Prompted by reference 1, we went on to think about the philosophical concept of moral status. And from thinking about what sort of things might have moral status, we went on to think about some set theory flavoured basics. Basics which might provide some better quality foundations for moral status.
We introduce the terms inner universe, outer universe, name, function and factorisation.
Contents
- Inner universes, outer universes and names
- Examples of things in universes
- Numbers
- Parts of speech
- Functions and values
- Functions and their factorisation
- An example of a function
- Membership
- Conclusions
We think here of collections or sets which we shall call universes, roughly in the sense of universes of discourse. These sets are worth writing about here because of their heterogenous membership: we are not writing of sets of apples, but of sets of apples, pears, oranges and all kinds of other things, some of which may not even be real – in the everyday sense of the word – at all.
A universe is finite and we could probably set an upper bound, typically more than 50,000 (a guess based on vocabulary size), probably less than a million.
The things in our universes are the sort of thing to which we might give house room in our brains, perhaps the sort of thing represented or at least labelled by some particular collection of neurons, thinking here of the fact in some peoples’ brains there will be a small number of neurons which light up when you say ‘Marilyn Munroe’ or allude to her in some other way, perhaps by showing a picture of her. We suppose that distinct things are represented by disjoint collections of neurons, that is to say no neuron appears for more than one thing, in more than one such collection. We sometimes call these collections, these things in the brain neural things, to distinguish them from things in the world outside. Collectively, an inner universe.
Note that a neural thing is more by way of a pointer than of data proper. It is a place from which one can get to the data proper. But probably not a place in the sense of a position with coordinates; so the neurons standing for Marilyn Munroe might be tightly bound together, as might those standing for Alexander the Great, but these two collections might well overlap in space, on the cortical sheet.
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| Figure 1 |
So such things in the brain, such neural things, usually have causes in the world outside the brain, including here the body beyond the brain as well as things further afield. Causes which stimulate the nervous system – with some causes stimulating one or more of our neural things in a reliable way. So, for example, seeing a zebra stimulates two or three zebra flavoured neural things. Perhaps including, if the brain is feeling wayward, a zebra fish.
We note that what we have shown here as a simple loop with a decision diamond, is described in a rather different way, as a more complicated interaction between top-down and bottom up processes, in the work of Friston & Hohwy. Ask for ‘predictive coding’ or ‘predictive mind’.
It seems likely that there are going to be plenty of neural things which we never get to know about consciously, never get to name or to identify causes, although causes there certainly are. But other neural things are conscious and can often be captured by an utterance. We see a zebra and we say to ourselves ‘zebra’ or perhaps ‘horse like animal with black and white stripes’ if we have not seen one before. We can report the zebra to others or attempt to define a zebra for the benefit of someone who has never seen one.
This gives rise to the idea of an outer universe, a dictionary which is to a large extent shared by some more or less large group of people, a dictionary in which each entry has one or more short names and a long definition, all in the form of a finite strings of characters, probably but not necessarily mainly letters and spaces. We might have a prejudice against compound names, preferring to separate them into their constituents, on which point see reference 4. In any event, names are not often unique, even in combination, but they do serve as a device for indexing our dictionary and they do give us a way in. For example, the entry for Queen Victoria of England may have two names, ‘queen’ and ‘victoria’. The first of which will be shared with all the Queen Streets in the universe, the second with all the Victorias and the combination with all the Queen Victoria Streets and any other queens of that name that there may be.
We can compare these names with those used in Wikipedia, which, as it happens, includes many of the examples which will be listed below. Their rule seems to be that the name is some sensible combination of letters, numbers (aka numerals), underscores, hyphens and round brackets – in which underscore takes the place of the space. The first character is usually, but not always, a capital letter. These names are not necessarily unique, in that the entry may be or may provide what it calls disambiguation in the form of pointers to further entries. So we have an entry for ‘shipmate’ which, before getting onto shipmates, provides a pointer to ‘shipmate(magazine)’, the United States Naval Academy's Alumni Magazine of the same name. Indeed, the names used for Wikipedia entries nicely illustrate many of the problems involved in providing names for things.
But while the names of something in a universe do not amount to an identifier, in the sense that a computer might use that word, but they do tell us quite a lot about what kind of a thing we have. A thing with the names queen and Victoria is most likely a queen of that name as all the other sorts of thing with those names will usually have additional names like street, glacier or lake. Names of this sort which describe, but which do not identify, are sometimes called descriptors.
