The introduction of symbols for negative numbers must have been a further source of difficulties, however; negative numbers seem somehow to be numbers that are not there, unreal ghosts of numbers – so is it legitimate to call them numbers? In modern times the introduction of symbols for imaginary numbers excited similar doubts. Even if we admit the legitimacy of talk about negative numbers, is it correct to speak of the square root of minus one as if it were a number? Wouldn't it be more honest just to say that minus one has no square root?
Philosophical puzzlement about the various kinds of numbers was much reduced4 thanks to the work of nineteenth century mathematicians who developed a unified theory of numbers. Their very important achievement consisted in showing how the mathematical theories concerning more sophisticated kinds of numbers can be “reduced to”, “constructed from”, a theory concerning only the basic kind of numbers. That is, they showed how each of the more sophisticated kinds of number, together with the operations (such as addition and multiplication) performable on numbers of that kind, can be defined in terms of the whole numbers and the operations performable upon them. They showed that this can be done in such a way that the laws which govern these more sophisticated kinds of numbers can then be deduced from the laws that govern the numbers.
This development is called the arithmetization analysis, because it is concerned with showing how those parts of mathematics that go under the heading of analysis, can be reduced to the elementary part of arithmetic (or elementary number theory, as it is called), when that is supplemented by certain notions that we shall mention.
This unified theory of numbers enables us to regard the various kinds of numbers as belonging to a single family, all springing from a single parent kind and all governed by laws that are strict deductive consequences of the laws governing that simple parent kind. If we accept this unified theory of numbers, we no longer need feel any special doubts about the more sophisticated kinds of number; any doubts that remain will be focused solely upon the numbers of the kind used in counting.
The numbers 0, 1, 2, 3, etc., will serve as our basic kind of numbers; they are called natural numbers (unfortunately that term has a slight ambiguity, for some writers include zero among the natural numbers while others do not but let us count it in). Now, our intuitive idea of the natural numbers is that they are all those numbers, each of which can be reached by starting from zero and adding one as often as necessary.
The Italian mathematician Peano was the first to organize the fundamental laws of these numbers in axiomatic form; his set of five axioms is notable. Let us consider these axioms so that we can feel more at home with the natural numbers before we go on to see how other kinds of number can be reduced to them. Expressed in words, Peano's axioms are:
1) Zero is a natural number.
2) The immediate successor5 of any natural number is a natural number.
3) Distinct natural numbers never have the same immediate successor.
4) Zero is not the immediate successor of any natural number.
5) If something holds true of zero, and if, whenever it holds true of a natural number, it also holds true of the immediate successor of that natural number, then it holds true of all natural numbers.
These axioms contain three undefined terms: “zero”, “ immediate successor”, and “natural number”. The axioms by themselves do not show us what these terms are supposed to mean (though they do connect together whatever meanings these terms may have), nor do they give us any evidence that the terms do refer to anything real.
If we wish to accept the axioms as true we must supply that understanding and that evidence for ourselves. Underlying the use of these terms in the axioms are the tacit assumptions that “zero” does refer to some one definite entity among those under discussion, and that for each entity among those under discussion there is just one entity among them that is its immediate successor.
It follows from the axioms that the immediate successor of zero, its immediate successor, and so on and on, all are natural numbers; and (by the fifth axiom) that nothing else is a natural number. From the axioms it follows that there must be infinitely many natural numbers, since the series cannot stop, nor can it circle back to its starting point (because zero is not the immediate successor of any natural number).
The fifth axiom is especially important, for it expresses the assumption which underlies mathematical induction. We can picture how reasoning by mathematical induction works if we imagine a series of dominoes standing in a row. Suppose we know that the first domino will fall and that whenever any domino falls the adjoining one also will fall; then we are entitled to infer that all the dominoes will fall; no matter how many there may be.
In the same spirit, if we know that something holds true of zero and that whenever it holds true of a natural number it also holds true of the immediate successor of that natural number, then we can infer that it holds true of every natural number. On the basis of Peano's axioms, we can introduce the names of further numbers: “one” by definition names the immediate successor of zero, “two” by definition names the immediate successor of one, and so on.
