phil math

7+5=12

Kant divides propositions of the subject-predicate form into two groups: the analytical ones and synthetic ones. Analytic ones are those where subject concept involves the predicate concept by definition. The rest -which are not analytic- are called synthetic. According to Kant very few mathematical concepts are analytic and most of them are synthetic. In these cases (when the argument is synthetic), by analyzing the mathematical concept you cannot come up with its predicate and you need intuition to figure out mathematical properties of concepts. For example, to see that the shortest path between two points is the straight line from one point to the other, it is not enough to analyze the concepts: points, line and shortest path but you need some other tools -like perceptual experience- to figure it out. Kant says that mathematical objects are modes of representing individual objects and arithmetic –which is about properties of mathematical objects- can be only known by the intuition that is used to comprehend that individual objects. Analytic argumentation does not give new information while synthetic ones do. This means that intuition is the only way of getting to know anything new. Kant considers most of mathematical propositions saying things that are not part of the definition of the mathematical objects they are referring to and therefore they are saying something new. Then they should be coming from intuition. He also says that this intuition is exactly the way human beings perceive the world. On the other hand he believes that human beings are able to capture a-priori truths via intuition because “pure intuition” is about the forms of possible perceptions and is aware of spatial-temporal conditions that are involved in perception. But I will not go into further discussion of it. When it is in particular, the arithmetic: he considers arithmetic to be a natural result of human’s perception of objects, distinguishing them and counting them. In a more broad way it can be concluded as follows: he says that we do have number concept attached to group of things because our eyes can distinguish between no objects, one object, two objects and so on. Number is a concept attached to a group of objects. Since we get to know about those objects by physical experience we know about their number in the same way. Arithmetic follows from counting- associating a number- to the resulting set after some arithmetical operations. This implies that if concept of number is synthetic and only can be known by means of perception, so is arithmetic.
Now lets see Kant’s claims on the proposition “7+5=12”. Most arithmetical results seem to be analytic since we are so comfortable with them and sure about their truth that arithmetic has come to be considered as part of logical argumentation. However the comfort we have gained- probably by experience- can be deceiving. Kant argues about how we do know that 7+5=12. What kind of properties do numbers 7 and 5 have that they lead to number 12, is there a way of dissecting the left hand side of the equation and obtain the right hand side? What do number 7 mean but 7 objects together and similarly 5? Kant says that one need the intuition of number 7and number 5 so that by adding 5 units to seven units one will get 12 units, number associated to which will be 12. In short Kant says that addition of numbers is bringing together two groups of objects of given numbers and finding the number of resultant set. Since we have to think about sets of objects to do addition, he says that it is intuitive therefore synthetic. However he skips the part about how do we know about numbers, might not it be the case that knowing about numbers of objects involves an analytic process? He says that one needs to think of five and seven objects instead of the numbers 5 and 7, but how does one know that it is a set of five objects but not a set of six objects.
My main opposition to Kant’s argument would be exactly at this point. Do we get to know about all numbers intuitively and can we distinguish between 5 and 6 at one sight? We should be able to do that if we know about numbers only by visual perception, but that is not the case, each time I have to count people in the room, to be able to bring exact many tea cups so that everyone will get one and I will not bring unnecessary cups, although they are sitting in front of me. This means that even there is a good possibility that knowing about number of objects has an intuitive aspect it is not enough to say that the number concept is totally intuitive unless we show counting is intuitive. There might be an analytic process following the intuition and I am not sure whether Kant would call it analytic or synthetic. Another problem with Kant’s argument with the example 7+5=12 is that analyzing concept 5 and 7 and saying that they do not have concept twelve in them does not disprove that this statement is analytic as he is not talking about the addition operation between them. He says that bringing 5 and 7 is nothing but adding 5 units to 7 units, which can be only understood intuitively. But what “+” represents in the context of arithmetic and therefore what 5+7 stands for? Is not “5+7=12” is exactly the representation for Kant’s argument. When we say “5+7 =?” it is asking “what will be the number of units when you bring together 5 units and 7 units?” Therefore problem is not really about going back to five objects when we see the representation “5” but how we add things and the previous problem I invoked: how do I know about number of a group of objects. This says that everything reduces to counting operation. Now we ask: How does one learn about concept of number and counting?
I will try to save Kant’s argument by claiming that we know about concept of numbers and addition operation because we can count by means of the intuition from perception. I know 5+7 is twelve because I can bring five points and 7 points together and then recount them. To do this I need to be able to know one set has 5 points and the other has 7 points. In short, I should be able to distinguish between sets of different number of elements. However once I know how to count I will also know about number concept and be able to distinguish between two sets with differently many objects. To be able to decide whether the proposition “5+7=12” is synthetic or analytic I will try to decide whether counting is an analytic thing or an intuitive thing. As promised, I will claim that counting is an intuitive thing to Save Kant’s argument. From there it will follow that some arithmetical propositions regarding addition and subtraction are synthetic.
How do I know about one object? By my visual skills, which can distinguish one yellow point among a set of black points. Once I know one object I can put another yellow point at a place different than my initial yellow point and now I say I have “2” of them. This is quite possible, as my eyes can perceive these yellow points and their locations. If I continue adding another point in a non-overlapping way with previous points each time, my eyes will lose their sensitivity to distinguish the last picture from the previous but I will know that they are different because there is a new point coming at each step. In the time framework I can order the steps and give a name to each step. These names are nothing but numbers. And the difference between two consecutive steps will be a yellow point that did not exist in the former one. I will give the name “1” to the initial condition, “2” to the one following step 1, “3” to the one following the step called as 2 and so on… Moreover when there is a similar process going on we can rename the steps in the same way. Independent of objects contained and their locations, similarly named steps of two different processes will have something in common which will be called as their “number”. It is also possible to go backwards in this process. Say you are at step 6 and by removing a yellow point (assuming an object is grasped just by intuition no matter how many alike are around). Going forward and backwards will just be intuitively possible since I can add one distinct object and remove one among those given. Now lets assume that I have two number series say A and B, which I was calling as processes at the beginning. Now I want to add 5 to 7. To do this, I will be going one move forwards from step 7 in A for each move to backwards from step 5 in B. When there is nowhere else to move (no objects left to remove in B) I will stop moving forwards in A. The number of step I ended up in A will be the sum of 7 and 5 and it will be 12 (result taken from previously done experiment). I am saying 12 without hesitation, but this is not because I can imagine 7 and 5 objects together and distinguish it from a different number of objects but that addition of any two numbers can be reduced to something else (adding one by one and knowing where to stop) which is conceivable by perception. So, counting is an algorithm where each step follows by intuition.
To conclude, I would hesitate to say that number concept is intuitive because of the worries above (that I need to follow an algorithm that will take me where I want to go). On the other hand I would go with Kant’s claim that there is an intuitive aspect of arithmetic as the objects of concern are to some extent grasped intuitively and arithmetical operations can be reduced to adding things one by one in which there is a lot of intuition involved.


Reference:
Shapiro S. (2000), Thinking about mathematics. (Oxford, Oxford University Press)