Someone doing that still puts faith on the computer, and the person who made the computer program to check the rules. Essentially, he has strong feeling that A holds because the computer program said so. He still has to rely on his “intuition” or “belief” that the computer program gives true statements.
Some people (mostly young children, though some adults as well) believe that the ratio of a circle’s circumference to its diameter should be an integer, or at worst a rational fraction. Most other people, however, do not believe this to be the case.
If mathematical truths are subjective as you claim, then a person who believes that pi == 3 should be able to build himself a 5-foot wide hula-hoop using exactly 15 feet of plastic tubing. Do you think this is actually the case ?
Maybe he is able to construct some sort of an abstract hula-hoop in his mind, which he believes to have those properties, but of course he isn’t able to do it in the physical reality. Strong intuition suggests that it isn’t possible.
However, we should not forgot that mathematical models of physical reality and mathematics itself are separate things. We can use mathematics to understand nature, but nature cares very little about anyones mathematical truths. Well, I think it’s safe to say so anyway.
Ok, so consider what happens when this person does indeed attempt to construct a physical hula-hoop. After failing a few times, assuming he doesn’t give up altogether, he’ll be forced to accept (however provisinally) that pi is not 3, but approximately 3.14159265 (in our current physical reality, at least). He now has two conflicting models in his mind: one of an abstract hula-hoop made with pi == 3, and another one made with pi ~= 3.14159265. Which one will he “have a strong feeling / intuition / belief” about, do you think ?
I think he will have a strong feeling that pi is about 3.141… . Like I said, in my definition truth is subjective and may chance since it’s tied to the person’s beliefs / feelings. This may not seem beatiful to everyone, but I can live with that.
I think he will have a strong feeling that pi is about 3.141...
Why ?
Like I said, in my definition truth is subjective and may chance since it’s tied to the person’s beliefs / feelings.
Hmm, well, if you truly believe that truth is subjective, then there’s nothing I can do to dissuade you, by definition—since my subjective opinion is as good as yours. Now if you’ll excuse me, I’ve got to go build some hula-hoops, and then maybe take to the skies by will alone.
“Listen,” Darwin says, more kindly now, “I have a simple notion for resolving your dispute. You say,” says Darwin, pointing to Mark, “that people’s beliefs alter their personal realities. And you fervently believe,” his finger swivels to point at Autrey, “that Mark’s beliefs can’t alter reality. So let Mark believe really hard that he can fly, and then step off a cliff. Mark shall see himself fly away like a bird, and Autrey shall see him plummet down and go splat, and you shall both be happy.”
Doesn’t seem to apply here, because Randolf admits that reality doesn’t care what nonsense he believes. The only problem is he seems intent on describing that nonsense as ‘truth’ and refusing to label what it is that reality is doing, which is what everyone else is calling ‘truth’.
Hehe, I knew someone would pick up on my reference, I just didn’t realize how fast it would happen :-)
But my point was this: if Randolf really does believe that truth is subjective, and that it is arrived at mostly through feelings and intuitions, then he has effectively removed himself from rational debate. There’s nothing I can say that will persuade him one way or another, because there’s no useful mechanism by which my subjective beliefs can influence his subjective beliefs. So, there’s little point in arguing with him on this (or any other) topic.
Randolf, my apologies if I seem to be putting words in your mouth; the above paragraph is simply my personal interpretation of your claim, taken to its logical conclusion.
No, I think you understood pretty well what I meant. However, even though I may not be a rationalist myself, I think I can still take part in rational debate by embracing the definition of rational truth during that debate. Same way a true Christian can take part in a scientific debate about evolution, even if he doesn’t actually believe that evolution is true. Rational talk, just like any talking, can also change my feelings and intuitions and hence persuade me to change my subjective beliefs.
However, I now realise this wasn’t exactly the right place to tell about my idea of subjective truth. Sorry about that.
