The Quantum World = The Mathematics

 First things first.

This essay may appear to advance two mutually-contradictory positions. On the one hand, it argues that without the mathematics (or, more correctly, without the mathematical formalism/s), there would be no quantum world — or at least no quantum mechanics. Yet, on the other hand, this essay also argues against Pythagoreanism — at least as it applies to this specific issue.

The main anti-Pythagorean argument in the following is that the world (or Nature) isn’t literally mathematics (whatever that may mean) or “made up” of numbers. It’s simply that, in quantum mechanics at the very least, without the mathematics, we’d have (almost) nothing.

Pythagoreanism: Things are Numbers

Pythagoreans believe that the world literally is mathematical. Or, perhaps more accurately, they believe that the world literally is (without the suffix “cal”) mathematics. I make this either/or distinction because if a physicist argues that “the world is mathematical”, then that may only mean that the world can be accurately — even very accurately — described by mathematics. The Pythagorean, however, states that “things are numbers”. Such a person therefore establishes a literal identity between the maths and the world (or parts thereof).

Yet we don’t need to accept the latter.

Having said all the above, it may well be the case that this essay does indeed advance a Pythagorean position — at least when it comes to quantum mechanics. That’s because it’s sometimes hard to tell what the Pythagorean position actually is. For one, it’s hard to make sense of the locution that the world is mathematical or that it’s made of numbers.

So the question which must now be asked is this:

What is it for “things” to be “numbers”?

To repeat: the Pythagorean position isn’t to only to argue that mathematics can describe (or model) things — it’s to argue that things literally are numbers. But what does that actually mean? And, as a consequence of that, it can now be asked if the statement “All things are numbers” is to be taken poetically or literally. Taken literally, it hardly makes sense. Taken poetically, it still requires much interpretation.

One interpretation of the Pythagorean position is that if things are numbers, then it’s no surprise that — for example — string theory is on top of things when it comes to describing reality. What I mean by that is this:

i) If things are numbers,
ii) and numbers are also used to describe (or model) things (which are numbers),
iii) then numbers are describing (or modelling) numbers.

That would mean that we never escape from numbers. Who knows, perhaps that’s precisely the result which Pythagoreans want!

To change tack a little.

The physicist John Archibald Wheeler (1911 — 2008) provided the best riposte to Pythagoreanism in physics. (I’m not entirely sure if this was his intention.)

It’s often been said that Wheeler used to write many arcane equations on the blackboard and stand back and say to his students:

“Now I’ll clap my hands and a universe will spring into existence.”

According to Pythagoreans, however, the equations are the universe.

And, after that comment, Steven Hawking (1942–2018) trumped Wheeler with an even better-known quote. He wrote:

“Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe?”

The American science writer Kitty Ferguson (1941-) offered a (possible) Pythagorean answer to Hawking’s question. She suggested the possibility that “it might be that the equations are the fire”. Alternatively, could Hawking himself have been “suggesting that the laws have a life or creative force of their own?”. Again, is it that the equations are the fire?

So what, exactly, does “breathe[] fire into the equation [to] make a world”?

John D. Barrow

Mathematics is an extremely useful tool. The English cosmologist, theoretical physicist and mathematician John D. Barrow (1952–2020), however, went one step beyond that truism. Barrow actually put his — arguably — Pythagorean position in the following way:

“By translating the actual into the numerical we have found the secret to the structure and workings of the Universe.”

Of course almost everyone can happily accept the Universe and its parts are assigned numbers… Or are described by numbers… Or are captured by numbers… Or are explained by numbers… Or are (to use Barrow’s own words) translated into numbers. The thing is, that’s not actually a Pythagorean position.

So, as a consequence of all the words above, it’s no wonder that so many people have believed that through maths (as Barrow puts it) “we have found the secret to the structure and workings of the Universe”. Yet even here there must be a non-Pythagorean (as it were) remainder. What I mean by this is that maths finds the secret of things which already and separately exist — in this case, the “structure and workings” of the world. Surely it doesn’t also need to be argued that these structures and workings are literally mathematics (or literally numbers).

… Or perhaps it does.

To repeat: to the Pythagorean, the world and its parts are actually mathematical. This means that it isn’t that maths is simply helpful for describing the world — the world itself is mathematical. Indeed one must take this literally. Here’s Barrow again on the Pythagorean position:

“[The Pythagoreans] maintained ‘that things themselves are numbers’ and these numbers were the most basic constituents of reality.”

Barrow then became ever clearer when he continued in the following manner:

“What is peculiar about this view is that it regards numbers as being an immanent property of things; that is, number are ‘in’ things and cannot be separated or distinguished from them in any way.”

Moreover:

“It is not that objects merely posses certain properties which can be described by mathematical formulae. Everything, from the Universe as a whole, to each and every one of its parts, was number through and through.”

