Chapter 8 Chapter 7 "Principles of Mathematics": Philosophical aspects

In the years from 1900 to 1910, Whitehead and I devoted most of our time to what became Principia Mathematica.Although the third volume of this work was not published until 1913, our work in this work (minus proofreading) was completed in 1910, when we handed over the entire manuscript to Cambridge University Press. The "Principles of Mathematics", which I finished on May 23, 1902, turned out to be a crude and very immature draft of the later work.However, the difference between "Principles of Mathematics" and "Principles of Mathematics" is that "Principles of Mathematics" contains debates with other mathematical philosophies.

There are two kinds of problems we want to solve: philosophical and mathematical.Broadly speaking, Whitehead left the philosophical questions to me.As for the mathematical problems, the notation is mostly invented by Whitehead, (except for those who quote Peano).Most of the work on series was done by me, the rest by Whitehead.But that's just the first draft.Each part is done three times.When either of us makes a first draft, he sends it to the other, who usually revises it considerably.Then, the person who originally wrote the first draft finalizes it.There is hardly a line in these three volumes that is not the product of a collaboration.

The main purpose of Principia Mathematica is to show that the whole of pure mathematics follows from purely logical premises and uses only concepts stated in logical terms.This is of course the exact opposite of Kant's teaching.At first I thought the book was an interjection to refute "that preposterous philistine," as George Kanter called Kant. To be more specific, Kantor added: "He doesn't know much about mathematics."But then the book went in two different directions.In mathematics a whole new subject arises, including a new notation by which it is possible to treat in symbols what was formerly treated in loose and rough ordinary language.In philosophy there are two opposite developments, one pleasant and one unpleasant.Happily, the set of logical mechanisms required turned out to be smaller than I had imagined.In particular, it turns out that classes are unnecessary.Much of Principia Mathematica discusses the distinction between classes of one and classes of many.The whole discussion of this, and many of the complex arguments in that book, proved unnecessary.As a result, after the book was written, it seemed that it lacked profound philosophy, and incomprehension is the most obvious characteristic of profoundness.

That unpleasant aspect is very unpleasant indeed.Logicians of every school since Aristotle seem to be able to infer some contradictions from their accepted premises.This indicates that something is wrong, but does not indicate how to correct it.In the spring of 1901 one of these paradoxical discoveries interrupted the logical honeymoon I was enjoying.When I told Whitehead of this misfortune, I was not comforted by his quote: "There will be no more pleasant and confident mornings." Kanter proves that there is no largest base.I discovered the above-mentioned contradiction after thinking about Kantor's proof.I was simple enough to think that the number of all things in the world must be the greatest possible number.

I'll apply his proof to this number and see how it goes.This approach leads me to consider a special class.I'm thinking along lines that seemed appropriate before, and I think a class is sometimes and sometimes not an item of its own.For example, the class Spoon is not another Spoon.But this class of things that are not spoons is one of those things that are not spoons.It seems that some examples are not negative: for example, all classes of this class are a class. Applying Kantor's argument leads me to consider classes that are not my own terms.It seems that these classes must be a class.I asked myself if this class is an item of its own.If it is an item of its own, it must have a distinct property of the class which is not an item of the class.If the class is not an item of its own, it must not have the distinct properties of the class, and so must be an item of its own.In this way, no matter which one of the two goes to its opposite, there is a contradiction.

At first I thought there must be some little error in my reasoning.I examined each step under a logical microscope, but I could see nothing wrong.I wrote Frege a letter and told him about it.He replied that arithmetic had faltered, and that he saw that his fifth law could not hold. This contradiction troubled Frege so much that he gave up the attempt to deduce arithmetic from logic, to which he had devoted his life until that time.Like the disciples of Pythagoras when they encountered irrational numbers, Frege fled to geometry, apparently thinking that up to that time his life's work had been misguided.As for me, I think the fault lies in logic, not in mathematics, and logic must be reformed.This opinion of mine is confirmed by the discovery of a secret by which an almost infinite number of contradictions can be produced.

Philosophers and mathematicians have reacted differently to this situation.Bangley doesn't like mathematical logic. He once criticized mathematical logic, thinking that it can't have results.He said happily: "It's not impossible to have results, it has created contradictions." These words are indeed very good, but they cannot solve the problem.Some other mathematicians who disapproved of George Kanter adopted the March Hare solution: "I'm tired of this, let's change the subject."I don't think that's appropriate either.But then some people seriously tried to solve this problem, people who understood mathematical logic and knew that it was necessary to use logic to solve it.The first of them was F. p.Lemousse.

