Chapter 7 Chapter 6 Logical Techniques in Mathematics

I think it is necessary to have departments in universities, but the results are unfortunate.Logic was regarded as a branch of philosophy, and had been dealt with by Aristotle, so it was thought that this subject could only be discussed by those who knew Greek.As a result, mathematics is only discussed by people who do not understand logic.From the time of Aristotle and Euclid to this century, this division has done much damage. When the International Philosophy Conference was held in Paris in 1900, I realized the importance of logical reform for mathematical philosophy. I realized this by listening to the discussions of Peano from Turin and some other philosophers present at the meeting.Before that, I didn't know what he had done.But I deeply feel that in every discussion, he is more precise and logical than others.I went to see him and said to him, "I want to read all your books, do you have them with you?" He did.I read all his works at once.It was these writings that prompted me to make my own claims about mathematical principles.

Mathematical logic is not a new discipline.Leibniz tried it, but was held back by his respect for Aristotle.Boole published his "Laws of Thought" in 1854, and came up with a whole set of calculation methods, mainly talking about the inclusion of classes.Pierce once created a relational logic.Schroeder has published a book, divided into three volumes, summarizing previous results.Whitehead devoted the first part of his Universal Algebra to Boole's method of calculation.Most of the above-mentioned works were familiar to me at that time.But I do not find these works to be of any help in understanding the basic principles of arithmetic.I still have the original manuscript of my essay on this subject which I wrote just before I went to Paris, and I have now read it again, and I find that it does not contain even the preliminary None of the solutions were done.

Peano inspired me mainly from two purely technical advances.The importance of these two advances is not easily appreciated by a man who has not spent years trying to understand arithmetic as I have done.Both advances were made by Frege in an earlier period.I suspect Peano didn't necessarily know this, and I didn't know until later.With difficulty, I will try my best to explain what these two advances are and why they are important.Let me first talk about what these two advances are. The first advance consists in separating propositions of the form "Socrates is mortal" from propositions of the form "all Greeks are mortal".Aristotle and the accepted doctrine of the syllogism (which Kant thought could never be improved) held that there was no difference between the two forms of propositions, or else there was no great difference at all. .But, in fact, neither logic nor arithmetic can go far without seeing that the two forms are quite different. "Socrates is mortal" adds a predicate to a subject that is a name. "All Greeks are not immune to death" indicates the relationship between two predicates, that is, "Greeks" and "not immune to death", to say "all Greeks are inevitable" is, "For all possible values ​​of x, if x is a Greek, x is not exempt from death".Here is not a subject-predicate proposition, but a link between two propositional functions.If a value is assigned to the variable x, each of the two propositional functions becomes a subject-predicate proposition. The proposition "All Greeks are mortal" is not about the Greeks alone, but a proposition about everything in the universe.If x is a Greek, the proposition "if x is a Greek, x is mortal" holds true, and it holds true if x is not a Greek.

In fact, this proposition would hold true even if the Greeks did not exist at all. "All people in Lilliputian countries are inevitable from death" can be established, although people in Lilliputian countries do not exist. The reason why "all Greeks are mortal" is different from the proposition "Socrates is mortal" is that it does not specify any one person, but only expresses the connection between predicates and predicates.Its validity cannot be proved by enumeration, because (again) the x in question is not limited to those of the Greeks, but extends to the whole universe.However, although this proposition cannot be proved by enumeration, it can be understood.I don't know if there are winged horses, I have never seen such a horse, but I know that all winged horses are horses.In a word, every proposition containing the word "everything" is a proposition containing propositional functions, but not containing any particular value of these functions.

