Underestimating the inductive step

Quoting Mark Jason Dominus:

Ranjit Bhatnagar once told me the following story: A young boy, upon hearing the legend of Milo of Croton, determined to do the same. There was a calf in the barn, born that very morning, and the boy resolved to lift up the calf each day. As the calf grew, so would his strength, day by day, until the calf was grown and he was able to lift a bull.

“Did it work?” I asked.

“No,” said Ranjit. “A newborn calf already weighs like a hundred pounds.”

Usually you expect the induction step to fail, but sometimes it’s the base case that gets you.

A newborn calf actually ways more like 60 pounds, which is only 27kg. For an adult that’s not that hard to lift over your head, but for a young boy it probably is.

But a moderately strong adult is probably going to fail at the inductive step instead.

I can probably lift a newborn calf. If I lifted it every day, I would have a very annoyed calf. But also at some point I would stop being able to.

I offer two pieces of information in support of this: The first is that an adult cow weighs substantially more than any of the world weight lifting records, let alone for lifting something above your head. If this worked, someone would have beaten this record. Therefore nobody will be able to lift the cow when it is an adult, and therefore by the well ordering principle there was a first day in that cow’s life that they could not lift it even if they followed this program.

The second is: Remember when your parents said “Oof. You’re getting too heavy for me to lift”, despite probably having lifted you on a lot of days as a child.

The problem is that incremental improvement only keeps working if the amount of improvement you can get from the current level is larger than the increment to the next level. With something like the calf story it only works if the amount of improvement you can get from a single day at the current level is greater than the amount the calf will grow in that day. Calves grow fast.

This is on my mind a lot recently because I’m working on general fitness and strength again. I don’t plan to lift any calves (I do have access to sheep, but I don’t think they’d like me lifting them very much and I don’t fancy being mobbed by a flock of them), but the same principle of incremental improvement applies. Indeed the book I’m currently semi-following, Convict Conditioning (which has a bunch of good advice, though I find it very hard to take seriously at times), is very explicitly built on this.

But I’m pretty sure that I’m never going to get to the promised end result of being able to do a one handed hand stand push up. As I make it through the program my progress is going to slow. I’ll almost certainly hit and overcome multiple roadblocks en route where different tactics allow me to make progress once the previous ones have plateaued, but ultimately I’m going to hit some basic physical limits of the design of my body.

This isn’t just about physical limits either. The same happens with purely mental skills. I’m not very good at visual design. I would go as far as to say I am awful at visual design. I could definitely improve my design skills to be better than they currently are, and I could probably even arrange things so I improved a bit every day… but the skill level at which I mostly plateau is not actually going to be very high, owing to certain eccentricities and failure modes of how my brain works (mild forms of dyspraxia and aphantasia if you’re interested).

A lot is made of the Growth Mindset. I am… lets say sceptical of some of the claims, but lets take it at face value: You will improve more if you believe in an expandable notion of intelligence than a fixed one. It’s at least plausible.

But even if it’s true it’s still important to bear in mind that the amount of growth is bounded. And that means that over time growth is always going to get harder.

This might be a depressing thought for you. I like to think of it as just realistic, but the boundary between the two can be somewhat hard to find.

But depressing or realistic I think it’s important, and it can and should guide the decisions we make.

It’s easy to let optimistic thoughts like “We only need to grow 1% per day to succeed big!” (usually accompanied by the formula \(1.01^365 = 37.78\)) get in the way, but you can’t actually keep growing at 1% per day indefinitely. A plan to add one push up each day might result in you hitting 100 pushups, but most people find that at some point adding that extra push up becomes remarkably hard (I’ve yet to make it past 40, and that requires some cheating on my part).

Whenever you’re looking at an opportunity to improve, the inductive step lies to you because you’re starting quite near to the point where it’s easiest (it’s usually not easiest right at the beginning, but you make rapid gains to the point where it is). Incremental growth looks easy, but the height of the plateau is more telling.

So when you want to improve at something, a better question to ask is not “How much can I improve per day?” but “What is a level I would be happy to reach, and what would be a realistic plan to get me there?”. Incremental improvement is certainly going to be part of the answer to the second question, but if it’s the only answer you have then that goal should be pretty close to achievable already.

That’s all I have to say on the subject for now, but I will leave you with another story, due to Raymond Smullyan. You can think of this as the coinductive version of the problem if that’s your thing:

A certain man was in quest of immortality. He read many occult books on the subject, but none of them gave him any practical advice on how to become immortal.

Then he heard of a certain great sage of the East who knew the true secret of immortality. It took him twelve years to find the sage, and when he did, he asked, “Is it really possible to become immortal?”

The sage replied, “It is really quite easy, if you just do two things.”

“And what are they?” the man asked quite eagerly.

“First of all”, replied the sage, “from now on, you must always tell the truth. You must never make a false statement. That’s as mall price to pay for immortality, isn’t it?”

“Of course!” was the reply.

“Secondly,” continued the sage, “Just say ‘I will repeat this sentence tomorrow'”.

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