The Infinity Deception

Mathematics is supposed to be a tool to help us get things right. But have you ever wondered if some mathematical concepts might be helping us in some ways, but leading us astray in others, causing us to make poor decisions?

What about the concept of infinity, and it's reciprocal infinitesimal? Infinity means to be countless, to go on forever. Infinitesimal, which could be defined as the reciprocal of infinity, 1 / infinity, is not the same thing as zero but is vanishingly slight. The infinite and the infinitesimal are useful mathematical tools that have become ingrained in our thinking.

The trouble is that infinity and infinitesimal are mathematical conveniences that do not actually exist in the "real world". Everything is really finite, expressible in real numbers between the infinitesimal and the infinite. Infinity, and thus also it's reciprocal infinitesimal, are not really numbers. They are convenient to us only because of our own scale perspective on the universe.

This can lead us into error if we confuse what is really finite with being either infinite or infinitesimal.

Consider the following simple question. There is a drawer with ten gloves inside, five right and five left gloves. The gloves are well-mixed. Without looking, you reach in and take out two gloves. What are the odds that you have a matching right-left pair, and not two right or two left gloves?

Think carefully.

If you are like many people, you would have reflexively answered that there is an even chance of having a matching right-left pair of gloves, a 50% chance since there is an equal number of right and left gloves.

That would be correct if you took only one glove, an even chance that it was either right or left. But when you took the first glove, whether it be right or left, that left nine remaining gloves. Upon taking the second glove, five of the nine would have resulted in a matching right-left pair.

The odds are thus 5 / 9, which is better than even.

Only if there had been an infinite number of gloves, with each glove thus being infinitesimal, would there be a 50 / 50 chance of taking out a matching pair. The reason that the reflexive answer would get this wrong is that the set of gloves is finite but we react as if it is infinite.

An important tool of geometry is the point. A point is an infinitesimal dot that is defined only by it's location, so that we might say "From point A to point B. But, once again, such an infinitesimal point, as ingrained as it is in our thinking, is a mathematical convenience that does not actually exist. It is so easy to think in terms of the infinite and the infinitesimal but it can lead us into error.

Suppose that a spacecraft is in orbit around the center of gravity of a planet. That would be easy to describe mathematically because the center of gravity of the planet is fixed point, right?

Actually wrong. The center of mass of the planet is fixed but the center of gravity, from the perspective of the spacecraft, is not. The location of the center of gravity depends on the distance of the spacecraft from the planet.

With the spacecraft at a finite distance from the planet, the closer side of the planet has a stronger gravitational effect on the spacecraft than the further side. This means that the center of gravity of the planet, from the perspective of the spacecraft, has to be closer to the spacecraft than the planet's center of mass, and is further from the center of mass the closer the spacecraft is to the planet.

Only if the spacecraft were at an infinite distance from the planet would the center of gravity be exactly the same as the center of mass. Once again, it is easy to reflexively confuse the finite with the infinite and infinitesimal. The reason here is that the planet is not an infinitesimal point. It has a volume of it's own so that the near side is closer to the spacecraft than the further side.

Here is another question. Over the course of a year the length of day and night must even out so that each average 12 hours, right?

It is tempting to answer "yes", but if we define day as being when the sun is visible over a flat horizon then we again encounter the difference between finite and infinitesimal.

If the sun were an infinitesimal mathematical point then the answer would be "yes". But the sun has a diameter of it's own. It's angular diameter is about 1 / 2 of a degree. This means that, with the visibility of the sun being the definition of a day, day must be about four minutes longer than night.

So much about how the difference between the infinitesimal and the finite can lead us into error. What about the difference between the infinite and the finite?

In mathematics, infinity is a mystical place where parallel lines are defined as finally meeting and odds are obligated to even out. What I mean by odds "evening out" is that, if the odds of something happening on each try is 1 / 20, the results are obligated to be exactly that only if we go through an infinite number of tries.

The trouble with gambling is that infinity is a mathematical convenience that does not actually exist. If a gambler has played a game 19 times, without a win, and it is known that the odds of winning the game are 1 / 20, then the gambler may feel "due" for a win.

Casinos must absolutely adore people who are sure that they are 'due" for a win. But in a game of pure chance, with no skill involved, the established odds are only obligated to "even out" if there is an infinite number of tries. In any finite number of tries, there is no obligation whatsoever for the odds to "even out".

Suppose that the intrinsic odds of winning at a game are 1 / 20. A gambler has played the game 19 times without a win. The gambler must thus be "due" for a win on his or her next try, right? This is commonly referred to as the "Gambler's Fallacy".

If you answered "yes" then you are the kind of person that casinos were made for. But there is no obligation for the odds to "even out" so that the twentieth try will be a win. No matter how many losses there have been in a row, the odds of winning the next one are still only 1 / 20. That is true for any finite number of tries. Only if the game were played an infinite number of times would one out of every 20 tries have to be a win.

Unfortunately, infinity is a mathematical convenience that does not really exist. But there are people in every casino who really believe that the rules of the infinite must somehow apply to the finite.

I have used gambling as a simple example. But the confusion of the finite with the infinite or it's reciprocal the infinitesimal goes far beyond gambling. Suppose that an employer is very selective of potential employees and overall only one out of every twenty applicants that are interviewed is hired.

One day the human resources manager is amazed at his good fortune. He has just interviewed three outstanding candidates in a row who clearly qualify for the job. This has been much better than interviewing for an entire day or more just to find one good candidate.

But then the human resources manager stops to think. How can this be possible? If thus far only one out of 20 candidates has qualified for the job then how can I possibly get three in a row? This is against all the odds and so I must have had poor judgement and have been too lenient with the candidates.

This is, of course, the reverse of the Gambler's Fallacy. There may well have been three qualified candidates in a row. Even if the overall odds are established as 1 / 20, there is no obligation whatsoever for those odds to "even out" in any finite set of candidates. Only in an infinite set of candidates would exactly one in 20 have to be qualified. Once again we see how confusing the infinite with the finite may lead to poor decisions.

Here is an old trick question, don't feel bad if you get it wrong because you will have plenty of company. When we flip a coin it has an equal chance of coming up as heads, H, or tails, T. If we flip the coin 8 times, which of the following sequences are most likely to occur?

HHHHHHHH

THHTHTHT

THTTTHTH

The answer is none of them. All three sequences, and all other possible sequences, have an equal chance of coming up. In any finite sequence there is no obligation for there to be an equal number of heads and tails.

Next question. You operate a business in a city state that prints it's own currency. What would be a good way to estimate the amount of currency in circulation?

Record the serial number of every bill that comes through the business, in order of when they came through. Count how many bills pass through before the same serial number comes up again. Multiply that by two because once you have the serial number, and send the bill back into circulation, the odds exceed 50% that you will get the bill again when half the bills in circulation have passed through the business. Multiple readings taken, and then averaged together, would give a more accurate picture of the number of bills in circulation than just one reading.

But there is still the chance that a bill will come through the business and give an estimate that is well out of the scope of how much currency is actually in circulation. This is what I refer to as either a coincidence, making it seem that there is less money in circulation than there actually is, or a reverse coincidence, making it seem as if there is more money in circulation than there actually is. Only with an infinite number of such measurements would a truly accurate picture of the currency be certain.