Mathematical Intuitionism and Anti-Realism Compared and Contrasted


L.E.J. Brouwer and Michael Dummett

Mathematical intuitionism (which is a subset of mathematical constructivism) has it (or had it) that mathematics is purely the result of the mental activities of human beings; not the discovery of mathematical entities which exist in an “objective” (or Platonic) realm.


In terms of L. E. J. Brouwer’s original intuitionism (i.e., of the first three decades of the 20th century). Brouwer believed that a mathematical statement is a subjective claim which is verified in terms of the validity of a particular mental “construction”; which is, in turn, dependent on human intuition.


As for anti-realism


Anti-realism is an epistemological position which was first explicitly articulated by Michael Dummett in the early 1960s. Taken more broadly, various (weak, strong or vague) forms of anti-realism (though not the term itself) can be said to date back to the ancient Greeks. Indeed it can even be convincingly argued that the Copenhagen interpretation of quantum mechanics is anti-realist.


Anti-realism is a position one can take on metaphysical, mathematical, semantic, scientific and moral issues. Importantly, this means that one can take an anti-realist position on one of these domains; though not necessarily on all the others.


More technically, Michael Dummett’s anti-realism has it that the truth of a statement rests on its demonstrability or verification. This is in opposition to the realist notion of truth in which a statement rests on its “correspondence” to an external, mind-independent reality.

In terms of this piece itself.


The following is about the anti-realist’s problems with mathematical realism (or mathematical Platonism); as well as about how — or if — intuitionist mathematics ties in with anti-realism.


The focus will partly be on Michael Dummett’s take on intuitionism; as well as on his position on mathematical realism. And, understandably within this context, the relation between mathematical truth and mathematical proof will also be discussed.


Dummett on Mathematical Realism and the Platonic Realm


The English philosopher Michael Dummett (1925–2011) once asked a (fictional) mathematical realist the following question:

“What makes a mathematical statement true, when it is true?”

The realist’s answer was:

“‘The constitution of mathematical reality.’”

Of course if Dummett hadn’t told us that he was talking to a mathematical realist, then the words “mathematical reality” might have meant anything. After all, mathematical inscriptions on a page could be deemed — by some — to be mathematical reality.


As it is, the realist’s mathematical reality exists regardless of minds, before minds existed and will exist after the extinction of all minds.


The anti-realist can now ask the mathematical realist these three questions:

(1) How do we gain access to this mathematical reality? (2) How can we know (or determine) that we have accessed mathematical reality? (3) How do we know that our mathematical statements match up with (aspects of) mind-independent reality?

Wouldn’t it need to be the case that the mathematician’s decision-procedures and resultant proofs would also be required to match parts of this (Platonic) mathematical reality? No, not according to (most) realists. The decision-procedures and resultant proofs are for us. Such things are the way that mathematicians determine the truths of their mathematical statements. However, the results of these decision-procedures and proofs are indeed in the Platonic realm; even if the procedures and proofs aren’t.


Having said all that, it may be hard to understand what all that means.


For one, it’s hard to accept this Platonist separation of truth (or mathematical results) from decision-procedures and proofs.


Now take the mathematical intuitionists.


They believe (or believed) that there are no truths without decision-procedures (or “constructions”) and proofs because such things quite literally “bring about” those truths — and even the numbers themselves. In other words, mathematical truths wouldn’t exist without such things.


What about other anti-realist problems with mathematical realism?


It can be said that a verifiable statement is also decidable statement. That is, whether it is true or false is a decidable matter. Thus it’s the case that unobservable (in principle) and observable states will be, respectively, unverifiable and verifiable.


Does all this also apply to mathematics?


Platonist mathematicians and philosophers say that mathematics doesn’t concern the observable. We can, of course, observe mathematical equations on paper or on the blackboard. We can even introspect certain mathematical symbols. Despite these qualifications, in the Platonist view we’re actually observing the symbolic representations of numbers and mathematics generally — not numbers and equations themselves. In addition, a Platonist will happily accept — and even emphasise — the fact that mathematics and numbers can be applied to the world; or used as the basis of our structural descriptions of the world.


Dummett on Truth and Proof


Michael Dummett often went into the technicalities of anti-realism without actually mentioning mathematics. In other words, what he argued is also the case when it comes to all (or at least most) of the domains which rely on the notions of truth and reality.


