What is intuitionistic mathematics?
What are its claims, and what are their justifications?
1a. Intuitionism as a philosophy.
L. E. J. Brouwer is credited as the originator of intuitionistic mathematics. A primary source in which he expounds his view, and perhaps the closest one can get to a definitive, formal, and technical definition, is here:
Luitzen Egbertus Ian Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind.
(Note: I don’t see any materials about intuitionistic philosophy except as a philosophy of mathematics, so this statement sounds oddly like there is some kind of intuitionistic philosophy that wasn’t originally about mathematics. Are there other intuitionistic fields? Is intuitionistic ethics related?)
The gradual transformation of the mechanism of mathematical thought is a consequence of the modifications which, in the course of history, have come about in the prevailing philosophical ideas, firstly concerning the origin of mathematical certainty, secondly concerning the delimitation of the object of mathematical science.
Brouwer makes it clear that his philosophical vision is to do just that: determine the origin of mathematical certain, and delimit the scope of what mathematics is and can be about, and can do.
In this respect we can remark that in spite of the continual trend from object to subject of the place ascribed by philosophers to time and space in the subject-object medium, the belief in the existence of immutable properties of time and space, properties independent of experience and of language, remained well-nigh intact far into the nineteenth century.
To make this clear, Brouwer seems to be saying that although modern philosophy posits time and space more as products of the human mind than before (á la Kant), the retrograde beliefs were still lurking in various corners, embedded in the thought of his day.
To obtain exact knowledge of these properties, called mathematics, the following means were usually tried: some very familiar regularities of outer or inner experience of time and space were postulated to be invariable, either exactly, or at any rate with any attainable degree of approximation. They were called axioms and put into language.
Brouwer says that humans observed properties of time and space, and then formulated those properties as laws, or axioms - like Newton’s laws of physics, or the idea of space and time being modeled in a Cartesian place with real number coordinates.
Thereupon systems of more complicated properties were developed from the linguistic substratum of the axioms by means of reasoning guided by experience, but linguistically following and using the principles of classical logic. We will call the standpoint governing this mode of thinking and working the observational standpoint, and the long period characterized by this standpoint the observational period. It considered logic as autonomous, and mathematics as (if not existentially, yet functionally) dependent on logic.
Thus, according to Brouwer, logic appeared to be taken as self-evident, and math was completely subservient to and arising from logic. (This sounds to me like something Bertrand Russell would support.)
For space the observational standpoint became untenable when, in the course of the nineteenth and the beginning of the twentieth centuries, at the hand of a series of discoveries with which the names of Lobatchefsky, Bolyai, Riemann, Cayley, Klein, Hilbert, Einstein, Levi-Cività and Hahn are associated, mathematics was gradually transformed into a mere science of numbers; and when besides observational space a great number of other spaces, sometimes exclusively originating from logical speculations, with properties distinct from the traditional, but no less beautiful, had found their arithmetical realization.
I do not understand why Brouwer thinks the observational point of view on space became untenable once space was reduced to a science of numbers.
Consequently the science of classical (Euclidean, three-dimensional) space had to continue its existence as a chapter without priority, on the one hand of the aforesaid (exact) science of numbers, on the other hand (as applied mathematics) of (naturally approximative) descriptive natural science.
I think this might be saying that on the one hand, Euclidean space could be derived from certain numerical properties or systems, and yet, scientifically, people were assuming that it also described the properties of the actual physical, spatial medium / world they lived in. Brouwer thinks this is philosophically weak, like not taking a clear position, a meaningless compromise.
In this process of extending the domain of geometry, an important part had been played by the logico-linguistic method, which operated on words by means of logical rules, sometimes without any guidance from experience and sometimes even starting from axioms framed independently of experience.
I don’t fully understand this, but clearly, Brouwer seems to think math can not just be any set of axioms you choose - he is looking for the right axioms, which describe our actual world - and he believes that this is evident by analyzing human thought itself (given that fundamental categories of the world and existence, like space and time, are properties of the mind, in his view, like Kant).
Encouraged by this the Old Formalist School (Dedekind, Cantor, Peano, Hilbert, Russell, Zermelo, Couturat), for the purpose of a rigorous treatment of mathematics and logic (though not for the purpose of furnishing objects of investigation to these sciences), finally rejected any elements extraneous to language, thus divesting logic and mathematics of their essential difference in character, as well as of their autonomy.
