mathematics

Function and Concept

Function and Concept is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Function and Concept rather than just read about it. In short: "Function and Concept" (German: "Funktion und Begriff") is a lecture delivered by Gottlob Frege in Jena at a meeting of the Jenaische Gesellschaft für Medicin und Naturwissenschaft on 9 January 1891, and published the same year as an article. The lecture involves a clarification of his earlier distinction between concepts and objects.

Key takeaways

Function and Concept belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Function and Concept to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Function and Concept from memory before moving on to harder problems.

Reference excerpt

"Function and Concept" (German: "Funktion und Begriff") is a lecture delivered by Gottlob Frege in Jena at a meeting of the Jenaische Gesellschaft für Medicin und Naturwissenschaft on 9 January 1891, and published the same year as an article. The lecture involves a clarification of his earlier distinction between concepts and objects. It The original article was reprinted in 1962.

Overview In general, a concept is a function whose value is always a truth value (139). A relation is a two place function whose value is always a truth value (146). Frege draws an important distinction between concepts on the basis of their level. Frege tells us that a first-level concept is a one-place function that correlates objects with truth-values (147). First level concepts have the value of true or false depending on whether the object falls under the concept. So, the concept F {\displaystyle F} has the value the True with the argument the object named by 'Jamie' if and only if Jamie falls under the concept F {\displaystyle F} (or is in the extension of F). Second order concepts correlate concepts and relations with truth values. So, if we take the relation of identity to be the argument f {\displaystyle f} , the concept expressed by the sentence:

∀ x ∀ y f ( x , y ) → ∀ z ( f ( x , z ) → y = z ) {\displaystyle \forall x\forall yf(x,y)\rightarrow \forall z(f(x,z)\rightarrow y=z)}

correlates the relation of identity with the True. The conceptual range (Begriffsumfang in Frege 1891, p. 16) follows the truth value of the function:

x 2 = 1 {\displaystyle x^{2}=1} and ( x + 1 ) 2 = 2 ( x + 1 ) {\displaystyle (x+1)^{2}=2(x+1)} have the same conceptual range.

Translations "On Function and Concept" in Michael Beaney, ed., The Frege Reader, Blackwell, 1997, pp. 130–148

References

External links "Logical Constants" "Chronological Catalog of Frege's Work" List of English translations

Worked examples

Example 1 — a first encounter with Function and Concept

Start with the simplest possible case. Write down what Function and Concept claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Function and Concept before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Function and Concept ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Function and Concept

In research: Function and Concept appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Function and Concept in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Function and Concept is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1891 essays, Essay stubs, Linguistics stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Function and Concept outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Function and Concept in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Function and Concept means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Function and Concept out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Function and Concept in simple terms?

Why does Function and Concept matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Function and Concept?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Function and Concept.