The Limits of Understanding....on Not Pa
“Either mathematics is too big for the human
mind or the human mind is more than a machine. ”
This is Part 1 (an Introduction) to a much more extensive document
which is currently under revision for considerable and hopefully provocative
expansion. ...but we welcome the reader to come along with us ... we urge
freely commenting on areas where you think we might have gone astray)
Part I. Introduction: Pointing to Gödel's Finger
As a preface to the video, roundtable discussion by a panel of
renowned scholars and scientists of the significance and further implications of
Gödel work "Limits of Understanding" that we discuss down below , they
present us with an introduction which, as we might legitimately presume, is
meant to catch our attention. The Participants include mathematician
Gregory Chaitin, author Rebecca Goldstein, astrophysicist Mario Livio and
artificial intelligence expert Marvin Minsky and they can be truly said to have,
among them, a formidable array of knowledge and experience. The Video has
this as the introduction:
"This statement is false. Think about it, and it makes your head hurt. If
it’s true, it’s false. If it’s false, it’s true. In 1931, Austrian logician Kurt
Gödel shocked the worlds of mathematics and philosophy by
establishing that such statements are far more than a quirky turn of
language: he showed that there are mathematical truths which simply
can’t be proven. In the decades since, thinkers have taken the brilliant
Gödel’s result in a variety of directions–linking it to limits of human
comprehension and the quest to recreate human thinking on a
computer."
!1
The video gathering/discussion itself is woefully misnamed, as we see
it, since Gödel's proofs in no way point to, or demarcate the "Limits of
Understanding" and, if anything, the mathematician's video account of it as the
"Limits of Mathematics" is much more apt and less prone to the kind of
rampant philosophical rumination we have seen in academia for nearly the past
century. The videos, entitled, The Limits of Mathematics" (part 1 and part 2 )
do provide a straightforward "bare bones" introduction to the mathematical
issues and a good context within the setting of the history of mathematics" in
which to see how and why Gödel's proofs arose.
We include here two
background video/lectures on
Gödel's work which are
readily accessible and attempt
to place it terms allowing us
to gauge its impact and
significance. Most probably we
won't be able to fully do
that. That is because most of
the impact of Gödel's work has been directly felt by those in philosophy and
mathematics and science Regrettably, even, within that group, many prefer to
just "shut up and work" and not get find themselves in the tempest of issues
that due considerations of Godel "incompleteness proofs' might raise.
What Gödel's proofs point to and demonstrate even more vividly,
rather, are the limits and utter incompleteness of our own and our own
culture's understanding of the concept of "understanding" as in any way
!2
related to the pedestrian expression of
formal logical/language based systems of
notations or symbols.
This "burden" which we carry
when we talk and when we talk about
talking, we shall note below, goes all
the way back to Aristotle...and likely
before his time
Indeed the odd and, to us,
hastily contrived and inappropriate,
entitling of the video in which we find
Rebecca Goldstein participating is very much an example of the kind of
misinterpretation to which she alerted us in that early, much more interesting
interview.
She, herself, in a very engrossing Interview or 'Talk" with Edge,
entitled promisingly GODEL AND THE NATURE OF MATHEMATICAL
TRUTH, also adds considerable historical sensibility to the emergence of
Gödel and his proofs as well
as more broadly considering
the mathematical/
philosophical
significance. However despite
the promising nature of her
interview as we note below
the notion of "truth" itself is
never truly examined. Instead
!3
as rather symptomatic of the situation which we sketch out below, there is
some talk about mathematics and its relation to "reality".
But without a confrontation with the issue of "truth", the question of
'incompleteness" is left without grounding in a sufficiently clear and distinct
notion of just what it is that might be "limited" or not be "limited" in virtue of
his proof of the "incompleteness' of the formal systems upon which science
and philosophy depend to give themselves a feeling that they are engaged in
"understanding" in the best sense of that word.
This dilemma, face by all who wish to "know' while they engage in
their efforts at 'knowing' was presented to those past two thousand years of
thinkiers in the following way: The notion of 'Truth" along with its
perplexities , as Tarski (1944) and others have suggested, is captured in the
slogan from Aristotle's Metaphysics , “to say of what is that it is, or of what is not
that it is not, is true” .
‘What is’, it is natural enough to say, is a fact, and while this turn of
phrase has become so natural that it is very much taken for granted to have
!4
some "meaningful" in our use , it cannot be used without recognition on our
part that we make some effort to be clear about of just what we are talking
about when we talk about the "Truth" . Even worse is to not recognize that
need and just go on talking...and talking...and talking.
