Mathematicians and machines
Many people think that the construction of mathematical machines? and necessarily computers? only the engineers contributed. This is not true, mathematicians have contributed to this work from the very beginning. And these are those who basically have only theory. Indeed, did some of them have the slightest idea that their discoveries would someday be used in the same mundane business as the creation of accounts?
Today I will tell you about two mathematicians from earlier times. Another one (that is, John von Neumann), without whose work and ideas computers would not have been created at all, I leave for later; it is too big and too important to be combined with others in one story. I also connect these two because they were close friends, although there was a certain age difference between them.
Alternative and union
But these two are also no less worthy than Neumann. However, before we move on to their biography, I offer a simple task. Consider any sentence consisting of two subordinate clauses connected by a union (such a sentence, who does not remember, is called alternative). Let's say:. The challenge is to refute this proposal. So what does this mean:
Well, the rule is this: we will replace the union with and contradict compound sentences, therefore:.
Not difficult. Well, let's try to object to a sentence consisting of two sentences connected by a union (again, who does not remember the term: Conjunction). For example: A similar rule, i.e. replacement by compound sentences? i deny so we get:, means exactly the same as
Usually: (1) the negation of an alternative is a conjunction of negations, and (2) the negation of a conjunction is a conjunction of negations. These ? extremely important? two de Morgan's laws for propositional calculus.
Fragile aristocrat
August de Morgan, the first of the mathematicians mentioned at the beginning, the author of these laws, was born in India in 1806 in the family of an officer in the British colonial army. In 1823-27 he studied at Cambridge? and immediately after his graduation he became a professor at this wonderful university. He was a weak young man, shy and not very rich, but extremely capable intellectually. Suffice it to say that he wrote and published 30 books on mathematics and more than 700 scientific articles; it's an impressive legacy. Were there many of his students at that time? how would we say today? celebrities and prominent figures. Including the daughter of the great Romantic poet Lord Byron? known Ada Lovelace (1815-1852), considered today the first programmer in history (she wrote programs for Charles Babbage's machines, which I will discuss in more detail). By the way, is the popular programming language ADA named after her?
The work of de Morgan (he died comparatively young in 1871) marked the beginning of the consolidation of the logical foundations of mathematics. On the other hand, his rules mentioned above found a beautiful electrical (and then electronic) implementation in the design of logic gates that underlie the operation of each processor.
By the way. If we negate the sentence: we get the sentence: In the same way, if we negate the sentence:, we get the sentence: These are also De Morgan's laws, but for the quantifier calculus. Interesting ? is there anywhere to show it? is this a simple generalization of de Morgan's laws for propositional calculus?
Hellishly gifted shoemaker's son
More or less today, another of our heroes lived with de Morgan, that is, George Bull. The Boules were a family of small farmers and traders from the North East of England. The family was nothing special before the arrival of John Bull? Who? although he was just an ordinary shoemaker? fell in love with mathematics, astronomy and? music to the point that like a shoemaker? went bankrupt. Well, in 1815, John had a son, George (that is, George).
After the bankruptcy of his father, little George had to be taken away from school. Maths? how was it successful? his father himself taught him; but this was not the first subject that little Yurek learned at home. First there was Latin, then languages: Greek, French, German and Italian. But the most successful was the boy's teaching of mathematics: at the age of 19, the boy published? in the Cambridge Journal of Mathematics? ? my first serious work in this area. Then the next ones came.
A year later, George, having no formal education, opened his own school. And in 1842 he met de Morgan and became friends with him.
De Morgan had some problems at the time. His ideas were ridiculed and sharply criticized by professional philosophers who could not imagine that a mathematician began to say something in a discipline hitherto considered a branch of pure philosophy, i.e. in logic (by the way, most modern scientists today consider that logic is just one of the branches of pure mathematics, which has almost nothing to do with philosophy, of course, it revolts philosophers almost the same as in the time of de Morgan?). Buhl, of course, supported a friend? and in 1847 he wrote a little work entitled. This essay is groundbreaking.
De Morgan appreciated this work. A few months after its release, he learned of a vacant professorship at the newly established King's College, University of Cork in Ireland. Buhl competed for the position but was eliminated and the contest was not allowed. After some time, a friend helped him with his support? and Boole, however, received a chair of mathematics at this university; having absolutely no formal education in mathematics or any other field?
A few years later, a similar story happened to our brilliant compatriot Stefan Banach. In turn, his studies before joining a professorship in Lviv were limited to undergraduate and one semester of a polytechnic?
But back to booleans. Expanding on his ideas from the first monograph, he published in 1854 his famous and today classic work? (the title, in keeping with the fashion of the time, was much longer). In this work, Boolev showed that the practice of logical reasoning can actually be reduced to the rather simple? albeit using a bit of weird arithmetic (binary!)? Accounts. Two hundred years before him, the great Leibniz had a similar idea, but this titan of thought did not have time to complete the matter.
But who thinks that the world has fallen on its knees before Boole's work and marveled at the depth of his intellect? not right. Although Boole had already been a member of the Royal Academy since 1857 and a widely respected and famous mathematician, his logical ideas were long considered curiosities of little importance. In fact, it was not until 1910 that the great British scientists Bertrand Russell i Alfred North Whitehead, by publishing the first volume of their brilliant work (), they showed that Boolean ideas - and not only have an essential relation to logic? but even are logics. Beyond the ideas of George Boole, is classical logic simple? with a bit of exaggeration? does not exist at all. Aristotle, the classic of logic, became only a curiosity of history on the day of publication.
By the way, one more interesting piece of information: about half a century later, all fat theorems have been carefully proven by Boolean calculus for many years? in eight minutes it turned out to be a less powerful computer, expertly programmed by the Chinese American genius Wang Hao.
By the way, Boole was a little lucky: if he had overthrown Aristotle from the throne three centuries earlier, he would have been burned at the stake.
And then it turned out that the so-called Boolean algebras? this is not only an extremely important and rich area of mathematics, which is still developing today, but also the logical basis for the construction of mathematical machines. Moreover, Boolean theorems, without any changes, apply not only to logic, where they describe the classical propositional calculus, but also to binary calculus (in a number system that uses only two digits - zeros and one, which is the basis of computer arithmetic), but they are also used in set theory developed much later. It turns out that in this theory the family of subsets of any set can be treated as a Boolean algebra.
boolean value? how is de morgan? he was in poor health. Let's also be honest that he didn't care about this health at all: he worked too hard and too hard, and he was extremely industrious. October 24, 1864, when was he going to lecture? He was terribly wet. Not wanting to delay classes, he did not change or undress. The result was a bad cold, pneumonia, and death a few months later. He died at the age of only 49.
Boole was married to Mary Everest, the daughter of a famous British explorer and geographer (yes, yes? the one from the highest mountain in the world) 17 years his junior. Romance? ended in an extremely successful marriage? started with? tutoring in acoustics given by a scientist to a beautiful young girl. He had five daughters with her, three of whom earned the title of outstanding: Alice became a great mathematician, Lucy was the first professor of chemistry in England, Ethel Lillian was recognized in her time as a writer.