Peano’s axioms express in a very clear way the essential principles about the natural numbers. However, they do not by themselves constitute a sufficient basis to permit the reductions of other higher kinds of numbers – assuming, that is, that we continue to restrict ourselves to the same comparatively low-level logical principles that are employed for deducing theorems in geometry. There are two reasons for this.
For one thing, Peano’s axioms, do not by themselves provide us even with a complete theory of the natural numbers. If we limit ourselves just to Peano's three primitive terms and to his five axioms, it is impossible for us (using only normal low-level logical principles) to define addition and multiplication in their general sense for these numbers.
So we could not even express within the system, let alone prove within it, such laws as that the sum of natural numbers x and y always is the same number as the sum of y and x, or that x times the sum of y and z always is the same number as the sum of x times y and x times z. We do not even worry about subtraction and division, since these are not operations freely performable on the natural numbers.
Furthermore, in order to carry out this reduction of higher kinds of number we need to employ two other very important terms, “set” and “ordered pair”, which Peano of course did not include among his primitives.
Notes
1. while geometry was being handed down – в то время как геометрия дошла
2. the mathematics of number was passed along – математика числа пришла к нам в виде
3. this uneasiness was gradually soothed – это неудобство постепенно сгладилось
4. philosophical puzzlement... was much reduced – философские сомнения... были в основном разрешены
5. the immediate successor – непосредственный последующий элемент
THE FACULTY OF BIOLOGY
What Is a Mutation?
The body is like a Chinese puzzle1 box. It consists of organs, such as liver, legs, eyes. The organs consist of tissues, such as bone, muscle, nerve. The tissues consist of cells. The cell contains a nucleus. The nucleus contains chromosomes. The chromosomes carry the genes. Mutations are changes in chromosomes and genes.
The cell and the nucleus can be seen under the microscope, but the chromosomes cannot always be seen. They become visible only at certain stages in the life of the cell, namely2, when the cell divides to give two daughter cells. They then appear as rod-like or dot-like structures which, in thin tissue slices (слой, срез), can be stained with certain dyes which they take up more readily than the rest of the cell. The genes are too small to be seen even with a high-power microscope. The genes are arranged linearly along the chromosomes. Some particularly big chromosomes show a visible subdivision into smaller units, so that they look like strings of beads, or like ribbons with a pattern of cross-bands? These beads and bands are much too big to be the genes themselves, but they indicate the position of the genes on the chromosomes.
The number of chromosomes in the nucleus is characteristic for each species. Man has 46, the mouse (мышь) 40, the broad bean plant (боб) 12, maize (кукуруза) 20. Each chromosome carries hundreds of thousands of genes. It has been estimated4 that the chromosomes in a human cell carry at least 40,000 genes, possibly twice as many. This seems a large number, but it is not so large when we consider that the genes between them are responsible for all that is inborn and inherited in us. Genes determine whether we belong to blood-group A or О, whether we are born with normal vision or not, whether we have brown, blue or hazel eyes, whether on a rich diet we grow fat5 or remain slim (стройный), whether musical education makes virtuosi of us or we are unable to distinguish one tune (мелодия) from another, and so on through the thousands of details which together make up our physical and mental personalities.
Every time, before a cell divides, each chromosome makes another chromosome just like itself with the same genes in the same order. Then, when two cells arise from one, the old chromosomes separate from their new-formed duplicates and both "daughter cells" receive exactly the same numbers and types of chromosomes and genes.
The human body develops from a single cell, the fertilized egg, which contains 46 chromosomes. The egg divides to form two cells; these divide again to form four cells, and so it goes on until the whole body with its billions of cells has been formed. Before every cell division, chromosomes and genes are duplicated. Every cell therefore contains the same 46 chromosomes carrying the same genes.
The process by which chromosomes and genes are duplicated is remarkably accurate. It results in millions and billions of cells with exactly the same genes. But sometimes, perhaps once in a million times, something goes wrong6. A gene undergoes a chemical change, or the new gene is not exactly like the old one, or the order of the genes in the chromosome has been changed. This process of change in a gene or chromosome is called a mutation. Its result, the altered gene or chromosome, is also often called mutation, but to avoid confusion7 it is better to speak of a mutated gene and a re-arranged chromosome, and reserve the term mutation for the process which produced them. The individual, which shows the effect of a mutated gene or re-arranged chromosome, is called a mutant.
