What Does the Question “What is it like to type these words?” Mean?

 

Here’s a passage from the David Chalmers’ well known and important book The Conscious Mind: In Search of a Fundamental Theory. In this passage, the phrase “something it is is like” is used:

“We can say that a being is conscious if there is something it is like to be that being, to use a phrase made famous by Thomas Nagel. Similarly, a mental state is conscious if there is something it is like to be in that mental state. To put it another way, we can say that a mental state is conscious if it has a qualitative feel — an associated quality of experience. These qualitative feels are also known as phenomenal qualities, or qualia for short. The problem of explaining these phenomenal qualities is just the problem of explaining consciousness. This is the really hard part of the mind-body problem.”

The phrase “something it is like” is the main theme of this piece. I’ll tackle that by taking each sentence of the passage above one at a time.

There’s Something it’s Like to Type These Words

Firstly, we have this opening sentence:

“We can say that a being is conscious if there is something it is like to be that being, to use a phrase made famous by Thomas Nagel.”

The American philosopher Thomas Nagel wrote his famous paper called ‘What is it Like to Be a Bat?’ in 1974. That paper inspired mountains of responses — both positive and negative. Nagel’s paper is widely thought to capture the main problem with any “physicalist” or scientific accounts of subjective experiences — whether those of bats or of human beings.

I can now ask readers the following question:

What is it like to be you?

And any reader can, in turn, ask me this question:

What is it like to be you [i.e., me]?

Indeed I can now ask myself this question:

What was it like to be me yesterday or even ten minutes ago?

I can even ask myself the following question:

What is it like to type these words — right here and right now?

So, on first glance, the words “something it is like” (or “what is it like to be…”) are hard to make concrete. That said, it’s of course the case that we intuitively seem to understand what it means. After all, isn’t there something it is like for me — right here and right now — to type these words? And isn’t there something it is like for you to read these words?

But what is this something it is like to type these words?

It’s definitely something very particular to me. And that, from the start, seems to make it an unscientific matter. (Perhaps non-scientific or even a-scientific would be a better term.) It may also be the case that the what it is like to type these words is incommunicable to other people (more of which later). I may of course attempt to do so in poetry, prose or in a subjective language. However, such descriptions (if that’s what they truly are) will never truly capture for someone else what it is like for me to to type these words. The best that can happen is that the person who reads (or hears) my accounts can place himself in some kind of “imaginative sympathy” with what it is that I’m describing. However, in that case it won’t be my words (or descriptions) alone which are truly capturing — for that other person — what it is like for me to type these words. This will basically mean that the hearer of my words will simply imagine himself typing his own words right here and right now. It won’t be my words alone which capture for him what it’s like for me to type these words.

Here’s another take on this.

Surely another person would literally need to be me in order to know what it’s this like for me to type these words. And that, surely, is impossible.

That said, let’s just accept that a sci-fi fusion is possible (if only for philosophical reasons). In other words, let’s accept that it’s logically possible that such a thing could happen. (See David Lewis’s ‘Survival and Identity’; which deals with both the fusion and fission of persons. Fission is when one person becomes two persons.)

Yet even if there could be some kind of fusion between myself and someone else, that fusion wouldn’t result in that other person experiencing what it’s like for me to type these words. Instead, it would result in this me-and-another-person fusion experiencing what it’s like to type these words. And even then, this person-fusion may still not be able to communicate “its” what-it-is-like-to-type-these-words experience to other people or even to itself.

What is it For a Mental State to be Conscious?

“Similarly, a mental state is conscious if there is something it is like to be in that mental state.” - David Chalmers

Again, there’s something it is like to be in this mental state of typing these words. That is, I’m experiencing and feeling things right here and right now while writing these words.

What can be said about these experiences?

Nothing very precise or exact. Or, in the words of a physicist, I may be able to capture something qualitative; though not something quantitative. That said, perhaps nothing determinately qualitative can be captured either!

And it’s this very inability to communicate — or offer precise “verbal reports” — about this particular mental state (or sequence of states) which makes it suspect to philosophers like Daniel Dennett. However, the fact that these mental states (or experiences) can’t be adequately communicated — or communicated at all — doesn’t mean that these experiences aren’t real (or actual).

