In all fairness, Frege was not the one responsible for interpreting truth
with
respect to model-theoretic objects. But, the current state of affairs is
a
result of logicist thinking that built upon his early work. Note,
however, this
excerpt from "The Frege Reader,"

"[In a diary that he kept from 10 March to 9 May 1924,
Frege wrote, on 23 March: 'My efforts to become clear
about what is meant by number have resulted in failure.
We are only too easily misled by language and in this
particular case the way we are misled is little short
of disastrous.' In the last eighteen months of his life,
Frege attempted to come to terms with this, and drew
the conclusion that arithmetic had to be given a
geometrical rather than logical foundation.
mitch

"mitch" <mitchs@rcnNOSPAM.com> wrote in message
news:3FE272C6.6BB5715@rcnNOSPAM.com...


Immortalist wrote:

"mitch" <mitchs@rcnNOSPAM.com> wrote in message
news:3FD85571.676C24B@rcnNOSPAM.com...


Yuck. You know, I'm just a guy who likes math and knows the
difference
between
a Boolean algebra (Frege and logicism) and a Boolean ring (Kant and
intuitionism).


Can you explain that in a coherent way?


Kant distinguished between analytic and synthetic on the basis of
"identity."
But, this is a somewhat ambigous statement in light of modern
developments.

In this particular instance, I mean that "Every Boolean algebra of sets is
a
Boolean ring of sets" follows from topological closure properties rather
than
the arithmetic properties. The arithmetic difference is in the "group
addition."

Addition in a Boolean algebra of sets is set-theoretic union:

A cup B


Addition in a Boolean ring of sets is set-theoretic symmetric difference:

(A - B) cup (B - A)


Classical model theory requires that the universal quantifier is
interpreted
with a set. This forces model theory to deal with Boolean algebras and
forces
set theory to deal with an infinite regress or a chaotic tree (?) of
models.

Goedel's constructible universe is a "class model." It does not require
one to
define truth in terms of an object. Its axiomatic assertion, V=L,
respects the
mathematical structuralism implicit to Kant's delineation between the
Transcendental Aesthetic and the Transcendental Logic.

In all fairness, Frege was not the one responsible for interpreting truth
with
respect to model-theoretic objects. But, the current state of affairs is
a
result of logicist thinking that built upon his early work. Note,
however, this
excerpt from "The Frege Reader,"

"[In a diary that he kept from 10 March to 9 May 1924,
Frege wrote, on 23 March: 'My efforts to become clear
about what is meant by number have resulted in failure.
We are only too easily misled by language and in this
particular case the way we are misled is little short
of disastrous.' In the last eighteen months of his life,
Frege attempted to come to terms with this, and drew
the conclusion that arithmetic had to be given a
geometrical rather than logical foundation. [...] Frege
distinguishes three sources of knowledge: sense perception,
the logical source of knowledge, and the geometrical and
temporal sources of knowledge. Frege continues to insist
that 'For mathematics on its own, we do not need sense
perception as a source of knowledge' but he now also
rejects logic as providing the sole source of knowledge
of arithmetic. [...]]"


Is he saying that we can get rid of sense perception or that we never needed
it to be able to function without it. Seems irreducable but still observable
if function is uninterupted.

"mitch" <mitchs@rcnNOSPAM.com> wrote in message
news:3FE272C6.6BB5715@rcnNOSPAM.com...
In all fairness, Frege was not the one responsible for interpreting truth
with
respect to model-theoretic objects. But, the current state of affairs is
a
result of logicist thinking that built upon his early work. Note,
however, this
excerpt from "The Frege Reader,"

"[In a diary that he kept from 10 March to 9 May 1924,
Frege wrote, on 23 March: 'My efforts to become clear
about what is meant by number have resulted in failure.
We are only too easily misled by language and in this
particular case the way we are misled is little short
of disastrous.' In the last eighteen months of his life,
Frege attempted to come to terms with this, and drew
the conclusion that arithmetic had to be given a
geometrical rather than logical foundation.
mitch

The idea that geometry (did he mean analytical geometry or Euclidian?) is a
suitable basis for arithmetic is strange. This seems to prefer the notion of
number as extension (interval) to number as counter. It seems to me that
'continuity' is inherent in the primitive idea of number as extension and
'discontinuity' in the idea of number as point. Both point and extension
occur in geometry which renders it paradoxical (the problem of continuity).

Provided that a logical foundation can be freed from the ambiguities of the
meta-language (English, German, French etc) there seems no reason to prefer
geometry to logic.

Indeed, I find it difficult to understand how there could
be a geometrical foundation for logic, beyond the topology of sets (say).

