James R Meyer
I am interested in how little attention is paid to the limitations of the language when it is used to make statements that are supposedly logical. The consequences of this are particularly evident in mathematics, where there are theories that are based on the philosophy that numbers and other mathematical concepts are ‘actual’ things that exist independently of any physical reality. Such beliefs are commonly held on an almost subliminal level; most people have never taken the time to carefully examine the basis and the consequences of such beliefs. It is because of such beliefs that detailed considerations of language are ignored - the ‘actual’ non-physical reality is considered all-important - with the result that a detailed evaluation of the possibility of errors due to limitations of language is generally considered unnecessary.
Every statement has to be stated in some language. If assumptions are made that ignore some aspects of the language of the statement, then how can we be sure that the statement is entirely logical? In particular, when a statement refers in some way, either implicitly or explicitly, to some language, whether it is the language of the statement itself or some other language, there is a significant possibility of confusion.
Unless every aspect of such statements is very carefully analyzed, a statement that superficially appears to be logical may actually contain subtle errors of logic. In my work, I show how such errors can occur and how we can avoid such errors by careful analysis of language.
The Infinity Delusion
by James R Meyer
The book asks why people might believe that numbers ‘exist’, rather than simply being a concept of our minds? In particular, why should we believe that numbers that consist of the sum of an infinite number of other numbers added together exist? This book presents a convincing argument against the independent 'existence' of such concepts.
The Shackles of Conviction
by James R Meyer
If you encountered something that you thought had to be wrong, what would you do? Would you try to prove that it was wrong? That’s what happens when Ralph McNeil encounters Kurt Gödel’s proof of Incompleteness. He sets out to prove it wrong. This is the story of how he tries to prove it wrong. It is also the story about the inner turmoil of the man who wrote the proof - Kurt Gödel.
on March 25, 2016
Unfortunately, Spinosa has jumped in at the deep end, and it is clear that he has failed to do the necessary research and has jumped to conclusions that are completely wrong due to his failure to understand the basics of what he is talking about. Two examples will suffice to demonstrate this:
Spinosa uses Hirzel’s English translation of Gödel’s proof. In Hirzel's translation, words in all capitals such as 'VARIABLE', 'FORMULA', 'AXIOM', etc do not actually designate variables or formulas or axioms of the formal system, but they denote natural numbers, where the natural numbers correspond (by Gödel numbering) to expressions of the formal system; and relations between these numbers correspond (by Gödel numbering) to relationships between expressions of the formal system. This is explicitly explained in Hirzel’s translation on his page 6, just before section 2.3, and the distinction is indicated by capitals (Note: in Meltzer’s translation, the distinction is indicated by italics.) Spinosa has completely failed to comprehend this distinction between expressions of the formal system and numbers that correspond to such expressions, and his article is full of examples of this misunderstanding.
Spinosa also fails to understand that in Hirzel’s paper, there are two completely different functions that have the same name: "subst", and he manages to completely confuse the two (This is partly why I recommend Melzer's translation over Hirzel's, since Meltzer uses two different names, as does Gödel’s original paper). It is true that Gödel does not assist the reader by his assertion that his relation 31 is the concept Subst that he referred to previously, but had Spinosa understood the distinction referred in the above paragraph, this would have not presented a problem.