So does all that mean that the Dennetts of this world are demanding that at least something — or indeed everything — about this particular mental state of typing (right here right now) must be communicable? And if it isn’t, then is that the sole reason why such people reject — both from a scientific and a philosophical perspective — subjective (or “private”) mental states?

All the above means that the following is the bottom line:

If that something it is like to type these words is essentially incommunicable, then either it isn’t real or it serves no purpose in either science or philosophy. (Dennett can be read as arguing that x isn’t real precisely because it’s incommunicable and/or verifiable.)

Yet the feels I experience when typing these words clearly do exist (or are actual/real)… at least to me. That — again - is the problem. These feels only has an existence (or reality) to me and to me alone. And that’s why the Dennetts of this world have a problem with such things.

It’s All About Feels

“To put it another way, we can say that a mental state is conscious if it has a qualitative feel — an associated quality of experience.” - David Chalmers

This seems to be suggesting — or stating — that a mental state must have a qualitative feel (or a set of qualitative feels) if it’s deemed to be conscious. In other words, can we make sense of a mental state without a qualitative feel?

More particularly, can we make sense of a mental state which involves, for example, mathematical calculations having no qualitative feels?

In this particular case, it doesn’t matter if mathematics and equations have nothing to do with mental states — let alone with qualitative feels. This is about what happens when mathematical calculations are carried out by human beings.

Of course, it can now be asked why anyone would care about the very particular — and (seemingly) subjective — “accompaniments” to mathematical calculations. However, that again doesn’t matter because the main argument here is that such subjective accompaniments to mathematical calculations can’t be described, grasped or (as it were) owned by any science.

Nonetheless, what mathematicians say about what happens when they calculate is amenable to scientific description or study. That is, the verbal reports of mathematicians are out there in public space. Thus, scientists and philosophers can indulge in heterophenomenological descriptions of the mathematical calculations which occur in human minds.

The word “heterophenomenological” has just be used.

The term heterophenomenology was coined by Dennett. He uses it to describe a third-person approach to the study of mental phenomena. This approach consists mainly in studying the verbal reports of subjects — such as mathematicians — and then making scientific and philosophical sense of them. And that is done in order to make mental states — such as mental calculations — scientifically acceptable. (See my ‘Against Daniel Dennett’s Heterophenomenology’.)

Meet Qualia

“These qualitative feels are also known as phenomenal qualities, or qualia for short.” - David Chalmers

At first glance, saying that qualitative feels are phenomenal qualities doesn’t help us very much. We can now simply ask:

What are phenomenal qualities?

Well, they’re qualitative feels!

Now we can also say that both phenomenal qualities and qualitative feels are qualia:

So what are qualia?

Well, they’re phenomenal qualities and qualitative feels…

Another problem here is that many accounts of qualia simply say what qualia aren’t, not what they are. And how could anyone truly say what qualia are anything other than that they’re phenomenal qualities and qualitative feels?

So here again we come up against a brick wall.

Can we describe such qualia/phenomenal feels/phenomenal qualities in a way that will take us anywhere that’s acceptable from a scientific point of view? More relevantly, can we say anything of philosophical substance?

Again, there is something it is like for me to type these words right here and right now. And, on that, I’m with the qualiaists. However, what can I say about that something? Nothing much — except in the guise of my own verbal reports. Yet, arguably, such reports don’t truly capture the supposedly private qualia I’m reporting. And, on that, I’m with Dennett in that he argues that there isn’t something “behind” the reports which can count in science or even in philosophy.

Explaining Qualia

“The problem of explaining these phenomenal qualities is just the problem of explaining consciousness. This is the really hard part of the mind-body problem.” - David Chalmers

Firstly, David Chalmers assumes that there is what he calls a “hard problem”. There may well be. However, that should never be simply assumed.

Secondly, what does Chalmers mean by the the word “explaining” (as in “explaining consciousness”)? That is, what is it to explain qualia or to explain consciousness?

What would an explanation of phenomenal qualities look like — even if there is one? This isn’t to say that there is — or there isn’t — an explanation: it’s to ask what such an explanation would look like. More particularly, I suspect that no explanation would ever satisfy David Chalmers… or Thomas Nagel for that matter. And that could quite possibly be because there can be no explanation which pleases everyone— at least not of the kind that Chalmers demands. 

