Auguste De Morgan

Childhood

Born from a father colonel in the army with the service of the company of the Eastern Indies, his/her mother was downward of James Dodson, which draws up a table of antilogarithms, i.e. the corresponding numbers to extract the Logarithme S. Because of the revolts, the colonel sent her family in England when Auguste had seven months. Consequently he regarded himself as a British to the full extent.

De Morgan loses his/her father the year of his ten years. With his/her mother who wanted to make of her son a monk, they will live in various cities in the south-west of England what made him often change school. Its mathematical talents were discovered when it was fourteen years old. A family friend found it making an elaborate diagram of Euclide with a rule and compasses and explained its logic to him.

Auguste De Morgan had a problem in an eye what made more difficult the practice of the sport to him and made it prone to mockeries. He claimed that he perceived nevertheless the distance and solidity.

University education

In 1823, it enters to Trinity College (Cambridge) to Cambridge, where its tutors, George Peacock and William Whewell become her friends. He will become an excellent player of flute, also. In the mathematical contest interns, it reaches the statute of Wrangler, which enables him to study for a vat of Article But for that, it is necessary for him to pass a test of theology. What he will refuse, although knowing the matter. Towards 1875 this requirement will be abolished.

London university

As no career is opened to him, it decides to make studies of right to London, where a university is in creation, in particular accommodating those which were not religious or refractory majority. It will become mathematics professor there.

At the time of an argument between the professor of anatomy and the hierarchy, De Morgan took party for his/her colleague. He was congédié but its successor drowned a few years later. It was invited to return and kept this station during thirty years.

It took part and was one of most active with the Société for the dissemination of the useful information. In 1837, he marries Sophie Elisabeth, one of the girls of his friend Frend.

The London university where De Morgan was a professor was an institution different from the Université of London, which was founded approximately ten years later by the government. That obliged the first to change its name into university college, London.

De Morgan had much success teaching. It made a one hour course then gave certain a number of problems and illustrated examples. The students were to give the results which it produced corrected to the following meeting.

Auguste had son, George, which followed it in the career which instituted a mathematical company, where the new articles was not only listened, but discussed.

Reprocess

In 1866, the pulpit of mental philosophy of the university college becomes available. Doctor Martineau, a unitarian monk and professor of this matter, was recommended by the senate of the council. Because of the opposition of a monk of the school of Bath, Spencer obtained the station. De Morgan considered that the principle of religious neutrality was violated and resigned, at the sixty years age. Its pupils ensured a pension of 500 dollars to him, but misfortunes followed. Two years later, his/her George son, " the Bernoulli" young person; as he liked to call it, dies, then comes the turn from one of his/her daughters. He will die of nervous prostration.

Mathematics

Auguste was a writer shining and pleasant, including in his correspondence, in particular with William Hamilton.

He did not like the countryside and while its family benefitted from the edge of the sea, the scientists had good time in clubs and remained to him in dusty bookstores of the metropolis. He never voted with an election or nor did not visit the monuments.

The best presentation of its design of the Algèbre is in trigonometry and algebra doubles published in 1849. The following stage would have being the “triple” algebra and if $a+b \ sqrt {- 1}$ represents really a line in a given plan it should be possible to find a third term which added to the precedent would represent a line in space. Argand and some others guessed that it was $a + B \ sqrt {- 1} + C \ sqrt {- 1} \, ^ {\ sqrt {- 1}}$ although that contradicted the truth established by Euler that $\ sqrt {- 1} \, ^ {\ sqrt {- 1}} =e^ {- \ frac {1} {2} \ pi}$ . De Morgan and well of others worked hard with the problem but nothing left there until the problem is taken by Hamilton. We see the reason now clearly: the symbol of the double algebra indicates not a length and a direction but a multiple and an angle . In that Ci the angles are confined in a plan; thus the following stage should be a algebra quadruples when the axis of the plan becomes variable. And that gives the answer to the first question; the double algebra is not only of analytical trigonometry of plan, and this is why it was found to be the natural analysis for alternate currents. But De Morgan never went also far.

When the study of mathematics was reactivated at the university of Cambridge it was also the case of logic. Dynamics came from Whewell. In formal logic of Auguste is interesting on the development of the syllogism. The aristotelicians say that starting from two particular proposals as a few M are has and a few M are B nothing is not obtained from necessary in the relation of has and B. But if they go further and say that any relation in connection with has and of B must follow by need the joint term must be taken universally in one of the premises. De Morgan announced that starting from the majority of the M are has and the majority of the M are B one deduces that some has are B and he formulated the numerical syllogism which puts this principle in a precise quantitative form. Suppose that the number of M is $m$ , of the M which are has is $a$ , and of the M which are B is $b$ ; then there is at least $(a+b-m)$ has which is B. For example with 1000 people on a boat of which 500 are in the living room and 700 die it is obligatory that at least 700+500-1000, therefore 200 of those in the living room were victims.

Here then De Morgan made an advance by introducing the quantification of the terms. At this time Hamilton taught in Edinburgh doctrines of the quantification of the predicate and a correspondence was done. However De Morgan quickly realized that the quantification of Hamilton was of another nature; that he wanted to say for example, in substituent the two forms all has is very B and all has is part of B for the form aristotelician All has is B . The philosophers generally have a great share of intolerance thinking that they have the totality of the truth. Hamilton thought that it had placed the keystone in the construction industry aristotelician although it would have been strange that she existed during two millenia without this crucial factor. As consequence it did not have any place for the innovations of Morgan which it showed to be a plagiarist and the argument continued between them during years.

Budget of paradoxes

In connection with the concept of paradox he explains why it is by comparison with established knowledge (accepted by the authorities). The budget is a kind of anthology.

The law of duality

De Morgan is recognized especially for its redécouverte law of duality between the sum and the product, where “the opposite of an aggregate (logical sum) is the compound (produced logical) of the opposites of incorporating; the opposite of a compound is the aggregate of the opposites of the components”. Here thus the expression of this law of duality:

~ (X + there) = ~x X ~y
~ (X X there) = ~x + ~y

or:

¬ (X ∧ there) = ¬x ∨ ¬y
¬ (X ∨ there) = ¬x ∧ ¬y

This law is also applicable to the calculation of the proposals under the terms of isomorphism with the calculation of the classes. There will be thus a principle of duality between the conjunction and disjunction, being expressed as follows:

~ (statement Q) º ~p. ~q
~ (p. Q) º ~ statement ~q

Sources

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