Bertrand Russells defintion of pure mathematics
Russells Definition
Classes and Sets
Set theory is the logic of classes. A class is a term that can be used interchangeably with a set. Therefore, set theory is the logic of sets and their relationships with one another. These sets, or collections of sets, can be finite or infinite. A set is a member of a class that is also a set. A proper class is one that is not a member of any other class.
For example, consider the set of all lions on Earth. This set is also a member of the set of all mammals on Earth. In contrast, consider the set of mammals that contain exactly one member. By definition, if the set contains more than one member, then it is not in the set of mammals that contain exactly one member.
Principles of mathematics not being entirely controversial
The definition of mathematics in the text seems to be fully justified. The problem analyzes the concept of mathematics that we have in our minds when we use the term mathematics, even unconsciously. This analysis is considered philosophical, as it moves from something complex to something simple. Most of the questions involved are no different from philosophical questions. In this way, many of the questions that have been the subject of philosophical debate can now be answered because the questions are reduced to pure logic. Concepts like infinities, the nature of numbers, and time can now have answers because the questions are reduced to pure logic.
All pure maths follows formally from twenty premises
Mathematics seems to follow from deductions. However, some of the older forms of logic, such as Aristotelian syllogistic logic and symbolic logic, were not able to capture the full range of mathematics. The thinking of Kant relied on this idea, which was that mathematical reasoning is not formal, but always used intuitions. This was Kant's a-priori knowledge of space and time. Progress in symbolic logic, especially from work by Peano, showed that mathematics does have formal reasoning. There are 10 rules of deduction and 10 rules of implication that allow all of mathematics to be deduced.
The Assertion of formal implication
The idea is basically that from logical principles, you can deduce logical principles. Leibniz advocated for such a system, for example. Things like Euclidean axioms were found, but they don't just come from the logical principles which Kant built upon. But since non-Euclidean geometries have come along, what we now hold is that both the Euclidean and non-Euclidean geometries are equally true. Whether or not they hold in some real-world case is something for the task of applied mathematics and science. Nothing is affirmed in Euclidean or non-Euclidean, except the implications themselves. The assertion that mathematics makes is that propositions can be asserted in relation to some set of entities. And some other propositions can also be asserted from those entities. But we cannot say that the propositions are true or false separate from the entities. We only assert a relation between the propositions, which is called formal implication.
Use of variables
Mathematical propositions have implications, but they also have variables. Something like "" seems like it is not concerning variables or an implication. But it can be shown that it says "if is one, and is one, and differs from , then and are two." Now it contains both propositions and variables. Something is a constant if it is dealing with a specific number in some equation, for example, one specific line or one specific time . Otherwise, even if we use something like , we are still using variables. The variables are and , while the constants are , , and . However, the constants can still represent any variable until their instantiation in reality. So, 1, 2, 3, dog, cat, and Koko the gorilla are all constants.