Thomas Nagel on “Darwinian Imperialism”, Naturalism and Mind (Part Two)

 Thomas Nagel fights “Darwinist imperialism” and Naturalism

Thomas Nagel being burned at the stake by satanic “Darwinian” and “materialist” analytic philosophers in red robes.

“The total world-view of modern man [has] let itself be determined by the positive sciences [which has resulted in] an indifferent turning-away from the questions which are decisive for a genuine humanity.”

— Edmund Husserl (see source here)

Skip the following square-bracketed introduction if you’ve already read my ‘Thomas Nagel as Philosopher-Priest and New Mysterian (Part One)’.

[This essay was written quite some time ago. The style is somewhat rhetorical, literary and (as it were) psychologistic. That said, I still agree with much of its philosophical content. However, if I were to write it today, the style would be a little different. Indeed most analytic philosophers would probably regard this essay as one long ad hominem against the American philosopher Thomas Nagel (1937-). Sure; there is an element of the ad hominem in the following. Yet hopefully it will be shown that there’s more to the essay than that.

In fact I chose to write in a rhetorical style partly in response to the clear and prevalent rhetoric and “psychologising” I found in Thomas Nagel’s own book, The Last Word.

In addition, the prefix “new” in “new mysterian” is a little dated (i.e., new mysterianism is no longer new) because the term was first coined in 1991 by the American philosopher Owen Flanagan (1949-). (See Flanagan’s The Science of the Mind.) What also needs to be said is that I’m using Flanagan’s word “mysterian” more widely and more literally than he does. That is, in Nagel’s case I’m stressing Nagel’s mysterianism and mysticism across the board. Flanagan, on the other hand, only (really?) had mysterian positions on consciousness in mind. That said, the term “new mysterianism” has also been applied to the wider position that there’s far more than one “hard problem” (i.e., the “hard problem of consciousness”). It also includes the belief that science is intrinsically and fundamentally limited in many respects. This is a self-conscious kind of anti-scientism.]

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Thomas Nagel’s General Anti-Naturalism

In the following I shall discuss Thomas Nagel’s desire to keep the human person (or “man”) securely away from any attempts to naturalise him or her.

This desire, of course, has a long tradition dating back to the ancient Greeks. A whole host of characteristics of the human subject have been put forward as truly distinguishing us from not only all the other animals; but also from the rest of the universe.

Originally, man was “the only rational animal” (laughter was also offered up by Aristotle). Then we had, in no particular order, the mind, the soul, the love of God, language, free will, “complex social structures”, music and so on. More recently, the distinguishing features of man which have been cited (at least by philosophers) have been meaning and intentionality. These characteristics are meant to distinguish man from everything else in the universe. The problem with that enterprise is that although there may well be features of human persons (or “mankind”) which distinguish them from everything else in the universe, every other thing in the universe probably has distinctive features that differentiates itself from human persons — and indeed from other non-humans too. (For example, having a hundred legs, being able to survive a mile under water, being able to live in freezing temperatures, having intrinsic spin, travelling at the speed of light, being able to penetrate solid walls… ad infinitum.)

There’s another manoeuvre which is of more recent origin. This is the stressing of aspects of man that can’t be naturalistically reduced to straightforwardly physical features. Here we have intentionality again, semantics, the “the intrinsic nature of the first-person perspective” and qualia.

The main threat against Nagelian mysticism or mysterianism is posed by what Nagel calls “Darwinist imperialism” (see here and here). What Darwinism does, according to Nagel, is

“eliminate purpose, meaning, and design as fundamental features of the world”.

My immediate reaction to that is that the vast majority of Darwinians probably don’t have anything at all to say about “meaning”. So perhaps Nagel is actually talking here about (as it were) Darwinist philosophers like Daniel Dennett.

So does “naturalistically reducing” automatically mean elimination?

The old religious account of “purpose” may well be eliminated; though what of meaning? (There are very few “naturalist” semantic eliminativists.)

As for “design”: it depends on what Nagel means. And, in any case, even many Christians have finessed their views on the world’s design by taking into account Darwinism. That said, perhaps I’m misinterpreting Nagel’s reference to the world’s “design”. After all, he does frequently stress (e.g., in almost every interview) that he’s an atheist.