We might say to Aristotle, 'Easy enough for you to say!" And indeed
this idea of the "truth" was a very nice one indeed, quite marketable to the
world, as we can from more than two millennia later it still holds sway over
everything we say and indeed, everything we say about our own saying.
But how in the world (no pun intended) are we to determine that we
have indeed "said of what is that it is". That little bit of fluffery that is
overlooked in our culture's adoption of the Aristotelian metaphysics is very
much a dangling thread that hangs from our otherwise elaborately woven
intellectually frameworks, that we all can see dangling but the prospect of the
entire framework coming undone if we try to pull that thread allows the
"Truth" to dangle out there as loose as it is.
We believe that there is much more of a story to be learned and to
be told than is presented in either the educational videos we have here or in
the rambling discussion by a few of the more venerable members of the
science community. If we don't make the effort to understand what we mean
by or what the grounds of 'understanding" itself are, then surely we, as are
these folks in the video, are going to talk ourselves into circles as philosophers
have done for thousands of airs, and most of the rest of would have to do if
we chose to persist in that kind of talk as a professional calling.
Unsurprisingly, during the decades since Gödel proofs were
presented in the early 1930's, now almost a century ago, various thinkers
!5
engaged in all sorts of thinking have taken the brilliant Gödel’s result in a
variety of directions–on the one hand linking it to an espousal by Gödel of the
limits of human comprehension and on the other hand as the inspiration for
today's ongoing quest to recreate human thinking on a computer.
Some “pop” writers with a more mystical bent seem to believe that
Gödel, in the course of his proofs ot those notorious incompleteness
theorems in the 1930s, set out to prove that there are limits to how much of
reality mathematical logic can grasp — something many "science bashers" and
holistic evangelists have claimed to have intuited but none has been able to
substantiated other than through endless rhetoric.
On the contrary, what we see is mathematics flowering as
organically as any living form on the planet through an exquisite selforganizational burst we have witnessed in the past two centuries, even it's
weeds seem to eventually become recognized as flowers. Within its own
expressions and formulations it has found the roots expanding its grasp
stunningly.
!6
This. to us, is not a sign that there is a problem of its' reach exceeding
its grasp, or being bounded in some confining, almost claustrophobic way by
any quaint and odd notion that there is some vast finite or even infinite
reality out there beyond the reach of our eyes and ears that we can never
"understand" that provides any basis for concern of fretting about our
own boundedness and the limits of mathematics tools of exploration.
Goldstein says very nicely, in her interview with The Edge, Godel
and the Nature of Mathematical Truth,
"Gödel mistrusted our ability to communicate. Natural language, he
thought, was imprecise, and we usually don't understand each other.
Gödel wanted to prove a mathematical theorem that would have all
the precision of mathematics—the only language with any claims to
precision—but with the sweep of philosophy.
"He wanted a mathematical theorem that would speak to the issues
of meta-mathematics. And two extraordinary things happened.
One is that he actually did produce such a theorem.
The other is that it was interpreted by the jazzier parts of the
intellectual culture as saying, philosophically exactly the opposite of
what he had been intending to say with it."
To fret over the inability of mathematics to somehow capture the full
scope of 'reality' (however vast the vastness of that also rather quaint historical
notion may be) is akin to being concerned over the infinite array of
integers being confined in some awfully anxiety provoking way because there
may be and are further infinite domains of integers which can be seen as
generated from any given array of infinite integers
!7
Ironically, this very difficulty with conceiving of "infinities" adequately
and of appreciating that there are hierarchies of infinities and indeed infinite
infinities that may be considered, created such a controversy (still ongoing)
that 'understanding" the concept of infinity is what was behind the historical
events leading to Gödels' work aimed at understanding a mathematics itself
one of whose key
concepts was infinity.
As many of the
mathematicians pointed
out, the debate itself
reveals a lack of human
intuition regarding the
concept of infinity
If anything, if we do
manage to arrive via our discussion here at a beginning of an understanding of
the notion of "understanding", his proofs in fact, we hope will be regarded by
readers as pointing, via their demonstration of the incompleteness of many of
our formal systems of pedestrian stepwise trudging via
logical connectors not to a pessimism about our limits as 'homo sapiens",
but rather that the incompleteness demonstrated by was his way of pointing
to the limitless and unbounded nature of our understanding..when that notion
is properly understood.