Here Brouwer says that formalists fully embraced the idea of math and logic as a symbolic game, valid under any chosen rule-set - but they forgot to try to characterize the actual world we live in, so their math wasn’t helpful for natural science (I think), or deep results about the truths in our actual world, even mathematical ones. I find this to be a kind of mathematical realism, the idea that proofs in math are not just hypothetically relative to their axioms, but that there is a singular truth we actually wish to know; thus, we should reject other false axiom systems, since they are not really “real”.
However, the hope originally fostered by this school that mathematical science erected according to these principles would be crowned one day with a proof of its non-contradictority was never fulfilled, and nowadays, after the logical investigations performed in the last few decades, we may assume that this hope has been relinquished universally.
(Note: this was in the 1940’s and 50’s, post-Gödel’s incompleteness theorem.)
Perhaps this is a critique of my belief that mathematics and logic is “necessarily necessary”: the incompleteness theorem ruins the hope for a perfect, pure theory of completely sufficient and necessary and universal principles of reason?
Of a totally different orientation was the Pre-intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the introduction of natural numbers, for the principle of complete induction, and for all mathematical entities springing from this source without the intervention of axioms of existence, hence for what might be called the 'separable' parts of arithmetic and of algebra.
For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language and logic and regarded its non-contradictority as certain, even without logical proof. For the continuum, however, they seem not to have sought an origin strictly extraneous to language and logic. On some occasions they seem to have contented themselves with an ever-unfinished and ever-denumerable species of real numbers' generated by an ever-unfinished and ever-denumerable species of laws defining convergent infinite sequences of rational numbers.
However, such an ever-unfinished and ever-denumerable species of real numbers is incapable of fulfilling the mathematical function of the continuum for the simple reason that it cannot have a positive measure. On other occasions they seem to have introduced the continuum by having recourse to some logical axiom of existence, such as the 'axiom of ordinal connectedness', or the 'axiom of completeness', without either sensory or epistemological evidence.
In both cases in their further development of mathematics they continued to apply classical logic, including the principium tertii exclusi, without reserve and independently of experience. This was done regardless of the fact that the non-contradictority of systems thus constructed had become doubtful by the discovery of the well-known logico-mathematical antonomies.
In point of fact, pre-intuitionism seems to have maintained on the one hand the essential difference in character between logic and mathematics, and on the other hand the autonomy of logic, and of a part of mathematics. The rest of mathematics became dependent on these two. Meanwhile, under the pressure of well-founded criticism exerted upon old formalism, Hilbert founded the New Formalist School, which postulated existence and exactness independent of language not for proper mathematics but for meta-mathematics,
which is the scientific consideration of the symbols occurring in perfected mathematical language, and of the rules of manipulation of these symbols. On this basis new formalism, in contrast to old formalism, in confesso made primordial practical use of the intuition of natural numbers and of complete induction. It is true that only for a small part of mathematics (much smaller than in pre-intuitionism) was autonomy postulated in this way. New formalism was not deterred from its procedure by the objection that between the perfection of mathematical language and the perfection of mathematics itself no clear connection could be seen. So the situation left by formalism and pre-intuitionism can be summarized as follows: for the elementary theory of natural numbers, the principle of complete induction and more or less considerable parts of arithmetic and of algebra, exact existence, absolute reliability and non-contradictority were universally ac-knowledged, independently of language and without proof. As for the continuum, the question of its languageless existence was neglected, its establishment as a set of real numbers with positive measure was attempted by logical means and no proof of its non-contradictory existence appeared. For the whole of mathematics the four principles of classical logic were accepted as means of deducing exact truths. In this situation intuitionism intervened with two acts, of which the first seems to lead to destructive and sterilizing consequences, but then the second yields ample possibilities for new developments.
1b.
Intuitionism as a set of axioms (?)
It is said that in intuitionistic
1. It is said that in intuitionistic mathematics, a mathematical statement is true if and only if one can construct a proof of it:
Intuitionism…is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true…
This contrasts with “classical mathematics”, because it contains the formation rule if ¬¬ϕ ⊨ L, then ϕ ⊨ L. This formation rule implies that if you can show ϕ ⊭ L, then ¬ϕ ⊨ L. This requires a more precise meaning of the word “proof”. The above is considered a proof, in classical mathematics. But it is not, in intuitionistic mathematics. Would it be more accurate to say that the two types of math have different formation rules, rather than that “a term has to be proven true, to be true”, which is true in both intuitionism and classical math?