As stated earlier, it’s hard to grasp what the sentence “things themselves are numbers” even means. Can we really argue that reality and its parts are mathematics (as in the “is of identity”)? Can we really argue that reality and its parts are literally made up of numbers or equations? And can we even argue that reality and its parts somehow instantiate maths, numbers or equations?

Max Tegmark

The physicist and cosmologist Max Tegmark (1967-) also puts the contemporary case for Pythagoreanism in the following very concrete example:

“[If] [t]his electricity-field strength here in physical space corresponds to this number in the mathematical structure for example, then our external physical reality meets the definition of being a mathematical structure — indeed, that same mathematical structure.”

To spell out the passage above.

Max Tegmark isn’t simply arguing that mathematics is perfect for describing the “electricity-field strength” in a particular “physical space”. He’s arguing that the electricity-field strength is a “mathematical structure”. That is, the mathematics we use to describe the electricity field is one and the same thing as the electricity field. Thus, if that’s really the case, then the so-called “miracle of mathematics” is hardly a surprise! And that’s because — as already stated above — we essentially have a situation in which maths is describing maths. And if maths is describing maths, then the word “describing” is surely not the right word to use in the first place.

Tegmark gives us more detail on his position when he tells us that

“there’s a bunch of numbers at each point in spacetime is quite deep, and I think it’s telling us something not merely about our description of reality, but about reality itself”.

It can be argued that Tegmark contradicts himself in the above.

At one point Tegmark argues that a field “is just [ ] something represented by numbers at each point in spacetime”. Note here that we have the two words “something [my italics] represented”. Yet elsewhere Tegmark also argues that the field “is just” (or just is) a mathematical structure — the latter two words implying that all we have is number. To repeat: Tegmark argues that the field is “represented” by “three numbers at each point in spacetime”. Yet he doesn’t (in this passage at least) also say that the field is a set of numbers (or even a “structure” which includes numbers).

So perhaps there’s a difference between arguing that (as the original Pythagoreans did) “things themselves are numbers” and arguing that the world is mathematical. (I may be drowning in a sea of grammar here.) The latter may simply state that the world exhibits features which are best expressed (or described) by mathematics. The former, on the other hand, states that the world literally is mathematics.

Now take the case of string theory.

Michio Kaku and String Theory

Not only is string theory seemingly more dependent on mathematics than all the other areas of physics (though, of course, that can be debated), it seems that some physicists even see string theory as being a “branch of pure mathematics”.

The string theorist Michio Kaku (1947-), for example, doesn’t hide from this when he quotes a “Harvard physicist” saying as much. In Kaku’s own words:

“One Harvard physicist has sneered that string theory is not really a branch of physics at all, but actually a branch of pure mathematics, or philosophy, if not religion.”

After Kaku puts the Pythagorean position, he then quotes Albert Einstein (i.e., as backup) stating the following:

“‘I am convinced that we can discover by means of purely mathematical construction the concepts and the laws… which furnish the key to the understanding of natural phenomena.’”

Einstein went deeper when he added these words:

“‘Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it.’”

Now all the words above do indeed sound Pythagorean (or, more broadly, Rationalist) — at least on the surface. And Einstein seems to more or less come clean about this in his final sentence. Thus:

“‘In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.’”

The strange thing is that Kaku also seems to offer us a mutually-contradictory account (in his book Beyond Einstein) of Einstein’s position on mathematical physics. Kaku writes the following words:

“Einstein revealed a clue to the way he arrived at his great discoveries: he thought in physical pictures. The mathematics, no matter how abstract or complex, always came later, mainly as a tool by which to translate these physical pictures into a precise language.”

Indeed elsewhere in the same book Kaku also writes:

“[] Feynman, and other great scientists []thinks in terms of pictures that express the essential physical concept. The math comes later.”

If we return to Kaku’s own position.

As for the charge (if it is an charge) of Pythagoreanism against Kaku, don’t take my word for it: take the words of the man himself. Firstly Kaku lays out the essential Pythagorean position in this way:

“Not surprisingly, the Pythagoreans’ motto was ‘All things are numbers.’ Originally, they were so pleased with this result that they dared to apply these laws of harmony to the entire universe.”

Then Kaku continues by arguing that “with string theory” what we have is “physicists [] going back to the Pythagorean dream”.

