Putting the green back in the greenery

......There is certainly some greenery in the world. And the green of it seems to be out there in the external world, with that greenery. And yet, what is out there are surfaces sending electromagnetic waves toward our eyes. That is a scientific fact. The colours are in the pictures of the world that our brains construct from data obtained via our sensory organs. And maybe the shapes are too (maybe the world is made of 10-dimensional strings, with our brains imposing upon the numerous sensations that come from our 10-dimensional sensory organs the 2- and 3-dimensional shapes that we see in the world around us). Perhaps we should not be surprised that some philosophers think that ordinary objects don’t exist in reality. Does the world (what we usually mean by ‘the world’) only exist in our heads? There are certainly some ordinary objects in reality: what we mean by ‘reality’ is whatever space it is that includes the people whose language includes such expressions. You and I are two people, whatever we are made of.
......And there is a sense in which the greenness really is out there, on the surfaces of such objects as leaves: we learn the meaning of ‘green’ by being shown various green objects (or pictures of green objects) and being told that they are all green. It hardly matters whether you and I have the same sensations when we see them. Green is something that ordinary objects can be. Still, green is also a sensation. And a very mysterious one: do you have the same sensation as I do, when we both look at the same leaf? We have similar eyes and brains, but we know nothing about how sensations arise in our brains. Still, something is green if its surface is such that under normal lighting conditions, it would give rise to the same sort of sensations in those looking at it as they had when they learnt the meaning of ‘green’ whatever those sensations are. If your sensation, when you see green, was exactly the same as mine, when I see blue, I would be wrong if I thought that you were seeing blue.
......Or would I? It seems to me that the meaning of ‘blue’ is the sensation that I have when I see blue things. Could I be wrong and right? Well, there is something like an equivocation here, for all that it is probably unavoidable: our references to things in our external world are only possible via our sides of our interactions with those things. Would it be any different in the land of the blind? Maybe they use sticks there, to feel their way around, so suppose that one of their sticks hits an unexpected obstacle. The person holding that stick might be able to tell that the other end of it had hit a bouncy object. And their word for such bounciness in an object might be the same as their word for the way that that stick had felt in that hand. But they would of course not think of the world as being full of such feelings. They would think of it as being full of objects, mysterious objects, some of them bouncy (in some non-visual sense of ‘bouncy’).

When a tree falls in a forest at night, and there is no one around to hear it, does it make a sound?

I suppose that it does not. And those trees do not look green: it is dark, and there is no one there to see them anyway. But they are green in the daytime. They are green trees. Are they green in the dark, with no one there? They are green trees. Suppose that you have a colour photograph of those trees on your wall. You do not think that those trees are there, in your room, but the green of those trees is there. Is it still there when you turn the light off and leave the room? Is it still a green photograph? Suppose that you return with a strange little light bulb: you change the bulb, turn the light on, and in that light the photograph looks blue. Is it a green photograph that looks blue? Is it still green, even though it looks blue in the strange light? And did other people learn words like ‘green’ in such a way that they would give similar answers to such questions?

What is clear is that a green alarm clock that goes off in a vacuum makes no sound.

And if the clock is painted black, then it is no longer green. And such rooms exist in houses that are quite distinct from each other. Your house and my house are two houses, in a very precise way. When we reason logically about the world, our thoughts are as complicated as our relationships with the world. But the simplest thing to reason logically about ought to be arithmetic.

