Is Cantor's work on transfinite numbers linked to philosophy and religion?

In summary, Georg Cantor argued that mathematics, philosophy, and religion are all linked and that his work on transfinite numbers was communicated to him by God.
  • #1
sigurdW
27
0
Lets begin with the Philosophy of Cantor.

Here is a quote from wikipedia:http://en.wikipedia.org/wiki/Georg_Cantor#Philosophy.2C_religion_and_Cantor.27s_mathematics

"Philosophy, religion and Cantor's mathematics

The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's. He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one. To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications—he identified the Absolute Infinite with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.

Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence. Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism.
Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.
Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set. Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all."
Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set."

Questions? Comments?
 
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  • #2
Please do not re-start a closed topic.
 

1. What is the purpose of studying the philosophy of mathematics?

The study of the philosophy of mathematics aims to understand the nature, foundations, and implications of mathematical concepts and theories. It also examines the relationship between mathematics and other areas of knowledge, such as logic, science, and language.

2. What is the difference between mathematics and philosophy of mathematics?

Mathematics is a branch of science that deals with the study of numbers, quantities, and shapes, while the philosophy of mathematics is a field of philosophy that focuses on the philosophical foundations and implications of mathematical concepts and theories.

3. What are the main schools of thought in the philosophy of mathematics?

There are several different schools of thought in the philosophy of mathematics, including Platonism, Formalism, Logicism, Intuitionism, and Structuralism. Each of these schools has a different perspective on the nature of mathematical objects and their relationship to the physical world.

4. Can mathematical concepts and theories be proven to be true or false?

The answer to this question depends on which school of thought one subscribes to. Some schools, such as Platonism and Formalism, believe that mathematical concepts and theories are objectively true and can be proven through logical reasoning. Other schools, such as Intuitionism, argue that mathematical statements are only true if they can be verified through direct experience or intuition.

5. How does the philosophy of mathematics relate to other areas of philosophy?

The philosophy of mathematics has connections to many other areas of philosophy, such as metaphysics, epistemology, and logic. It also has implications for the philosophy of science, as mathematics plays a crucial role in scientific theories and explanations. Additionally, the philosophy of mathematics has connections to the philosophy of language, as it examines the nature of mathematical language and its relationship to reality.

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