This latter approach would soon tax the intuition beyong the third
dimension.

The use of purely symbolic means (axiomatics) seems more
practical.

On Thu, 18 Dec 2003 21:38:46 -0600, mitch <mitchs@rcnNOSPAM.com
wrote:



Immortalist wrote:

"mitch" <mitchs@rcnNOSPAM.com> wrote in message
news:3FD85571.676C24B@rcnNOSPAM.com...


Yuck. You know, I'm just a guy who likes math and knows the difference
between
a Boolean algebra (Frege and logicism) and a Boolean ring (Kant and
intuitionism).


Can you explain that in a coherent way?


Kant distinguished between analytic and synthetic on the basis of "identity."
But, this is a somewhat ambigous statement in light of modern developments.

You are talking about Immanuel Kant, 18th century Critical
philosopher, correct? Because that sure doesn't sound like the "Kant"
myself and others are familiar with.

HPO Jury = Malenoid wrote:

On Thu, 18 Dec 2003 21:38:46 -0600, mitch <mitchs@rcnNOSPAM.com
wrote:



Immortalist wrote:

"mitch" <mitchs@rcnNOSPAM.com> wrote in message
news:3FD85571.676C24B@rcnNOSPAM.com...


Yuck. You know, I'm just a guy who likes math and knows the difference
between
a Boolean algebra (Frege and logicism) and a Boolean ring (Kant and
intuitionism).


Can you explain that in a coherent way?


Kant distinguished between analytic and synthetic on the basis of "identity."
But, this is a somewhat ambigous statement in light of modern developments.

You are talking about Immanuel Kant, 18th century Critical
philosopher, correct? Because that sure doesn't sound like the "Kant"
myself and others are familiar with.


Malenor,

If you think I have mistated my quotes in the other reply, please let me know.

I wish I had your knowledge practical reason and judgement. Your input on things
where I went too far afield would add a lot to Immortalist's questions. I know my
limitations.

On the other hand, Kant enjoyed a certain resurgence in the nineteenth century as
a philosophy of science and was embroiled in the twentieth century disputes on the
foundations of mathematics. I ended up where I am because of "reverse
engineering" things I had been taught until I concluded that Kant's explanation of
mathematics as synthetic a priori knowledge was the only reasonable conclusion.

I never expect anyone to understand the mixture of my discussions anymore. As I
understand it from casual reading of newsgroup posts, Kant thought the sciences
should not confuse their subject matter. But, most of them reverted to
mathematics as a means of justification.

Mathematics is *not* a justification for knowledge.

As you have noted with respect to Einstein and general relativity, there are
teleological principles at work. The introduction of formal languages and
deductive calculi have motivated analytical philosophers to a form of reductionism
wherein epistemic logic is a variant of modal logics. In turn, completeness in
modal logics is closely related to the formalisms used to investigate temporal
logics. In fact, there is only one altered axiom between the epistemic logic of
philosophers and the epistemic foundatin used for temporal logic in computational
systems. On the other hand, Frege tried to use his formal system to ground
arithmetic, and, this program of logicism was cemented by Russell and Whitehead.
Of course, Kant had associated arithmetic with temporality.

In the meantime, Husserl observed that the development of probability theory
should be viewed as evidence for rejecting the idea that mathematics was simply
the science of number. While Russell and Whitehead pushed forward with a calculus
intended to ground arithmetic, physicists pursued explanation of phenomena
associated with spatio-temporal intuition along two lines of evidence. Einstein's
commitment to the continuum can be captured through Hilbert's geometry and later
work by Alexandrov in combinatorial topology. The issue is decomposition of space
into tetrahedra and the fact that a tetrahedral simplex has a relative metric with
the same polynomial form as an expectation in probability theory (actually, that
is true of any relative metric). The other line of investigation was, of course,
quantum mechanics. Most people are aware of its close association with
probabilities.

But, very few people are aware of the close association of geometric sphere
packing problems with the error correction codes used in analog-to-digital
quantization. Moreover, the geometric representations involved are tetrahedra
whose apex is the point at infinity for a spherical geometry, whose spherical
triangle is the intersection of the tetrahedral frontier with the fixed sphere,
and whose planar triangle is the intersection with the plane corresponding with
the spherical geometry.

I suppose we should also keep in mind that Frege did not read a great deal of
Kant's philosophy before grounding modern analytical philosophy. The idea of
cardinal number and its representation in set theory is closely associated with
the notion of objects under a concept.

Really. How long do you think it would be before someone would read Kant and
decide that the ambiguities of modern set theory were just plain nonsense?