In sum, the members of universes are things that would be recognised by a reasonable man. So man A might have thing B is his inner universe and he could explain to some other man C what this thing B was. Man C might not really understand, might not agree with A’s position on B, but he would allow it as a thing in a universe. So it has to be the sort of thing that can be encapsulated, at least to some extent, in language and shared, at least to some extent, with someone else.
The soft box diagram
We start with a technical digression.
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| Figure 2 |
That there is a many to many relation between real world cause and stimulation. That is to say that any one cause can result in a variety of stimulations and some stimulations can be the result of more than one cause – with, for example, a number of writers having suggested various interesting real world causes to account for what looks like a tiger behind a picket fence.
That there is a many to many relation between stimulation and neural thing. First because several people may get the same stimulus. Second because stimulation may not be that reliable, sometimes activating one neural thing, something another. Or we might allow the activation of more than one.
But we do like these linkages to have a property which we call coherence. That is to say, generally speaking the stimulations associated with any one neural thing in a person’s brain are associated with the same real world cause, and vice-versa, that is to say the stimulations associated with any one real world cause are associated with the same neural thing in any one brain. We don’t expect our world to be completely neat and tidy, just reasonably so.
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| Figure 3 |
Real world is grey because it is not really a proper box at all, because there is only one real world.
Dictionary is grey because while there is apt to be more than one of them, there are not all that many: in the UK might settle for four, one for each of our stronger languages.
The outer universe is a dictionary which is shared by a number of people. Each person has his own inner universe, made up of the neural things in their brains.
A neural thing may be activated by any one of a number of stimulations, stimulations which are the result of things happening, or of things being out in the world. What we have here called real world causes.
People can say what is in their mind, this is utterances. Utterances which will vary with time and place. The same neural thing can give rise to more than one utterance. And an utterance can be related to a dictionary entry which describes that thing in their mind. Such an utterance may or may not involve a name. With the complication that more than one utterance can be associated with any one dictionary entry, and any one utterance can be associated with more than one dictionary entry. Somehow our brains have to sort all this out.
Implicit in all this is a many to many link between cause and name, shown dotted because they are inferred, artificial rather than natural. The idea is that generally speaking the same cause will result in much the same utterance in most people and that we can link cause to entry through name. But this requires culture, language and work; it is not given to us on a plate.
Examples of things in universes
Many members of universes are well defined material things, like lions, tigers, bridges and houses. Things which have a life cycle, which come and go, but which do persist for months and years and which are apt to persist for longer in our universes than they do in real life. They do have an afterlife of sorts there. But we also allow other kinds of things like places, gods, events and even theorems.
Possible members (with some issues flagged up):
- Alexander the Great. A regular sort of thing to have in a universe, with the only catch being that he is long dead
- The Battle of Hastings. An event which took place a long time ago, rather than a thing, but an event which can, nevertheless, certainly be shared with someone else. No problem about giving it a name and a definition
- The death of Achilles. A fictional event which took place a long time ago. But again, no difficulty about its inclusion
- Any of the Victorias of reference 3
- A modest number of numbers. We say more about these in what follows
- Joe Doe. A name used in the US to signal an ordinary man in the street. A concept rather than a particular person
- John Smith. A name used in the UK to signal an ordinary man in the street. A concept rather than a particular person. Different to Joe Doe in that what passes for ordinary in the UK is not the same as what passes for ordinary in the US
- Bṛhaspati. A god from India
- Mount Everest. A mountain named for a colonial surveyor, a one-time Surveyor General of India, a large thing, apt to fall foul of any size rule that might be applied (see below). And things might get sticky if one wanted a precise geographical specification of what exactly you meant by the phrase ‘Mount Everest’. Fortunately, the average brain is not troubled by such considerations
- Kentish Town. A place in north London
- A packet of corn flakes. A particular packet of corn flakes, perhaps the one from which I was feeding on the morning of Sunday 5th August and in which, for some reason, I took an interest. Perhaps there was an important plastic toy inside
- Ford C-max cars. The generality of such cars, not the rather battered example which we drive around in
- The Mistral. A cold north wind in the south of France. A thing which persists, an event which happens many times during its mainly winter season, which is known to travel guides everywhere, but which cannot be tied down to composition, to place or anything very much at all, beyond it being a wind blowing down from the mountains onto the resorts of the French Côte d'Azur
