basic constructions:
strong axioms
further
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
Broadly speaking, constructive mathematics is mathematics done without the principle of excluded middle (or other principles, such as the full axiom of choice, that imply it). Sometimes one adds further restrictions or alternatively adds axioms that contradict excluded middle but are otherwise consistent; these variations may be seen in the list of schools below. However, it is probably best to use more precise terms (predicativism, intuitionism, etc) in this case. Others emphasize the necessity that constructive theories be proof relevant, with denial of the excluded middle’s universality subordinate to this requirement; see (Harper 2013).
Constructivism is the philosophy that such mathematics is useful, or (more strongly) that non-constructive mathematics is wrong. Historically, constructive mathematics was first pursued explicitly by mathematicians who believed the latter. However, many modern mathematicians who do constructive mathematics do it not because of any philosophical belief about the wrongness of non-constructive mathematics, but because constructive mathematics is interesting in its own right. In the ‘pluralist’ approach to the foundations of mathematics, a constructive proof (when it exists) is better because it is valid in more versions of mathematics, but a classical proof remains valid for classical mathematics.
Another motivation for modern mathematicians —especially category theorists like those on the nLab— is that the study of constructive mathematics has potential applications to non-constructive mathematics. For example, even if one believes the principle of excluded middle to be true, the “internal” version of excluded middle in many interesting categories is still false; thus constructive mathematics can be useful in the study of such categories, even if mathematics is “globally” non-constructive. This is the neutral motivation for constructive mathematics from the nPOV.
Here we write mostly about the mathematics, trying to be mostly neutral philosophically.
During the “foundational crisis” in mathematics around the beginning of the 20th century, a number of mathematicians espoused philosophies that are generally grouped together and labeled constructivist. The common feature of these philosophies was a rejection of axioms and principles of logic that lead to nonconstructive proofs. There was much talk at the time about potential vs absolute infinity, but from an axiomatic perspective, it turns out that (if one stops short of finitism) the two main culprits are
Thus, constructivists (including many still active today) reject proofs that make use of either of these. (In fact, it was realized in 1975 by Diaconescu that the axiom of choice implies the principle of excluded middle. Excluded middle is precisely the “finitely indexed” axiom of choice; see excluded middle for a proof.)
There are, however, differences among constructivists as well, even discounting the pluralists.
Some, like Errett Bishop, simply remove choice and excluded middle from classical mathematics with nothing to replace them. However, this makes it difficult to define a satisfactory notion of continuous function, even from the real line to itself, without using locales; see (Waaldijk 2003).
Others add “non-classical” axioms which contradict choice or excluded middle, but which are consistent in their absence, such as “all total functions $[0,1] \to \mathbb{R}$ are continuous” (the continuity axiom? of the “intuitionistic” school of L. E. J. Brouwer) or “all partial functions $\mathbb{N} \to \mathbb{N}$ are computable” (the computability axiom? of the “Russian” school of A. A. Markov, which is also called “constructive recursive analysis”).
Most accept weaker versions of choice, such as countable choice or even dependent choice. Toby Bartels argues that the intuition behind accepting these justifies the (yet stronger) presentation axiom (that the category of sets has enough projectives). On the other hand, Fred Richman rejects even these limited forms of choice; see (Richman 2000).
Still others, following Hermann Weyl, go even further and refuse to allow “impredicative” constructions; see predicative mathematics. Most of the work done by the schools of Brouwer, Bishop, and Markov (but not Richman) is also predicative, even though those founders were not adamant about it; as a result, predicativism often appears in the foundations of constructive mathematics (in particular, in those of Aczel and Martin-Löf, but not those of Friedman or Coquand).
Very extreme constructivists, like Doron Zeilberger, reject the existence of infinite sets; see finitism. (Kronecker, the ‘father of constructivism’, is often considered a finitist, but as he came before the foundational crisis, it is difficult to classify him accurately.) Ironically, this means that excluded middle and choice, in their naïve forms, may become acceptable, or even true. (Even constructivists believe that they are true in the category FinSet of finite sets.)