When a chromosome on which a mutation has occurred makes a duplicate of itself in preparation for the next cell division, it copies the mutated gene or the new gene arrangement as accurately as it copies the unaltered portions. In this way a mutation is inherited and becomes perpetuated9 exactly like the original gene from which it arose. The enormous variety of genes which are found in every living species results from mutations, many of which may have occurred millions of years ago.
Notes
1. Chinese puzzle – неразрешимая загадка
2. namely – а именно
3. ribbons with a pattern of cross-bands – ленты с поперечными полосами
4. estimate – подсчитывать
5. grow fat – толстеть, полнеть
6. go wrong – разладиться, испортиться
7. to avoid confusion – чтобы избежать путаницы
8. perpetuate – сохранять навсегда, увековечивать
Evolution and Heredity
More than a hundred years ago people believed that plants and animals have always been as they are now. They thought that all the different sorts of living things, including men, had been put here by some mysterious (таинственный) power.
It was Charles Darwin, born at Shrewsbury in February, 1809, who showed that this was just a legend. As a boy Darwin loved to walk about the countryside collecting insects, flowers and minerals. He enjoyed helping his elder brother at chemical experiments in a shed (сарай) at the far end of their garden.
These hobbies interested him much more than Greek and Latin, which were his main lessons at school. His father, Dr. Robert Darwin, sent Charles to Edinburgh University to study medicine. But Charles disliked the medical career. He spent a lot of time with a zoologist friend watching birds and other animals in their natural state and collecting insects in the surrounding countryside.
Then his father sent him to Cambridge to become a clergyman1. But Darwin did not care for lectures. He did not want to be a clergyman. At 22 he graduated from Cambridge University and soon was offered an unpaid post as naturalist on the ship "The Beagle".
The young naturalist asked himself whether all forms of life always existed just as they are now. This was what everyone believed and what he had been taught, but he doubted it very much. Three and a half years travelling around the world on the British ship "The Beagle" convinced Darwin that his doubts were justified. He returned from his travels convinced that man and all the living creatures on earth today are related. All have grown from earlier types, and those from earlier ones in an unbroken line back to a primitive one-cell creature.
More than a thousand million years ago, a small blob of jelly1 floated on the shallow seas of the young earth. It and others like it were the only life on earth. In half a milliard years that blob of jelly had become different kinds of sea worms (червь) and sea scorpions, sea weeds (морские водоросли) and other simple sea plants.
During the next half milliard years some of this life crawled (ползти) onto the barren (бесплодный) land. The first land animals were "amphibians", equally at home on land and in the water, like present-day frogs. There were also primitive scorpions, the descendants of which became insects or spiders (паук). From the seaweeds that took root on shore came ferns (папоротник) and mosses (мох). The amphibian became reptiles. For one hundred million years they ruled the earth. Out of them came birds and mammals. Gradually the mammals changed into all the different kinds we have today, including man. Each of these changes was very gradual and took thousands of years.
What makes you and your brothers and sisters look somewhat alike? What makes all of you look like your father and mother, and yet also a little different? The answer is to be found in the laws of heredity.
Gregor Mendel, son of an Austrian farmer, wanted to be a scientist but couldn't afford the university. He became an Augustinian monk3 and, in the years between 1843 and 1865, he became a great scientist. In the garden of the monastery he raised garden peas – pure tails, pure dwarf (карликовый) and so on. Then, when he was sure he had pure strains, he began crossing them. He did the same with green and yellow peas. In all he raised and studied more than 10,000 specimens.
From the way these peas transmitted and inherited various traits such as height or colour, Mendel worked out the laws of heredity. They have been found to be true for all types of plants and animals, including man, and have been widely used in the improvement of flowers and agricultural crops and the breeding of dogs and livestock.
Notes
1. clergyman – священник
2. blob of jelly – студенистая капля (комочек)
3. Augustinian monk – монах-августинец
Animal Behaviour
Wherever people have a chance to watch animals – at a zoo, park, pet store, or circus – it is evident that animal behaviour is a source of fascination1 for most humans. As they watch animals at play and at rest, feeding or protecting themselves, and tending to their young, frequently marvel (восхищаться) at the similarities between animal and human behaviour. These similarities are, in fact, one important reason for studying the activities of animals: that is, their implications for the better understanding of human behaviour.