'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 11 Apr 2010 18:17 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> David Bernier wrote: > >>> Do you see problems with starting with a false premise, not(P) ? >> Yes. I'd break "the Principle of Symmetry": if we could start with not(P), >> we could start with P. > > And so you could, so that's not a problem. > > But what would you prove? > Assuming you understand I said 'It'd break "the Principle of Symmetry"...' and assuming you meant such a symmetry breaking isn't a problem, then a) one (you or I for example) would prove ordinary mathematical theorems through rules of inference, as may people have been able to do so; b) the Principle is actually not about what you can or can't know how to prove through rules of inference. So your question here is moot; c) in light of rules of inference and syntactical proofs are finite, it's an incorrect attitude, as people in your school of thought seem to have, to insist the knowledge required in reasoning is entitlement-based instead of endowment-based. We're human beings with ability to comprehend only finite mathematical properties, not god-like super beings who would comprehend "omega" properties, such as the syntactical consistency of PA. Frankly speaking, to think otherwise is just an illusion.
From: Nam Nguyen on 11 Apr 2010 18:19 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> David Bernier wrote: > >>> Do you see problems with starting with a false premise, not(P) ? >> Yes. I'd break "the Principle of Symmetry": if we could start with not(P), >> we could start with P. > > And so you could, so that's not a problem. > > But what would you prove? > Assuming you understand I said 'It'd break "the Principle of Symmetry"...' and assuming you meant such a symmetry breaking isn't a problem, then a) one (you or I for example) would prove ordinary mathematical theorems through rules of inference, as may people have been able to do so; b) the Principle is actually not about what you can or can't know how to prove through rules of inference. So your question here is moot; c) in light of rules of inference and syntactical proofs are finite, it's an incorrect attitude, as people in your school of thought seem to have, to insist the knowledge required in reasoning is entitlement-based instead of endowment-based. We're human beings with ability to comprehend only finite mathematical properties, not god-like super beings who would comprehend "omega" properties, such as the syntactical consistency of PA. Frankly speaking, to think otherwise is just an illusion.
From: Marshall on 11 Apr 2010 18:37 On Apr 11, 2:46 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Apr 11, 8:25 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > >> Marshall <marshall.spi...(a)gmail.com> writes: > >>> I don't see any reason to pay much attention to anyone's > >>> intuition, my own included. > >> This is a sober attitude. > > > I am mostly a sober person, in that I am drunk less > > than half the time. > > >>> "Intuition" is just a fancy word for "hunch." > >> "Intuition" can mean pretty much anything, from a vague hunch to > >> something very specific, as in e.g. Kant's thought. > > > Indeed so, which is exactly what makes it a poor choice > > when used in contexts such as this newsgroup. > > Then, it also looks like a poor choice of using the _intuition_ about > the naturals as a foundation of reasoning, as the school of thought AK, > TF seem to have subscribed to, would suggest. I reject your claim that intuition, of whatever kind, plays any part in our thinking about the naturals. Marshall
From: Nam Nguyen on 11 Apr 2010 18:51 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Nam Nguyen wrote: >>> Alan Smaill wrote: >>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>>> >>>>> David Bernier wrote: >>>>>> Do you see problems with starting with a false premise, not(P) ? >>>>> Yes. I'd break "the Principle of Symmetry": if we could start with >>>>> not(P), >>>>> we could start with P. >>>> And so you could, so that's not a problem. >>>> >>>> But what would you prove? >>>> >>> It was a typo, I meant "It'd the Principle of Symmetry". >>> Would you still have any question then? >> I meant "It'd break the Principle of Symmetry". > > How? > > You *can* start by supposing P; > and if you can derive a contradiction, you get a proof > on "not P". > > What's the symmetry that's broken, according to you? Let's recall this conversation started by David Bernier's suggestion to a way of obtaining an absolute truth, in responding to my suggestion the nature of mathematical reasoning is subjective and relative. Let's also recall that the Principle as suggested in this thread to safeguard against _incorrect assumption_ of an absolute truth of ANY non-tautologous, non-contradictory formulas. Choosing a formula to be absolutely true is to break this Principle's safeguard of correcting reasoning.
From: Nam Nguyen on 11 Apr 2010 18:54
Marshall wrote: > On Apr 11, 2:46 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Apr 11, 8:25 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >>>> Marshall <marshall.spi...(a)gmail.com> writes: >>>>> I don't see any reason to pay much attention to anyone's >>>>> intuition, my own included. >>>> This is a sober attitude. >>> I am mostly a sober person, in that I am drunk less >>> than half the time. >>>>> "Intuition" is just a fancy word for "hunch." >>>> "Intuition" can mean pretty much anything, from a vague hunch to >>>> something very specific, as in e.g. Kant's thought. >>> Indeed so, which is exactly what makes it a poor choice >>> when used in contexts such as this newsgroup. >> Then, it also looks like a poor choice of using the _intuition_ about >> the naturals as a foundation of reasoning, as the school of thought AK, >> TF seem to have subscribed to, would suggest. > > I reject your claim that intuition, of whatever kind, plays > any part in our thinking about the naturals. So is the naturals collectively a finite syntactical notion *to you*, since you'd reject the idea they're an intuition notion? |