Why Empty Logic Leads to the Liar Paradox

i) Introduction

ii) Empty Syntax

iii) Roger Penrose on Mathematical Formalism

iv) The Logical Syntax of Sentences and Premises

v) The Liar Sentence

vi) Conclusion

Many philosophers and logicians have analysed the forms of arguments and shown us that the same forms occur across many different cases. They’ve also shown us that when it comes to what are called argument-forms, the semantic content of the premises doesn’t (really) matter as long as the argument-form (which involves the proper use of the symbols for the premises and logical operations) is adhered to. 

This piece also deals with the form (or syntax) of the premises (or sentences) themselves. It attempts to show the philosophical consequences of premises (or sentences) being taken to be without semantic content. This is to deal with something more than the simple fact that the semantic content of premises (or sentences) can be ignored by (formal) logicians: it’s an argument that in some cases (such as the sentence “This sentence is false”) the premise (or sentence) doesn’t have semantic content in the first place (i.e., even after that content is analysed or determined). More specifically, an attempt is made to show that if a sentence in an argument has no semantic content, then that may be at least one reason why it can lead to a semantic paradox. (Note the irony of the adjective “semantic”.)

The first section deals with a purely syntactic logic as it applies to argument-forms. In the second section, Roger Penrose’s position on mathematical formalism is discussed in order to see if it can shed light on what may be deemed to be logical formalism. The third section deals with the syntax of premises (or sentences) mainly as they are found in logical arguments. It also comments on the philosophical consequences of dealing with premises (or sentences) in purely syntactical terms. The Final section pays exclusive attention to the “liar sentence” and defends the position that because it can be taken to have no semantic content at all (therefore it is treated purely syntactically), then that may be at least one reason why it leads to a paradox.

Empty Syntax

Noam Chomsky’s well-known example of a syntactically- and grammatically-correct surreal sentence.

The broad position of this piece is at least partly expressed by Stephen Read in the following:

“What must be acknowledged is that belief that every valid argument is valid in virtue of form is a myth, and exclusive concentration on the study of pure forms of argument does a disservice both to logic and to those who can be helped by it.”

That’s Stephen Read’s broad philosophy-of-logic position. And in the following piece a similar way of thinking is also applied to the premises (or sentences) actually found within arguments. 

Read then goes on the apply his (as it were) non-formalist position to the actual technical detail of valid arguments. He continues:

“Validity is a question of the impossibility of true premises and false conclusion for whatever reason, and there are arguments which are materially valid and where that reason is not purely logical.”

It can be argued that Read goes too far in the first passage above. That passage is a clearly a philosophical take on logic. Yet one can fully accept a purely syntactical (or formal) logic because such a thing clearly has its place. However, it’s also true — at least in part — that such a logic is indeed (to use Read’s words again) “a disservice to those who can be helped by it”. That is, for those interested in the philosophy of logic and in using logic in their philosophical arguments (i.e., logical arguments which aren’t about logic), the purely syntactical approach may seem odd or even pointless. On the other hand, syntactic (or formal) logic has had a great impact on metamathematics and mathematics itself. In addition, the purely formal way of looking at logic has been of great help to the development of programmes, computers and other technologies.

So one doesn’t need to reject syntactic (or formal) logic. One simply needs to accept that a purely syntactic logic is one thing, and a logic with important or relevant semantic content is another. Nonetheless, there’s more to it than that. If we recognise the important difference between syntactic logic and semantic logic, then that will help us explain much about the so-called semantic paradoxes and sentences like “This sentence is false”. (It may even help us develop a position on Gödel sentences.)

Stephen Read makes the formalist position on logic very clear when he states the following:

“Logic is now seen — now redefined — as the study of formal consequence, those validities resulting not from the matter and content of the constituent expressions, but from the formal structure.”

We can now ask: 

What is the point of a logic without material, semantic or relevant content? 

Wouldn’t all the premise, proposition, predicate, etc. symbols — not the purely logical symbols —used simply be self-referential in nature? (Thus all the p’s, q’s, x’s, F’s, G’s etc. would be autonyms.) And what would be left of logic if that were the case? Clearly we could no longer really say that it’s about argumentation at all— or could we? That is, we can still learn about argumentation from argument-forms which are purely formal in nature. The dots don’t always — or necessarily — need to be filled in.

Example

The following is both a valid and sound argument:

Football is a sport.
 — — — — — — —
∴ Snow is white.

To the layperson the above will seem both wrong and silly. However, classical logic allows premises and conclusions to be completely unrelated. That is, it is the form (hence argument-forms) of the argument that matters to logicians.

Now form is just as much a matter of premises (or sentences) as it is about arguments. As stated, the argument-form above is both valid and sound. But what of the premise (or antecedent sentence) itself? This premise is true. So is the conclusion. But what about this argument? -

This sentence is false.
 — — — — — — — — — 
∴ Football is a sport.

The premise above is perfectly acceptable — according to classical logic. The conclusion is also true. But is the premise also true… or false for that matter? If it is true, then the argument is valid. If it is false, then the argument isn’t sound. 

What if the premise is both true and false? Then, according to the principle of explosion, “anything and everything follows from it”. Thus the conclusion must also follow from it.

But what if the premise is neither true nor false, or both true and false? And what if its status as a proposition (i.e., as used as a premise) is questionable in the first place?

