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Comment by gavinray | Hacker News Reader

Human brains have difficulty reasoning about exponential growth.

They keep saying that. I'd say it is more like human brains that don't remember high school math have trouble with it.

If something at rest is accelerating at 9.8 m/s^2, how long in seconds will it take to reach 10% of c? Answer to the nearest order of magnitude - will it take approximately 1000, 10k, 100k, 1000k seconds?

I’m sure you know this is an exponential growth question but have no intuition of the answer.

That is a linear growth problem whose answer is very easy to intuit.

It’s not a linear growth problem. Acceleration is quadratic.

As per your intuition, how many seconds later will you hit 10% of c?

Acceleration is not quadratic with respect to speed/velocity, and c is a speed/velocity.

You must be joking. A high schooler with a few hours of physics classes can intuit the answer.

Until special relativity kicks in to completely invalidate whatever intuition you have about this problem.

How is this a joke?

Knowing exponents and how to apply it is not the same as having any intuition about what the actual value of a certain exponential function will be at a certain point and when it crosses a threshold.

There’s no exponential function involved, which is why this is easy to intuit. A true exponential function would indeed be difficult to intuit.

You should have asked something simpler.

Let's say you have infinite ships traveling at light speed originating from earth trying to colonize the entire universe.

All of this is funded by borrowing capital from earth and earth expects a 5% annual return in perpetuity.

The space ships must pay interest to earth and if they fail to pay it, they are not allowed to perform further colonization.

Will earth manage to colonize the entire universe? Aka, can the infinite number of at light speed traveling ships outrun the interest payments?

After answering that question this one should be easy:

What if the universe was infinitely large and infinite growth was possible?

Reasoning is a strong statement here. But it is fair to say that it usually is not intuitive to us.

An example is if I gave you a huge sheet of thin paper (huge so that folding isn’t an issue) - how many times could you fold it in half until you couldn’t physically do it anymore? Could you do at least 10? Try this with random people and you’d be surprised how many say they could do 10 easily.

Or the chess board question. Works to rather get the financial equivalent of starting with a penny and then doubling it for every square on the board or a million dollars for each square? Again, if you ask people to pick one without giving them the time to work it out they will usually pick the million dollar per square.

But you're not reasoning yourself here. In face you're literally parroting what an LLM would do in this situation: you already know the answer so you think your comprehension of this is better than that if others, when in fact it's just simple pattern matching. It's not a measure of intelligence, it's a measure of memory.

I actually started by saying "reasoning" is a strong statement because as you note, it's really not reasoning. It's about intuition. And yes, if you've seen it before then intuition doesn't play a role. But if you haven't had sufficient experience with it, even if you know "double each square", and even if I walk you through the first ten squares on the chess board "OK, now we're at $10.24 cents... and we're at $10 million dollars with the other method -- it's not looking too good for the doubling method is it!?".

But a philosophical question is, with sufficiently large memory -- does almost everything just reduce to a measure of memory?

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