Quantum Mechanics

Philip Ball

The science writer Philip Ball (1962-) argues (in his book Beyond Weird: Why Everything You Thought You Knew about Quantum Physics Is Different) that the mathematics of quantum mechanics “doesn’t say anything about the ‘real world’”. Many physicists — and some philosophers — have also echoed that sentiment. Yet that may appear to be an odd position. It’s odd because if the mathematics of quantum mechanics is extraordinarily successful when it comes to predictions, applications, engineering, technology and whatnot, then (perhaps almost by definition) surely it simply must be about the real world…

Yet what work is the word “real” actually doing here? Does it imply that we (or the maths) must mirror the world? But how does — or would — that work? And even if the maths perfectly describes physical phenomena in terms of their magnitudes, values, strengths/charges, velocities, spatial dimensions/positions, etc., then is all that actually a case of mirroring the world itself? Surely if the maths of quantum mechanics mirrored the world, then it would look — and even be — the same as that world. In that case, what purpose would such mirroring actually serve? (Think here of the often-made claim: The best model of x is x itself.)

So it can be argued that maths can’t — literally — mirror the world.

Despite saying that, one thing is still certainly the case.

As stated in the introduction, without the maths, we’d have almost (or even literally) nothing to say about the quantum world — real or otherwise. When it comes to the quantum world, the usual (as it were) means of ownership aren’t available to us. That is, we can’t observe, feel, smell or (often) even imagine the quantum world. Thus the maths is all we’ve got.

All this is excellently expressed in the following passage from the science writer John Horgan (1953-):

“[M]athematics helps physicists definite what is otherwise undefinable. A quark is a purely mathematical construct. It has no meaning apart from its mathematical definition. The properties of quarks — charm, colour, strangeness — are mathematical properties that have no analogue in the macroscopic world we inhabit.”

Thus if maths is all we’ve got, then it’s not really a surprise that many physicists (i.e., the more philosophical ones) argue that quantum mechanics doesn’t really say anything about the real world. (This has been said since Niels Bohr in the 1920s.) Or, at the very least, everything important — or even relevant — that’s said about the quantum world is said by the maths.

So when Philip Ball also writes that Richard Feynman could only do “quantum theory” (i.e., the maths), then that’s not a surprise. That’s because it can be argued that the maths is all we’ve got and all Feynman had. Indeed when we stray beyond the maths into interpretation, then we (perhaps by definition) can’t help but get things wrong.

Or at least that’s one (sceptical) scenario we must consider.

Again, it’s not a surprise that — even — Feynman didn’t “know what the maths means”. That may be because the words what the maths means are — almost — meaningless. At the very least, there’s a hint here that we can’t go beyond the maths. Yet it’s still the case that so many philosophers, and a somewhat lesser number of physicists, believe that the maths is only second best to something far… deeper.

John Gribbin

The British science writer and astrophysicist John Gribbin (1946-) appears to agree with these conclusions. That is, as a consequence of much of what’s been said above, it can be concluded that all the imagery, picture painting, metaphors, analogies, etc. we find in the popular accounts — and even the technical interpretations — of the quantum world are simply (to use Gribbin’s own words)

“crutches to help us imagine what is going on at the quantum level and to make testable predictions”.

Indeed Gribbin also believes that

“none of [the quantum mechanical interpretations] is anything other than a conceptual model designed to help our understanding of quantum phenomena”.

Indeed Gribbin also talks about the interpretations of quantum mechanics:

“I stress, again, that all such interpretations are myths, They are not, any of them, uniquely ‘the truth’; rather, they are all ‘real’, even where they disagree with one another.”

Many will read Gribbin’s words as being very radical — and even deflationary — when it comes to quantum mechanics. Yet, despite all the above, Gribbin also happily acknowledges that all these quantum interpreters genuinely believe that their very own interpretations are true. He writes:

“[T]he interpreters and their followers will each tell you that their own favoured interpretation is the one true faith, and all those who follow other faiths are heretics.”

And that passage comes straight after Gribbin had told us that

“[a]t the level of equations, none of these interpretations is better than any other”.

Thus, logically, “none of the interpretations is worse than any of the others, mathematically speaking”. That said, all this hinges on precisely how we’re supposed to take the phrases “at the level of equations” and “mathematically speaking”.

Gribbin also becomes very psychological (or aesthetic) when he concludes (as the very end of one of his books) that we are

“free to choose whichever one gives you most comfort, and ignore the rest”.

Again, there may well be an argument that all the interpretations of quantum mechanics are superfluous when it comes to predictions, tests, experiments, technology, etc. However, that certainly doesn’t mean that all these interpretations are “equally good”. They may all be equally bad in the sense that they don’t make the slightest bit of difference when it comes to to mathematical theory, predictions, (quantum) technology, etc. However, are they all equally good in literally every other respect?

Despite all Gribbin’s words above, he still stresses the importance of what he calls a “physical model” of “mathematical concepts”. He writes (in his Schrodinger’s Kittens and the Search for Reality) that “a strong operational axiom” tells us that

“literally every version of mathematical concepts has a physical model somewhere, and the clever physicist should be advised to deliberately and routinely seek out, as part of his activity, physical models of already discovered mathematical structures”.