The most extreme position of all is to deny even the existence of very large finite sets. This is called ultrafinitism. This has been studied largely by Alexander Yesenin-Volpin? and (more recently) Edward Nelson. Zeilberger also claims sympathy with ultrafinitist philosophy, although his actual work fits well into a (merely) finitist framework.
Many constructivists (like many classical mathematicians) believe in an absolute mathematical sense of “truth,” and that in this sense choice and excluded middle are simply wrong. (Some constructivists, using classically false axioms, can even refute them; others merely claim that no possible correct reasoning could ever prove them. See Truth versus assertability below.) To most mathematicians, this makes them seem quite strange. Other constructivists adopt a wait-and-see attitude, or even a relative notion of truth (which can seem strange in another way).
With the invention of topos theory in the second half of the 20th century, a new sort of constructivism arose. It was observed (by Lawvere and others) that any topos with a natural numbers object has an internal logic which is powerful enough to interpret most of mathematics, but that this logic in general fails to satisfy choice and excluded middle. This meant that even for a mathematican who likes to use choice and excluded middle (and a fortiori for one who believes them to be “true”), there is a reason to care about what can be proven without them, because only if a proof is constructive can it be interpreted in an arbitrary topos with NNO.
Even starting from a “completely classical” world of mathematics, many interesting toposes arise naturally (such as the category of sheaves on any topological space, or more generally any site) whose internal logic is not classical logic. Then by internal reasoning in such a topos (which is constructive reasoning), one can prove various useful facts, which can then be reinterpreted as external statements about the behavior of the topos itself. For example, the constructive theorem that every one-sided real number that is both a lower real and an upper real must be located (which, classically, is an utter triviality) becomes the theorem that any semicontinuous function (from any topological space to the real line) that is both upper and lower semicontinuous must be continuous (which is at least somewhat nontrivial).
By now it is known that many of the non-classical axioms used by the early constructivists have natural models in particular toposes. For instance, in the topos of sheaves on the real numbers $R$, the continuity axiom? that “all total functions $R \to R$ are continuous” holds. And in the effective topos, the computability axiom? that “all partial functions $N\to N$ are computable” holds.
However, there are no non-classical (or classical) axioms beyond “pure constructivism” that are true in all toposes with NNO. In particular, there is a free topos with NNO such that a statement is true in the free topos precisely when it is provable in pure (Richman-school) constructive mathematics. This means that for an argument to apply in all toposes, even mild assumptions such as countable or dependent choice are unacceptable (but impredicativity is fine). However, topos theory has also provided ideas that solve many of the problems with pure constructivism. For example, a well-behaved notion of “continuous function” can be recovered by using locales instead of topological spaces, which was first discovered in the context of toposes and is closely related to them in any case.
Sometimes, all that is necessary to make a piece of mathematics constructive is careful use of language. It is common in classical mathematics to define things with an unnecessary amount of negation. This doesn’t work so well constructively, since a statement can be not false without being true, but we can sometimes do perfectly well by just removing unnecessary double negations.
For example, classically one often speaks about “nonempty” sets, meaning a set which “does not have no elements.” Constructively it is much better to say that a set “does have at least one element”; constructivists often call such a set inhabited to remind themselves that it is a “positive” notion to replace the negative one of “nonempty”. Others continue to use the word “non-empty” but understand it as a term of art that really means “inhabited”.
On the other hand, differences in axiomatization or definition that make no difference classically can result in actual differences in behavior constructively. Therefore, classically equivalent notions often “bifurcate” (or “trifurcate” or worse) into multiple inequivalent constructive ones. This tends to happen whenever a concept involves negation and, to a lesser degree, disjunction and existential quantification. In some cases there is a “correct” constructive version of the definition, although it may take some thought to uncover it, but in many cases more than one of the resulting concepts is important and useful. For example:
There are multiple inequivalent constructive definitions of a field, because of the axioms “every nonzero element has an inverse” and $0 \neq 1$.
In pure constructive logic, the Dedekind real numbers and the Cauchy real numbers need no longer coincide. The Cauchy reals sit inside the Dedekind reals, but in general not every Dedekind real need be approximable by a sequence of rationals. From a topos-theoretic viewpoint, the Dedekind reals are usually the “correct” notion to study (if not the locale of real numbers as a whole). However, either excluded middle or countable choice suffices to ensure (via weak countable choice) that every Dedekind real is a Cauchy real, and hence that the two notions coincide; see (Bridges et al, 1998).