I think I can still take part in rational debate by embracing the definition of rational truth during that debate
I don’t think it will work in this case, because we’re debating the very notion of rational truth.
However, I now realise this wasn’t exactly the right place to tell about my idea of subjective truth.
I personally didn’t mean to give you that impression at all, I apologize if I did. Just because I happen to think that using reason to debate with someone who does not value reason is futile, doesn’t mean that I want to actively discourage such debate. After all, I could be wrong !
Hmm, well, if you truly believe that truth is subjective, then there’s nothing I can do to dissuade you, by definition—since my subjective opinion is as good as yours. Now if you’ll excuse me, I’ve got to go build some hula-hoops, and then maybe take to the skies by will alone
Oh, you probably could. I’m not so fond on this definition. It’s just something I have found most satisfying so far but it’s still subject to chance (How ironic!).
I think he will have a strong feeling that pi is about 3.141…
That’s the key issue. Reality is doing something here. And you know, in advance what his model will move to. You don’t think he will succeed at his event. At the end of the day, you are pretty sure that there’s something objective going on.
More starkly, I can give you mathematical examples where your intuition will be wildly at odds with the correct math. Some of those make fun games to play for money. I suspect that you won’t be willing to play them with me even if your intuition says that you should win and I shouldn’t.
That’s a bit differend from what I’m trying to say. My word choosing of intuition was clearly bad, I should have talked about mental experiences. My point is that when I do the mathematics, when I, for example, use the axioms and theorems of natural numbers to proof that 1+1 is 2, I have to rely on my memories and feelings at some point. If I use a theorem proven before, I must rely on my memories that I have proven that theorem before and correctly, but remembering is just another type of vaque mental experience. I could also remember axioms of natural numbers wrong, even if it would seem clear to me that I remember them correctly. I have to rely on the feeling of remembering correctly.
This is why I define truth as what you truly believe. Once you have carefully checked that you used all the axioms and theorems correctly, you will truly believe that you made no mistake. Then you can truly believe that 1 + 1 is 2, and it’s safe to say its the truth.
my beliefs are always the outputs of real-world embodied algorithms (for example, those associated with remembering previously proven axioms) and therefore not completely reliable.
there exists a non-empty set S1 of assertions that merit a sufficiently high degree of confidence that it is safe to call them “true” (while keeping in mind when it’s relevant that we mean “with probability 1-epsilon” rather than “with probability 1”).
I would also say that:
there exists a non-empty set S2 of assertions that don’t merit a high degree of confidence, and that it is not safe to call them true.
the embodied algorithms we use to determine our confidence in assertions are sufficiently unreliable that we sometimes possess a high degree of confidence in S2 assertions. This confidence is not merited, but we sometimes possess it nevertheless.
Would you agree with both of those statements?
Assuming you do, then it seems to follow that by “what I truly believe” you mean to exclude statements in S2. (Since otherwise, I could have a statement in S2 that I truly believe, and is therefore definitionally true, which is at the same time not safe to call true, which seems paradoxical.)
Assuming you do, then sure: if I accept that “what I truly believe” refers to S1 and not S2, then I agree that truth is what I truly believe, although that doesn’t seem like a terribly useful thing to know.
May I recommend “Godel, Escher, Bach” to you? It discusses the issue of what proof is at a rigorous but accessible level, including that a proof is just a well-formed finite string.
Yes, I believe that proof is just a well-formed finite string, but I take that a little bit futher because one can always ask that “what a well formed finite string is?”. Basically, I tell that person to use his honest intuition to check which things are “well-formed finite strings”.
Thesequestionshavesimpleanswers. Please explain what part of carrying out a proof-checking procedure—which can be by hand if need be—requires intuition.
