Dr. R. May
PHI 178: Frege
17 March 2010
Response to ‘Discuss Frege’s conception of logicism. What is a number and how are they defined?’
Frege’s conception of logicism is beautiful in its simplicity. He set out to prove a link between logic and arithmetic; arithmetic meaning the branch of mathematics that is concerned with counting. More specifically he wanted to capture the Peano postulates of arithmetic in logic, prove that all arithmetical functions can be shown true in purely logical proofs. Frege’s concerns at the beginning of the project included several frustrations about the seemingly superficial way in which many of his contemporaries were exploring mathematical proofs in general. In his search for logicism, Frege managed to give definitions to mathematical concepts that, until he defined them, were ambiguous at best—at least in definition.
Both as a professional mathematician and as a philosopher Frege considered it scandalous that the foundations of ‘the most exact of all the sciences,’ mathematics, should be treated in the cursory and superficial manner…He embarked, therefore, upon a programme of investigation which was designed to fulfill two basic theoretical needs: to explain the concepts, and to secure the truths of arithmetic and analysis. (Bell 141)
Despite such huge mathematical leaps being made during Frege’s life there were still many mysteries surrounding the “elementary” mathematics of arithmetic. In setting out to find a link between logic and arithmetic that Frege was certain existed, the managed to give more depth to the definitions, and led to a revolution in the way mathematicians and philosophers alike approached math, science, and language.
One of the fundamental questions that had not been under scrutiny before Frege was the question ‘What is a number?’ Arithmetic is taught to school children every day with the emphasis that counting and arithmetical functions are easy to understand, with little to no explanation beyond what those functions do to manipulate numbers. Beginning with the premise that arithmetic reduces to logic, Frege took the first step in showing that every step in logic, or mathematics, must follow the highest standard of logical precision and proof. If something “follows logically” from (a) premise(s) then it is being held to the highest certainty in order to be called “knowledge.” This is the same standard that science adheres to in its theoretical postulates. Mathematics, until this time, had not this standard of logical certainty, and instead allowed themselves the flexibility of semi-ambiguous inference and even unscientific “intuition” to permeate its claims:
Since it is arbitrary what reference one wants to give to a sign, it follows that the content of the sign will have these or those properties, depending on the particular choice made…Mathematicians…are very peculiar people; instead of investigating the properties a thing really has, they don’t care about them on iota, but using so-called definitions, ascribe all sort of properties to a thing that have absolutely no connection with the thing itself, and then investigate these properties. (Frege “Formal Theories” 115-116)
In being critical of his colleagues’ intuitive claims about the definition of number, Frege realized that the numbers have a seemingly arbitrary quality: for example an army could be called “one” army, “ten” regiments, or “five-thousand” men, depending on the “concept” that is in question. His realization allowed him to make a distinction between the name of the number and the number itself. He outlines this in Sinn and Bedeutung when he makes the distinction between the morning star and the evening star. Since the term “morning star” and the term “evening star” each refer to the same point in the sky, they both represent the planet Venus, there is something significant that is to be learned from which term is thus used. The claim “the morning star is the morning star,” however, seems trivial; Frege then asks the question: what about the statement “the morning star is the evening star” gives information when the the statement “the morning star i the morning star” is deemed trivial? The answer lies in the what the sign is referencing: the signified. In logic this is represented through the use of identity; a = a is trivial because it is obvious (under normal conditions) that a thing is equivalent to itself. The claim a = b, however, gives information beyond the trivial: it signals to the thinker that there are two objects that are identical, and that those objects have a sign that is not the same—the significance in the difference of the sign is what makes the information important instead of trivial.
Numbers work the same way: the name “two” refers to a number, but that number has no meaning unless it refers to a concept that can be linked to the number itself. Numbers, then, are concepts—not objects with properties—and those concepts have different levels that signal the thinker to the meaning of the concept. An example is given in the Stanford Encyclopedia of Philosophy: “The concept being an author of Principia Mathematica falls under the concept being a concept under which two objects fall” (Malpas). This claim involves the concept two, much like the statement “There are two authors of Principia Mathematica” but the emphasis in the first claim distinguishes the conceptual nature of “two.” “Two” is not an object with properties that can be pointed out in space. Instead, “two” acts like a column on a table whereby claims such as “being an author of Principia Mathematica” might be filed under as being true when placed under “two.” Concepts themselves have properties, but, to quote Frege,
By properties that are asserted of a concept I do not, of course, mean the marks that make up the concept. These are properties of the things that fall under concept, not of the concept…In this respect existence is similar to number. Affirmation of existence is indeed nothing other than denial of the number zero. (Frege “Foundations” 103)
The properties of numbers are such that they are attributed to the objects within the concept. To know a number is not just a case of pointing to it: you would have to know something about it. For example five is the sum of two and three it is the second prime number. Each of these examples tells what “five” is, but each one is also a relation to other numbers. In other words, numbers are only “known” through their relation to other numbers, such as the successor function which orders numbers according to the number that comes before it.
When Frege conceived of logicism as a way to link logic and arithmetic, but his idea, despite its flaws, was so well-formulated that it affected philosophy of mathematics and language in ways he never intended.
Frege’s…’logicism’ essentially involves establishing (i) that the primitive concepts of arithmetic and analysis are logical concepts, i.e. that numbers, arithmetical functions, and operations can be reduced to, or defined exclusively in terms of, purely logical notions; and (ii) that the truths of arithmetic are logical truths, analytical statements which can be known a priori. (Bell).
In looking at the very foundations of “simple” arithmetic with such a scrutinizing eye, Frege helped to define things that, until he attempted it, did not have definition. Part of what Frege discovered in his exploration of logicism is that language is inherently flawed in describing his “most exact of all sciences” and that in order to accurately portray logicism, arithmetic, or logic he would have to build his own system in order to give the kind of accuracy he demanded. His attention to detail and high standards for proof led to a revolution in the way philosophers and mathematicians considered language, be it natural, unnatural, or mathematical. Despite his failure to prove a link between arithmetic and logic, the standards to which he held himself managed to reshape modern though: his short-term failure to prove logicism was a long-term success in that this rigorous standards paved the way for a higher mathematical standard that raised the bar for philosophy of mathematics, as well as creating new philosophical movements such as analytical philosophy and the philosophy of language. The fact that his seemingly narrow goal has shaped so much modern though has become more evident over time.
Works Cited
Bell, Davis. “Appendix: Frege’s Philosophy of Arithmetic” from Frege’s Theory of Judgment. From PHI 178 course website. Winter 2010.
Frege, Gottlob.
“On Formal Theories of Arithmetic.” From PHI 178 course website. Winter 2010.“The Foundations of Arithmetic.” The Frege Reader. Ed. Michael Beaney. Malden, MA: Blackwell Publishing, 1997.
Malpas, J., “Donald Davidson.” The Stanford Encyclopedia of Philosophy (Winter 2003 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/win2003/entries/davidson/
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