The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 12 May 2010 23:04 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> So everything boils down to define term such as "intuition" in a >>>> technical debate? Great! >>> Great! But as you have told us, anything and everything except knowledge >>> of specific formal derivations is "intuitive". >> So, what would you call reasoning by rules of inference? Counter intuitive >> reasoning? > > Just as worthwhile as the rules of inference and axioms in question are -- > not to be trusted on the sole grounds that there is a formal system; > and so open to dispute. Just like reasoning about natural numbers. > > Why should I think that every statement of the form "A or not A" > is true? Do you think that all such statements are true? Not sure why you're asking me all those questions. My "attack" here is that the definition of what'd constitute a truth value for a formula is vague, which I'm not sure in what way it has anything to do with your questions. > You said that LEM "appears to be true" IIRC. If my memory isn't bad I think that was about that if there's a context in which a formula is truth assigned-able then LEM would be applicable. But there are cases where you can't make the truth assigment to the formula since there's no firm context. In other word there's no absolute truth. So I'm not sure I ever said LEM would be _always_ applicable. >
From: Nam Nguyen on 13 May 2010 00:59 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> So everything boils down to define term such as "intuition" in a >>>> technical debate? Great! >>> Great! But as you have told us, anything and everything except knowledge >>> of specific formal derivations is "intuitive". >> So, what would you call reasoning by rules of inference? Counter intuitive >> reasoning? > > Just as worthwhile as the rules of inference and axioms in question are -- > not to be trusted on the sole grounds that there is a formal system; > and so open to dispute. My English comprehension is weak here in that I'm not sure what you'd like to convey. Could you clarify your point? > Just like reasoning about natural numbers. Reasoning about the naturals numbers on their own merit is actually NOT a logical reasoning. What is said in Shoenfield's "Mathematical Logic": "The conspicuous feature of mathematics, as opposed to other sciences, is the use of proofs instead of observations." "A mathematician may, on occasions, use observation; for example, he may measure the angles of many triangles and conclude the sum of the angles is always 180 [degree]. However, he will accept this as a law of mathematics only when it has been proved." The "intuition", the "truth", about the naturals numbers is no more worthy of reasoning than "measure", or "observation" that is mentioned in this book, and which shouldn't be the basis for reasoning.
From: herbzet on 13 May 2010 01:34 Nam Nguyen wrote: > In other word there's no absolute truth. That's absolutely true. -- hz
From: Nam Nguyen on 13 May 2010 01:55 herbzet wrote: > > Nam Nguyen wrote: > >> In other word there's no absolute truth. > > That's absolutely true. Relatively speaking.
From: Nam Nguyen on 13 May 2010 03:00
Nam Nguyen wrote: > herbzet wrote: >> >> Nam Nguyen wrote: >> >>> In other word there's no absolute truth. >> >> That's absolutely true. > > Relatively speaking. All truths are relative, including this truth. (Kind of rings the bell, doesn't it?) |