More broadly, what the Darwinian picture offers us is a denial of Nagel’s assertion that “mind is an irreducible and nonaccidental part [of the] cosmic order”. I suspect that few Darwinists (certainly not all physicalists) believe that the mind is in fact (fully?) reducible (see non-reductive physicalism). However, all most certainly do believe that mind is an “accidental” result of biological evolution and all the cosmic physical processes that preceded it. This means that Nagel’s fusing of the irreducible and the nonaccidental is suspect and somewhat arbitrary.

Again, bearing in mind that Nagel claims to be an atheist, what can he mean by saying that the mind is a “nonaccidental part [of the] cosmic order”? And it also depends on what he means by “reduction” or the mind being “reducible”. Perhaps the mind can’t be reduced in the way a table can’t be reduced (i.e., to atoms and molecules) without losing its essential (as it were) tablehood. (This simply means that physicists wouldn’t be the ones who explain to us that tables are for eating or for playing tennis on.)

Nagel offers us another criticism of the Darwinian story of mind. He lets Mark Johnston articulate it in the following way:

“‘Why is the natural order such as to make the appearance of rational beings likely?’”

According to Nagel, the only acceptable answer to the above question must be a “teleological” one. And I can’t help thinking that whenever teleological explanations are brought to bear on the appearance of mind (or Soul), then that telos is invariably God. (Keep in mind. Nagel’s earlier reference to “design”.) Nagel will certainly believe that any mention of “design”, “teleology”, the “nonaccidental appearance of mind” and its “irreducibility” within a theistic/ Christian framework would make many other analytic philosophers take him less seriously. That said, the analytic philosophers Richard Swinburne, Alvin Plantinga, Antony Flew, William Alston, Linda Zagzebski, Paul Moser, etc., on the other hand, are explicit about their “Christian theism” and other religious or Deistic positions. (Take Richard Swinburne’s position on personal identity: it’s an explicit commitment — if not a simple reworking - of the Christian position on the soul. And Alvin Plantinga slips God into many of his papers.)

Again, it is in Nagel’s attitude to reason that the philosopher’s quasi-mysticism or mysterianism can be seen.

Reason, Mind and Intentionality

So why does Nagel feel the need to say that the “basic methods of reasoning we employ are not merely human”? What could that possibly mean other than being a religious or mystical statement? To put it starkly: Nagel must surely believe that reason and mind are not “human” — or at least they have no relation to the biological and historical. He must believe, them, that reason is a faculty of a non-corporeal soul.

Another mantra of Nagel’s is that human subjects are not objects. This position is clearly seen in certain debates in the philosophy of mind. More precisely, Nagel vehemently rejects all attempts to what may be called scientise the mind. This can be seen in the on-running third/first person debate. Nagel believes that there is something about the first-person perspective that could never — even in principle! - be caught by the third-person language of science. Nagel also famously believes that no scientist could ever tell us “what it is like to be a bat”; or, indeed, what it is like to be Thomas Nagel.

Most of Nagel’s anti-naturalistic fire has been aimed at those who have attempted to naturalise (or reduce or eliminate) intentionality and qualia. He wrote:

“Intentionality cannot be naturalistically analysed…nor can it be given naturalistically sufficient conditions. It is not to be captured by either physical or phenomenological description.”

Nagel’s argument also seems to be that whatever we say about the evolutionary heritage of intentionality (i.e., if we accept such a thing), or the physics and neurobiology which underpins it, we could never indulge in (as it were) a priori reasoning to satisfactorily arrive at the effect (i.e., intentionality and qualia) exclusively from the causes (i.e., natural conditions). And even if we work backwards with purely a posteriori reasoning from the phenomena of intentionality and qualia, natural conditions may be necessary; though they would never be sufficient for a satisfactory explanation of intentionality. In fact with intentionality, it’s believed that human subjects have something that the rest of the universe simply doesn’t have. Intentionality sticks out like a sore thumb both in nature and in the minds of non-naturalist and mysterian philosophers. Indeed Nagel believes, as Brentano believed over 100 years ago, that “aboutness” is irreducible.

So Nagel argues that he can place intentionality firmly outside the causal nexus. He claims that

“it is possible to accept a worldview that does not explain everything in terms of quantum field theory without necessarily believing in God”.

