Thursday, September 30, 2010

Putting the green back in the greenery

The external world of objective reality clearly contains objects of various kinds, shapes, colours and so forth. E.g. there are conifers (an evergreen). Many students of philosophy (following Locke) learn to distinguish between primary and secondary properties of ordinary objects. The former exist out there, in the objects themselves, e.g. their shapes (and natural kinds). The latter exist only as we perceive such objects, e.g. their colours; and so we soon reach the philosophical problem of perception: We see a tree as green, out there in the world, but the green exists only in our heads (so to speak). The modern scientific picture of the world has it that where we see the green tree are really just various biochemicals, reflecting certain electromagnetic waves toward our eyes. The green seems not really to be where we can clearly see that it is—out there, in the leaves of the tree—but to be only in the pictures of the world that our brains construct.
......Indeed, since such pictures are what we’ve been calling ‘the world’, some might think of the world as in their heads (which is one way to put the green back in the greenery). Many philosophers (following Kant) take the shapes of ordinary objects to be, not primary properties (as Locke thought), but also secondary. The world might really be composed of atoms composed of 10-dimensional strings, for example, with our brains imposing, upon the numerous sensations that come from our sensory organs, the usual 2 and 3-dimensional shapes that we see in the world around us. Indeed, since our brains may even be imposing the basic structure of a number of objects upon our sensory input, some philosophers conclude that ordinary objects just don’t exist in reality (e.g. see Jackie’s comments on my previous post, Chairs Exist). But what do we mean by ‘reality’? Surely we could only mean whatever space it is that includes the people whose language includes such expressions. So there are certainly some objects out there, i.e. other people.
......And similarly, I think, there is a sense in which the greenness that we see really is objectively out there, on the surface of such objects as leaves. That is because we learn the meaning of ‘green’ (as part of learning the concept of colour) by being shown various green objects or pictures (and red ones, etc.) and being told that they are all green (red etc.). Green is therefore something that ordinary objects can be. Basically, 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’. So when it comes to something being green—to it being true to say of it that it is green—it is irrelevant what those sensations are, whether they are the same for one person as for another (although they are probably very similar, in view of our similar physiologies); the objective greenness that we see via those subjective sensations is, by definition, less subjective than they are.
......Of course, you usually take the meaning of ‘green’ to be just such sensations as you would call ‘green’, because that is how you learnt to use that word. Indeed, we all do, and so that is also part of the meaning of ‘green’. That equivocation usually goes unnoticed—except in such philosophical contexts as the problem of perception—precisely because it is irrelevant what such sensations are (how they differ between people). And of course, the problem of perception is hardly a mistake of the order of a misperception. How else could we possibly refer to things in an external world, except via our side of our interactions with it? Perception is never a view from nowhere. Even scientific observations are careful perceptions of the external world. And when it comes to predicating existence of something, our most certain knowledge comes from some of us seeing that it is.
......Incidentally, a surprisingly good analogy for the problem of perception is a blind person, e.g. using a white stick to check her picture of where she is. Suppose her stick hits an unexpected obstacle in her path. Just from how her stick reacts to hitting it—how the other end of it feels in her hand—she can tell that it’s a bouncy, light, smoothly rolling object... presumably a child’s ball. She can knock it out of the way and carry on; but in the land of the blind, her word for such bounciness in an external object may well be the same as her word for the way her stick felt in her hand. Nevertheless, she would hardly be tempted to think of the world as full of feelings. It would be full of objects that feel one way with a stick and another to the touch, and in other ways via gloves (or other skin, hair, etc.).

Saturday, September 18, 2010

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.

Wednesday, September 15, 2010

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

Wednesday, September 08, 2010

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.

Tuesday, September 07, 2010

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.

Monday, September 06, 2010

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}}.

Thursday, September 02, 2010

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.