Subject
Re: More help for otherworldly logic (DN responds on VN and
mathematics)
mathematics)
From
Date
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> Can we be sure that VN, who was extremely careful
> not to overstep the boundaries of his own expertise, was "sadly out of
> tune with _real_mathematics" if [this is how] we see [the truth] today:
> "The laws of nature do not determine uniquely the one world that
> actually exists."(Hermann Weyl)? As for the crack about royalty
> percentages, that, actually, was my mother's bailiwick until a
> professional accountant took over.
>
> Dmitri Nabokov
Weyl's quote comes at the end of the Becerra/Barnes paper on The Evolution
of Mathematical Certainty which I recommended to N-Listers. I also cited my
own ACM column on the pi-calculus where I warn that the Becerra/Barnes paper
is "A splendid but _pessimistic_ summary from Thales to Cook via Leibniz,
Kant, Turing, Cohen, et al."
The 'commonsense bean-counting mathematics' that VN is knocking is
undoubtedly boring & yawningly useful, but has no connection with the wild,
creative, counterintuitive magic that makes PURE mathematics
King/Queen/Knave of the sciences. Pure Mathematics works in its own
'reality' beyond any of Weyl's Laws of Nature and any possible 'actual
existing worlds' (here one could digress for millennia on why most real
mathematicians are closet Platonists). Weyl's statement is about the
limitations of Mathematical Physics -- a strange domain that borrows
symbols/concepts from earlier branches of Pure Mathematics that were
formerly considered totally 'abstract' and beyond the sordid sweat/grunt of
worldly applications.
One of my earliest N-L posting wondered whether VN had encountered G H Hardy
at Cambridge -- Hardy's "Mathematician's Apology" praises Pure Mathematics
for its very lack of applications -- alas, his then arcane work on Prime
Numbers now plays a major role in Military Cryptography. Other examples:
Non-euclidean geometries and the tensor calculus were pure 'mental'
constructs until Einstein borrowed them to clothe his General Relativity. VN
lived through several major paradigm shifts in pure & applied mathematics
(Cantorian sets; Hilbert spaces; quantum mechanics; relativity;
Goedel/Turing etc) yet they had little or no impact on his world-view. And
WHY should this bother us FANS? His novels might have had a few more
eclectic allusions. John Shade's Treatise on Electricity might have bristled
with the sheer beauty of Maxwell's partial-differential equations -- a
handful of symbols that spell out all the classical properties of
electromagnetic waves.
On the rare occasions when VN seems to reference mathematics (and I can't
yet claim an exhaustive search), there does seem to be a conceptual
dissonance to my tripos-mind -- an opinion venturing outside the
knowledge-justified domain
1. "In this divinely absurd world of the mind, mathematical symbols do not
thrive." (p 374 VN's Lectures on Literature)
Here's SKB's proof that ALL numbers are INTERESTING:
By reductio-ad-absurdum: Suppose otherwise. Then the set of all
NON-INTERESTING numbers is NON-EMPTY. Take the smallest member N of this
set. N is the smallest non-interesting number which makes N quite
INTERESTING. Contradiction! Similarly, we can prove that ALL numbers are
DULL.
(Hardy compares this type of proof with a GAMBIT in Chess -- except
mathematicians are willing to SACRIFICE the whole GAME, not just a piece or
two.)
2. "When commonsense is ejected together with its calculating machne,
numbers cease to trouble the mind." (ibid)
Non-mathematicians are usually unaware that there are MANY LOGICS to pick
from. And many DEDUCTIVE SCHEMATA, too. Russell, he say: "In mathematics we
do not know which objects we are discussing -- and we don't care whether
what we say about them is true." Less flippantly: there's a BIG IF sitting
up front. We list some AXIOMS (assumed TRUE, not to be argued over), pick
some DEFINITIONS (purely optional -- they just save time), select our LOGIC
(binary, n-valued, ...), select the DEDUCTION RULES (law of the excluded
middle, ...). We then see what THEOREMS follow. We are NOT committed to any
particular formal system. We don't have to BELIEVE our axioms. We can
axiomatize geometries without having any numbers at all.
3. "Space thrives on surds" (Ada?)
Hard to pin down the meaning & decide if it's VN or the narrator speaking.
The word 'space' has many technical varieties. Archives show some
discussion on aleph-0 etc. If we mean the locally Euclidean space we
inhabit, the usual metric is called R3 and 'surd' is not strictly correct.
Surds are a subset of the real-number-continuum. Space precludes!
4. "The hyphen in space-time is stupid" (Strong Opinion paraphrased from N-L
posting?)
This needs an essay. Briefly, an EVENT a la mode d'Einstein is determined by
E = (x, y, x, it) -- where and when -- four parameters 3 of space, one of
time -- hence the perfectly valid term: 4-dimensional space-time [adjective]
continuum. The noun is often written as spacetime! Note the magic 'it' where
i-is the sqare-root of minus one. Rather than somehow bundling space and
time together, 'it' helps to keep the spatial and temporal SEPARATE during
our calculations.
Finally: I meant no offence re-royalties. We all must acquire basic boring
numeracy to survive. I had just been re-reading BB's VN-TAY -- pp 300-301
sees VN litigating with Girodias over that lousy 5%.
