Gottlob Frege · Mathematics
In <em>Die Grundlagen der Arithmetik</em> (1884) Frege asked a question no one had answered rigorously - what <em>is</em> a number? - and argued that a statement of number is an assertion about a <em>concept</em>, that numbers are objective logical objects, and that arithmetic is reducible to logic. This is <em>logicism</em>.
Mathematicians used numbers every day, proved deep theorems about them, and yet, Frege observed with astonishment, not one of them could say what a number actually is. Is the number three a thing? A property? A collection? A mark on paper? A feeling in the mind? In his 1884 book Die Grundlagen der Arithmetik (The Foundations of Arithmetic) - written almost entirely in plain prose, with scarcely a symbol - Frege set out to answer this scandalously neglected question with total rigour. He proceeded by demolition first: patiently clearing away the wrong answers before building the right one. Numbers are not properties of external things the way colour or weight are; they are not heaps or aggregates of objects; and they are emphatically not ideas in anyone’s head. The book is a masterpiece of philosophical prose, and it remains the single most influential work ever written on the nature of number.
Frege’s two great targets were the empiricist and the psychologistic accounts of number. The empiricist, exemplified by John Stuart Mill, held that arithmetic is a very general natural science: numbers are properties of physical aggregates, and ‘2 + 3 = 5’ is an inductive generalisation from our experience of pebbles and beads. Frege found this almost comic. What, he asked, is the physical fact corresponding to the number 0, or to very large numbers no one has ever counted out - and what pile of things is the number 703? Arithmetic cannot rest on observation, because its truths are certain, exceptionless, and apply even to things (thoughts, proofs, other numbers) that are not physical at all. The psychologistic view - that numbers are mental constructions, ideas we form - fared no better. If the number three were my idea, then your three and my three would be different threes, and arithmetic would be a report on the contents of minds rather than a body of objective truth. Against both, Frege insisted: numbers are objective and non-physical.
Frege insisted, against much of the tradition, that numbers are genuine objects - self-subsistent things, not mere properties or aspects of other things. His argument turned on how numbers behave in language. In the sentence ‘the number of Jupiter’s moons is four,’ the phrase ‘the number of Jupiter’s moons’ functions as a proper name, picking out a particular object, and the sentence states an identity: this object is the very same object as four. We speak of the number four, we say numbers are equal or unequal, we quantify over them - all the marks of objecthood. But numbers are a special kind of object: logical objects, not found in space or time, grasped by reason rather than perception, yet fully objective and the same for everyone. This is a form of platonism about mathematics - numbers are real, mind-independent abstract entities - and Frege’s defence of it is still the reference point for every debate about whether mathematical objects exist.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Frege attacked both the view that numbers are physical heaps and the view that they are subjective ideas, argued that a statement of number is an assertion about a concept, defined 0 and successor logically, and founded logicism (arithmetic is derivable from logic). Explain these moves in your own wor…
Leads to Immanuel Kant.
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