Chairs Exist

The basic contrast is with imaginary objects: Pixies don’t exist, electrons do; epicycles don’t exist, bicycles do. We learn the meaning of ‘exist’ in a world of tables and chairs, trees and cars, and so when we say that electrons exist we mean that they exist like chairs do. We can spray them onto surfaces, for example, much as we might throw chairs into a van. We can catch chairs and electrons, but not pixies.
......If we doubted that chairs exist, what could we mean by ‘exist’ if we said that electrons exist? That they are in our best theory of reality? But the thing about epicycles is not only that they aren’t fundamental objects, in our best theory. It is that they don’t exist, to be further analysed, and therefore shouldn’t have been in our best theory. Of course, pixies exist within fictions, so they exist fictionally, but that is also to say that they don’t really exist.
......Some philosophers think that chairs are imaginary, that only the atoms that make them up exist, but how could that be right? A chair made of Lego bricks would still be a chair. It would still exist, wholly composed of Lego bricks. Had it been made one brick at a time, with one brick not being a chair, and with no addition of one brick making a chair out of a non-chair, it would exist. Consider how, even though orange fades smoothly into yellow and red, with no sharp boundary, that doesn’t mean that carrots are not orange.

Do Chairs Exist?

Yes they do:

A physicist will tell me that this armchair is made of vibrations and that
it’s not really here at all. But when Samuel Johnson was asked to prove
the material existence of reality, he just went up to a big stone and kicked
it. I’m with him.

......David Attenborough

No they don't:

Like many philosophers, I don't believe that tables and chairs are fundamental
objects. Like a much smaller number of philosophers, I like to say that I don't
think tables and chairs exist. I have good reasons for my denial. For instance,
it does not appear that there is an exact moment at which a table comes into
existence.

......Alexander Pruss

Euler’s ‘2’ postscript

Since our reductionists (see previous post) assume that we can refer to abstract objects, let us see if any of the ways in which such reference might occur support their view of the referent of Euler’s ‘2’. It seems not; for suppose, for example, that reference to abstract objects is correctly described by Fictionalism. Then the view in question is like taking most twentieth century utterances of ‘Sherlock Holmes’ to refer to the character recently played by Benedict Cumberbatch on the BBC. Suppose instead that Gödelian platonism is true, so that we have something like a perception of abstract objects. Then the only choice we should make in our reference to them is the choice of their names. Between those two possibilities lies a Full-Blooded platonism, according to which all possible abstract objects exist. But that position is hardly available to those who don’t want—but don’t consider impossible—non-set-theoretic numbers. So in conclusion, it seems that our reductionists are quite eccentric after all.

Euler’s ‘2’ continued

For a more eccentric analogy (see previous post for previous analogy), let a family of cooks in some dull country be introduced to the meaning of ‘orange’ by means of some imported carrots. Since our cooks want to refer only to ordinary objects, not to such things as properties, which seem to them hardly things at all, they reduce talk of orange things to talk of their carrots. But they would clearly be wrong to take us to be referring to their carrots with our uses of ‘orange’. Indeed, we are not even referring to the colour of their carrots, which might turn yellow.
......In view of the way things are—e.g. the cells of the human retina—a more realistic reduction would reduce orange to the two primary colours red and yellow. And to do something similar for the referent of ‘2’ would take us, not to standard set theory but to psychology. The letters ‘M’ and ‘N’ are angular and black and are, collectively, 2 letters, and it is by means of such examples that we came to know what ‘2’ means. Much as shapes and colours are predicated of ordinary objects, the natural numbers are predicated of finite sets.
......Could such properties be collections? Well, there is a philosophical tradition of reducing properties (e.g. orange) to extensions of properties (the class of all orange things), but there is a well known problem with reducing 2 to the class of all pairs. Set-theoretic paradoxes show that such classes are, if absolutely general (not just the class of pairs of ordinary objects), indefinitely extensible. There is no pre-existing class of all pairs to reduce 2 to. Our reductionists therefore reduce 2 to a particular pair-set. Not being eccentric, they don’t reduce it to something concrete, like a pair of carrots (although that has obvious reductionist benefits), but to something as abstract as numbers are thought to be.
......However, it is not so much a discovery as a technicality to use {Ø, {Ø}} rather than {{Ø}}, or at least, such is the choice not to use some other set, class or category theory, or indeed, constructive mathematics. The set-theoretic axiom of infinity, in particular, is true by definition of all standard sets, but is not so much a discovery as a guess about the natural numbers. Now, that assertion only makes sense insofar as numbers are not sets, but that just means that our reductionists risk losing the ability to assert that the axiom of infinity is only a guess; about what would it be a guess? Such reductionists therefore put themselves in the position of those nineteenth century scientists who, for good reasons, took ‘space’ to mean Euclidean space. Those reasons were just not good enough; and note that various supertasks and other paradoxes currently give us cause to question the truth of the axiom of infinity, construed as a property of the non-set-theoretic natural numbers.