Yet even in Gribbin’s case and on a final quasi-Pythagorean reading, it’s still clear that a “mathematical concept” comes first and only then is a physical model found to square with it.

[I can be found on Twitter here.]

Colin McGinn’s Important Paper, ‘Can We Solve the Mind-Body Problem?’

Our brains are made of matter: neurons, biochemicals and other physical stuff. All these things are (as it were) “flesh and blood”. So, the British philosopher Colin McGinn (1950-) asks, how can “technicolour phenomenology arise from [this] soggy grey matter?”. In other words, why does the brain produce, cause or give rise to something so different from matter itself? What makes the brain so different to all the other human organs? More specifically, what is it, exactly, about billions of neurons that gives rise to consciousness or experience?

These questions were asked (if not in these precise ways) in Colin McGinn’s well-known and important paper, ‘Can We Solve the Mind-Body Problem’. That paper was published in 1989 in the journal Mind.

Mind and Brain

Despite all the above, the main issue of McGinn’s paper is not one of how or why consciousness arises from the the brain. It raises the possibility that the link between the brain and the mind (or, more accurately, consciousness/experience) may be — permanently? — closed off to us.

McGinn calls this “cognitive closure”. And the “property” responsible for that cognitive closure he names P.

McGinn focuses on our (possible) cognitive limitations.

But firstly let’s take different species of animals. McGinn tells us that they are

“capable of perceiving different properties of the world and no species can perceive every property things may instantiate”.

Think here of bats and echolocation. Dogs too can hear sounds which we can’t even register and they have a far better sense of smell than human beings. So what’s true for other animal species will also be true for human beings.

Yet all these examples only display sensory — i.e., not cognitive — limitations and extensions.

So let’s get back to cognitive closure.

McGinn’s (or our) P is a real (or actual) property. It may even be concrete (e.g., part of the brain). Thus McGinn doesn’t believe that he’s (to use his own term) “irrealist” about P. That said, McGinn does cite the possibility that P is noumenal. As with Kant, noumena are permanently unknowable — by (Kantian) definition. (Kant also believed — living before Darwin - that the structures of the mind-brain would remain static for… well, the rest of time.)

It’s the very nature of the mind-brain link which renders P permanently closed off to us. Yet despite that possibility, P could still be — at least according to McGinn — part of a respectable naturalistic theory. Again, we shouldn’t be irrealist or mysterious about P. (Many philosophers do see Colin McGinn as a mysterian.) That said, one can’t help but see x as being mysterious if that x is also believed to be permanently closed off to us.

So does McGinn actually think in terms of permanent cognitive closure?

Hume, Locke and Kant

McGinn discusses another case of closure which occurred in the philosophy of David Hume (1711–1776).

To Hume, our perceptual limitations determine our cognitive limitations. (Hume was a thoroughgoing empiricist.) More clearly, Hume believed that because our “ideas” are always copies of sensory “impressions”, then that also meant that our concept-forming system itself must always rely on— in whichever ways — those impressions. Thus nothing can transcend the information provided by impressions. (It can be asked here what was Hume’s philosophical position — as an empiricist — on such things as the existence of distant stars, the past, other minds, mathematics, etc.)

Now for Locke.

According to McGinn, the English philosopher John Locke (1632–1704) also believed that “our ideas of matter are quite sharply constrained by our perceptions”. To Locke, this also meant that (to use McGinn’s words) “the true science of matter is eternally beyond us”. Mind-free matter is a (to use Locke’s own well-known phrase) “something-I-know-not-what”. In concrete and specific terms, this means (or in the 17th century it meant) that we can never know what, say, solidity “ultimately is”. (Why does McGinn feel the need to use the word “ultimately”?) Again, this doesn’t mean that Locke believed that nature “is itself inherently mysterious”. (It can be asked how McGinn knows that Locke didn’t think this.) According to McGinn’s Locke, the mystery simply

“comes from our own cognitive limitations, not from any objective eeriness in the world”.

Kant, on the other hand, believed that knowledge begins with impressions (or, more accurately, with “phenomena”). However, he also believed that impressions aren’t the sole source of our knowledge. That’s primarily because Kant postulated innate a priori concepts and categories which (as it were) get to work on phenomena.

Back to McGinn

Colin McGinn explicitly states his position with regards to P. To repeat: he claims to resolutely shuns what he calls “the supernatural”. P must be “natural”. (How does he know this?) He goes on to say that

“it must be in virtue of some natural property of the brain that organisms are conscious”.

McGinn then concludes:

“There just has to be some explanation for how brains subserves minds.”

Again, why must P be natural? Is this a case of McGinn having (as it were) faith in naturalism? Alternatively, is McGinn’s theory of P naturalist in the first place?

Alan Turing on Intuition and Human-Machine Computation

What is a computation?

According to Alan Turing (writing in 1936/7), when it comes to human beings, a computation is the following:

A computation occurs when the human mind carries out a mental action according to a rule.