Roger Penrose on Formalist Mathematics

Roger Penrose and Plato.

A parallel to this issue of a purely formal logic can be found in mathematics — or at least in the philosophy of mathematics (or in metamathematics). For example, here’s the mathematical physicist and mathematician Roger Penrose on mathematical formalism:

“The point of view that one can dispense with the meanings of mathematical statements, regarding them as nothing but strings of symbols in some mathematical system, is the mathematical standpoint of formalism.”

Penrose has a serious problem with the “point of view” that is (or was) mathematical formalism. He goes on to say that “[s]ome people like this idea, whereby mathematics becomes a kind of ‘meaningless game’”. Penrose concludes:

“It is not an idea that appeals to me, however. It is indeed ‘meaning’ — not blind algorithmic computation — that gives mathematics its substance. Fortunately, Godel dealt formalism a devastating blow!”

One doesn’t need to be a Platonist to understand how formalism can also be a problem for logic. As it is, one can believe that both mathematics and logic can indeed have a purely formal aspect. Indeed one can see maths (though not logic) as always being purely formal (i.e., the contrary of Penrose’s position). However, in logic (unlike in mathematics) we have logical arguments expressed in a natural language. And these arguments contain sentences which include names and predicates. That is, they contain premises (or sentences) which have extensional, referential and existential import. None of this is true of maths — unless, that is, one believes that numbers, sets and even functions (as abstract objects) are the referents or extensions of numerals and other symbols!

Penrose on Mathematical Syntax

In the above the word “syntax” has been used a few times. So what exactly do I mean by the word “syntax”? Let Penrose again (though he’s talking about mathematics) explain:

“[‘S]yntactically correct’ essentially means ‘grammatically’ correct — i.e. satisfying all the notational rules of the formalism, such as brackets being paired off correctly, etc. — so that P has a well-defined true or false meaning.”

Indeed this “formalism” is given even more importance by Penrose because he places it within the context of David Hilbert’s quest for a systematic and secure grounding of all mathematics. Penrose continues:

“If Hilbert’s hope could be realized, this would enable us to dispense with worrying about what the propositions mean altogether! P would just be a syntactically correct string of symbols.”

Now Penrose rejects this formalism for mathematics. And here I am applying it to logic. Prima facie, if formalism can be challenged in the case of maths, then it seems to be far easier to challenge it when it comes to logic. After all, logic (among other things) makes explicit use of the words “true” and “false”. In addition, logic’s Ps and Qs are meant to stand in for propositions, which themselves can be (or are) expressed in natural-language sentences which contain names, predicates and suchlike. None of this can be found in maths. Indeed in maths the words “true” and “false” are rarely used in the symbolism itself. Any use of the words “true” and “false” that can be found are usually found in the domains of metamathematics and the philosophy of mathematics, not in mathematics itself.

So, as with logic, maths can be seen purely as a “game” (to use Penrose’s word just quoted above). Alternatively, maths can be given an interpretation or a philosophical account. Thus it can be said that formalists are like the shut-up-and-calculate brigade in quantum mechanics who’re only concerned with the mathematical formalism and how that formalism helps with experiments, predictions and observations. However, other physicists are also keen on the “interpretation” of quantum mechanics. That means that we can conclude by saying that Penrose is keen on the interpretation of maths. (His interpretation is essentially Platonic — see here.)

As for logic, rather than maths.

Robert S. Tragesser (who discusses Gottlob Frege) explains logical formalism in a similar way to which Penrose explained mathematical formalism. He wrote:

“Frege believed that the principle virtue of such formal-syntactical reconstructions of inferences — as validly moving on the basis of the meanings of the signs for the logical operations alone — was that it eliminated dependence on intuition and let one see exactly on what our inferences depended.”

The important point here being that it’s only “the signs for the logical operations alone” that have “meanings”. In other words, it doesn’t matter what conditions, events, facts, etc. A and B (in this case) stand for (or, in other cases, which natural-language sentences the symbols P and Q are meant to stand for), or what the conditional sign ⊃ “means” (i.e., beyond its syntactic role), what matters is the “formal-syntactical reconstructions of inferences” which (according to Frege at least) allow us to bypass any use of “intuition”.

The Logical Syntax of Sentences and Premises 

One problem with standard (or classical) logic is that the content of a premise is irrelevant. What matters is its form. So just as form is important in logical arguments, so form is also important when it comes to what are taken to be the premises (or sentences) found in logical arguments. This last aspect of logical formality hasn’t been discussed as much as the former.

Of course all premises are taken to have a semantics by formal logicians. If this weren’t the case, then they wouldn’t be seen to be premises at all. Yet the precise semantics isn’t the concern of the formal logician (if there is such a pure being). He doesn’t himself establish the references and extensions of the premises he uses. That is, premises are taken to have already-established extensions and references — otherwise they couldn’t be taken to be either true or false in the first place.

So logic always has a semantics or an interpretation.

In addition, there’s the (sorta) semantics of the symbols and their operations themselves. That is, a semantics of the symbols qua symbols and of the operations qua operations. In this case, the symbols and operations — taken in and of themselves — are taken to have “meanings”.

