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.
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.