# Wittgenstein's Critique of Gödel and Russell: A Philosophical Inquiry

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## Chapter 1: Unpublished Insights from Wittgenstein

In 1956, posthumous writings of Wittgenstein, which he never published during his lifetime, were disclosed to the public. These texts were compiled into the book *Remarks on the Foundations of Mathematics*. Within these pages, Wittgenstein expressed dissatisfaction with how philosophers, logicians, and mathematicians interpreted paradoxes, providing several controversial arguments against accepting Gödel’s incompleteness theorems.

## Wittgenstein's Perspective on Logical Paradoxes

Here’s what Wittgenstein had to say regarding troublesome paradoxes in logic and arithmetic:

"If a contradiction were to emerge in arithmetic, it would merely demonstrate that an arithmetic containing such a contradiction could still function effectively. It is more advantageous for us to adjust our understanding of the certainty required, rather than claim it could never be a legitimate arithmetic. 'But surely this isn't ideal certainty!' — Ideal for what purpose? The rules of logical inference are merely rules of the language-game." (RMF, Wittgenstein).

Consider this: when determining the square root of 4, we encounter two potential answers: 2 and -2. Does this ambiguity detract from the integrity of arithmetic? Certainly not. Both natural numbers and integers are employed in economics without issues.

A similar situation arises in Russell’s Paradox. Here, we grapple with whether the “set of all sets that are not members of themselves” is or isn't a member of itself, leading us to a formal limitation akin to the square root of 4, which results in ambiguity. Is it irrational or impossible for mathematics to deal with such ambiguity? Absolutely not! As Wittgenstein articulated in the *Tractatus*:

"It is as impossible to express in language anything that 'contradicts logic' as it is to depict a figure in geometry that contradicts the laws of space, or to specify the coordinates of a non-existent point." (Wittgenstein, Tractatus 3.032).

Thus, the ambiguity inherent in Russell’s paradoxical set appears to reveal a different category of set—one that cannot define its own members, much like our inability to resolve the square root of 4.

## The Wittgenstein-Gödel Dispute

The ongoing debate between advocates of Wittgenstein and Gödel merits closer examination. For those seeking to delve deeper into this conflict, I recommend the following articles: [1], [2], [3], which are cited at the conclusion of this text.

First, it is essential to recognize that Wittgenstein largely disregarded Gödel’s theorems. He made considerable efforts to overlook them. However, a student once posed a question to him: “Could there not be true propositions expressed in this (Gödelian) symbolism that are not provable within Russell’s system?” Wittgenstein’s response was, “Why should propositions—such as those in physics—be articulated in Russell’s symbolism?” This indicates a rather limited and unusual reaction to Gödel’s first incompleteness theorem.

To grasp the complexities involved, we must first understand how propositional variables function within Gödel’s symbolic language—a system adapted from Russell and Whitehead’s *Principia Mathematica* (1910), which illustrates the implications for every closed formal system, including those utilized in physics.

According to Nagel and Newman in *Gödel’s Proof* (1958), Gödel’s first theorem states that if ‘S’ signifies a formula, its formal negation, non’S’, also qualifies as a formula. Therefore, we arrive at the following scenario: if “p is a non-demonstrable formula,” it follows that “p is a demonstrable formula.” When operating within a system containing both formulas, we ultimately cannot ascertain whether p is a non-demonstrable or demonstrable proposition.

But is this truly the case? Let’s consider an illustration involving a liar who asserts:

"Everything I say is a lie."

Before we declare this statement true or false, it’s vital to understand what this self-proclaimed liar is actually lying about. We need examples of statements that can be tested against reality for their truth, falsity, or ambiguity. Without such examples, we are left with mere indeterminacy.

Wittgenstein's point in response to Gödel’s work is clear: we require propositions that can be verified in “reality” to assess their truth, falsity, or ambiguity. Since propositional variables often assert the existence of valid negations of fundamental propositions without supporting evidence, it can be argued that Gödel assumed the conditions of undecidability and incompleteness as inherent in the foundational rules of the language of his theorem. Nevertheless, Gödel's assertions were not erroneous.

Indeed, Wittgenstein’s pragmatic mindset led him to question whether Gödel’s incompleteness could apply to the realm of physics. The answer to this inquiry is both “yes and no”: incompleteness can, and cannot, be applicable to physics. This is because the rules of physics could evolve if the universe’s rules were to change (suggesting that physics might never achieve complete status). However, all successful theories in physics today are grounded in facts. Consequently, these theories will invariably remain the successful ones, as they accurately relate to facts (indicating that they will always correspond to fact-based truths).

Check here the distinction between Platonic completeness and pragmatic completeness.

[1] A Note on Wittgenstein’s “Notorious Paragraph” about the Gödel Theorem (2000). By Juliet Floyd and Hilary Putnam in *The Journal of Philosophy*, Vol. 97, No 11 (2000), pp. 624–632.

[2] Misunderstanding Gödel: New Arguments about Wittgenstein and New Remarks by Wittgenstein (2003). By Victor Rodych in *Dialectica*, Vol. 57, No 3 (2003), pp. 279–313.

[3] Wittgenstein and Gödel: An Attempt to Make "Wittgenstein’s Objection" Reasonable (2018). By Timm Lampert in *Philosophia Mathematica*, Vol. 26, No 3 (2018), pp. 324–345.

## Chapter 2: Engaging with Gödel’s Theorems

**Russell vs. Wittgenstein: Judgment or Representation?**

In this video, we explore the contrasting philosophies of Russell and Wittgenstein, particularly focusing on their differing views regarding judgment and representation in logic.

**Wittgenstein versus Gödel part 1**

This video provides an in-depth analysis of the conflict between Wittgenstein and Gödel, examining their perspectives on logic, mathematics, and the implications of Gödel’s theorems.