Gaza Proportionality – A mathematical argument

By Paulo dos Santos Ferreira *

Whenever Israel undertakes a more incisive and forceful action to stop terrorist attacks, we hear the angry cries of condemnation for those who only seek to defend themselves and the cries of pity for those who attack, the poor Palestinian massacred.

Among the main charges and condemnations there is a topic that is always touched upon, the lack of proportionality between the Israeli reaction and the action which caused it. It’s a cruel joke.

The mathematical concept of proportionality is expressed by the equality between two ratios: a/b = c/d

Even in everyday language we do not speak of proportionality until after two ratios are compared. When we say, for example, that the weight of a person is (or is not) proportional to its height, it means that the quotient of dividing weight by height is (or is not) equal to that which is regarded as the standard for the people of his or her age. Without this standard it would be impossible to talk about proportionality.

It is obvious that such a standard does not exist when it pertains to the means to put an end to the deadly and systematic attacks. Humanity has not yet come to establish such a standard, despite its long history of conflict and war, international organizations are powerless to stop terrorist attacks. It is necessary to stop their agents, at all costs. The opposite would be to continue to kill until total annihilation. Nobody can demand such a sacrifice of an entire people in the name of preserving an imaginary indefinable proportionality.

But let’s see what the Israeli reaction would have to be, in order for it to be proportional to the terrorist attack, as the world demands.

Consider the relationship between the measurements of the intensity of this reaction, we will call it “y”, and the intensity of the terrorist attack, we represent here by “x”. Without a standard to be used as reference to make the comparison, we will compare the ratio to that between the same “x” and “z” defined as the value of the intensity of the Israeli action which, by definition, would have given rise to the terrorist attack. Proportionality would be expressed by the equality between these two ratios:

But as the assault was arbitrary and does not correspond to any action that could be the cause, we have z = 0, from whence we then conclude that .

Or in common language: If we want to maintain proportionality, the intensity of the reaction to an unprovoked assault has no limitations.

Long live “Proportionality”!

* Paulo Ferreira dos Santos, Brazilian mathematician.


1. Text originally published in Portuguese (Brazil) as a letter to the editor, translated into French by the author and translated from French into English by Israel Bonan.

2. Text published here with permission of the author.

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