If there is any doubt about that, we only have to look at the
incredible flood of new mathematical formulations for which Godel's
work planted the seeds.
!8
What we have here is the archetypal case of the story of 'the man
who pointed to the moon, while we, most of the rest of us, have thus far
pointed to his finger".
We could hardly stop from standing up and saying “bravo” when we
read Jim Holt, hin his delightfully deep and profoundly delightful essay in the
New Yorker, What Were Einstein and Godel Talking About", characterize Godel
in a way which seemed to have escaped most of his commentators…including
all those in video roundtable, as well Holt writes of Godel,
when “significance of Gödel’s theorems began to sink in, words like
“debacle,” “catastrophe,” and “nightmare” were bandied about.
“It had been an article of faith that, armed with logic,
mathematicians could in principle resolve any conundrum at all—
that in mathematics, as it had been famously declared, there was
!9
no ignorabimus. Gödel’s theorems seemed to have shattered this
ideal of complete knowledge.”
“That was not the way Gödel saw it” (and we add “amen”)
“He believed he had shown that mathematics has a robust reality that
transcends any system of logic. But logic, he was convinced, is not the
only route to knowledge of this reality; we also have something like
an extrasensory perception of it, which he called “mathematical
intuition.”
Rebecca Goldstein talks
about the friendship between
Godel and Einstein, and she tells
us that what bound them so
closely for so many years, among
other things that we hope to
explore, was a common dream.
Einstein, during the later years of
his life at Princeton, remarked
that at that stage of his life, his
own work did not mean that much, but that he stayed at Princeton only for the
privileged of taking his walks home with Godel. “Einstein and Gödel”
She writes,
'What bonded them was that, first of all, they were so keenly
interested in the meta-questions of their respective fields, those
interpretive questions regarding what is it that these fields are really
doing and how is it that they manage to do it
!10
” Both Einstein and Gödel who had a legendary friendship when they
were at the Institute for Advanced Study—could not have been
more committed to the idea of objective truth.”
They were united by how much they shared their feeling of being
disaffected from the popular fashions and fads in their fields of and how the
"ultimate questions" were not being addressed by their colleagues ,
But as they walked and as they talked
all those years, although they had a
common dream and grand dream,
that perplexing notion of "truth" was
such that it encompassed two very
different ways of reaching for
its' "understanding".
!11
mind or the human mind is more than a machine. ”
This is Part 1 (an Introduction) to a much more extensive document
which is currently under revision for considerable and hopefully provocative
expansion. ...but we welcome the reader to come along with us ... we urge
freely commenting on areas where you think we might have gone astray)
Part I. Introduction: Pointing to Gödel's Finger
As a preface to the video, roundtable discussion by a panel of
renowned scholars and scientists of the significance and further implications of
Gödel work "Limits of Understanding" that we discuss down below , they
present us with an introduction which, as we might legitimately presume, is
meant to catch our attention. The Participants include mathematician
Gregory Chaitin, author Rebecca Goldstein, astrophysicist Mario Livio and
artificial intelligence expert Marvin Minsky and they can be truly said to have,
among them, a formidable array of knowledge and experience. The Video has
this as the introduction:
"This statement is false. Think about it, and it makes your head hurt. If
it’s true, it’s false. If it’s false, it’s true. In 1931, Austrian logician Kurt
Gödel shocked the worlds of mathematics and philosophy by
establishing that such statements are far more than a quirky turn of
language: he showed that there are mathematical truths which simply
can’t be proven. In the decades since, thinkers have taken the brilliant
Gödel’s result in a variety of directions–linking it to limits of human
comprehension and the quest to recreate human thinking on a
computer."
!1
The video gathering/discussion itself is woefully misnamed, as we see
it, since Gödel's proofs in no way point to, or demarcate the "Limits of
Understanding" and, if anything, the mathematician's video account of it as the
"Limits of Mathematics" is much more apt and less prone to the kind of
rampant philosophical rumination we have seen in academia for nearly the past
century. The videos, entitled, The Limits of Mathematics" (part 1 and part 2 )
do provide a straightforward "bare bones" introduction to the mathematical
issues and a good context within the setting of the history of mathematics" in
which to see how and why Gödel's proofs arose.
We include here two
background video/lectures on
Gödel's work which are
readily accessible and attempt
to place it terms allowing us
to gauge its impact and
significance. Most probably we
won't be able to fully do
that. That is because most of
the impact of Gödel's work has been directly felt by those in philosophy and
mathematics and science Regrettably, even, within that group, many prefer to
just "shut up and work" and not get find themselves in the tempest of issues
that due considerations of Godel "incompleteness proofs' might raise.
