Thursday, 7 September 2017

On the logical form of maximal inclusivity

It strikes me as obvious, even a truism, that people find it important to feel that they belong to a larger group or community of some sort or another.  I will, at any rate, take that as given.  So, a sense of inclusion is psychologically valuable to many.  There are, no doubt, those who prefer to be alone, and we all have times when we need solitude, but overall, one wants to be part of a community.

It seems, further, that in many cases the sense of belonging is engendered by a shared characteristic of some sort: all who believe that P, or accept that Q, or who are R, bond over P, Q, or R, and form a community on that basis:

“All of us, here, are P/Q/R”.

As valuable as such congregating may be in creating a sense of inclusion, it also entails exclusion, for:

“You must be F to belong here”

is logically equivalent to:

“If you are not-F, you do not belong here”.

The risk as that those who are F will clash with those who are not-F, over resources, or legislation, or simply broader social norms.  On risk is the urge to segregate: let the Fs live here, the non-Fs there.  This is the dark side of the craving for inclusion.  For example, suppose a child grows to reject the religious upbringing of her parents but still wants to feel a sense of belonging to her family and community.  It would be a shame for her to have experience rejection on that basis, so we want families, for example, to have a broader sense of belonging/inclusion that a shared religious belief or practice.  Accordingly, I think it worthwhile ask what greater inclusion would look like.

Which leads me to wonder what the maximally inclusive community would be.  Is there a conception of inclusion that is sufficiently broad that it could include everybody?  I will eave aside for the moment whether such a broad definition of community would satisfy the psychological need for a sense of belonging.  My question is whether it is logically possible to define a community that would not be exclusive, something along the lines of:

            “All are welcome here”.

Right of the bat, this seems like a paradoxical formulation.  Suppose we had a community defined by this principle: would it welcome those who actively wish to destroy the community?  Could they really be welcome there?  It is hard to see how a community could, in general, be robust enough to survive if it had to invite in those who wished to destroy it, as they could not be integrated into the community as a whole.

What is nice, however, about “all are welcome here” is that it is a, relatively, content-free criterion of inclusion.  Any community built on a specific belief, practice, or physical trait, is inherently alienating because many will not share that belief, practice, or trait.  So a content-free, or formal criterion of community would likely be the most inclusive.  We can start with the idea of a community that is welcome to anyone who doesn’t wish to destroy the community itself:

(Ax)(Ac)[~D(x, c) à W(x, c)]

D(x, y) = x works to destroy y
W(x, c) = x is welcome in community c

This seems necessary but not sufficient.  After all, the most exclusive community imaginable could still satisfy this requirement: e.g. a very religiously intolerant group would still count as maximally inclusive so long as it was unwelcome to those who are religiously tolerant.

Okay, so let us focus on the concept of welcoming, which is a naturally inclusive idea.  Consider a community that is open to anybody who is him/herself maximally welcoming, or, put differently, a community that welcomes any who welcomes any:

(Ax)(Ay)(Ac)[W(x, y) à W(x, c)]

Of course, such a community must only include those who are maximally welcoming, so let's make this a biconditional:

(Ax)(Ay)(Ac)[W(x, y) <-> W(x, c)]

So long as “x welcomes y” entails that x does not try to murder, humiliate, cheat, or otherwise harm y”, then it seems that such a community could be defined without welcoming in murderers, thieves, and those who otherwise wish to destroy communities.

Thus, perhaps it is logically coherent to imagine a community that is maximally welcoming without immediately inviting self-destruction.  Such a formal principle is useful because it provides a criterion for determining when a community is as open as it can be.

Wednesday, 14 June 2017

Philosophy, Science, and the "Core"

There is a clear sense in which modern science is a progression away from an anthropocentric worldview toward one in which human kind and human experience are no longer viewed as central to our understanding of the general features of the universe.  The scientific revolution ushered in a period of human decentering, which of course continues to this day. 

Aristotelian physics, as sophisticated as it was, revolved around the human experience of the world.  For example, according to him: the Earth occupies the center of reality with the far off starts utterly alien to anything we encounter here; just as we govern ourselves by making decisions about what is best to do, so nature governs itself in accordance with its proper purposes; just as biological species seem distinct and categorically different from each other, so they are; and so on.

