Mathematics, due to its absolute certitude, was for Kant the epitome of science. And furthermore, since it was for him unquestionably a synthetic science, it served as the proof par excellence for his own theory of epistemology. Hence it is worthwhile for the student of Kant to take a moment to consider his argument with regard to mathematics, for it should then shed light on his entire system of thought.

[* Kant's treatment of arithmetic is presented in the first few paragraphs of Section V of the Introduction to his Critique of Pure Reason.]

In order to make his point concerning the foundation of mathematics very clearly, Kant limits himself to a simple proposition of arithmetic, namely: 7+5=12. And then in order to destroy the notion that this relationship can be ascertained analytically, Kant asserts that anyone who thinks carefully about what is actually thought in the concepts of 7 and 5 and that of their unity in a single number will never find the 12.* **

[* It seems that the only purely logical way of dealing with numbers is to identify them in terms of each other, e.g., 7 is the first number past 6, or the 2nd number before 9, etc. This can be translated into a so-called definition, e.g., that 7 is 6+1 or 9-2. The drawback with this is that we are given an infinite count of definitions for any one number, and would have no way of knowing if all these were consistent with each other. And furthermore this would also not help us find the answer to our problem, as the following footnote indicates]

[** Some may think that because 12 and the 7 and 5 can each be reduced to an equal count of 1's (through machinations of such definitions as 12=11+1 and 11=10+1) and matched up and scratched off in a mutual and coincident annihilation, that the solution of the problem has no need of a synthetic method. But they forget that in such a way the 12 is conceived of as precedingly given, and, as we shall shortly see, this is a gratis and unwarranted assumption.***]

[*** There is another problem with this match and scratch method which is more interesting perhaps than helpful with regard to our problem, namely: for it to be valid we would have to accept as a universal principle that if the parts of any two wholes are interchangeable, then the wholes are interchangeable; but this is not true with regard to two, scalene triangles of the same size and shape and located on a sphere and on the opposite side of a common base where their shorter sides share a common end point. For here, even though the parts (the sides) of the triangles can be substituted the one for the other, the two triangles cannot be, for they are like the left and right hands. And so therefore since the principle does not hold for these triangles it obviously does not hold of all things in general, and so is not universal, and so its application to the elements of numbers would also have to be justified synthetically and cannot simply be assumed.****]

[**** Now as a matter of fact the principle does hold of numbers, even if not of all things in general; but there you have it! it is a matter of fact! It is not a function of ideas, as the Scotish philosophyer, David Hume, assumed, but a matter of fact and so therefore a so-called look-see is require, but one which is not empirical but a priori. Therefore it is only after the establishment of applicability of this principle to numbers that we see that arithmetic is subject to the law of contradiction; and so therefore Hume was right with regard to the rigor of mathematics, but not with regard to the origin of this rigor.]

The only other approach is the synthetic a priori* where we utilize an envisagement (Anschauung) for assistance. By means of the fingers (or some other grouping of undifferentiated objects to stand for sheer spaces without reference to their content) we are able to go out from the 7 (of our task) to new territory by looking at the 1 through 5 of the hand now in a different way so that the 8 through 12 appear in their place**, and whereby then we come to discover the 12 to be the required answer.

[* A discussion of the possibility of an empirical solution is presented in an appendix to this essay.]

[** Which is very much akin to the phenomenon of seeing some otherwise commonplace stars as elements in a single object, e.g., the Big Dipper or Bear; except that in seeing the 8 through 12 in place of the 1 through 5 we are dealing with a pure envisagement without the least empirical component, for it is not the shape that suddenly appears, but merely the positioning of the elements in a single row. The five fingers are a constellation which (with the palm) we call a hand, and it is a further way of looking at them to see them not only as the first five positions in a row, but then also as a series of five beginning from the 7.]