I am not Randolf, but I’ve met people who would answer this question thusly:
Ultimately, you are still relying on faith, intuition, or some other objective criterion in order to construct all of these logical proofs. I could choose different axioms and construct some proofs of my own, which would differ from yours. Furthermore, the very value you place on axioms and logic is subjective; I, on the other hand, place a much higher value on feelings and intuitions. Therefore, even though your arguments may be entirely logical and therefore important in your subjective worldview, they hold very little value in mine (though the reverse is also true).
I don’t think it’s possible to use logic to convince someone of the importance of logic, unless he happens to be convinced already.
You can use naive logic to convince people of the importance of more rigorous logic, though, and I suspect that most of the people decrying logic, axiomatic systems, etc. aren’t objecting to reasoning in general so much as certain levels of formality, or certain attitudes surrounding them. I’ve met a lot of people claiming to put more stock in gut feelings than clever reasoning, but I’ve never met one such that didn’t have a handy store of justifications for their beliefs—which seems to point to a certain trust even if it’s unacknowledged.
I suspect that most of the people decrying logic, axiomatic systems, etc. aren’t objecting to reasoning in general so much as certain levels of formality...
Or, in my experience, specific topics. For example, such a person would say that reasoning does apply to topics such as deciding which car to buy, or which stock to invest to, or what the sum of the angles in a triangle is. Reasoning does not, however, apply to other topics such as deciding what to eat for lunch, which deity to worship (if any), whom to date, and which topics are subject to reason in the first place.
The above is a real example, BTW (assuming I understood the person’s position correctly).
(nods) I generally summarize this as “reason is useful only for those topics where I’m confident I’m right or am willing to be corrected if wrong.” To which my response is typically “how very convenient for you that it works out that way.”
Yes, I wouldn’t have bothered if he had said something like that; the thing is from the above that didn’t seem to be the objection he was making. Since he now says it essentially is, I think I’ll step out of this argument. (Well, the first two sentences are easily answerable, but I’ll let someone else do that if they really want.) Also apparently by “intuition is required”, he means “brains cannot carry out an algorithm 100% reliably, and 100% reliability is required (or something like that)”. Which would I suppose make him the first person I’ve heard to actually (effectively) endorse “ordinary person reasoning”, where only chains of reasoning of a bounded (and very short!) length are valid! (I seem to recall this being discussed somewhere here before… can’t find it right now, though.) Anyway, I won’t bother commenting on this any further.
Yes, that’s pretty much what I would say. Also, a simple answer to the question would also be:
At least the part where you use feelings to verify you didn’t make an error. After writing the proof, you remember that you checked every part carefully that you didn’t make an error. But this remembering is a mere feeling.
My world view used to be differend until I read the following pharse somewhere. That moment I realised I can only be as sure as my feelings let me.
Not even mathematical facts necessarily hold since there could always be a magical demon blurring your mind, making you make errors and making you blind at them.
I still have a great interest in mathematics and am hoping my studies and everything goes well so I can bear the title of mathematican one day. Maybe my beliefs change when I get less green.
Not even mathematical facts necessarily hold since there could always be a magical demon blurring your mind, making you make errors and making you blind at them.
That’s a much weaker statement than the one you originally stated. This new statement says, basically, “you can never be 100% sure of anything”, whereas before you seemed to be saying, “there exist no objective standards of truth at all, any story is as good as any other”.
Whetever it is a weaker statement or not isn’t the point. I only brought it up because it made me change the way I think about mathematics and the world.
While I don’t know what you mean by “any story is as good as any other”, I do not believe that it is possible to give truth a honest definition which would leave no open questions about the very nature of truth, while still being entirely objective.
While I don’t know what you mean by “any story is as good as any other”
Well, let’s say I believe that I can fly by will alone. You, on the other hand, believe that I cannot fly by will alone. Which one of us is right ? If truth is entirely subjective, then we’re both “right”, in the sense that we both have some sort of a story in our heads regarding flight, and in our respective worldviews this story makes perfect sense, and since there’s no objective standard for truth (at least, none that we can access in any way), the stories are all that matters. Thus, all stories are equally true, just by the virtue of being stories.
