Why do some mathematical ideas seem counter-intuitive?

Question

Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the first flip, you win $\$1$. If you flip heads on the second flip, you get $\$2$, on the third flip $\$4$, and so on. Now, if a casino were to host this games, how much should they make their buy-in?

Intuition says not much, but mathematically they should make it as high as they want. Why? Because the payout is infinite. The probability of flipping heads on the first flip is $\frac 12$, which gives a $\$0.50$ average payout. The probability that you flip heads on the second flip (which also means heads on the first flip) is $\frac 12\times\frac 12=\frac 14$, which also pays out $\$0.50$ on average. Continuing like this gives you a payout of $\sum^{\infty}\$0.50=\$\infty$ every time you play the game! Not such a bad thing, but it leads to my main question (shortly after).

Suppose you hold a party with $30$ people in it, and you want to find the probability that any two of them will have a birthday on the same day. Do you expect that to happen, or not?

Again, common everyday intuition says it seems unlikely that any two people out of thirty will have a birthday on the same day, but, again, mathematically, it is more likely than not. The exact probability is $1-\frac{365!}{365^n(365-n)!}\approx 0.7063$. So is it time to ask the question?

Why do some mathematical ideas seem counter-intuitive? Mathematics isn't based off of physical observations; it's an abstract concept, so shouldn't it explain our world better, not worse?

The above game (which I was told is St. Petersburg paradox) is only an example of what I mean when I say "counter-intuitive". Among others, ones I can name off the top of my head are the Monty Hall problem, Benford's Law, and the Banach-Tarski paradox. Those all have specific aspects to which a normal non-mathematician would turn their heads in confusion.

I really hope my question isn't too philosophical for this site.

This question has been in my head for as long as I can remember, so I decided to post some of my thoughts. Mathematical laws don't just hold for our world or our universe. It holds for all universes. For example, maybe the Banach-Tarski paradox makes perfect sense in $34$ dimensions. Or maybe the second dimension finds the concept of $\pi$ being irrational hard to grasp, whilst we find it easy. The most important thing to note is that mathematics is always right. It doesn't matter what we think. We're stupid. But in the long run, math has and always will get out on top.

Is the reasoning in the previous paragraph correct? The answers so far are good, but they don't really address counter-intuitivity in general, instead specific problems. Several answers below state something along the lines of "some ideas seem counter-intuitive because we've adapted to it; that is to say, it is best for the human race". Can any of you think of a practical application of counter-intuitive ideas in the evolution of humankind? I certainly can't.

So what do you think? I know my question doesn't have a solid answer, and I know it might be put on hold because of it (please don't though!). I just want to put my question out there, and hope it gets answered.

Thanks for reading!

Answer 1

Counterintuitive results come from our inductive thinking. We tend to think that if all, or even most of the objects we have met so far have a certain property, then all objects with similar characteristics will have that property.

If you look at the 19th century mathematicians, they first thought that a continuous function is differentiable with at most countable exceptions, but then a nowhere differentiable continuous function was defined -- and now we know that amongst the continuous functions, most of the functions are nowhere differentiable.

As you proceed in mathematics you learn that the objects that interest us are often the pathological objects, if you consider the broader picture. Most functions from $\Bbb R$ to itself are not continuous, of the continuous ones, most are not differentiable anywhere, of the differentiable, most are not continuously differentiable, and so on. Similarly of the subsets of $\Bbb R$ most of them are not Lebesgue measurable, of the Lebesgue measurable, most of them are not Borel measurable. And similarly for the real numbers, most of them are not rational, or even algebraic.

This is why you run into "normal" and "regular" terms in mathematics. We model the basic axioms of an object based on a smidgen of intuition (which may or may not be a well-developed mathematical intuition), but then we learn that there are other objects as well, so the original objects are added an extra hypothesis and we call them "normal" or "regular". And then we develop better mathematical intuition, and the cycle continues to grow.