Nagel continues:

“[I]t seems hardly credible that its [the mind’s] appearance should be a natural accident, like the fact that there are mammals.”

So if the mind’s (or soul’s) appearance isn’t a “natural accident”, then is it a supernatural… what?

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See my: ‘Thomas Nagel as Philosopher-Priest and New Mysterian (Part One)’.

To follow: ‘Thomas Nagel on Good and Bad Philosophy (Part Three)

Thomas Nagel as New Mysterian and Philosopher-Priest (Part One)

 

Thomas Nagel on Transcendence and Ineffability

“There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical.”

- Ludwig Wittgenstein (from his Tractatus Logico-Philosophicus)

[This essay was written quite some time ago. The style is somewhat rhetorical, literary and (as it were) psychologistic. That said, I still agree with much of its philosophical content. However, if I were to write it today, the style would be a little different. Indeed most analytic philosophers would probably regard this essay as one long ad hominem against the American philosopher Thomas Nagel (1937-). Sure; there is an element of the ad hominem in the following. Yet hopefully it will be shown that there’s more to the essay than that.

(In fact I chose to write in a rhetorical style partly in response to the clear and prevalent rhetoric and “psychologising” I found in Thomas Nagel’s own book, The Last Word.]

In addition, the prefix “new” in “new mysterian” is a little dated (i.e., new mysterianism is no longer new) because the term was first coined in 1991 by the American philosopher Owen Flanagan (1949-). (See Flanagan’s The Science of the Mind.) What also needs to be said is that I’m using Flanagan’s word “mysterian” more widely and more literally than he does. That is, in Nagel’s case I’m stressing Nagel’s mysterianism and mysticism across the board. Flanagan, on the other hand, only (really?) had mysterian positions on consciousness in mind. That said, the term “new mysterianism” has also been applied to the wider position that there’s far more than one “hard problem” (i.e., the “hard problem of consciousness”). It also includes the belief that science is intrinsically and fundamentally limited in many respects. This is a self-conscious kind of anti-scientism.]

(i) Introduction

(ii) Nagel’s Metaphilosophy and His Moral Hopes

(iii) Nagel on Transcendence and Ineffability

(iv) Is Nagel’s Position Mystical?

(v) Is Nagel an Atheist?

Introduction

“There are desires and sentiments prior to reason that it is not appropriate for reason to evaluate.”

-Thomas Nagel, The Last Word (1997)

My reason for taking the following approach to Thomas Nagel’s philosophical positions comes from certain comments he made in his books The View from Nowhere (1988) and The Last Word. In those books he states that his philosophy is driven by his moral feelings. Now some people may say that there’s nothing strange with that in the field of ethics itself. However, what if Nagel’s “desires and sentiments” are responsible for his metaphysical and epistemological positions too?

Take Nagel’s (following) pronouncements and compare them with Michael Ayers comments on John Locke. In that book Ayers writes that it’s “absurd” to think of

“a particular moral, political or religious orientation’s motivating a theory of knowledge or being”.

So this essay isn’t really an analysis of Nagel’s positions within metaphysics and epistemology. It’s an examination of why Nagel may hold such positions in the first place. It’s not to the point, therefore, to say that I don’t offer any good critical analyses of Nagel’s positions on necessity, objectivity or “pre-linguistic concepts”. Such analyses are not the purpose of this essay. Instead, I want to explain why Nagel holds such positions; not whether his positions have any argumentative power.

This means the work is also somewhat speculative and psychological. And that’s primarily the case because — to state the obvious — I haven’t actually got access to Nagel’s (non-verbalised) mind.

Nagel’s Metaphilosophy and His Moral Hopes

In Nagel’s case there seems to be a kind of pragmatism in the air. In fact precisely the same kind of pragmatism that Nagel argues against when displayed by philosophers like Richard Rorty (see Nagel on Rorty here) and Hilary Putnam (1926- 2016) (see Putnam on Nagel here). What is meant by this is that one could take Nagel’s opening quote to be an acknowledgment that one’s metaphysics and epistemology should simply be the handmaidens of one’s metaphilosophical and moral hopes.

Indeed this was what Kant more or less argued (in his Critique of Pure Reason) about “moral philosophy” and its relation to metaphysics and epistemology. It’s also what Aquinas argued (in his Summa Theologica) about logic’s and philosophy’s relation to theology.

