Philosophy Now Forum

[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.

Veritas Aequitas wrote: ↑Wed May 01, 2024 4:21 am Godelian harped and dogmatically insisted on Peano & & Dedekind's Axioms as Platonic realities independent of the human conditions, but actually Peano & & Dedekind's Axioms Ultimately Experienced, i.e. contingent to the human conditions.

https://en.wikipedia.org/wiki/Philosophy_of_mathematics

Psychologism in the philosophy of mathematics is the position that mathematical concepts and/or truths are grounded in, derived from or explained by psychological facts (or laws).

John Stuart Mill seems to have been an advocate of a type of logical psychologism, as were many 19th-century German logicians such as Sigwart and Erdmann as well as a number of psychologists, past and present: for example, Gustave Le Bon. Psychologism was famously criticized by Frege in his The Foundations of Arithmetic, and many of his works and essays, including his review of Husserl's Philosophy of Arithmetic. Edmund Husserl, in the first volume of his Logical Investigations, called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it. The "Prolegomena" is considered a more concise, fair, and thorough refutation of psychologism than the criticisms made by Frege, and also it is considered today by many as being a memorable refutation for its decisive blow to psychologism. Psychologism was also criticized by Charles Sanders Peirce and Maurice Merleau-Ponty.

Veritas Aequitas wrote: ↑Wed May 01, 2024 4:21 am In addition, Godelian insisted Kant got is wrong with Mathematics and their axioms.

https://en.wikipedia.org/wiki/Analytic% ... istinction

"All bachelors are unmarried" can be expanded out with the formal definition of bachelor as "unmarried man" to form "All unmarried men are unmarried", which is recognizable as tautologous and therefore analytic from its logical form: any statement of the form "All X that are (F and G) are F". Using this particular expanded idea of analyticity, Frege concluded that Kant's examples of arithmetical truths are analytical a priori truths and not synthetic a priori truths.

Veritas Aequitas wrote: ↑Wed May 01, 2024 4:21 am 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?

Veritas Aequitas wrote: ↑Wed May 01, 2024 4:21 am 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.

godelian wrote: ↑Wed May 01, 2024 5:49 am

Veritas Aequitas wrote: ↑Wed May 01, 2024 4:21 am Godelian harped and dogmatically insisted on Peano & & Dedekind's Axioms as Platonic realities independent of the human conditions, but actually Peano & & Dedekind's Axioms Ultimately Experienced, i.e. contingent to the human conditions.

I reject Psychologism as an ontology for mathematics, if only because animals and computers can also do it:

https://en.wikipedia.org/wiki/Philosophy_of_mathematics

Psychologism in the philosophy of mathematics is the position that mathematical concepts and/or truths are grounded in, derived from or explained by psychological facts (or laws).

John Stuart Mill seems to have been an advocate of a type of logical psychologism, as were many 19th-century German logicians such as Sigwart and Erdmann as well as a number of psychologists, past and present: for example, Gustave Le Bon. Psychologism was famously criticized by Frege in his The Foundations of Arithmetic, and many of his works and essays, including his review of Husserl's Philosophy of Arithmetic. Edmund Husserl, in the first volume of his Logical Investigations, called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it. The "Prolegomena" is considered a more concise, fair, and thorough refutation of psychologism than the criticisms made by Frege, and also it is considered today by many as being a memorable refutation for its decisive blow to psychologism. Psychologism was also criticized by Charles Sanders Peirce and Maurice Merleau-Ponty.

Some relatively simple mathematics is built into the biological firmware, i.e. "intuition", of humans and animals for reasons of sheer survival. That does not mean that it is the origin or the true nature of mathematics. If it were, then only living beings could do it. In that case, how do you explain that computers can do it too?

Veritas Aequitas wrote: ↑Wed May 01, 2024 4:21 am In addition, Godelian insisted Kant got is wrong with Mathematics and their axioms.

Kant wrote that mathematics is synthetic a priori. Frege quite successfully argued that it is analytic at priori.

The only saving grace for Kant's take on the matter, is that Kant's distinction between analytic and synthetic is in fact ambiguous and quite ill-defined. Kant's examples in natural language such as "All bachelors are unmarried" are not precise enough for use in mathematics. It does not answer the question whether the non-logical axioms of a mathematical system are part of the definitions and logical rules or rather external and therefore synthetic.

https://en.wikipedia.org/wiki/Analytic% ... istinction

"All bachelors are unmarried" can be expanded out with the formal definition of bachelor as "unmarried man" to form "All unmarried men are unmarried", which is recognizable as tautologous and therefore analytic from its logical form: any statement of the form "All X that are (F and G) are F". Using this particular expanded idea of analyticity, Frege concluded that Kant's examples of arithmetical truths are analytical a priori truths and not synthetic a priori truths.

The analytic-synthetic distinction is not only ambiguous, it is also irrelevant. What could it possibly add to the discussion on the nature of axiomatic systems?

Another problem is that Kant's non-arithmetic examples are entirely based on Euclidean geometry. None of his conclusions transpose correctly to the algebraic version of geometry. When geometry is conducted by figuring out the roots of multivariate polynomials, Kant's explanation completely falls apart. In that incarnation of geometry, it no longer has anything to do with solving visual puzzles.

Gotlob Frege actually wrote quite extensively on the problem that Kant's views on mathematics do not withstand scrutiny.

Veritas Aequitas wrote: ↑Wed May 01, 2024 4:21 am 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?

What about animals and computers?

Veritas Aequitas wrote: ↑Wed May 01, 2024 4:21 am 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.

So, computers also have a mind and also construct natural numbers through a priori intution? I don't think so.

Veritas Aequitas wrote: ↑Thu May 02, 2024 6:02 am Philosophers of mathematics generally fall into one of two camps: mathematical realists, or mathematical anti-realists. These essentially correspond to Platonism and nominalism, respectively.[7]

Is philosophical realism or antirealism[Kantian] realistic.
Since philosophical realism* is grounded on an illusion,
and that your Godel's argument is based on realism
therefore Godel's argument is based on an illusion.

https://www.hawking.org.uk/in-words/lec ... of-physics

What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

In the standard positivist approach to the philosophy of science, physical theories live rent free in a Platonic heaven of ideal mathematical models. That is, a model can be arbitrarily detailed and can contain an arbitrary amount of information without affecting the universes they describe. But we are not angels, who view the universe from the outside. Instead, we and our models are both part of the universe we are describing. Thus a physical theory is self referencing, like in Godel’s theorem. One might therefore expect it to be either inconsistent or incomplete. The theories we have so far are both inconsistent and incomplete.

https://en.wikipedia.org/wiki/Pythagoreanism

Pythagoras, in his teachings focused on the significance of numerology, he believed that numbers themselves explained the true nature of the Universe.