Finally, since mathematics is not based on physical observation, I don't see why it should describe physical reality "better" or "worse". It shouldn't describe physical reality at all. It can be used to model reality, but since mathematics require infinite precision, and our senses can give us a very limited bound of input, we can never truly model the physical reality via mathematics, since we don't fully grasp it.

Answer 2

In your first example, the St. Petersburg Paradox, the expected value of the pay-off is infinite. That is a firm mathematical result following from the axioms of probability.

However, the buy-in amount (or the decision to play given a specified buy-in) depends upon other considerations -- for example, your risk preferences. This requires a behavioral model and is not a purely mathematical inference. Risk preferences could be modeled mathematically using a utility function and depend on other mathematical results such as probability of ruin, variance of the pay-off, etc.

If you continue to play and bet all of your accumulated gains on subsequent turns, the probability of ruin approaches 1. Would you still play for the infinite expected gain?

Answer 3

"Why do some mathematical ideas seem counter-intuitive? Mathematics isn't based off of physical observations;..."

The bold portion above could be paraphrased as asking how come mathematics can do things our intuition cannot.

The bold portion below could be paraphrased as asking shouldn't mathematics be able to do things our intuition cannot.

"... it's an abstract concept, so shouldn't it explain our world better, not worse?"

$$$$

Contrary to what your intuition is telling you, these two ideas are not in contradiction.

Answer 4

There are several reasons that mathematics seems counter intuitive.

• For several issues our intuition models the universe badly. For example;
  • We tend to discount very rare events as impossible and we also tend to value a high impact higher than we should compared to high frequency impacts.
  • We fail to model scale correctly. See for example this less wrong article
  • We tend to model infinity as a really big number.
  • We tend to overemphasise effects that are close to us at the expense of things far away. This is particulary noticable in that we tend to discount long term effects.
• The mathematical model can be of something different than what our intuition is modeling. This is the case in your example.
  • The model assumes that the entity backing the bet can give infinite money. Our intuition will tend to assume that the entity behaves like a concrete legal person with bounded cash.
  • The model operates on cash value, our intuition might just as well operate on expected utility.

Answer 5

Other people seem to agree with your initial claim that the payoff is infinite. Maybe it's because I have a cold, but I'm not seeing it. First off, I'm assuming that the money won stacks (in other words, if you get $10$ heads in a row and quit, you get $1+2+4+8+\cdots+512=1023$ dollars, versus just getting $512$). In that case, your expected payout is $2^{-10}\cdot1023$ which is just slightly less than $1$ ($1023/1024$)... and for the case of an infinite games, your expected payout would be $1$. (Note that as written right now, your problem doesn't appear to be identical to the St. Petersburg Paradox; not sure if this is an edit issue or something else)

Answer 6

Short answer: Because we're not good at intuiting results in infinite situations (whereas practical, finite versions tend to make more intuitive sense), and we can be deceived by probability too. I harbour a suspicion that this latter one happens when we're working without much information or context; the kind of thing that our evolutionary psychology hasn't prepared us for, I suppose.

Two of your examples revolve around infinite quantities: your lottery game (the St Petersburg paradox) and the Banach-Tarski paradox. We're not good at intuitions around infinity, which makes a certain amount of sense. And if you reduce the St Petersburg lottery to finite possible winnings (or finite time to play), it actually becomes quite in line with intuition.

I like to think of the Monty Hall paradox as a bit of sleight-of-hand played with information. You may intuitively realise that your first guess is probably wrong (2/3 chance), but you think that, since picking any door would have led to the same conclusion, the "switch" door is no better. But really, since your first guess is probably wrong, that means the right door is probably not yours--and the host has conveniently eliminated the only other wrong door! (To put it another way, you're being offered a chance to play a new guessing game with a 50-50 chance of winning, instead of the original game, which you probably lost.)

As for Benford's law, I'm not really seeing what's unintuitive, but I only just read about it for the first time, thanks to your question.