The Paradox of the Barber Who Shaves Everyone Who Doesn't Shave Himself

There are actually some (as it were) false paradoxes: arguments or situations which seem paradoxical until they are seen not to be paradoxical – simply false. Some logicians claim that this is one.



Well, does this barber shave himself? He must do because he shaves all those who don't shave themselves. But if he shaves himself, he can't be a member of [the class of those who don't shave themselves]. Yet his job is to shave all those who don't shave themselves – and he only becomes a member of that class when it's seen by himself that he doesn't shave himself.

Thus we have a contradiction. As a person who doesn't shave himself, he must shave himself. If he did shave himself, then he wouldn't be a member of [the Class of Those Who Don't Shave Themselves]. But the resultant situation, in both cases, is that he shaves himself. If he shaves himself, then obviously he shaves himself. However, if he doesn't shave himself, and he must shave all those who don't shave themselves, he must also end up shaving himself.

Why on earth should we assume that there isn't a barber who shaves all and only those who don't shave themselves? It's certainly not illogical if he excludes himself. But perhaps that's precisely the problem – he can't exclude himself! However, even if he can't exclude himself, the paradox doesn't appear to disappear: talk about not assuming that there could be a barber who only shaves those who can't shave themselves seems to be to sidestep it the problem.



The philosopher Roy A. I believe that Sorensen makes the mistake of saying “we should not assume that it is possible for there to be a barber who shaves all and only those he does not shave”. That locution doesn't appear to make sense. It's not paradoxical – just senseless.

The end result of this possible paradox is that we have a barber who shaves all and only those he does not shave. But I still can't work out how you get there. Therefore is may well be a mistake – perhaps just a typing mistake. The thing is, I don't know!

So perhaps Sorensen's locution is correct after all. The end result of this possible paradox is that we have a barber who shaves all and only those he doesn't shave. However, I still can't work out how Sorensen gets there. Therefore may well be a mistake – perhaps just a typing mistake.

David Hilbert

David Hilbert embarked on an enormous enterprise: the reduction of the whole of mathematics to a set of axiomatic systems. Clearly these systems must have been interrelated in many ways in order to determine and guarantee their mutual consistency. We can distinguish each axiomatic system by their different axioms. In addition, in order to move from these axioms to theorems, we must utilise something that is shared with logic: the rules of inference. Again like logic, one such rule of inference is the well-known modus ponens. This shared interest in the rules of inference is partly accounted for by the fact that "they are properties of reasoning as such". That is, both logic and mathematics must use the inferences which belong to every rational mind and even every thought or act of reasoning. They were, therefore, as foundational and fundamental as Aristotle’s ‘laws of thought’.

However, there must be some things, other than numbers, which distinguish mathematics from logic. For example, a prime candidate for this difference is the nature of the axioms in mathematics. They are different because they describe space, time and measurement in all its forms. These applications, to space, time and measurement clearly distinguish mathematics from logic, at least from pure formal logic but not, for example, quantificational logic.

Hilbert was a Platonist. He didn’t reject numbers in his systems. We cannot "eliminate the idea of number from the axioms". They are, then, fundamental to maths precisely because they are used in its many axioms. Hilbert expressed his Platonism in an even more platonistic way. Like Plato himself, "we must therefore suppose that numerical expressions stand for objects, which have a reality independent of our calculations". He was clearly not, therefore, a mathematical constructivist or a mathematical Wittgensteinian, and neither was he a mathematical Kantian. In addition, although these number-objects are "known to us through proof, but which are entities over and above the proofs by which we discover them". So Hilbert accepts that they are only known to us through proof, which is an operation and perhaps a psychological operation, these number-objects do not need us in order to exist. The numbers can then be called evidence-transcendent, at least in the case of those numbers and operations that can never be known, or are not known now. They are also mind-independent objects, even though they can be ‘known’ by minds.

In some cases, then, some object-numbers and operations may never be known as a matter of necessity. Take ‘Golbach’s theorem’ which cannot be proved (see Kripke’s 1971). These positions seem very counterintuitive to many minds, not only non-philosophical minds. Indeed they seem even stranger in the context of that branch of mathematical constructivism known as intuitionism. In the case of intuitionists, if a mathematical statement has not been proved or disproved it is, in fact, neither true nor false, mind-independently or otherwise. In fact, unproven statements have another ‘truth-value’: indeterminate. Incidentally, this is also the case with ‘future contingents’ – statements about the future. They too are neither false nor true, but they are indeterminate instead. Perhaps this is because a future-statement cannot be proved either, almost by definition.

