I understand that Kant remarked that Space & Time are forms that the intuition take. Would he also say that of the integers? Are they judgements? That is they lie within his Category of Quantity.
Its impossible to predict what Kant would say exactly.
However the concept of Integers is not to be confused with sense of space,time and scale/quantity:
To intuitively observe and acknowledge the existence of an entity does not require a basis or prior memory of it, like there is this strange thing on the table. This is natural intuition of acknowledging existence of something.
However, to comprehend the quantity of such entities , a prior understanding (memory) of such entities is required. like now there are more of those strange things on the table. Else you might mistake a collection as a single object when you notice it for the first time (intuition explained in point 2 below will eventually resolve this). This is a composite natural intuition of quantity. Being composite means it depends roughly upon following intuitions:
- Acknowledge: Accepting and memorizing the existence of an object.
- Remember: Noticing the similarity of all those objects with, accepted definition. This also helps when we are capable of observing similar parts of an object and proceed to divide it further.
- Compare: Have a sense of contrast, a scale, that is differentiating between more and less, one and many. (let's say general readings from our biological sensors)
All these natural intuitions work together to make us capable of counting. Similar interplay of memory and intuitive acknowledgements and comparisons also brings us sense of time passing by , and a sense of distance/space. Like now those strange things on the table are becoming more and more, and now they cover the entire table.
Notice how our intuition works fine with the still unknown strange thing. Let's call these strange things Integers. Let them represent our sense.
Nobody can tell what Kant would say.
With regard of integer numbers being forms that the intuition take, I have to say that they aren't.
Neither space or time.
They are only physical properties, or mathematical entities.
Both physics and mathematics, are universal fields of study.
If some day we make contact with an extraterestrial civilization, I'm sure they will have physics and mathematics as fields of study.
We need integers to say how many instances there are of an idea within a given context, what ever that idea is.
Many interactions between physical entities, can only be described using integers.
A cell replicates itself an integer number of times.