Cantor's Diagonal?
in [Math]
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From: Nam Nguyen on 10 Feb 2010 22:21 Aatu Koskensilta wrote: > Transfer Principle <lwalke3(a)lausd.net> writes: > >> Consider the thread in the link above. In this thread, Nguyen used >> mostly symbolic language, which according to criterion III above, >> suggests that he is standard. > > You do realize that in passages such as this you come off as extremely > silly? > I'm glad that I'm neither crank nor standard. You must be a standard then?
From: Jim Burns on 11 Feb 2010 11:19 Aatu Koskensilta wrote: > Gc <gcut667(a)hotmail.com> writes: >> I think you need at least borel measure to define >> a Hilbert space which is used in Quantum mechanics, >> but in the other hand Hilbert space probably >> contains functions which have no physical meaning, >> not differentiable anywhere etc. Anyway you probably >> need some axiom of choice like stuff in functional >> analysis, operator algebras etc. > > This is where the tedious coding and massaging, the > chiseling off of spurious generality, comes in. > There's loads of stuff on this in the literature on > reverse mathematics. As to the axiom of choice, its > use in physical predictions and such like is always > eliminable. This encoding and massaging you refer to brings a different but similar-sounding question to my mind, one that may not have a mathematical answer available. Is it possible that these deeper mathematical principles, the ones that could be chiseled off, are nonetheless necessary, if not in a logical sense, then in order to create effective theories that are also comprehensible to the merely genius? Could even Einstein have seen that the massaged and encoded version of Riemannian geometry could lead to a description of physics that does not use a gravitational force? Perhaps Hilbert space and operators and all that could be expressed and reasoned about in a much emptier conceptual landscape, but would we have been able to develop them in that landscape? A crude way of putting it is that I have a suspicion that we would have taken a lot longer to invent the wheel if we lived in a world without round things. In a world without Hilbert spaces, would the time needed to invent quantum mechanics also be "a lot longer", maybe even "forever"? Jim Burns
From: Aatu Koskensilta on 11 Feb 2010 11:20 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > I'm glad that I'm neither crank nor standard. You must be a standard > then? Why do you think I must be a standard? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: T.H. Ray on 11 Feb 2010 02:23 Jim Burns wrote > Aatu Koskensilta wrote: > > Gc <gcut667(a)hotmail.com> writes: > > >> I think you need at least borel measure to define > >> a Hilbert space which is used in Quantum > mechanics, > >> but in the other hand Hilbert space probably > >> contains functions which have no physical meaning, > >> not differentiable anywhere etc. Anyway you > probably > >> need some axiom of choice like stuff in functional > >> analysis, operator algebras etc. > > > > This is where the tedious coding and massaging, the > > chiseling off of spurious generality, comes in. > > There's loads of stuff on this in the literature on > > reverse mathematics. As to the axiom of choice, its > > use in physical predictions and such like is always > > eliminable. > > This encoding and massaging you refer to brings a > different > but similar-sounding question to my mind, one that > may > not have a mathematical answer available. > > Is it possible that these deeper mathematical > principles, > the ones that could be chiseled off, are nonetheless > necessary, if not in a logical sense, then in order > to > create effective theories that are also > comprehensible > to the merely genius? > Sorry to enter this discussion in the middle. There may have been some coherent explanation somewhere of "... principles that could be chiseled off ..." for physical theories, but it makes no sense to me as it stands. Among physical theories, there are those that are called mathematically complete, such as Einstein's special and general theories of relativity -- i.e., those theories that follow from first principles and predict results independent of physical observation -- and those theories such as the standard model of particle physics that are formalized after the fact of observation (Thomas Young's 1803 2-slit experiment being the starting point). In modern times, string theory is mathematically complete (though lacking so far any predictions that differ from the results of classical physics). > Could even Einstein have seen that the massaged and > encoded version of Riemannian geometry could lead to > a description of physics that does not use a > gravitational force? > I don't know what that means, but it follows from the geometry of the Riemann surface that a finite but unbounded spacetime (Minkowski space) has the property that Einstein cited as "physically real" in his introduction to general relativity in The Meaning of Relativity (Princeton, 1956): "... having a physical effect but not itself influenced by physical conditions." Thus, no force can inhere in continuous spacetime itself (as would be the case in Newtonian physics, where space and time are independent and force is transmitted by the ether) but gravity would be completely described in spacetime geometry and the ether cannot be differentiated from the vacuum. Tom > Perhaps Hilbert space and operators and all that > could be > expressed and reasoned about in a much emptier > conceptual > landscape, but would we have been able to develop > them in that landscape? > > A crude way of putting it is that I have a suspicion > that > we would have taken a lot longer to invent the wheel > if we lived in a world without round things. In a > world > without Hilbert spaces, would the time needed to > invent quantum mechanics also be "a lot longer", > maybe even "forever"? > > Jim Burns >
From: Frederick Williams on 11 Feb 2010 12:48
Aatu Koskensilta wrote: > > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > > > I'm glad that I'm neither crank nor standard. You must be a standard > > then? > > Why do you think I must be a standard? Perhaps he's seen you climbing up a flagpole. -- .... A lamprophyre containing small phenocrysts of olivine and augite, and usually also biotite or an amphibole, in a glassy groundmass containing analcime. |