![]() They extrapolate this idea to set theory, showing that two sets are equivalent in cardinality, or “size,” if this one-to-one correspondence exhausts itself without any leftover elements in any set. Counting consists of identifying numbers with objects, such as one’s fingers, and building up a sequence through correspondence. To do this, they borrow from the thinking of Gottlob Frege, a German mathematical philosopher, and define the number as an object we reach via counting, rather than via abstraction. Then, they provide a formal definition of number in a non-circular manner. They first spend time creating a “theory of types” defined by their axiomatic language. Next, Russell and Whitehead develop their theory of mathematics as an extension of logic. He qualifies that they need not necessarily be true when applied to real objects. For his own system, he makes sure that his axioms are ones that humankind considers self-evident when applied to sets and classes. In general, he argues that a formal system must start with a finite set of axioms, which he also calls assumptions. From the expressions we can make by linking these symbols together, Russell posits, we can deduce other logical expressions that are true because they stem from earlier, logically true statements. He thinks of propositions and other logical operators as things we can represent using simple symbols. Russell’s first effort along these lines is to describe propositional logic and attach it to a formal system, the former built on a finite number of logical axioms. Russell states the project of his book is to show that this effort can be extended into the domain of math. ![]() Russell traces logical form back to Aristotle, the first great example of an effort to give linguistic shape to these natural logical forms. He takes a few examples of logical laws we have recognized to exist in basic human reasoning that existed even before we had a logical language, or even an alphabet, for describing them. In addition, logic is a natural human faculty, which seems to extend out of the rational mind. ![]() He elevates logical truth above all other truths that humans can create, arguing that it has its own form, unlike other truth-statements, which primarily state a semantic content. Russell, the principal writer, begins with the argument that it would be easier to “prove” the validity of mathematics if we could show its logical soundness. The book is considered a seminal contribution to that which we now know about mathematics and its relation to the describing activities of human behavior, as well as to the reality outside the bounds of human consciousness. Russell holds that logic, by nature, is the most accurate general language with which to describe reality. Principia Mathematica endorses a thesis introduced by modern logicians arguing that mathematical language can be broken down into a more fundamental logical language. It was intended to extend an earlier work by Russell called Principles of Mathematics, but ended up being a longer, more nebulous endeavor, undergoing several editions. Principia Mathematica (1910) is a work of philosophy in the domains of logic and mathematics by Bertrand Russell, with contributions from a contemporary mathematician, Alfred North Whitehead.
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