The heart of the problem can be seen when it comes to the logician himself. Specifically, it’s the distinction most (formal) logicians make between a sentence which is written down (or spoken), and the proposition that the written (or spoken) sentence is said to express. Such logicians are only interested in the proposition. Thus the proposition can be seen as being “underneath” or “behind” the sentence. Or, less metaphorically, the proposition is seen as being the idealisation of the sentence. 

This emphasis on the proposition is but a means to an end. Put simply, propositions are much easier to logically manipulate than natural-language sentences (as with sentences/statements which are given Gödel numbers).

All this will depend on what we take a proposition to be. And that may end up being partly — or even wholly — a stipulational matter. For example (as we shall see), it can be argued that if the sentence “This sentence is false” is without reference and extension, then perhaps it can’t be a genuine proposition at all. Having said that, in logic, it is taken to be a bona fide proposition.

The (as it were) formalist position on sentences (i.e., not on arguments) is put very well by Bryson Brown in the following:

“These rules are based on the syntactic structure of the sentences, that is the symbols and how they are arranged in each sentence, rather than on an account of their truth conditions.”

So here, rather than concentrate on the syntactic structure of an argument form, Bryson Brown talks of the “syntactic structure of a sentence”. In other words, the sentences (or premises) in this particular “consequence relation” can be seen as being purely syntactical too.

So why this stress on syntax? Bryson continues:

“One advantage of this approach to consequence relations is that it focuses our attention on the process of reasoning, rather than on ‘meanings’ that are taken to lie behind that process.”

This means that one can reason without a semantics; just as one can do arithmetic without worrying about the meanings — or the ontology — of the numbers one is using. 

So let’s tackle one logically-acceptable (though still philosophically problematic) sentence.

The Liar Sentence

The following is the liar sentence:

This sentence is false.

One problem is that the liar sentence certainly looks like other sentences. Grammatically, it appears to be in very good shape. It’s in good shape in a similar way to which Noam Chomsky’s well-known “colorless green ideas sleep furiously” (see image above) looks like a grammatically-acceptable sentence. (There are, of course, clear distinctions which can be made between the liar sentence and Chomsky’s surreal sentence.)

But what about the following? -

This sentence is true.

By inference, if one accepts the sentence “This sentence is true”, then one must also accept its negation — namely, “This sentence is false”. Yet it’s the case that the sentence “This sentence is true” is less problematic than the sentence “This sentence is false”. That is, the former doesn’t engender a paradox. Nonetheless, it’s still as empty as the sentence “This sentence is false”.

But firstly, let’s get something out of the way.

Take the liar sentence again:

This sentence is false. 

Is that really the logical form of the following? -

“All Cretans are liars.”

Aren’t there obvious differences between “This sentence is false” and “All Cretans are liars”? That may not matter because the former is still taken to be the logical form of the latter.

We can summarise the sentence “All Cretans are liars” in the following ways:

1) The sentence is spoken by a human person.
2) It includes the extension that is the set Cretan liars.
3) It is spoken by a member of the set Cretan liars.
4) It includes a reference to the psychological (behavioural) act of lying.

The sentence “This sentence is false”, on the other hand, has no extension or reference. So in what sense can it really be said to be the logical form of the sentence “All Cretans are liars”? If anything, the sentence “All Cretans are liars” is more acceptable than “This sentence is false” because the latter has no content and is paradoxical; whereas the former both has content and is paradoxical. 

Of course one can argue that the extension (or reference) of “This sentence is false” is the sentence itself (i.e., even if the sentence itself has no references or extensions). Thus the sentence “This sentence is false” can only have an empty inscription as its reference or extension. But an empty inscription can’t be a proposition (or statement). It’s a mere collection of words with an acceptable grammar or syntax. That is, a grammar or syntax without any semantic content.

Here’s another take. One possibility is that the sentence

“All Cretans are liars.”

should actually be something like the following:

“Except for myself, all Cretans are liars.”

The above is a more natural — and less problematic — expression of the words “All Cretans are liars”. However, the universal quantifier “all” ( or ∀ in logic) is negated by the proceeding clause “Except for myself”. This also has the consequence of making the entire sentence no longer paradoxical or self-contradictory.

Of course all this hinges on the quantifier “all” and the problems self-reference throw up. In logic, it’s often agreed that quantifiers nearly always have a restricted range (or domain) which is determined by specific contexts. Does that mean that the word “all” in “All Cretans are liars” also has a restricted range? Is the speaker of the words “All Cretans are liars” that very restriction (or exception) himself? Yet if we take the word “all” literally, then he can’t be. However, if we take the word “all” contextually or as a quantifier with a restricted range, then the Cretan liar may well be that very exception. After all, in natural-language terms (therefore also in terms of context), many people would be happy to accept that when a person says that “All people are evil”, then he may well be exempting himself from that universal generalisation. Indeed if someone were to say (out loud) that “All people always remain silent”, then (by definition) he must be an exception to his own universal generalisation.

Following on from all that, it’s also the case that the liar sentence can be seen as being neither true nor false. But here again it depends on which version we’re talking about. It seems easier to believe that the sentence “All Cretans are liars” is neither true nor false than the sentence “This sentence is false”. Why is that? Because it hardly makes sense to say of an empty sentence that it is neither true nor false. On the other hand, it makes more sense to say that the sentence “All Cretans are liars” is neither true nor false in that this may make sense of its paradoxical nature.

