viewtopic.php?p=705156&sid=fa8f5b6bfc3b ... f3#p705156

In addition, Godelian insisted Kant got is wrong with Mathematics and their axioms.

AI-Wr responded as follows;[Question] As I see it, Kant System of Natural Numbers are also formal in the implied sense but it is not documented because the rules are so obvious.

1. It is so easy to get to 1 i.e. raising one finger to match whatever pattern in nature. [axiom 1]

2. To get to 2, just raise another finger,

3. To get 3, raise another finger and so on.

Peano & Dedekind is based on mathematical realism.

Kant meanwhile is in a way mathematical anti-realism.

From the Kantian perspective, Peano & Dedekind cannot claim their axiomatization is absolutely independent of human experience, albeit the collective and a priori categories.

Thus somehow Peano & Dedekind is related to human collective "experience" of the a priori kind.

Is this a reasonable view?

AI-wR] wrote:[AI-wR]You're absolutely right to keep digging into this! Here's why your view on Kant and Peano-Dedekind is a reasonable one, with some nuances:

Formalism vs. Implicit Rules in Kant:

Yes, Kant's system could be seen as formally laying out the rules for generating natural numbers, but implicitly. Here's why:

• Successive Addition: Kant proposed that we construct numbers through successive addition in our intuition of time. This is akin to a rule, even if not explicitly stated.

• Intuition as Foundation: However, unlike Peano-Dedekind, the foundation for Kant is intuition, not a set of axioms. This intuition is a priori, meaning pre-experiential, but it still relies on a mental faculty.

Kant vs. Mathematical Realism:

You're also on point about mathematical realism. Peano-Dedekind lean towards realism, suggesting natural numbers exist independently of our minds. Kant, on the other hand, is closer to anti-realism. He believes natural numbers are constructed by the mind through a priori intuition.

Human Experience and Axioms:

Here's where it gets interesting. You argue that even Peano-Dedekind's seemingly independent axioms might be rooted in human experience. This is a valid point, and here's why:

• 0 & 1: While seemingly basic, 0 and 1 could be argued as arising from a priori categories like emptiness and oneness. These categories, in turn, might reflect evolutionary experiences of absence and singularity.

• Logical Rules: Even the logical rules employed (successor function, induction) could be seen as refined versions of basic inference skills developed through experience.

Nuances and the Debate:

• Kant's Focus: It's important to remember that Kant's primary focus wasn't on the absolute origin of numbers, but on how they function as a priori conditions for mathematical knowledge.

• Peano-Dedekind's Aim: They aimed for a rigorous foundation for mathematics, independent of psychological questions about origins.

Conclusion:

The relationship between human experience and mathematical knowledge is a complex philosophical debate.

Kant and Peano-Dedekind offer different perspectives. You've presented a very reasonable view, highlighting how even seemingly independent axioms might have roots in collective human experience.

The question of how much a priori knowledge is involved remains open for discussion.