In the case of Nagel, if certain metaphilosophical and moral “desires and sentiments” are “prior to reason” and should therefore not be “evaluated”, then they should also be defended all the way down the line of philosophical speculation and analysis. And, as is clear, Nagel’s comments on logic, mathematics and various givens may simply the groundwork for his positions on metaphilosophy and morality which he deems to be much more important. This may mean that the metaphysical and epistemological positions advanced in (for example) The Last Word are simply the substructure which Nagel feels is needed to support his larger metaphilosophical and moral edifice.

As already hinted at, there’s nothing new here.

For example, Kant the transcendental idealist professor ended up believing exactly the same things about God, immortality, freedom and morality as Kant the North German Pietist who preceded him. Indeed it can easily be argued that Kant’s whole philosophical enterprise was an attempt to “limit reason in order to make room for faith”. Bishop Berkeley’s philosophical system was also an attempt to limit the pretensions of the advancing materialist and deterministic Newtonian science. And Descartes was at his most dishonest when he smuggled in God to tighten up his system and guard Christianity from 17th century sceptics.

So Nagel (via his anti-naturalism or mysterianism) is attempting to limit philosophical naturalism (which, in his view, includes pragmatism, positivism and the late Wittgenstein) in order to make room for (amongst other things) the ineffable, the necessary and for non-linguistic “thought”.

Nagel on Transcendence and Ineffability

It will become clear that Nagel is [or was] offering some kind of mystical or quasi-religious line on philosophy. Take Nagel’s own words on this in his book The Last Word:

“Rationalism has always had a more religious flavour than empiricism.”

So it’s clear (to me at least) that Nagel is precisely the kind of rationalist he refers to above.

So let’s cite some examples.

Nagel wants to downgrade the analysis of language (or at least language use). The analysis of language is, after all, an empirical enquiry. Nagel, instead, emphasises the point that his own own “concepts” predate experience in a manner which roughly compares with Kant’s a priori “concepts” or Frege’s notion of what he called Thoughts.

Nagel is a rationalist in the Cartesian sense too.

To such rationalists, the human subject can unhook himself from the rest of the empirical world (including his own body) and then philosophise in order to gain truth. Indeed not even the empirical data needed for such cogitations are of much importance when compared to the (to use Nagel’s words) “order of reasons” which we must “submit to” rather than “create”.

Thus Nagel is searching for the necessary, the absolute and those aspects of the human mind which we all share. Such a search is essentially Kantian in spirit, if not in actual content.

Yet what is it, precisely, that’s mystical or quasi-religious about all the above?

Is Nagel’s Position Mystical?

As just stated, Nagel downgrades experience, language and contingency. He’s essentially stressing those facets of reason and mind which human beings come into the world with. That is, those things which are built into us as human subjects in a manner which Kant would recognise. (Note that Nagel attacks Kant’s “idealism”.) And if these things were there before language and all other interactions with society, then they’re closer to God than they are to the world of departmental seminars, raging hormones, book deals, voting and tax forms. (A world in which Nagel, presumably, lives and breathes.) By tapping into these givens (or simply by using or understanding them through realist analytic philosophy), we’re also acknowledging God’s — rather than society’s — gifts to us.

Yet even if Nagel is not — rather immodestly — placing himself and selected other philosophers closer to God, he is in effect celebrating the things which God has placed within us. This effectively means that by distancing himself from (as Nagel puts it) “time and chance”, Nagel becomes closer and closer to the Eternal and the Necessary — which are God’s works. Contingency, on the other hand, is the Devil’s work.

This is also the Platonist strain in Nagel.

Yet there are some things about Nagel’s philosophical positions which are clearly more mystical than outrightly religious.

Take his frequently reiterated position on the “view from Nowhere”.

Firstly, when it comes to Nagel’s view of the world and its relation to scientific claims, things don’t appear particularly mystical. However, when Nagel says that he wants to “get outside” of himself in order to “reach a descriptive standpoint from which the first person has vanished”, he does indeed sound mystical.

Of course it needs to be asked if readers are meant to take Nagel literally or metaphorically when he utters these phrases. Yet it seems that there are good reasons to take him literally. In any case, how would a metaphorical reading of his words be cashed out? Perhaps not even Nagel could tell us.