So there’s no actual truth-gap when it comes to the sentence “This sentence is false” because — arguably — the issue of truth or falsehood can’t arise in the first place for an empty sentence. Again, a truth-gap result may make much more sense for the sentence “All Cretans are liars” than it does for “This sentence is false”.

****************************

Strictly speaking, then, if any approach to logic is purely syntactical in nature, then truth and falsehood (at least as most people see such things) simply aren’t the issue. Having said that, if the symbols and operations in such a logic are correctly adhered to, then why can’t we call its results (or conclusions) “true” and “false”? After all, since what’s called “the nature of truth” has created controversy throughout the history of philosophy, then why can’t we be a little logically ad hoc about this matter?

Kitty Ferguson asks: Why these constants of nature? Why this logic & mathematics?

Kitty Gail Ferguson was born in 1941. She’s a very successful and excellent American science writer who’s written books on Stephen Hawking, Tycho Brahe, Johannes Kepler and Pythagoras.

Why These Constants of Nature?

Kitty Ferguson asks various questions about the constants of nature (as found in her brilliant book The Fire in the Equations). She asks us why the “fundamental forces” are the way they are. (This question also ties in with other questions about their necessary or contingent nature.) Namely:

“[N]o scientific theory we have at present can tell us why the speed of light and the strengths of the fundamental forces of nature are what they are.”

So let’s reformulate that passage as a simple question:

Why is the speed of light, the strengths of the fundamental forces of nature, etc. the way that they are?

Can we say that these values and strengths just are the values and strengths that they are? They have to be of some value and strength. The fact that they have the values and strengths that they have is entirely contingent. However, if the values and strengths were necessary, then perhaps there would be answers to Kitty Ferguson’s questions. And, of course, all her questions are tangentially — as well as tacitly — linked to the anthropic principle; and, in some cases, the existence of God.

Take this more specific question:

Why is the speed of light 186,000 miles per second?

The speed of light is 186,000 miles per second because if it were slightly less (it can’t be more), then it wouldn’t be light. It would be something else. That is, if the speed of light were something else, then it wouldn’t be light that’s being discussed.

Now take this question:

Why does gravity have the strength and value that it has on earth and elsewhere?

Gravity has the force and value that it has because it wouldn’t be gravity if it had a different strength and value. It would be something else. The same goes for this question:

Why is the charge of an electron -1.6 x 10–19 coulomb and its mass 9.11 x 10–31 kilograms ?

The mass and charge of all electrons is x and y because if they weren’t that mass and charge, then they wouldn’t be electrons. They would be something else.

Nonetheless, all this seems to be truer of all electrons than of gravity and the speed of light. That is, it seems easier to conceive of a different speed of light, or gravity with a different strength and value, than it is to accept electrons with different masses and charges.

Why These Symmetries?

Ferguson says the same kind of thing about symmetries in the universe. She writes:

“[W]e might ask whether there are underlying reasons why this symmetry and not another should be the one to apply in our universe.”

We can also rephrase the passage as a simple question:

What are the underlying reasons why this symmetry and not another one should be the one which applies in our universe?

Again, there had to be some kind of symmetry — that is, if there’s any kind of symmetry at all. Sure, other kinds of symmetry might have been instantiated. They weren’t. (Again on the anthropic view, we can say that this question couldn’t have been asked without the given symmetries.) Thus perhaps the question is whether or not there was something before our symmetries; something responsible for our symmetries; or something more basic than our symmetries.

Why This Mathematics and That Logic?

Finally, Ferguson asks similar questions about mathematical logic. Thus:

“It’s a question of profound importance whether mathematical consistency required an Inventor. I’ve heard it asked at the end of public lectures on physics: ‘Is mathematical consistency as we know it the only way it COULD be — or is it conceivable it could be something different?…’ If the lecturer is a scientist or mathematician, he or she may answer that mathematical consistency just is.”

This too can be put as a simple question (though this time using Ferguson’s own words). This is her question:

“Is mathematical consistency as we know it the only way it COULD be — or is it conceivable it could be something different?”

The idea that mathematical consistency would need an inventor may strike some (or even many) people as ridiculous. Nonetheless, it may still be a good (or legitimate) question.

As for the possibility of alternative mathematics.

Some say that even the question can’t be constructed without begging the question. (This, of course, doesn’t rule out complementary or even rival mathematical systems, incompleteness, inconsistency, etc.)

As for logic.

If we must start with the basic building blocks of logic (first forcefully stated as long ago as Aristotle), then the very idea of truly independent logics is rejected by some (or many) logicians and philosophers — even by by some of those who accept paraconsistent and dialethic logics. Or as the philosopher Dale Jacquette once put it:

“Even paraconsistent logics that tolerate logical inconsistencies without inferential explosion, that accommodate contradictions but do not authorize the logically valid deduction of any and every proposition, do so according to strict rules, as strict as the rules that govern Aristotle’s syllogisms.”

Now the passage above isn’t saying that the “strict laws” of paraconsistent logic are the same as Aristotle’s. However, it is saying that there are rules in paraconsistent logic which — seemingly — aren’t even called into question. They are simply accepted.

So now perhaps we can say that same about the constants of nature; the symmetries which exist in our universe; and the basic “rules” of logic and mathematics.

