The End of an Era - That of The Natural Numbers
in [Math]
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From: Alan Smaill on 16 May 2010 16:45 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > 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? You make a big distinction between meta-theoretic results (which are apparently OK to cite as evidence, eg relationship between decidability and truth of purely universal mathematical statements); while objecting to statements of basic arithmetic properties. In one case, you trust the traditional apparatus, in the other you object to just about everything. Why the big difference? >> Just like reasoning about natural numbers. > > Reasoning about the naturals numbers on their own merit is actually NOT > a logical reasoning. not *just* logical. > 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." Fine. But still he has axioms for arithmetic; where do they come from? In fact, he has more than one axiomatisation. Are these random sets of formulas, perhaps? > 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. This applies just as much to the meta-reasoning itself that Shoenfield uses thoughout. -- Alan Smaill
From: J. Clarke on 17 May 2010 09:13 On 5/16/2010 11:35 AM, Nam Nguyen wrote: > J. Clarke wrote: >> On 5/16/2010 2:50 AM, Nam Nguyen wrote: >>> J. Clarke wrote: >>> >>>> >>>> Have you ever taken an abstract algebra course? If not you might want >>>> to. After you have completed it you should understand how vacuous your >>>> whole line of argument is. >>> >>> Like you could technically demonstrate how abstract algebra would >>> lead to the conclusion our knowledge of the naturals is not just >>> an intuition! (Iow, if you really knew what you were talking about, >>> you utterance above doesn't show it!) >>> >>> [Btw, I took 2 semesters of Abstract Algebra as an undergraduate math >>> major.] >> >> And yet you continue to blather about how "our knowledge of the >> naturals is just an intuition". >> >> Define the naturals and the operations on them in terms of what you >> learned in those classes (assuming you learned anything) and then get >> back to us on how it's "just an intuition". >> > > You're clueless in the matter of abstract algebra: the first few chapters > and the key subject of discussion in abstract algebra are about the concept > of a group. And the naturals as you yourself intuit it is NOT a group. One would have hoped that in two semesters you'd have gone beyond the first few chapters. But apparently not. There's a whole world there beyond groups. > Jesuz! You're lecturing people about the knowledge of natural numbers and > abstract algebra? It's a dirty job but somebody needs to do it.
From: Nam Nguyen on 18 May 2010 01:18 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> 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? > > You make a big distinction between meta-theoretic results > (which are apparently OK to cite as evidence, eg relationship > between decidability and truth of purely universal mathematical > statements); while objecting to statements of basic arithmetic > properties. In one case, you trust the traditional apparatus, > in the other you object to just about everything. Why the big difference? In a nutshell, it has something to do with the essence of what President Reagan once said: "trust but verify". In _all_ cases of reasoning through rules of inference you could verify the agreed correctness of reasoning: by finite proofs - no trust is required. In some critical cases of reasoning about certain infinite concepts, such as in basic arithmetic, there's only trust and no verification is possible. > >>> Just like reasoning about natural numbers. >> Reasoning about the naturals numbers on their own merit is actually NOT >> a logical reasoning. > > not *just* logical. > >> 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." > > Fine. > > But still he has axioms for arithmetic; > where do they come from? In fact, he has more than one axiomatisation. > Are these random sets of formulas, perhaps? The axiom systems for arithmetic come from our intuitive concepts of arithmetic and they (the systems) are formal (syntactical) descriptions of the arithmetic concepts, and as such there might be more than one system describing about the same set of concepts. So no, they're not random in that sense. But what would all that have to do with the certainty of syntactical proof as a verification tool and with arithmetic concept as an intuitive tool and not verification tool, in reasoning? > >> 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. > > This applies just as much to the meta-reasoning itself that Shoenfield > uses thoughout. Perhaps some parts of his book, but in general not every meta reasoning is based on untrusted knowledge of the naturals.
From: Nam Nguyen on 18 May 2010 01:33 Daryl McCullough wrote: > Nam Nguyen says... > >> Btw, that's not my definition as you mentioned above. Although he >> used a slightly different word "valid", the definition could be >> found in Shoenfield's book: >> >> "A formula is valid in T if it is valid in every model of T" >> (Pg. 22) > > I think you misquoted. Isn't it "A formula is valid in T if it is *true* > in every model of T (or "is satisfied by every model of T"). Why don't you get a copy of his book and verify for yourself whether or not I had misquoted him. (I read his sentence there word by word I could see a misquote). > > Anyway, in light of the completeness theorem, that definition > is equivalent to "A formula is valid in T if it is provable from > formulas in T". So any formula provable from pure first-order logic > is valid in every theory. > >> Again, hopefully by now you understand x=x is false in my inconsistent >> T. > > No, that's wrong. Before you say "wrong" or "right" here, what is your definition of a formula being true or false in an inconsistent formal system?
From: Nam Nguyen on 18 May 2010 01:34
Nam Nguyen wrote: > Daryl McCullough wrote: >> Nam Nguyen says... >> >>> Btw, that's not my definition as you mentioned above. Although he >>> used a slightly different word "valid", the definition could be >>> found in Shoenfield's book: >>> >>> "A formula is valid in T if it is valid in every model of T" >>> (Pg. 22) >> >> I think you misquoted. Isn't it "A formula is valid in T if it is *true* >> in every model of T (or "is satisfied by every model of T"). > > Why don't you get a copy of his book and verify for yourself whether > or not I had misquoted him. (I read his sentence there word by word > I could see a misquote). I meant "I couldn't see a misquote" > >> >> Anyway, in light of the completeness theorem, that definition >> is equivalent to "A formula is valid in T if it is provable from >> formulas in T". So any formula provable from pure first-order logic >> is valid in every theory. >> >>> Again, hopefully by now you understand x=x is false in my inconsistent >>> T. >> >> No, that's wrong. > > Before you say "wrong" or "right" here, what is your definition of a > formula > being true or false in an inconsistent formal system? |