Wednesday, 16 November 2016
Anti-Realism, Intuitionism, and Mathematical Statements
The following piece is essentially about the anti-realist's problems with mathematical Platonism; as well as about how - and if - intuitionist mathematics ties in with anti-realism.
The focus will be on Michael Dummett's specific take on intuitionism; as well as his position on mathematical realism (i.e. Platonism). Understandably, at least within this limited context, the relation between truth and proof will also be discussed.
Dummett on the Platonic Realm
Michael Dummett asks a (fictional) Platonist the following question:
“'What makes a mathematical statement true, when it is true?'”
The Platonist's answer is:
“'The constitution of mathematical reality.'” 
Of course if Dummett hadn't told us that he was talking to a (fictional) Platonist, the words “mathematical reality” could mean anything. After all, mathematical inscriptions on a page could be deemed to be, by some, mathematical reality. As it is, the Platonist's mathematical reality exists regardless of minds, before minds existed and will exist after the extinction of minds.
The anti-realist, along with many other philosophers of mathematics and mathematicians, will now ask:
How do we gain access to this mathematical reality?
More relevantly to the anti-realist's verificationism, the anti-realist will also ask:
How do we determine that we've accessed mathematical reality and how would we know that our mathematical statements match up with that reality?
Indeed wouldn't it need to be the case that even the mathematician's decision-procures and resultant proofs would also be required to match parts of this Platonic mathematical reality? No, not according to (most?) Platonists. The decision-procedures and resultant proofs are for us here in the non-platonic world. They're the way that mathematicians determine the truths of their mathematical statements. However, the results of these decision-procedures and proofs are indeed in the/a platonic realm; even though the procedures and proofs aren't.
Having said all that, and in the hope of capturing the Platonist position, I find it hard to know what all that means. That is, it's hard to understand or accept this Platonist separation of truth (or mathematical results) from decision-procedures and proofs. The intuitionist, of course, believes that there are no truths without decision-procedures (or “constructions”) and proofs because they quite literally bring about the truths and even numbers themselves. Mathematical truths wouldn't exist without them. (Despite that, one may also have problems with the Platonist construal of a mathematical reality which aren't particularly anti-realist or intuitionist is nature: e.g., what about our causal links to the platonic realm?)
What about other anti-realist problems with Platonism (i.e., realism)?
It can be said that a verifiable statement is also decidable statement. Thus it may/will be the case that unobservable (in principle) and observable states will be, respectively, unverifiable and verifiable. Does this 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 a number or mathematics itself. (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 structural descriptions of the world.)
Dummett on Truth and Proof
Michael Dummett goes into the technicalities of anti-realist truth without actually mentioning mathematics. In other words, what he says is the case about all/most domains of truth.
In terms of the lack of proof of a statement (although proofs aren't really applicable to verifiable non-mathematical statements), Dummett says 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 is a slight problem with that. The realist - against whom Dummett is arguing - doesn't say that he knows that a statement is true or false regardless of proof (or of “an effective method of finding one”). His position, in this respect, is effectively the same as the anti-realist's. However, the statement under consideration still has a determinate truth-value: regardless of proofs. The realist, sure, can't say that the statement is true. And he can't say that the statement is false. Though he can say that the statement is either determinately true or determinately false regardless of proofs. So, yes, even the realist, in Dummett's words, “cannot assert” (at this juncture) that “it is either true or false”. Instead he can simply say: It is either true or false.
Having said all that, I see very little point in saying that a mathematical statement is determinedly true or false regardless of decision-procedures and their resultant proofs. What does this claim amount to? Where does it get us? It's effectively equivalent to Bertrand Russell's teapot flying around somewhere in a distant galaxy. Yes, there could be such a thing. However, we can never establish that there is such a thing. Therefore what, exactly, is the point of saying that “there's a flying teapot somewhere in a distant galaxy”?
Dummett (in his own way) puts the gist of the last paragraph in more circumspect and, indeed, Dummettian prose. On the proof-independent mathematical statement, Dummett says 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”. 
Yes, what's the point of “suppos[ing] a true statement as yet unproved”? As it is, however, there are indeed unprovable truths in mathematics. Dummett himself puts that 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 is most certainly applicable to non-mathematical statements; whether about distant galaxies, the past, other minds and other problematic areas. This is even more the case when it comes to statements about the exact number of people in a given aeroplane at a given time; or, more mundanely, about whether or not Jesus H. Corbett is dead at time t.
