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Blue Scallop
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Is it possible for a math to be non-algorithmic? Can you give example?
.Scott said:Most math is non-algorithmic.
In general, math makes declaratory statements.
In contrast, algorithms make imperative statements.
For example:
N = N + 1;
That is an imperative statement. When executed, it will cause a variable "N" to be incremented.
Taken as a Math statement, it is a declaration which is false.
Since you asked for a specific example of a Math statement that is non algorithmic, her is one:
$$ x^2 + y^2 = r^2 $$
.Scott said:I did read "The Emporer's New Mind", but not recently enough to remember clearly.
As I recall, "algorithmic" in that context referred to any process that was equivalent to how conventional (non-quantum) computers process information. He was arguing against the generation of consciousness by "algorithmic" information processing by arguing that all such processing was the equivalent of reading the algorithm from a printed document and working it out manually. Since that printed document exercise does not elicit its own "consciousness", neither would any other mechanical or electronic equivalent.
Quantum computers operate on information in a fundamentally different way that conventional computers. And therefore, the potential for creating consciousness through a QM process is not affected by Penrose's analogy to reading a printed description of the algorithm.Blue Scallop said:You meant Penrose was saying quantum computers were non-algorithmic yet still describable by math? I'm trying to search his book now where he strings the word "math" and "non-algorithmic" together..
It seems a bit of nitpicking admittedly, and that is not my intention. More or less what you have written is correct.Gigaz said:There is however widespread belief that the Church-Turing thesis is valid and that computers and human brains are insofar equally powerful that any non-computable function for one is also non-computable for the other, and that computers can solve any mental task that a human can solve.
.Scott said:Quantum computers operate on information in a fundamentally different way that conventional computers. And therefore, the potential for creating consciousness through a QM process is not affected by Penrose's analogy to reading a printed description of the algorithm.
The only reason I am not answering your question with a simple "yes" is that there are QM algorithms (ex, Shor's, Grover's, annealing). So if Penrose said that quantum computers are non-algorithmic, then he was using the term "non-algorithmic" is a very context-dependent sense.
And yes, what quantum computers can be described by Math.
Non-computable denotes that the algorithm cannot complete, cannot complete in a finite amount of time, or cannot complete with all available resources in the universe. It refers to a task set to run on a Turing machine or other classical computer.Blue Scallop said:We posted the replies at the same time so maybe you didn't see the message above it. I was saying I searched all instances of the words "Non-algorithmic" in Penrose book and saw the following: "In Chapter 10, I shall try to present arguments to show that the action of our conscious minds is indeed non-algorithmic (i.e. non-computable)".
I first read the book about 10 years ago and my first impression was non-computable means can't be solved by math. So I thought non-algorithmic means it can't be solved by math. But then in your understanding, can you give some examples of maths that are non-computable in the context of Penrose's? Thanks a lot Scott!
Non-algorithmic math, also known as non-computable math, is a branch of mathematics that deals with problems that cannot be solved by an algorithm or computer program. These problems involve complex and unstructured data that cannot be easily broken down into a set of rules or steps.
Some examples of non-algorithmic math problems include the halting problem, the tiling problem, and the four color theorem. These problems involve tasks that cannot be completed by following a set of instructions or algorithms, and often require creative thinking and intuition to solve.
Non-algorithmic math is important because it allows us to explore and understand problems that cannot be solved through traditional computational methods. It also helps us develop new approaches and techniques for solving complex problems, leading to advancements in various fields such as computer science, physics, and biology.
Non-algorithmic math is used in various fields such as cryptography, artificial intelligence, and theoretical physics. For example, in cryptography, non-algorithmic math is used to develop encryption methods that are resistant to hacking and other attacks. In artificial intelligence, it is used to create algorithms that can learn and adapt to new situations. In theoretical physics, it is used to understand and model complex systems such as black holes and the behavior of subatomic particles.
Non-algorithmic math can be challenging to understand, as it often deals with abstract concepts and requires a high level of mathematical knowledge. However, with proper training and practice, it can be understood and applied effectively. It is also a fascinating and rewarding field for those who are passionate about problem-solving and critical thinking.