The second important advance I heard from Peano was that a class of a term is not equal to that term. For example, "Satellites of the Earth" is a class that has only one item, the Moon.But equating a class with its only items raises completely insoluble problems in the logic of sets, and therefore in the logic of numbers, for numbers apply to sets.Once pointed out, it is easy to see that it is inappropriate to equate the "satellites of the earth" with the moon.The phrase "satellite of the Earth" would not change its meaning if it were discovered that the Earth had a second satellite; nor would the phrase be lacking in meaning to a person who knows astronomy but does not know that the Earth has a satellite.On the other hand, if we can take "the moon" as a name, the proposition about the moon is meaningless except to those who know the moon.For those who don't know the moon, if you don't explain "the moon"

Equivalent to the phrase "the only satellite of the earth," "the moon" is but a meaningless sound; and if this interpretation were replaced, there would be no proposition about the moon when we say: "The moon is bright tonight" when you and It makes sense to me.A person does not need to describe, he is connecting concepts together, not directly contacting the sensory world.A person says: "The moon is bright", but he is in direct contact with the sensory world.On this point, the difference we are discussing now is somewhat similar to the difference we said earlier between "Socrates is not immune to death" and "all Greeks are not immune to death".

The reader may think that the above distinctions are nothing but pedantic posturing and pedantry.I cannot but think now that this is not the case. Authors before Frege got the philosophy of arithmetic wrong.The mistake they made was a natural one.They think numbers come from counting.They are stuck in an unsolvable dilemma because what counts as one also counts as many.Take, for example, the question: "How many football clubs are there in the UK?" When answering this question, you treat each club as one, but you can also ask: "How many members does a certain football club have?" In that way, you treat this club as many.Moreover, if Mr. A is a member of one of these clubs, although he counts as one at first, it is equally justified for you to ask, "How many molecules is Mr. A made of?" Then Mr. A counts as many.So, clearly, what makes something one from a computational point of view is not the physical constitution of the thing, but the question "A concrete instance of what is this?"

The numbers you get from calculations are some kind of collective numbers.Before you count the collective, it has any number. The collective is many only in terms of many instances of something.This collective is an instance of something else, and when counting, it counts as one by instance.We are thus forced to face the question: "What is a collective?" and "What is an instance?" Both cannot be understood without propositional functions.A propositional function is a formula, which contains a variable, once a value is given to the variable, the formula becomes a proposition.For example, "x is a person" is a propositional function.If we substitute Socrates or Plato or anyone else for x, we get a proposition.We can also replace x by something that is not a person, and we still get a proposition, although in this case it cannot be true.A propositional function is just a formula.It doesn't mean anything by itself.It can be part of a sentence which does assert something, which may or may not be true: "x was an apostle" is meaningless.But "x has twelve values, so 'x is an apostle' holds" is a complete sentence.Similar words can be used for the concept of instance.When we treat something as an instance, we are treating it as a possible value of a variable in a propositional function.If I say: "Socrates is an instance of man", I mean that Socrates is a value of x, so "x is a man" is true.The scholastics had a dictum which meant that one and being are synonymous.As long as this adage is believed to be true, there is no way to clarify the meaning of 1.The truth of the matter is that existence is a useless word.Moreover, the kind of thing to which the person who misuses the word applies it can be either one or often many. ?One is not a property of things, but a property of certain propositional functions, that is, those that have the property that there is an x ​​that makes the function true, and that this x is such that if If y makes this function true, y and x are identical.This is the definition of a function of one variable. The number 1 is a unary property, which is possessed by certain functions.Likewise, a zero function is a function that is false for all values ​​of x, being a zero function, whose property is zero.

The old doctrines of numbers always encountered difficulties beyond 0 and 1. What impressed me at first was Peano's ability to deal with these difficulties. But it was many years before I reached the full conclusions of this new view.It is convenient to think of "classes" in mathematics.For a long time I thought it necessary to distinguish classes from propositional functions.However, I finally came to the conclusion that, except as a technical means, this distinction is unnecessary. The words "propositional function" may sound scary, but there is no need to be afraid.There are many times we can use the word "feature" instead.So we can say that every number is a property of certain properties.But, except for the final analysis, it might be easier to keep using the word "class".

The definition of number that I have come to for the above-mentioned reasons, Frege has already come to it sixteen years before me. But about this, I didn't know until about a year after I rediscovered this definition.I define 2 as all pairs, 3 as all triples, etc.A pair is defined as a class which has terms x and y, x and y are not equal, and if z is a term of this class, z is equal to either x or y.Generally speaking, a number is a group of classes, and this group of classes has a characteristic called "similarity". This can be defined as follows: Two classes are similar if there is a way to match their items one-to-one.For example, in a monogamous country, you can know that there are as many married men as there are married women, without knowing how many of them there are (I refer to widows and widowers). except).