It is also said that in intuitionism, a mathematical object exists if and only if one can provide a construction of it. How does this differ from classical math? In classical math, how do you prove something exists, without constructing it? Do you prove a logical statement of the form, “There exists x such that…”? Do you prove that by contradiction, by showing the inverse to be false (by contradiction)?
2. What does “intuitionism” have to do with intuition? Is the idea that a sense of what is true or false does not need further justification than human intuition? Why would the term “intuitionism” come to be associated with a view on math which seems to say, “It is only true if you provide a rigorous, explicit proof.” This seems constructivist - which to me, is formalist - it depends on principles outside the human mind. It is explicitly rule-governed.
3. What is the philosophical reason for why intuitionism must reject the law of the excluded middle? Can it be proven that it would not follow, or is there merely an informal argument for why it will be left out?
This view on mathematics has far reaching implications for the daily practice of mathematics, one of its consequences being that the principle of the excluded middle, (A∨¬A), is no longer valid.
How does that follow? Is it because Brouwer thought that it was more “intuitive” that “not not A” did not necessarily mean “A”?
Indeed, there are propositions, like the Riemann hypothesis, for which there exists currently neither a proof of the statement nor of its negation. Since knowing the negation of a statement in intuitionism means that one can prove that the statement is not true, this implies that both A and ¬A do not hold intuitionistically, at least not at this moment.
What? Is this saying that even though intuitionism rejects the law of the excluded middle, it would still never be the case that one could know both A and not-A at the same time? But then, why would intuitionism reject the law of the excluded middle, if it espoused that?
4. How do considerations of “time” lead to intuitionism’s claims? What is the argument?
The dependence of intuitionism on time is essential: statements can become provable in the course of time and therefore might become intuitionistically valid while not having been so before.
Meaning that the truth of a statement is dependent on time, so A could be true now, false later? This sounds completely unlike any kind of mathematics I have encountered before.
5. Why does intuitionism think that math is “languageless”?
…the communication between mathematicians only serves as a means to create the same mental process in different minds…Besides the rejection of the principle of the excluded middle, intuitionism strongly deviates from classical mathematics in the conception of the continuum, which in the former setting has the property that all total functions on it are continuous. Thus, unlike several other theories of constructive mathematics, intuitionism is not a restriction of classical reasoning; it contradicts classical mathematics in a fundamental way. According to Brouwer mathematics is a languageless creation of the mind.
How can it exist structurally and yet not be a language? Isn’t the language of mathematics meant to just be an isomorphic structure to the structure of mathematics and reason themselves?
Time is the only a priori notion, in the Kantian sense. Brouwer distinguishes two acts of intuitionism: The first act of intuitionism is: Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognizing that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as 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 twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics.
What two things?
6. What is intuitionism’s conception of the continuum, that is so essential?
As will be discussed in the section on mathematics, the first act of intuitionism gives rise to the natural numbers but implies a severe restriction on the principles of reasoning permitted, most notably the rejection of the principle of the excluded middle. Owing to the rejection of this principle and the disappearance of the logical basis for the continuum, one might, in the words of Brouwer, “fear that intuitionistic mathematics must necessarily be poor and anaemic, and in particular would have no place for analysis”. The second act, however, establishes the existence of the continuum, a continuum having properties not shared by its classical counterpart. The recovery of the continuum rests on the notion of choice sequence stipulated in the second act, i.e. on the existence of infinite sequences generated by free choice, which therefore are not fixed in advance.
Is this the better answer, then - behind the hand-waving, intuitionism basically rests on two precise characterizations of a) natural numbers and b) the continuum?
The second act of intuitionism is: Admitting two ways of creating new mathematical entities: firstly in the shape of more or less freely proceeding infinite sequences of mathematical entities previously acquired …;
That is, a “successor” function.
…secondly in the shape of mathematical species, i.e. properties supposable for mathematical entities previously acquired, satisfying the condition that if they hold for a certain mathematical entity, they also hold for all mathematical entities which have been defined to be “equal” to it…
The idea of “classes”, i.e., the ability to establish properties on a type of object.
Already from these basic principles it can be concluded that intuitionism differs from Platonism and formalism, because neither does it assume a mathematical reality outside of us, nor does it hold that mathematics is a play with symbols according to certain fixed rules. In Brouwer’s view, language is used to exchange mathematical ideas but the existence of the latter is independent of the former.
