The Paradox of the Heap

In Oppositions and Paradoxes John L. Bell explores a variety of mathematical and scientific paradoxes with philosophical precision, while retaining a great sense of humour in his investigations. In this excerpt, Bell formulates and works through “The Problem of the Heap,” asking: how many grains of sand does one need to make a heap, exactly?

The paradox of the heap or sorites paradox (from the Greek sōritēs “heap”) - attributed to the ancient Greek philosopher Eubulides of Miletus - arises from the vagueness of certain predicates in ordinary language. In a typical formulation, we consider a heap of sand, from which grains are removed one by one. The paradox arises when one considers what happens when the process is repeated sufficiently many times. For suppose we make the natural assumption that, if we remove a single grain from a heap, we are still left with a heap. Then eventually just a single grain remains: is it still a heap? Or are even no grains at all a heap? If not, when did the heap change into to a non-heap?

We can turn the paradox on its head by starting with a totally bald man, and, noting that any man with just one more hair than a bald man is still bald, conclude that every man must be bald. For a man with no hair is bald, so a man with just one hair is bald, and thus a man with two hairs is bald, … whence a man with any number of hairs is bald.

A related formulation of the paradox is to suppose given a set of coloured chips such that the variation in colour of two adjacent chips is too small - a difference in wavelength of 1 nanometre say - for the human eye to be able to distinguish between them. Suppose that the first chip is coloured violet, which has a wavelength of about 400 nanometres, and the last chip is coloured red, with a wavelength of 650 nanometres. If we assume, as in the case of the bald man, that a chip whose colour differs in wavelength by one nanometre from a violet coloured chip would still be seen as violet, then the ‘bald man’ argument leads to the conclusion that the red chip would also have to be seen as violet.

The paradox can be reconstructed for a variety of predicates—all of which can be seen to be vague - for example, with “short,” “poor,” “young,” “red,” and so on.

A natural response to the paradox is to introduce a “fixed boundary” to the concept of heap by defining a “heap” to be a set of grains containing at least a certain fixed number - 10000, say - of grains. In that case, a set of 9999 grains is not a heap but one of 10000 is. This seems unnatural since there would appear to be little significance to the difference between 9999 grains and 10000 grains. Wherever the boundary is set, it remains arbitrary. A more acceptable, if radical solution would be to call any collection of two or more a heap!

~ From The Paradox of the Heap, from John L. Bell’s Oppositions and Paradoxes - Broadview Press

If you can’t be a good example – be a warning

No one put things more starkly and more bleakly that the German philosopher Arthur Schopenhauer, often referred to as the “philosopher of pessimism”.

In early youth, as we contemplate our coming life, we are like children in a theater before the curtain is raised, sitting there in high spirits and eagerly waiting for the play to begin. It is a blessing that we do not know what is really going to happen. Could we foresee it, there are times when children might seem like innocent prisoners, condemned not to death, but to life, and as yet all unconscious of what their sentence means. Nevertheless every man desires to reach old age; in other words, a state of life of which it may be said, “It is bad today, and it will be worse tomorrow— and so on till the worst of all.”

If you try to imagine as nearly as you can what an amount of misery, pain, and suffering of every kind the sun shines upon in its course, you will admit that it would be much better if on the earth as little as on the moon the sun were able to call forth the phenomena of life; and if, here as there, the surface were still in a crystalline state.

Again, you may look upon life as an unprofitable episode, disturbing the blessed calm of nonexistence. And in any case, even though things have gone with you tolerably well, the longer you live the more clearly you will feel that, on the whole, life is a disappointment, nay, a cheat.

If two men who were friends in their youth meet again when they are old, after being separated for a lifetime, the chief feeling they will have at the sight of each other will be one of complete disappointment at life as a whole; because their thoughts will be carried back to that earlier time when life seemed so fair as it lay spread out before them in the rosy light of dawn, promised so much — and then performed so little. This feeling will so completely predominate over every other that they will not even consider it necessary to give it words; but on either side it will be silently assumed, and form the groundwork of all they have to talk about.

He who lives to see two or three generations is like a man who sits some time in the conjurer’s booth at a fair and witnesses the performance twice or thrice in succession. The tricks were meant to be seen only once, and when they are no longer a novelty and cease to deceive, their effect is gone.

If children were brought into the world by an act of pure reason alone, would the human race continue to exist? Would not a man rather have so much sympathy with the coming generation as to spare it the burden of existence? Or at any rate not take it upon himself to impose that burden upon it in cold blood.

- From Studies in Pessimism (1913)

The problem with being on time is that nobody is there to appreciate it

Try to imagine a life without timekeeping.
You probably can’t. 

You know the month, the year, the day of the week. 

There is a clock on your wall or the dashboard of your car. 

You have a schedule, a calendar, a time for dinner or a movie. 

Yet all around you, timekeeping is ignored. 

Birds are not late. 

A dog does not check its watch. 

Deer do not fret over passing birthdays. 

Man alone measures time. 

Man alone chimes the hour. 

And, because of this, man alone suffers a paralyzing fear that no other creature endures. 

A fear of time running out.

- Mitch Albom, The Time Keeper

Spiral of Silence

The spiral of silence theory (Maibach et al., 2016) posits that humans are less willing to express their opinions when they believe that they are not shared by others, and this unfortunately influences public opinion.

Specifically, the perception that one’s opinion is unpopular tends to inhibit or discourage one’s expression of it, while the perception that it is popular tends to have the opposite effect. Developed by German survey and communication researcher Elisabeth Noelle-Neumann in the 1960s and 1970s, the spiral of silence theory more broadly attempts to describe collective opinion formation and societal decision making regarding issues that are controversial or morally loaded.

According to the spiral of silence theory, most people have a natural and mostly unconscious fear of social isolation that prompts them to constantly monitor the behaviour of others for signs of approval, or disapproval. People also issue their own “threats” of isolation—mostly unconsciously—through behaviour such as criticizing someone, turning away from someone, scowling at someone, laughing at someone, and so on. To avoid isolation, people tend to refrain from publicly stating their views on controversial matters when they perceive that doing so would attract criticism, scorn, laughter, or other signs of disapproval. 

Conversely, those who sense that their opinions will meet with approval tend to voice them fearlessly and at times vociferously. Indeed, speaking out in such a way tends to enhance the threat of isolation faced by supporters of the opposing position, reinforcing their sense of being alone. Thus a spiraling process begins, the dominant camp becoming ever louder and more self-confident while the other camp becomes increasingly silent.

Some are just better able to deal with the daily punches of life

Pain or damage don’t end the world, or despair, or f**king beatings.
The world ends when you’re dead, until then, you got more punishment in store.
Stand it like a man, and give some back.

- Al Swearengen (Ian McShane) on HBO show Deadwood; S02 E07.