In terms of the lack of proof of a statement (although proof isn’t applicable to verifiable — i.e., nonmathematical — statements), Dummett said that

“we cannot assert, in advance of a proof or disproof of a statement, or an effective method of finding one, that it is either true or false”.

There’s a problem with that.


The realist (against whom Dummett is arguing) doesn’t claim that he knows that statement P is true or false regardless of proof (or regardless of “an effective method of finding one”). So, in this respect at least, his position is effectively the same as the anti-realist’s. However, the realist believes that the statement under consideration still has a determinate truth value — regardless of proof. Sure, the realist can’t say that statement P is true. And he can’t say that statement P is false. However, he can say this:

Statement P is either determinately true or determinately false — regardless of proof.

So, yes, even the realist “cannot assert” (at this juncture) that P is true or that P is false. Instead, he can simply say that P is either true or false.


Having said all that, there may be very little point in saying that mathematical statement P is “either determinedly true or determinately false” regardless of decision-procedures and their resultant proofs. Indeed what does this claim amount to? Where does it get us?


And this is the main reason why the intuitionists — and many others — rejected the law of excluded middle. That is, they rejected this logical law (i.e., P ∨ ¬P) when it came to these troublesome mathematical statements or theorems. Dummett himself said that

“we shall be unable to conceive of a statement as being true although we shall never know it to be true, although we can suppose a true statement as yet unproved”.

As it is, many do indeed believed that there are unprovable truths in mathematics. Dummett himself put that position in this way:

“A Platonist will admit that, for a given statement, there may be neither a proof nor a disproof of it to be found.”

Despite that, what Dummett has just said about mathematical statements is most certainly (at least partly) applicable to various non-mathematical statements too; whether they’re about distant galaxies, the past/future, “other minds” or whatever.


Proof, Truth and Truth-conditions


Clearly it’s problematic to think in terms of truth-conditions when it comes to mathematical statements. (A Platonist may think in terms of such truth-conditions being abstract entities in an abstract realm; all of which can be accessed through “intuition”, “direct insight” or mental “seeing”.) More concretely, it seems odd to demand truth-conditions for the statement 5 x 56 = 280. Strictly speaking, it has no truth-conditions. So what does it have? According to intuitionists, it has a mathematical procedure which produces a definite result.


Is is that a decision-procedure is also the proof of a mathematical statement? Or is the way of determining the truth of a mathematical statement also a proof of it? Yes; though truth may still not be proof. The intuitionist believes that a proof demonstrates the truth of a mathematical statement. (Without proof, there is no truth.) Nonetheless, isn’t it still the case that a proof of statement P is still not the same thing as P’s truth?


When true, mathematical statements are decidable; though not verifiable (i.e., because unobservable). Despite that, the decidability of mathematical statements does the job that verifiability does when it comes to statements about things which can be observed (even if only observable “in principle”). So, in mathematics, proof is somewhat like the verification of statements about the observable realm.


In reference to all the above, an intuitionist would say that mathematical truth depends on mathematical proof. And mathematical proof is itself a question of decidability.


So the word “truth” (or “true”) may refer to the end result of a mathematical procedure which works as a way of saying that a statement has been proved.


Still, what does the word “true” actually refer to or mean? To the proof itself? To the construction of the proof itself? Again, perhaps all we have this:

truth = proof

Or alternatively:

a decision-procedure = (a) truth

Following all that, it must now be said that the realist will obviously ask about the situation in which there is no proof of a mathematical statement or theorem. He will follow that question by stating that such a statement is still true (or false) regardless of whether or not it had been proved. Indeed this is the case with, for example, both Goldbach’s conjecture and Fermat’s Last Theorem. (The latter was proved by Andrew Wiles.)


Intuitionism


Objectivity in mathematics isn’t a question of (literal) objects. (It’s worth reading Hartry Field’s paper ‘Mathematical Objectivity and Mathematical Objects’ within this context.) Instead, objectivity can be said to be about the objectivity of the procedures and proofs which lead to truth. On the other hand, if this were a question of truth-conditions, then it may be the case that objects enter the equation.


Can’t we be realists about numbers, functions, sets, etc?