The words above (which aren’t Turing’s own — exact — words) don’t mean that people know that they’re following a rule. (Therefore people don’t know what that rule is.) Human grammar, after all, is also rule-governed; though not all children — and even many adults — can formulate the grammatical rules they employ. Nonetheless, almost everyone still follows these rules. (This is similar to both the philosopher’s notion of tacit knowledge and Noam Chomsky’s theory of innate universal grammar.)

It was said that computability is the mind’s following a rule. Well, Turing believed that this is usually the case. So does that mean that there can be computability (or computations) without the following of rules?

More specifically, when machines (or computers) modify their own behaviour, is that an example of not following a rule?

It’s certainly the case that computers can do things which weren’t predicted by their programmers (or designers). So does that mean that such computers aren’t following any rules? After all, they could be following new rules which they have (as it were) created themselves. So not following the programmers’ rules doesn’t automatically mean that computers aren’t following any rules at all.

We can even say that such computers have genuinely learned (forget the semantics for now) something which wasn’t fed into them by their programmers (or designers). However, they may still be following rules. In fact the new rules may be the logical/mathematical consequences of the programmers’ old rules.

That said, how does all this stuff about rule-following computers directly connect with the brains and minds of human beings?

Alan Turing did think that the human brain is a machine… Or at least he thought that many — perhaps all — of the functions of the brain are that of a machine. Nonetheless, he also believed that the brain is so complex that it can give us the impression of not following a rule.

Now it seems clear that it’s the complexity of the brain that generates only an “appearance” of the brain not following a rule (see here). So that basically means that even though the brain appears not to be following a rule, it may still be doing so. It’s just that the brain is so complex that the investigator — or even the (as it were) owner of the brain — couldn’t know all the rules which the brain is following. Similarly, the complexity of the brain may also generate the belief that it is an “indeterministic machine”. Yet if the brain were truly indeterministic, one could also question its status as a “genuine machine”. (Douglas Hofstadter, for example, seems to have believed — at least at one point in his career — that if a “machine” does go beyond the rules, then, by definition, it can’t actually be a machine — see here.)

Following on from all this, does that mean that if the computer has learned something — or has created its own rules — that it’s displaying (or showing) what’s often called “genuine intelligence”? After all, the computer has gone beyond what the programmer programmed. It can be supposed that all this depends on the semantics of the word “intelligence” (as with the word “learned” earlier). If not following the rules of the programmer is a case of genuine intelligence, then the computer is displaying genuine intelligence. Nonetheless, is not following the rules of the programmer really genuine intelligence or is it something else? In that case, what exactly is it?

Intuition

So what is intuition?

It depends on how the word is used and in which context it’s being used. In Alan Turing’s case, we (or mathematicians) use our intuition when seeing the truth of a formally unprovable Gödel sentence. That’s because Gödel sentences can’t be proved. Nonetheless, they’re true and they’re taken to be true.

How do we know they’re true without mathematical proof?

According Kurt Gödel himself, it’s through the use of human intuition (see here).

And if a Gödel sentence can’t be formally proved, then it can’t be shown to be true through “mechanical” methods. Again, it can’t be proved because proof is deemed to be a mechanical process (at least in this respect).

Another way of looking at intuition is with another of Turing’s ideas: the “oracle”.

In the case of a Gödel sentence, the mathematician (or the oracle) simply “has an idea” that the Gödel sentence is true. That is, he doesn’t use a mechanical method to establish its truth. He has an idea (or an intuition) that it is true. This hints at the brain (not the mind) working in ways which are way beyond conscious thought. That is, intuition is a result of the brain (not the mind) indulging in unconscious processes.

You may now ask how something can be established as true — especially in mathematics — without proof. You may also ask how truths can be established — especially in maths — only on the flimsy basis of a mathematician’s (or even of hundreds of mathematicians’) intuition or his simply “having an idea”.

Computer scientists — and the philosophers of mind who focus on computer/brain comparisons (or who even see the brain-mind as a literal computer) — will like Turing’s conclusion (of 1945/6) that algorithms are enough to account for all mental activity. Bearing in mind the previous comments about intuition, Turing believed that algorithms also encompassed non-mechanical intuition.

Just as intuition follows algorithms (therefore rules), Turing believed that what he called “initiative” didn’t require uncomputable steps either. In other words, both human and computer initiative is also a mechanical process. (That would make the idea of computer’s showing initiative less problematic for the simple reason that what it’s doing is still a computable — or mechanical - process.) However, as stated earlier when it was mentioned that computers may go beyond the rules (or programmes) created by the programmer, even if a computer departs from the computations which were programmed by the programmer, it would still be following a (new) rule, indulging in computations, or following mechanical processes.

Indeed — what else could a computer (or machine) be doing?

