Friday, 30 December 2016

Post-analytic Philosophy?




The term 'post-analytic philosophy' was first used in the mid-1980s. At that time it referred to those philosophers who were indebted to analytic philosophy but who nonetheless believed that they'd moved on from it (for whatever reasons).

The term 'post-analytic philosophy' seems, prima facie, odd. After all, how can philosophers be 'post'- or anti-analysis? Surely even most examples of post-analytic philosophy will contain analyses of sorts.

Thus the term must instead refer to the tradition of analytic philosophy. But which aspects of that tradition? Which particular philosophers? Did all analytic philosophers have an philosophical essence in common? And let's not forget that philosophical analysis occurred well before the analytic tradition got under way. (What is it that Hume, or Hobbes, or Aquinas, etc. were doing if it's not  - at least partly - analysis?)

The above are all prima facie problems which, to some extent, subsides once the history and use of the term 'post-analytic philosophy' is studied.

However, it is indeed analysis that some philosophers seem to have a problem with. Or, rather, perhaps it's more accurate to say 'philosophical analysis' rather than the simple 'analysis'. This is obviously the case because the words 'philosophical analysis' are more particular than 'analysis' and it may/will contain assumptions as to what philosophical analysis actually is.

Objective Truth?

If we want to put meat on what post-analytic philosophers see to be the problem (or simply a problem) with analytic philosophy, it's best to consult late-20th century and contemporary American pragmatism. This school is itself seen as being part of the post-analytic movement (i.e., which isn't a determinate school).

Many would say that such American pragmatists have a problem with the very notion of objective truth. This is something they see as being an idée fixe throughout the history of philosophy. This, indeed, was no less the case when it came to 20th and 21st century analytic philosophy.

My prima facie personal objection to this is that I've hardly read a single analytic philosopher mention - or use - the words “objective truth”. (I have read, however, Peter van Inwagen's 'Objectivity'.) Then again, it can easily be argued that a philosopher needn't use the actual words “objective truth” in order for him to be committed to the notion of objective truth. In other words, perhaps he simply calls it by another name.1

In any case, the position that objective truth doesn't exist, or that it's not a worthy aim in philosophy, goes alongside a stress on the contingency of cognitive activity, the importance of convention and utility and, indeed, the idea that human (or social) progress can never be ignored – not even in philosophy. Nonetheless, here again I don't see how there's an automatic or prior problem with accepting all this and still engaging in analytic philosophy (or in philosophical analysis). In very basic terms, for example, one could offer a philosophical analysis of philosophical analysis (or some part thereof); and then, as a result, see philosophical problems with such philosophical analysis. Despite that, such a philosopher would still be in the domain of analytic philosophy (or of philosophical analysis).

Strangely enough, Rorty seems to agree with this position. Or, at the least, he says something similar. In an interview conducted by Wayne Hudson and Win van Reijen, Rorty states:

"I think that analytic philosophy can keep its highly professional methods, the insistence on detail and mechanics, and just drop its transcendental project. I'm not out to criticize analytic philosophy as a style. It's a good style. I think the years of superprofessionalism were beneficial." [2005]

I said the position is “similar” to the one advanced by Rorty. It's similar in the sense that an analytic philosopher needn't “drop [his] transcendental project”. That is, an analytic philosopher may be fully aware of Rorty's positions/arguments (or the general positions of post-analytic philosophers) and still be committed to the transcendental project. (Of course we'd need to know what Rorty means by the words “transcendental project”.)

Philosophy Must be Political?

It seems that the position of many post-analytic philosophers is primarily political - or at least primarily social – in nature. Putnam (1985), for example, has said that analytic philosophy has “come to the end of its own project—the dead end”. That can be taken to mean that philosophy should connect itself more thoroughly with other academic disciplines. Or, more broadly, that analytic philosophy should connect itself with culture or society as a whole.

The problem with this is that, on and off, analytic philosophy has already connected itself to many other disciplines. Admittedly, that's been more the case since the 1980s and the rise of cognitive science. However, to give just a couple of examples, the logical positivists connected themselves to science (or at least to physics). And, to give one other example, philosophers in the 19th century connected themselves to logic and mathematics and, again, to science. This non-ostentatious “interdisciplinary” nature of philosophy has been true, in fact, throughout the history of philosophy.