I don’t have a clear idea of what mathematics prior to language is. It seems to ask for what Brouwer’s definition or conception of a “language” is, in this case.
The distinction between intuitionism and other constructive views on mathematics according to which mathematical objects and arguments should be computable, lies in the freedom that the second act allows in the construction of infinite sequences.
The ability to declare that classes of entities share a certain property gives you more freedom in constructing infinite sequences? What is an example of this? What is an example of a type of constructive mathematics that cannot do this?
Indeed, as will be explained below, the mathematical implications of the second act of intuitionism contradict classical mathematics, and therefore do not hold in most constructive theories, since these are in general part of classical mathematics.
The article says above that the “second act” is about 2 ways of constructing mathematical ideas: the successor functions, and I believe, universal comprehension (specifying a set of elements sharing a property). I do know that universal comprehension is not a part of classical mathematics; one reason is it can be used to construct Russell’s paradox. I believe the Axiom of Choice was introduced as a controlled from of comprehension? (And I read ZFC was originally much simpler, but once they took out I think comprehension and added choice, a lot of the other axioms had to be discovered and fit in, to make the totality work - I think so, anyway.) But I thought earlier in the article similarities between intuitionism and constructivism were highlighted - ie, truth requires construction of a proof. Yet here they say constructive theories are generally a part of classical math? I can’t understand that - I thought they were starkly opposed to one another, since classical math does not require a construction, to prove something I’d do (for example, by contradiction).
Thus Brouwer’s intuitionism stands apart from other philosophies of mathematics; it is based on the awareness of time and the conviction that mathematics is a creation of the free mind, and it therefore is neither Platonism nor formalism.
I would like more detail on specifically why it is different from Platonism and formalism.
It is a form of constructivism, but only so in the wider sense, since many constructivists do not accept all the principles that Brouwer believed to be true. The two acts of intuitionism do not in themselves exclude a psychological interpretation of mathematics.
Why would they?
Although Brouwer only occasionally addressed this point, it is clear from his writings that he did consider intuitionism to be independent of psychology.
So Brouwer felt it was independent of whatever persuasions one has about the nature of the human mind. But is it? It seems to be emphasized that intuitionism is very “Kantian”. Furthermore, what is the virtue of “a psychological interpretation of mathematics”, in addition to intuitionism? In other words, we could have, say, a George Lakoff-type view of the psychology of math, and be intuitionists, or some other psychological view? What other psychological views are there?
Brouwer’s introduction of the Creating Subject as an idealized mind in which mathematics takes place already abstracts away from inessential aspects of human reasoning such as limitations of space and time and the possibility of faulty arguments. Thus the intersubjectivity problem, which asks for an explanation of the fact that human beings are able to communicate, ceases to exist, as there exists only one Creating Subject. In the literature, also the name Creative Subject is used for Creating Subject, but here Brouwer’s terminology is used. In (Niekus 2010), it is argued that Brouwer’s Creating Subject does not involve an idealized mathematician. For a phenomenological analysis of the Creating Subject as a transcendental subject in the sense of Husserl see (van Atten 2007).
Brouwer used arguments that involve the Creating Subject to construct counterexamples to certain intuitionistically unacceptable statements. Where the weak counterexamples, to be discussed below, only show that certain statements cannot, at present, be accepted intuitionistically, the notion of the idealized mind proves certain classical principles to be false. An example is given in Section 5.4 on the formalization of the notion of the Creating Subject. There it is also explained that the following principle, known as Kripke’s Schema, can be argued for in terms of the Creating Subject:
(KS) ∃α(A↔∃nα(n)=1).
In KS, A ranges over formulas and α ranges over choice sequences, which are sequences of natural numbers produced by the Creating Subject, who chooses their elements one-by-one. Choice sequences and Kripke’s Schema are discussed further in Section 3.4.
In most philosophies of mathematics, for example in Platonism, mathematical statements are tenseless. In intuitionism truth and falsity have a temporal aspect; an established fact will remain so, but a statement that becomes proven at a certain point in time lacks a truth-value before that point. In said formalization of the notion of Creating Subject, which was not formulated by Brouwer but only later by others, the temporal aspect of intuitionism is conspicuously present.
Important as the arguments using the notion of Creating Subject might be for the further understanding of intuitionism as a philosophy of mathematics, its role in the development of the field has been less influential than that of the two acts of intuitionism, which directly lead to the mathematical truths Brouwer and those coming after him were willing to accept.