Another way of looking at a computer’s — or a Turing machine’s — ability to follow its own rules (or to show initiative) is for the programmer to engineer an element of randomness into the computer (or into the programme). That was what Turing did with his Manchester computer of 1948/50. That means that such randomness is deemed to (as it were) bring about intuition (or initiative). However, it would still be intuition (or initiative) that’s grounded in computation or in mechanical processes. The randomness, therefore, would simply be a result of the computer not abiding by the programmer’s rules (or programmes). It wouldn’t — or doesn’t — mean that the computer has gone beyond rules or computations.

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G.E. Moore and Norman Malcolm on Certainty

The philosophers Bertrand Russell (1872–1970) and A.J. Ayer (1910–1989) once argued — along with many others — that no empirical statement could ever be certain. As a consequence of that position, Russell particularly targeted the use of the word “certain” in ordinary language.

The American philosopher Norman Malcolm (1911–1990) put what he called “the philosopher’s position” in this way:

“[I]t never has been and never will be right, for any person to say ‘I know for certain that p’, where p is a material-thing statement, is that he regards that form of speech as improper. He regards it as improper in just the same way that the sentence ‘I see something which is totally invisible,’ is improper… in the sense in which every self-contradictory expression is improper.”

Certainly at first glance it does not seem that the sentence

“I know for certain that I have two hands.”

is linguistically “improper” and/or indeed “self-contradictory”. The sentence

“I see something which is totally invisible.”

on the other hand, can be seen as linguistically improper and/or self-contradictory. That said, it’s not even clear that the word “improper” should be used about the above. After all, I understand it. Why not simply say, instead, that it’s false or that it does indeed say something which is self-contradictory. It’s just odd (or even pompous) to argue that it’s linguistically improper (or a “misuse” of language). Indeed even Noam Chomsky’s well-known sentence

“Colorless green ideas sleep furiously.”

can be deemed to be linguistically proper in that it’s grammatically correct. What’s more, perhaps Ludwig Wittgenstein (1889–1951) might have argued that there could be a language-game in which the sentence above is perfectly proper. Thus the same may go for “I see something which is totally invisible”. That sentence can still be said to be (simply) false and/or self-contradictory.

Again, the sentence

“I know for certain that I have two hands.”

is only improper if one knows about — and is committed to — the philosophical position which renders it improper and/or self-contradictory. Yet ordinary language isn’t really (or at least always) a philosophical language. So why should the utterer have prior knowledge of Bertrand Russell’s — or any other philosopher’s — philosophical position before he/she can say something about his/her hands? Indeed what if another philosopher says something different about that very same sentence? For that matter, what if a philosopher sees it as perfectly proper — as G.E. Moore himself did?

We can’t escape the fact that even if this sentence is improper and/or self-contradictory, then it is so only according to a prior philosophical theory which we may have no knowledge of. And even if we have knowledge of — or a commitment to — it, then we can still separate it from our ordinary-language locutions and commitments. Yet Bertrand Russell and A.J. Ayer suggested that we use the language of philosophers — and not ordinary language — when we’re tempted to say things like “I know for certain that I have two hands”. They argued (at least at one point in their careers) that only in logic and mathematics can statements be certain. (No empirical statement can ever be certain.) Thus one must conclude that anything we say about the two hands in front of us can never be certain.

G.E. Moore on Certainty

As a riposte to all that, Norman Malcolm quotes the English philosopher G.E. Moore (1873–1958) in this way:

“‘It is a proper way of speaking to say that we know for certain that there are several chairs in this room, and it would be an improper way of speaking to say that we only believe it, or that it is only highly probable!’”

“The philosopher” says that Moore’s way of speaking is “improper” and Moore (who was a philosopher) says that the philosopher’s way of speaking is improper. Moore’s way is improper, according to the philosopher, because we can never be certain about the empirical matter of there being two chairs in a room. The philosopher’s way is improper, according to Moore, because he’s using the word “certain” in its strictly logical sense — i.e., not how it’s used in ordinary language. Yet both the philosopher and Moore are talking about the correctness or otherwise of “linguistic statements”; not about the truth or falsehood of an empirical fact. According to Moore, “when we sat in a room seeing and touching chairs” it would simply be wrong (according to the dictates or ordinary language — if such things even exist) to say that

“we believed there were chairs but did not know it for certain, or that it was only highly probable that there were chairs”.

Quite simply, phrases like “highly probable” aren’t (often?) used in ordinary language. Or, if they are, then they aren’t used in contexts like this. Instead they’re used in sentences such as this:

"It is highly probable that the Loch Ness monster does not exist."

Not in sentences like this:

“It is highly probable that these chairs in front of me exist.”

So we can now say:

“I believe that the Loch Ness monster exists.”

