Is it logical to conclude that any mathematical hypothesis that deals with infinite series of numbers can not have a formal proof but can only have a conjecture at the best?
The Reality
No. It is illogical and also false. Proof is proof, and theorems once proven stay proven, including proofs about infinities. The current understanding in mathematics is that not all hypotheses involving infinite series or infinite sets cannot have formal proofs. Indeed, many theorems concerning infinite series and cardinality of infinite sets have been rigorously proven.
Rigor in Mathematics
The definitions of concepts such as limit and one-to-one correspondence are rigorous, allowing for formal proofs. For example, the claim that the sum of an infinite series can be finite, as in the case where n 1/2 1/4 1/8 1/16... and n 1, is well-substantiated and has been proven correct. This is because the series converges to 1 based on standard calculus principles.
Examples of Provable Hypotheses
1. Summation of Infinite Series
Propositions concerning the sum of an infinite series rely on a precise definition of the concept of a limiting process. The limit of a sequence is a fundamental concept in calculus and analysis, and it provides a rigorous framework for proving such propositions. Therefore, conclusions drawn about the sum of infinite series can be established through formal proof.
2. Cardinality of Infinite Sets
Propositions concerning the cardinality of infinite sets, which involves the concept of one-to-one correspondence, also rely on formal definitions. These propositions can be proven formally. For instance, the fact that the set of even numbers is equinumerous (of the same cardinality) to the set of all natural numbers is a proven result. This is shown by the function f(x) 2x, which maps each natural number to a unique even number and vice versa.
3. Defying Common Sense
Even propositions that seem counterintuitive, such as the proposition that the sum of all positive integers is -1/12, can be proven within the context of specific mathematical frameworks. This is an example from the theory of analytic continuation in complex analysis, where series with negative sums are assigned meaningful values based on the extension of functions.
Theoretical Foundations of Mathematics
Set Theory and Infinity
The axioms of set theory, such as ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice), explicitly deal with the concept of infinity. Within these frameworks, it is possible to prove the existence and properties of infinite sets. The concept of infinity is not an arbitrary idea but a well-defined construct that can be rigorously studied.
Large Cardinal Axioms
In addition to the axioms of ZFC, mathematicians have introduced large cardinal axioms to explore and define even larger infinities. These axioms allow for the creation and study of more complex mathematical objects and structures, further supporting the idea that formal proofs can be arrived at and are essential in understanding the properties of infinite sets.
Intuitionist Mathematics
While some schools of thought, notably Intuitionism, take a more constructivist approach to mathematics, even they do not reject all infinities. Constructivist mathematicians like L.E.J. Brouwer and Arend Heyting worked within frameworks that allowed them to handle constructible infinities such as the natural numbers and the real numbers. The concept of infinity in Intuitionism is restricted to those that can be explicitly defined and constructed.
Historical Perspectives
Even opponents of Cantor's set theory, such as Leopold Kronecker, did not reject all forms of infinity. Similar to Intuitionism, Kronecker's criticism was limited to what he considered to be non-constructive mathematics, particularly Cantor's hierarchy of transfinite numbers.
Conclusion
The claim that all mathematical hypotheses concerning infinite series of numbers cannot have formal proofs but can only have conjectures at the best is a mischaracterization of the field of mathematics. Formal proofs are indeed possible and are a fundamental part of the study of infinities. The historical and theoretical development of mathematics has provided a robust framework within which such hypotheses can be rigorously analyzed and proven.