[PREV - IMPOSSIBLE_GEOMETRIES] [TOP]
MATHISM
January 16, 2019
September 20-30, 2019
Rev: October 25, 2019
_Principia Mathematica_
was published 1910-1913:
After 1910 or
Whitehead and Russell made a so, I gather it
game attempt at rebuilding was mostly
all of mathematics from Russell working
scratch, logically deriving on this-- Alfred
everything from a minimal Whitehead
number of postulates. dropped out of
the project,
THE_DOCTRINE_OF_POSTULATES
because Russell
had an affair
with his wife.
Russell was a
big advocate of
free love in
those days-- his And so that's all
personal life another piece of the
was complicated. scene, at the beginning
of the 20th century,
This is now widely just before WWI.
regarded as a dead end,
having been shot down by DUSTY_MYSTERY
Goedel's Incompleteness
Theorems in 1931. No formal system
can be complete,
I'm beginning to wonder if or so they say. It's not clear that
it might not have been Russell ever really
fundamentally silly for understood Goedel's
other reasons: a lot of Incompleteness (for
effort for little benefit. that matter, I'm not
Perhaps: indulging in an sure that I do).
intellectual vice.
Russell's comment on
hearing about it was
Even before Goedel there something along the lines
were troubles, they of "I'm glad I'm not
needed some way to deal working on that project
with "Russell's Paradox": right now."
Is the set of all sets Perhaps you could defend the _Principia_
that do not belong to a effort in the usual way that pure math is
set a member of itself? always defended: fields or mathematical
research often begin just as intellectual
Setting up that kind of self- curiosities but have practical applications
contradiction is obviously that are discover only later.
bad, and the solution is, of
course, "don't do that"-- but EXCURSIONS_IN_THEORY
don't do what?
Re-factoring all of mathematics
They experimented with various might not have an obvious point, but
kinds of what I would call logical it doesn't mean it has no point.
hacks to try to cover for this;
they were trying to contrive a set I gather part of the
of rules that would forbid these I don't idea was to develop
contradictory self-reference think they a new foundation of
situations in a way that didn't got to math that would be
seem too ugly or unreasonable... inventing easier to extend:
"weaken". algorithmic
generation of new
theorems and
automated proofs.
Over the years, Russell changed his
mind about which approach was best, and
it seems that there are people still
playing around with these hacks, though I have seen a remark by
I gather that ZFC, a variant of Goedel that claimed that
"axiomatic set theory" is essentially once you understood the
the standard view. But then, one also the solution it wasn't
hears about "type theory", "category any big deal.
theory", and so on, which seem to me to
be competing theories (or at least I don't really know
related fields?). which solution
Goedel had in mind.
At present, I only have the vaguest "Axiomatic Set
sense of how these work, but I would Theory", perhaps.
gather the idea is something like you
treat "meta-sets" as a different order [link]
of thing than a "set"... but that
would need a few more pieces to get it Wildberger says the
to work... consensus settled on
Zermelo-Frankel plus
The metaset of all sets that do not belong the Axiom of Choice:
to some other metaset is not a member ZFC.
of itself, because it's a metaset not a set.
WILDBERGER_MATH
The metaset of all sets that do not belong
to any metaset would not be a member
of itself for the same reason, but there's There are sets that should
the question of whether any set could be added to it, but if
belong to it without contradiction. they're added, they should
be removed.
Some rule about sets not being
allowed to refer to themselves in
their definition might do it-- but
I would guess that could rule out ZFC has a rule against sets
other things we'd rather see being members of themselves--
allowed.
The notation it makes it look a
But then, if metasets are allowed little more complex than that,
to be members of other metasets, though: I think they were going
then you've got a parallel to after indirect self-reference
Russell's paradox at a different as well.
level-- you'd need the concept of
meta-meta-* sets ad infinitum.
So much for my naive
attempt at fixing
naive set theory.
There are also other paradoxes kicking [link]
around, such as the Burali-Forti. Patrick
Suppes, the author of "Axiomatic Set
Theory" (1960) makes the point that the
hacks that cover one paradox don't
*always* cover the others.
In any case, as I was looking over Russell's MODEST_PROPOSAL
philosophy of mathematics, I started
developing a funny feeling-- familiar from
other fields-- that I was looking at a smart It's a pretty common pattern
guy plunging off into some highly intelligent for a field to make great
madness... success early on using a
heavily mathematical approach,
I'm used to feeling this way about only to hit a wall and
"computer science" which seems to be reluctantly fall back on
dominated by people who want it to be empirical approaches.
all about math, though to most
practicing programmers the work seems Fluid Mechanics
like something much different... Crystal Growth
Mathematical elegance just doesn't Computer Science...?
matter all that much for the practical
problems of getting software working. MATH_SIMPLE
Now I found myself wondering if There's often a lingering
"mathematical elegance" really made sense of mathism in these
all that much sense for mathematics fields-- papers with masses of
itself. impenetrable equations are
always awarded respect.
There's a habit of mind of
mathematicians that leads them to
start thinking things like "hm, GENERAL_RUSSELL
Euclid's fifth postulate seems
kind-of clunky, wouldn't if be cool
if we could prove it in terms of the "... in mathematics, every new
first four? Then there'd be only axiom diminishes the generality of
four postulates!" the resulting theorems, and the
greatest possible generality is
before all things to be sought."
