The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 2 May 2010 11:46 Daryl McCullough wrote: > William Hughes says... > >> Clearly. I do not understand what is meant by >> >> "There are infinitely many examples of GC" >> >> I take it to mean there are infinitely many even >> integers greater than 4 that are the sum of two primes. >> This follows immediately from the fact that there are >> an infinite number of primes (just add 3 to every prime, >> if the prime is not 2 you get an x for which GC(x) is true). >> What do you mean by the statement? > > You're completely missing the point by bringing up actual > mathematics. This is logic! First of all there was already corrections in the arguments about "There are infinitely many examples of GC" so whatever you intended by referencing the quoted is late and rather misleading, if not irrelevant at this point. Secondly, what do _you_ think "actual mathematics" be, when you can't define it or don't know (and look you'd never know) the truth of *only a handful _real_ mathematical formulas* such as (1) or the like? In the name of "actual mathematics" the cranks have brought in many "rational" conclusions, have they not?
From: Nam Nguyen on 2 May 2010 12:00 Nam Nguyen wrote: > Daryl McCullough wrote: >> William Hughes says... >> >>> Clearly. I do not understand what is meant by >>> >>> "There are infinitely many examples of GC" >>> >>> I take it to mean there are infinitely many even >>> integers greater than 4 that are the sum of two primes. >>> This follows immediately from the fact that there are >>> an infinite number of primes (just add 3 to every prime, >>> if the prime is not 2 you get an x for which GC(x) is true). >>> What do you mean by the statement? >> >> You're completely missing the point by bringing up actual >> mathematics. This is logic! > > First of all there was already corrections in the arguments about > "There are infinitely many examples of GC" so whatever you intended > by referencing the quoted is late and rather misleading, if not > irrelevant at this point. > > Secondly, what do _you_ think "actual mathematics" be, when you > can't define it or don't know (and look you'd never know) the > truth of *only a handful _real_ mathematical formulas* such as (1) > or the like? In the name of "actual mathematics" the cranks have > brought in many "rational" conclusions, have they not? Also, I suppose many decades ago truth-equals-provability mathematics was the "actual mathematics" of the time. A kind of "soup-du-jour mathematics" isn't it? [Actually Shoenfield gave a hint this was (and is) the case!]
From: Alan Smaill on 2 May 2010 12:14 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > William Hughes wrote: > >> >> If we do not assume T is sound, then there is no connection >> between provable and true. Now we have to phrase things: >> (1) is true iff cGC is false. > > >> However, "cGC is false" is a >> perfectly reasonable intuition, so even if we don't assume >> T is sound, observation 1 is false. > > I'm sorry: that simply doesn't cut it. Anyone else could equally > say "cGC is true" is a perfectly reasonable intuition! So who > would be correct: you or they? People differe on their intuitions; surely you admit that? And diverging intuitions can both be reasonable. > And why would that _intuition_ > be correct at the expense of others? Why is *your* claim about intuitions on arithemetic better than other people's? -- Alan Smaill email: A.Smaill at ed.ac.uk
From: Alan Smaill on 2 May 2010 12:28 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > In addition I already gave a caveat: > >>>> It's not true such imprecise knowledge of the naturals would >>>> mean we know nothing about them: we do know know some formula >>>> being true. It's just that we wouldn't be able know the truth >>>> status of all formulas! > > which means by standard acceptance of mathematicians and logicians > the truth of SOME formulas should be in such knowledge, such as the > non-induction axioms of PA, which means even by the vague notion > of "intuition" in this context we don't have a carte-blanche to > claim anything whatsoever. [Schoenfield's book implicitly suggests > some formulas that it'd be questionable whether or not they'd be > true in the (purported) natural numbers]. > > In summary, you and I each should give, as William Elliot suggested, > "insights" to _defend_ our different positions on this issue about > the knowledge of the naturals. And though "insight" is not a precise > word, that doesn't at all mean unreasonable insights such as there > are only finite number of the naturals could be used to further > the arguments. Hmm, calling those who disagree unreasonable is a tactic that other have used before. "Only those in the school of Nam may enter here!" Are all (arithmetical) instances of the law of the excluded middle true, in your opinion? If so, why? -- Alan Smaill
From: Alan Smaill on 2 May 2010 12:31
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Also, I suppose many decades ago truth-equals-provability mathematics > was the "actual mathematics" of the time. A kind of > "soup-du-jour mathematics" isn't it? > [Actually Shoenfield gave a hint this was (and is) the case!] Given his acceptance of G�del's incompleteness theorem, your claim is implausible. Which "hint" of his do you have in mind? -- Alan Smaill |