One can also say that philosophy can connect itself to other disciplines - and even culture as a whole - and still remain analytic philosophy. Philosophers can still practice philosophical analysis. (This, again, raises the question as to what analytic philosophy - or philosophical analysis - actually means in either narrow or broad terms.)

A philosopher may also ask why he should connect himself to other disciplines, never mind to something as vague or as broad as culture. In other words, a philosopher must have philosophical reasons as to why this would be a good thing; just as a philosopher must have philosophical reasons as to why it's a bad thing. That means that there'll be philosophical angles to this very debate; though, it can be added, those angles needn't always be philosophical in nature.

Another slant on this philosophy-society binary opposition is that analytic philosophy is too professional and therefore too narrow. In other words, analytic philosophers are over-concerned with very tiny, narrow and specialised problems which have almost zero connection to society as a whole.

More technically and philosophically, it can also be argued that certain central commitments and assumptions of analytic philosophy have been shown to be indefensible. (Hence Putnam's own words quoted earlier.)

All disciplines can be said to concerned with narrow or specialised issues or concerns. Yet this is an accusation more often than not aimed more at philosophy than any other academic subject. 

Richard Rorty

I would say that Richard Rorty appears to be talking about analytic philosophy as it was in the past (say, the 1950s to the 1970s), not as it is today or as it's been since, say, the 1980s.

Take the view that analytic philosophy has as its primary aim a form of knowledge which, as it were, grounds all other forms of knowledge. This is odd. It's true that much traditional philosophy has placed various philosophical domains as what used to be called First Philosophy. (It was once metaphysics, then epistemology, then language, then mind...) However, in the 20th century this has been far from the case. Indeed philosophers - throughout the 20th century - have argued against the nature of a first philosophy. Take naturalists for a start (e.g., the logical positivists then Quine): they placed science or physics in the role of first philosophy. (Although such naturalists saw physics as being primary, that isn't in itself a commitment to also seeing it as some kind of first philosophy.)

It can be said that just as it can be said that Rorty's post-philosophy is a philosophical position, so too is the Wittgensteinian attempt to “dissolve” and then disregard philosophical problems (if not philosophy itself). This position can be said to be held by Putnam and McDowell, as well as by Rorty. And a more specific example of this would be the “problem” of how mind and language are connected to the world.2

In any case, there's just as strong argument for saying that Rorty's later position was more a case of post-philosophy than post-analytic philosophy. In other words, like Heidegger and Derrida, he had a problem with the whole damn show that is philosophy. And, here again, it can be argued that Rorty's position was more political or social, rather than strictly philosophical. Though, of course, a position that rejects philosophy in toto can't help being philosophical – in some or many ways – itself; as Rorty would have no doubt happily admitted. (Jacques Derrida did admit this.)

Notes

1 For example, if you say that an argument or a single statement is “warranted and therefore assertible”, is that a case of being wedded to the notion of objective truth? Or is the notion of warranted assertibility a different species entirely?

2 I put the word “problem” in scare quotes because the very stance of seeing such problems as problems means - according to Rorty, Derrida, Heidegger, etc. - that we've fallen prey to particular philosophical “style of thinking”. But that too would be a philosophical position!

References

Putnam, Hilary (1985) ‘After Empiricism’ in Rajchman and West's Post-analytic Philosophy.



Wednesday, 7 December 2016

Is Constructive Empiricism's Use of Counterfactuals Illicit?


Bas van Fraassen


Possible Worlds

At first glance it may seem odd that James Ladyman (2000) argues that constructive empiricist talk of observability may well require a commitment to possible worlds. Thus it would also seem - again on the surface - that any kind of empiricist must shun possible worlds. 

Possible worlds - to state the obvious – are neither empirical entities and nor are they observable (not even in principle). Thus, as a consequence, constructive empiricists would require unobservable entities in order to legitimise or justify their talk of observable entities.

Let's put the observability-in-principle position thus:

x is observable iff observers were in a suitable place, then they could observe x.

