History of Western philosophy

Take the series of integers starting from 1. How many numbers are there? It is clear that this number is not finite. Up to a thousand, there are a thousand numbers; up to a million, there are a million numbers. No matter what finite number you come up with, there are obviously more numbers than that, because from 1 to that number, there are so many numbers, and then there are other larger numbers. Therefore, the number of finite integers must be an infinite number. But now it is a wonderful fact that the number of even numbers must be as great as the number of all integers. Try the following two rows: 1?2?3?4?5?6?…… 2?4?6?8?10?12?…… For every item in the upper row, there is a corresponding item in the lower row; so the number of items in both rows must be equally large, although the lower row is made up of only half of the items in the upper row. Leibniz noticed this and thought it was a contradiction, so he concluded that although there are infinite groups, there are no infinite numbers. Georg Cantor, on the other hand, boldly denies that this is a contradiction. He's right; it's just a curiosity. Georg Cantor defines an "infinite" group as one that has a partial group that contains as many terms as the whole group. On this basis he was able to establish a very interesting mathematical theory of infinite numbers, thus bringing into the scope of strict logic a whole field previously abandoned to mystery, fantasy,interactive flat panel display, and confusion. The next important figure was Frege, who published his first work in 1879 and his definition of number in 1884; but, in spite of the epoch-making nature of his discoveries, he remained completely unrecognized until 1903, when I drew attention to him. It is worth noting that before Frege, all the definitions of numbers put forward by everyone contained basic logical errors. Conventionally, "number" and "pluralism" are always regarded as the same thing. However,smart boards for conference rooms, the concrete instance of "number" is a specific number, such as 3, and the concrete instance of 3 is a specific triple. A triple is a multivariable, but the class of all triples, which Frege thinks is the number 3 itself, is a multivariable made up of some multivariables, and the general number of which 3 is an example is a multivariate made up of some multivariables. By confusing this plurality with the simple plurality of a known triad, this basic grammatical error is made, and the result is that all of Frege's previous philosophy of numbers is nonsense, "nonsense" in the strictest sense. It can be inferred from Frege's work that arithmetic, and pure mathematics in general, smart board whiteboard ,75 inch smart board, is nothing but an extension of deductive logic. This proves that Kant's theory that arithmetic propositions are "comprehensive" and contain time relations is wrong. Whitehead and I detailed how to develop pure mathematics from logic in our book, Principia Mathematica. It has come to be understood that a large part of philosophy can be reduced to something that can be called syntax, but the word syntax has to be used in a slightly broader sense than has hitherto been used. Some people, especially Carnap, have put forward a theory that all philosophical problems are actually syntactic problems, and that as long as syntactic errors are avoided, a philosophical problem is either solved or proved to be unsolvable. I think this is an overstatement, and Carnap now agrees with me, but there is no doubt that the utility of philosophical syntax in traditional problems is very great. I would like to briefly explain the so-called theory of description to illustrate the utility of philosophical syntax. By "impersonation" I mean a phrase like "the current President of the United States," which refers to a person or thing not by name, but by some assumed or known characteristic of him or it. Such phrases have caused a lot of trouble. Suppose I say, "Gold Mountain does not exist," and suppose you ask, "What does not exist?" If I say "it is the mountain of gold", then it is as if I attribute some kind of existence to the mountain of gold. Obviously, when I say this, it is not the same statement as saying that "circles and squares do not exist.". This seems to mean that gold is one thing and the square is another, although neither of them exists. The theory of imitations is intended to cope with this and other difficulties. According to this theory, a statement containing a phrase in the form of "theso-and-so" would, if properly analyzed, result in the absence of the phrase "theso-and-so". Take, for example, the statement that Scott was the author of Waverley. The imitative theory interprets this statement as saying: "One man, and one man only, wrote Waverley, and that man was Scott." Or, to put it more precisely: "There is an entity C such that if X is C, the statement that X wrote Waverley is true, otherwise it is false; and C is Scott." The first part of this sentence, the part before the word "and", is defined as meaning "that the author of Waverley exists (or has existed, or will exist)." Therefore, the meaning of "Jinshan does not exist" is: "There is no entity C such that when X is C, 'X is gold and a mountain' is true, otherwise it is not true." With this definition, the problem of what it means to say that "Jinshan does not exist" is eliminated. According to this theory, "existence" can only be used to assert a description. We can say that the author of Waverley exists, but it is ungrammatical,interactive panel board, or rather unsyntactic, to say that Scott exists. This clarifies the confusion of "existence" for two thousand years, which began with Plato's Theaetetus. hsdsmartboard.com

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