Perhaps this is, after all, a bogus problem in that either it isn’t a problem at all; or because no explanation would ever satisfy those philosophers who’re demanding an explanation. (See my David Chalmers’ Unanswerable “Hard Question” About Consciousness | by Paul Austin Murphy | Predict | Medium.)

More technically, would such an explanation of phenomenal properties be an explanation from a first-person/subjective/phenomenological point of view? Well, that wouldn’t satisfy most scientists and many philosophers.

Okay. Would such an explanation be a neuroscientific/behaviourist/functionalist explanation? Well, that wouldn’t satisfy Chalmers, Nagel and many other philosophers.

So what about uniting the physical/functional accounts with first-person accounts? 

Is that even possible? 

Yes; in the particular respect that a person may verbally report his experience/s of particular phenomenal properties and that may satisfy Dennett and others. However, such reports would still be solely verbal reports of qualia — not accounts of qualia themselves (whatever that may mean). Indeed, what would an account of a single quale even look like? And perhaps even if there were such an account, then it still couldn’t — almost by definition — be united with a scientific account.

Thus, there’s both a definitional gap and an “explanatory gap” between scientific accounts of phenomenal properties and subjective/first-person/phenomenological accounts of the (supposedly) very same things — and never the twain shall meet.

The upshot here is that David Chalmers will never be satisfied with the accounts of Daniel Dennett and (many) scientists. And Dennett and these scientists will never be satisfied with the accounts of Chalmers and the “mysterians”. Thus, again, there’s a chasm between the two positions. And even those much-advocated “structural correlations” (i.e., between neural states and conscious states) will certainly never bridge that chasm. 

Perhaps nothing will.

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Note: A Single Mental State and its Set of Qualia?

It’s worth questioning if there is such a thing as a perfectly circumscribed and delineated single mental state. Don’t mental states “flow” into each other? And if mental states do indeed flow into each other, then there may not be single mental states at all.

Similarly, how many qualia make up a single mental state? Does that question make sense? Can it be answered? What would an answer look like? And even if a single mental state had a determinate number of qualia, how would we know what that number is?

So it seems that the full nature of “qualia” can’t be expressed or described. And that, again, is why the Dennetts of this world have a problem with such things.

The Obsession With Essence and Necessity Began With Aristotle

 
The ancient and historically important philosophical notions of essence and necessity were vital to Aristotle’s overall philosophy. And, ever since Aristotle’s day, these philosophical notions have played a vital role in Western philosophy, theology, science and even in political theory.

It’s commonly agreed that the notion of essence began with Aristotle. However, a philosophical prototype can also be found in, for example, Plato’s Euthyphro. In that book Plato argued that physical entities acquire their essential being when they instantiate (or exemplify) what he called Forms. For Plato this meant that Forms are abstract universals which exist before any (concrete) particulars instantiate them. Plato therefore saw Forms as the paradigms of the particulars (i.e., things) which we experience in everyday life.

Aristotle himself departed from Plato in that he attempted to discover the (non-capitalised) form (or nucleus) of an individual physical entity. (Aristotle called such a thing ousia or substance.) In other words, Aristotle believed that Forms (or universals) must be instantiated in order to have being.

In more detail.

Aristotle used the Greek phrase τὸ τί ἦν εἶναι; which means “the what it was [or is] to be”. (The equivalent scholastic term is quiddity.) He also used the shorter phrase τὸ τί ἐστι; which means “the what it is”. (This is equivalent to the scholastic term haecceity). Then, in turn, these Greek phrases were rendered into the Latin term essentia (i.e., “essence”).

More relevantly to the following piece: Aristotle believed that gaining knowledge about the world is essentially about discovering what is essential and necessary to any given x.

The Purpose and Natural Place of Flames

Firstly, let’s take the often-cited example of Aristotle’s explanation of why fire (or flames) goes upwards. This is Aristotle on the subject:

“The ‘simple’ bodies, since they are four, fall into two pairs which belong to the two regions, each to each: for Fire and Air are forms of the body moving towards the ‘limit,’ while Earth and Water are forms of the body which moves towards the ‘centre’.”