There are at least three different constructive notions of ordinal number; see (Taylor 1996) and (Joyal–Moerdijk 1995).
Without the axiom of choice, functors and anafunctors become distinct, and often the latter is more appropriate; see (Makkai, 1996).
Perhaps most disturbingly of all to the classical mathematician, one must distinguish between finite sets, subfinite sets, finitely-indexed sets, and even subfinitely indexed sets; see finite set for definitions. However, in practice it is usually either finite or finitely-indexed sets that are important, and with practice a little bit of thought suffices to show which is the relevant concept.
This allows one to translate classically valid theorems into intuitionistically valid theorems. See double negation translation.
Already in classical mathematics, there is a difference between saying that something is true and saying that something is provable. If you adopt ZFC because you believe it correct, then presumably you believe that ZFC is consistent even though (assuming that you're aware of certain theorems) you also know that you cannot prove it so. In that case, you also believe that the continuum hypothesis (for example) is either true or false; you may or may not have an opinion on which it is, but in any case again you know that you cannot prove it either way.
A constructive mathematician can be even subtler. If you belong to the Bishop school, then you accept no classically false axioms, and anything that you can prove is valid also in classical mathematics. Even so, you can believe that the principle of excluded middle is false (even though, of course, you know that you cannot prove it false). So you can really confuse the classical mathematicians by saying, on the one hand, that they can safely accept all of your theorems as valid, and then saying, on the other hand, that you know excluded middle to be false. The resolution, of course, is that you never claimed that this was a theorem.
This way of talking has even been formalised in (Bishop, 1967) with a convention adopted by many (but not all) of his followers. In this convention, the word ‘not’, used normally in a vernacular sentence whose mathematical content would otherwise be the proposition $p$, changes to the content to $\neg p$, or $p \to \bot$, as usual. (This follows all of the rules of intuitionistic logic; in particular, any statement that is ‘not’ true is false (just as in classical logic).) However, the word ‘not’, in italics as shown here, changes the content to $p \to x$, where $x$ is some statement that is known (although not proved!), according to Bishop, to be false. Now a statement that is ‘not’ true may be false, but this may be unknown, and it is even possible that it is also ‘not’ false (so $\neg p \to x$, possibly for a different $x$).
Bishop gives in his introduction several statements, suitable for use as $x$ above, including:
At one point in his book, while discussing the possibility of a pointwise-continuous function $[0,1] \to R$ that is not uniformly continuous (a Specker function?, whose existence Markov's school claims as a theorem), Bishop seems to assert that this theorem is both not true and not false; he does not put it this way, so this may not be exactly what he meant, but there is no contradiction if it is. (But it is a contradiction, even in intuitionistic logic, if a statment is both not true and not false; indeed, a definition of ‘false’ may be taken to be ‘not true’.)
This practice can be understood through a careful distinction between object language and metalanguage. A mathematical statement $p$ (such as the continuum hypothesis, or that a Specker function exists) belongs to the object language, as does $\neg p$ or $p \to x$ for any specific mathematical statement $x$. But the statement that $p$ is provable in some formal system belongs to the metalanguage (although it can also phrased internal to any object language suitable for mathematics), and as such may be written $\vdash p$. The metalanguage has its own logic (the metalogic, which for the sake of argument we may even assume to be classical), but notice that $\neg \vdash p$ and $\vdash \neg p$ are different; the first claims that $p$ cannot be proved, while the second claims that $p$ can be refuted, which (in any consistent? formal system, even a classical one) is strictly stronger. Although Bishop did not commit to any formal system, if we assume for the sake of argument that he settled on one, then we may write $\vdash \neg p$ as our interpretation of his meaning when he asserts ‘not $p$’ but $\neg \vdash p$ as his meaning when he asserts ‘not $p$’.