If we pick Copernicus as a natural starting point, we can see the scientific revolution as a rather systematic dismantling of this kind of perspective.  For example: not only is the Earth not the center of the universe, it isn’t even the center of the solar system; the universe operates by causal laws devoid of teleological residue; far from the universe being somehow made for us, we have adapted to it through eons of variation and selection (and the difference between species is better understood as one of degree rather than kind); and so on.

Now, one difference between the humanities and the natural sciences is that while the latter continue to produce a picture of a world that is largely indifferent to our viewpoint, the humanities remain centered on the human experience.  In studying such human creations as art, literature, music, sports, and society, the humanities have as their mission the understanding of the ways in which our endeavours impact ourselves.  The sciences have as their mission the understanding of the impersonal laws that underlie all phenomena, human or otherwise (this even when studying specifically human phenomena such as art or communities).

None of this is meant as a complaint against either the sciences or the humanities.  Indeed, I think it is important and appropriate that we engage in a study of both sides of the anthropocentric vs. impersonal viewpoint dichotomy.

All of which brings me to philosophy.

Philosophy is a unique discipline in that it has one leg firmly planted in the spirit of the scientific worldview and the other in the spirit of the humanistic one.  Much of philosophy is the struggle to understand and, if possible, reconcile this divide: how can an impersonal universe, as uncovered by scientific inquiry, contain such things as meaning, consciousness, qualia, morals, counterfactual truth, mathematical truth, truth at all, and so on?  This is part of what makes philosophy difficult and worthwhile: it is a bridging discipline that offers insight from a unique vantage point.

All of which brings me to the so-called “core” of philosophy: metaphysics, epistemology, logic, and the philosophies of language, mind, and science (I take this usage of "core" from here).

We can, very roughly, divide philosophy into impersonal and personal, or scientific and humanistic, sides.  Philosophy includes, for example: the attempt to understand both the structure of causation and also causal talk; the attempt to understand the nature of time and our experience of time; the nature of moral properties and the nature of our reactive attitudes; the nature of aesthetic properties as well as the human experience of art; and so on.  Indeed, without paying attention to both sides of these sorts of dichotomies, it is hard to imagine the result being anything like philosophy at all. 

All of this is a big part of the reason why philosophy has to remain in conversation with contemporary science: the psychological and cognitive sciences will help us to understand the human experience while the natural sciences help us to understand the world.  Without both of these, our philosophizing will certainly be impoverished.

However, all of this also points to something that often gets overlooked both about and within philosophy: we need the “core” of philosophy precisely because it is the entry point and locus of the impersonal point of view into philosophy.  Value theory leans, rightly, toward the human experience, while M&E lean toward the impersonal point of view.  If the latter is impoverished, so is the discipline of philosophy as a whole.

Think, for example, of Bertrand Russell’s argument that the temporal present is an artefact of human psychology, not a feature of time itself; or David Lewis’s argument that our world is just one of infinitely many, equally real worlds; or Quine’s argument that the most intimate part of our experience – the meanings of our thoughts and utterances – are actually devoid of determinate content because even a perfect scientific observer of language users would be unable to determine such content; or Mackie’s argument against the existence of moral properties; or Putnam’s arguments that cognition can be understood on analogy to computation; and so on.

What all of these landmarks of 20th century “core” philosophy have in common is the attempt to reconcile the human perspective with the detached, scientific one.  Similar examples could easily, of course, be produced from the works of Hume, Nietzsche, Descartes, Locke, Plato, and other notables.

None of this is to argue that the “core” of philosophy is more valuable or important than any other area.  It is just to argue two things: (1) the “core” remains essential to philosophical inquiry; and (2) the function of injecting the detached, impersonal perspective into philosophy is inherently interesting and valuable precisely because our understanding of ourselves and reality is greatly enhanced by the attempt to reconcile – or, if that’s not possible, understanding the gap between – the two perspectives.  If we are going to engage properly in the human project of self-understanding, then the impersonal perspective is essential (this may sound paradoxical, but I hope to have given reason to think it in fact isn’t). 

So long as we think it is important and valuable to have a proper understanding of ourselves and the universe we inhabit, the “core” of philosophy remains something worth doing and something worth pursuing.  In short “core”, theoretical philosophy has value.

This argument is directed not only at academics outside of philosophy, university administrators, lawmakers, and the general public, but also at fellow philosophers, some of whom tend to dismiss “core” philosophy for the usual reasons: the Hawking argument that the questions addressed there should just be addressed by the sciences; the positivistic one that they all amount to meaningless speculation; the postmodern one they disguise an oppressive, harmful ideology; and so on. 

