to your belief. Ex: I'm 70% sure that "George Washington signed these bills in the late 1700s". Practically speaking we can't calculate these probabilities but they're there implicitly (normally we speak in terms of "how sure" we are rather than using actual numbers).
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Don't "factual beliefs" and "moral beliefs" differ in type- independent of why or how I come to believe that George Washington was president, he either was president or not i.e. there is an underlying fact of the matter. On the other hand, moral beliefs, to the extent that they are decoupled from one's culture/environment/religion, tend to be centered on some premise (e.g. a moral action is one that maximizes happiness). However, unlike in the "factual" case, I don't think such premises can be meaningfully assigned a truth value.
quote:Originally posted by Tresopax: Raymond, My original argument was in response to KoM who said: "I have been quite careful to distinguish moral and factual beliefs. Moral beliefs at some point boil down to primate instinct, or axiom, or something else un-provable." He said "factual" beliefs don't rely on unprovable axioms. I argued that they do, and that moral beliefs are simply a type of factual belief. So the argument is really about whether moral beliefs are fundamentally different in the way they are justified than other "factual" beliefs.
Fugu, Let's say I have a bunch of bills signed by George Washington that could serve as evidence. My senses, if I trust them, can tell me that those bills exist and have a George Washington signature on them. However, I can't infer anything about history from those bills without another axiom (or more) - my senses cannot even say whether or not they all simply popped into existence at this particular moment. In order to take the step from "I have a bunch of bills that have a George Washington signature in front of me" to "George Washington signed these bills in the late 1700s" I'd need a premise my senses can't provide.
posted
Tres: I have seen bills before, and they have existed over a period of time (per my senses, which I can axiomatically trust). Per reality being consistent, these bills have also existed over a period of time. No other axioms needed.
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quote:You can account for any uncertainty that you have by assigning a probability to your belief. Ex: I'm 70% sure that "George Washington signed these bills in the late 1700s". Practically speaking we can't calculate these probabilities but they're there implicitly (normally we speak in terms of "how sure" we are rather than using actual numbers).
How would you logically deduce such a probability without an axiom stating that you can do so?
quote:Tres: I have seen bills before, and they have existed over a period of time (per my senses, which I can axiomatically trust). Per reality being consistent, these bills have also existed over a period of time. No other axioms needed.
Firstly, you'd need an assumption saying you can trust your memory as well as your senses, in order to say you've seen bills before (unless we are counting memory as part of our senses, which is questionable, given memory is often less trustworthy). Secondly, if we accept "reality being consistent" and "I've seen bills before existing over a period of time", I don't think that by itself allows us to logically deduce that the George Washington bills existed 200 years ago. For instance, we don't know that the bills we saw before have the same properties as the George Washington bills, unless we assume similar things have similar properties, or something along those lines. I've also seen faked documents, raising the question of how we can assume these particular bills are real and not faked.
That's how even simple "factual" beliefs end up requiring a complicated array of assumptions.
quote:Don't "factual beliefs" and "moral beliefs" differ in type- independent of why or how I come to believe that George Washington was president, he either was president or not i.e. there is an underlying fact of the matter. On the other hand, moral beliefs, to the extent that they are decoupled from one's culture/environment/religion, tend to be centered on some premise (e.g. a moral action is one that maximizes happiness). However, unlike in the "factual" case, I don't think such premises can be meaningfully assigned a truth value.
Why not? I think the 9/11 attacks were wrong; I believe I understand the meaning when I assign a truth value to that statement.
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Not at all. I have current experiences of my memory via some sort of sense, so that's handled by my senses being trustworthy (as I already mentioned).
And I merely dealt with the specific issue you brought up. Do you have another issue that would be brought up? I should think it would be quite obvious at this point how to first deduce the likely existence of things 200 years ago, then start filling in the details. For instance, sensing the existence of written records, then testing various sorts against more direct uses of the senses, et cetera.
None of that "similar things have similar properties" silliness is needed; it is completely handled by noticing via the senses (which are trustworthy) that similar things have similar properties, then extending it to other things via the consistency axiom.
Where is another axiom needed?
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quote:You can account for any uncertainty that you have by assigning a probability to your belief. Ex: I'm 70% sure that "George Washington signed these bills in the late 1700s". Practically speaking we can't calculate these probabilities but they're there implicitly (normally we speak in terms of "how sure" we are rather than using actual numbers).
How would you logically deduce such a probability without an axiom stating that you can do so?
Of course you need an axiom for that. However, the axiom doesn't have to be remotely related to George Washington.
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quote:Don't "factual beliefs" and "moral beliefs" differ in type- independent of why or how I come to believe that George Washington was president, he either was president or not i.e. there is an underlying fact of the matter. On the other hand, moral beliefs, to the extent that they are decoupled from one's culture/environment/religion, tend to be centered on some premise (e.g. a moral action is one that maximizes happiness). However, unlike in the "factual" case, I don't think such premises can be meaningfully assigned a truth value.
Why not? I think the 9/11 attacks were wrong; I believe I understand the meaning when I assign a truth value to that statement.
That the 9/11 attacks occurred is "factual belief". That these attacks were wrong is an a posteriori value judgment based on some moral theory. It is the underpinnings of this moral theory that I was speaking of.
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