Also, if a man has no missing leg, you can probably know with certainty that he has the same number of shoes on his right foot as he has on his left. In a meeting, if everyone has a chair and there are no empty chairs, there must be as many chairs as there are people sitting on them. In these examples, there is a so-called one-to-one relationship between those items in one class and those in another class.Similarity is the very definition of the existence of this one-to-one relationship. The number of any class is, so to speak, all those classes similar to it. This definition has several advantages.It can handle all the problems that have arisen with 0 and 1 in the past. 0 is the class of those classes that have no items, that is, it is a class whose only item is a class with no items. 1 is a class of classes whose characteristic is that they are made of anything equal to an x ​​term.The second strength of this definition is that it overcomes the difficulties concerning ones and manys. Since the term computed is computed as an instance of a propositional function, the ones involved are the ones of the propositional function. The oneness of this propositional function in no way contradicts the manyness of instances.But even more important than these two advantages, we stop treating numbers as metaphysical entities.In fact, numbers are just a linguistic convenience, no more than "wait" or "that is" more content.Kronek studied the philosophy of mathematics and said: "God made integers, and mathematicians made other mathematical devices."What he meant by this was that every integer must have an independent existence, but that is not necessary for other kinds of numbers.With the previous definition of numbers, this privilege of integers disappears.The fundamental apparatus of the mathematician is reduced to purely logical terms? Or, not, all, some, etc.For the first time I felt the usefulness of Occam's razor in reducing the number of ambiguous terms and unproved propositions that are required in a branch of knowledge. The above definition of numbers has another advantage, which is extremely important.That is, this definition removes the difficulty about infinite numbers.As long as the number is obtained by counting the terms one by one, it is not easy to imagine the number of groups that cannot be counted all at once.For example, you cannot count finite numbers to the end.No matter how long you count, there are always bigger numbers behind. So, as long as numbers are derived from counting, it seems impossible to talk about the number of finite numbers.But it seems that counting is only a way of knowing how many items are in a group, and it can only be used for those limited groups.The logic of counting that fits this new doctrine is this: For example, suppose you are counting gold pound notes.You work hard in your mind to make a one-to-one relationship between these banknotes and the numbers 1, 2, 3, etc., until the banknotes are counted.According to our definition, you know that the number of banknotes is the same as the number you have read. And, if you start at 1 and keep going without missing anything, the number that is one of those numbers that you have read is the last number you have read.You cannot use this method for infinite collectives, because human life is not long enough.But since counting doesn't matter anymore, you shouldn't care. Now that the integer image has been defined above, it is not difficult to extend its meaning to meet the needs of mathematics.Rational fractions are ratios between integers derived from multiplication.The real numbers are groups of rational numbers consisting of everything above zero up to a certain point.For example, the square root of two are all rational numbers whose square is less than two.I believe I am the inventor of this definition.It solves a riddle to which no mathematician since the time of Pythagoras has been able.A complex prime number can be regarded as a double real number. The meaning of "double" is that there is a first item and a second item in it, that is to say, the order of the items is very important. Besides the items I have mentioned, there is something else that I like about the work of Peano and his disciples. I like the way they develop geometry without figures, which means that Kant's intuition is not needed.I also like Peano's curve, which is common across a whole range.Before I met Peano, I was well aware of the importance of relationships.So I immediately set about dealing with relational logic with symbols, to complement the work done by Peano.I met him at the end of July.In September I wrote an article discussing the logic of relations, which was published in his journal. I spent October, November, and December of the same year writing Principia Mathematics.Now the third, fourth, fifth and sixth parts of that book are almost exactly the same as what I wrote in those months.However, the first, second and seventh parts I later re-wrote.I finished the first draft of "Principles of Mathematics" on the last day of the nineteenth century, December 31, 1900.The months after June of that year were a honeymoon of my intellect, such as I had never tasted before or since.Every day I find that I understand something that I didn't understand the day before.I thought all difficulties were solved, all problems were over.But the honeymoon didn't last.At the beginning of the second year intellectual sorrows fell upon me in full force.