According to a weaker interpretation of your statements, however, one of us is probably closer to the truth than the other. More specifically, it is very likely that my belief about my ability to fly by will alone is false. It’s still not 100% likely, of course—there’s always that chance that we live in the Matrix, or that I’m a superhero, or that by “flight” I really mean “pretending to fly without physically moving”, etc. -- but such chances quite small. Thus, for all practical purposes, we can say, “Bugmaster’s belief about flight is false”, with the understanding that we can never be 100% sure.
There could be other interpretations of your claims, of course; these are just the two I could come up with. I could support the second interpretation, though whether it applies to math or not is highly debatable. However, if you support the first interpretation, or if you don’t place any significant value on reason, then any further discussion on the topic is pointless—because, by definition, there’s nothing I can say that will make any difference to you.
Checking whether mathematical rules are satisfied does not require intuition; it can be done by a computer program (and often is).
Someone doing that still puts faith on the computer, and the person who made the computer program to check the rules. Essentially, he has strong feeling that A holds because the computer program said so. He still has to rely on his “intuition” or “belief” that the computer program gives true statements.
Some people (mostly young children, though some adults as well) believe that the ratio of a circle’s circumference to its diameter should be an integer, or at worst a rational fraction. Most other people, however, do not believe this to be the case.
If mathematical truths are subjective as you claim, then a person who believes that pi == 3 should be able to build himself a 5-foot wide hula-hoop using exactly 15 feet of plastic tubing. Do you think this is actually the case ?
Maybe he is able to construct some sort of an abstract hula-hoop in his mind, which he believes to have those properties, but of course he isn’t able to do it in the physical reality. Strong intuition suggests that it isn’t possible.
However, we should not forgot that mathematical models of physical reality and mathematics itself are separate things. We can use mathematics to understand nature, but nature cares very little about anyones mathematical truths. Well, I think it’s safe to say so anyway.
Ok, so consider what happens when this person does indeed attempt to construct a physical hula-hoop. After failing a few times, assuming he doesn’t give up altogether, he’ll be forced to accept (however provisinally) that pi is not 3, but approximately 3.14159265 (in our current physical reality, at least). He now has two conflicting models in his mind: one of an abstract hula-hoop made with pi == 3, and another one made with pi ~= 3.14159265. Which one will he “have a strong feeling / intuition / belief” about, do you think ?
I think he will have a strong feeling that pi is about 3.141… . Like I said, in my definition truth is subjective and may chance since it’s tied to the person’s beliefs / feelings. This may not seem beatiful to everyone, but I can live with that.
Why ?
Hmm, well, if you truly believe that truth is subjective, then there’s nothing I can do to dissuade you, by definition—since my subjective opinion is as good as yours. Now if you’ll excuse me, I’ve got to go build some hula-hoops, and then maybe take to the skies by will alone.
Doesn’t seem to apply here, because Randolf admits that reality doesn’t care what nonsense he believes. The only problem is he seems intent on describing that nonsense as ‘truth’ and refusing to label what it is that reality is doing, which is what everyone else is calling ‘truth’.
Hehe, I knew someone would pick up on my reference, I just didn’t realize how fast it would happen :-)
But my point was this: if Randolf really does believe that truth is subjective, and that it is arrived at mostly through feelings and intuitions, then he has effectively removed himself from rational debate. There’s nothing I can say that will persuade him one way or another, because there’s no useful mechanism by which my subjective beliefs can influence his subjective beliefs. So, there’s little point in arguing with him on this (or any other) topic.
Randolf, my apologies if I seem to be putting words in your mouth; the above paragraph is simply my personal interpretation of your claim, taken to its logical conclusion.
No, I think you understood pretty well what I meant. However, even though I may not be a rationalist myself, I think I can still take part in rational debate by embracing the definition of rational truth during that debate. Same way a true Christian can take part in a scientific debate about evolution, even if he doesn’t actually believe that evolution is true. Rational talk, just like any talking, can also change my feelings and intuitions and hence persuade me to change my subjective beliefs.