In addition, being constructivists, intuitionists do not believe that numbers are objects either, whether abstract or concrete. We ‘construct’ numbers by the operations we carry out on them. They are not found, either, via platonic ‘intuition’ or Husserl’s ‘direct insight’. If there were no minds, there would be no numbers and no mathematical operations on these numbers.

'What do materialists make of love & justice?'

“How do you consider concepts (like love and justice) to be physical/material? Where do they exist? To say something like a number (which is abstract) is physical is misplaced concreteness.” - Wololo

I'm not sure that anyone who has sympathy with materialism, or physicalism, would say that love and justice were material in any strict sense. Nonetheless they may well give a physicalist or a naturalist account of such things. In depends on whether or not you have a Platonic or quasi-Platonic view on love and justice. If you do, then, by definition, they would be non-physical universals (or 'Ideas' in Plato's parlance).

However, if you aren't a Platonist of some kind, then love and justice can be given a naturalist, if not a physicalist, explanation which may or may not be successful. For example, love - rather than Love - can be explained in terms of human biology, history, human and social relationships, etc. – all of which a naturalistic in nature. It need not be the case that love and justice are 'reduced', strictly speaking, to such things. However, love and justice must be dependent on natural things and not run free, as it were, of them.

So, yes, both justice and love exist wherever there are examples or expressions of love or justice. Love can exist where acts of love occur and the same with justice. (For now we can forget the different positions people adopt on both love and justice because this question is about whether or not they are natural, or even physical, phenomena.)

No scientist today would ever say that numbers are physical or material. And only a very small numbers of philosophers would argue against numbers being abstract in nature. Despite that, these latter philosophers don't think that numbers are concrete either. Some philosophers argue that numbers, or equations, are basically invented. They are a product of conventions and symbols and the rules which are applied to those conventions and symbols. (Perhaps this does mean that numbers are concrete, in a certain sense.) In other words, there is no need for abstract numbers as such. These few philosophers, and even fewer mathematicians, argue that numbers are created, or invented, by the mathematical procedures that bring them about. That is, a new number is created when a new procedure brings that number into existence in a similar way in which a new concept or word (such as 'nerd') is brought into being. Or, if not words/concepts, numbers are created in the way that, say, a new design for a building is created. (I'm not saying I agree with any of this; only that these views exist.)

'Does reality really exist?'

“Let's take my apple example. I can see it, feel it, taste it and smell it. But if I'm dreaming or under hypnosis or drug induced, the apple then isn't real. And being awake may not matter if the apple is an illusion or if my senses are altered to create that apple.

“So does reality really exist? When can we trust our senses with 100% reassurance?” - Philosophy Explorer

You can say that instead of the question 

"Does reality really exist?"

the question should be: 

How do I know that I'm not dreaming, etc.? 

After all, such scenarios (as is often the case) seem to be written into questions such as "Does reality really exist?" In other words, the sceptic is, in a way, begging the question or at least assuming some preliminary questions and answers.

The question is, then, about knowledge: 

How do I know that reality exists? 
How do I know I'm not dreaming? 
How do I know that this apple isn't a Ford Escort?

If you miss out the word "know" from the question, then you're certainly feeling, tasting and smelling something that you think is an apple. Those experiences are real even if dreamed. Those experiences are real even if simulations of the real thing.

I personally wouldn't ask a question like "Does reality exist?". I'd stick to something traditional like: 

"How do you know your hand in front of you exists?"

Science’s Communal Spirit & Philosophy

An essential part of science is what various commentators have called its “community spirit”. That is, scientific truths aren’t confirmed, justified or accepted (rather than simply “discovered”) intuitively, in isolation or through meditation. (This isn’t to discount independent scientific theories, original speculations and the reality of scientific genius.) Scientists don’t pluck out truths from the air above their heads. (One can philosophically dispute the use of the word “truth” in science; though this isn’t the place to do so.) As Bertrand Russell once put it:

“A body of individually probable opinions, if they are mutually coherent, become more probable than any one of them would be individually. It is in this way that many scientific hypotheses acquire their probability. They first fit into a coherent system of probable opinions, and thus become more probable than they would be in isolation.” 

