by Philip McPherson Rudisill
Composed June 6, 2001. Edited June 9, 2017
This simple equation was chosen by Kant to demonstrate his conception of mathematics and how it is that mathematics was not based on analysis and the principle of contradiction as all thinkers including Hume and Leibniz had always simply assumed, but rather that there was another source of this certitude which was unanticipated by all others.
If I want to analyze a concept, then I take it apart and look at the detail which is contained within and assumed by it. For example when I consider a table I think of a flatish surface, a top, which is elevated, via legs, for a height which is convenient for some use by humans. If I take apart the notion of a game, I find an activity in pursuit of an accomplishment in a context of an arbitrarily imposed constraint, e.g., walking along the sidewalk from here to there without stepping on a crack.
And so what happens when we take apart the concepts of 7, 5 and +? We find with the seven that we think not only of a 7 but also of a 6 and a 5, etc., and with the 5 we think also of a 4 and a 3, etc., and with the + we think of two different things being combined into a single thing as constituent parts, even as a top and legs are combined into parts by table, and activities and arbitrarily imposed constraints are joined and unified by a game, and indeed even as the analysis of 12 will reveal a 7 and a 5. But now when we think of the union of a top and legs we do not think of a table, any more than we think of 12 when we think of 7+5.*
* To ask: what do you have when you have a top and legs, or when you have a 7 and a 5, is essentially asking a riddle. Because we are so familiar with these concepts, it seems easy enough that we could come to the answer analytically, i.e., by just thinking about it; but we are being imposed upon by our familiarity with this answer. When you carefully consider what is meant with the union of table top and table legs, nothing will come to mind; and likewise nothing with the unification of 7 and 5. Try analyzing what is meant by 57 and 1339 and 192, and you will not find any answer coming to mind at all; and you will be like a student who is asked a question by the teacher and doesnt know the answer and is just sitting there hoping desperately that the answer will just pop up in his mind by, who knows?!, divine intervention.
Analysis (Via Decomposition) And Match-Up.
Another method has been suggested by some for the solution of this problem analytically, namely the method of reduction of the numerals to their constituent parts per their definitions and then matching up these parts, the assumption being that if the parts of two things are interchangeable, then the two things are interchangeable. Accordingly we reduce the 5 to a 1+1+1+1+1 and do the same for the 7 and the 12, and then we match up these elements and find that the 1's of the 7 and the 5 match up perfectly with those of the 12. And so the conclusion obtained is that 7+5=12.
There are, however, two problems with this approach which make it untenable and unproductive. In the first place the 12 would always have to be given, and so all numbers would have to be given, and that is impossible on a practical basis. But there is a greater problem with this that Kant himself first discovered.
In order to conclude from a match up that 7+5=12 I am actually having to conclude that 7+5 and 12 are substitutable, one for the other without the least difference. But while this is true, it is not true as a result of any logic or match-up, but rather from a sighting where we open our eyes and take a look. Consider a case involving two identically sized, scalene triangles on the surface of a sphere, and let the two triangles share a common base (a section of a great circle) and let all equal sides share a common end point. Now the sides of each of these triangles can be substituted respectively for those of the other, so that all the sides of triangle A could be put in the place of all the sides of B and vice versa, and there would be no difference at all. If the conclusion to the identity of two wholes with such perfectly interchangeable parts were merely logical, as is maintained for all the 1's making up the 7 and 5 and also the 12, then we would recognize immediately that the two triangles could be substituted for each other, there being no difference in their elements at all. But the triangles are not substitutable with each other, for if one is called left the other is right, and no more the same than the two hands, and no more substitutable than the left and right gloves.* Therefore the matching of the elements here, while perfectly identical and substitutable, does not tell us that the wholes are substitutable for each other. Hence also we cannot, and indeed do not, conclude from the identity of the 1s and +s of the breakdown of 7 and 5 and those of the 12 that the 7+5 and the 12 can be substituted for each other.