What Gödel's proofs point to and demonstrate even more vividly,
rather, are the limits and utter incompleteness of our own and our own
culture's understanding of the concept of "understanding" as in any way
!2
related to the pedestrian expression of
formal logical/language based systems of
notations or symbols.
This "burden" which we carry
when we talk and when we talk about
talking, we shall note below, goes all
the way back to Aristotle...and likely
before his time
Indeed the odd and, to us,
hastily contrived and inappropriate,
entitling of the video in which we find
Rebecca Goldstein participating is very much an example of the kind of
misinterpretation to which she alerted us in that early, much more interesting
interview.
She, herself, in a very engrossing Interview or 'Talk" with Edge,
entitled promisingly GODEL AND THE NATURE OF MATHEMATICAL
TRUTH, also adds considerable historical sensibility to the emergence of
Gödel and his proofs as well
as more broadly considering
the mathematical/
philosophical
significance. However despite
the promising nature of her
interview as we note below
the notion of "truth" itself is
never truly examined. Instead
!3
as rather symptomatic of the situation which we sketch out below, there is
some talk about mathematics and its relation to "reality".
But without a confrontation with the issue of "truth", the question of
'incompleteness" is left without grounding in a sufficiently clear and distinct
notion of just what it is that might be "limited" or not be "limited" in virtue of
his proof of the "incompleteness' of the formal systems upon which science
and philosophy depend to give themselves a feeling that they are engaged in
"understanding" in the best sense of that word.
This dilemma, face by all who wish to "know' while they engage in
their efforts at 'knowing' was presented to those past two thousand years of
thinkiers in the following way: The notion of 'Truth" along with its
perplexities , as Tarski (1944) and others have suggested, is captured in the
slogan from Aristotle's Metaphysics , “to say of what is that it is, or of what is not
that it is not, is true” .
‘What is’, it is natural enough to say, is a fact, and while this turn of
phrase has become so natural that it is very much taken for granted to have
!4
some "meaningful" in our use , it cannot be used without recognition on our
part that we make some effort to be clear about of just what we are talking
about when we talk about the "Truth" . Even worse is to not recognize that
need and just go on talking...and talking...and talking.
We might say to Aristotle, 'Easy enough for you to say!" And indeed
this idea of the "truth" was a very nice one indeed, quite marketable to the
world, as we can from more than two millennia later it still holds sway over
everything we say and indeed, everything we say about our own saying.
But how in the world (no pun intended) are we to determine that we
have indeed "said of what is that it is". That little bit of fluffery that is
overlooked in our culture's adoption of the Aristotelian metaphysics is very
much a dangling thread that hangs from our otherwise elaborately woven
intellectually frameworks, that we all can see dangling but the prospect of the
entire framework coming undone if we try to pull that thread allows the
"Truth" to dangle out there as loose as it is.
We believe that there is much more of a story to be learned and to
be told than is presented in either the educational videos we have here or in
the rambling discussion by a few of the more venerable members of the
science community. If we don't make the effort to understand what we mean
by or what the grounds of 'understanding" itself are, then surely we, as are
these folks in the video, are going to talk ourselves into circles as philosophers
have done for thousands of airs, and most of the rest of would have to do if
we chose to persist in that kind of talk as a professional calling.
Unsurprisingly, during the decades since Gödel proofs were
presented in the early 1930's, now almost a century ago, various thinkers
!5
engaged in all sorts of thinking have taken the brilliant Gödel’s result in a
variety of directions–on the one hand linking it to an espousal by Gödel of the
limits of human comprehension and on the other hand as the inspiration for
today's ongoing quest to recreate human thinking on a computer.
Some “pop” writers with a more mystical bent seem to believe that
Gödel, in the course of his proofs ot those notorious incompleteness
theorems in the 1930s, set out to prove that there are limits to how much of
reality mathematical logic can grasp — something many "science bashers" and
holistic evangelists have claimed to have intuited but none has been able to
substantiated other than through endless rhetoric.
On the contrary, what we see is mathematics flowering as
organically as any living form on the planet through an exquisite selforganizational burst we have witnessed in the past two centuries, even it's
weeds seem to eventually become recognized as flowers. Within its own
expressions and formulations it has found the roots expanding its grasp
stunningly.