But what exactly would that
get you? They're "playing golf" with Bertrand Russell,
Euclidean geometry, sinking the hole "The Study of Mathematics"
in the fewest number of strokes, but
still getting to the same place.
Now, as it happens, Russell But showing that the 5th axiom is
had some issues with Euclid-- redundant would hardly increase the
evidently there are some range of application of what's been
implicit assumptions scattered derived.
throughout the work outside
of the explicit 5 postulates.
But that shows exactly the I can see how hypothetically that a
sort of thing I'm talking reduction in the number of premises
about... why wouldn't you could reduce the number of things
just tack the additional you'd need to check to make sure the
postulates on to the list of math is applicable to a given
givens and call it done? situation.... but it'd need to be a
big reduction to make a practical
And that indeed has difference (to be precise: a big
been done by modern reduction in the difficulty of
mathematicians-- checking the postulates, not
necessarily their total number).
[link]
Russell had an And in practice, I doubt even
extremly dismissive that really matters. When a
attitude toward scientist or engineer tries to
Euclid, something like use some math, they hardly pay
"Oh that guy was such any attention to the
a *poseur*, he was so mathematician in the back
overrated-- no way Those were waving their hands and shouting
should we be teaching the days, "But you haven't established
Euclid to school when you continuity!". The physicists
children. could gripe Just Try It, and if it seems to
about work they keep using it, and
Russell's take schools get someone to stuff a gag in
was that the that had the mathematician's mouth.
first *real* kids read
math book was original
Boole's "Laws texts like
of Thought" Euclid.
from 1854.
Actually, I can think of one way
this could matter in mathematical
research: if you could show that
Could it be that there's a one of your postulates was
certain nuttiness about this redundant, you'd stop worrying
entire drive to minimalism? about examining alternative
possibilities.
In the case of Euclid's 5th,
it actually isn't redundant,
and there are variant forms
of geometry that drop it
("hyperbolic geometry").
Bertrand Russell, in "The Study of Mathematics"
objects to justifying the study of math on And as is not unusual,
practical grounds ("facilitates the making of whenever you find someone
machines") or on pedagogic ones ("trains the promoting a weirdly
reasoning faculties"), instead he passionately idealized view of the
declares it should be studied for it's own sake: world, there you will find
a fan of Plato:
"Mathematics, rightly viewed, possesses
not only truth, but supreme beauty-- a "Plato, we know, regarded
beauty cold and austere, like that of the contemplation of
sculpture, without appeal to any part of mathematical truths as
our weaker nature, without the gorgeous worthy of the Deity; and
trappings of painting or music, yet Plato realised, more
sublimely pure, and capable of a stern perhaps than any other
perfection such as only the greatest art single man, what those
can show." elements are in human
life which merit a place
This passionate aesthetic fascination-- in heaven."
and an explicit denial that practical
applications matter-- starts
to look like evidence of mania...
Now, going after the entire discipline of
mathematics, is shall we say, ambitious,
but let's look this over:
Russell's ideas aside, the usual
defense of "pure math"-- itself a
revealing name, perhaps-- is that it
often later turns out to be useful.
The classic example is
non-euclidean geometry: an EXCURSIONS_IN_THEORY
intellectual curiosity before
general relativity convinced
us it was reality.
An example I came across recently:
Number theory and digital encryption CERTAINTY
This one is interesting because
it's a technological application,
not a scientific one.
There's a famous essay by I think some have trouble seeing
the physicist Eugene what Wigner was getting at, I've
Wigner "The Unreasonable heard remarks like:
Effectiveness of
Mathematics", which "Math theories are conceptual tools
ponders why it is that created by humans so why would it
understanding physical seem strange that they're useful?"
reality is so tractable
to mathematic approaches. The point is that Math is often
invented with *no expectation* that
There's an apropos it will be of any use. It's often
quip by Russell himself, literally a matter of playing
to the effect that around with abstract concepts.
math is simple, but
then so are we.
MATH_SIMPLE
Now, granted that these logical games
sometimes turn out to be useful in
retrospect, we don't really know if there
might be some other approach to math that
would work as well or better.
SCARCE_RIGOR
Pure Math might be compared to flinging stones
blindly over a wall, and demanding respect because
they occasionally land in useful configurations.
In a few places Russell speaks dismissively
about Kant's philosophy of mathematics, WILDBERGER_MATH
because it allows room for something like
intuition, and a path toward understanding
besides logical deduction.
Russell would have none of this, of course:
his attitude was something like "if you're
going to allow things like that you might
as well not bother at all".
But Russell's critique of Kant implicitly
acknowledges that there was a time when
mathematics was done differently-- that's
what Kant was describing.
Though for some, that contradicts
The fact that we changed once always their faith that we always advance:
suggests to me that we may change the the new ways *must* be better or
way we do things again. else why did we switch?
WHIG_OUT
We're already in an era where
math differs from the way it was
done in Russell's day:
The first of these was the proof
Modern mathematical proofs are often of the four color theorem.
complex, elaborate things that may
even be software assisted--
And it's not unusual for a long There's an interview with Conway
proof to be more-or-less accepted, where he complains about this
even though it's acknowledged it attitude-- he evidently think it's
probably has mistakes in it, and a justification for intellectual
further work will be needed to sloppiness that a good mathematician
bridge the gaps. would be able to avoid.
--------
[NEXT - THE_DOCTRINE_OF_POSTULATES]