'When Are Relations Neither True Nor False?
in [Math]
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From: Alan Smaill on 11 Apr 2010 19:10 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > 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. I'm not addressing that question. > 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. What I'm querying is your claim that your objection to reductio ad absurdum is based on a failure of *symmetry*. You say "if we could start with not(P), we could start with P", as though there is a problem with starting with P in particular. If "P" is problematic (non tautology, non contradictory), then I expect "not P" to be likewise problematic, by symmetry, and your principle of symmetry is not broken -- you reject both sorts of argument, I take it. So my question remains: what symmetry do you think is broken? -- Alan Smaill
From: Marshall on 11 Apr 2010 19:22 On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > 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? I reject the idea that the naturals are anything particular *to me* that they are not to anyone else. Other than that, and as much as I hesitate to accede to any formula of yours given how unreliably you use terminology, my answer is "yes." We can completely capture enough about the naturals to uniquely characterize them up to isomorphism. For example, we have many syntactic representations of natural numbers available to us that can represent any natural up to resource limits, and we have simple algorithms on those representations that can compute successor, addition, etc. of any naturals up to resource limits. These things are entirely mechanical, and free of any vague or intuitive aspects. I am under no delusions that you will agree with me, but please spare me the umpteenth repetition of your GC counterargument. Marshall
From: Nam Nguyen on 11 Apr 2010 19:52 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> 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. > > I'm not addressing that question. > >> 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. > > What I'm querying is your claim that your objection to reductio ad > absurdum is based on a failure of *symmetry*. You say "if we could > start with not(P), we could start with P", as though there is a problem > with starting with P in particular. > > If "P" is problematic (non tautology, non contradictory), > then I expect "not P" to be likewise problematic, by symmetry, > and your principle of symmetry is not broken -- > you reject both sorts of argument, I take it. > > So my question remains: what symmetry do you think is broken? I don't think you've paid attention to the conversation. Whatever I had said here that your question was about has a context: my responding to DB's alluded a formula being _absolutely true_. And my point here is that he (or any of us) could forget about a formula [neg(P) or P] being absolutely true, since that would break symmetry: the Principle would stipulate both formulas must necessarily be of relative truth, to be symmetrical in reasoning! To claim some formula being absolutely true (or false) is to destroy the notion of such symmetry.
From: Nam Nguyen on 11 Apr 2010 20:09 Marshall wrote: > On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> 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? > > I reject the idea that the naturals are anything particular *to me* > that they are not to anyone else. The question was simply asking you to explain from your mathematical knowledge what the natural number be. That's all it was asked of you. Whether your concept of the naturals is the same as mine, or TF's, or anybody else' isn't relevant to the question. Can you share with us your resounding, firm, answer to that simple question? After all, you seem to have said you know the naturals number well enough to reject the idea they're of an intuition notion! > > Other than that, and as much as I hesitate to accede to any > formula of yours given how unreliably you use terminology, > my answer is "yes." We can completely capture enough > about the naturals to uniquely characterize them up to > isomorphism. For example, we have many syntactic > representations of natural numbers available to us that > can represent any natural up to resource limits, and we > have simple algorithms on those representations that > can compute successor, addition, etc. of any naturals > up to resource limits. These things are entirely > mechanical, and free of any vague or intuitive aspects. I'm sorry: "up to resource limits" isn't an intuition or technical notion of the natural numbers. So you don't seem to understand the notion of the natural numbers, despite your having claimed otherwise. > > I am under no delusions that you will agree with me, > but please spare me the umpteenth repetition of your > GC counterargument. Until you demonstrate precisely what you meant by the naturals, while asserting they're not of intuition notion, I wouldn't have a reason not to repeat my GC counter argument.
From: Alan Smaill on 11 Apr 2010 20:15
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> 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. >> >> I'm not addressing that question. >> >>> 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. >> >> What I'm querying is your claim that your objection to reductio ad >> absurdum is based on a failure of *symmetry*. You say "if we could >> start with not(P), we could start with P", as though there is a problem >> with starting with P in particular. >> >> If "P" is problematic (non tautology, non contradictory), >> then I expect "not P" to be likewise problematic, by symmetry, >> and your principle of symmetry is not broken -- >> you reject both sorts of argument, I take it. >> >> So my question remains: what symmetry do you think is broken? > > I don't think you've paid attention to the conversation. > Whatever I had said here that your question was about has a context: > my responding to DB's alluded a formula being _absolutely true_. I'm simply not addressing the whole conversation. > And my point here is that he (or any of us) could forget about a formula > [neg(P) or P] being absolutely true, since that would break symmetry: the > Principle would stipulate both formulas must necessarily be of relative > truth, to be symmetrical in reasoning! I just agreed in my previous post with the point that symmetry says both are of relative truth, by symmetry. > To claim some formula being > absolutely true (or false) is to destroy the notion of such symmetry. Do you agree that symmetry is only broken if the cases "P" and "not P" are treated differently? -- Alan Smaill |