Unfortunately he died early and did not finish his work.But in the years before Principia Mathematica was published, I was unaware of subsequent efforts to solve this problem. I was actually alone there wondering. There are older paradoxes (some of which were known to the Greeks) which I feel raise similar questions, although some of my later authors have considered them to be of a different kind.The most famous of these is the paradox concerning Epimenides of Crete.He said that all Cretans are liars.This makes people ask whether he is not lying when he says this.This paradox is manifested in its simplest form if one says, "I am lying."If he is lying, then his lying is a lie, so he is telling the truth; but if he is telling the truth, he is lying because that is what he says he is doing.Thus, contradictions are unavoidable.St. Paul once mentioned this paradox.But he was not interested in the logical side of the paradox.What interests him is that this paradox proves that heretics are bad.But mathematicians can dismiss these elusive problems as irrelevant to their subject, though they cannot ignore the question of whether there is a largest cardinal number or a largest ordinal number, both of which All bring them into conflict.The contradiction about the largest ordinal number was discovered by Bradley Forte before I discovered mine.But his case was much more complicated, so I thought there was some slight error in reasoning.In any case, since his contradictions are far less simple than mine, it seems at first glance that the destructive force is not so great.However, in the end I have to admit that the severity is the same.

In Principia Mathematics I did not openly say that I had found a solution.I said in the preface to that book: "Publishing a book that contains so many unresolved controversies, my interpretation is that, after research, the contradictions discussed in Chapter 10 I do not see that have been properly addressed recently. There is no hope of a solution, no hope for a deeper insight into the nature of classes. Some solutions have given me momentary gratification. They have often been found to be wrong. This discovery makes one feel, well, It seems that after thinking for a long time, some superficially satisfactory theories may be obtained. With these theories, the problem will not be revealed. Because of this reason, it is better to just talk about the difficulties than to wait until I believe a person. There is truth in a doctrine that is almost certainly false, as if it were better."

At the end of the chapter on contradictions I said: "The contradictions mentioned above involve no particular philosophy. The contradictions are of direct origin in common sense. The only solution to this contradiction is to abandon some assumption of common sense. Only by Hegel's philosophy, which feeds on contradictions, can not care, because it encounters problems like this everywhere. In any other doctrine, such a positive challenge requires you to make an answer, otherwise you admit that there is no way. Fortunately, , so far as I know, there is no other similar difficulty in any other part of "Principles of Mathematics." In the appendix at the end of the book, I proposed that the theory of types can give a plausible explanation.In the end I was convinced that this theory would solve the problem, but while I was working on Principia Mathematics I only made it rough.