- The wild boars in the forest. The generality of such
- The plum tree in some particular garden. A particular plum tree, an individual. No need to fuss here about the biological niceties about the status of suckers, still tethered to the mother tree
- A new born child. The generality of such. A thing which is important in the context of the moral status with which we started
- The God Augustus. A god from the early Empire. Very few adherents these days
- The word ‘draft’, as a word in the dictionary rather than as something in the real world. All its various meanings. Its shape and sound
- The word ‘yes’, as a word in the dictionary
- The theorem of Pythagoras
- Chelsea Football Team. This being a shorthand for the first men’s team, the club running half a dozen or more different teams. A good example of a thing which persists, but whose composition is constantly changing
- Chelsea Football Club. Of west London, now the property of a rich Russian, a good friend of President Putin
- Stamford Bridge. This one needs disambiguation, as there used to be a brook and there used to be a Stamford Bridge over it. But this part of the brook is now a railway line and the phrase is generally used as a name for the stadium of the Chelsea Football Club. There is also a street sign with this name lurking in a front garden in Stamford Green, Epsom
- The River Thames. Another good example of a thing which persists, but whose composition is constantly changing. The water in the river is changing all the time, the tide comes and goes and river, in slower time, might well changes its course. There is much scope for argument about what exactly it is that the river consists of, what space exactly it is that it occupies. The sort of argument that makes much more sense for a river than it would, for example, for a zebra
- Thibaut Courtois. A member, at the time of writing, of the Chelsea first team. But there will be other times when he is not a member
- Cheese. Cheese in general. What sort of information will it come with? A conflation of information about cheese that the owner of the universe has come across over the years?
- Lincolnshire Poacher. A particular sort of cheese
- A piece of cheese. A particular piece of cheese, which has the plus of being a material object in a way that cheese in general is not and the minus of having a fairly short life. And much scope for argument about exactly when this piece came into existence and when it goes out of existence. How big a change amounts to termination?
- Zebras. Zebras in general. Probably no need to get into a tangle about animals at the margin of zebra-hood. Enough to concentrate on the central concept
- Zebra No.2346 kept in enclosure AK in London Zoo. A particular zebra, a thing which persists for a reasonably long time, which is clearly separate from the rest of the world and which is of a sensible size. An example of an ordinary sort of thing, the sort of thing that we think of as a thing.
Consider, for example, the property ‘material’, in the sense that something is material if it is made of matter – the word material being given around a page in OED, a word with many and various meanings. So a child’s wooden brick is material and the Holy Spirit is not. The Battle of Hastings is not material because it is an event. The Duke of Wellington is material because while he has been long gone, he was material in his time. So we include things which were material, even if they are no longer. We exclude ideas, like that of a square or the theorem of Pythagoras. But what about a book, for example the Iliad? There are lots of copies of the book around the world, there have been lots of copies of the book in the past, and all these books are or were material enough. But what about the Iliad, the poem itself, quite possibly composed before it could be written down in a book? Does that not transcend its material manifestations? What about heat? Heat which is expressed in the jostlings and vibrations of the atoms and molecules which make materials. It is real enough, it has a physical basis, but perhaps we can call those jostlings and vibrations an event, so excluding heat from the material universe. So what started out as a simple binary choice has rapidly become a bit of a mess – and reference 2 very probably has something to say about it.
Some of these things will have names. Sometimes the name will be unique. It is quite possible, for example, that just the one thing is named ‘the God Augustus’, that the first emperor of Rome was the one and only Augustus to acquire divinity. Much more often the name will only be locally unique, it will be unique on its home ground, in its context. So to a person living in Cambridge, talking to someone in Cambridge about some local matter, there is only one possible ‘Victoria Avenue’, even though there are dozens of them scattered up and down the country, with still more overseas. While according to reference 3 about 1% of girls are called Victoria, so there are probably hundreds of them in Cambridge alone. In this case consideration of context is not always going to be enough and sometimes one is going to add a qualifier, such as the family name.
Other things which might lodge quite happily in my head will not have names at all, my head can manage without. We are quite happy to know about the red brick house with a funny turret at the corner of Windmill Lane and East Street, without ever troubling ourselves about the house number, the house name or even the names of the occupants.
But none of that matters: they are all things which would make for a perfectly respectable entry in a dictionary or an encyclopaedia – or in one of our universes.