Now that's a bone fide counterfactual. Thus an old-style empiricist may now ask:

What are the truth-conditions of the statement “x is observable” (i.e., the statement above)?

Perhaps the constructive empiricist would say:

x is observable-in-principle because it has truth-conditions(-in-principle).

Why should the old-style empiricist accept either observables-in-principle or indeed truth-conditions-in principle? The whole point of observables is that they can be seen, smelt, heard, touched, etc. at the present moment in time (or at least they can... in principle!).

The constructive empiricist, thus, may require his truth-conditions to exist at possible worlds. Nonetheless, the truth-conditions can't exist at this moment in time because possible worlds don't exist at this or at any moment in time – at least not according to the empiricist.

Context-dependence

I'm not sure if I understand Bas van Fraassen's reply to such points. It involves a strong use of the notion of “context-dependence”. In basic terms, when we “fix the context” of a counterfactual claim about observability, that somehow stops the claim from being modal in nature.

What is fixed is the epistemic community – all the “suitably constituted observers” who're relevant to the counterfactual claim of observability. This, itself, is supposed to make the counterfactual conditional non-modal in nature. That is, once we explicate the nature of (ideal) observers in (ideal) situations, it is these things which we can empirically investigate (Monton and van Fraassen 2003, 413-414). Nonetheless, we still have the non-Humean move from what's empirically observable at this present moment in time and in this place at this time, to that which is not empirically observable at this moment in time or in any place at this moment in time.

Thus statements about what's true at this moment in time - and at this place at this time - slide into what would be the case at other times and at other places. There's still a non-Humean jump that's not eased with these technical additions.

Van Fraassen adds extra detail to this.

He argues that the principle of observability can be cashed out in terms of the objective properties of the world. Moreover, we can use our best scientific theories in order to determine the truth (or content) of “x is observable” (Monton and van Fraassen 2003, 415-416). But, again, there are hidden modal assumptions in all this technical detail and even, as James Ladyman argues, hidden commitments to possible worlds.

Take this statement:

If Bertrand Russell's teapot in Andromeda showed itself to observers (suitably prepared, etc.), then they could observe it.

It seems like a sleight of hand to say that to understand the statement above is - even though a counterfactual - entailed by the facts or the phenomena of the observable (empirical) world. Presumably that must be a reference to teapots and observers which and who exist at this moment in time and can be observed at this moment in time. However, wouldn't Hume have argued that a bone fide empiricist couldn't jump from teapots in our solar system to teapots in Andromeda without begging a few questions or assuming a few facts?

I mentioned facts in the last sentence. Ladyman (2004, 762) talks, instead, about “laws”. That is, he says that

unless we take it that the specification by science of some regularities among the actual facts as laws … is latching onto objective features of the world”.

Wouldn't that mean, in our case, that the laws and objective features of Andromeda are assumed to be like the laws and objective features of the earth and our solar system? Yes, the laws of Andromeda are the same as the laws of the earth and our solar system. And, I assume, Russell's teapot would behave in a pretty similar way to how it would behave if it were floating near the moon or even in the sky above us.

Nonetheless, Ladyman does go on to say that only objectively-existing laws (rather than “pragmatically selected empirical regularities”) can justify (or warrant) our claims about the nature of Russell's teapot or any other phenomena of Andromeda. So it's not the constructive empiricist's claims about the nature and behaviour of Russell's teapot in Andromeda that are problematic per se. It's that in order to justify (or warrant) those claims the constructive empiricist would need to commit himself to entities which aren't kosher from an empiricist's point of view: viz., objective laws. Again, the only thing that a constructive empiricist can rely on are pragmatically-selected empirical regularities, not objective laws. Indeed the acceptance of objective laws commit one to a metaphysics that's not empiricist in nature.


References

Monton, B., and van Fraassen, B., (2003) “Constructive Empiricism and Modal Nominalism”, British Journal for the Philosophy of Science, 54: 405–422.
Ladyman, James. (2000) “What's Really Wrong With Constructive Empiricism? Van Fraassen and the Metaphysics of Modality”, British Journal for the Philosophy of Science, 51: 837–856.
- (2004) “Constructive Empiricism and Modal Metaphysics: A Reply to Monton and van Fraassen”, British Journal for the Philosophy of Science, 55: 755–765.