To place Kant's solution in a proper light we need to understand that he has conceded in his example only the knowledge of the 1 through 11 and not that of the 12. From the subjective standpoint, as the numbers are given, they are all known in a thoroughly empirical way, for when the 7 is given we have no way of even guessing what is coming next (or even that anything is coming next), and likewise also not upon the 8 the next number.* It is only upon being informed that a 10 follows the 9 that we are awakened to the possibility of a pattern, for while the 0 is new, the 1 is an old friend. We are now in the position of the on-looker to the pantomimic description of a circle by the mime artist and we have noticed that heshe has not only paused (in the mid-air, description of a circle), but that he/she has paused again, and indeed not only again, but in precisely the same position as heshe had paused earlier (in what we later come to call the Noon or 12 o'clock position). Given now this state of my reception of information with regard to the numbers it is now possible for me to at least guess that something else is coming (and which would be analogous to a situation where something followed the Z namely AZ or ZA). The most reasonable guess, I guess, would be that following the 10 we would come to a 20 and then to a 30 and so on.

[* Empirically speaking it is already a mental jump to think that there is a next number. There is no next letter after Z in the alphabet. And so it is clear that we are in an empirical environment at this stage, for the hallmark of the empirical is an inability to say what comes next, or even that there is a next.]

Now we come to the dramatic moment, for once the next number is given to us, namely the 11, we are suddenly able to fathom all the counting numbers that could ever be given in any length of time, and we know with a certitude that the next number is 12,* and for which reason we are able then to dispense with the teacher and any further input, for we have ourselves essentially invented mathematics.**

[* There is one gratis underpinning to this, of course, and that is that the world makes sense, i.e., that existence is orderly, namely: orderly in time and space. The alternative is David Hume's world of how-do-you-know-there's-a-next and how-could-you-know-what-it-will-be?]

[** It is at this point, perhaps, that we can be said to have achieved to some precursor of adulthood, for now we can speak authoritatively and on our own; for now we are the masters of arithmetic.]

And so then, having been given merely the 1 through 11, when I begin to count 8 9 10 11 from my little finger and on to my ring finger and finally come to the thumb, I hold it up and point to it as the answer, even when I don't have the name; and so I require a name; but I not only require it, I even find it through the recognition of the logic of the pattern which is given in the 1 through 11, and so I am able to come to the answer, 12, entirely synthetically and entirely a priori.* ** ***

[* It does not take long for me to understand the logic of the decade system in counting, for I see how the 12 tells me to stack a row of two units above a row of ten and in such a way that the one and two of each of the two rows correspond.]

[** It is significant, of course, that we are dealing with numbers as locational devices, e.g., where the 12 tells us to find the tenth space or item and then to find the second past that. For if we were dealing with numerals as names, then we would just as easily use such as the English language terms, i.e., one, two, three, etc., and so would never come to see the pattern, for the eleven and the twelve (instead of the one-teen, two-teen) forever destroy that, for they match nothing in the twenties or the thirties or anywhere else.]

[*** As of late November, 1999, I am thinking also that in order for the synthetic mode of Kant to be valid with regard to sumation, we must look upon the numbers as totals to begin with, where for example the 7 would be thought of as encompassing all the stars of the Big Dipper, for example, and not referring merely to the last star counted, i.e., the 7th star, i.e., the star you end up with when you follow a path and say "1 2 3 4 5 6 7" as you go. In this way then it is easy to see that since the 7 merely culminates a particular summation, this summation could then also be continued for 5 more places or spaces. In this way the necessity of the identity of 7+5 and 12 arises to sight, for the 7 and 5 can now be seen as partitions of 12. Here we make appeal to the category of totality; and so whle the 12 is seen as the solution to the addition of 7 and 5 by means of the envisagement, it is only by means of the category of total that we can understand that the 12 refers not to the 12th finger, when counting on our fingers, but rather on all of the12 fingers.****]

[**** This reconciles what many think to be a contradiction in Kant's thinking between his treatment of space in the Aesthetic of the Critique of Pure Reason and that in second (B) version of the Transcendental Deduction of the Categories.

