Kant says all mathematical judgments are synthetic. To clearly understand why this is the case one need only apply analysis to the proposition 7 + 5 = 12. In doing so, without introducing any new concepts, it is evident that the analytic response is merely the union of the two numbers 7 and 5, or 75. However, by applying synthesis “we have to go outside these concepts, and call in the aid of the intuition,” in order to create a concept not present in the judgment. But what is this intuition that we rely upon to complete the synthetic judgment? “Intuition is that through which [a mode of knowledge] is in immediate relation to [an object], and to which all thought as a means is directed.” By means of sensibility representations of objects are given to us, and it is this very process that yields intuition, the thoughts of which give rise to concepts. Our ability to intuit new concepts to the synthetic mathematical judgment allows us to arrive at 12, the answer to the question of 7 + 5. By the aid of intuition synthesis is made possible. Mathematical judgments are not only synthetic but they are also judgments of a priori knowledge. A priori is the complete absence of experience; experience gives rise to empirical judgments. Mathematical judgments are a priori because “they carry with them necessity, which cannot be derived from experience.” Necessity and strict universality are criteria for all a priori knowledge. Therefore, all mathematical propositions are synthetic judgments a priori because they could not have been otherwise; that is to say, mathematical judgments are metaphysically necessary because there are no possible worlds in which 7 + 5 ≠ 12. In calculating the mathematical formula 7 + 5 = 12, we add to the concept of 7, only with the aid of intuition, such as 5 fingers, 5 blocks, or any other combination of objects that amount to the number 5, and the successive total produces the number 12. Now it is evident that the concept 12 has been added to the concept of 7 + 5, when no amount of analysis could produce such a concept; only by way of intuiting the concept 5 to the concept of 7 can we arrive at our destination. 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… For starting with the number 7, and for the concept of 5 calling in the aid of 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 5, and with the aid of that figure [the hand] see the number 12 come into being. It is only by way of intuition that one arrives at a construction of the concept 12. By adding the fingers one to another our synthesis makes possible a new concept, and our product of 12 follows a priori. It is the presence of the 5 fingers in Kant’s example that affords him the ability to intuit the new concept and arrive at the sum of 12. In other words, the fingers fulfill a necessary requirement in the process of the mathematical judgment. They allow for the successive adding of numbers one to another, at a given space throughout a given period of time. Only as the fingers are counted one by one do they allow us to arrive at the sum of 12; if the fingers did not occupy a given point in space during a successive period of time then no amount of calculation would yield the sum of 12. In Kant’s example the fingers are the intuition that makes the mathematical calculation possible thereby making the mathematical judgment 7 + 5 = 12 a necessary and strictly universal truth. That we are able to know synthetic judgments a priori follows from what has already been said on the subject. How we are able to know is the more interesting and still more difficult question. Precisely how are we able to gain knowledge from synthetic judgments a priori, how is it possible? We already know that intuition is responsible for making synthesis possible, but what are the other factors that contribute to this grand knowledge arising from pure reason? Kant says, “Time and space, taken together, are the pure forms of all sensible intuition, and so are what make a priori synthetic propositions possible.” Clearly the answer to what makes synthetic judgments a priori possible is the conjunction of time and space, that is, together the two make possible all pure forms of sensible intuition. At first glance this might appear siple, but the complex notions of time, space, and even pure sensible intuition must now be made clear in order to fully understand Kant’s answer. Space is that thing which gives rise to all objects outside us; also it is that which makes our own existence possible, while also providing a construct for representing things outside us. All objects reside in space, and ther’s only one all encompassing, simultaneous space, which results in all objects standing in relation one to another. Kant says, “Space is a necessary a priori representation” of our outer sensibility. It should be remembered that space is one form of pure intuition. As sch we interpret space, and the objects therein, with a subjective sensibility. We perceive objects in space according to the way in which our sensibility represents them. However, “objects in themselves are quite unknown to us.” This statements suggests that Kant believes while we derive our sensible representations from space, the true nature of such is unattainable. We are left with building our foundation of knowledge on appearances rather than the true nature of things. If this holds true one could argue that the error at the beginning of our foundation of knowledge might