This doctrine is incompetent under the circumstances.The conclusion I reached at that time is expressed in the last paragraph of this book: "In summary, it appears that the particular contradiction in Chapter 10 is resolved by the theory of types. However, at least one very similar contradiction probably cannot be solved by this doctrine. It appears that all objects of logic, or all propositions, as a whole involve a fundamental logical difficulty. What the satisfactory solution of this difficulty I have not discovered; but since it affects reasoning foundations, and I earnestly hope that all those who study logic will pay attention to it." After writing Principia Mathematica, I was ready to resolve to find a solution to these paradoxes.I feel like it's almost a personal challenge, and if I had to, I'd spend my entire remaining years fighting it.But I find this extremely unpleasant for two reasons.First, I feel that the whole issue is trivial.I'm extremely reluctant to focus on something that isn't necessarily really interesting.Second, no matter how hard I try, I'm not making progress.During the whole period of 1903 and 1904 I devoted myself to this task almost exclusively, without success.My first achievement was the narrative theory in the spring of 1905.I will discuss this doctrine below. On the surface, this has nothing to do with these contradictions, but then an unexpected relationship emerged.In the end, it became clear to me that some form of typology is crucial.I will not now focus on the particular form of the doctrine presented in Principia Mathematica.But I am still fully convinced that these paradoxes cannot be resolved without some form of the doctrine. While I was searching for a solution, it occurred to me that three conditions must be met if the solution is to be at all satisfactory.The first of these is absolutely necessary, namely, that these contradictions must disappear.The second condition is preferable, though not logically necessary, that the solution should leave the mathematics intact as much as possible. The third condition is not easy to say correctly, that is, the solution should cater to what we might call "logical common sense", that is to say, it should finally look like what we have been expecting.Of these three conditions, the first is of course recognized.The second, however, is denied by a large school which holds that a large part of the analysis is incorrect.Those who are complacent with good use of logic think the third condition is unimportant. For example, Professor Kui Yin has produced some systems.I admire the ingenuity of these systems, but I cannot consider them satisfactory, because they seem to have been created for this purpose, and even the most ingenious logician would not have thought of them if he had not been aware of these contradictions. of these systems.However, there has been a large and profound literature on this issue, and I will not say more about its subtleties. Leaving aside difficult technical details, we can give an outline of the theory of types.Perhaps the best way to investigate this doctrine is to examine what it means to be a "class".Let's illustrate with a trivial example.Suppose the host who invites you to eat after a meal asks you to choose among three sweets, and asks you to eat one or two, or all three, as you like.How many ways can you do it?You can decline all of them.This is one way.You can take one of the sweets. There are three different possible approaches to this, so you have three options again.You can choose two of the sweets.There are three possible ways of doing this.Or all three sweet treats you want.This gives you one final possibility.So the total number of possibilities is eight, or 23.It is not difficult to generalize this procedure into general rules.Assume that there are n so many things in front of you, and you want to know how many options there are in total if you don’t choose one of them, or choose a few, or both.You have to know that the number of ways is 2n.In the language of logic: a class with n items has as many subclasses as 2n.This proposition is still true if n is infinite.What Kanter proved was that even in this one example, 2n is greater than n.If we apply this to everything in the universe, as I do, we come to the conclusion that there are more kinds of things than things.Classes are therefore not "things".But since no one quite knows what the word "thing" means in this sentence, it is not easy to say exactly what we have proved. The conclusion I can't help but come to is that classes are just a convenience when speaking.When I was writing "Principles of Mathematics", I already felt that there was no way to solve the problem of classes.However, the language I used to express my meaning at that time, I think now, should not be so realistic (realism is the meaning of learning from the philosophy of the school). I said in the preface of that book: "Discussing the undefinable (which accounts for the main part of philosophical logic) is trying to see these entities clearly, and also to make others understand these entities. In this way, our psychology Perhaps there is a kind of knowledge of these substances, like the color of red or the taste of pineapple. Whenever we acquire indefinable things mainly as necessary residues from the analysis process (as in the example we are speaking of), knowing There must be such entities which are often easier than actually perceiving them; there is a process which is similar to the discovery of Neptune, with the difference that the mental telescope is used to seek that which has been deduced. out of the entity, this final stage is often the most difficult part of doing it. Regarding the class example, I have to confess that I don't see any concept that satisfies the necessary conditions for the concept of a class. In the tenth The contradictions discussed in the chapter prove that something is not quite right, but what exactly it is I have not been able to see." I should now say something different about the matter.I should say that given any propositional function, say fx, then there is a considerable range of values ​​for x in terms of which the function is "meaningful," that is, either true or fake.If a is within this range, fa is a proposition, and this proposition is either true or false.Apart from substituting a constant for the variable x, there are two things that can be done about a propositional function: one is to say that it is always true; the other is to say that it is sometimes true. The propositional function "If x is a man, x cannot escape death" is always true; the propositional function "x is a man" is sometimes true.So there are three things to do with a propositional function: first, substitute a constant for the variable; second, assert all values ​​of the function; third, assert some value, or at least one value. A propositional function itself is just a formula.It does not affirm or deny anything.Likewise, a class is nothing more than a formula.It's just a convenient way to talk about the values ​​of the variables that make