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Welcome to the home page of PAIAS, the Palo Alto
Institute for Advanced Study.
Our Primary Mission is to increase awareness of serious
“... oversights” in science
and mathematics, in and among not only the scientific and mathematical communities, but also the
general public. Numerous expository-exploratory-critical essay-like substances
are aimed at targets in specific areas of science and mathematics, such as
relativity and set theory. Anyone who enjoys science should have no problem
following the threads of reasoning.
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Main Section)
Presentation-Paper:
The Vanishing Remainders Paradoxes and Other Overlooked Paradoxes of Infinity
(pps slides, ~340KB)
The Vanishing Remainders Paradoxes and Other Overlooked Paradoxes of Infinity
(pdf, ~700 KB, last minor “update”, May 4, 2005)
was presented at the
2004 Phoenix Joint Mathematics Meeting.
(minor revisions have been made since, the most recent as of Mar 16, 2005)
Abstract
The rational 1/3 is transformed into the
real (infinite decimal expansion) 0.333... by successively multiplying 1 by 10,
dividing that 10 by 3 getting a quotient of 3 with a remainder of 1 which
becomes the 10 for the next decimal place, and so on... But that remainder of 1
vanishes from theoretical view when we reach the Cantorianly completed countable
infinity of decimal places. What happens to it? Does it somehow become an
absolute zero after an infinite number of divisions? A variant asks: what
happens to the remainder of 1 when it is divided by a natural number
n as n “goes
to infinity”? Does it somehow become an absolute zero when “divided by
infinity”? These are some of the paradigmatic questions for the “Vanishing
Remainders Paradoxes”. If the remainder does not become an absolute zero, then
1/3 cannot be represented by 0.333... (they cannot be strictly equal) and
therefore 1/3 can not truly be a
real number; and if it does, this violates an as yet
unrecognized implicit conservation-type law, opening the door to inconsistency
in addition to paradox. E.g., if the remainder of 1 when 1 is “divided by
infinity” becomes an absolute zero, one can derive that the reals are countable.
There are many other as yet overlooked paradoxes in mathematics, some of which
offer a basis for their joint resolution, and for the problem of renormalization
in physics.
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Oversights and Other Issues in:
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PAIAS: |
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Science
Mathematics
Philosophy
Religion (working on it)
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Dedication
Our Mission in more detail
Our Philosophy
Our Intended Audience
Institute Profile
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Highlights of... Oversights in
Science: |
Highlights of... Oversights in
Mathematics: |
Lighter and heavier bodies
actually fall at different rates, unless that falling is measured in an absolute
Newtonian frame of reference (with a fascinating exception involving
Lagrange’s Trojan points).
Newton
overlooked this,
Einstein
overlooked this...
Thermodynamic entropy does not increase monotonically. In fact, no
ergodic system can have a strictly monotonic state function.
And physicists have also overlooked that Brownian motion applies to entropy,
i.e. the direction of the change in thermodynamic entropy is
“almost always” reversing. Also overlooked is that
increasing entropy does not mean increasing disorder; order, it turns out,
is a purely subjective concept.
Comets actually
come from local swarms of proto-comets (in the Kuiper belt and the Oort cloud) that every so often eject one of their
members due to gravitational interactions, much like the sling-shotting that
is used to propel space-crafts more quickly to the outer reaches of the
Solar System. What about “passing stars”... ??? Stars just don’t “pass”
anywhere near often
enough and/or close enough to explain the time distribution of old versus new comet sightings.
“Creationism” and
“Evolutionism” — better termed
“Anti-Evolutionism” and “Anti-Creationism” — are both guilty of
serious... oversights.
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Set Theory, and any theory for which it is fundamental (e.g.
Real Number Theory), is plagued with paradoxes, some of which will
eventually be accepted as “too paradoxical”.
The Good Shepherd’s Paradox gives a fun approach to Set Theory’s...
oversight that is accessible to non-mathematicians and mathematicians alike.
The Bijection Permutation Paradox gives a rigorous formal approach to
the same issue (Set Theory’s... oversight).
It shows how one can permute bijections in such a way as to maintain
bijectivity invariant while subtracting out all elements that the 2 sets of
the bijection have in common. It may seem intuitively obvious to even the
most casual observer that one can do this, but Set Theory depends on
bijections between sets and proper subsets of themselves (such bijections
are considered definitional of transfinite sets). This means that Set Theory
has heretofore unrecognized “Paradoxical Bijections” from non-empty sets onto the empty set. This can definitely be
considered “too paradoxical”.
The Vanishing
Remainders Paradoxes are particularly approachable. Anyone who knows how to
construct the real number 0.333... from the rational number 1/3 will be able
to appreciate the significance of the continuing remainder of 1 that...
Vanishes.
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