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Anything you can do with a calculator is obviously trivial. Is there a clear point of departure when AI can no longer handle mathematical reasoning?

by daoboy 3y ago

devit 3y ago
It's problem solving (i.e. finding a proof of a statement or finding one or more mathematical objects that have some given properties and proving they have them and possibly that no other objects do), not computation.
- daoboy 3y ago
  That distinction seems really obvious now that you said it. Thank you.
edf13 3y ago
The IMO are a bit more involved... e.g. IMO 2023: > Problem 1. Determine all composite integers n > 1 that satisfy the following property: if d1, d2, . . . , dk are all the positive divisors of n with 1 = d1 < d2 < · · · < dk = n, then di divides di+1 + di+2 for every 1 ⩽ i ⩽ k − 2 Although ChatGPT 4 makes a pretty decent start on this already: > To determine all composite integers \( n > 1 \) that satisfy the given property, we need to closely examine the condition set forth: for a composite number \( n \) with its divisors \( d_1, d_2, \ldots, d_k \) where \( 1 = d_1 < d_2 < \cdots < d_k = n \), the property \( d_i \) divides \( d_{i+1} + d_{i+2} \) must hold for every \( 1 \leq i \leq k - 2 \). Let's break down the problem: 1. *Composite Number Requirement*: We are looking for composite numbers, which are integers greater than 1 that are not prime. These numbers have more than two distinct positive divisors. 2. *Divisor Property*: The property \( d_i | (d_{i+1} + d_{i+2}) \) means that \( d_{i+1} + d_{i+2} \) is a multiple of \( d_i \). 3. *Systematic Examination*: We will analyze this property systematically for composite numbers starting from the smallest composite number (4) and going upwards. Let's start the analysis: - For \( n = 4 \), the divisors are \( 1, 2, 4 \). Here, \( 1 | (2 + 4) \), so 4 satisfies the condition. - For \( n = 6 \), the divisors are \( 1, 2, 3, 6 \). Here, \( 1 | (2 + 3) \) and \( 2 | (3 + 6) \), so 6 satisfies the condition. - For \( n = 8 \), the divisors are \( 1, 2, 4, 8 \). Here, \( 1 | (2 + 4) \), but \( 2 \nmid (4 + 8) \), so 8 does not satisfy the condition. Let's implement a systematic method to identify such composite numbers up to a reasonable limit (let's say up to 100 for a start) and check if they satisfy the given condition. The composite integers greater than 1 and up to 100 that satisfy the given property are: 4, 8, 9, 16, 25, 27, 32, 49, 64, and 81. Each of these numbers has the characteristic that for all their positive divisors \( d_1, d_2, \ldots, d_k \) (where \( 1 = d_1 < d_2 < \cdots < d_k = n \)), the condition \( d_i \) divides \( d_{i+1} + d_{i+2} \) is true for every \( 1 \leq i \leq k - 2 \).
- singularity2001 3y ago
  Tangential is there a list of Unicode characters allowed on hacker news? ﷽!
  - edf13 3y ago
    Yes - apologies for the poor copy-paste here!
    - Recursing 3y ago
      You can just ask GPT-4 to rewrite it using unicode characters: Rewrite this post using unicode characters: > The IMO are a bit more involved... Answer: > The IMO are a bit more involved... e.g. IMO 2023: > Problem 1. Determine all composite integers n > 1 that satisfy the following property: if d₁, d₂, ..., dₖ are all the positive divisors of n with 1 = d₁ < d₂ < ... < dₖ = n, then dᵢ divides dᵢ₊₁ + dᵢ₊₂ for every 1 ≤ i ≤ k − 2 Although ChatGPT 4 makes a pretty decent start on this already: > To determine all composite integers 𝑛 > 1 that satisfy the given property, we need to closely examine the condition set forth: for a composite number 𝑛 with its divisors d₁, d₂, ..., dₖ where 1 = d₁ < d₂ < ... < dₖ = 𝑛, the property dᵢ divides dᵢ₊₁ + dᵢ₊₂ must hold for every 1 ≤ i ≤ k − 2. Let's break down the problem: 1. Composite Number Requirement: We are looking for composite numbers, which are integers greater than 1 that are not prime. These numbers have more than two distinct positive divisors. 2. Divisor Property: The property dᵢ | (dᵢ₊₁ + dᵢ₊₂) means that dᵢ₊₁ + dᵢ₊₂ is a multiple of dᵢ. 3. Systematic Examination: We will analyze this property systematically for composite numbers starting from the smallest composite number (4) and going upwards. Let's start the analysis: - For 𝑛 = 4, the divisors are 1, 2, 4. Here, 1 | (2 + 4), so 4 satisfies the condition. - For 𝑛 = 6, the divisors are 1, 2, 3, 6. Here, 1 | (2 + 3) and 2 | (3 + 6), so 6 satisfies the condition. - For 𝑛 = 8, the divisors are 1, 2, 4, 8. Here, 1 | (2 + 4), but 2 ∤ (4 + 8), so 8 does not satisfy the condition. Let's implement a systematic method to identify such composite numbers up to a reasonable limit (let's say up to 100 for a start) and check if they satisfy the given condition. The composite integers greater than 1 and up to 100 that satisfy the given property are: 4, 8, 9, 16, 25, 27, 32, 49, 64, and 81. Each of these numbers has the characteristic that for all their positive divisors d₁, d₂, ..., dₖ (where 1 = d₁ < d₂ < ... < dₖ = 𝑛), the condition dᵢ divides dᵢ₊₁ + dᵢ₊₂ is true for every 1 ≤ i ≤ k − 2.
criddell 3y ago
> Is there a clear point of departure when AI can no longer handle mathematical reasoning? I'm more interested in the point where AIs are presenting proofs far beyond human capability. I'm imagining a day when an AI says it has solved some interesting problem and when we ask for the proof it spits out a 4 million page document. What are we supposed to do with that? What's the role of humans in that world?
lupire 3y ago
Yet AI "models" ar terrible at calculation.

