Heap of Sand Paradox

Sand from Pismo Beach, California via Wikipedia

The Heap of Sand Paradox is a problem posed in philosophy, originally attributed to the Greek thinker and harsh critic of Aristotle, Eubulides (4th century BCE).

Eubulides conceived of many different types of paradoxes, including the Liar paradox, which is similar to the well-known Cretan Liar paradox.

In a nutshell, Eubulides’ heap of sand paradox asks at what precise point a heap of sand no longer is a heap when a single grain of sand is repeatedly removed from a pile.

This is similar to Zeno‘s grain of millet paradox.

The heap of sand paradox may seem trite to some but it raises important questions about moral judgments based on the quantity or intensity of a characteristic or action.

For instance, one could ask at what precise point art becomes pornography, jesting becomes abuse, poking becomes assaulting, hugging becomes groping, and so on.

The heap of sand paradox also poses potentially complicated questions about the nature of valid reasoning and the often ambiguous role of representational symbols (such as numbers, characters and language) in logical problem solving.¹

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¹ See for instance, the discussion here: http://en.wikipedia.org/wiki/Sorites_paradox

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Temporal Paradox

Time Machine Clockwork by Pierre J.

Temporal Paradox

In the TV science fiction series Star Trek, which involves the idea of time travel, the following hypothetical problem arises:

If one must choose among two or more possible historical outcomes while time traveling, how would one be certain which choice is right?

One might think that the best ethical choice would be right. But even for those of us stuck in the present, ‘right ethics’ are not always easy to determine or assess, and fictional scenarios such as time travel make the issue even more complex.

One hypothetical solution to the problem includes the idea that every time we make a choice, a new universe branches off in a new direction, creating a potentially infinite number of universes for each choice.

» Free will, Prime Directive, Jane Roberts

On the Web:

• http://en.wikipedia.org/wiki/Temporal_paradox

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Zeno

Paradox Cafe

Originally uploaded by jamieca

Zeno (c. 495 BCE) Zeno was a Stoic philosopher best known for his four ‘paradoxes.’

The two most popular paradoxes are:

1) Zeno asks how many grains of millet must fall before a sound is heard. One fallen grain makes no sound on impact, therefore it accounts for ‘nothing.’ A second grain (a second ‘nothing’) added to the first might also make no sound. But suppose a third grain (a third ‘nothing’) is added to the two grains and this does make a sound. This would result in a ‘something’ (audible sound) being made out of three ‘nothings.’

2) The great runner Achilles can never catch a slower tortoise in a race if the tortoise begins ahead of Achilles. By the time Achilles reaches the tortoise’s starting point, the tortoise has moved to a new position. And by the time Achilles reaches the tortoise’s new position, the tortoise has vacated it and moved on to another position. The distances between the two may become increasingly small but the tortoise always remains a fraction ahead of Achilles.

Philosophers still debate the import of the Achilles paradox but its solution might be simple. The problem seem to arise from Zeno’s use of logic divorced from observation.

The student of vectors will observe that a higher-velocity object gaining on and moving in the same direction as a lower-velocity object will at some point overtake the slower moving object.

Not so complicated.

But Zeno imaginatively ‘stops motion’ to observe the competitors in a series of equally imaginative points to say that Achilles will never reach the tortoise’s position. And this sheer act of imagination doesn’t correspond to what actually happens in observable reality.

Among other things, Zeno’s paradoxes illustrate how thinking about problems and their apparent solutions can be influenced, constrained and distorted by our use of a symbol system, such as language, logic or mathematics–especially when divorced from empiricism. » Achilles, Heap of Sand Paradox, Semiotics, Signifier, Signified, Stoicism

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