Even without the notational convention of the Bishop school, it is important when reading constructive mathematics to remember the difference between what can be refuted (proved false) and what merely cannot be proved true. Although Brouwer's and Markov's schools will sometimes claim to prove statements that are (classically) outright false (such as that every function $[0,1] \to R$ is pointwise-continuous), it is more common (and can happen in any school) that a constructive mathematician makes a claim that merely sounds false, when what they really mean is only that they don't accept what you say as true. Often modal phrases like ‘not necessarily’ will be used where Bishop would use ‘not’, as a clue that we're shifting to a metalanguage, or at least merely remaining agnostic rather than outright disagreeing.
The distinction between object language and metalanguage exists even in classical mathematics, but it seems that most classical mathematicians are not used to remembering it, although it is not entirely clear why this should be so. One possibly relevant observation is that even if $P$ is a statement which is neither provable nor disprovable (like the continuum hypothesis), in classical mathematics it is still provable that “$P$ is either true or false.” Moreover, classical model theory often restricts itself to two-valued model?s in which the only truth values are “true” and “false,” although classical logic still holds in Boolean-valued models, and in such a case the truth value of $P$ may be neither “true” nor “false,” although the truth value of “$P$ or not $P$” is still “true.” Certainly when talking about classical truths which fail constructively, such as excluded middle, it is important to remember that “fails to be provable” is different from “is provably false.”
See also intuitionistic mathematics and realizability and computability.
Concepts that usually arise in constructive mathematics, often because they are classically trivial:
Some of these are also useful internally or even classically.
Topics relevant to the foundations of constructive mathematics:
Other articles with content relating to constructive mathematics (rather incomplete):
In principle, every article could explain how it applies to constructive mathematics, but that will probably never happen.
There is also
Gentle introductions include
Douglas Bridges, Introducing constructive mathematics, Lecture Notes, Nis 2013 (pdf)
Andrej Bauer, Five Stages of Accepting Constructive Mathematics, talk at IAS March 18, 2013 (video), Bull. Amer. Math. Soc., 2016 (publisher)
Fred Richman, Interview with a constructive mathematician, Modern Logic 6 (1996), no. 3, 247–271 (MathSciNet)
Ingo Blechschmidt, Double-negation translation and CPS transforms, 2015 (pdf)
Stanford Encyclopedia of Philosophy, Constructive mathematics
An more technical introduction to constructive reasoning in mathematics is in the introduction of
Other references include the following.
Errett Bishop (1967). Foundations of Constructive Analysis. Rewritten with Douglas Bridges in 1985 as Constructive Analysis.
Douglas Bridges, Fred Richman, Peter Schuster (1998). A weak countable choice principle. Available from Fred Richman’s Documents.
Michael Makkai (1996). Avoiding the axiom of choice in general category theory.
Fred Richman (2000). Constructive mathematics without choice.
Paul Taylor (1996). Intuitionistic Sets and Ordinals. Available (with several other references) at Induction, recursion, replacement and the ordinals.
André Joyal and Ieke Moerdijk (1995). Algebraic set theory.
Frank Waaldijk (2003). On the foundations of constructive mathematics - especially in relation to the theory of continuous functions (PDF).
For an emphasis on proof relevance, see:
Most books on topos theory include some discussion of toposes' internal constructive logic. One good reference is:
A historical account is in
The relation to realizability and computability is discussed in
See also the references at intuitionistic mathematics for more.
In
the following comment about mathematical proof (in section 12 Historical and mathematical proof of the Preface) might be read as being a complaint about the traditional non-constructive concept of proof and about the traditional lack of proof relevance:
All the same, while proof is essential in the case of mathematical knowledge, it still does not have the significance and nature of being a moment in the result itself; the proof is over when we get the result, and has disappeared. The process of mathematical proof does not belong to the object; it is a function that takes place outside the matter in hand.
footnote 42: Mathematical truths are not thought to be known unless proved true. Their demonstrations are not, however, kept as parts of what they prove, but are only our subjective means towards knowing the latter. In philosophy, however, consequences always form part of the essence made manifest in them, which returns to itself in such expressions.
See also earlier conceptions of proofs expressing the ‘cause’ of a theorem, where reductio proofs in particular were taken generally to fail. Such an idea goes back to Aristotle for whom a proper answer to the question “Why is the angle in a semicircle a right-angle?” gives its cause.