Anything you can do with a calculator is obviously trivial. Is there a clear point of departure when AI can no longer handle mathematical reasoning?

by daoboy 3y ago

Anything you can do with a calculator is obviously trivial. Is there a clear point of departure when AI can no longer handle mathematical reasoning?

devit 3y ago

It's problem solving (i.e. finding a proof of a statement or finding one or more mathematical objects that have some given properties and proving they have them and possibly that no other objects do), not computation.

daoboy 3y ago
That distinction seems really obvious now that you said it. Thank you.

edf13 3y ago

The IMO are a bit more involved... e.g. IMO 2023: > Problem 1. Determine all composite integers n > 1 that satisfy the following property: if d1, d2, . . . , dk are all the positive divisors of n with 1 = d1 < d2 < · · · < dk = n, then di divides di+1 + di+2 for every 1 ⩽ i ⩽ k − 2 Although ChatGPT 4 makes a pretty decent start on this already: > To determine all composite integers \( n > 1 \) that satisfy the given property, we need to closely examine the condition set forth: for a composite number \( n \) with its divisors \( d_1, d_2, \ldots, d_k \) where \( 1 = d_1 < d_2 < \cdots < d_k = n \), the property \( d_i \) divides \( d_{i+1} + d_{i+2} \) must hold for every \( 1 \leq i \leq k - 2 \). Let's break down the problem: 1. *Composite Number Requirement*: We are looking for composite numbers, which are integers greater than 1 that are not prime. These numbers have more than two distinct positive divisors. 2. *Divisor Property*: The property \( d_i | (d_{i+1} + d_{i+2}) \) means that \( d_{i+1} + d_{i+2} \) is a multiple of \( d_i \). 3. *Systematic Examination*: We will analyze this property systematically for composite numbers starting from the smallest composite number (4) and going upwards. Let's start the analysis: - For \( n = 4 \), the divisors are \( 1, 2, 4 \). Here, \( 1 | (2 + 4) \), so 4 satisfies the condition. - For \( n = 6 \), the divisors are \( 1, 2, 3, 6 \). Here, \( 1 | (2 + 3) \) and \( 2 | (3 + 6) \), so 6 satisfies the condition. - For \( n = 8 \), the divisors are \( 1, 2, 4, 8 \). Here, \( 1 | (2 + 4) \), but \( 2 \nmid (4 + 8) \), so 8 does not satisfy the condition. Let's implement a systematic method to identify such composite numbers up to a reasonable limit (let's say up to 100 for a start) and check if they satisfy the given condition. The composite integers greater than 1 and up to 100 that satisfy the given property are: 4, 8, 9, 16, 25, 27, 32, 49, 64, and 81. Each of these numbers has the characteristic that for all their positive divisors \( d_1, d_2, \ldots, d_k \) (where \( 1 = d_1 < d_2 < \cdots < d_k = n \)), the condition \( d_i \) divides \( d_{i+1} + d_{i+2} \) is true for every \( 1 \leq i \leq k - 2 \).