!6
This. to us, is not a sign that there is a problem of its' reach exceeding
its grasp, or being bounded in some confining, almost claustrophobic way by
any quaint and odd notion that there is some vast finite or even infinite
reality out there beyond the reach of our eyes and ears that we can never
"understand" that provides any basis for concern of fretting about our
own boundedness and the limits of mathematics tools of exploration.
Goldstein says very nicely, in her interview with The Edge, Godel
and the Nature of Mathematical Truth,
"Gödel mistrusted our ability to communicate. Natural language, he
thought, was imprecise, and we usually don't understand each other.
Gödel wanted to prove a mathematical theorem that would have all
the precision of mathematics—the only language with any claims to
precision—but with the sweep of philosophy.
"He wanted a mathematical theorem that would speak to the issues
of meta-mathematics. And two extraordinary things happened.
One is that he actually did produce such a theorem.
The other is that it was interpreted by the jazzier parts of the
intellectual culture as saying, philosophically exactly the opposite of
what he had been intending to say with it."
To fret over the inability of mathematics to somehow capture the full
scope of 'reality' (however vast the vastness of that also rather quaint historical
notion may be) is akin to being concerned over the infinite array of
integers being confined in some awfully anxiety provoking way because there
may be and are further infinite domains of integers which can be seen as
generated from any given array of infinite integers
!7
Ironically, this very difficulty with conceiving of "infinities" adequately
and of appreciating that there are hierarchies of infinities and indeed infinite
infinities that may be considered, created such a controversy (still ongoing)
that 'understanding" the concept of infinity is what was behind the historical
events leading to Gödels' work aimed at understanding a mathematics itself
one of whose key
concepts was infinity.
As many of the
mathematicians pointed
out, the debate itself
reveals a lack of human
intuition regarding the
concept of infinity
If anything, if we do
manage to arrive via our discussion here at a beginning of an understanding of
the notion of "understanding", his proofs in fact, we hope will be regarded by
readers as pointing, via their demonstration of the incompleteness of many of
our formal systems of pedestrian stepwise trudging via
logical connectors not to a pessimism about our limits as 'homo sapiens",
but rather that the incompleteness demonstrated by was his way of pointing
to the limitless and unbounded nature of our understanding..when that notion
is properly understood.
If there is any doubt about that, we only have to look at the
incredible flood of new mathematical formulations for which Godel's
work planted the seeds.
!8
What we have here is the archetypal case of the story of 'the man
who pointed to the moon, while we, most of the rest of us, have thus far
pointed to his finger".
We could hardly stop from standing up and saying “bravo” when we
read Jim Holt, hin his delightfully deep and profoundly delightful essay in the
New Yorker, What Were Einstein and Godel Talking About", characterize Godel
in a way which seemed to have escaped most of his commentators…including
all those in video roundtable, as well Holt writes of Godel,
when “significance of Gödel’s theorems began to sink in, words like
“debacle,” “catastrophe,” and “nightmare” were bandied about.
“It had been an article of faith that, armed with logic,
mathematicians could in principle resolve any conundrum at all—
that in mathematics, as it had been famously declared, there was
!9
no ignorabimus. Gödel’s theorems seemed to have shattered this
ideal of complete knowledge.”
“That was not the way Gödel saw it” (and we add “amen”)
“He believed he had shown that mathematics has a robust reality that
transcends any system of logic. But logic, he was convinced, is not the
only route to knowledge of this reality; we also have something like
an extrasensory perception of it, which he called “mathematical
intuition.”
Rebecca Goldstein talks
about the friendship between
Godel and Einstein, and she tells
us that what bound them so
closely for so many years, among
other things that we hope to
explore, was a common dream.
Einstein, during the later years of
his life at Princeton, remarked
that at that stage of his life, his
own work did not mean that much, but that he stayed at Princeton only for the
privileged of taking his walks home with Godel. “Einstein and Gödel”
She writes,
'What bonded them was that, first of all, they were so keenly
interested in the meta-questions of their respective fields, those
interpretive questions regarding what is it that these fields are really
doing and how is it that they manage to do it
!10
” Both Einstein and Gödel who had a legendary friendship when they
were at the Institute for Advanced Study—could not have been
more committed to the idea of objective truth.”
They were united by how much they shared their feeling of being
disaffected from the popular fashions and fads in their fields of and how the
"ultimate questions" were not being addressed by their colleagues ,
But as they walked and as they talked
all those years, although they had a
common dream and grand dream,
that perplexing notion of "truth" was
such that it encompassed two very
different ways of reaching for
its' "understanding".
!11