science

Identity type

Identity type is a science 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 Identity type rather than just read about it. In short: In type theory, a branch of mathematics, the identity type represents the concept of equality. It is also known as propositional equality to differentiate it from "judgemental equality".

Key takeaways

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

Reference excerpt

In type theory, a branch of mathematics, the identity type represents the concept of equality. It is also known as propositional equality to differentiate it from "judgemental equality". Equality in type theory is a complex topic and has been the subject of research, such as the field of homotopy type theory.

Comparison with Judgemental Equality The identity type is one of 2 different notions of equality in type theory. The more fundamental notion is "judgemental equality", which is a judgement.

Beyond Judgemental Equality The identity type can do more than what judgemental equality can do. It can be used to show "for all x , x + 1 = 1 + x {\displaystyle x,x+1=1+x} ", which is impossible to show with judgemental equality. This is accomplished by using the eliminator (or "recursor") of the natural numbers, known as "R". The "R" function lets us define a new function on the natural numbers. That new function "P" is defined to be "(λ x:nat . x+1 = 1+x)". The other arguments act like the parts of an induction proof. The argument "PZ : P 0" becomes the base case "0+1 = 1+0", which is the term "refl nat 1". The argument "PS : P n → P (S n)" becomes the inductive case. Essentially, this says that when "x+1 = 1+x" has "x" replaced with a canonical value, the expression will be the same as "refl nat (x+1)".

Versions of the Identity Type The identity type is complex and is the subject of research in type theory. While every version agrees on the constructor, "refl". Their properties and eliminator functions differ dramatically. For "extensional" versions, any identity type can be converted into a judgemental equality. A computational version is known as "Axiom K" due to Thomas Streicher. These are not very popular lately.

Complexity of Identity Type Martin Hofmann and Thomas Streicher refuted the idea that type theory required all terms of the identity type to be the same. A popular branches of research into the identity type are homotopy type theory and its cubical type theory.

References

Worked examples

Example 1 — a first encounter with Identity type

Start with the simplest possible case. Write down what Identity type claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Identity type 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 Identity type 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 Identity type

In research: Identity type appears in science 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 Identity type 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: Identity type is common in secondary-school and first-year university syllabi. It links to neighbouring topics Type theory, so understanding it makes those chapters shorter.
In everyday life: Look for Identity type 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 Identity type in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Identity type 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 Identity type out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Identity type in simple terms?

In type theory, a branch of mathematics, the identity type represents the concept of equality. It is also known as propositional equality to differentiate it from "judgemental equality".

Why does Identity type matter?

Because it connects several science 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 Identity type?

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 Identity type.