What Rose Petals Teach Us about Induction | Hacker News Reader

What Rose Petals Teach Us about Induction

oranlooney.com

41 pointsolooney10 comments

It’s interesting to note that in regular mathematics, induction on the natural numbers is axiomatic.[1] That is, you can’t get there from within the framework you have built up to then so you need to build an extra axiom into the system to make it work. Both the mathematical technique of induction and the use of the word induction as per the article to mean deducing knowledge from experience are due to Descartes I believe.[2]

[1] I’m slightly simplifying. Some people teach “the principle of mathematical induction” as a “principle” (whatever that is, but it sounds like an axiom to me), and some people teach it as a theorem like any other theorem but derive it by adding an axiom to set theory called the “well-ordering principle” (that every non-empty set of natural numbers has a smallest element) that turns out to be equivalent to induction. If you don’t accept induction as an axiom and don’t take it to be axiomatic that you can choose the smallest element from a set of natural numbers then as far as I know you can’t do induction in a well-founded way.

[2] https://www.jstor.org/stable/43154967

In first-order logic, it's an infinite axiom schema, which means you can take a finite subset of them, like assuming induction axioms hold for the formulas you're actually using and not in any other case.

The problem of induction doesn't need to be solved because the problem refutes itself when time is brought into the picture.

Let's assume Hume's premise "there can be no demonstrative arguments to prove, that those instances, of which we have had no experience, resemble those, of which we have had experience" is true.

Ok, then how do we know this premise will stay true in the future? Because it is true right now? Because every moment in the future will eventually become the present? But how do we know that? Because it has always happened this way. Time has only ever moved forward, never backward, at least in my experience so far. Oops, this is inductive.

If the premise might not stay true in the future, then that means we've managed to acquire a piece of information about the future: the premise of the problem might not be true by then. We've just gained a bit of knowledge about the future which the premise claims we can't possibly do!

Knowing that a premise will stay true in the future and knowing that it will become false in the future are both knowing something about the future.

Compare with knowing whether a program will terminate.

Thanks to your comment, I realized I've not successfully refuted Hume. The possibility of a truth value change is not a new piece of info about the future. We did arrive at it by deduction, but that info is also an innate part of the propositional logic system, so we already had access to it beforehand. And even though we've never observed Hume's premise change its truth value in the past, the possibility of change has always been there. So Hume's premise does not refute itself, and it still stands. All I can say about it is that the premise does not rule out the possibility that it itself may become false in the future, but this is more of an epistemic boundry technicality rather than an useful insight.

Also, I may have jumped the gun with that comment. I didn't read the whole article the first time around because I thought it was just going to be a hit piece on scientific method when the problem of induction came up. But that isn't the case and they're talking about AI research.

I only read up to the "Petals around the Rose" exercise. I found it pretty easy to deduce (induce?) but the author is right, there's no general algorithmic method to do it .. except for intelligence.

So, the solution is "AI", and even then it's only a "probably".

The exercise didn't necessarily feel easy to deduce within the 20 minutes I tried, but after asking LLMs (spoilers alert!):

  1. How big of a hint is the name
  2. How big of a help will choosing numbers manually be
  3. How important is the number of dies
  4. How familiar do I have to be with flowers

The answer to the riddle, and then to the questions I asked were immediately obvious. I feel kind of ashamed, haha.

> The exercise didn't necessarily feel easy to deduce within the 20 minutes I tried,

I admire your breakdown, and I felt your pain.

My frustration was with opening a new tab for each sample because I was too lazy to transcribe rolls (for review) -- so once I'd opened and revisited about 16 tabs I got rather annoyed and just started rapid rolling looking for high values ...

After about 30 seconds of this pique I rolled a Yahtzee 55555 value 20 and went: Doh!

My takeaway is that using "induction" to illuminate Human ~vs LLM may be wrong framing when Eureka moments are possible.

Matrix Reloaded (2003) ...

  Neo: Hiya fellas.
  Agent 1: It’s him.
  Agent 2: The Anomaly.
  Agent 3: Do we proceed?
  Agent 2: Yes.
  Agent 3: He is still...
  Agent 1: ...only human.
  Neo: Hmm--*upgrades* [beats all three nonetheless]

1*0=0

I seriously thought this article would be about induction top stoves and roses. Little did I know...

Reply on news.ycombinator.com