The ancient Greeks did not really address the problem of negative numbers, because their mathematics was founded on geometrical ideas. Lengths, areas, and volumes resulting from geometrical constructions necessarily all had to be positive. Their proofs consisted of logical arguments based on the idea of magnitude. Magnitudes were represented by a line or an area, and not by a number like 4.
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In this way they could deal with 'awkward' numbers like square roots by representing them as a line. For example, you can draw the diagonal of a square without having to measure it see note 2 below. Negative numbers did not begin to appear in Europe until the 15th century when scholars began to study and translate the ancient texts that had been recovered from Islamic and Byzantine sources. This began a process of building on ideas that had gone before, and the major spur to the development in mathematics was the problem of solving quadratic and cubic equations.
As we have seen, practical applications of mathematics often motivate new ideas and the negative number concept was kept alive as a useful device by the Franciscan friar Luca Pacioli - in his Summa published in , where he is credited with inventing double entry book-keeping.
In the 17th and 18th century, while they might not have been comfortable with their 'meaning' many mathematicians were routinely working with negative and imaginary numbers in the theory of equations and in the development of the calculus. Negative numbers and imaginaries are now built into the mathematical models of the physical world of science, engineering and the commercial world. There are many applications of negative numbers today in banking, commodity markets, electrical engineering, and anywhere we use a frame of reference as in coordinate geometry, or relativity theory.
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All rights reserved. Although the first set of rules for dealing with negative numbers was stated in the 7th century by the Indian mathematician Brahmagupta, it is surprising that in the British mathematician Francis Maseres was claiming that negative numbers " In BCE the Chinese number rod system see note1 below represented positive numbers in Red and Negative numbers in black.
An article describing this system can be found here.
These were used for commercial and tax calculations where the black cancelled out the red. How small, before it is infinitely small? Physics and astronomy are full of numbers that are unbelievably enormous. Think of the dimensions of the universe. But infinity, as Wallace is quick to point out, is not a number.
Infinity | perroaplesca.ml
It's a concept. There are no numbers bigger than infinity, but that does not mean that infinity is the biggest number. And some infinities are definitely bigger than other infinities. It all depends on what you are talking about. Mathematicians have struggled with the notion of infinity for more than two-and-a-half millennia. The history of infinity is the story of a battle to tame the infinite.
The idea was to show that our reasoning about infinity could be conducted like our reasoning about ordinary mathematical objects. Many inspired attempts have produced important mathematical advances. But infinity has always evaded control. As you would expect from someone known for his intellectually ambitious fiction, Wallace tells the story well. It all began with Zeno of Elea, who propounded paradoxes.
Their intent was to show that our reason is not self-contained or complete. When we scrutinise our arguments about change, or about classification, we find contradictions. The famous paradox about Achilles and the tortoise imagines a race in which the tortoise gets a head start. In one leap, Achilles halves the distance between them. In the next leap, he halves it again.
But how do we describe his catching up? We cannot specify that last leap; after each leap, there is still the halved distance remaining. Trying to solve the paradox only confuses us further. There's a lot of history after that. But the last engagement in the long battle was set off by the German mathematician Georg Cantor. His work produced some quite astounding results about infinite collections. Cantor came up with infinite sets that were countable, and infinite sets that were uncountable.
Foundations of mathematics
His work proved the opposite of his intention. He found that the set of all "real numbers", the points on a line, cannot be counted. After Cantor, philosophers and mathematicians struggled heroically to rescue the "foundations" of mathematics. But their own tools were turned against them. We now accept that mathematical knowledge is quite arbitrary. While Everything and More is good on narrative, it is abysmal on analysis.
Wallace is too clever by half, and too obsessed with demonstrating his own mathematical dexterity. He shows that the attempts to prove Zeno wrong have enormously enriched our understanding of our concepts and, perhaps, ourselves too. But his rather pedestrian treatment of infinity fails to draw the obvious conclusions. There are at least two important lessons to be learnt from the history of infinity. First: knowledge is not logical, but its genuineness comes from its roots in practice. Mathematics works because it is rooted in experience. Second: any abstract system of ideas will generate its own contradictions.
Or, to put it another way, Zeno was right. You can find our Community Guidelines in full here. Want to discuss real-world problems, be involved in the most engaging discussions and hear from the journalists? Try Independent Premium free for 1 month.
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