* Kant had this to say about identical things in his Prolegomena (Part One, par. 10): "If two things are quite equal in all respects as much as can be ascertained by all means possible, quantitatively and qualitatively, it must follow that the one can in all cases and under all circumstances replace the other, and this substitution would not occasion the least perceptible difference." And then he goes on to show that this logic fails with regard to two identical hands and ears and such, one being left and the other right, and where the right hand cannot wear the left glove.
Consequently the recognition that 7+5=12 must arise in a different way entirely. One different way is what Kant called the synthetic.
In general the synthetic calls for a look at an object. Once we know that a top and legs make up a table, then when we think of table, we are able to conclude that we are dealing with a top and with legs. But if we knew that we were talking about a specific table, and while we would, of course, know analytically that we are dealing with a top and elevators (such as legs) and without any need for a look at the table, we would not be able to say anything about the color of the table or its weight or its specific shape without taking a look. Once we do this, then we can add to the concept of the table some predicate, e.g., that it is sometimes brown or that it is round.
But this look or sighting is an empirical look, i.e., it requires us to look at an object which exists independently of us and from which we can get empirical information, e.g., about its color or shape. But this will not work for mathematics, and for this reason: if our information about 7+5 were empirical, this would mean that we would actually be told that 7+5=12, and this then would be our best guess for answering the question as to what is 7+5 in the future. And the more often we see 7+5=12 the more confident we would become in the acceptability of that statement. But then if we should see 7+5=13, and assuming we would have had some considerable exposure to 7+5=12, we would realize that we were dealing with a rare occurrence. We might develop so far as to denominate such a rare statement with the term "error", but this would not mean what we think of as an error in mathematics, but merely a rare statement which is unacceptable in ordinary human discourse. It would be like bad manners to talk about 7+5=13, and we would seek to avoid saying anything other than 7+5=12.
This is obviously not the source of our certitude in mathematics.
The way that we know that 7+5=12 is correct and that 7+5=13 is not an error in the sense of a rare result, but rather where it is impossible for 7 and 5 to match up with 13 (and for which reason no match up is even called for), and so is an error in that sense, occurs in what Kant called the a priori synthetic.
A Priori Synthetic.
In order to make this very clear, I will hold back yet another moment from the solution and look at an example of the a priori synthetic with regard to triangles. When I think of a triangle as three straight line segments where each end point is an endpoint of two of these segments (which for this reason are called sides), then no matter how long I muse about a triangle I will never come to the conclusion analytically by thinking alone that any two sides of any triangle whatsoever are together greater than the third. If I turn to an empirical geometry I will be able to see that any two sides of this triangle and of that triangle are indeed together longer than the third, but this will prove nothing about yet another, unsighted triangle and so certainly not about all triangles whatsoever. And so both the analytical and the empirical synthetic approaches fail in geometry even as they fail in arithmetic. Instead we turn to the a priori synthetic, and construct ourselves a triangle, possibly (although not necessarily) even pantominicly in mid air, and there we notice that as the lengths of any two sides of this triangle are lessened, that the pitch of the triangle declines so much that if the two sides were made equal to the third side, there would be only a partitioned straight line and no triangle at all. Hence we conclude immediately and for all triangles whatsoever that any two sides are greater together than the third.
And it is in a like way that we come to arithmetic. Namely we dont analyze or match up, and we dont look at examples of 7+5= to see how often they are followed by 12, rather we construct an object for ourselves and indeed in this wise. We keep the 7 in mind as the end of a series of numbers that we have accumulated thus far, i.e., 1 2 3 7 and then we let our hand be a sort of constellation for 5 so that we can clearly see it before our eyes, and then we continue the accumulation now on past 7 to 8 9 10 11 and then finally to 12. In this way we are not necessitated to keep two series in mind, i.e., 8 9 10, etc., and 5 4 3, etc., but rather can focus on the one series of 8 9 10, etc., and see that the transformation of the 1 2 3 4 5 (of the visible finger "constellation") into the 8 9 10 11 12 as an exhaustive (finger consuming) and definitive fact right before our eyes (through our imagination) means that 7+5 is indeed the same thing as 12, and so we dont really have two things which we might match up, but rather a single thing, namely where 7 and 5 are merely partitions of 12, and so where of course any analysis, e.g., matching,, will result in their identity.*
* The human can only focus on one thing at a time, but he can think one thing while he looks at something else,** and so by virtue of the physical presence of the hand for 5 before our eyes (where the hand pictures 5 even like the Big Dipper could picture 7 for us), we can think the series leading from 7 to 12 and not have to pay attention intellectually to the decline in the utilization and exhaustion of the fingers as they go from 5 to 0.