So if we take Nagel literally, shades of (amongst others) Plotinus can be seen. However, whereas Plotinus wanted to achieve harmony with what he called “the One”, Nagel would like to achieve harmony with the perspective-free external world — the world we discover when we cleanse ourselves of what Nagel’s calls the “first person”.

The world (as it’s often put) as it is in itself is like God — or is God — to Nagel. (This may show the influence of Spinoza.) Similarly, whereas Plotinus rages against “plurality”, Nagel’s devil is contingency. (Wasn’t “plurality” a consequence of contingency for Plotinus?)

To finish off on this line of reasoning, here’s a poetic (i.e., for Kant) passage from the Critique of Pure Reason (1781/1787):

“[] Plato, abandoning the world of sense because of the narrow limits it sets to the understanding, venture[s] upon the wings of ideas beyond it, into the void space of pure intellect.”

It need hardly be said that Nagel is similarly advocating “abandoning the world” of empirical (or experiential) reality and language because he deems such things to be inadequate to the task of providing us with necessity, certainty, objectivity and other such holies. So perhaps Nagel too ends up in the “void space of pure intellect”.

Now there’s large problem for my position as advanced in all the above: Nagel has often called himself an “atheist” (see here)!

Is Nagel an Atheist?

Even if Nagel’s honesty can’t be doubted (as Nagel himself doubts the honesty of Alasdair MacIntyre and what he calls the “ritualist relativists and subjectivists”), his atheism isn’t really a big problem to the central thesis. That’s because the religious impulse (or instinct) can survive quite easily without the God of monotheistic religions. Indeed all sorts of psychologists and sociologists have written thousands of words on this very phenomenon. These people have shown us (if we needed to be shown) how hard it is to throw off the baggage placed upon us by our history, society and our parents.

Thus Nagel has finessed mystical/religious positions and made them seem more modern and sophisticated. That is, he’s expressed mystical/religious positions in the language game of late 20th century analytic philosophy. Thus it’s mainly a case of old wine in new bottles.

The early Wittgenstein (i.e., unlike the late Wittgenstein) also provides Nagel with ammunition to fight for the ineffability and transcendence cause. Nagel sums up Tractarian Wittgenstein as claiming that

“what couldn’t [] be said was much more important than what could be”.

For Wittgenstein at this point, these (as it were) ineffables were (among other things) “logical objects”, logical relations and the relations between empirical propositions and the possibilities they represented. However, to Wittgenstein these things were simply (at least on my reading) the logical entities which grounded much more important ineffables — such as the human subject. This subject is outside the world and also the foundation of moral and aesthetic value judgements — judgements which the subject simply intuits or feels (i.e., subjects not unlike Kant’s noumenal self).

Again like the early — and late — Wittgenstein, Nagel doesn’t like too much talk. Too much talk ruins or destroys what it is that’s talked about. So let’s try not to talk about, say, neurons and synapses when we talk about the self or mind. Let the self or mind, unlike every other thing in this gigantic universe, spring free of all causal mechanisms. Nagel believes that we intuitively know this to be true. And talk could undo that truth. Indeed only positivists and “scientistic philosophers” want to talk about everything. Such people, Nagel seems to believe, don’t recognise the soul (or the soul’s calling) quite simply because they don’t recognise that they have souls.

This interest in the ineffable often goes alongside a belief in philosophical depth.

Nagel believes that that real (or important) philosophical problems are intrinsically deep. He even tells us that an old teacher of his (i.e., J.L. Austin) distrusted depth; though “depth was what I thirsted for”. This seems to mean that it was depth in itself (or depth for itself) that interested Nagel at that time — and still does; not the fact that depth is often simply an accidental byproduct of certain (i.e., not all) philosophical problems.

So what is “philosophical depth”?

Those words can’t be a synonym for “philosophical complexity”. Is it more, well, spatial than that? Is it the Reality under the Appearances? The Noumenal Self under the empirical self? The Logical Form under the grammatical form? The Essence under the accidents? The Truth under the (to use Nagel’s word) “simulacra”?

Depth is something priests like to give us. It’s something they believe to be beyond trivialities and superficialities. And when we reach the depths such priests have hinted at, then we will find “the meaning of life” and “what it’s all about”. In the case of Nagel, we also find essence, necessity and objective truth: all seen from the non-perspectival and incredible View From Nowhere.

