# 10 - The Nature of Mathematics - [ ] [The Mathematicians’ Misconception](The%20Mathematicians’%20Misconception.pdf) ## TODO * Update [Is Logical Consistency a greater governing force than The Laws of Physics?](Is%20Logical%20Consistency%20a%20greater%20governing%20force%20than%20The%20Laws%20of%20Physics?.md) * [Reach and Constraints](Reach%20and%20Constraints.md) * [Any physical experiment can be regarded as a computation, and any computation is a physical experiment](Any%20physical%20experiment%20can%20be%20regarded%20as%20a%20computation,%20and%20any%20computation%20is%20a%20physical%20experiment.md) * [Computations and Physical Processes Represent Abstract Concepts](Computations%20and%20Physical%20Processes%20Represent%20Abstract%20Concepts.md) * [Problems Create a Logic of Reasoning](Problems%20Create%20a%20Logic%20of%20Reasoning.md) * Pg 224 - reality of abstractions based on [Dr Johnsons Criteria](Dr%20Johnsons%20Criteria.md) * Pg 233 - Severing the link. * "But by severing the link between their version of the abstract ‘natural numbers’ and the intuitions that those numbers were originally intended to formalize, intuitionists have also denied themselves the usual explanatory structure through which natural numbers are understood." * Severing the link: By denying the existence of an infinite sequence of natural numbers, intuitionists essentially sever the link between the abstract concept of natural numbers and the intuition that each number has a successor1. This link is part of the usual explanatory structure through which we understand how numbers relate to each other. * Loss of explanatory structure: This rejection means they lose the standard explanatory framework that connects our intuitive understanding of counting to the abstract, formal definition of natural numbers. This leaves their concept of natural numbers without a clear explanation of how to progress from any given number to the next number. * A concrete example of something intuitionists struggle to explain, due to their rejection of the standard explanatory structure of natural numbers, is the progression to the next number in the sequence, particularly after any given large number * Standard Intuitive Explanation: The typical understanding of natural numbers involves the concept that for any given number, there is a next number obtained by adding one to the current number. This process can be repeated indefinitely, leading to an infinite sequence. This is directly linked to our experience of counting, where we always know how to get to the next number. - Intuitionists' Limitation: Intuitionists reject the idea of an infinite sequence of numbers and therefore can't use the standard explanation that numbers are generated through this "add one" process. They believe only in finite numbers, meaning at some point, there is no next number. - The Problem of Succession: If there are only finitely many natural numbers, at some point there must be a largest number. The question then arises of how the intuitionist explains the existence of this largest number without reference to a successor function. If there is a largest number, say, _n_, what explains why _n_ is the largest and not _n_ + 1?. - Deutsch's Criticism: According to Deutsch, they cannot provide a satisfactory explanation of this progression because they have severed the link between the abstract numbers and the intuitive notion of each number having a successor. By rejecting the infinite sequence, they have removed the underlying mechanism for understanding how to move from one number to the next, thus losing a key part of the explanatory structure. - Lack of Explanation: Deutsch says that the intuitionists introduce further unexplained complications such as denying the law of the excluded middle. This denial makes it impossible to explain how many numbers there are, or how there is a last number. They can't explain the move to the next number, or why the next number is not a valid number. - Specific Example: The source mentions that intuitionists "believe in the reality of the finite natural numbers 1, 2, 3, …, and even 10,949,769,651,859". They accept that a large number like 10,949,769,651,859 exists, but they cannot explain why we cannot go to the next number. The standard explanation would be that there is a next number by adding one. The intuitionists have denied themselves the use of that explanation, and they do not have an alternative. - In essence, while intuitionists accept the existence of individual natural numbers, they cannot coherently explain how the sequence of these numbers is generated or what would stop that process, as they have removed the concept of an infinite sequence of successors. They are left with the unexplained idea that, at some arbitrary point, the process stops for no clear or consistent reason. They are stuck with the finite and cannot explain how or why there are only a finite number of numbers. * We cannot prove the laws of physics are consistent The sources suggest that **it is not possible to definitively prove that the laws of physics are consistent,** due to limitations inherent in both mathematics and physics. Here's a breakdown of why: - **Proofs are Physical Processes**: The sources emphasize that proofs, whether in mathematics or physics, are not abstract logical exercises, but rather physical processes. They involve manipulating physical objects (like symbols on paper, or computations in a computer) according to certain rules. Therefore, the validity of a proof depends on the physical behavior of these objects, and our knowledge of that behavior depends on the laws of physics. - **Dependence on Physical Laws**: Mathematical truths are indeed independent of physics, but our ability to _prove_ them depends on the laws of physics. The physical laws determine which abstract entities and relationships are modeled by physical objects such as mathematicians' brains, computers and sheets of paper. - **No Abstract Proving**: There is no such thing as abstractly proving something, just as there is no such thing as abstractly knowing something. All knowledge is generated by physical processes, and its scope and limitations are conditioned by the laws of nature. - **Limitations of Logic**: Logic alone cannot prove substantive truths about the world. The only propositions that logic can prove without recourse to assumptions are tautologies—statements that assert nothing. - **Fallibility of Reasoning**: Logical reasoning is a physical process and is therefore inherently fallible. The laws of logic are not self-evident. - **The Problem of Infinite Regress**: Any attempt to justify the laws of deduction logically must lead either to circularity or to an infinite regress. We rely on the laws of deduction because no explanation is improved by replacing them, but we cannot prove them with absolute certainty. - **Uncertainty in Mathematical Knowledge**: Mathematical knowledge does not have a privileged status with guaranteed certainty. Mathematical truths are absolutely necessary and transcendent, but our knowledge of those truths is derived from physical processes and thus not certain. The reliability of our mathematical knowledge depends entirely on our knowledge of physics. - **The Role of Intuition**: Mathematical intuition is not a self-evident source of justification but a set of theories about how physical objects behave. - **Gödel's Incompleteness Theorem**: Gödel's theorems demonstrate that for any set of rules of inference, there are valid proofs that the rules will not designate as valid, and that it is impossible to prove that the rules of inference are consistent within the system. The new way in which Gödel managed to prove general assertions about proofs depends on certain assumptions about which physical processes can or cannot represent an abstract fact in a way that an observer can detect and be convinced by. - **Quantum Computing**: Quantum computation has revealed that a proof cannot always be represented as a sequence of statements. Some mathematical calculations can be performed by quantum computers that cannot be represented by a classical proof. This demonstrates that our traditional notions of what constitutes a valid proof are not complete. Therefore, instead of seeking proof of consistency, the sources suggest that our confidence in the laws of physics should be based on: - **Explanatory Power**: The laws of physics are useful for explaining and predicting phenomena in the physical world. The strength of a theory comes from how well it explains what is actually happening. - **Testability**: We should focus on creating good explanations that are testable and falsifiable. If a theory is inconsistent, this is likely to manifest as contradictory predictions. - **Universality**: The laws of physics are assumed to be universal, applying everywhere. - **Seeking the deepest explanations**: The emphasis is not on finding absolute certainties, but on seeking the deepest explanations. - **Interconnectedness**: The goal is to achieve a unified conception of reality, recognizing that understanding any aspect requires understanding its relationships with other aspects. In conclusion, **we cannot definitively prove that the laws of physics are consistent** because proofs are physical processes, and our knowledge of the physical world is not absolute. Instead, we should focus on developing theories that provide the best explanations and are consistent with observations and experimental evidence. --- Date: 20241217 Links to: Tags: References: * []()