Perhaps the prime distinction (at least traditionally)between science and both philosophy and religion is that scientists deal with what’s called “probable opinions”; whereas philosophers and religious thinkers/leaders deal with truths. In that sense, an individual scientist wouldn’t think that he has found a/the truth in a situation of “splendid isolation”. However, if there were a general consensus within the/a scientific community on his probable opinion, then perhaps the honorific “truth” could then be applied to his opinion.

And just as a philosophical coherentist compares an individual belief with all the other beliefs within the given system it is part of, so a scientist needs to place his probable opinion within his own scientific community. In addition, just as the individual scientist relies on his own scientific community, so too does a particular scientific community rely on other scientific communities (i.e., those which may be focussing on different areas of research or investigation).

And what Ludwig Wittgenstein argued about “private knowledge” can now be applied to the situation of the lone scientist. Indeed the term “lone scientist” is almost a misnomer when taking into account the history of science and how science is actually practiced. That is, there can’t really be genuinely lone scientists; just as Wittgenstein argued that there couldn’t really be lone epistemologists or people with private truths of private knowledge. Sure, there have been many highly original and sometimes unacknowledged scientists. However, their work only became acceptable science when legitimated by the scientific community (or communities in the plural) as a whole. (A good contemporary example of this is the physicist Julian Barbour who works outside the academic system of physics and who yet still influences that system.)

The scientific approach is antithetical to the philosophical or religious/spiritual approach. In many instances philosophers and religious thinkers/leaders worked in complete isolation. Indeed there’s a sense that because of the nature of philosophy (loosely, its — as it were — a priori method), then clearly philosophers don’t need to cooperate in the way that scientists cooperate. Indeed that almost solipsistic attitude was challenged — by a philosopher — in the 19th century. 

Take the 19th century American philosopher C.S. Peirce. He believed that philosophers should learn as much as they can from science and scientists. (This was primarily because of Peirce’s penchant for science and his many years in the laboratory.) He even thought that philosophers should actually use scientific methods. (Peirce also believed that philosophers and even logicians should study the way scientists reason.) According to Peirce, the idea that a single individual could arrive at the truth entirely on his own is a complete mistake. Yet although philosophers obviously read and analyse the works of other philosophers, they’re still doing so (arguably) within the context of their own intellectual autonomy.

Having said all that, it’s nonetheless argued that the “analytic tradition” of philosophy has (to some extent at least) been a cooperative endeavour in which philosophers not only learn from each other; but, in many instances, they actually work with each other too. (Think here of the Vienna Circle or W.V.O. Quine writing a paper alongside Nelson Goodman — see here.) Of course what really makes analytic philosophy a cooperative effort is the shared vocabulary and set of technical tools/terms — i.e., the shared philosophical and logical toolsterms that are utilised in all areas of analytic philosophy and research. And because of that, both analytic realists and analytic anti-realists, analytic dualists and analytic anti-dualists, for example, use the same tools and belong to the same philosophical tradition. Of course there will be peripheral disputes on terms and definitions (as well as on the reality of that tradition); though such disputes usually still occur within the context of a generally cooperative environment.

Perhaps we could say here that if philosophers don’t even share a vocabulary, then the conversation couldn’t even get started. Philosophers would be debating at cross-purposes. Indeed isn’t that what actually happens when, say, a analytical logical empiricist debates with a Parisian deconstructor? And that is a lesson that philosophers should (or must) learn from science.

Against the Science of Mind

"The reality of the external world to which science points has no psychic depth, no depth of being. It is a plastic mass of events. When scientists study Man, they want to prove that the mind, the psyche, the being of Man, is the effect of bodily existence and thus an effect of matter. They conclude that if the mind is caused by matter, then it is basically unreal, secondary, not a primary reality." - Granth

I'm not sure if there is such a consensus in science on the mind. Even in the limited domain of 'materialist' philosophy of mind I don't think that there is such a consensus.