The early Wittgenstein (again) similarly “thirsted for depth”. That said, he came to believe that philosophers had (in a sense) been mislead by this spatial metaphor (amongst other things). Or, as the American philosopher Terence Parsons once put it (if about a slightly different subject):

“[As] half the rest of the [analytic] philosophical world have been telling us, you can’t trust the ‘surface’.”

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See my: ‘Thomas Nagel on Darwinian Imperialism, Naturalism and Mind (Part Two)’.

[I can be found on Twitter here.]

Bertrand Russell’s Set of Everything

 

First things first.

The following essay isn’t going to tackle Bertrand Russell’s famous (or infamous) paradox (i.e., Russell’s Paradox). Instead, it’s going to deal with some of the set-theoretical and philosophical details which led up to the paradox. Primarily, it asks various questions about the philosophical (i.e., ontological) nature of sets. Specifically, questions about the Universal Set — or the Set of Everything.

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Georg Cantor

Rather bizarrely, the English philosopher Bertrand Russell (1872–1970) once set out to calculate how many things there are in the universe. And after Russell had done that, he was led to conclude that there must (or simply could) be a set which includes literally everything (or every… thing).

(Of course the practical problems with such an act of “counting” are irrelevant because this was a purely mathematical or set-theoretical enterprise.)

This set came to be called the Universal Set. And that set, in turn, led to a well-known paradox — Russell’s Paradox.

Bertrand Russell at roughly the same age he was when he wrote about the Set of Everything.

Russell originally reasoned in this rather odd way in order to counteract Georg Cantor’s view that there is no largest number. Russell himself said — when looking back in 1919 — that he’d “attempted to discover some flaw in Cantor’s proof that there is no greatest cardinal”.

Russell claimed that if he could (as it were) discover a set which included literally every… thing, and that the nature of sets determines the nature of numbers (or that sets literally are numbers), then that set would be the largest number.

In short and on the resultant paradox.

In the 1890s Georg Cantor himself had already come to realise that his positions on sets would lead to a contradiction. Indeed he communicated this finding to David Hilbert (1862- 1943) and Richard Dedekind (1831–1916). That said, Cantor didn’t technically formulate the paradox: Ernst Zermelo (1871–1953) did. Moreover, in 1903 Hilbert (again) wrote a letter to Gottlob Frege (1848–1925) about Russell’s paradox in which he wrote the following words: “I believe Dr. Zermelo discovered it three or four years ago.” However, Zermelo (like Cantor before him) never published his finding.

Sets and Numbers

Clearly with Russell’s (early) position we have the notion that the members of a set determine that set. And, if that’s the case, then there must be a largest number — at least if we follow Russell’s reasoning at that point in his career. (In set theory, the actual nature of numbers and sets is much disputed — see later.)

According to Russell and Frege before him, numbers are sets. Therefore what makes up a set determines or (as it were) constitutes a number. So if literally everything (or every thing) is included in a set, and that set is finite, then it also determines the finitely largest number.

But what did Russell mean by the word “everything”? Did he mean only every concrete or empirical thing? What about minds? Indeed what about abstract entities such as universals, properties, possible worlds… and numbers? Yet if Russell had included abstract objects (or entities) in his (to use German/Platonic capitals) Set of Everything, then surely it couldn’t have included everything because numbers, for one, are infinite…

Of course that statement raises two issues:

(1) If sets determine (or constitute) numbers, then numbers clearly can’t be the members of sets.

(2) Is there an assumption (from the outset) here that there are (or are not) actual infinities?

As just hinted at, one can’t call in numbers here because it’s the nature of sets (at least in this theory) which determines the nature — and even existence — of numbers.

More simply, that Set of Everything can’t have numbers as members because sets themselves are deemed to be numbers.

But are Numbers Really Sets?

The problematic nature of (in the singular) set theory is displayed by statements such as the following:

“Many set theories do not allow for the existence of a universal set.”

Yet some set theories do allow for the existence of a universal set.

This multiplicity of set theories hints at either the complexity of the issue or that there’ll never be a definitive (or conclusive) position on the nature (rather than the mathematical everyday use) of sets and numbers. And perhaps the main reason for this is one of the following: 1) Sets are abstract. 2) Sets (at least if seen as Platonic entities) don’t exist at all.