have a multiplier effect that extends even into the realm of a priori knowledge. This cannot be the case though, as a priori knowledge is subject to necessity and strict universality, such as been proved of the mathematical formula 7 + 5 = 12. One further comment on space is that interestingly we cannot conceive of the absence of space but only the absence of things in space. Kant said that space is unified, or simultaneous. Time, however, is not simultaneous but successive. True there is only one dimension of time; nonetheless, each additional moment of existence is another succession in time. Time is unlimited, thus necessary and universal, and absolutely a priori. “Time is the formal a priori condition of all apperances whatsoever.” Time is a priori because it is prior to the representation of all inner intuitions. It is only in time that our intuitions are capable of being represented. Space makes possible all outer intuitions, but time, insofar as representations belong in themselves, make both outer and inner intuitions possible, and Kant is therefore justified in his claim that time is the formal a priori condition of all appearances whatsoever. To make completely clear the notion of time as a pure form of a priori inner intuition it is necessary to understand that time exists in and of itself. “[Time] has objective validity only in respect of appearances, these being things which we take as objects of our senses.” If all sensibility, or sense perception, were removed then likewise time would vanish. This can be understood insofar as our intuitions, both inner and outer, reflective of both time and space, are subject to our peculiar condition of sensibility. Our pure form of sensibility is tantamount to pure intuition, which as previously noted is a condition for a priori knowledge. All that has been said regarding space and time is critical to understanding just how intuitions allows us to calculate synthetic mathematical judgments a priori. To make perfectly clear the way one arrives at the sum of 12 when adding 5 to the number 7 it will be beneficial to the reader to participate in an interactive example. Lending to the attentive and close scrutiny the reader has taken in contemplating this essay it is safe to assume he has noticed the multiple errors by way of grammatical contractions and misspellings. Those errors were done intentionally in hopes that the reader would make some mark at each of the instances of infraction. The condition of the markings is unimportant. that the reader has made some notation is of importance; whether it is underlining, circling, crossing-out, or any other intentional attempt to highlight the infraction. Now if the reader will kindly turn back and count the errors in this essay. At this point it should be obvious the vehicle of intuition used to arrive at the total of 4 (assuming there are no other errors beyond those intentionally performed) are the markings themselves. In other words, the markings just are the forms of intuition used in that given space during the successive period of time that allows the reader to reach the sum of 4. The exercise of this example should make crystal clear the way in which one calculates a mathematical judgment. Again, by use of the markings as aids of intuition one is able to add the errors one to another, through a period of time, thereby making necessary connection with the markings until arriving at the sum of 4. Kant’s example of 7 + 5 = 12 is calculated by the same method of using objects as aids of intuition to create a new concept not expressly thought in the mathematical judgment. With close inspection it becomes clear that Kant uses circular reasoning to explain the possibility of synthetic a priori knowledge. His reasoning follows this pattern: (1) label time and space as the factors that make synthetic judgments a priori possible, (2) define pure sensible intuition as the combination of space and time, (3) then he claims, “The pure form of sensible intuitions…must be found in the mind a priori.” Therefore, space and time, which are the pure forms of sensible intuition and must be in the mind a priori, are responsible for making synthetic judgments a priori possible. Now we have gone full circle in search for what makes synthetic judgments a priori possible. In other words, what makes knowledge from pure reason possible just are forms of a priori knowledge, namely space and time. Viewed this way it is obvious why Kant labels the support of understanding what makes synthetic judgments a priori possible as the “unknown = X.” For that which makes it possible is further still a priori knowledge, and even the layman knows you cannot use that which you are defining in your explanation or definition; but this is precisely what Kant does. However, the reason he does this is because he recognizes the antecedent nature of time, which in itself makes possible all notions of space wherein objects reside and are subject. Specifically, it is the successive nature of time that affords us the opportunity of any knowledge, and makes possible the celebrated synthetic a priori knowledge form the exercise of pure reason. It is only when things, such as fingers or markings on a paper, come into time that one is able to use them as aids of intuition. It is the successive nature of time that allows us to arrive at the necessarily true and strictly universal concept of 12.

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