Constructivity

Constructivity

One of the most striking

differences

between classical

foundations

and

type theory

is the idea of

proof relevance

, according to which mathematical statements, and even their proofs, become first-class mathematical objects.
In type theory, we

represent

mathematical statements by types, which can be regarded simultaneously as both mathematical constructions and mathematical assertions, a conception also known as

propositions as types

.
Accordingly, we can regard a term

a

:

A

as both an element of the type

A

(or in

homotopy type theory

, a point of the space

A

), and at the same time, a proof of the

proposition

A

.
To take an example, suppose we have sets

A

and

B

(

discrete spaces

),
and consider the statement “

A

is

isomorphic

to

B

.”
In type theory, this can be rendered as:

𝖨𝗌𝗈

(

A

,

B

)

:

≡

∑

(

f

:

A

→

B

)

∑

(

g

:

B

→

A

)

(

(

∏

(

x

:

A

)

g

(

f

(

x

)

)

=

x

)

×

(

∏

(

y

:

B

)

f

(

g

(

y

)

)

=

y

)

)

.

Reading the type constructors

Σ

,

Π

,

×

here as “there exists”, “for all”, and “and” respectively yields the usual formulation of “

A

and

B

are isomorphic”; on the other hand, reading them as sums and

products

yields the

type of all

isomorphisms

between

A

and

B

! To prove that

A

and

B

are isomorphic, one

constructs

a proof

p

:

𝖨𝗌𝗈

⁢

(

A

,

B

)

, which is therefore the same as constructing an isomorphism between

A

and

B

, i.e., exhibiting a pair of

functions

f

,

g

together with

proofs

that their composites are the respective

identity maps

. The latter proofs, in turn, are nothing but

homotopies

of the appropriate sorts. In this way,

proving a proposition is the same as constructing an element of some particular type.

In particular, to prove a statement of the form “

A

and

B

” is just to prove

A

and to prove

B

, i.e., to give an element of the type

A

×

B

.
And to prove that

A

implies

B

is just to find an element of

A

→

B

, i.e. a function from

A

to

B

(determining a

mapping

of proofs of

A

to proofs of

B

).
This “constructive” conception (for more on which, see

[

3

]

,

[

10

]

,

[

11

]

) is what gives type theory its good
computational character.
For

instance

, every proof that something exists carries with it enough

information

to actually find such an object; and from a proof that “

A

or

B

” holds, one can extract either a proof that

A

holds or one that

B

holds.
Thus, from every proof we can automatically extract an algorithm; this can be very useful in applications to computer programming.

However, this conception of logic does behave in ways that are unfamiliar to most mathematicians.
On one hand, a naive translation of the

axiom of choice

yields a statement that we can simply prove.
Essentially, this notion of “there exists” is strong enough to ensure that, by showing that for every

x

:

A

there exists a

y

:

B

such that

R

⁢

(

x

,

y

)

, we automatically construct a function

f

:

A

→

B

such that, for all

x

:

A

, we have

R

⁢

(

x

,

f

⁢

(

x

)

)

.

On the other hand, this notion of “or” is so strong that a naive translation of the

law of excluded middle

is

inconsistent

with the univalence axiom.
For if we assume “for all

A

, either

A

or not

A

”, then since proving “

A

” means exhibiting an element of it, we would have a uniform way of selecting an

element

from every nonempty type — a sort of

Hilbertian

choice operator

.
However, univalence implies that the element of

A

selected by such a choice operator must be

invariant

under all self-equivalences of

A

, since these are identified with self-identities and every

operation

must respect

identity

.
But clearly some types have

automorphisms

with no

fixed points

, e.g. we can swap the elements of a two-element type.

Thus, the logic of “proposition as types” suggested by traditional type theory is not the “classical” logic familiar to most mathematicians.
But it is also different from the logic sometimes called “intuitionistic”, which may lack

both

the law of excluded middle and the axiom of choice. For present purposes, it may be called

constructive logic

(but one should be aware that the terms “intuitionistic” and “constructive” are often used differently).

The computational advantages of constructive logic imply that we should not discard it lightly; but for some purposes in classical mathematics, its non-classical character can be problematic.
Many mathematicians are, of course, accustomed to rely on the law of excluded middle; while the “axiom of choice” that is available in constructive logic looks superficially

similar

to its classical namesake, but does not have all of its strong

consequences

.
Fortunately, homotopy type theory gives a finer

analysis

of this situation, allowing various different kinds of logic to coexist and intermix.