If we have a universe A, then we can also think of the power set of A, the set of all subsets of A and call it P(A). For present purposes we do not need to trouble about proper and improper subsets, where by improper we mean the empty set (Ø) and the entire set (A). A distinction which is important to mathematicians – but not to us.
We might, however, find to convenient to put restrictions on particular universes. We may, for example, require members of a universe to be both tangible and material and we may require them to be alive. We might have lower and/or upper limits of their size. So we might say that anything which can be entirely contained within a 2mm sphere is excluded – with a possible purpose of such an exclusion being to exclude things like grains of sand, spores, bacteria and viruses.
Numbers
A universe will probably contain a modest number of numbers. Perhaps zero and the first 100 or so whole numbers. Perhaps some approximations to a small number of real numbers of special interest.
Perhaps rather more real numbers known by their names or defined by their properties, rather than spelt out in digits – thinking here of numbers like π (3.141592653589793…), e (2.71828182845904523…) or the melting point of glass.
But a defining feature of numbers is that you can do sums with them. Most people can say what whole number follows another whole number. Some people can do some computation using one or more input numbers to produce an output number.
We do not consider that numbers generally are members of any universe, there are far too many of them. Their home, when we need them, is more in working memory, most of the contents of which do not make it to a permanent home in a universe.
Similar remarks can be made about those strings of alphabetic characters which are not words – that is to say the vast majority.
Parts of speech
People who do grammar and languages do parts of speech, they have categories like noun, verb and adjective and proceed to classify all the words in a given language to one or more of those categories – here allowing for all those words which can function in more than one way, for example ‘house’ which is usually a noun but may also be a transitive verb.
Parts of speech works well for lots of words. But there are always other words which do not seem to fit so well. What about, to give just one example, all the numbers of the previous section? So parts of speech is another of those properties, discussed above, which start straightforward enough and rapidly get complicated.
However, it is also the case that a large proportion of the names arising from things in our universes are going to be nouns. Houses, bricks, trees, people and horses. Even theorems. A rather smaller proportion are going to be values of properties, so ‘red’, a value of the property ‘colour’ is an adjective. But something which might, nevertheless, be a thing in our inner universe. While the name of the property itself, ‘colour’, is a noun. In the same way verbs like ‘run’ or ‘kill’ might be things in our inner universe. And even a grammar word like ‘the’ will probably feature in our inner universe, even if no more than as a frequently occurring word which one is supposed to know about.
One might argue that this sort of ‘the’ is really a noun, but that is a nice distinction. It is not a noun in the everyday sense of the word.
Functions and values
Let U be one of our universes.
Then a function F is a map from U to some set of values, V. For every member A of U there is a value of the function F, usually written F(A), a member of V. Other words for function include property and attribute. Sometimes it is helpful to think of the rows of an Excel worksheet as members of one of our universes and of the columns as functions, properties or attributes.
Given that U may well be heterogenous, sometimes it is convenient to allow a function to take a value which stands for not defined or not applicable. This device allows a function to be defined on the whole of U when, in reality, it is only defined on a part of U, with the function not being applicable to the rest of U. So, if, for example, our U was a set of apples and oranges, a function ‘thickness of rind’ would not be applicable to apples which have skin rather than rind.
Given the many imperfections of data collection exercises, sometimes it is convenient to allow a value which stands for not stated. That is to say the thing in question does have or should have a value for the function in question, other than not applicable, but we do not know what it is.
We distinguish sets of values which are ordered (totally ordered in the jargon of mathematicians), and which can be considered to be numbers, and sets of values which are not ordered, and which can be considered to be short strings of letters. There is a natural order on such strings, as used in a dictionary, but that order has no meaning here. While in the case of values which are numbers, we can say that the value of the function for this object is greater than, equal to or less than its value for that object and we suppose that this order has some meaning, some significance out in the real world. So price, for example, would have such significance, and if my oranges cost less than your apples, I might well be prepared to do a swap.
Sets of values which are not ordered will generally be quite small, say less than twenty or so.
Demographic functions like place of birth, sex and marital condition take values of this sort. It makes little sense to order them, at least to the modern, western mind other than for the purposes of presentation. One place of birth is not more than another place of birth, although one place of birth may well be further north than another place of birth.
The smallest possible set, consisting of just one and zero, true and false, is here considered to be unordered. Example: a function called ‘smoker’ might take the value true for smokers (defined in some way or another; no free lunch here either) and false otherwise.