Really. How long do you think it would be before someone would read Kant and
decide that the ambiguities of modern set theory were just plain nonsense?

I don't know. But is it very difficult for a bricklayer to find a
publisher for these ideas?

"mitch" <mitchs@rcnNOSPAM.com> wrote in message
news:3FD55D82.B3D9F2AF@rcnNOSPAM.com...


Orb wrote:

So, I'm writing my final paper for my Kant class, and I've decided to
write on whether or not Kant could conceive of a being that existed
outside of time. I know he had some of notion of a god, but I'm really
more concerned with his view of time (as explained in the Critique of
Pure Reason).

Assuming God exists, and assuming that Kant's view of time and space is
true, what would the nature of such a God be like?

Any ideas?


From my understanding, Kant's basic premise in "Critique of Pure Reason"
is
that the answer to such a question is not meaningful. He does, however,
discuss related issues in the ideal of pure reason--a later section of the
book.

You might consider posting your question to alt.philosophy.kant. There
are
one or two knowledgeable individuals that might be able to help you more
substantially.


Hey Mitch what do you think of what these Yahoo's gotta say about Kant, God
& Time?

For Kant the Christian could have faith in God, and this faith would be
consonant with reason and the categorical imperative. Given that human
beings have the autonomy to create moral values, it would not be irrational
to believe in a God who gives purpose to the moral realm.

S?ren Kierkegaard (1813-1855) agreed with Kant that the existence of God
could not be proven by reason. However Kierkegaard did not think that it was
rational to believe in God, rather one should have faith in God even if this
seems to reason to be absurd. To put it another way reason has no place in
faith. God is beyond reason.

http://www.philosopher.org.uk/god.htm

---------------------------------

But if the mind actively generates perception, this raises the question
whether the result has anything to do with the world, or if so, how much.
The answer to the question, unusual, ambiguous, or confusing as it would be,
made for endless trouble both in Kant's thought and for a posterity trying
to figure him out. To the extent that knowledge depends on the structure of
the mind and not on the world, knowledge would have no connection to the
world and is not even true representation, just a solipsistic or
intersubjective fantasy.

Kantianism seems threatened with "psychologism," the doctrine that what we
know is our own psychology, not external things. Kant did say, consistent
with psychologism, that basically we don't know about
"things-in-themselves," objects as they exist apart from perception. But at
the same time Kant thought he was vindicating both a scientific realism,
where science really knows the world, and a moral realism, where there is
objective moral obligation, for both of which a connection to external
existence is essential.

And there were also terribly important features of things-in-themselves that
we do have some notion about and that are of fundamental importance to human
life, not just morality but what he called the three "Ideas" of reason:
God, freedom, and immortality. Kant always believed that the rational
structure of the mind reflected the rational structure of the world, even of
things-in-themselves -- that the "operating system" of the processor, by
modern analogy, matched the operating system of reality.

But Kant had no real argument for this -- the "Ideas" of reason just become
"postulates" of morality -- and his system leaves it as something
unprovable. The paradoxes of Kant's efforts to reconcile his conflicting
approaches and requirements made it very difficult for most later
philosophers to take the overall system seriously.

http://www.friesian.com/kant.htm

-------------------------------------

Kant locates the order of nature in reason. Reason does for the
understanding what understanding does for the manifold of intuition - "the
understanding is an object for reason, just as sensibility is for the
understanding." Reason's regulative capacity renders the unconditioned
totality of objects systematic. There are three ideas of reason: self, world
and God. God is the Ideal of Reason, whose concept

aims . . . at complete determination in accordance with a priori rules.
Accordingly it thinks for itself an object which it regards as being
completely determinable in accordance with princi ples,

that is, in accordance with universal a priori cognition. This ideal of the
ens realissimum, of the universal concept of a reality in general, is then
thought of as containing the being of all beings. But as an idea of reason,
the ens realissimum is never met with in appearances. The Ideal of Reason
does not satisfy the transcendental conditions and so cannot be considered
objectively real. As such, Kant holds that the existence of God cannot be
proved by speculative reason.

Kant argues that there are three, and only three, possible ways in which
speculative reason can argue for the existence of God, characterized as the
Ideal of Reason. But all fail to prove God's existence. Reason, according to
Kant's analysis, can attempt to prove God's existence by either an empirical
or a transcendental path, both of which involve going beyond the scope of
reason to the transcendental concept. Consequently, Kant first discusses the
transcendental proof of God's existence, arguing that the other proofs
ultimately depend on it. The ontological argument moves from idea to
existence, arguing that the essence of a supreme and perfect being involves
existence. Kant argues that this type of proof fails to recognize that
existence is not a predicate, and that to call it so is to claim that there
is a quality in the world corresponding to it.