Stan Kelly-Bootle
Search the archive: http://listserv.ucsb.edu/archives/nabokv-l.html
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> not to overstep the boundaries of his own expertise, was "sadly out of
> tune with _real_mathematics" if [this is how] we see [the truth] today:
> "The laws of nature do not determine uniquely the one world that
> actually exists."(Hermann Weyl)? As for the crack about royalty
> percentages, that, actually, was my mother's bailiwick until a
> professional accountant took over.
>
> Dmitri Nabokov
Weyl's quote comes at the end of the Becerra/Barnes paper on The Evolution
of Mathematical Certainty which I recommended to N-Listers. I also cited my
own ACM column on the pi-calculus where I warn that the Becerra/Barnes paper
is "A splendid but _pessimistic_ summary from Thales to Cook via Leibniz,
Kant, Turing, Cohen, et al."
The 'commonsense bean-counting mathematics' that VN is knocking is
undoubtedly boring & yawningly useful, but has no connection with the wild,
creative, counterintuitive magic that makes PURE mathematics
King/Queen/Knave of the sciences. Pure Mathematics works in its own
'reality' beyond any of Weyl's Laws of Nature and any possible 'actual
existing worlds' (here one could digress for millennia on why most real
mathematicians are closet Platonists). Weyl's statement is about the
limitations of Mathematical Physics -- a strange domain that borrows
symbols/concepts from earlier branches of Pure Mathematics that were
formerly considered totally 'abstract' and beyond the sordid sweat/grunt of
worldly applications.
One of my earliest N-L posting wondered whether VN had encountered G H Hardy
at Cambridge -- Hardy's "Mathematician's Apology" praises Pure Mathematics
for its very lack of applications -- alas, his then arcane work on Prime
Numbers now plays a major role in Military Cryptography. Other examples:
Non-euclidean geometries and the tensor calculus were pure 'mental'
constructs until Einstein borrowed them to clothe his General Relativity. VN
lived through several major paradigm shifts in pure & applied mathematics
(Cantorian sets; Hilbert spaces; quantum mechanics; relativity;
Goedel/Turing etc) yet they had little or no impact on his world-view. And
WHY should this bother us FANS? His novels might have had a few more
eclectic allusions. John Shade's Treatise on Electricity might have bristled
with the sheer beauty of Maxwell's partial-differential equations -- a
handful of symbols that spell out all the classical properties of
electromagnetic waves.
On the rare occasions when VN seems to reference mathematics (and I can't
yet claim an exhaustive search), there does seem to be a conceptual
dissonance to my tripos-mind -- an opinion venturing outside the
knowledge-justified domain
1. "In this divinely absurd world of the mind, mathematical symbols do not
thrive." (p 374 VN's Lectures on Literature)
Here's SKB's proof that ALL numbers are INTERESTING:
By reductio-ad-absurdum: Suppose otherwise. Then the set of all
NON-INTERESTING numbers is NON-EMPTY. Take the smallest member N of this
set. N is the smallest non-interesting number which makes N quite
INTERESTING. Contradiction! Similarly, we can prove that ALL numbers are
DULL.
(Hardy compares this type of proof with a GAMBIT in Chess -- except
mathematicians are willing to SACRIFICE the whole GAME, not just a piece or
two.)
2. "When commonsense is ejected together with its calculating machne,
numbers cease to trouble the mind." (ibid)
Non-mathematicians are usually unaware that there are MANY LOGICS to pick
from. And many DEDUCTIVE SCHEMATA, too. Russell, he say: "In mathematics we
do not know which objects we are discussing -- and we don't care whether
what we say about them is true." Less flippantly: there's a BIG IF sitting
up front. We list some AXIOMS (assumed TRUE, not to be argued over), pick
some DEFINITIONS (purely optional -- they just save time), select our LOGIC
(binary, n-valued, ...), select the DEDUCTION RULES (law of the excluded
middle, ...). We then see what THEOREMS follow. We are NOT committed to any
particular formal system. We don't have to BELIEVE our axioms. We can
axiomatize geometries without having any numbers at all.
3. "Space thrives on surds" (Ada?)
Hard to pin down the meaning & decide if it's VN or the narrator speaking.
The word 'space' has many technical varieties. Archives show some
discussion on aleph-0 etc. If we mean the locally Euclidean space we
inhabit, the usual metric is called R3 and 'surd' is not strictly correct.
Surds are a subset of the real-number-continuum. Space precludes!
4. "The hyphen in space-time is stupid" (Strong Opinion paraphrased from N-L
posting?)
This needs an essay. Briefly, an EVENT a la mode d'Einstein is determined by
E = (x, y, x, it) -- where and when -- four parameters 3 of space, one of
time -- hence the perfectly valid term: 4-dimensional space-time [adjective]
continuum. The noun is often written as spacetime! Note the magic 'it' where
i-is the sqare-root of minus one. Rather than somehow bundling space and
time together, 'it' helps to keep the spatial and temporal SEPARATE during
our calculations.
Finally: I meant no offence re-royalties. We all must acquire basic boring
numeracy to survive. I had just been re-reading BB's VN-TAY -- pp 300-301
sees VN litigating with Girodias over that lousy 5%.
Stan Kelly-Bootle
Search the archive: http://listserv.ucsb.edu/archives/nabokv-l.html
Contact the Editors: mailto:nabokv-l@utk.edu,nabokv-l@holycross.edu
Visit Zembla: http://www.libraries.psu.edu/nabokov/zembla.htm
View Nabokv-L policies: http://web.utk.edu/~sblackwe/EDNote.htm