Sets of values which are ordered include:
- The set of integers in some small range, for example between and including zero and seven, written [0..7]. Example: household annual income banded in eight bands
- The set of real numbers in some small range, for example between and including zero and one, written [0, 1]. Example: proportion of children of school age in part-time employment
- The set of positive real numbers. Example: household annual income in USD.
In other contexts, functions are little more than a presentational or lexical device for dividing our population into groups. So we give one group of objects the value A, another group the value B and the third group the value C. But all we know is that all the members of any one group have something in common and the function does not help us with what that something might be. That requires statistical analysis.
We leave aside for now the complication that functions themselves have names and might appear in universes. Universes really are universal, but we hope that this will not lead to the paradoxes and problems which vexed philosophers and others in the first half of the last century.
Functions and their factorisation
Suppose we have a function F from C1 to C2, where both C1 and C2 are complicated sets. Where the function F is complicated and difficult to work with.
But suppose we take the unit real interval [0, 1], called I. That we have a function F1 from C1 to I and a function F2 from I to C2. Both relatively simple because I is relatively simple. And suppose it so happens that our original function F is simply the composition of F1 and F2, so that F = F2 ○ F1, the successive application of F2 to the result of F1.
Then we have broken the complicated function F down, factored the function F into two relatively simple functions F1 and F2, much easier to grapple with than the complicated whole.
Let us suppose further that C2 is a power set of some sort, the set of all subsets of some third set. The inclusion relation between those subsets gives us a partial order on C2 and for some members A and B of C2 we will have it that A includes B.
We might then have it that the function F2 respects that partial order in the sense that if A > B in I, then F2(A) includes F2(B). That F2 captures an important aspect, an important regularity of the original function F. We might call such a function transitive.
And if we are lucky, there may be some other regularity about all this which makes it easy to define F1. If, for example, C1 was the set of all species of vertebrates, if might turn out that F1 is a simple consequence of the number of toes on their hind legs. So given F2, we can now compute F for some species of vertebrate when we know about toes, easy enough to sort out by inspection.
An example of a function
In the case of moral status, we might take C1 to be all the animals in the world which cannot be included in a sphere which is 1cm in diameter. We will deal with the complications lurking inside the innocuous looking word ‘all’ in due course.
While C2 is the set of actions by humans which are permitted on such animals. Keeping them as pets, shooting them for fun, rearing them for food, removing their eggs and so on.
But it makes things easier if we capture this problem in a concept called moral status, a concept which is itself captured by a map F1 from C1 to I, a function which might only take half a dozen or so values, perhaps defined by the number of toes on their front legs or the number of their cheek teeth. Or maybe the weight of their brain. We then get the philosophers to define a transitive F2 for us, and the job is done, neatly broken into the two halves, F1 and F2. We now know what we are allowed to do to any particular animal. No need to scratch our heads at all.
Membership
Sometimes a universe will contain things which are in a membership relation. So above we had both Thibaut Courtois and the Chelsea Football Team, a shorthand for the first men’s team. However, the team is greater than the parts. Chelsea Football Team is not identical to the set of the members of the men’s first team.
Membership relations are complicated by often being time dependent. The life cycles of Thibaut Courtois and the Chelsea Football Team are quite different and only overlap, intersect for a relatively short time, perhaps a few years, compared to their individual lives of the order of a hundred years or more.
These membership relations may be qualified in other ways. One might have honorary members, reserve members, occasional members and amateur members.
Conclusions
We have set out some machinery which we expect to use in posts to come. We have introduced the terms inner universe, outer universe, name, function and factorisation.
PS: cabbages and kings, a phrase from a well know ditty in ‘Through the Looking Glass’, is used as a title for its splendid heterogeneity. Note the carpenter’s paper hat, just the sort of thing that Eric Gill of reference 6 liked to affect.
References
Reference 1: http://psmv3.blogspot.com/2018/07/fishy-pain.html.
Reference 2: https://plato.stanford.edu/entries/grounds-moral-status/. The Stanford Encyclopedia of Philosophy (SEP), which as of March 2018, has nearly 1600 entries online.
Reference 3: https://psmv3.blogspot.com/2018/08/supplemental.html.
Reference 4: https://psmv3.blogspot.com/2018/08/counting-words.html.
Reference 5: https://en.wikipedia.org/wiki/The_Walrus_and_the_Carpenter.
Reference 6: https://psmv3.blogspot.com/2018/07/arts-crafts-2.html.
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