The foundational crisis in mathematics along with roughly four decades following it, was likely the most fertile period in the history of logic and studies in the foundations. After discovering the set-theoretic paradoxes, such as the paradox of the set of all sets, together with the logical ones, like Russell’s paradox, mathematicians, realizing that the naïve set teory, or Frege’s Grundlagen der Arithmetik cannot do for a consistent basis for mathematics, began to seek for another, more solid foundation. This came also with the justified skepticism in regard to the means employed by Georg Cantor and Richard Dedekind in their work in set theory and analysis respectively. In this article I set out to tell a story of various paths taken by mathematicians in the first half of the 20th century in an attempt to set boundaries on the hitherto unrestricted Cantor’s paradise. My aim is to distinguish as clearly as I can between the tags like “constructivism”, “finitism”, “intuitionism”, since they have often been conflated and I struggled for some time with understanding their boundaries myself. For example, in the first half of the last century the terms “intuitionism” and “finitism” were used interchangeably, so if we want to rely only on the original publications, they will surely confuse us sometimes.

I believe that Georg Cantor can be called, paradoxically, the grandfather of explicit constructivist thought. Although this attitude is as old as mathematics itself, it was Cantor’s work that directly provoked the plain-spoken constructivism of some mathematicians of his time and later generations. And it is his work in transfinite set theory that remains the main target of any constructivist critique. But what was the flashpoint of the entire dispute?

The Motivation

In mathematics, when proving some statement, one can work on a special case; arbitrarily high, but finite instance; or the general case. I should add that the general case is not always the infinite one: it is always restricted to the domain of definition in question, which can be limited by assumptions that exclude the infinite case. It was Cantor who began to treat infinity as a completed unity and started working on the infinitude of reals. In his early work in trigonometric series, he was able to form new methods and concepts, when existing approaches to some problems failed. This lead him to his own definition of a rational number and later of a number in general. His inventiveness and independency gave birth to set theory, which originated in the philosophical nature of his mind: Cantor came up with the theory when looking for further generalisations of his theorems. But, even before the creation of the theory of transfinite numbers, his drive towards higher generalisations had enemies. Even among his co-workers, like in the case of Kronecker. Cantor brought about new methods which implied the existence of mathematical objects not only controversial for methodological and philosophical reasons, but also bearing contradictory effects. The discussion about the foundations of mathematics that ensued showed roughly three strategies to counter his ideas: (1) “Cantor’s theory isn’t how maths should be done” (methodological reasons); (2) “Mathematical objects do not exist in the way they are established by Cantor” (methodological and philosophical reasons); (3) “Infinity understood as a completed totality is unintelligible and should be kept out from mathematical considerations” (philosophical reasons). We will see how each constructivistic stance tackles the issue.

Constructivism

“Constructivism” has two meanings, a general and a more technical one. The former pertains to the set of loosely connected views that share the distrust towards treating infinity as a completed whole. All of the below positions fall under that category. But they vary in reasons for this distrust: some have certain methodological claims that imply such account (“technical” constructivism, predicativism and some accounts of finitism), others repudiate Cantorian style of doing mathematics more for philosophical reasons (intuitionism, other accounts of finitism).

“Constructivism” in its technical meaning directly refers to a method in which mathematics should be done. It claims that an assertion about the existence of some object should by its proof give us a method for constructing such an object. Objects should be constructed or computed, and we should know how to “find” them in the mathematical realm, not only know that they exist somewhere out there without our having a clear example of their instantiation. Hence, a proof by contradiction is repudiated as a proof method. That is because it simply does not tell us how to construct an object, but merely that its non-existence would becontradictory.

It can be stated without controversy that throughout its development, mathematics has remained mainly constructive in the above meaning right until the last quarter of the 19th century, with the notable exception of geometry where proof by contradiction was universally accepted and applied even before Euclid.

But the reflection about the meaningful limits of mathematics was for the most part of the history in the subconscious. Regarding the question about the nature of infinity, it is known that the Greeks always understood it potentially and not actually, i. e. as a completed whole. Maybe the first known mathematician to explicitly reflect upon those two types of construing infinity was none other than Gauss. He famously said that

I must protest against the use of an infinite magnitude as a completed quantity, which is never allowed in mathematics. The Infinite is just a manner of speaking, in which one is really talking in terms of limits, which certain ratios may approach as close as one wishes, while others may be allowed to increase without restriction.

He seemed to be aware of the problems that bringing infinity to mathematics might cuse. But Cantor went in the downright opposite direction. And it was maybe his results that induced a more explicit account of constructivism.