However, I now realise this wasn’t exactly the right place to tell about my idea of subjective truth. Sorry about that.
I don’t think it will work in this case, because we’re debating the very notion of rational truth.
I personally didn’t mean to give you that impression at all, I apologize if I did. Just because I happen to think that using reason to debate with someone who does not value reason is futile, doesn’t mean that I want to actively discourage such debate. After all, I could be wrong !
Yes, I agree, it doesn’t work on this case. It was an interesting talk though, thank you for that. Now I must sleep over this..
Oh, you probably could. I’m not so fond on this definition. It’s just something I have found most satisfying so far but it’s still subject to chance (How ironic!).
That’s the key issue. Reality is doing something here. And you know, in advance what his model will move to. You don’t think he will succeed at his event. At the end of the day, you are pretty sure that there’s something objective going on.
More starkly, I can give you mathematical examples where your intuition will be wildly at odds with the correct math. Some of those make fun games to play for money. I suspect that you won’t be willing to play them with me even if your intuition says that you should win and I shouldn’t.
That’s a bit differend from what I’m trying to say. My word choosing of intuition was clearly bad, I should have talked about mental experiences. My point is that when I do the mathematics, when I, for example, use the axioms and theorems of natural numbers to proof that 1+1 is 2, I have to rely on my memories and feelings at some point. If I use a theorem proven before, I must rely on my memories that I have proven that theorem before and correctly, but remembering is just another type of vaque mental experience. I could also remember axioms of natural numbers wrong, even if it would seem clear to me that I remember them correctly. I have to rely on the feeling of remembering correctly. This is why I define truth as what you truly believe. Once you have carefully checked that you used all the axioms and theorems correctly, you will truly believe that you made no mistake. Then you can truly believe that 1 + 1 is 2, and it’s safe to say its the truth.
FWIW: I agree with you that:
my beliefs are always the outputs of real-world embodied algorithms (for example, those associated with remembering previously proven axioms) and therefore not completely reliable.
there exists a non-empty set S1 of assertions that merit a sufficiently high degree of confidence that it is safe to call them “true” (while keeping in mind when it’s relevant that we mean “with probability 1-epsilon” rather than “with probability 1”).
I would also say that:
there exists a non-empty set S2 of assertions that don’t merit a high degree of confidence, and that it is not safe to call them true.
the embodied algorithms we use to determine our confidence in assertions are sufficiently unreliable that we sometimes possess a high degree of confidence in S2 assertions. This confidence is not merited, but we sometimes possess it nevertheless.
Would you agree with both of those statements?
Assuming you do, then it seems to follow that by “what I truly believe” you mean to exclude statements in S2. (Since otherwise, I could have a statement in S2 that I truly believe, and is therefore definitionally true, which is at the same time not safe to call true, which seems paradoxical.)
Assuming you do, then sure: if I accept that “what I truly believe” refers to S1 and not S2, then I agree that truth is what I truly believe, although that doesn’t seem like a terribly useful thing to know.
Yes, I think you managed to put my thoughts into words very well here. Probably a lot more clearly than I.
May I recommend “Godel, Escher, Bach” to you? It discusses the issue of what proof is at a rigorous but accessible level, including that a proof is just a well-formed finite string.
Yes, I believe that proof is just a well-formed finite string, but I take that a little bit futher because one can always ask that “what a well formed finite string is?”. Basically, I tell that person to use his honest intuition to check which things are “well-formed finite strings”.
These questions have simple answers. Please explain what part of carrying out a proof-checking procedure—which can be by hand if need be—requires intuition.
I am not Randolf, but I’ve met people who would answer this question thusly:
I don’t think it’s possible to use logic to convince someone of the importance of logic, unless he happens to be convinced already.
