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Are the propositions of arithmetic "synthetic a priori" ?

 If the Kantian criterion that distinguishes synthetic from analytic propositions, whether they are axioms or postulates, is the non-inclusion of the synthetic ones versus the inclusion in the subject of the analytic propositions, an analytic proposition could not be reduced or transformed to a synthetic one, nor vice versa, since something is or is not included in something else. Well, the very concept of inclusion is not exactly clear, modern set theory has opted for an extensional definition, that is, giving a list of the things that are included in another given. Inclusion thus conceived is less problematic, although it is not without paradoxes. Let's skip this for a moment and focus on the Kantian distinction.

   Kant gives us his well-known example of a priori synthetic proposition, capable of giving us knowledge independently of experience:

7+5=12

But we can transform this equality into this other:

7=12-5

And then into:

7=7

That is an analytical proposition.

And conversely, from 

7=7

We can construct:

7+5=7+5

by the second Euclidean axiom (analytic), and from there, by means of a change of analytical notation (that is, we put in a compact form -which is not a conceptual unification- and in base 10 the fifth successor of 7), we go to:

7+5=12

We have gone so from synthetic to analytical propositions (and back) using only arithmetic rules.

Can we then exchange synthetic for analytical propositions? In what sense can a numeric predicate be included or not included in a number? Is 12 a unification of the numbers 7 and 5 -as Kant says in the Critique of Pure Reason (B15)? But we can obtain the number 12 by adding infinitely many whole numbers (that is, adding negatives and positives), does this mean that in the number 12 those infinite numbers are unified? And if they were unified, does an infinite synthesis have something to do with intuition? According to what Kant proposes, in 1 + 1 = 2, the two ones would unify in the two, but how could the ones unify? Are they not rather de-unified to be two? What do these questions mean? We enter very swampy metaphysical terrain.

This does not make any sense, not even within the Kantian system. Doesn't arithmetic depend on the counting process, an algorithm that, although based on the internal sense, requires the analytical concept of a number without a predecessor? A sum, such as 7 + 5, or 22 + (- 10), is an algorithm, that is, a procedure. Such an algorithm is not a process of unification but of determination: through a systematic process, we establish the relationships between three numerical signs. None of them is thought of separately, but rather as part of a single numerical system that has historically been developed with great difficulties.

  Contrary to what Kant thought, it is precisely the sum of large numbers, as Frege already noted, which shows that arithmetic generates analytical propositions. In the sum:

123.456.789 + 213.456.789 = 336.913.578

we can know that the proposition is true without needing any intuition about any of the numbers. However, and contrary to what Frege thought, they are not a priori propositions (not a posteriori either) but formal systemic propositions, the result of a conceptual elaboration in which elements of everyday experience and analytically defined elements intervene. What makes a mathematical proposition is its possibility of inclusion in a system of generalization of its results. Mathematics, as Stanislav Dehaene's research seems to corroborate, has its roots in animal biology, in our neurophysiological structure, but on that basis, it has later been constructed according to both analytical and synthetic principles until reaching the formal constructions of science. which is today. The impossibility of a total formalization, proved by Gödel's theorems, is nothing but a reference to its intuitive, animal basis, which is conditioned by the way in which we experience space-time.


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