Concepts of Numbers? Consequences for the Synthetic A Priori?

Just what 'is' the concept of a number? Further, what is the relation of this concept to a picture image of the quantity that corresponds to the concept? 

When it comes to the number 3, one can easily picture some image representing the quantitative value of 3. But let's try 37. That's harder. However, if I am familiar with numbers, I can churn out an image representing the value. I can place 37 dots on the paper, for instance. Here, we have a relationship between some intellectual idea and a physical 'phantasm' as it were, which we can generate. The paper will have better memory than I; hence, I need paper or a slate. Whatever this idea of the number is, then, comprised therein is the 'rule' for creating the phantasm. (Here, let phantasm have its impression on a physical medium.) 

Now, it seems to me that in the rule regarding the construction of the phantasm for 37 is any set of rules for the generation of, say, factors and sets of numbers equalling the number 37. If so, included in the rule for the construction of an image of 37 is the rule by which I can judge that concept from which I can construct the image of 13 added to the image of 24, equals 37. The latter rule seems included in the very rule by which I churn out the phantasm for 37. So, if the number were 36, I'd include in the rules included in 36 also those of its interesting factors (those besides 1 and 36). 

Now, to say that in the concept 37 I do not see the concept "13 plus 24" seems correct at first sight. If it is correct and yet our judgment of its truth is necessary, it seems that we have a synthetic a priori judgment. 

However, I suggest that whatever darkness lies between the concept 37 and the concept 13 + 24 is similar to the darkness that lies between the concept 37 and the very rule whereby I construct the image of 37. Just what is this latter relationship? 

In short, if it is correct that the concept "13 plus 24" is not included in 37, then, similarly, the rule for generating the image of 37 is not in the concept 37. But is it not obviously false that the rule for generating the image of 37 is not in the idea of 37, whatever an idea of 37 is? Would not all agree that the rule for generating the image of 37 is most certainly in the idea of 37? The alleged difficulty of finding in the concept 37 the concept 13+24 is really indistinguishable from the difficulty of finding the phantasm of 37 without the process of executing the rule. From the concept 37 I cannot perceive at once the image representative of 13+24.

However, I clearly do grasp from 37 the rule for the construction of the image of that quantity. Similarly, I grasp the various sets of rules tucked in the number; or I can acquire the habit of such knowledge; or I can work it out case by case, just as I work out case by case the image of the quantity 37 or 43 or 317. 

What does this matter? If it is claimed that the way I grasp the necessity of the rules regarding the parts of 37 is that of a 'synthetic a priori judgment', I respond by saying that the way I grasp the necessity of the rule regarding the creation of its phantasm is a 'synthetic a priori judgment.' But would anyone grant that one grasps the relation of a rule to the idea of a number by way of synthetic judgment? If few would, why would not few also agree that the relations of the concepts need not be grasped by synthetic judgment but rather that analytic judgment is what occurs? Further, if the relation of my concept of a number to the rule generating its image is grasped by synthetic judgment, what in fact would be linked in the judgment except a symbol and a rule? My concept becomes simply a symbol. Would all concepts vanish? Perplexity. What are the relations between concept, symbol, and the various rules? What is the concept of a determinate number?

The constructive character of arithmetic here certainly includes the relationship of concept to phantasm. Insofar as phantasm is required for insight, one can say that this constructive character is constitutes a dispositional condition for the growth in ideas, as one enters the science. I think the science of classical geometry follows a similar pattern.