...In his two earlier works, Kant asks the question 'How is the realm of
possibility possible?', whereas in the Critique the question of possibility
has become an epistemological problem rather than an ontological one. The
problem is now seen in terms of a structure of conditions related to the
possibility of cognition, not as a problem requiring the proof of a single
actual ground. Kant's argument that existence is not a predicate is given
new force by his transcendental analysis, which shows that God's existence
is not an issue in the realm of reason. If the transcendental analysis is
accepted, and we therefore see that all existential propositions are
synthetic, we cannot, without contradiction, maintain that existence is a
predicate.

...Kant's rejection of all possible proofs of God's existence and his moving
God out of the sphere of ontology, rules out the traditional ground of a
systematic universe. He therefore must provide some other explanation of how
we perceive the world as systematic and purposive.26 Some principle of
systematicity is necessary to account for the interconnectedness or
coherence we perceive in nature. Kant explains systematicity in terms of
epistemology. For Kant, systematicity is the creative organization of our
cognition of nature in accordance with certain regulative principles. These
principles are the ideas of reason. Reason prescribes a logical principle
that assists

the understanding by means of ideas, in those cases in which the
understanding cannot by itself establish rules, and at the same time ...
give[s] to the numerous and diverse rules of the understanding unity or
system under a single principle.

At the level of the understanding there is no system. An idea, as a
regulative principle, supplies a rule for systematization. Kant
distinguishes this from the 'Ideal of Reason', which supplies the notion of
an 'archetype' or individual ground for systematization.28 This too must be
seen as only regulative, as it has no content, that is, 'God' does not
correspond to the concept of God. It is the regulative ideal of nature that
makes possible the unity of nature itself. The Ideal of nature, as
regulative, has a purely methodological status.

Kant moves from systematicity to purposiveness. He argues that saying that
all things are part of a system is tantamount to saying that all things have
a purpose within that system. To see things as systematic, for Kant, is to
see them as in "seemingly purposive arrangements." A move cannot be made
from purposive order to God, because God has no matter of intuition, that
is, God is not 'given' to sensibility and so does not conform to the
transcendental conditions. But there is an analogical correspondence between
systematicity and God. They are logically equivalent and "it must be a
matter of complete indifference to us, when we say we perceive . . . unity,
whether we say that God in his wisdom has willed it to be so, or that nature
has wisely arranged it thus." From a philosophical point of view, God as
creator corresponds to system. The principle of God is not needed to explain
nature. All that is needed is the principle of systematicity. Analogy
between teleology and theology gives meaning or content to our idea of God.
But to designate what the relationship is between God and the world in
itself is completely outside the conditions of possible experience.

In the Critique, it no longer makes sense to ask whether natural laws form a
systematic whole. The ideal of reason is simply the principle of systematic
unity. This unity has no ontological significance, since it does not exist
empirically. Rather it is "a mere fiction," or "a mere illusion," with "no
object that can be met in any experience." When we act according to the
principle of systematicity, we presuppose that nature is a completed system.
Despite this status, Kant claims that we need the idea of systematic unity,
that we "must" assume it. This is, in Kant's eyes, the strongest possible
argument for God - not for the existence of God in the constitutive or
formal sense, as no such proof is possible, but for the meaningfulness of
the idea of God in the cognitive sphere.

IMMORTALIST:
[Therefore it takes time to systematize cognitively?]

The Critique of Pure Reason, then, moves God out of the realm of ontology
and into that of epistemology. The concept of God is involved in cognition,
but is merely an analogical image. From the standpoint of speculative
reason, God has no objective reality. Yet Kant posits two types of reality,
the cognitive and the moral. These two points of view are tied together by
reason. The concept of sensation is not simply a negative boundary to stop
us from bringing up: something that lies behind sensation. Kant wants us to
leave this something completely unspecified in the realm of cognition. God
is indeterminable in the sphere of understanding, determinable in the sphere
of reason, and determinate in the sphere of moral experience. There are then
two view points, the cognitive, which includes understanding and reason, and
the realm of moral experience. Kant retains the reality and determinability
of God in the sphere of moral experience. Only the reality of God makes
morality possible. In terms of empirical cognition, however, we cannot go
beyond the bounds of the a priori conditions. Only Kant's analysis of moral
experience lets us go beyond the analogical level.

God in Kant's Critique of Pure Reason
http://www.mun.ca/phil/codgito/vol3/v3doc1.html


:-)

mitch