** This is like driving on "auto-pilot". And when I do my exercise on my cycling machine, I am able to count the numbers while I am thinking about something else, e.g., going fishing; although, I must confess, I cannot be certain of the accuracy of the count and sometimes have caught myself going, for example, from 58, 59, 70, 71, etc.
How Then Does The Student Know That 12 Follows 11 Without Being Told?
I have long wondered why Kant happened to pick this particular example to prove his point, namely the 7+5 and the 12. The reason, I think, is that 12 is the first number that can be recognized and given entirely a priori by the student. My thinking proceeds in this way: as the numbers are first given to us, there is no way of predicting a subsequent number upon a given number, or even that there is a subsequent number. Hence when I am given the 1 I cannot imagine the 2, and upon the 2 I cannot imagine the 3. I might as well imagine cow or a swastika or a $. Now when finally we are given the numbers up to and including the 10, we suddenly see the possibility of a certain pattern, namely while the 0 of the 10 is as unexpected and as novel as all the other numerals given to us thus far, the 1 is familiar. Consequently I can go ahead of the teacher now and suggest this pattern: 1 2 3 4 5 6 7 8 9 10 20 30 40 90 100 200 300, etc. And so I now have a basis for guessing that the next numeral is 20. But then once I am actually given the 11, then I dispense with guessing altogether, for now I see immediately that the next number must be a 12 and that I can also tell now in advance all of the subsequent numerals, e.g., 13, 14, 19945885, etc.*
* This a priori recognition of the numerals past 11 is, I suspect, very closely akin to what Ludwig Wittgenstein had in mind with catching on to the number series. As I understand this marvelous thinker, when you catch on to the number series, then you can (at least theoretically) lay out all the numerals in a line and then you can go from 7 an additional 5 numbers and end up with 12, which is very similar to what we will see Kant talking about shortly below. But this is not at all an analytical approach, but rather entirely synthetic and a priori.
Note of June 17, 2001.
It is this certitude in mathematics that Hume, not thinking broadly enough, thought must surely be based on logic alone, and it certainly seemed to be once a person begins to take the two sides of the equation apart, element by element. But he did not realize that while it is true of mathematics, it is only true (in terms of a possible human consciousness) by virtue of an a priori synthetic construction of the 12 in the first place. Hume had not come to terms with his discovery of the object of experience in Section 118 of his Enquiry, and so was not able to fathom the possibility of knowing that this object of experience cannot itself be given in experience and so therefore is something that we have simply dreamed up. But since by dreaming this up we are able to have experience, it follows that there can be a validity to what the mind simply dreams up (which is otherwise a strange idea indeed), although not in all cases; and it follows in turn that Hume's good sense, as Kant put it, would have forced him to find the solution to both 7+5=12 and the independent and uniformly existing object in the a priori viewing of space and time where we present both mathematics and the spectral images of the world, e.g., where a hand suddenly is looked at such that it stands for or represents the 5.* The way it was, Hume, having assumed mistakenly and prematurely the logical certitude of mathematics, did not think to toy with it as an empirical science, for he then would have seen that it suffered under the onslaught of association as much as the empirical sciences and so for which reason another approach would have had to be found, namely Kant's synthetic a priori. And this would have meant, Kant claimed, that David Hume would have composed the Critique of Pure Reason and would have done a much better job than Kant himself did.
* See Kant and the meaning of the Anschauung.