Platonist Roger Penrose "Sees" Mathematical Truths

i) Introduction

ii) Does Penrose Go Beyond Mathematics?

iii) Gödel, Turing, Penrose

iv) Beyond Mathematical Platonism?

v) Seeing Truths

vi) Do Platonic Truths Stand Alone?

vii) Penrose’s Rationalism

viii) Laurence BonJour: A Rationalist of the 21st Century

vix) Conclusion: Is Roger Penrose Really a Platonist?

The following piece is about what the theoretical physicist and mathematician Roger Penrose has said about “seeing” certain mathematical truths. Penrose’s overall Platonic position is also discussed. Indeed the question as to whether Penrose actually has an overall Platonic position is also asked.

But before all that, let’s place Penrose’s position in some kind of context.

ii) Does Penrose Goes Beyond Mathematics?

Penrose has been very open-minded about (as it were) Platonic seeing when it comes to such things as “beauty” and “goodness”. Despite that, he’s never done any detailed work on any of these strictly philosophical issues. Much of what Penrose has said has been the result of interviewers pressing him on subjects which aren’t his speciality . (In most cases these have been attempts — by such interviewers — to get Penrose to backup their own prior religious or spiritual views — see here for a perfect example of this.)

One aspect of the wider context of Penrose’s Platonism concerns his deflationary position on “algorithmic thinking”. And, as a consequence of that position, his critical position on the possibility of strong artificial intelligence; as well as his controversial views on human consciousness.

And this is where the Austrian mathematician and logician Kurt Gödel inevitably enters the picture.

Let Penrose himself tell you about his first encounter with Gödel’s work and what effect it had on him. He wrote:

“Why did I believe that consciousness involves noncomputable ingredients? The reason is Gödel’s theorem. I sat in on a course when I was a research student as Cambridge, given by a logician who made the point about Gödel’s theorem that the very way in which you show the formal unprovability of a certain proposition also exhibits the fact that it’s true.”

What follows is perhaps more relevant to this piece. Penrose continued:

“[A]s long as you believe in the rules [of “any system of rules”] you’re using in the first place, then you must also believe in the truth of this [Gödel] proposition whose truth lies beyond those rules.”

And now for the relevant (as it were) clincher:

“This makes it clear that mathematical understanding is something you can’t formulate in terms of rules.”

This is where Platonic seeing comes to the rescue. Penrose believes that no machine (or computer) can have Platonic vision. That is, “we have a proposition that we can see [Penrose’s italics], by the use of insight, must actually be true”. However, “the given algorithmic action is not capable of telling us this”.

Penrose then goes into greater detail when he asks us this question:

“Why, then, can one not simply get a computer also to follow this Gödel argument and itself ‘see’ the truth of any new Gödel proposition?”

What’s more:

“The catch lies in ‘seeing’ that the Gödel argument, in any specific realization, has actually been correctly applied. The trouble is that the computer does not have a way of judging truth; it is only following rules. It does not ‘see’ the validity of the Gödel argument. It does not ‘see’ anything unless it is conscious!”

It may of course be asked how literally Penrose wants his readers to take his words “seen”, “see”, “seeing” and “vision”. After all, Penrose himself used scare quotes in the passage above. However, scare quotes can sometimes be somewhat ambivalent. On the one had, they can be used ironically or questioningly. (As Heidegger and Derrida had it, words can be used sous rature — i.e., “under erasure”.) On the other hand, even though some writers use scare quotes, that doesn’t automatically mean that their words are meant to be taken ironically (or not to be taken seriously). In Penrose’s case, it can be argued that the scare quotes simply signify that he’s aware that words like “see” are controversial within this precise context. However, that doesn’t mean that Penrose believes that they’re unacceptable or that they’re in some sense, say, metaphorical. (In all the other places I’ve read Penrose use these words, he doesn’t use scare quotes.)

iii) Gödel, Turing, Penrose

Of course this philosophical angle of Penrose is not all his own invention. Far from it. Gödel himself made philosophical comments about his own theorems. (This is to disregard the Gödel Industry; which applies his theorems to everything under the sun.)

In his ‘What is Cantor’s Continuum Problem’ of 1947, for example, Gödel explicitly made something (or much) of the fact that there’s a way of contacting “reality” other than through “sense perception”. (That said, even empiricists have acknowledged this; if with many additional clauses.) More precisely, Gödel used the words “mathematical intuition” to account for our (as it were) Platonic receptor.

So Alan Turing is a good counterweight to bring in here.

Although Turing accepted Gödel’s theorems, he didn’t also accept some of the things which were said to be consequences of those theorems. (Many see Turing’s Halting problem as the “computational equivalent” of Gödel’s first incompleteness theorem.) More specifically, Turing accepted that what Gödel called “intuition” is used in order to see the truth of a formally unprovable Gödel sentence. What he didn’t accept was that the brain and mind go beyond the “mechanical”. That is, Turing might well have asked Gödel how it is possible that non-mechanical intuition is carried out by a physically-embodied brain. (See Turing’s ‘Computing Machinery and Intelligence’ — 1950 — for a very clear account of Gödel’s theorems and why Turing believed that they’d been somewhat overstretched.)