Aren’t these things abstract objects?


And if numbers, etc. are abstract objects, then don’t we have truth-conditions because we have objects of some abstract kind?


Yet what if — specifically — numbers are simply “free creations”; as Richard Dedekind believed? In that case, numbers are created (or constructed) by the mathematician. The free creation of a number would still be the creation of a determinate something. So that would be similar to when a person makes — or constructs - a toy dog whose physical nature then becomes determinate or definite.


In further terms of an intuitionist position on free creations.


There must still be decision-procedures which can come up with definite results even if numbers are constructed. Take the toy dog just mentioned. This toy dog was indeed constructed. Yet we still have various ways of deciding (or determining) its nature. That is, like a constructed number (or theorem), we have ways of determining the — physical — nature of the toy dog. The metaphysical (rather than mathematical) realist, on the other hand, may now need to say that any truths about the toy dog would hold even if no one could ever gain access to any data or information about that toy.


Brouwer and Heyting on Maths as a Human Activity


The Dutch mathematician and philosopher L.E.J. Brouwer thought of mathematics as an “activity rather than a theory”. In that simple sense, truth-conditions (or a Platonic reality) can’t give mathematics a realist foundation. According to Dirk van Dalen and Mark van Atten:

“Mathematical truth doesn’t consist in correspondence to an independent reality.”

This means that such a construal of mathematics is somewhat beyond the anti-realism/realism debate in that verification (or observation) isn’t even possible when it comes to mathematics.


The basic intuitionist position on mathematical truth was also put by the Dutch mathematician and logician Arend Heyting. Michael Dummett put Heyting’s position in this way:

“[T]he only admissible notion of truth is one directly connected with our capacity for recognising a statement as true: the supposition that a statement is true is the supposition that there is a mathematical construction constituting a proof of that statement.”

In terms of mathematics, that “capacity for recognising a statement as true” would depend on a decision-procedure (or construction) for determining a proof of that statement. As can be seen, everything in the quote above seems to refer to human actions — even if human cognitive (or mental) actions. A mathematical construction is a cognitive activity. A proof is also a result of cognitive activity. Indeed recognising a statement to be true is also a cognitive activity. So what we don’t have (in the quote above) is any reference to anything outside these cognitive actions (such as truth-conditions, states of affairs, facts, etc.). Indeed there isn’t even any mention of abstract objects.


Dummett’s Problems With Intuitionism


It may seem odd that the intuitionists’ rebellion against “mystical Platonism” should rely, instead, on the happenings which go on in the privacy of mathematicians’ heads. At least that’s how Dummett saw it. (Dummett’s position is Wittgensteinian.) Yet, in a qualified sense, that’s what mathematicians do. That is, even if we supply an externalist (or broad content) account of the meaning/s, etc. of the symbols used in those mental constructions, and also argue that they have externalist/broad features (even during the private acts of mental construction), at the initial stage these mathematical mental activities are still individualistic (or private) in nature. The symbolised functions, numbers are, of course, public (or communal). Nonetheless, the mental (or cognitive) acts of construction are — in an obvious sense — private.


So, on the one hand, it remains the case that mental “objects” and actions aren’t really private in that the meanings, senses, extensions or whatever of the symbols used (or thought about) are communally (or externally) determined. Nonetheless, it’s still the case that the mental constructions of numbers (or of mathematical statements) are private. (Early intuitionism came before “late Wittgenstein”.)


It’s still the case that private mental constructions of an individual mathematician can’t be the subject of a decision-procedure by other mathematicians. This of course ceases to be the case once the mental constructions are written down (or notated) in some way. In other words, when the mathematician uses communal (or public) mathematical symbols.

On a Wittgensteinian reading, then, it can now be said that even the symbols (“in the head”) which the mathematician uses in “private” mental constructions are still essentially communal (or public) in nature. That is, they have no meaning (or nature) outside of the communal (or public) practices of mathematicians and of the larger community.


And that is the fundamental Wittgensteinian — and indeed antirealist — point about the nature of mathematics.


Main References

  • Dalen, Dirk van and Mark van Atten (2007). ‘Intuitionism’.

  • Dummett, M. (1982). ‘Realism’.

  • Dummett, M. (1978). Truth and Other Enigmas.

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