However, we can’t say:

“I believe that these chairs in front of me exist.”

Similarly, many people would say that they aren’t certain that God exists; though they are certain that the two chairs in front of them exist. Another way to put this is as Moore did. He argued that if a child were to say (not that he would) “that it was ‘highly probable’ that there were chairs there”, then “we should smile, and correct his language”.

We can now ask, then, whether or not all these disagreements are really just about language, semantics or language use; rather than being about whether or not the chairs really do exist or whether we can be certain that they exist.

So doesn’t all that raise questions against Moore’s thesis?

What Norman Malcolm wrote next suggests that this isn’t just a question of language or semantics. It is, in fact, about what is expressed by “empirical statements”. Malcolm continued:

“Moore’s reply constitutes a refutation of the philosophical statements that we can never have certain knowledge of material-thing statements… and so shows us that Ayer is wrong when he says that ‘The notion of certainty does not apply to propositions of this kind.’…”

So Moore wasn’t just arguing that we’re allowed to use the word “certain” in “material-thing statements” (i.e., not just in logic and maths). He was arguing that we’re certain that what those statements claim is in fact the case. In this case, we’re certain that the chairs in front of us exist. After all, “material-thing statements” are about… well, material things. And Moore argued that we’re certain about the existence of (certain) material things such as chairs and our hands. Indeed what would be the point of being able to use the word “certain” if we weren’t (really) certain about material things? (Similarly with the word “know”.)

Having said that Moore’s position isn’t just a question of correct usage or semantics, Malcolm does then go on to make a distinction between the phrase

“I know for certain…”

as it’s used in — or about — a priori statements; and as it’s used in — or about — empirical statements. In each case, the word has a different (to use Malcolm’s word) “sense”. Malcolm wrote:

“The truth is, not that the phrase ‘I know for certain’ has no proper application to empirical statements, but that the sense which it has in its application to empirical statements is different from the sense which it has in its application to a priori statements.”

We’ll need to know what sense the word “certain” (or the phrase “I know for certain”) has when applied to an empirical statement. It clearly can’t have the sense that “the negative of it is self-contradictory”. That sense, then, must be that if we can see and touch our hands or the chairs in front of us, then we have the right to be certain that such things exist. The “negative” of such empirical statements will indeed not be self-contradictory. That’s because these statements aren’t supposed to be a priori or logical in nature — i.e., they don’t even open up the possibility of any contradictions or self-contradictions. However, couldn’t we say that it’s indeed self-contradictory to claim the following? -

“I can see and feel these chairs in front of me but I don’t know for certain that they exist.”

At least according to the dictates of ordinary language that sentence is contradictory. In this example, then, it wouldn’t (only) be a question of “correcting the language” of the person who uttered the above. In other words, if you can see, feel and kick the chair in front of you, then what more do you want or need? You can’t have logical or a priori certainty about this claim because it isn’t a logical or an a priori statement. So let us, Wittgenstein might have said, keep our language-games apart. And by keeping them apart, we can also stop all this silly fuss and bother about certainty or the correct use of the word “certainty”.

References

Malcolm, Norman. (1952/1964) ‘Moore and ordinary language’.
Moore, G.E. ‘Certainty’ (1959).

[I can be found on Twitter here.]

A Short Take on the Indeterminacy of Meaning

The American philosopher Ernest Lepore (1950-) put the case that if the (general) indeterminacy of meaning thesis (which is broader than — and a consequence of — the indeterminacy of translation and the indeterminacy of reference theses) were the case, then

“you and I mean different things when we say ‘dog’”.

The (perhaps) counterintuitive and/or dismissive response to that statement is to ask this question: Is that really such a bad thing? Indeed do the implications of Lepore’s statement actually amount to much?

Ask yourself what it means — yes, means — when someone states the following:

You and I mean the same thing by the word “dog”.

Of course we can both say that we both mean dog by the word “dog”. However, that won’t get us very far. (This is somewhat like Alfred Tarski’s seemingly vacuous T sentence. Namely: “The sentence ‘Snow is white’ is true if and only if snow is white.”)

Lepore’s claim (i.e., “you and I mean different things when we say ‘dog’”) wouldn't seem quite so outlandish if we were referring, instead, to a word like “democracy”. Thus:

You and I mean different things when we say “democracy”.

Most of us would happily accept the fact that what different people mean by the word “democracy” may be — or even will be — different (if only to varying degrees). So why couldn’t the same — or something similar — also be the case with the word “dog”?

Of course the word “democracy” is an abstract noun; whereas the word “dog” is a concrete noun. But does that difference really make much of a difference? Sure, it’s easier to refer to — or pick out — a dog than it is to do the same with democracy. Yet that ease of referring to — or picking out — a dog with the word “dog” doesn’t necessarily mean that you and I mean the same thing — or exactly the same thing — when we use that word. Both you and I always — or nearly always — successfully pick out (or refer to) dogs with the word “dog”. Again, it doesn’t follow from this that we do so because we mean the same thing by the same word.