One other argument Turing offered was that because the brain is so complex, it simply appears to transcend its mechanical (or rule-following) nature. He also argued that (what he called) “initiative” doesn’t require uncomputable steps. (See Andrew Hodges’ ‘The Logical and the Physical’.) However, a machine’s (or computer’s) computations could still go beyond any programmer’s programme. (See also Turing’s “randomizer”.)

All this meant that Turing didn’t conclude (i.e., from Gödel’s theorems) what Penrose later came to conclude about the human mind or consciousness. And it also meant that Turing was happy with the idea of what later came to be called (firstly in 1956) “artificial intelligence”.

iv) Beyond Mathematical Platonism?

I was fairly suspicious about the quote above because it was doing the rounds in the quote zoo on the Internet and I couldn’t find the source. (That is, the editors of such quotes websites were quoting other quotes websites in which they’d seen this quote… Then I came across this interview. The exact quote above can’t be found within it. However, Roger Penrose does come very close to saying it.

Despite the restricted scope (i.e., mathematics) of what has been said above, Penrose also seems to go beyond purely mathematical Platonism when he stated the following:

“[I] find words almost useless for mathematical [Penrose’s italics] thinking. Other kinds of thinking, perhaps such as philosophizing, seem to be much better suited to verbal expression. Perhaps this is why so many philosophers seem to be of the opinion that language is essential for intelligent or conscious thought!”

Like Plato, Penrose appears to glory in this escape from contingency. In this case, it’s the contingency of “words” and (no doubt) their lack of precision that must be escaped from… Or at least this is the case when it comes to mathematics.

Yet language, surely, is essential for Platonists too. That is, in order to become the Platonists that they are, language itself must have led their way in most/all of their philosophical directions. Indeed that’s even the case when it comes to mathematics.

Here again we also need to get our heads around what Penrose means by the word “visually”. It can be argued that Penrose would never have adopted and used these non-linguistic “concepts” if they weren’t first described to him in “words”. Sure, as with the a priori, we need to learn what the a priori is — and also to learn the words used in a priori statements — in order to have a priori “thoughts”. So this must mean that Penrose must be talking about what happens after the basic words and concepts are acquired. And what happens after is (he argues) something that’s completely non-verbal (or non-linguistic). So, yes, one has to know a posteriori what the words and symbols in the equation 2 + 2 = 4 mean. However, after that, the non-linguistic status of this truth remains unchanged.

As has just been stated about the a priori: we firstly need to learn the terms involved in this debate. However, once Penrose and other Platonists have acquired these words and symbols from a natural language, then perhaps they can “float free of the moorings” (to quote Kant on Plato) and rise into the Platonic realm.

v) Seeing Truths

So Roger Penrose often uses the words “see”, “seen” and “visualised” when it comes to certain mathematical truths (as well as, perhaps, other things). That is, he believes that many mathematical truths are seen to be true without being proved to be true. (In that simple sense at least, he’s simply putting Gödel’s position.)

Along with “seen”, Penrose also uses the words “insight” and “intuition”. For example he writes:

“[A] specific Gödel proposition — neither provable not disprovable using the axioms and rules of the formal system under consideration — is clearly seen [Penrose’s italics], using our insights into the meanings of the operations in question, to be a true [ditto] proposition!”

A useful and indeed apt technical term which captures Penrose’s claims is Paul Boghossian’s “flash-grasping”. The American philosopher defines his own term in the following way:

“Flash-Grasping: We grasp the meaning of, say, ‘not’ in a flash — prior to, and independently of, deciding which of the sentences involving ‘not’ are true.”

It may well be unfair to apply claims about the epistemology of logical terms (such as “not”, “and”, “or”, etc.) to what Penrose claims about certain mathematical truths. We’ll see later whether it’s possible to glide smoothly over from this area (as well as with the epistemology of the a priori) to Penrose’s seeing of mathematical truths.

Of course Penrose isn’t the only one to use words like “see” in the context of mathematical truths. For example, in the specific case of number theory and the Gödel sentence, G, the philosopher of logic Alasdair Urquhart uses the word “perception” (although it too is in scare quotes) in the following:

“Since we do seem to have a ‘clear and distinct perception’ of the notion of truth in number theory, it has often been argued that this demonstrates a clear superiority of humans over machines.”

And in the following paragraph Urquhart continues:

“[We], standing outside the formal system, and using our mathematical insight, can see that the sentence G is true, and so we can surpass the capacity of any fixed machine.”

However, in the above it can be said that Urquhart is (at least in part) putting other people’s positions. And since I’ve just quoted Urquhart, it’s also interesting that he questions Penrose’s claim that he can see that a Gödel sentence is true. He writes:

“The problem with the Lucas/Penrose argument … is that the key premise asserting that we can see the Gödel sentence to be true, remains undemonstrated. In fact, there are good reasons for thinking it to be false.”

In addition to the above, it also needs to be said that people may disagree as to exactly what it is they see. That is, one person may (platonically) see that p is true and another person may see that (the same) p is false. So even if we accept that there is Platonic seeing in both cases, that seeing alone doesn’t — and can’t — guarantee truth (or “truth without proof”).

So what about Penrose himself?

What does Penrose mean by “seen” here? Does he simply mean understand? Is it that we see the “meanings of the operations” simply because we understand them? That said, Penrose also stresses the fact that he “visualises” these things. So do people visualise meanings?