On the contrary, I argue, we need half of philosophy to consist in the regular input of the impersonal perspective into semantics, the philosophy of mind, ethics, political philosophy, aesthetics, and so on.  Without this, there can only be philosophically impoverished humanistic projects.

These reflections were stimulated in part as a result of some interesting comments by Marcus Arvan (see here) in which he suggests that we work to raise the public profile and financial security of the profession of philosophy by emphasizing and selling the practical, naturalistic, and inter-disciplinary aspects of the field, which, he argues, are more likely to attract public support and funding, and then use this funding to support the more theoretical research of the core of the discipline.  This argument is based on the model provided by the natural sciences which, he argues further, garner widespread financial and moral support on the basis of their practical benefits, but which use this support to fund the pure research wings of the disciplines.  

I think this is an interesting suggestion, but I have a worry, which is that the relationship between the “core” and the more practical, value theoretic sides of philosophy are not like the relationship between applied and theoretical science.  In the sciences, as much as there is plenty of jocular put-downs of the opposing side, the theoreticians and the experimentalists generally respect each other and recognize the other as engaged in worthwhile, genuinely scientific work (one side may be *more* valuable, but both are valuable).  In Philosophy, there are too many examples of thinkers accusing their opponents of not even doing worthwhile work.  This is not, contrary to one narrative one often encounters, just a matter of analytic metaphysicians telling practical ethicists or Heidegger scholars that what they are doing is “not philosophy”, but also of thinkers telling analytic metaphysicians that what they are doing is pointless drivel or actively harmful.

I think we need to stop this and until we do I think that a strategy such as Arvan’s could inspire increased in-fighting as different philosophical camps view a renewed focus on practical philosophy as just a grab for an increased share of limited resources that won’t lead to increased support for “core” research, since the latter isn't respected by the practical side.

Yes, I am advocating for the importance of the “core” of philosophy, where I happen to work.  I do not mean to denigrate or complain about other areas of philosophy.  As I argue above, we need both sides of the divide for philosophy to continue to be the special and important discipline that it is. 

Tuesday, 13 June 2017

Gödel’s Incompleteness Theorem – An Informal Explanation Of The Argument

Gödel’s first Incompleteness Theorem involves some significant technical achievements but is actually based on a very straightforward argument.  Let’s examine that argument.

In brief, what Gödel argues is that any formal language that is sufficiently rich that it can:

(i)            Enumerate formulas;
(ii)          Refer to formulas by their numbers;
(iii)         Express the predicate ‘x is provable’; and
(iv)         Express negation;

will have the following property:

If the system is sound, i.e. contains no falsehoods, then there will be formulas expressible in the system that are not provable in that system. 

Gödel shows that the formal language of Russell and Whitehead’s Principia Mathematica, which can produce the formulas of arithmetic, is just such a language.  Accordingly, he concludes that arithmetic can contain unprovable formulas.  This is a more technical part, so I will just focus on how the core argument works.

To begin, suppose we can enumerate, i.e. list, all the formulas of our system.  For example:

(1) 0 = 0
(2) 1 + 0 = 1
(3) 2 + 2 = 2 + 1 +1
(4) 1 + 1 > 0 + 1

Okay, now a question arises: how shall we order this list?  It isn’t really important how so long as we have a 1-1 correspondence between numbers and formulas in our list, so we will choose something such as the following: we list our formulas in ascending order of number of terms in the formula; where there is a tie, we break it alphabetically, e.g. 1 = 1 comes before 2 = 2.

So, let’s put our list in order:

(1) 0 = 0
(2) 1 + 0 = 1
(3) 1 + 1 > 0 + 1
(4) 2 + 2 = 2 + 1 + 1
Okay, that’s (i) above.  Now consider the following addition to our list:

(1) 0 = 0
(2) 1 + 0 = 1
(3) 1 + 1 > 0 + 1
(4) 2 + 2 = 2 + 1 + 1
(5) Formula (4) has more terms than formula (3)

What we have done in formula (5) is pick out, or refer to, other formulas by their numbers, i.e. location in the list.  This is straightforward, so that’s (ii).  Next, let’s continue our list but add a new resource, the predicate ‘x is provable’:

(1) 0 = 0
(2) 1 + 0 = 1
(3) 1 + 1 > 0 + 1
(4) 2 + 2 = 2 + 1 + 1
(5) Formula (4) has more terms than formula (3)
(6) Formula (4) is provable.