His teacher from Berlin, Leopold Kronecker (1823–1891) can be described as the first conscious constructivist. For him, only the natural numbers were “God-given” and all other objects should be explained in terms of naturals. Assertions about existence must be backed by constructions, and the properties of numbers should be decidable in finitely many steps (in the spirit of finitism). Kronecker vitally opposed Cantor’s set theory, but he demonstrated his attitude more by example than by a clear discussion of his tenets.

Finitism

Finitism can be described as a project aiming at interpreting mathematics in terms of the most tangible terms or translating the abstract in terms of concrete. Its base is the concept of the natural number, a finite, concretely representable structure. This approach of founding mathematics then entails the acceptance of definition by recursion and proof by induction — the structure of natural numbers is after all produced by applying the simple successor function to the last element.

It is clear why finitists don’t like quantifiers: statements like ∀x φx or ∃x φx do not really provide us with a meaningful example of an object x that has the property φ. The repudiation of quantified notation is where the two tenets of constructivism meet: the existential quantifier is done away with because it does not give any construction of the object whose existence is claimed, whereas the universal quantifier is rejected because it is (at most occasions) ranging over the domain unacceptable to be understood as a completed whole, i. e. the infinite set of sentences or objects. Fortunately, finitism has found its way around that issue.

It would not be an overstatement if I said that the greatest pioneer and champion of finitistic methods was the Norwegian mathematician, Thoralf Skolem (1887-1963). His greatest contribution to the field (but by no means the only one) is probably the method for elimination of quantifiers from the statements of the first order predicate logic, now called Skolemization. Thanks to his technique, statements with quantifiers can be interpreted in terms of quantifier-free statements. In this way, ∃x φx is understood only as a partial communication that is to be supplemented by showing an exact x that satisfies φ. ~∀x φx finitistically means: providing a particular x such that φx does not hold.

Moreover, Skolem showed further that a substantial part of arithmetic can be done in the calculus without quantifiers. Functions in such calculus are introduced by primitive recursion. Since Skolem did not present his results in a formal context, we had to wait until H. Curry and R. Goodstein formulated a purely equational formalism, the primitive recursive arithmetic (PRA), in 1941. It can be called a first conservative finitistic formal system.

Skolem’s most significant results in foundations of mathematics can be found in the following publications:

• Skolem, Th. (1920), “Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theorem über dichte Mengen“, Skrifter utgit av Videnskapsselskapet i Kristiania. I, Matematisk-naturvidenskabelig klasse, no. 4, pp. 1­–36.
• Skolem, Th. (1923), “Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich“, Skrifter utgit av Videnskapsselskapet i Kristiania. I, Matematisk-naturvidenskabelig klasse, no. 6, pp. 1–38.
• van Heijenoort, J. ed. (1967) “From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931”, Harvard University Press.

Finitism is not so limiting as one might think. Goodstein took finitism beyond Skolem’s achievements and showed how some parts of analysis can be done through finitist means. Furthermore, it has been shown in the second half of the 20th century that large parts of mathematics have a finitistic foundation. The results were of the form: given a particular weak formal system S with strong expressive power, S can be shown to be consistent by methods expressible in PRA, which implies that S is conservative over PRA.

Skolem was repeatedly cited by Hilbert in the latter’s publications on his Program for a particular reason. Hilbert showed an ambitious approach towards the problem of finding a plausible foundational theory for mathematics. He believed that set theory is the best one and should not be rejected. He did not want to throw out any methods from the mathematician’s toolkit, but aspired to found the transfinite mathematics on the more reliable, so to speak, finite mathematics. He said that

No one shall expel us from the paradise that Cantor has created for us.

He understood and recognized the significance of the problems stemming from the primary form of the theory, but did not want to throw the baby out with bathwater, rather find a way to convince other, more skeptic mathematicians that it offered valuable conceptual tools for a “universal language of mathematics”. He criticized Kronecker as a dogmatist (for taking the concept of an integer as a primitive, undefinable dogma), and praised Frege, Dedekind and Cantor for their, as he said, sagacious approach (1904) to the issue, while pointing out that it still needs clarification and rescue from the lurking paradox. His strategy was as follows.

First he wanted to take mathematics into the higher level of abstraction by “translating” its contents into a formal system (axiomatizing it, that is), so that no meaning of the signs comprising mathematical statements would be taken into consideration. He wanted to check the reliability of mathematical tools on the most concrete level — no metphores, no interpretation, no semantics, only the sign and its occurrence in proofs. He then wanted to show the consistency of such a system in the finitistic means.

