# How did non-Euclidian geometry affect other fields?

The relationship between space and geometry changed forever in people's minds, thanks to non-Euclidian geometry. The question arose of whether space itself was curved. This made the whole of geometry seem hypothetical and led some to question the possibility of *a priori* knowledge. That is, if space is not necessarily Euclidian and there are other unknown geometries of space, then what does it mean to say that we have "a *priori* knowledge of space?" Also, the idea of the curvature of space was conducive to Albert Einstein's theory of relativity, thus influencing physics and our concept of the universe.

# What are Venn diagrams?

British philosopher and logician John Venn (1834-1923) invented the system of logic diagrams named after him, which consisted of the overlapping circles. They can be used to test and demonstrate the validity of inferences. Venn diagrams illustrate collections of sets and their relationships to each other, which are useful in logic theory.

# Who was Jules Henri Poincare?

Jules Henri Poincare (1854-1912) was a mathematician, physicist, and philosopher of science. He responded to the discovery of non-Euclidian geometry by suggesting a modification of Immanuel Kant's (1724-1804) claim that we have synthetic *a priori* knowledge of the world (that is, certainly true knowledge that applies to reality, which is not based on experience).

His proposal was what became known as "conventionalism," namely that physicists will retain Euclidian geometry because it has the simplest geometrical conventions and is therefore appropriate for them. This proposal was short-lived in mathematics, because Albert Einstein was to show in his General Theory of Relativity that the curvature of space obeyed the principles of non-Euclidian geometry. However, the broader principle of conventionalism, namely that truth in science depends on agreement about specified rules, was to be revived as an idea of scientific truth in the twentieth century.

**Bertram's model of n-dimensional hyperbolic geometry in which points are represented by the points in the interior of the n-dimensional unit ball (or unit disk, in two dimensions, in this schematic) and lines are represented by the chords or straight line segments with endpoints on the boundary sphere (here, it is the circumference of the two-dimensional disk.)**

**A Venn diagram of sets A, B, and C. Where one or more sets overlap, it means that they have members in common. It can be seen by the overlapping in this diagram that some things are A, B, and C, some things are A and B, some things are B and C, and some things are A and C.**

# Who was Gottlob Frege?

Gottlob Frege (1848-1925) was a professor of mathematics at the University of Jena, who thought that Immanuel Kant (1724-1804) was mistaken in claiming that mathematical truth is syntheticâ€”that is, about reality. (Kant had claimed that mathematical truths were synthetic *a priori,* which is to say both true of the world and known independently of experience of the world.) His task was to show how the concepts of mathematics could be defined in terms of logic alone, so that the theorems of mathematics would then appear as logical truths. If mathematics could be reduced to logic in this way, it would be shown that mathematics was merely true by definition, meaning that it had no empirical content, so that it could not be about the world. Mathematics would thereby be *a priori,* but not also synthetic, as Kant had insisted.

# What was Gottlob Frege's main innovation in the philosophy of logic?

Frege treated predicates as functions and subjects as arguments. Thus "Socrates is mortal" becomes "function 'mortal' is applied to argument 'Socrates.'" In his *Conceptual Notation* (1879), Frege also introduced a simple way to treat words and terms such as "all" and "there is" as logical quantifiers. Logical quantification is a notational system that connects a variable with what is being talked about. For example, in the sentence "Every person alive today will die some day," "person alive today" is being talked about and "every" is the quantifier. This treatment of Frege's still stands today.

# What was Gottlob Frege's landmark insight about meaning?

Frege's theory of language was set forth in three essays: "Function and Concept," "On Concept and Object," and "Sense and Reference." He noted that some identity statements are true and informative. For example, the sentence "Venus is Venus," does not tell me anything, but the sentence, "The Morning Star is the Evening Star," is informative, although it means the same as "Venus is Venus," because Venus is in fact both the Morning Star and the Evening Star.

How can this be? Frege's explanation was that there is a difference between "sense" and "reference." Reference is the actual planet Venus, in this case. But sense is *how* the planet is referred to by the term "Morning Star" (i.e., a bright object in the eastern sky before sunrise). Thus, "The Morning Star" does not stand for Venus itself, but for the sense of how Venus is presented. This is why the two sentences that appear to be equivalent really are different. It explains why it is not informative to say that Venus is Venus or that The Morning Star is the Morning Star, but it is informative to say that Venus is the Morning Star.

# How did Gottlob Frege attempt to reduce mathematics to logic?

In his *Foundations of Arithmetic* (1884), Frege argued that logic, or the laws of thought, are not descriptive of how we think and that words do not have meaning in isolation but only within context. Then in his two-volume *Basic Laws of Arithmetic* (1893 and 1903), Frege began his project in earnest by showing that every predicate determines a class that can be described logically. For example, red is a predicate and red determines a class of red things.

# Did Gottlob Frege succeed in reducing mathematics to logic?

Alas, no. When the second volume to Frege's *Basic Laws of Arithmetic* (1893) had been sent to the printer, he received a letter from British philosopher, historian, and mathematician Bertrand Russell (1872-1970) in which Russell introduced his famous paradox: "Is the class of all classes that are not members of itself a member of itself or not?" The question is coherent but it entails a contradiction, so it has no answer.

Frege had to admit that he had no foundation for his reasoning: "A scientist can hardly encounter anything more undesirable than to have the foundation collapse just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was almost through the press." The great irony in this is that Russell embarked on his own project to reduce mathematics to logicâ€”and failed!