In what way is a priori knowledge apodictic that makes it impossible for empirical knowledge to provide the same universality and necessity? — darthbarracuda
The way in which pure
a priori knowledge is apodeitic is becaiuse it arises from the understanding alone, an internal cognitive faculty, thereby granting sufficient causality for certainty, in that there is no other influence on it. That which arises from itself, cannot be other than it is, which holds the same value as truth, but truth restricted to the very domain from which it is given. Now “not other than it is” may eventually be shown to be false, but at the originating time of it, the certainty is not questionable, and if eventually shown to be false, it cannot be of the same domain from which it originated, for in such case, there is an outside influence. It is clear from this stipulation, the only possible pure
a priori knowledge is in the form of principles, or the laws derivable from them, either with respect to the physical domain, which is properly science, “...the science of what is...”, or with respect to the metaphysical domain, which is properly morality, “...the science of what ought to be...”. Again, a furtherance of the intrinsic Kantian epistemological dualism.
Universality and necessity are principles that cannot apply to anything empirical, because they are overturned by, subsumed under, the more powerful Principle of Induction, which makes explicit experience is always contingent: undeniable observational proof that what’s true today may not be true tomorrow, re: determinations of the nature of the observable Universe. What reason seeks, on the other hand, is that which is never contingent, or, which is the same thing, never self-contradictory, itself just conventional speech for seeking the unconditioned, the ideal, the irreducible. The question then becomes....does reason ever reach that state of affairs, and the Kantian speculative metaphysics proves it does not, and it cannot.
Given what reason cannot do, it remains to be determined what reason can do, the controlling condition being the LNC, which immediately suggests the entire human cognitive system is inherently logical. This, in turn, makes it impossible to demonstrate how logic itself comes about, but instead, must simply be granted as being the case. Otherwise, no theoretical sciences of any kind that are predicated on logical propositions can facilitate knowledge, which means we can never claim knowledge of anything at all.
That 7 + 5 = 12 is a synthetic a priori true proposition is certainly plausible to me, but that it is necessarily and universally true that 7 + 5 = 12 is not. — darthbarracuda
Truth here is irrelevant. Synthetical propositions denote nothing but the relation of conceptions to each other, with no judgement as to the truth of the proposition being enabled. Again...dualism, in that synthetical is only to differentiate a kind of relation of conceptions from its complement, the analytical. It is identity, not truth, which makes these relational determinations. Analytical propositions are those in which the conceptions hold similar identities, synthetical propositions are those in which identity does not hold. Identity herein meant to indicate only that the conception in the predicate of a proposition can be found in the subject of that same proposition. In synthetical propositions, then, the conception in the predicate cannot be found in the subject.
From that it follows that while 7 + 5 = 12 is synthetical, in that neither of the numbers to the left, in and of themselves, can give the number on the right.....
“....The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, and have recourse to an intuition which corresponds to one of the two—our five fingers, for example, or like Segner in his Arithmetic five points, and so by degrees, add the units contained in the five given in the intuition, to the conception of seven. For I first take the number 7, and, for the conception of 5 calling in the aid of the fingers of my hand as objects of intuition, I add the units, which I before took together to make up the number 5, gradually now by means of the material image my hand, to the number 7, and by this process, I at length see the number 12 arise....”
(Added in B16, not found in A)
.....the fact that the proposition is true only arises from empirical proofs, in which it is found that it is impossible for this particular arithmetic operation to give a different result, and thereby sets the stage for establishing the criteria for any mathematical entailment, and in turn, establishing the possibility and the validity of pure
a priori conditions in general. In this way, all logical propositions determined by reason are in logical form only, the content be what it may. From the simplest analytical proposition, A = A, to the most complex abstract synthetical mathematical calculus, the proofs of all logical forms depend on empirical conditions.
For what reason do I have to believe that it may not be different in the future? — darthbarracuda
According to Kantian metaphysics, you don’t. Knowledge destroys belief, so if you know without the possibility of refutation that mathematical propositions are the mark of absolute certainty, because you can prove all of them to yourself, you have no reason whatsoever to doubt them. It behooves one, nonetheless, to keep in mind such certainty is only determinable under the auspices of the very system from which the the ground for it is given. In such case, not only is it impossible to doubt this certainty, but it is just as impossible to think of what form the doubt would have.
The justification of all this, is in the categories, the “...pure conceptions of the understanding...”, from which are given the schema of “quantity”, first in the form of numbers, and thereafter in the form of unity, the manifestations of the permissible connectedness of numbers. Because it is the case, at least in this particular epistemological theory, that the categories are absolutely essential, and given that the schema of the categories are always the same, it becomes impossible to arrive at different conclusions for any one proposition predicated on them, assuming internal logical consistency is met, the primary condition of the system as a whole. Still, justification is not proof, which, as already shown, is entirely dependent on empirical conditions.
Universality and necessity, in fact any terminology of any kind, the categories, even reason itself, if developed by humans, only applies to humans. Mathematical propositions will therefore be true, iff a human is responsible for them. They will be true wherever and whenever there is a human to think them, but not necessarily otherwise. To a rational agent with other than a intuitive/discursive cognitive system, nothing about mathematical truths, or any truths at all, can be said. Does 1 + 1 = 2 to an elephant? Or a resident of a planet we don’t even know about? Not only can we not say, but we don’t even have the means to understand how to ask.
And Nagel thought himself the first to wonder. No reason for it, really, for the answer had already been given, fully 200 years before he even thought about it.