Did Euler’s ‘2’ refer to {Ø, {Ø}}?

The foundation of mainstream mathematics is standard set theory, within which ‘2’ usually refers to {Ø, {Ø}} (when we are considering the natural numbers, see comments below).
......Alexander Paseau (2009: ‘Reducing Arithmetic to Set Theory,’ in Otávio Bueno & Øystein Linnebo, New Waves in Philosophy of Mathematics, Palgrave Macmillan, pp. 35–55) thinks that those reducing arithmetic to set theory in such a way—perhaps they want their ontology to include only sets, not also non-set-theoretic numbers—may also take most mathematicians, past and present, to have been referring to the standard set {Ø, {Ø}} with their ‘2’s. He (2009: p. 42) made the following analogy: ‘When the ancient Greeks spoke about the sun, they spoke, unknowingly, about a hydrogen-helium star that generates its energy by nuclear fusion.’
......By contrast, anyone taking ‘say, a carrot to be the referent of “2” in Euler’s mouth’ should, he (2009: p. 43) thinks, ‘be an error theorist about Euler’s claims involving “2”,’ and so risk taking too many—according to Hartry Field (2001: Truth and the Absence of Fact, Clarendon Press, p. 214)—of Euler’s words to be untrue. According to Paseau (2009: p. 43), our ‘less eccentric reductionists need not interpret Euler’s arithmetical claims error-theoretically and may respect his intended truth-values.’
......They could, he thinks, take that view even if ‘2’ referring to the standard set {Ø, {Ø}} was not so much a discovery about 2 as a technical convention. As Paul Benacerraf (1965: ‘What Numbers Could Not Be,’ Philosophical Review, 74, pp. 47–73) famously observed, another possible referent is {{Ø}}. So for another analogy, suppose some chromatographers took ‘orange’ to refer to wavelengths of light within some definite range, in order to avoid vagueness and because such a stipulation was sufficient for their scientific needs. They would surely need further reasons to take us to be referring to such wavelengths with our uses of ‘orange’. And similarly, if our less eccentric reductionists have not so much discovered the referent of ‘2’ as accepted a useful technicality, then it seems to me that they should not be taking our ‘2’s—nor Euler’s—to be referring to the standard set {Ø, {Ø}}.

What Rainbows Cannot Be

Rainbows are clearly not ordinary objects, being more like mirages than oases. But they are just as clearly not fictional objects. One’s conception of a rainbow may well contain false presuppositions, but in that way rainbows do resemble ordinary objects. Consider a red apple. Perhaps what is really there is a 10-dimensional collection of particle-strings within a 4-dimensional block universe, with the red and the spheroid existing only in the minds of certain kinds of potential perceivers of that collection. (That modern scientific hypothesis is not a million miles away from theistic idealism, of course.) Anyway, suppose some fictional meteorologists defined ‘rainbow’ to be a specific sort of event. They might do so because such a technical convention suited their scientific needs better than the rather vague ordinary meaning. And of course, an event is a kind of object (especially in a 4-Dimensionalist world). Now, some relatively arbitrary choices may well have been made as they specified their referent of ‘rainbow’. But that does not mean that rainbows cannot be objects. They are, after all, intentional objects, over which we might quantify. And of course, our intuitions that rainbows are not objects all derive from the fact that they are not ordinary objects. I mention this because it strikes me as similar to Benacerraf’s famous argument about what numbers cannot be.