You can use naive logic to convince people of the importance of more rigorous logic, though, and I suspect that most of the people decrying logic, axiomatic systems, etc. aren’t objecting to reasoning in general so much as certain levels of formality, or certain attitudes surrounding them. I’ve met a lot of people claiming to put more stock in gut feelings than clever reasoning, but I’ve never met one such that didn’t have a handy store of justifications for their beliefs—which seems to point to a certain trust even if it’s unacknowledged.
Or, in my experience, specific topics. For example, such a person would say that reasoning does apply to topics such as deciding which car to buy, or which stock to invest to, or what the sum of the angles in a triangle is. Reasoning does not, however, apply to other topics such as deciding what to eat for lunch, which deity to worship (if any), whom to date, and which topics are subject to reason in the first place.
The above is a real example, BTW (assuming I understood the person’s position correctly).
(nods) I generally summarize this as “reason is useful only for those topics where I’m confident I’m right or am willing to be corrected if wrong.” To which my response is typically “how very convenient for you that it works out that way.”
Yes, I wouldn’t have bothered if he had said something like that; the thing is from the above that didn’t seem to be the objection he was making. Since he now says it essentially is, I think I’ll step out of this argument. (Well, the first two sentences are easily answerable, but I’ll let someone else do that if they really want.) Also apparently by “intuition is required”, he means “brains cannot carry out an algorithm 100% reliably, and 100% reliability is required (or something like that)”. Which would I suppose make him the first person I’ve heard to actually (effectively) endorse “ordinary person reasoning”, where only chains of reasoning of a bounded (and very short!) length are valid! (I seem to recall this being discussed somewhere here before… can’t find it right now, though.) Anyway, I won’t bother commenting on this any further.
Oh, I found it. It wasn’t a discussion here, it was a post on Scott Aaronson’s blog: http://www.scottaaronson.com/blog/?p=232
Yes, that’s pretty much what I would say. Also, a simple answer to the question would also be:
My world view used to be differend until I read the following pharse somewhere. That moment I realised I can only be as sure as my feelings let me.
I still have a great interest in mathematics and am hoping my studies and everything goes well so I can bear the title of mathematican one day. Maybe my beliefs change when I get less green.
That’s a much weaker statement than the one you originally stated. This new statement says, basically, “you can never be 100% sure of anything”, whereas before you seemed to be saying, “there exist no objective standards of truth at all, any story is as good as any other”.
Whetever it is a weaker statement or not isn’t the point. I only brought it up because it made me change the way I think about mathematics and the world. While I don’t know what you mean by “any story is as good as any other”, I do not believe that it is possible to give truth a honest definition which would leave no open questions about the very nature of truth, while still being entirely objective.
Well, let’s say I believe that I can fly by will alone. You, on the other hand, believe that I cannot fly by will alone. Which one of us is right ? If truth is entirely subjective, then we’re both “right”, in the sense that we both have some sort of a story in our heads regarding flight, and in our respective worldviews this story makes perfect sense, and since there’s no objective standard for truth (at least, none that we can access in any way), the stories are all that matters. Thus, all stories are equally true, just by the virtue of being stories.
According to a weaker interpretation of your statements, however, one of us is probably closer to the truth than the other. More specifically, it is very likely that my belief about my ability to fly by will alone is false. It’s still not 100% likely, of course—there’s always that chance that we live in the Matrix, or that I’m a superhero, or that by “flight” I really mean “pretending to fly without physically moving”, etc. -- but such chances quite small. Thus, for all practical purposes, we can say, “Bugmaster’s belief about flight is false”, with the understanding that we can never be 100% sure.
There could be other interpretations of your claims, of course; these are just the two I could come up with. I could support the second interpretation, though whether it applies to math or not is highly debatable. However, if you support the first interpretation, or if you don’t place any significant value on reason, then any further discussion on the topic is pointless—because, by definition, there’s nothing I can say that will make any difference to you.