Of course all this may boil down to the simple fact that Penrose isn’t using the word “see” literally. (I mentioned scare quotes earlier.) Yet that still raises two questions:

1) Why does Penrose use the word “see”?

2) What does he mean by “see”?

Penrose also uses the word “sensing”. (Don’t we also sense when we see?) In this instance, Penrose goes beyond seeing the truth of a Gödel sentence and starts using much more modal and clearly Platonic ways of speaking. Indeed he partly explains what he means by “seeing” here:

“[]I believe consciousness to be closely associated with the sensing of necessary truths — and thereby achieving a direct contact with Plato’s world of mathematical concepts.”

There is a reason (at least within this specific context) why Penrose stresses Platonic sight. As stated in the introduction, it’s because he believes that “[sensing necessary truths] is not an algorithmic procedure”. 

vi) Do Platonic Truths Stand Alone?

In terms of looking at things epistemologically again, is it the case that Penrose sees Platonic truths because they “derive their evidence from themselves” (as Laurence BonJour puts it about a priori claims — see later)? In other words, is the proposition/equation/sentence itself all that’s required and nothing more? Yet surely this doesn’t work in terms of a mathematical theorems and especially not for a Gödel sentence. That’s because such things come at the end of a lot of reasoning, deductions, inferences, etc. which involve other propositions/equations/axioms/etc.

So it should be said (in strict accordance with the Penrose quote above) that what is seen is not, for example, the truth of Gödel sentence (G) itself. Instead, Penrose tells us that “the meanings of the operations in question” are seen. That is, firstly the meanings of the operations are seen, and only then does that lead to also seeing the truth of the Gödel sentence. 

However, it can also be said that the truth of sentence G is seen precisely because the meanings of the operations (which led to it) were also seen. (Basically, if you see x, then you must also see y.)

One way of looking at this is the distinction made by philosophers between “relative” and “absolute” truths (or, more often, relative and absolute modalities) as found in logical deduction. 

Absolute truths are true in and of themselves (as in an a priori or analytic manner, which will be discussed later). Relative truths, on the other hand, are a consequence of a previous set of axioms/statements/theorems/etc. Thus if the truths which Penrose can see are merely relative truths (in this strict logical sense), then how can we make sense of seeing within this relative context? That is, what is it to see truth T if T is actually a consequence of a further set of axioms/statements/theorems/etc. which may also be taken to be true (though not necessarily — they can simply be taken as given)?

This must mean that such theorems don’t have the same status as so-called analytic truths, such as:

All married men are bachelors.

Sure, you need to know the meaning of the terms and the fact that “married men” and “bachelors” are synonyms. But apart from that, there are no explicit (though they may be tacit) processes which lead towards one’s knowledge that the statement above is true.

So what about this? -

2 + 3 = 5

Laurence BonJour (more of whom later) says that the truth of the above is “guaranteed by its content”. That means that “understanding the proposition is a sufficient condition for recognising its truth”. Basically, as BonJour adds, this proposition is not “made true by experience”; but only “by content alone”. 

vii) Penrose’s Rationalism

In many respects it would be just as accurate to classify Penrose as a rationalist (if with qualifications) as it would be to class himself as a Platonist. (Of course Platonism is a kind of rationalism.) And Penrose’s positions have a very rationalist feel to them. Indeed we can see similarities between Penrose’s position and statements found in Rene Descartes’ work — who was himself a rationalist.

So here’s Descartes himself putting his classic rationalist position:

“[I]f it could ever happen that a thing which I conceived so clearly and distinctly could be false… I can establish as a general rule that all things which I perceive very clearly and very distinctly are true.”

And elsewhere Descartes also writes:

“Clear and distinct perceptions are so coercive in their effect upon the mind that the mind cannot help assenting to them as true at the time it has such perceptions.”

It needs hardly be said that Descartes uses the words “perceive” and “perceptions” in the above. That is, he perceived certain truths. But to be fair (if that’s the right word) to Penrose, Descartes wasn’t talking about mathematical truths in the above. He was primarily talking about “the thing which thinks”. And, elsewhere, Descartes claimed to have “clearly and distinctly” perceived truths about God, the possibility of the mind existing without the body and suchlike. Penrose, of course, has never really ventured into any of these areas. (He has classed himself as an “atheist”.)

In addition to all that, it’s very clear that Penrose can’t be a strict (or complete) rationalist for the simple reason that he places a lot of emphasis on observations, scientific experiments, predictions, etc. Penrose even comes close to accusing string theorists of being Platonists in his Road to Reality and his most recent book Fashion, Faith and Fantasy in the New Physics of the Universe. He does so because such string theorists appear to have a complete trust in their “consistent” and purely mathematical theories — theories which aren’t backed up by (unique) experiments or predictions. 

Yes; Penrose is also a mathematical physicist. Then again, Descartes (along with Leibniz and Spinoza) too had a great respect for science and experiment and indeed his rationalist work was seen (by himself) to be but a means of securing the science of his day. 

viii) Laurence BonJour: A Rationalist of the 21st Century

Let’s bring things more up to date and move beyond Plato, Descartes and other dead rationalists. Take the American philosopher Laurence BonJour.