Sure, it’s easy to be (retrospectively) smug about Aristotle’s views on fire and on much else. (Thus displaying a kind of Whig history of philosophy.) But, at the time, he had no reason not to conclude that flames displayed purposive behaviour. In addition, Aristotle relied on observations to come to his conclusions. After all, all flames do indeed move upwards. So, unlike Plato, Aristotle did pay much attention to the transitory (physical) things around him.

So why did Aristotle believe that flames go upward?

Aristotle reasoned in terms of what he called first principles. In the particular case of fire (or flames), he believed that fire seeks it “natural place” above the earth. This also meant that Aristotle believed that flames must necessarily rise upward.

Aristotle also believed he could discover was the “final cause” of things. In the case of flames, that final cause is the end of their upward journey. More technically, Aristotle argued that the four elements rise or fall toward to their natural place. (More on what Aristotle meant by the words “natural place” can be found here.) They do so in the concentric layers which surround the center of the earth and which form the sublunary spheres. More relevantly, the natural place of fire is higher than that of air but below the innermost celestial sphere (which contains the Moon).

This is, of course, Aristotle’s teleological account of things. Thus the telos of flames is to rise upward toward the heavens. In that sense, flames display “purposive behaviour” — their purpose is to rise to the heavens.

Necessity and Essence

As just hinted at, necessity is built into many of Aristotle’s explanations.

In basic terms, Aristotle believed that given causes necessitate given effects.

So let’s take flames (or fire) again.

Aristotle would have argued that the flames under a pan necessarily boil the water in the pan. In contemporary terms, this relation between flames and the boiling water in a pan is one of causal determination.

To return to the opening theme of flames rising upward.

So why is it also necessary that flames flow upward? Aristotle believed that this is because the essence of flames is that they flow upward. More abstractly:

Is it necessary that x displays behaviour (or action) A because x instantiates essence E.

Of course there’s a danger of conflating (or confusing) essence and necessity here. Surely only a physical (or perhaps abstract) property can be essential to any given x, not a specific behaviour (or action) of x.

However, why can’t a specific behaviour (or action) be seen as a property — even an essential property — of x? This means that the meaning of the word “essence” is — at least partly — a definitional (or stipulational) matter. That is, the word “essence” doesn’t itself have an essential meaning (or definition).

So we can conclude by arguing the following position:

If any given x always behaves (or acts) in a specific way, then that behaviour (or action) can be seen as being essential to x.

We can also link essence to behaviour (i.e., rather than seeing a specific behaviour — or action — as itself being essential). In this sense, we can argue that x’s essence E (i.e., a single property or a set of properties) brings about behavior A.

All this can be summed up and and stated in the following way:

If x has essence E, then it must behave (or act) in way A.

Yet even if there is essence E of x, how does E itself bring about behaviour A? In other words, how does E necessitate A?

The question as to whether behaviour can be essential is also relevant when it comes to subatomic particles. The following discussion on particles also raises the point that the philosophical notion of essence can be jettisoned entirely.

Relationalism: Particles and Mass

Subatomic particles are almost entirely defined in terms of their “relational” properties: such as their interactions with fields, forces or with other particles. (See my ‘Carlo Rovelli’s Relational Quantum Mechanics’.) Even the mass and charge of particles can’t be defined as (what philosophers call) intrinsic properties in that they’re determined by each particle’s place within a quantum system (or systems). More technically, a particle’s mass is determined by its relation to fields, forces and to other particles. And charge is similarly relational (see here).

This is the philosopher James Ladyman on this subject. He writes:

“[A] particle that never interacts with anything else could [never] have any value whatever for its mass.”

Nonetheless,

“since real particles will always interact with something or other [we can] ignore this”.

It can also be said that mass is “defined operationally” in that “the ratio of the masses of two particles is a constant of proportionality”. This too is a broader way of putting the better-known example which states that the number of electrons in an atom is equal to the number of protons. (Technically, a proton has a positive charge equal in magnitude to a unit of electron charge.)

Interestingly enough, Ladyman takes a particle’s lack of a (non-relational) essence to have the consequence that it can’t be what philosophers call an “individual”. (In broad terms, an individual is any given x which is deemed to be — to a large degree at least — self-sufficient, determinate and circumscribed.)