The new insight that makes this possible is that the system of all types, just like spaces in classical

homotopy theory

, is “stratified” according to the

dimensions

in which their higher homotopy

structure

exists or collapses.
In particular, Voevodsky has found a purely type-theoretic definition of

homotopy

n

-types

, corresponding to spaces with no nontrivial homotopy information above dimension

n

.
(The

0

-types are the “sets” mentioned previously as satisfying Lawvere’s axioms.)
Moreover, with

higher inductive types

, we can universally “truncate” a type into an

n

-type; in classical homotopy theory this would be its

n

th

Postnikov section.

With these notions in hand, the homotopy

(

-

1

)

-types, which we call

(mere) propositions

,

support

a logic that is much more like traditional “intuitionistic” logic.
(Classically, every

(

-

1

)

-type is empty or

contractible

; we interpret these possibilities as the truth values “false” and “true” respectively.)
The “

(

-

1

)

-truncated axiom of choice” is not automatically true, but is a strong

assumption

with the same sorts of consequences as its counterpart in classical

set theory

.
Similarly, the “

(

-

1

)

-truncated law of excluded middle” may be assumed, with many of the same consequences as in classical mathematics.
Thus, the homotopical perspective reveals that classical and constructive logic can coexist, as

endpoints

of a

spectrum

of different systems, with an

infinite

number of possibilities in between (the homotopy

n

-types for

-

1

<

n

<

∞

).
We may speak of “

𝖫𝖤𝖬

n

” and “

𝖠𝖢

n

”, with

𝖠𝖢

∞

being provable and

𝖫𝖤𝖬

∞

inconsistent with univalence, while

𝖠𝖢

-

1

and

𝖫𝖤𝖬

-

1

are the versions familiar to classical mathematicians (hence in most cases it is appropriate to assume the

subscript

(

-

1

)

when none is given). Indeed, one can even have useful systems in which only

certain

types

satisfy

such further “classical” principles, while types in general remain “constructive.”

It is worth emphasizing that

univalent foundations

does not

require

the use of constructive or

intuitionistic logic

. Most of classical mathematics which depends on the law of excluded middle and the axiom of choice can be performed in univalent foundations, simply by assuming that these two principles hold (in their proper,

(

-

1

)

-truncated, form).
However, type theory does encourage avoiding these principles when they are unnecessary, for several reasons.

First of all, every mathematician knows that a

theorem

is more powerful when proven using fewer assumptions, since it applies to more examples.
The situation with

𝖠𝖢

and

𝖫𝖤𝖬

is no different:
type theory admits many interesting “nonstandard” models, such as in sheaf toposes, where classicality principles such as

𝖠𝖢

and

𝖫𝖤𝖬

tend to fail.
Homotopy type theory admits similar models in higher toposes, such as are studied in

[

9

]

,

[

8

]

,

[

7

]

.
Thus, if we avoid using these principles, the theorems we prove will be valid internally to all such models.

Secondly, one of the additional virtues of type theory is its

computable

character.
In addition to being a foundation for mathematics, type theory is a formal theory of computation, and can be treated as a powerful programming language.
From this perspective, the rules of the system cannot be chosen arbitrarily the way set-theoretic axioms can: there must be a harmony between them which allows all proofs to be “executed” as programs.
We do not yet fully understand the new principles introduced by homotopy type theory, such as univalence and higher inductive types, from
this point of view, but the basic outlines are emerging; see, for example,

[

5

]

.
It has been known for a long time, however, that principles such as

𝖠𝖢

and

𝖫𝖤𝖬

are fundamentally antithetical to computability, since they assert baldly that certain things exist without giving any way to compute them.
Thus, avoiding them is necessary to maintain the character of type theory as a theory of computation.

Fortunately, constructive reasoning is not as hard as it may seem.
In some cases, simply by rephrasing some definitions, a theorem can be made constructive and its proof more elegant.
Moreover, in univalent foundations this seems to happen more often.
For instance:

1.

In set-theoretic foundations, at various points in homotopy theory and

category theory

one needs the axiom of choice to perform transfinite constructions.
But with higher inductive types, we can encode these constructions directly and constructively.
In particular, none of the “synthetic” homotopy theory in

http://planetmath.org/node/87582

Chapter 8 requires

𝖫𝖤𝖬

or

𝖠𝖢

.

2.

In set-theoretic foundations, the statement “every fully faithful and essentially surjective functor is an

equivalence of categories

” is

equivalent

to the axiom of choice.
But with the univalence axiom, it is just

true

; see

http://planetmath.org/node/87583

Chapter 9.

3.

In set theory, various circumlocutions are required to obtain notions of “

cardinal number

” and “

ordinal number

” which canonically represent isomorphism classes of sets and

well-ordered sets

, respectively — possibly involving the axiom of choice or the

axiom of foundation

.
But with univalence and higher inductive types, we can obtain such representatives directly by truncating the

universe

; see

http://planetmath.org/node/87584

Chapter 10.

