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      <text>3.4.2 (History of proof)</text>
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      <text><![CDATA[<h3>History of mathematical proof</h3>
<p>The story of mathematical proof is the story of mathematics itself. As such, the history of proof is rather convoluted and incoherent. The idea of using plausibility arguments or heuristic devices such as analogies or pictures proceeded the formal idea of proof that we recognise today. The different types of proof that we have met all originated at different times, largely out of necessity. It is important to note that we still have concepts such as diagrammatical proof today, see the below images for examples.</p>
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<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Pythagoras%20diagram.png" alt="diagram proving pythagoras" width="273" height="260" /></p>
<p style="text-align: center;">This is a diagram showing Pythagoras' Theorem</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Proof%20of%20sum%20of%20odd%20numbers.png" alt="sum of odd numbers diagram" width="272" height="258" /></p>
<p style="text-align: center;">This diagram shows that the sum of the first \(2n-1\) positive integers is a perfect square of \(n^2\)</p>
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<p>The idea of a logical proof to demonstrate a conclusion began in the field of geometry, which was seen as the most important type of mathematics to ancient civilisations as it is immediately practical (for e.g. measuring land). The idea of proof was really pioneered by the ancient Greeks (although there is some evidence of proof appearing with the Babylonians or the ancient Chinese). At this point the mathematics of the Greeks became intertwined with the philosophy of the age, as questions such as 'what is a definition?' became important to philosophers and thinkers. It was around this time when Pythagoras (569–500 BCE.) proved his famous theorem. Pythagoras was somewhat of a mystic and he believed that numbers were a divine gift from the gods and that numbers ruled the universe. This is why he didn't believe that \(\sqrt{2}\) could be a real number. After Pythagoras, Eudoxus (408 BCE–355 BCE.) was the first mathematician to begin to formulate mathematics in terms of theorems. However the notion of an axiomatic proof would have to wait for another 100 years.</p>
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<p>The idea of proof was revolutionised by Euclid in his book 'The Elements'. Incidentally, this book is the second most widely spread western book of all time. Euclid's genius was to implement the axiomatic method of proof that we use today. He was the first to formally create definitions and axioms and then prove theorems using these definitions. Euclid's work was also limited by the Greek focus on geometry however it continues to have a considerable influence, not only in modern mathematics, but modern science as well.</p>
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<p>The application of axiomatic methods to areas of maths other than geometry came with the medieval Islamic empire, who introduced algebra and and arithmetic. As Europe left the dark ages, the next major advances came around the nineteenth century as mathematics continued to develop new disciplines . Modern mathematical proof has really been directed by the work of influential German mathematician, David Hilbert (1862-1943 CE) who in 1900 presented a list of 23 of the most important unsolved problems in mathematics. This had the effect of driving many mathematical developments as mathematicians sought to solve these problems. To this day, many remain unsolved and some come with substantial monetary prizes for those who can prove them such as the Riemann Hypothesis.</p>]]></text>
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