I
Immortalist
Guest
On Aug 23, 10:06 am, John Larkin
<jjlar...@highNOTlandTHIStechnologyPART.com> wrote:
I disagree with you idea that the higher level statements are
contradicted by new evidence, it seems that there are small changes
almost daily to sum assumed axioms.
Here Kant tries to show how adding two numbers is really a complex set
of entirely independent arguments.
V. IN ALL THEORETICAL SCIENCES OF REASON SYNTHETIC
A PRIORI JUDGMENTS ARE CONTAINED AS PRINCIPLES
1. All mathematical judgments, without exception, are synthetic. This
fact, though incontestably certain and in its consequences very
important, has hitherto escaped the notice of those who are engaged in
the analysis of human reason, and is, indeed, directly opposed to all
their conjectures. For as it was found that all mathematical
inferences proceed in accordance with the principle of contradiction
(which the nature of all apodeictic certainty requires), it was
supposed that the fundamental propositions of the science can
themselves be known to be true through that principle. This is an
erroneous view. For though a synthetic proposition can indeed be
discerned in accordance with the principle of contradiction, this can
only be if another synthetic proposition is presupposed, and if it can
then be apprehended as following from this other proposition; it can
never be so discerned in and by itself.
First of all, it has to be noted that mathematical propositions,
strictly so called, are always judgments a priori, not empirical;
because they carry with them necessity, which cannot be derived from
experience. If this be demurred to, I am willing to limit my statement
to pure mathematics, the very concept of which implies that it does
not contain empirical, but only pure a priori knowledge.
We might, indeed, at first suppose that the proposition 7 & 5 = 12 is
a merely analytic proposition, and follows by the principle of
contradiction from the concept of a sum of 7 and 5. But if we look
more closely we find that the concept of the sum of 7 and 5 contains
nothing save the union of the two numbers into one, and in this no
thought is being taken as to what that single number may be which
combines both.
The concept of 12 is by no means already thought in merely thinking
this union of 7 and 5; and I may analyse my concept of such a possible
sum as long as I please, still I shall never find the 12 in it. We
have to go outside these concepts, and call in the aid of the
intuition which corresponds to one of them, our five fingers, for
instance, or, as Segner does in his Arithmetic, five points, adding to
the concept of 7, unit by unit, the five given in intuition. For
starting with the number 7, and for the concept of 5 calling in the
aid of the fingers of my hand as intuition, I now add one by one to
the number 7 the units which I previously took together to form the
number, and with the aid of that figure [the hand] see the number 12
come into being. That 5 should be added to 7, I have indeed already
thought in the concept of a sum = 7 & 5, but not that this sum is
equivalent to the number 12. Arithmetical propositions are therefore
always synthetic. This is still more evident if we take larger
numbers. For it is then obvious that, however we might turn and twist
our concepts, we could never, by the mere analysis of them, and
without the aid of intuition, discover what [the number is that] is
the sum.
Just as little is any fundamental proposition of pure geometry
analytic. That the straight line between two points is the shortest,
is a synthetic proposition. For my concept of straight contains
nothing of quantity, but only of quality. The concept of the shortest
is wholly an addition, and cannot be derived, through any process of
analysis, from the concept of the straight line. Intuition, therefore,
must here be called in; only by its aid is the synthesis possible.
What here causes us commonly to believe that the predicate of such
apodeictic judgments is already contained in our concept, and that the
judgment is therefore analytic, is merely the ambiguous character of
the terms used. We are required to join in thought a certain predicate
to a given concept, and this necessity is inherent in the concepts
themselves. But the question is not what we ought to join in thought
to the given concept, but what we actually think in it, even if only
obscurely; and it is then manifest that, while the predicate is indeed
attached necessarily to the concept, it is so in virtue of an
intuition which must be added to the concept, not as thought in the
concept itself.
Some few fundamental propositions, presupposed by the geometrician,
are, indeed, really analytic, and rest on the principle of
contradiction. But, as identical propositions, they serve only as
links in the chain of method and not as principles; for instance, a =
a; the whole is equal to itself; or (a & b) a, that is, the whole is
greater than its part. And even these propositions, though they are
valid according to pure concepts, are only admitted in mathematics
because they can be exhibited in intuition.
2. Natural science (physics) contains a priori synthetic judgments as
principles. I need cite only two such judgments: that in all changes
of the material world the quantity of matter remains unchanged; and
that in all communication of motion, action and reaction must always
be equal. Both propositions, it is evident, are not only necessary,
and therefore in their origin a priori, but also synthetic. For in the
concept of matter I do not think its permanence, but only its presence
in the space which it occupies. I go outside and beyond the concept of
matter, joining to it a priori in thought something which I have not
thought in it. The proposition is not, therefore, analytic, but
synthetic, and yet is thought a priori; and so likewise are the other
propositions of the pure part of natural science.
3. Metaphysics, even if we look upon it as having hitherto failed in
all its endeavours, is yet, owing to the nature of human reason, a
quite indispensable science, and ought to contain a priori synthetic
knowledge. For its business is not merely to analyse concepts which we
make for ourselves a - priori of things, and thereby to clarify them
analytically, but to extend our a priori knowledge. And for this
purpose we must employ principles which add to the given concept
something that was not contained in it, and through a priori synthetic
judgments venture out so far that experience is quite unable to follow
us, as, for instance, in the proposition, that the world must have a
first beginning, and such like. Thus metaphysics consists, at least in
intention, entirely of a priori synthetic propositions.
http://www.arts.cuhk.edu.hk/Philosophy/Kant/cpr/
http://www.bright.net/~jclarke/kant/
http://en.wikipedia.org/wiki/Critique_of_Pure_Reason
<jjlar...@highNOTlandTHIStechnologyPART.com> wrote:
I agree, it kind of goes with the stereotype; "assume that A = X" But
I disagree with you idea that the higher level statements are
contradicted by new evidence, it seems that there are small changes
almost daily to sum assumed axioms.