Now more generally what we do synthetically and a priori in arithmetic and mathematics in general, we also do in experience. We are faced with constellations of spectral date (Erscheinung) and it is impossible to say for sure what is coming next, even as it was with the numerals less than 11. And yet this is precisely what we do, i.e., we say what is coming next, and this is proven in the general human agreement with regard to space, namely that 1. things about us are all without exception in space, even the rainbow, and 2. this space is entirely within our heads. Our agreement on this is proven via our conversation with regard to the split finger which can often be seen touching our nose, for this split finger is in space as much as the table or the rainbow or the rain, and yet we know that that "real finger", as we come to call it, does not split and so that while what we see is undeniably in space (as the immediate evidence of our sight tells us) still this space is entirely within our heads, and thereby are we able to engage in rational discourse with each other.*

[* For otherwise the term "small" when used by a person at some distance from me would simply have no relationship to the same term with regard to myself and to things about me, and hence would be no more meaningful than the "uhh" that we sprinkle our speech with anyway.]

In conclusion: it is right and proper that Kant presents his 7+5=12 as an example of the activity of the human mind in its more creative role, for it exemplifies a procedure which is mirrored by a conception of the object of experience whereby experience itself (and with it language) can become meaningful. For it is only by means of this a priori synthetic conception of the object as the empirical thing on its own that the specters (retinal objects) of the eyes (and sense organs in general) can be relegated to mere perspectives of that thing.

The mark of the empirical recognition is a contingent necessity, i.e., there is a necessity with regard to the recognition (for else it would not be a recognition), but there is the possibility of an exception, but then (consistent with the necessity) this exception must be explained. For example, my concept of a tree might be a trunk, branches and leaves. But then I see what looks like a tree, except it has no leaves. This is a contradiction and an explanation must be forthcoming. I find the explanation in the notion of the seasons of the tree, namely this tree with no leaves is dying or going dormant. And so synthetically I amend my concept syntheticly and yet conclusively and state that a tree is a trunk with branches and with leaves in season.

Now we want to imagine an empirical mathematics. Such an animal Kant never mentions, and rightly not, for it is totally contrary to the recognition we have of mathematics as an unconditioned necessity, i.e., where no exception is ever possible and so no need ever to even think of having to find an explanation for a deviation. But attempting to formulate an empirical mathematics we will imagine that a student learns about 7+5 through exposure and finds in repeated observations that 7+5= has always been followed by a 12.*

[* Now it is possible that the student has even learned to “count” 8 9 10 11 and 12 on his fingers to estimate what the answer would be, and much as someone might singsong his way along a ditty. This then would not be an a priori envisagement as we have discussed above in the body of this essay, but rather more of what we might call an alphabetic progression of A B C, etc., and where upon B A is completely forgotten and so also B upon C. And so finger counting does not render the least necessity in this case, but is merely a device for making a best guess, or in aid of the remembrance of a sing-song, e.g., one two buckle my shoe, etc.]

Now let 7+5= be followed on some occasion by a 13. This then, in keeping with the empirical conception, would not be impossible but rather merely a rare result. Consequently the term “error” would have a meaning of “a very rare result, and so rare that it is not considered polite to discuss it further”. And so if the student, upon writing down 7+5=13, were to be challenged, and then singsong his way from 7 by the tune of 5 and find 12 upon the competition, he would declare that the first result were an error, but only in the sense just described, i.e., a result so rare that it is not to be discussed. He would continue to mean that 7+5 occasionally does equal 13, and that this an error, for in the vase bulk of cases 7+5 equals 12, and in any case it is considered rude to insist that 7+5 sometimes equals 13.

And so, of course, an empirical arithmetic is totally absurd, exactly as Kant would have realized it to be, and so much so that it would have been an insult to his readers to drag them through an analysis of the concept.