So are these essentially philosophical — even stipulative — interpretations of sets and numbers simply vying for intellectual dominance?

In any case, if numbers are sets, then sets are numbers. Thus:

If sets = (i.e., are numerically identical to) numbers, then numbers = sets

Yet the often-seen phrase “sets of a number” (or “sets of numbers”) implies, by its very grammar, that sets can’t be (or equal) numbers. Specifically, the two words “sets of” implies that there are sets and there are numbers. (Alternatively, there is a set and there are that set’s members.) The same kind of reasoning also applies to phrases such as “numbers are represented — or modelled - by sets” and “the natural numbers can be constructed using sets”.

So some theorists argue that there are sets and there are the sets’ members. The two are very different. Analogically, there is a collection of sweets and the bag that the sweets come in. Clearly the bag and the sweets aren’t identical. Thus saying “There is no bag of sweets” (i.e., there is no set) and saying “The bag of sweets is empty” (i.e., the set is empty) are two very different statements.

Yet because we’re talking about abstracta here (i.e., sets and numbers, not bags and sweets), then that analogy may well break down.

In any case, any given x can’t also be y. Of course that’s only if the symbols x and y symbolise (or refer to) physical or abstract objects, not intensional terms, the content of psychological states… or, for that matter, sets. (Intensions basically “belong” to words/terms or psychological states, not to things. However, that’s the case even though such words or psychological states can actually be “about” things.)

As for sets. Jimmy the cat (x) belongs to the set of mammals (y). Yet if we’re talking about individual objects and not sets, Jimmy the cat can’t actually also be Johnny the dog — which is also a mammal (x ≠ y).

On the other hand, many (or even all) the arguments which state that numbers aren’t sets rely on the (Platonic) position that numbers are (eternal) abstract objects. That said, sets too are usually seen to be abstract in nature. That said, Bertrand Russell (again) once believed that all a set’s members (when taken together) literally constitute — or are — the said set. Yet it’s unclear if Russell’s view — even if true or correct — automatically stops sets from being abstract.

There’s also problems with the empty set (symbolised: {}) and sets which only have one member. (A set with one member is called a singleton or unit set -sometimes symbolised: { s }.) In the case of the empty set, there’s literally nothing (or no thing) to determine the nature of that set. And in the case of sets with only one member (whose one member can even be {null}), then such sets and “their” single members (or elements) must be one and the same thing.

So why the need to postulate such sets in the first place?

Conclusion

Finally, it can be said (if only in very broad terms) that mathematics works fine as it is: it’s only the philosophical nature of numbers and sets that’s problematic. Indeed this parallels — at least to some extent - the clear opposition between the mathematical formalism/s of quantum mechanics and the many and various interpretations of that theory.

To repeat: mathematics is a domain of much consensus. Yet the domain of the philosophy of sets and numbers isn’t. Indeed there are (to be melodramatic for a moment) an infinite amount of actual and possible positions to take on the ontological nature of sets and numbers.

[I can be found on Twitter here.]

The Verification Principle Is Not Self-Referentially Self-Refuting

 

It’s strange that in a philosophical world (circa the 1920s to the late 1940s) in which metalinguistic statements, metalanguages and other higher-order techniques were so popular that the Verificationist Principle (as well as Karl Popper’s falsificationism) came in for so much criticism. And from then on, of course, the Verification Principle’s ostensible self-refuting nature is usually the first thing that many people say about it. (Many people say it about logical positivism itself.)

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The English Philosopher A.J. Ayer (1910–1989) was just one philosopher who himself offered more than one version of the Verification Principle (see here). (This raises the question as to the use of the singular word “Principle”.)

There are many and various versions of Verification Principle, each of lesser or greater degrees of detail. That said, I’ll cite the following simple expression of it. Thus:

“Every statement must be either verifiable or tautological in order to be meaningful.”

Of course we can now ask:

What about that very statement? Is is either verifiable or tautological?

Isn’t it (almost) obvious that the verificationist statement is aimed at statements which (to put it broadly) refer to the world? If that wasn’t the case, then why did verificationists talk about such things as “evidence”, “testability” and “verification” in the first place? Yet if the Verification Principe is a statement about other statements (i.e., not a statement about the world), then is it at all apt to ask for a verification of the Principle itself?