Here Kant tries to show how adding two numbers is really a complex set
of entirely independent arguments.
V. IN ALL THEORETICAL SCIENCES OF REASON SYNTHETIC
A PRIORI JUDGMENTS ARE CONTAINED AS PRINCIPLES
1. All mathematical judgments, without exception, are synthetic. This
fact, though incontestably certain and in its consequences very
important, has hitherto escaped the notice of those who are engaged in
the analysis of human reason, and is, indeed, directly opposed to all
their conjectures. For as it was found that all mathematical
inferences proceed in accordance with the principle of contradiction
(which the nature of all apodeictic certainty requires), it was
supposed that the fundamental propositions of the science can
themselves be known to be true through that principle. This is an
erroneous view. For though a synthetic proposition can indeed be
discerned in accordance with the principle of contradiction, this can
only be if another synthetic proposition is presupposed, and if it can
then be apprehended as following from this other proposition; it can
never be so discerned in and by itself.
First of all, it has to be noted that mathematical propositions,
strictly so called, are always judgments a priori, not empirical;
because they carry with them necessity, which cannot be derived from
experience. If this be demurred to, I am willing to limit my statement
to pure mathematics, the very concept of which implies that it does
not contain empirical, but only pure a priori knowledge.
We might, indeed, at first suppose that the proposition 7 & 5 = 12 is
a merely analytic proposition, and follows by the principle of
contradiction from the concept of a sum of 7 and 5. But if we look
more closely we find that the concept of the sum of 7 and 5 contains
nothing save the union of the two numbers into one, and in this no
thought is being taken as to what that single number may be which
combines both.
The concept of 12 is by no means already thought in merely thinking
this union of 7 and 5; and I may analyse my concept of such a possible
sum as long as I please, still I shall never find the 12 in it. We
have to go outside these concepts, and call in the aid of the
intuition which corresponds to one of them, our five fingers, for
instance, or, as Segner does in his Arithmetic, five points, adding to
the concept of 7, unit by unit, the five given in intuition. For
starting with the number 7, and for the concept of 5 calling in the
aid of the fingers of my hand as intuition, I now add one by one to
the number 7 the units which I previously took together to form the
number, and with the aid of that figure [the hand] see the number 12
come into being. That 5 should be added to 7, I have indeed already
thought in the concept of a sum = 7 & 5, but not that this sum is
equivalent to the number 12. Arithmetical propositions are therefore
always synthetic. This is still more evident if we take larger
numbers. For it is then obvious that, however we might turn and twist
our concepts, we could never, by the mere analysis of them, and
without the aid of intuition, discover what [the number is that] is
the sum.
Just as little is any fundamental proposition of pure geometry
analytic. That the straight line between two points is the shortest,
is a synthetic proposition. For my concept of straight contains
nothing of quantity, but only of quality. The concept of the shortest
is wholly an addition, and cannot be derived, through any process of
analysis, from the concept of the straight line. Intuition, therefore,
must here be called in; only by its aid is the synthesis possible.
What here causes us commonly to believe that the predicate of such
apodeictic judgments is already contained in our concept, and that the
judgment is therefore analytic, is merely the ambiguous character of
the terms used. We are required to join in thought a certain predicate
to a given concept, and this necessity is inherent in the concepts
themselves. But the question is not what we ought to join in thought
to the given concept, but what we actually think in it, even if only
obscurely; and it is then manifest that, while the predicate is indeed
attached necessarily to the concept, it is so in virtue of an
intuition which must be added to the concept, not as thought in the
concept itself.
Some few fundamental propositions, presupposed by the geometrician,
are, indeed, really analytic, and rest on the principle of
contradiction. But, as identical propositions, they serve only as
links in the chain of method and not as principles; for instance, a =
a; the whole is equal to itself; or (a & b) a, that is, the whole is
greater than its part. And even these propositions, though they are
valid according to pure concepts, are only admitted in mathematics
because they can be exhibited in intuition.
2. Natural science (physics) contains a priori synthetic judgments as
principles. I need cite only two such judgments: that in all changes
of the material world the quantity of matter remains unchanged; and
that in all communication of motion, action and reaction must always
be equal. Both propositions, it is evident, are not only necessary,
and therefore in their origin a priori, but also synthetic. For in the
concept of matter I do not think its permanence, but only its presence
in the space which it occupies. I go outside and beyond the concept of
matter, joining to it a priori in thought something which I have not
thought in it. The proposition is not, therefore, analytic, but
synthetic, and yet is thought a priori; and so likewise are the other
propositions of the pure part of natural science.
3. Metaphysics, even if we look upon it as having hitherto failed in
all its endeavours, is yet, owing to the nature of human reason, a
quite indispensable science, and ought to contain a priori synthetic
knowledge. For its business is not merely to analyse concepts which we
make for ourselves a - priori of things, and thereby to clarify them
analytically, but to extend our a priori knowledge. And for this
purpose we must employ principles which add to the given concept
something that was not contained in it, and through a priori synthetic
judgments venture out so far that experience is quite unable to follow
us, as, for instance, in the proposition, that the world must have a
first beginning, and such like. Thus metaphysics consists, at least in
intention, entirely of a priori synthetic propositions.
http://www.arts.cuhk.edu.hk/Philosophy/Kant/cpr/
http://www.bright.net/~jclarke/kant/
http://en.wikipedia.org/wiki/Critique_of_Pure_Reason