Granth refers to "no psychic depth, no depth of being" of science. These technical terms seem to be taken from a specific philosophical tradition so it will be hard for people unfamiliar with that tradition to know what such locutions mean.

Granth also says that scientists (all of them?)

 

"want to prove that the mind, the psyche, the being of Man, is the effect of bodily existence and thus an effect of matter. They conclude that if the mind is caused by matter, then it is basically unreal, secondary, not a primary reality".

 

It doesn't follow that if a scientists argues (or shows) that "the mind is caused by matter" that he also believes that it's "unreal, secondary". A forest fire can be caused by a discarded cigarette; though the fire is still real even if it has causes. The mind and brain can even be acceptably different domains, according to scientists, and it still be the case that the brain (or something larger) is the "cause" of the mind. Scientists, on the whole, are no longer interested in erasing mind or consciousness from the equation. In fact only a few scientists ever were completely that way inclined.

As for "the Being of man" - that seems to be the technical language of a specific philosophical tradition which, presumably, not all people will be aware of even if they know much philosophy. What is "the Being of man"?

Karl Popper on the Infinite Variety of Facts

It is because there is an ‘infinite variety of facts’, and each one of this infinite variety has itself an ‘infinite variety of aspects’, which we have to approach the world of facts with presuppositions. It may be an ideology, a ‘scientific theory’, a prejudice, or whatever.


The ‘theory’ itself makes the choices in science. The ideology makes the choices in historical research and political knowledge.

So these theories, or even prejudices, act in a similar manner to the un-argued premises or axioms in arguments or logical/mathematical systems. That is, because of the infinite amount of facts that are possible candidates for comment and research, something prior is required to get the ball rolling; just as the sceptical doubt in the very act of doubting must presuppose certain things that are beyond doubt (at least within specific contexts). The theory, or prejudice, then, allows the theorist to do some ‘proper ignoring’ of certain facts or possible facts, as David Lewis put it (“within the context” of epistemology). The theory, in a certain sense, selects what facts should be concentrated upon and which should be ignored. It therefore determines the scope of the investigation and research. Without an a prior theory or prejudice, as it were, the theorist would be faced with a manifold of possible facts. Such a bombardment would serve no purpose in either science or philosophy.  As Popper puts it:

“… a science is not merely a ‘body of facts’. It is, at the very least, a collection, and as such it is dependent upon the collector’s interests, upon a point of view. In science, this point of view is usually determined by a scientific theory; that is to say, we select from the infinite variety of facts, and from the infinite variety of aspects of facts, those facts and those aspects which are interesting because they are connected with some more or less preconceived scientific theory.” (259)

The scientist must have ‘interests’ and ‘points of view’. More than that: his interests and points of view help determine the direction in which his investigations go. Indeed without such interests and perspectives, the scientist would not in effect go in any direction in his investigations. He would be in a continual state of cognising more and more facts, without a hope of connecting them or making sense of them. This is also some kind of selection process. Not all possible facts are open to investigation or analysis. Many must be disregarded and the rest are selected. Without such selection, the scientific project itself would not even begin. The facts that will be selected, therefore, are ones that help legitimise the theory which itself is responsible for determining the selection process. It could be said, therefore, that there is a certain circularity involved here. That is, the theory selects the facts to investigate, and those very facts may help to legitimise the theory that is doing the selection.

The above could easily be rewritten in this way:

In politics and history, our point of view is often determined by an ideology. That is to say, we select from the infinite variety of political and historical facts, and from the infinite variety of aspects of these political and historical facts, those facts and those aspects which are useful because they backup and are connected with our preconceived ideologies.

So just as in science, so too in politics and even everyday life, we must in a sense select the facts and bits of information that will help us make sense of the world around us. We cannot be open to the manifold or the infinite. And even if we were, such a state wouldn’t get us anywhere. It would result in some kind of endless list and description of an indefinite amount of facts and bits of information. But what point would that serve? It was just be a glorified list with no practical or theoretical purpose; just as the ‘accumulation of facts’ theory of science is a complete parody of actual science. Something needs to guide us, whether a theory, an ideology, a prejudice, an interest, a bias, or mixture of these.

In the end, we build up a securer and securer ideological edifice. The problem is, of course, that the foundation of this ideological edifice may be insecure because it is based on prejudice and/or falsehoods.