Truth-conditions, Proof and Truth
Clearly it appears to be problematic to think in terms of truth-conditions when it comes to mathematical statements. (A Platonist, perhaps, could think in terms of such truth-conditions being in an abstract realm; which are accessed through “intuition”, “direct insight” or - metaphorical - “seeing”.) More concretely, it seems odd to demand truth-conditions for the statement 5 times 56 = 280. Strictly speaking, it has no truth-conditions. So what does it have? According to anti-realists or intuitionists, there is a procedure which can result in a determinate result. Is that decision-procedure also the proof of the mathematical statement? Is the way of determining the truth of a mathematical statement also a proof of the statement? Yes, but truth is not proof. The intuitionist believes that a proof leads to the truth of a mathematical statement. Without proof there is no truth. Nonetheless, a proof isn't the same thing as a truth.
Mathematical statements, when true, are decidable, not verifiable (because unobservable). Despite that, the decidability of mathematical statements (in a way) does the job that verifiability does when it comes to statements which can be observed (or only observed “in principle”). That is, there's a decision-procedure for deciding the truth of mathematical statements.
In reference to all the above, an anti-realist philosopher of mathematics would say that mathematical truth depends on mathematical proof. And mathematical proof is itself a question of decidability. That is, from proof comes truth. And proof, in mathematics, is very much like verification when it comes to the observable realm. That means that proof, in intuitionist mathematics, satisfies the anti-realist position.
Nonetheless, the realist would ask about the situation in which there is no proof of a mathematical statement or equation. He would follow that by saying that such a statement would still be determinately true regardless of whether or not it had been proved. (This is/was the case with Goldbach's Conjecture and Fermat's Theorem. The latter was proved by Andrew Wiles.)
Objectivity in mathematics isn't a question of objects. 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 have been the case that objects do enter the equation. As it is, according to the intuitionist, this isn't the case for mathematics.
Can't we be realists about numbers, functions,sets, etc.? Aren't they abstract objects? And if they are abstract objects, then don't we have truth-conditions (of some kind) because we have objects (of some kind)?
On the other hand, what if 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; just as when a person makes (or “constructs”) a toy dog whose nature becomes determinate.
In terms of an intuitionist/anti-realist position on free creations. There must still be decision-procedures which can come up with definite results even if numbers are “constructed”. The toy dog just mentioned was indeed constructed. Still, we have various ways of deciding its nature. That is, like the constructed number or equation, we have ways of determining the nature of the toy dog. The realist, on the other hand, would say that any truths about the toy dog would hold even though no one could ever gain access to it (say, after the creator died, etc.).
Brouwer/Heyting on Maths as a Human Activity
L.E.J. Brouwer, according to Dirk van Dalen and Mark van Atten [2007, 513], thought of mathematics as an “activity rather than a theory” . In that simple sense, truth-conditions or a Platonic reality don't give maths a realist foundation. More importantly, perhaps, “[m]athematical truth doesn't consist in correspondence to an independent reality”. In a certain sense, this means that such a construal of mathematics is beyond the anti-realism/realism debate in that verification or observation isn't even possible in principle.
The basic intuitionist position on mathematical truth was also put by Arend Heyting. Michael Dummett puts Heyting's position this way:
“... the 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 actions. A mathematical construction is a cognitive activity. A proof is also a result of a cognitive activity. Indeed recognising a statement to be true is a cognitive activity. 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 a mention of abstract objects as such. However, that doesn't automatically mean that abstract objects aren't illicitly or tacitly referred to.
For example, the word “truth”may refer to the end result of a mathematical construction which works as a proof that a statement is true. But what does the word “true” refer to or mean? To the proof itself? To the construction of the proof itself? In that case, perhaps we have:
truth = proof.
a construction (or decision-procedure) = (a) truth
Dummett's Problem 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 a mathematician's head. At least that's how Dummett saw it. Dummett's position is, of course, Wittgensteinian [Dummett, 1978, 215-247]. Yet, in a sense, that's exactly what intuitionists or mathematicians do. That is, even if we supply a retrospective externalist (or “broad content”) account of the meaning/s, etc. of those mental constructions, and also argue that they have externalist/broad features (even during the private acts of mental construction), at the initial stage this mathematical mental activity is still individualistic (or private). The functions, numbers and symbols are of course public or communal. Nonetheless, the mental/cognitive acts of construction are, in an obvious sense, private. So it remains the case that, on the one hand, mental objects and actions aren't really private in that their meanings, senses, extensions or whatever are externally determined. Nonetheless, it's still the case that the mental constructions of numbers or mathematical statements are private. And surely no intuitionist would have denied any of that. (Perhaps we can say said that Dummett was trying too hard to be a Wittgensteinian.)
It is still the case that private mental constructions can't be the subject of a decision-procedure by other mathematicians (even if the mathematical symbols, functions, etc. are bone fide externalist items). This of course ceases to be the case once the mathematical statements are written down or notated in some other way.
Dalen, Dirk van and Mark van Atten (2007) 'Intuitionism'.
Dummett, Michael. (1982) 'Realism'.
- (1978) Truth and Other Enigmas.