We should note that Hilbert was not a constructivist himself. According to him, the sole condition of existence in mathematics was freedom from contradiction, not having a construction. He wanted to show the consistency of, for example, Peano Arithmetic (the most popular formalization of basic arithmetic) exactly to establish the mathematical existence of that system. Similarly, he believed that fundamental concepts of set theory, in particular Cantor’s alephs, have a consistent existence.

So, not being a finitist, Hilbert initiated a fertile research in finitistic means of proving. Proof theory, theory of recursion, incompleteness theorems, “Reverse Mathematics”— all have been initiated or inspired by Hilbert’s program.

Actualism or Ultrafinitism

It should be noted that some mathematicians disbelieved even the intelligibility of the notion of a natural number. The claim was that, as far as we can conceptualize the number four, it is much harder to get to grips with much bigger numbers, like 9^9 ^9. They claimed that even the idea of a natural number involves a strong idealization of “concretely representable”. Such idealization is implicit in the assumption that all natural numbers are constructions of the same kind. In fact, we cannot handle understanding such big numbers without a general concept of exponentiation. Those mathematicians, calling themselves actualists, ultra-finitists or ultra-intuitionists, claimed that mathematics should restrict itself only to numbers that can be actually realized (in our imagination or in nature).

The first to argue for the actualist program was Alexander Yessenin-Volpin, who sketched out a project of proving the consistency of ZF obly through ultra-finitistic means. But the program in general has not seen much development since, since there appear to be inherent difficulties with actualism.

However, the idea has motivated some to investigate the basic arithmetical operations in terms of their actualist, finitist, or transfinite nature. R. Parikh has indicated in 1971, by technical results, that there is a substantial difference in character between addition and multiplication on the one hand and exponentiation on the other.

Predicativism

It is a school a bit older than the above two. It might be the first more specific offshoot of constructivism, and the forerunner of intuitionism. It is closely associated with the figure of the renowned French mathematician, Henri Poincaré. Poincaré’s input into the discussion about the foundations of mathematics is significant for two reasons.

First, he was one of the first to indicate the key role the intuition has in mathematics, in particular the “intuition of pure number”, as he called it. This intuition justifies the principle of induction on the naturals. This means that he believed that this principle is a priori synthetic, i. e. neither tautological (justified by pure logic), nor empirical. It stems, he believed, from our basic, intuitive understanding of the notion of number. Hence, Poincaré allowed for potential infinity of naturals. In this respect, he fully agrees with Brouwer.

Second, he had a sensible and interesting idea about the cause of the set-theoretic paradoxes. According to him, they were due to impredicative definitions, which trigger a vicious circle. An impredivative definition is such that defines an object O by referring to a totality S, whose O is simultaneously a member. His example of impredicativity in motion was the Richard’s paradox: let S be the totality of all real numbers given by infinite decimal fractions, definable in finitely many words. S is countable (because of the latter assumption), so by Cantor’s diagonal argument (neatly explained here) one can define a real number O that is not an element of S. But O has been defined in finitely many words! Here Poincaré indicates that the definition of O as an element of S refers to S itself and is therefore impredicative.

Predicativism may be described as constructivism with respect to definitions of sets (but not with respect to the use of logic): sets are constructed “from below”, not by singling them out from a greater totality conceived as existing previously.

Poincaré’s foundational ideas have inspired semi-intuitionists and can be found in the following publications (mainly in French):

• Mooij, J.J.A. (1966) “La philosophie mathématique d’Henri Poincaré” https://www.persee.fr/doc/rhs_0048-7996_1967_num_20_4_2545_t1_0394_0000_1
• Poincaré, H. (1902) Science and Hypothesishttps://www.amazon.com/Science-Hypothesis-Henri-Poincare/dp/1475091567

Intuitionism

If it wasn’t for the foundations of computer science originating form the research in finitism, intuitionism would probably be the most prolific and influential school of thought among the “constructivisms”. Nevertheless, it is a position with a serious philosophical background that attracted various logicians and mathematicians who sought for more than a methodological rationale for mathematical practice. And indeed, intuitionism prides itself on likely the most developed and consistent account among all of the discussed positions.

Luitzen Egbertus Jan Brouwer (1881–1966) was the founder of intuitionism. It seems to me that his foundational stance is a direct extension of his broader philosophical view. He expressed it in a short essay entitled “Life, Art and Mysticism” (published in 1905) articulating a deeply pessimistic worldview, soaked in solipsism, epistemic anti-realism and stressing the importance of the individual. In his work, he stated that mathematical enterprise is one of the ways to freeing oneself from the bonds of the mundane and wordly. The individual can commit to mathematical endeavor thanks to the intuition of certain mathematical objects, which underlies entire mathematics, he believed. Some critics of Brouwer argued that his idea of a cognizing subject — the idealizedmathematician — is described like some kind of a mystic, connecting to the realm of mathematics. And it is understandable, given Brouwer’s philosophical tenets in his philosophy of mathematics, which he called “intuitionistic acts” (doesn't this sound like a credo of sorts?):

Mathematics is not formal. The objects of mathematics are mental constructions in the mind of the (ideal) mathematician. Only the thought construction of the (idealized) mathematician are exact.