Thursday, 24 November 2016

Realism, Anti-realism, and Evidence-transcendent Statements



This piece deals with the nature of truth-valued statements which have semantic contents which are said to be “evidence-transcendent”. In less technical terms, the nature of unobservability and observability-in-principle are tackled within an anti-realist versus realist context.

The classic cases are covered: including the doubling in size of the universe, Bertrand Russell's flying teapot, Michael Dummett's organisms in Andromeda, past and future events, electrons, other minds and what it is to be bald.

Within these contexts, we'll also try to clarify what it is to understand statements which have evidence-transcendent content.

Realist Truth

The realist position on truth can appear strange, at least prima facie. Take this statement:

In 607 AD there were precisely one million people with ginger hair in Europe.”

According to the realist, that's either true or false. He may also say that it's determinately true or false (i.e., it's truth is fixed in time).

Similarly, for this statement:

Is is true [false] that Theresa May, at this precise moment, is dreaming about flowers.”

If Theresa May isn't asleep, it's false. That would be easy – in principle – to determine. Though what about if she is asleep at this precise moment? Is it still determinedly true or false that she's dreaming about flowers?

Despite that, it may well be the case that although one takes a realist position on this, one needn't take a similarly realist position on all other domains of discourse. (This is often said of anti-realism, not realism.)  More specifically, statements about Bertrand Russell's flying teapot or Dummett's organisms in Andromeda (both covered later) may well be determinately true. Nonetheless, is it automatically the case that a realist should also have exactly the same position, for example, on statements about the future? Perhaps a realist believes that statements about the future throw up problems which aren't encountered in these other domains.

Understanding Statements

Following on from all that, an anti-realist can ask a realist two questions:

i) If you understand the statement “It is true [or false] that that the universe sprang into existence just five minutes ago, replete with traces of a long complex past” [worded by Bob Hale], then how do you understand that sentence?

ii) What gives you the warrant to say that it's either true or that it's false?

The realist may now reply:

What do you mean by the word 'understand' [as in “understand that sentence”]?”

A standard picture is that in order to understand p, one needs to understand both p's truth-conditions and then somehow decide whether or not those truth-conditions obtain. So, in the case of the statement about the universe doubling in size, how would the truth-conditions for the universe being the same size differ from the truth-conditions of a universe which has doubled in size? [I'm assuming here that the philosophical puzzle of a doubled universe works. As it is, there are arguments against it.] Secondly, how would someone be warranted, or justified, in saying that the universe has or hasn't doubled in size?

The argument is that if the realist can't answer these questions, then his position is untenable. That is, he doesn't know what he's talking about. Or, less judgementally, he doesn't understand what he's talking about. That means that we have no means of understanding what a realist position on truth (at least as regards the doubled-universe scenario) amounts to.

The Doubled Universe

Since we've just mentioned the doubled-universe scenario, Bob Hale talks in terms of what he calls “chronically e-transcendent statements” [1999]. (The 'e' is short for 'evidence'.) He cites the doubled universe case:

Everything in the universe has doubled in size.”

As well as:

The entire universe sprang into existence just five minutes ago, replete with traces of a long and complex past.”

(These statements have, of course, been much discussed in philosophy; though not always in the context of the realism vs. anti-realism debate.)

If the universe had doubled in size, so the argument goes, then there'd be no way of telling that it had actually done so. Thus we couldn't say that it has or that it hasn't doubled in size. Nonetheless, isn't it the case that it either has or it hasn't doubled in size?! And if that's the case according to the realist, the statement is indeed determinately true or determinately false.

Unobservable Electrons

There are, of course, many problems for the anti-realist position too; especially if anti-realism is tied closely to acts of verification (or to verificationism).

Take the many unobservable phenomena of science (specifically of physics). Can it be said that statements about, say, electrons are similar in kind to statements about our doubled universe or flying teapots in distant galaxies? Certain anti-realists would say that even though electrons aren't observable, we're nonethless led to posit their existence because of the evidence supplied by phenomena which are indeed observable. Thus, although electrons are too small to be observed, we're led to them by observable phenomena (plus, of course, lots of theory). (Could the realist argue that he's led to his statements about determinate truth about the unobservable-in-principle by what is actually observable?)