BonJour’s rationalism is encapsulated in his position on the a priori. The following is what he says on that subject:

“[I]f we never have a priori reasons for thinking that if one claim or set of claims is true, some further claim must be true as well, then there is simply nothing that genuinely cogent reasoning could consist in. In this was, I suggest, the rejection of a priori reasons is tantamount to intellectual suicide.”

As it is, BonJour describes himself as a “rationalist”. And, clearly, he’s also well aware of the criticisms of rationalism. For example, in reply to the Australian philosopher Michael Devitt, Bonjour talks of Devitt’s

“allegations that rationalism is ‘objectionably mysterious, perhaps even somehow occult’…”

He concludes by saying that he find these allegations “very hard to take seriously”.

So what else does Devitt have to say on BonJour? The following:

“BonJour is an unabashed old-fashioned rationalist (apart from embracing the fallibility of a priori claims). He rests a priori justification on ‘rational insight’: ‘a priori justification occurs when the mind directly or intuitively sees or grasps or apprehends… a necessary fact about the nature or structure of reality’…”

Sure, it may not be such a good thing to quote Philosopher X on Philosopher Y — especially if they take diametrically opposed positions on the same issue. In any case, there are certain elements in the passages above that don’t entirely square with Penrose’s own positions.

For example, how strong and clear is Penrose himself on the “fallibility” of his seeings of mathematical truth? (Note: If BonJour accepts the possibility of a priori fallibility, then what about the possibility that all a priori claims are fallible and/or indeed false?)

In addition, are Penrose’s claims necessarily applicable to “the nature or structure of reality”? Or are they simply about mathematics/mathematical systems? Yes, it’s true that mathematics must be part of reality; though many of BonJour’s a priori claims are literally about the physical world itself.

So what does BonJour’s rationalism amount to? Take the following passage:

“[A]n intuition is a semi-cognitive or quasi-cognitive state, which resembles a belief in its capacity to confer justification, while differing from a belief in not requiring justification itself.”

And elsewhere:

“[I]n the most basic cases such reasons result from direct or immediate insight into the truth, indeed the necessary truth, of the relevant claim.”

So, like Penrose, BonJour uses the phrase “necessary truth”. That is, BonJour ties necessity to truth. Again, in BonJour’s own words:

“Devitt seems to me to be simply rejecting the idea that merely finding something to be intuitively necessary can ever constitute in itself a reason for thinking it is true…”

Indeed BonJour goes further by stating the following:

“[A priori] insights at least purport to reveal not just that the claim is or must be true but also, at some level, why this is and indeed must be so. They are thus putative insights into the essential nature of things or situations of the relevant kind, into the way that reality in the respect in question must be.”

There’s just been a lot of focus on the epistemology of seeing truth. So here’s another problem from epistemology on a priori (if not Platonic) seeing from the American philosopher James Van Cleve. He writes:

“If the foundationalist claims that his principles are immediately justified, then what it to prevent, let us say, a [religious] revelationist from claiming the same status for a principle to the effect that if S has an ostensible revelation that P, then S is justified in believing that P?”

(Of course it doesn’t help us much when Van Cleve concludes by saying that “[s]ome claims to immediate justification are spurious”. After all, as stated earlier in regards to a priori “fallibility”, if some cases of “immediate justification” are “spurious”, then perhaps all of them are.)

The question now is: 

How applicable is all the above to Penrose’s own seeings of mathematical truths?

ix) Conclusion: Is Roger Penrose Really a Platonist?

It can be argued (or seen) that Roger Penrose doesn’t really extend his Platonic vision much (or at all) beyond mathematics. Having said that, his strong and controversial positions on consciousness and artificial intelligence can indeed be seen to be going beyond the mathematics. Nonetheless, Penrose would no doubt argue that these positions are largely (or even wholly) derived from the maths. 

It’s often said that the general “Platonic position” on mathematics is the “consensus position” among mathematicians — and indeed among many other people outside mathematics. However, it must also be said here that most mathematicians don’t philosophise in this way about their own subject. (Though some mathematicians most certainly do.) In addition, the acceptance of Gödel’s theorems is almost universal. So too is the notion of mathematical intuition. (Here again it must be said that not many mathematicians will use these terms.) So it’s when Penrose extrapolates from these Platonic and Gödelian positions that he loses the consensus. For example, not many scientists or philosophers accept his very particular take on consciousness. And as for his stance against (strong) artificial intelligence, that too has received a large amount of criticism from many different quarters.

And from a strictly philosophical (or epistemological) point of view, Platonic seeing comes up against many hurdles; many of which (as discussed above) have been applied to the positive (rationalist) positions on the a priori in epistemology. In addition, cognitive scientists (of various kinds) will also have many things to say about Platonic seeing. 

Yet, despite all that, it’s still the case that many people’s intuitive (as it were) position on mathematical intuition will be to claim that we do indeed immediately and (platonically) see the truth of, say, the equation 2 + 2 = 4. But where does that possibility (or reality) take us?

Finally, it can be said that Roger Penrose can indeed be seen as a rationalist — but only in very limited respects (i.e., when it comes to his position on mathematics). Yet if Penrose is a rationalist only in such limited respects, then how can he be a rationalist at all? In other words, were Plato, Descartes, Leibniz and Spinoza rationalists only in limited respects? And, as a further consequence of this, can it also be said that Penrose is a Platonist only in limited respects? After all, Plato himself wasn’t a Platonist only in limited respects.