It now needs to be said that the word “world” is being used very broadly here. It’s simply meant in the sense of statements that are about — or refer to — phenomena which aren’t themselves other statements. And that includes statements about physical reality, metaphysical reality, “abstract reality” or whatever.

To repeat: clearly the sentence

“Every statement must be either verifiable or tautological in order to be meaningful.”

is not itself a statement about the world. Of course the linguistic expression above can be said to be in the world. (Where else could it be?) Yet it’s not about the world — it’s about other statements which are about (or refer to) the world. Again, linguistically- or vocally-expressed statements about the world are in the world too… That’s unless one is referring to the abstract propositions which are believed — by some philosophers — to be their “content”. In that case, then, the linguistic expression of the Verification Principle itself may have the status of expressing an abstract entity that isn’t in the world — i.e., it is abstract. In that case, it’s an abstract entity (that’s also expressed linguistically in space and time) about other statements which refer to — or are about — the world.

Clearly, then, the Verificationist Statement is not verifiable in the way in which statements about the world are verifiable. It’s of an entirely different logical order.

[It must be said here that if that much is granted, then perhaps many more statements can be given such a — as it were — special status.]

Take also the Testability Principle and the following expression of it:

“A hypothesis is testable if there is a possibility of deciding whether it is true or false based on experimentation by anyone.”

Now the above is strictly a scientific doctrine. Thus it’s not really meant to be applicable across the board; or applicable to, more relevantly, philosophical statements (as, arguably, the Verification Principle is). Nonetheless, let’s express it as a general principle:

Every statement about the world must be either testable or tautological if it is to be deemed meaningful.

On first glance, the statement directly above must refer to itself — primarily because of the quantifier “every” at the beginning. That is, it states that every statement must be either testable or tautological. Therefore it too must be either testable or tautological. However, the statement (or principle) can be taken normatively. In that case, it may not be correct to call it a “statement” at all. And that’s because it’s not evaluable in terms of truth or falsity in the first place (see later).

Take the formulation again:

Every statement about the world must be either testable or tautological if it is to be deemed meaningful.

Now the above can be taken to be a linguistic expression of an abstract content or proposition (i.e., that’s not strictly speaking in the world or about the world). It’s not, therefore, a reference to things/events/conditions/etc. which are in the world. Thus the testability statement doesn’t refer to the world; though the statements it refers to do refer to the world.

So here’s another formulation:

The testability statement expresses an abstract entity (i.e., it can be taken propositionally) which is about statements which refer to the world.

It follows that the testability statement (at least on the reading so far given) needn’t necessarily be either testable or tautological. We could see it, instead, in various other ways. It could be a normative, stipulative, prescriptive, foundational, axiomatic or simply a higher-order statement. Of course whether any of these possibilities are workable or acceptable is another matter. Nevertheless, the testability statement is certainly of a different logical order than statements which refer to the world.

The Normative and Modal ‘Must’

The Verification and Testability Principles state that meaningful statements must either be testable/verifiable or tautological in nature.

Now the word “must” can be taken normatively or modally (see here).

If the word “must” is taken normatively, then it can also be replaced by the word “should”. (This would make its normative character clearer.) If the word “must” is taken logically, on the other hand, then the testability/verificationist metastatement is claiming that all statements about the world are (or should be) either testable/verifiable or tautological otherwise they’re meaningless. It can also be a modal claim about world-directed statements.

In a sense, then, the modal and/or normative character of the Testability and Verifiability Statements can’t be clearly disentangled. That is, if it’s the case that all meaningful statements which refer to the world are either testable or verifiable, then the verificationist can (or should) also say that they should be either testable or verifiable (i.e., in order to be genuine statements about the world). The normative “must” (or “should”) is therefore born of the modal “must”. Or, more clearly, the normative “must” (or “should”) is born of the modal property must necessarily be the case.

Of course what’s necessarily the case doesn’t depend on what should necessarily be the case (or even on what simply should be the case). However, we’re talking about both a normative statement and a statement which also includes a modal claim about world-directed statements in all possible cases or situations.

Finally, even if the Verification and Testability Principles aren’t actually (as it’s often put) “self-referentially self-refuting”, then that doesn’t automatically mean that there aren’t other things wrong with them.

[I can be found on Twitter here.]