What we have done at this step is nothing other than what we have already done, except for the addition of the new predicate.  So that’s (iii).

Pause here for a moment to notice that the order in which we list our formulas is arbitrary: we could have chosen another order just as easily (maybe in descending order of number of terms, for example).  Accordingly, what has ended up here as formula (4) might have been formula (3,245) in another list.  We can think of it this way: suppose we write each formula on a card and instead of stacking the cards randomly, we put them in order according to our rule.  So the card with ‘2 + 2 = 2 + 1 + 1’ ends up as the fourth card in our stack, but wouldn’t have had we used another ordering rule.  Still, once cards are put in their place on the list, we can refer to them by their number.

What this means is that the card with ‘Formula (4) is provable’ written on it will be true or false depending on how we stack the cards: one ordering may put a card in slot (4) that is not provable; another one may put a card there that is.  Also, the card with ‘Formula (4) is provable’ written on it may occur in slot (6) as above, but it may otherwise have occurred in slot (1) or slot (1,000,000).

Okay, so let’s continue our list of formulas.  We will use the resources we have used so far with the addition of one more, negation:

(1) 0 = 0
(2) 1 + 0 = 1
(3) 1 + 1 > 0 + 1
(4) 2 + 2 = 2 + 1 + 1
(5) Formula (4) has more terms than formula (3)
(6) Formula (4) is provable
(7) It is not the case that formula (1) is provable

By employing negation, formula (7) tells us that the first formula on the list is unprovable (assume for now that this is true).  We have now made use of all of (i) – (iv) above.

So, let’s put all the resources together in one more addition to our list:

(1) 0 = 0
(2) 1 + 0 = 1
(3) 1 + 1 > 0 + 1
(4) 2 + 2 = 2 + 1 + 1
(5) Formula (4) has more terms than formula (3)
(6) Formula (4) is provable
(7) It is not the case that formula (1) is provable
(8) It is not the case that formula (8) is provable

Formula (8) is almost exactly like formula (7); though it refers to formula (8) rather than formula (1), there is nothing untoward in that.  Recall that the formulas could have been put in another order if we had so chosen; it is simply ‘luck’ that the card with ‘It is not the case that formula 8 is provable’ landed in spot (8) on our list – had we ordered things differently, it might have ended up in slot (2).  But, the point is that there is nothing fishy about this formula landing in slot (8) – if a formula can refer to a formula by number, it can refer to itself by its own number: it is very easy to create such a formula, as has just been done above, all in accord with (i) – (iv).

In sum, formula (8) refers to a formula by its number, employs the predicate ‘x is provable’, and makes use of negation.  It is a perfectly legitimate entry in any list that involves a language rich enough to include (i) – (iv). 

Okay, but now consider formula (8).  If (8) is provable, then (8) must be false.  Why?  Well, simply because (8) expresses the proposition that it is note the case that (8) is provable, which can only be true if (8) is not provable.  So, (8) can only be provable if it is false, in which case the system as a whole contains a false formula (namely, (8)).  But that is the argument in a nutshell: if we assume the system contains no falsehoods, and is rich enough to allow for (i) – (iv), then there will always be a formula expressible in the system – (8) in our little example – that is not provable. 

So, a sound system rich enough to allow for recursive enumeration, negation and the predicate ‘x is provable’ will always admit unprovable formulas.  That’s the gist of the first incompleteness theorem.  Once Gödel showed that arithmetic is a formal system in which all these conditions apply, he had proven that arithmetic is incomplete with respect to provability. 

Notice that we know that (8) is true.  If (8) were provable, then the system would contain a falsehood, so (8) must not be provable, which is precisely what it expresses.  Hence, something can be true without being provable. For Gödel, it is the assumption of universal provability that leads to trouble, not the assumption of soundness, i.e. universal truth, in a formal system.

What is particularly interesting, in my opinion, is Gödel’s employment of Gödel Numbering to show that formulas that express “such-and-such is a proof of so-and-so” can be rendered as ordinary arithmetic equations.  Hence, all of the foregoing can be done from within the language of mathematics itself.  The language of mathematics is suitable for meta-mathematics, at least in this instance.