this function true. Concerning the third of the three necessary conditions mentioned above for the solution of this problem, I have advanced a theory which seems to displease the other logicians.But in my opinion, this doctrine is still correct.This doctrine can be stated as follows: When I assert all the values ​​of a fx function, the values ​​that x can take must be explicit if what I assert is to be definite.That is, there must be a population of all possible values ​​of x. If I now go on to create new values ​​in terms of that totality, the totality seems thus enlarged, and the new values ​​related to it are thus related to the enlarged totality.However, since new values ​​cannot but be included in the population, the population can never catch up with these new values, and the process is as if you want to jump onto the shadow of your head.We illustrate this most simply with the paradox of the liar.The liar said: "Whatever I say is false".In fact, that's what he said in one sentence, but the sentence refers to the totality of what he said.It is only when the sentence is included in the totality that a paradox arises.We cannot fail to distinguish between propositions which refer to the totality of propositions and those which do not.Those propositions which refer to a totality of propositions can never be members of that totality.We may say that the propositions of the first degree are those propositions that do not involve the totality of propositions; the propositions of the second degree are those propositions that involve the totality of propositions of the first degree; and the rest are like this to infinity.So our liar has to say now: "Now we are affirming a false proposition of the first degree, which is false." But this itself is a proposition of the second degree. So he is not uttering any first-order propositions.So what he said is simply false, and the argument that it is also true is self-defeating.This kind of argument can be used for any higher-level proposition. We can see that in all logical paradoxes there is a reflexive self-reference which should be condemned on the same grounds.That is to say, it contains something about that totality (which is again a part of that totality).This kind of thing has definite meaning only if the whole has been fixed. I cannot but say frankly that this doctrine is not yet widely accepted.But I have not seen convincing arguments against this theory. The narrative theory mentioned above was first proposed in my article "On Instructions" published in the journal "Xin" in 1905.The editor at that time felt that this theory was unreasonable, and he asked me to reconsider it and not ask to publish it as it is.However, I believe this doctrine to be correct, and I refuse to budge. This doctrine was later generally recognized as my most important contribution to logic.Indeed, those who do not now believe in the distinction between names and other words have a reaction to this doctrine.But I think it's only among those who haven't dealt with mathematical logic.All in all, I do not see any validity in their criticism.I admit, however, that perhaps the doctrine of names is a little more difficult than I thought at one time. But I leave these difficulties aside for the moment, and speak of ordinary language in ordinary use. I have drawn for the purpose of my argument a comparison between the title "Scott" and the statement "author of Waverley." The proposition "Scott is the author of Waverley" expresses an identity, not a tautology. George Fourth wanted to know if Scott was the author of Waverley, but he didn't want to know if Scott was Scott.While this makes sense to everyone who has never studied logic, it is a mystery to logicians.Logicians hold (and may say formerly) that if two terms refer to the same thing, a proposition containing the one can always be replaced by a proposition containing the other without loss of value. True, if the original proposition was true, or without losing its falseness, if the original proposition was false.However, as we have already said, using "Scott" By substituting "the author of Waverley" you can turn a true proposition into a false one.This shows that a name cannot be distinguished from a statement: "Scott" is a name, but "the author of Waverley" It's just a narrative. Another important difference between names and narratives is that a name has no meaning in a proposition if it has no referent, whereas a narrative is not subject to this restriction.I had great respect for the work of the wheat farmer, but he could not see the difference.He once pointed out that we can come up with some propositions whose logical subject is "golden mountain", although the golden mountain does not exist.His contention is that if you say that the golden mountain does not exist, obviously there is something you are talking about that does not exist, that is, the golden mountain: so the golden mountain must exist in some kind of vague world in Plato's philosophy. , because, if this is not the case, your proposition that Jinshan does not exist is meaningless.I'll be honest, before I came up with the narrative theory, I found McNon's argument convincing.The main point of this theory is that although "golden mountain" can be the subject of a meaningful proposition grammatically, such a proposition, if analyzed correctly, does not have such a subject. The proposition "the golden mountain does not exist" becomes "the propositional function 'x is golden and is a mountain' is false for all values ​​of x". The proposition "Scott is the author of Waverley" becomes "For all values ​​of x, 'x wrote Waverley' is equal to 'x is Scott'." Here, The words "Author of Waverley" no longer appear. The doctrine also clarifies what it means to be. "The author of Waverley exists" means "there is a value of c for which the propositional function that x wrote Waverley' is always equal to 'x is c' is true. of." In this sense, existence can only be used to say a statement, and, after analysis, it can be seen that it is an instance of a propositional function, which is true at least with respect to one value of the variable.We can say "The author of Waverley exists", and we can also say "Scott is the author of Waverley", but "Scott exists" is not correct.This statement can at best be interpreted as meaning: "the one named Scott exists," but "the one named Scott" is a statement, not a name.Whenever a name is properly used as a name, it is incorrect to say "it exists". The main point of the narrative doctrine is that a phrase can contribute something to the meaning of a sentence that, by itself, means nothing at all.Narratively, there is precise proof of this: if "the author of Waverley" means something other than "Scott," "Scott is the author of Waverley" is a pseudonym. Yes, in fact this proposition is not false.If "the author of Waverley" means Scott, then "Scott is the author of Waverley" is a tautology, which is not the case.So "the author of Waverley" means neither "Scott" nor anything else.That is to say, "the author of Waverley" means nothing.Certified.