singularity2001 3y ago
Tangential is there a list of Unicode characters allowed on hacker news? ﷽!
- edf13 3y ago
  Yes - apologies for the poor copy-paste here!
  - Recursing 3y ago
    You can just ask GPT-4 to rewrite it using unicode characters: Rewrite this post using unicode characters: > The IMO are a bit more involved... Answer: > The IMO are a bit more involved... e.g. IMO 2023: > Problem 1. Determine all composite integers n > 1 that satisfy the following property: if d₁, d₂, ..., dₖ are all the positive divisors of n with 1 = d₁ < d₂ < ... < dₖ = n, then dᵢ divides dᵢ₊₁ + dᵢ₊₂ for every 1 ≤ i ≤ k − 2 Although ChatGPT 4 makes a pretty decent start on this already: > To determine all composite integers 𝑛 > 1 that satisfy the given property, we need to closely examine the condition set forth: for a composite number 𝑛 with its divisors d₁, d₂, ..., dₖ where 1 = d₁ < d₂ < ... < dₖ = 𝑛, the property dᵢ divides dᵢ₊₁ + dᵢ₊₂ must hold for every 1 ≤ i ≤ k − 2. Let's break down the problem: 1. Composite Number Requirement: We are looking for composite numbers, which are integers greater than 1 that are not prime. These numbers have more than two distinct positive divisors. 2. Divisor Property: The property dᵢ | (dᵢ₊₁ + dᵢ₊₂) means that dᵢ₊₁ + dᵢ₊₂ is a multiple of dᵢ. 3. Systematic Examination: We will analyze this property systematically for composite numbers starting from the smallest composite number (4) and going upwards. Let's start the analysis: - For 𝑛 = 4, the divisors are 1, 2, 4. Here, 1 | (2 + 4), so 4 satisfies the condition. - For 𝑛 = 6, the divisors are 1, 2, 3, 6. Here, 1 | (2 + 3) and 2 | (3 + 6), so 6 satisfies the condition. - For 𝑛 = 8, the divisors are 1, 2, 4, 8. Here, 1 | (2 + 4), but 2 ∤ (4 + 8), so 8 does not satisfy the condition. Let's implement a systematic method to identify such composite numbers up to a reasonable limit (let's say up to 100 for a start) and check if they satisfy the given condition. The composite integers greater than 1 and up to 100 that satisfy the given property are: 4, 8, 9, 16, 25, 27, 32, 49, 64, and 81. Each of these numbers has the characteristic that for all their positive divisors d₁, d₂, ..., dₖ (where 1 = d₁ < d₂ < ... < dₖ = 𝑛), the condition dᵢ divides dᵢ₊₁ + dᵢ₊₂ is true for every 1 ≤ i ≤ k − 2.

criddell 3y ago

> Is there a clear point of departure when AI can no longer handle mathematical reasoning? I'm more interested in the point where AIs are presenting proofs far beyond human capability. I'm imagining a day when an AI says it has solved some interesting problem and when we ask for the proof it spits out a 4 million page document. What are we supposed to do with that? What's the role of humans in that world?

lupire 3y ago

Yet AI "models" ar terrible at calculation.