Does it matter, for example, that you think of — or simply mean — creatures with wagging tails, and I think of — or mean — canine animals when we use the word “dog”? (Of course we may not even rely on any explicit — or conscious - definitions at all when we use the word “dog”.) Yet despite these (possible) differences, both you and I both still successfully refer to — or pick out — the same things when we use the word “dog”.

All this may even be true if we take W.V.O Quine’s well-known example of rabbits. The American philosophy Michael J. Loux (1942-) put it this way:

“If there is indeterminacy in our use of the term ‘rabbit,’ [or ‘dog’] it is an indeterminacy that is somehow rabbit-involving [or dog-involving]. While it may be indeterminate whether a given individual’s use of the term ‘rabbit’ [or ‘dog’] involves a reference to three-dimensional enduring substances, their temporal parts, their undetached spatial parts, their fusion, or some universal they instantiate, all the alternative reference assignments that are admissible here are tied together by being, in some rough sense, all rabbit-centered [or dog-centered].”

Alternatively, perhaps we don’t really mean anything when we use the word “dog”. Or perhaps it’s our causal and/or historical connections with dogs (i.e., rather than our internal meanings) which really matter. In addition, perhaps you (in homage to Quine again) pick out dog-parts, temporal dog-slices or doghood with the word “dog”; whereas I — being the sensible person that I am — pick out… dogs when I use the word “dog”.

Lepore goes on to say that

“this line of argument leads to such surprising claims as that natural languages are not, in general, inter-translatable”.

According to Lepore, Quine believed this. Yet Quine could speak fluent German — so how could he have believed this?… Of course I’m being slightly rhetorical here.

What Quine did believe is that the (in his own words) “facts of the matter” about, say, German meanings aren’t kept complete and intact in any translation. That’s because there are no facts of the matter when it comes to translation and therefore to meaning! (All of this is well captured in Quine’s paper ‘Ontological Relativity’.) That’s primarily because there can be no one-to-one correspondence between the language of translation and the object language — and that’s not only the case when it comes to “alien speakers”. Again, that’s primarily because there’s nothing determinate to translate — and therefore no determinate translation — in the first place. And this isn’t only because of the banal reason — among others — that there are certain German words which don’t (really) have an English translation (at least not a precise one). It’s because there are no abstract meanings, meanings as mental items or even meanings simpliciter — to correctly (or incorrectly) translate in the first place.

Quine himself argued that all we actually have (or need?) is “stimulus-synonymy” (as well as behavioural-response synonymy) when it comes to the translation of alien utterances. And that’s also the case — if to a lesser extent — when it comes to German speakers and even to our fellow English speakers! As for a circumscribed translation, even if all the physical and behavioural facts are available, various translations — perhaps indefinitely many — will always be possible. Thus it will be our choice of which “translation manual” to use — and that will largely determine the translation. Thus no single translation will capture the true meaning better than any other translation. Again, that’s because there’s no true meaning to capture. That said, perhaps some translations will be better or worse than others (say, for pragmatic or other reasons). We may — or will — also need to adopt the principle of charity when it comes to translating other speakers. Still, these qualifications (if that’s what they are) of the indeterminacy of meaning thesis may be hard to argue for.

The only way it would be possible to capture the true meaning of any utterance would be if there were true meanings to capture in the minds of the speakers of another language. Yet there are no such things — as Ludwig Wittgenstein and others have forcefully shown. (See the Private Language argument.) Indeed even if there are such private mental items, then how can we — as third parties — gain access to them? Thus the only way we can access them is when the translated person overtly expresses them. (Quine often used the words “overt behaviour”.) Consequently, if there’s no missing part when it comes to such behavioural expressions (in that they don’t really lose anything of the mental meanings), then perhaps such mental meanings can be entirely dropped from the picture. And that’s because such a (private) mental meaning (or item) will be a (to use two of Wittgenstein’s phrases) “wheel without a function” and a “beetle in a box”.

Lepore himself goes on to admit that the people he calls “mitigated holists” needn’t be too concerned with meaning indeterminacy in that to them

“a notion of similarity of meaning (of mental content) somehow replaces the notion of semantic identity”.

Thus perhaps we simply don’t require any identity of meanings in order to understand other people when they use the same words we use. And we especially don’t require it when an alien speaker uses a word which we believe is his own word for, say, dogs. However, if we can’t have identity of meaning, it can now be asked why we can have similarity of meaning. Perhaps if meaning identity isn’t available (or possible), then neither is meaning similarity. A further question is:

What would count as similarity of meaning and how would that in itself solve the problem of the indeterminacy of meaning?

[I can be found on Twitter here.]