Mathematics is independent of the experience in the outside world, and it is in principle also independent of language. Communication by language may serve to suggest thought constructions to others, but there is no guarantee that these constructions are the same.

Mathematics does not depend on logic; on the contrary, logic is a part of mathematics.

We see how radically Brouwer rejected both formalism and logicism: he explicitly repudiated the main principles of these positions. More than that, the separation goes deeper, as the “revised” formalism of the post-Gödelian times, which claimed that mathematics is just some sort of a language is also clearly anticipated and rejected; on the other hand, Platonism of any kind is similarly cut off, since the existence of mathematical objects depends on their being constructed (they are not eternal and independent of the mind).

When Brouwer formed his intuitionistic views he was still a practicing topologist and only a couple years later (probably about 1908) did he realize that his stance required the revision of classical mathematics along with its logic. The consequence was the rejection of

Law of Excluded Middle: p ∨ ~p

and its classical equivalent, the Principle of Double Negation: ~(~p) → p

And with them, of course, the method of proving by contradiction. The reason was that, say we know that the statement p does not hold. The intuitionistic claim is that with only that knowledge we cannot conclude that ~p holds, since we do not have a constructive proof of the latter.

Regarding the mentioned concept of intuition, Brouwer explains that our mathematical intuition comes from the intuition of time:

mathematics is an […] activity of the mind having its origin in the perception of a move of time, i.e. of the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the two-ity thus born is divested of all quality, there remains the empty form of the common substratum of all two-ities. It is this common substratum, this empty form which is the basic intuition of mathematics. (Brouwer, 1952)

And he adds:

This empty two-ity and the two unities of which it is composed, constitute the basic mathematical systems. (ibid.)

One could say that this idea of two-ity founds the structure of ever proceeding numbers, i. e. the naturals. So we see that here, infinity is also regarded only potentially. On the other hand, agreeing with Poincaré, the principle of induction (on naturals) is fully acceptable from an intuitionistic point of view.

When it comes to the real numbers, according to Brouwer human mind cannot ever construct a totality like continuum:

The continuum as a whole was intuitively given to us; a construction of continuum, an act which would create “all” its parts as individualized by the mathematical intuition is unthinkable and impossible. The mathematical intuition is not capable of creating other than countable quantities in an individualized way. (Brouwer, 1907)

The idealized mathematician, the key part of the philosophical landscape of intuitionism, is a curious thing. I believe that it is a very neat idea to reconcile realism and anti-realism in philosophy of mathematics. On the one hand, Brouwer stressed that mathematics is a free activity of the mind, and completely depends on the mental constructions of the subject, so it was common to hear realist critiques that his intuitionism relativises mathematical truth and leaves it to the whim of a mathematician. But, on the other hand, the idealizedmathematician is in a way an ultimately rational creature, or should be understood as mathematical knowledge of humanity as a whole, so, by that disclaimer, Brouwer ensures that the mental constructions of such idealized subject are determined to capture what’s mathematically true. Hence, summing up, we could describe the intuitionism’s cognizing subject to be fully free to construct whatever she wants, but she is “destined” to construct only the correct (whatever that means) constructions. Like a God who is volitionally free, but will only do what’s right. Well, though I believe I have some substantial literature to back this up, this is still my interpretation of Brouwer (so feel free to indicate possible inconsistencies with him).

However strange, captivating and deep intuitionistic philosophy is, it would probably not last longer than until the 40s, if it wasn’t for Brouwer’s doctoral student and developer of intuitionistic program, Arend Heyting. But, as this is supposed to be just a beginner’s guide, I will save his achievements for later.

Brouwer’s foundational writings may be found in the following publications (though it is not by any means a comprehensive list):

• Brouwer, L. E. J., Collected Works, Vol. I, Amsterdam: North-Holland, 1975.
• Brouwer, L. E. J., Collected Works, Vol. II, Amsterdam: North-Holland, 1976.

Throughout writing essay I have also strongly drawn from:

Troelstra, A. M. (1991) “History of Constructivism in the Twentieth Century”, ITLI Prepublications for Mathematical Logic and Foundations.