...ROTA: Philosophers are needed today more than ever to tell the A1
engineers some unpleasant truths. The philosopher's role has always
been that of stating facts that may have been on everybody's mind
but that no one dared state clearly. Eventually, engineers will reluctantly
acknowledge that what the philosopher says is the truth, but
they will then get rid of the philosopher.
Let me give you an example. In the early fifties engineers attempted
to build a machine that translated from one language to
another. They fell flat on their faces because they had unclear ideas on
language. A good philosopher would have said, "You must begin by
realizing that language is not what you thought it to be. You must
bring out your unverbalized prejudices and observe language objectively
without a screen of preconceived ideas on what it ought to be
like." Research is sometimes not so much discovering something
new as becoming aware of prejudices that stop us from seeing what is
in front of us. For example, a naive view ofwords states that, by and
large, they have a fixed meaning. Contemporary philosophy stresses
instead the variety of possible contextual senses. The problem of
meaning is the problem of describing the nature of the interaction
between the inherited meaning of a word and the variable contextual
senses it may have. For example, when someone utters a sentence,
you can understand it, because of your anticipation of what comes
next. This element of anticipation is essential in all grasp of meaning.
It's easy to write poems about it, but try to write down the formal
rules! This is precisely the task contemporary cognitive philosophy
has set itself.
...ROTA: The notion of axiomatic description that we have today is
engineers some unpleasant truths. The philosopher's role has always
been that of stating facts that may have been on everybody's mind
but that no one dared state clearly. Eventually, engineers will reluctantly
acknowledge that what the philosopher says is the truth, but
they will then get rid of the philosopher.
Let me give you an example. In the early fifties engineers attempted
to build a machine that translated from one language to
another. They fell flat on their faces because they had unclear ideas on
language. A good philosopher would have said, "You must begin by
realizing that language is not what you thought it to be. You must
bring out your unverbalized prejudices and observe language objectively
without a screen of preconceived ideas on what it ought to be
like." Research is sometimes not so much discovering something
new as becoming aware of prejudices that stop us from seeing what is
in front of us. For example, a naive view ofwords states that, by and
large, they have a fixed meaning. Contemporary philosophy stresses
instead the variety of possible contextual senses. The problem of
meaning is the problem of describing the nature of the interaction
between the inherited meaning of a word and the variable contextual
senses it may have. For example, when someone utters a sentence,
you can understand it, because of your anticipation of what comes
next. This element of anticipation is essential in all grasp of meaning.
It's easy to write poems about it, but try to write down the formal
rules! This is precisely the task contemporary cognitive philosophy
has set itself.
...ROTA: The notion of axiomatic description that we have today is
inadequate to
render this difference. All axiomatic descriptions of
"but"
have failed, not to speak of those of "nevertheless."
SHARP: Or
"meanwhile."
ROTA: The language we
speak is at odds with logic. It used to be
thought
that formal logic is a rendering of our reasoning process, but
after the A1 experience
we have realized that formal logic is just
another
flight from reality...
...ROTA: We must first look for the unstated wishes that we have in the
back of our minds when we ask for foundations of mathematics or
science. When you search deeply into the cravings of the Western
mind, you discover the craving that all things should be reduced to
one, that the laws of nature should all be consequences of one simple
one, that all principles should eventually be reduced to one principle.
It's a great Jewish idea. One God, one this, one that, one everything.
We want foundations because we want oneness.
SHARP: Isn't this another instance of reductionism?
ROTA: This craving for reducing all physical laws to one law may be
a delusion. Einstein was the last genius of oneness. Maybe he is right.
But more probably, we will have to get used to several sets of laws of
nature, existing together and irreducible to one another. The laws
that describe living systems, if any, will not be reduced to the laws of
physics or to the laws of cognitive behavior, if any...
ROTA: We need to understand how sense-making arises out of
staring at physical data that by themselves are meaningless. How can
you get sense by merely looking...
...ROTA: We must first look for the unstated wishes that we have in the
back of our minds when we ask for foundations of mathematics or
science. When you search deeply into the cravings of the Western
mind, you discover the craving that all things should be reduced to
one, that the laws of nature should all be consequences of one simple
one, that all principles should eventually be reduced to one principle.
It's a great Jewish idea. One God, one this, one that, one everything.
We want foundations because we want oneness.
SHARP: Isn't this another instance of reductionism?
ROTA: This craving for reducing all physical laws to one law may be
a delusion. Einstein was the last genius of oneness. Maybe he is right.
But more probably, we will have to get used to several sets of laws of
nature, existing together and irreducible to one another. The laws
that describe living systems, if any, will not be reduced to the laws of
physics or to the laws of cognitive behavior, if any...
ROTA: We need to understand how sense-making arises out of
staring at physical data that by themselves are meaningless. How can
you get sense by merely looking...
 |
| Gian-Carlo Rota
-----
[Godel] clung throughout his life to the presumption that Hilbert expressed in his 1900 address: Is the axiom of the solvability of every problem a peculiar characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable? In line with that presumption, Godel was not content merely to demonstrate the possibility of alternative models - to establish, that is, the consistency of the underlying theories; he sought to determine which model was the correct one. His consummate faith (so often vindicated) in the power of his own mathematical intuition led him to search for axioms from which the Continuum Hypothesis might be decided, to find a consistency proof for arithmetic based on constructively evident, though abstract, principles...What is most striking about Godel's philosophical stance is his resolute optimism in confronting the implications of his own results. Having demonstrated that the axiomatic method is fundamentally inadequate for number theory insofar as the truths of arithmetic cannot all be obtained as theorems within any fixed recursively axiomatized system, he saw no reason for despair. In contrast to Post and Turing, he construed the incompleteness theorems not as establishing limitations on the power of human reasoning, but rather as showing "that the kind of reasoning necessary in mathematics cannot be completely mechanized" and thus as affirming the role of the human intellect in mathematical research. Because of his belief that our minds are "not static, but constantly developing," he was confident that new mathematical insights [the "missing axioms"] - principles that "force themselves upon us as being true" - would continue to emerge. -John Dawson, Logical Dilemmas: the life and work of Kurt Godel, p.263 |