Kant: 7 + 5 = 12 and concepts

I have already read several posts on this topic but I have some other questions.

1. According to Kant,

5 + 7 = 12

is a synthetic a priori judgement.

I want to know what the judgement really is. Is it `5 + 7 = 12`, or `5 + 7` (which is judged to be 12) ? If it's the first case, we have to know 5 concepts (5,+,7,=,12) to determine if the judgement is true.

2. Kant says, nothing allows us to know 5 + 7 equals 12 without synthetic judgement. But that depends on how we see the word "concept". The number "5" has no meaning if it's not in a context (real number, etc.), and in real number context, we have axioms to know 5 + 7 = 12 (by ++ operator). We can't separate "+" and "5" in different contexts. For example, in a strong-type language, it is meaningless to add a float and an integer.

Thank you.

• I'm not sure I understand the second question- the concept "numbers" doesn't inherently include the concept "mathematics". Dec 13, 2018 at 15:03
• I would mediate a bit over the claim that the number "5" has no meaning if it's not in a context (real number, etc.). This is mathematical structuralism (Shapiro), but it's not without its problems. For me, 5 as natural number and 5 as real number, are the same 5. And the fact that strongly typed programming languages have problems with this isn't such a convincing argument IMHO. A computer is a finite state machine and can't represent all real numbers anyway. Jan 3 at 20:17
• Anyway, I guess 2. is a strong assumption in favor of mathematical structuralism, that was not in Kant's mind when he wrote that 5 + 7 = 12 is a synthetic judgement (btw, the whole formula is the judgment) . Jan 3 at 20:23

what the judgement really is ?

A judgement is a mental act. It is characterized by the attitude (of the actor making it) towards the represented object (its "content").

When we acknowledge or reject an representation (or the content of a thought), we make a judgement.

The linguistic act that corresponds to a judgement is an assertion.

In term of Kant's phylosophy, we may say (very roughly) that we have a judgement when two concepts (or representations) are united in thought .

According to Kant, the distinction between a priori and a posteriori is an epistemic one : it concerns the basis of the knowledge of a fact.

A priori knowledge is not based upon empirical observation but upon reason alone.

For Kant, all mathematical judgement (we today say: statements) are synthetic and a priori.

Obviously, Kant assumes that arithmetical concepts have meaning. Numbers are counting numbers, i.e. "number of ...".

And sum is the abstract concept related to juxtaposition [see CPR, B14-16 English transl. page 144]:

We have to go outside these concepts, and call in the aid of the intuition which corresponds to one of them, our five fingers, for instance, or five points, adding to the concept of 7, unit by unit, the five given in intuition. For starting with the number 7, and for the concept of 5 calling in the aid of the fingers of my hand as intuition, I now add one by one to the number 7 the units which I previously took together to form the number 5, and with the aid of that figure [the hand] see the number 12 come into being.

• Ok thank you. So if I understood well the judgement (statement) is 5 + 7 = 12. That contains 5 concepts. And I don't see why we can't get 12 out of 5 and 7 as long as we know the concept of "+" and "=" Dec 13, 2018 at 14:29
• So, what is a number? Is it something out of experience (in which they are a posteriori concept)? Is it something deduced from axioms? In that case, we aren't equating to different concepts of "7+5" and "12", and we do not have a synthetic proposition. Dec 13, 2018 at 22:03

Why is judgement or intuition required here? Perhaps for some things but ever since Russell and Whitehead certainly not for Mathematics. To save some typing please consider 1+2=3 which means s(0) + s(s(0)) = s(s(s(0))); which can be proven by repeated application of Peano's axioms for addition: s(0) + s(s(0)) = s(s(0) + s(0)) = s(s(s(0) + 0))) = s(s(s(0))) QED.