4.

In set-theoretic foundations, the definition of the

real numbers

as

equivalence classes

of

Cauchy sequences

requires either the law of excluded middle or the axiom of (

countable

) choice to be well-behaved.
But with higher inductive types, we can give a version of this definition which is well-behaved and avoids any choice principles; see

http://planetmath.org/node/87585

Chapter 11.

Of course, these simplifications could as well be taken as evidence that the new methods will not, ultimately, prove to be really constructive. However, we emphasize again that the reader does not have to care, or worry, about

constructivity

in order to read this book. The point is that in all of the above examples, the version of the theory we give has

independent

advantages, whether or not

𝖫𝖤𝖬

and

𝖠𝖢

are assumed to be available. Constructivity, if attained, will be an added bonus.

Given this discussion of adding new principles such as univalence, higher inductive types,

𝖠𝖢

, and

𝖫𝖤𝖬

, one may wonder whether the resulting system remains consistent.
(One of the original virtues of type theory, relative to set theory, was that it can be seen to be consistent by proof-theoretic means).
As with any foundational system,

consistency

is a relative question: “consistent with respect to what?”
The short answer is that all of the constructions and axioms considered in this book have a model in the

category

of Kan complexes, due to Voevodsky

[

2

]

(see

[

6

]

for higher inductive types).
Thus, they are known to be consistent relative to ZFC (with as many

inaccessible cardinals

as we need nested

univalent

universes).
Giving a more traditionally type-theoretic account of this consistency is work in progress (see,
e.g.,

[

5

]

,

[

1

]

).

We summarize the different points of view of the type-theoretic operations in Table 1.

Types

Logic

Sets

Homotopy

A

proposition

set

space

a

:

A

proof

element

point

B

⁢

(

x

)

predicate

family of sets

fibration

b

⁢

(

x

)

:

B

⁢

(

x

)

conditional proof

family of elements

section

𝟎

,

𝟏

⊥

,

⊤

∅

,

{

∅

}

∅

,

*

A

+

B

A

∨

B

disjoint union

coproduct

A

×

B

A

∧

B

set of pairs

product space

A

→

B

A

⇒

B

set of functions

function space

∑

(

x

:

A

)

B

⁢

(

x

)

∃

x

:

A

B

⁢

(

x

)

disjoint

sum

total space

∏

(

x

:

A

)

B

⁢

(

x

)

∀

x

:

A

B

⁢

(

x

)

product

space of sections

𝖨𝖽

A

equality

=

{

(

x

,

x

)

|

x

∈

A

}

path space

A

I

Table 1:

Comparing points of view on type-theoretic operations

References

1

Bruno Barras and Thierry Coquand and Simon Huber. A

Generalization

of Takeuti-Gandy

Interpretation

.

\url

http://uf-ias-2012.wikispaces.com/file/view/semi.pdf, 2013.

2

Chris Kapulkin and Peter LeFanu Lumsdaine and Vladimir Voevodsky. The Simplicial Model of Univalent Foundations, 2012.

\href

http://arxiv.org/abs/1211.2851/arXiv:1211.2851.

3

Andrey Kolmogorov. Zur Deutung der intuitionistischen Logik.

Mathematische Zeitschrift

, 35:58–65 1932

4

Licata,Daniel R. and Harper,Robert. Canonicity for 2-dimensional type theory. In emphProceedings of the 39th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages, pages 337–348ACM, .

5

Licata,Daniel R. and Harper,Robert. Canonicity for 2-dimensional type theory. In emphProceedings of the 39th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages, pages 337–348ACM, .

6

Peter LeFanu Lumsdaine and Michael Shulman. Higher inductive types. In preparation, 2013.

7

Jacob Lurie,

Higher topos theory

Princeton University Press,2009

8

Charles Rezk. Toposes and homotopy toposes.

\url

http://www.math.uiuc.edu/ rezk/homotopy-topos-sketch.pdf, 2005.

9

Bertrand Toën and Gabriele Vezzosi. Homotopical Algebraic Geometry I: Topos theory, 2002.

\href

http://arxiv.org/abs/math/0207028/arXiv:math/0207028.

10

Troelstra,Anne Sjerp and van Dalen,Dirk,

Constructivism in mathematics. Vol. I

North-Holland Publishing Co.,1988

11

Troelstra,Anne Sjerp and van Dalen,Dirk,

Constructivism in mathematics. Vol. II

North-Holland Publishing Co.,1988

Title

Constructivity

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