Large numbers. Little comprehension.
"The human brain has 100 billion neurons, each neuron connected to 10 thousand other neurons. Sitting on your shoulders is the most complicated object in the known universe." ~ Michio Kaku
For the most complicated object in the known universe, brains are quite bad at math. Logarithmic scales, exponential growth, and large numbers at scale are simply not intuitive. It's ironic (or perhaps telling?) that our intuitions fail us here in the most pure representation of reality that we have.
The opening quote tells us that your brain has 100 billion neurons... but I’m not convinced that the majority (hell - even a minority) of people who hear a number like '100 billion' actually have a conceptual grasp of its magnitude.
For example, there are 7.5 billion people on the planet… that’s a lot of people. If you added up all the people you have ever met in your entire life, it'd be in the ballpark of ten of thousand people. In reality, you really only have the cognitive real estate for about 150 people because of the Dunbar limit. But let's give you a picture-perfect memory for thought experiment- even with this superpower, you would have only met 0.07% percent of the world's population. You still haven’t met more than 99.9% of the people out there! There are a lot of people in the world. Like - a whole lot of people. And you’ve never met them!
So can we actually fathom the scale of 7.5 billion people? It’s kind of… too large. Once numbers hit a certain level, they all just get labeled as 'really big' in our mind. But there is so much size difference in that category we call 'really big'. A million billion is another 'really big' number. If I don't put much thought into it, it slips into the same category that 7.5 billion does. But these are drastically different numbers. Consider this: There are a million billion ants on earth. Yes - 1,000,000,000,000,000 ants. A hundred thousand ants for every human. The difference in scale between the two is like comparing the size of a human to the size of a bacteria. Those are two different universes of scale. It's insulting to even put a million billion in the same category as 7.5 billion. Yet, it is the norm for large numbers like these to subconsciously end up in the same category of ‘really big’.
'The most complicated object in the universe' misleads us on more than just scale. We're also intuitively bad at statistics.
Question 1: How many people do you need to have in a room before you’re more-likely-than-not to have two people with the same birthday? (i.e. more than 50% chance)
Commit to a number before reading on.
No seriously - choose a number.
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Don’t be that person who won’t commit to a number even when no one else is watching to see you get it wrong. Put that ego in its place and commit to a guess before continuing on.
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Answer: For a 50% chance you only need 23 people. That's right - you only need 23 people in a room before it more likely than not that there's a birthday match. For a 99% chance of there being a match, you only need 70 people. This birthday paradox highlights that we don't need large numbers to throw off our intuitions.
But now we’ve course-corrected. Mathematical witchcraft will not work on us again. We've seen the error in our ways and are wiser now.
Question 2: You shuffle a deck of cards extremely well. You shuffle it for hours via multiple shuffling techniques and even have a friend shuffle it a few times. The order of cards now is as random as can be. So, how many other times has a deck of cards with this exact same order of cards exsisted? You think about all the 24-hour casinos and families playing card games around the world. You consider the fact that the 52 card deck is older than the United States of America is. You want to make sure your estimation isn’t too low because you are aware of how poor your mental model is of the scale of things, but you also don’t want to estimate too high... What’s your conclusion? How many other times has a shuffled of deck cards ended up exactly in this order? Millions of billions? Billions? Millions? Hundreds? Never?
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Answer- Millions of times.
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Ok actually that’s a lie. Felt reasonable though right? Truth is that there are so many permutations of a deck of cards, that every well-shuffled deck is probably completely unique and never existed before. There are 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 different ways the cards in a deck could be arranged. Exponential growth is unintuitive. Need another example? Out of all the humans to have ever lived (the Egyptian civilization, the Mayan civilization, all the cowboys and pirates, Asian dynasties, the Roman Empire, etc.), 8 percent of them are alive right now. That's right, nearly 1 out of every 10 people who has ever lived is alive right now.
Resize and rescale.
To make sense of numbers and sizes that are not in our normal vocabulary, we need to turn them into a scale we better understand. Jeff Bezos, CEO of Amazon, has a net worth of more than 100 billion dollars. On July 20th, 2020 Amazon stock rose sharply giving Bezos his largest single-day increase in wealth ever - 13 billion. That amount of money is unfathomable. The US national median household income is 66,000$. We don't normally think about money on the scale of billions (if you do, please donate to this blog). But consider the following:
1 second is 1 second
A thousand seconds is 16 minutes.
66,000 (average US income) seconds is 13 hours
A million seconds is 11.5 days.
A billion seconds is 31.7 years
13 billion seconds is 400 years
100 billion seconds is 3100 years
When rescaled this way, the massive difference is better visualized. It becomes a lot more clear that a million (11.5 days) is a whole lot less than a billion (32 years). We probably shouldn't lump these together in the same 'really big' category. And the US average household income (13 hours) pales in comparison to these. You can intuitively feel the difference between having 13 hours to live and 11.5 days. It's not just numbers when felt this way. Go back and look at Bezos wealth with this scale in mind.
Rescaling like this can be done in (slightly) more realistic scenarios. Most people don't know what a 250,000$ income is like - it's out of scale for us. It's the 97% percentile of US income. In order to better grasp it, we need a change in perspective. For simplicity's sake, let's compare it to a 50,000$ annual income. When we look at the 250,000$ income, we shouldn't think of them having five times more money than the 50,000$ income. Doing that puts us in an unfamiliar scale of things. Instead, imagine everything being 5 times less expensive for them. This puts that higher-income into a perspective you are used to. This is the key point. When things are out of your normal scale, don't pretend you understand it: re-scale it!
Let's visualize this.
Going out to eat for $4? I think I'll get sushi more often.
The take-home point from the table is that everything is really cheap when you are rich. A seemingly obvious statement, that hopefully makes more intuitive sense when things are in a scale you can appreciate better. This is a simplified model of course, but the idea of scaling things into numbers you are familiar with is the point here.
Why does this matter? Our bad mathematical intuitions and mental models have real-life implications. What does it mean that the US Federal government spends 600 billion annually on the military? What does it mean that there are 100 billion alien planets in the Milkyway galaxy, and the Milkyway is one of a 100 billion galaxies? These are numbers - there should be next-to-no miscommunication here because numbers are as close to truth as we can get. We're making the Communication Chasm as small as possible here. But still, things are getting lost in translation.
Ultimately, a common theme I've picked up on in life is that everyone has really inaccurate perceptions of the world. It's not just math and scale. Misperception of others, the self, etc. all create a nightmare of miscommunication, divisiveness, and suffering that is (at least partially) avoidable. Numbers and math aren't the most glamorous misperception to try to fix, but it is some low hanging fruit.
"There are a million billion ants on earth. Yes - 1,000,000,000,000,000,000 ants." The number of ants is displayed in the article as 1E+18, shouldn't it be 1E+15 (1E+6 x 1E+9)?