Subject

Re: More help for otherworldly logic (DN responds on VN and

mathematics)

mathematics)

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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

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

> 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