Free Thinking in Numbers: How Maths Illuminates Our Lives by Daniel Tammet

identity would be known to his boon companions. Nor again was his death the result of a quarrel; they would not have been quarrelling at the dead of night or in a deserted spot. Nor did the criminal strike the dead man when intending to strike someone else; he would not in that case have killed master and slave together. As all grounds for suspecting that the crime was unpremeditated are removed, it is clear from the circumstances of the death themselves that the victim was deliberately murdered.
     
    The plaintiff renders the potential defence precisely as: ‘death either by (1) thieves (2) drunks (3) quarrel (4) accident’ in order to refute each part. But he does something more. Each scenario represents a miniature proof. Rejecting, for instance, the possibility that a thief committed the murder, his argument proceeds:
     
A thief (claims the defendant) killed the victim.
But thieves steal the victim’s cloak.
Thus a thief did not kill the victim.
     
Which self same structure we can find throughout the
Elements of Euclid.
     
CA and CB are each equal to AB.
But things equal to the same thing are also equal to one another.
Thus CA is also equal to CB.
     
    Written in the third century before the Christian era, it is hard to overstate the Elements’ importance to the history of intellectual progress, or the extent to which it embodied the flourishing of rhetoric and logic that permitted its creation. Proposition 21, in Book IX of the treatise (among its oldest pages, dating back to the Pythagoreans) illustrates the lawyerly manner that its author would adopt throughout.
     
If as many even numbers as we please be added together, the whole is even . . . For, since each of the numbers . . . is even, it has a half part; so that the whole [number] also has a half part. But an even number is that which is divisible into two equal parts; therefore [the whole number] is even.
     
    We might summarise this argument as:
     
Proposition: Even numbers (of any quantity) added together make an even number.
Clarification: Since even numbers have half parts, their sum will also have a half part.
Axiom: An even number is that which is divisible into two equal parts.
Conclusion: Therefore the sum of any quantity of even numbers is even.
     
Which echoes any number of the possible arguments
that would have passed through the Ancient Greek
courts.
     
Proposition: The accused stole my ox.
Clarification: Since he said nothing to me about the ox before taking it, the ox was taken without my permission.
Axiom: An item of property removed without the owner’s consent is stealing.
Conclusion: Therefore my ox was stolen.
     
    With their axioms – statements that we accept as being self-evidently true – the Greek plaintiff was able to methodically construct his case, the Greek mathematician, his theorem. Nowhere else on Earth had men thought to agree on what constituted the essence of such and such a thing. Uniquely, the Greeks tore themselves away from the word of rulers, gods or tradition, in favour of logical reasoning. What is wrongdoing? What is murder? What is theft? The Ancient Greeks were the first to ask themselves these kinds of questions. They were the first to distinguish a ‘criminal act’ from ‘misfortune’ or ‘error of judgment’. Definitions – concise, basic and unambiguous – entered the Athenian imagination, as Aristotle tells us.
     
It often happens that a man will admit an act, but will not admit the accuser’s label for the act . . . He will admit that he took a thing but not that he ‘stole’ it; that he struck someone first, but not that he committed ‘outrage’; that he had intercourse with a woman, but not that he committed ‘adultery’; that he is guilty of ‘theft’ but not of ‘sacrilege’, the object stolen not being consecrated; that he has encroached, but not that he has ‘encroached on State lands’; that he has been in communication with the enemy, but not that he has been guilty of