But firstly we need to see what this position is a reaction against. It’s largely a reaction against Leibniz. This is how Leibniz expressed his position on an individual

“The nature of an individual substance or of a complete being is to have a notion so complete that it is sufficient to contain and to allow us to deduce from it all the predicates of the subject to which this notion is attributed.”

On the other hand, James Ladyman and Don Ross — in relation to the physicist David Bohm’s position - say that

“[s]ince none of the physical properties ascribed to the particle will actually inhere in points of the trajectory, giving content to the claim that there is actually a ‘particle’ there would seem to require some notion of the raw stuff of the particle; in other words haecceities seem to be needed for the individuality of particles of Bohm theory too”.

The Irish philosopher Ernan McMullin also elaborated on this overall “relationist” position. He wrote:

“The use of namelike terms, such as ‘electron’, and the apparent causal simplicity of oil-drop or cloud-track experiments, could easily mislead one into supposing that electrons are very small localized individual entities with the standard mechanical properties of mass and momentum. Yet a bound electron might more accurately be thought of as a state of the system in which it is is bound than a separate discriminable entity… What is meant by ‘particle’ in this instance reduces to the expression of a force characteristic of a particular field[].”

Intuitive Knowledge of Essence and Necessity

We can now ask the following question:

How did Aristotle know about any given x’s essence or what is necessary about x’s behaviour?

An essence can’t be observed. And even if what is taken to be an essence could be observed, then that mere observation couldn’t tell a philosopher that such a property is essential. Properties don’t (as it were) broadcast their essential nature.

This meant that Aristotle couldn’t extract anything essential or necessary from flames (or any other given x) merely by observing them. Yet a similarly thing can be said about Isaac Newton and his discovery of the laws of gravity. In this case, too, Newton needed more than his apocryphal observation of a falling apple to conclude that it is gravity which accounts for such a thing. Newton also needed mathematical and theoretical insight — none of which could have been drawn solely from his observations.

Of course Aristotle himself would never have claimed that essences can be observed. Instead, essences are known through intuition.

(It can be questioned whether the word “intuition” is an accurate translation of anything Aristotle actually wrote. The word itself comes from the Medieval Latin intuitio; which can be translated as: “a looking at, [an] immediate cognition”.)

In the limited respect of his commitment to intuition, Aristotle was a Platonist.

Aristotle believed that intuition allows us to know (or “see”) things directly. That is, intuition allows us to make philosophical conclusions which go beyond observation, data or inductive support. However, unlike Plato, observation was still part of the overall story for Aristotle.

But what exactly is (philosophical) intuition?

Firstly, let’s ask the following question:

If we have direct access to the essence of any given x, then what would show the (philosophical) intuitor that he is wrong (or right) about what he concludes about x?

Interestingly enough, Aristotle is at one with those contemporary mathematical Platonists who similarly stress intuition. (See my ‘Platonist Roger Penrose Sees Mathematical Truths’.) Mathematical Platonists also claim to intuit (or “see”) those mathematical truths which haven’t — as yet — been proved.

So now, in turn, we can ask such a mathematician this question:

How do you know that some mathematical statements are true when they haven’t been proven to be true?

Alternatively:

What w/could tell you that you’ve made a mistake about a mathematical statement’s truth?

And now to move back to Aristotle:

What is it, exactly, to intuit the essence of any given x or to intuit the necessary causes (or behavior) of any given x?

It may now seem that Aristotle’s position can be seen as an example of “a priori theorising” or rationalist reasoning. Yet Aristotle’s approach could never have been completely rationalist because he did, after all, take pains to observe such things as flames and he meticulously noted how they behaved.

Conclusion

To finish: it’s now worthwhile noting that Aristotle’s positions on essence and necessity are opposed to what occurred during — and after — what came to be called “the scientific revolution”.

More clearly, most scientists came to disregard the search for necessary truths completely. In terms of essence, what became important was not essence; but the motions of objects and the particles which constitute them. In addition, qualitative (or subjective) descriptions were substituted with quantitative descriptions. The search for “final causes” was also largely abandoned and scientists concentrated instead on finding (still to use Aristotelian terms) “efficient” material causes. Finally, although Aristotle relied to some degree on observations (as stated above), he didn’t carry out any experiments.

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.