The idea that an electron is posited due to phenomena we can observe (along with theory) is parallel - or additional - to the idea of something's being observable-in-principle.

It could be said that something as tiny as the electron could be observable in principle; except for the large problem that it's deemed to be a “theoretical entity” anyway. That is, besides mathematical structure (as well as theory), there would be nothing to observe even if we could observe it. On the other hand, we can say that a distant something in our solar system could be observed in principle. That may mean that this something isn't a theoretical entity. Well, in a sense, it is a kind of theoretical entity in that it hasn't actually been observed. Though being, say, a teapot, it could be observed if we were able to travel to the distant place it inhabits. (Let's forget the science here!)

There's one clear problem for this observable/unobservable opposition. This is that there isn't always (or ever) a clear dividing line between observation-statements and theoretical statements. That can be because observation-statements involve theory and theoretical statements involve (elements derivable from) observation. Still, whatever problems there are here, they're not as problematic as those statements about unprovable mathematical statements; and certainly not as problematic as our doubled-universe scenario.

Other Minds

A similar problem arises (for anti-realism) when it comes to other minds. We can't observe the goings-on in other people's minds. Nonetheless, like electrons, we're led to acknowledge other minds because of the things we can indeed observe. However, in this case we still need to accept that behaviour (including speech and writing) isn't conclusive evidence for other minds.

There are many problems thrown up by other minds. Behaviourism, for one, was a response to these philosophical problems. And that's why certain types of behaviourist relied exclusively on behaviour (whether physical or verbal behaviour) in his experiments and musings. That meant that other minds ceased being a problem for behaviourists because minds in effect didn't exist. Or, at the least, they believed - at that time - that the mind wasn't a fit subject for science.

Is John Bald?

There's also the problem of statements which involve vague concepts or references to vague states of affairs (if there can be such a thing!). Take the well-known case of whether a certain person is bald.

To clarify with a statement: “John is bald.” This can certainly be said to have truth-conditions (which certain earlier examples didn't have). Nonetheless, in a certain sense, truth-conditions don't really help here. That is, we have access to John and to his head. What we don't have access to is whether or not it's true or false that he's bald. (I'm taking it here that someone can be bald even if they have a few hairs left.) Since it's already been said that truth-conditions aren't the problem, then perhaps we do have a problem with the “vague predicate” that is “bald”.

Here we encounter problems covered by a sorties paradox. Can we ignore them for now? Perhaps we can. It can be said, for example, that we can make a stipulation as to what makes someone bald. (This is deemed to be problematic if taken as a sorties paradox.) We can say that anything less than 100 hairs constitutes baldness in a given male. Consequently it can be said that it's determinately true that John is bald or not bald (i.e., post-stipulation).

What if we accept the sorties paradox? Then we'd be unable to decide (care of truth-conditions or anything else) whether or not John is bald. Nonetheless, the realist, yet again, would argue that it's a determinate fact which makes it the case that either John is bald or John isn't bald. The problem is that if we accept the paradox, we can't know either way.

Michael Dummett, for one, had a problem with this realist conclusion.

In terms of the word “bald”, that would mean that our use of words like that would have “confer[red] on them meanings which determine precise applications for them that we ourselves do not know”. Basically, that would mean that the world tells us if John is bald or not. Or, at the least, the world (including John's head) determines the truth or falsity of the statement “John is bald”. In addition, the world determines the truth regardless of whether or not we can ever determine it to be true or false. Yet surely whether or not someone is bald is something to do with what we decide. The world has no opinion on this or on anything else.

Still, this sorties paradox has an impact on the nature/reality of baldness even if we accept a conventional stipulation about baldness. That is, the logical process which leads from having, say, 1000 hairs to having a single hair is still ultimately paradoxical. That is, step by step we can move from the statement “A man has a thousand hairs is not bald” to the statement “A man with three hairs is not bald” without a hiccup.

Another way of looking at this is to say that if the realist is correct, then any indeterminacy there is has to do with our vague predicates or vague statements, not the world itself (or with John's baldness).

