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davepamn davepamn
 

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 M¢25  Funded By Mahalo ? |  November 05, 2009 08:47 PM

How could a computer program prove true or false an axiom?

Explain how a computer algorithm would determine, if an axiom is true of false.
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opher opher
 
 November 05, 2009 09:34 PM
By definition, an axiom is an assumption which can neither be proven nor disproven, and which one makes as the basis of an argument or a construction. No computer program, or person, can thus determine if an axiom is true or false.

For example, Euclidean geometry has a set of axioms (see e.g. http://www.bymath.com/studyguide/geo/sec/geo5.htm ). These axioms can neither be proved nor disproved. If one was to define a space in which one or more of these axioms could be demonstrated to be false, that space would not follow Euclidean geometry. However, that would not mean Euclidean geometry could not be true in a different space.
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• What is the purpose of identifying the axiom system?


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davepamn davepamn
 
November 05, 2009 10:06 PM
http://mitworld.mit.edu/video/149

I listening too the lecture by Stephen Wolfram at MIT about cellular automata and towards the end of his lecture, Wolfram started talking about a computer algorithm that could navigate an axiom node and determine true or false assumptions. The axiom network seemed to be very deterministic.

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opher opher
 
November 05, 2009 10:24 PM
I don't have over an hour and a half to listen to the lecture. However, you can read his blog here: http://blog.wolfram.com/2007/09/11/my-hobby-hunting-for-our-universe/ and get a sense of what he's trying to do (obviously there are nowhere near enough details to assess if his work is even plausible).

From what I read there, he's not claiming to be able to disprove axioms, merely to identify which axiom systems support a particular mathematics (in his blog the example he uses is that the 50,000th axiom system in his enumeration proved to be Boolean logic).

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davepamn davepamn
 
November 06, 2009 03:53 AM
Will you give an example of how a computer algorithm could identify which axiom systems support a particular mathematics?

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davepamn davepamn
 
November 06, 2009 03:58 AM
Wolfram says, "So I started enumerating axiom systems, and trying to discover where in the space of possible axiom systems our familiar areas of mathematics show up."

"One might think this was crazy–like searching for our universe in the space of possible universes."

Is Wolfram connecting axiom systems which relate and calling the network, a searchable binary system?

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opher opher
 
November 06, 2009 04:13 AM
Due to the site maintenance work I did not see notifications of the further comments until now. To address them briefly, I do not know the purpose of identifying the axiom system, beyond the possibility that Wolfram wanted to know if his algorithm works, and finding the axiomatic basis of Boolean logic supports the possibility that it does. I have no idea what his algorithm is so I cannot venture to answer your other two questions, namely "Will you give an example of how a computer algorithm could identify which axiom systems support a particular mathematics?" and "Is Wolfram connecting axiom systems which relate and calling the network, a searchable binary system?"

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davepamn davepamn
 
November 06, 2009 03:22 PM
Stephen Wolfram proposes:

1. There is complexity in nature. Perhaps, simple algorithms can create a new form of science primitives for understanding nature, as miniature universes of complexity
2. Mathematicians and scientist have used Math to explain nature. For discrete systems math seems to accurately describe nature (the beauty of partial differential equations). However, partial differential equations can be come extremely complex to calculate and discover. Perhaps, it would be better to find a simple algorithm that simulates the behavior of the system.
3. Cellular automa programs have been used to describe complex systems.
4. Other simple programs like binary search, bifurication, fluid dynamics, and Turing machines create complex patterns. Wolfram ran simple programs and observed how they behaved. For rule 110, Wolfram discovered complex random behavior which statistically proved to be random. The simple program had created complicated behavior.
5. Computer languages were designed to help people create primitives to describe what they wanted. Perhaps, there are primitives that describe nature. Language would reflect the rules for describing the natural constructs.

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