The Teapots and Organisms of Andromeda

Michael Dummett offers us this statement:

'There are living organisms on some planet in the Andromeda galaxy.'”

That statement, according to Dummett's realist, is “determinately true or false” [1982].

In response, the anti-realist adds an extra dimension to this case in terms of the aforementioned idea of observability-in-principle. Dummett expresses the anti-realist's (as well as, I suppose, the realist's) position in this way:

'If we were to travel to the Andromeda galaxy and inspect all the planets in it, we should observe at least one on which there were living organisms.'”

Basically, because the science and the practicalities are so far-fetched in this case, we can't do anything else but forget them. In other words, we need to give the anti-realist the scientific benefit of the doubt. The problem here, though, is that if we give the anti-realist the benefit of the doubt about this currently unobservable situation (which is nonetheless supposedly observable-in-principle), then we can - or must – do the same in the countless other cases of unobservable phenomena in science (particularly in physics). Having said all that, these provisos may not be to the point here.

In any case, if the aforementioned organisms are observable-in-principle, then perhaps they can't be (fully) theoretical entities. Or, less strongly, if the Andromeda organisms are theoretical entities at the present moment, then they needn't remain theoretical entities simply because they can be observed in principle. (Though, again, perhaps the atomic and subatomic world may one day be observed; though not if the entities concerned are simply “theoretical posits” and/or mathematical structures.)

Statements About the Future

Dummett also brings up another example of something that's unobservable-in-principle: a future event. How can we deal with truth-valued statements about future events?

My prima facie position is twofold. One, such statements are neither true nor false. Two, if such statements are neither true nor false, then they serve little purpose.

The realist, of course, believes that statements about future events are determinately true or false. According to Dummett, the realist believes that “there is [ ] a definite future course of events which renders every statement in the future tense determinedly either true or false” [1982].

I find realism towards statements about the future even more difficult to accept than realist claims about other domains. I would agree with Dummett when he says that the only way that a future-tensed statement can be true or false at this moment in time would “only [be] in virtue of something that lies in the present”. This is surely Dummett hinting at some form of determinism in that what is the case at this moment in time will have a determinate affect on what will be the case in the future. (Try to forget arguments against determinism here; as well as references to quantum mechanics, backwards causation, action-at-a-distance, etc.)

Let's take that deterministic position to be the case. That is, a future-tensed statement is true or false at the present moment in time because of what is the case at the present moment in time. That's the case even though the event referred to is in the future. That's fair enough; though it's clear that the realist would have no way of knowing whether or not it's true or false at the present moment. Nonetheless, we've already seen that the realist happily and willingly accepts his position of epistemic deficiency.

Instead of using the word “determinism”, Dummett talks about “physical necessity” instead (577).

Dummett picks up on an interesting consequence of what was said in the previous paragraph. What the realist must do, Dummett argues, is tell us what are the truths about the statement (or situation) at the present moment in time and how these truths bring about the truth of a statement about a future event. That means that only known truths at the present moment can contribute to truths about future events – at least within this context of “physical necessity” or determinism.

Dummett spots a double problem with the realist's position here. The realist can neither determine the present-time truths which would bring about truths about the future. And, by definition, he can't determine - as a consequence of this - a statement about the future that's true at the present moment. That means that although the realist acknowledges his lack of a means to determine the truth of a statement about a future event, he hasn't even got a way of determining the present-time truths which will determine - or cause by virtue of physical necessity – the truth of statements about future events. Thus, in order to make sense of his realism, the least we should expect from the realist is the truths of statements about current situations which would cause - or determine - the truths of statements about future events. Without all that, realism towards statements about the future make little sense.

References

Dummett, Michael. (1982) 'Realism'.
Hale, Bob. (1999) 'Realism and its Oppositions'.




Wednesday, 16 November 2016

Anti-Realism, Intuitionism, and Mathematical Statements


Michael Dummett

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.'” [1982]

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”.  [1982]

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”.      [1982]

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.” [1982]

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.)

Intuitionism

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.” [1982]

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.

Or alternatively:

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.

References

Dalen, Dirk van and Mark van Atten (2007) 'Intuitionism'.
Dummett, Michael. (1982) 'Realism'.