Naturalists are analytical philosophers who take the natural sciences to be giving us the best foundation for our thoughts. As philosophers, they don’t usually trust scientists to interpret their own discoveries, however, so I wonder what Naturalists contribute to the scientific enterprise. Is it logical rigour and coherence, or just the opposite?

......Naturalists emphasise the very careful observations that scientists make. And indeed, it is important that we be sure of our facts. But when it comes to the most certain thing in the world—that we are conscious entities—they favour materialistic theories that reduce such things away. They may justify their preference on the grounds of ontological parsimony, or overall theoretical elegance, or the fact that natural scientists don’t usually include subjective stuff in their scientific observations. But Naturalists rightly regard as unscientific those who would explain away such less obvious facts as the fossil record on similar grounds (e.g. not needing all those aeons of inhumanity, and there being no mention of such things in a favoured account of reality).

......Naturalists criticise religion for giving too-easy answers. And it is indeed important to get the right answers to the questions the world raises, not just the most convenient ones. But Naturalists like theories that are based on standard logic, which is simple and popular. And standard logic takes a timeless view from nowhere, which is a bit unrealistic. E.g. consider how the Copenhagen interpretation of quantum mechanics, by having an observer-shaped hole in its description of reality, is less tidy but clearly more realistic than those no-collapse interpretations that would have us doing everything it was physically possible for us to do by splitting into lots of different people all the time.

## Saturday, October 30, 2010

## Friday, October 29, 2010

### Valid Enough

The essence of analytical philosophy is the presentation of a valid argument. But often the result is a lot of boring nonsense, many of us find. Why? Well, the reason may be that we are encouraged to work with an absurd definition of ‘valid’. A technically valid argument is, for example, since x and y, therefore x.

......But what about, since the sky is blue, and there’s little wind, I won’t need my umbrella? Technically, that’s invalid because it’s not impossible that it suddenly clouds over and rains. Some philosophers would therefore call it an induction. But I don’t see any generalisation over lots of observations there. And while such a generalisation may well be one of the argument’s many implicit premises, surely it is all the obvious implicit premises that make the argument sufficiently valid for human communication.

......Or, for a more philosophical example, consider Moore’s argument: Since that looks like a tree, therefore that is a tree. Now, we usually make such a deduction subconsciously, but nevertheless, surely such arguments are usually valid enough. And even in more rigorous contexts, how else are we to do science except by taking our readings to be as we read them? What would make such arguments invalid is something like bad lighting, not the mere possibility that we’ve just been taken into the Matrix.

......Indeed, even if we had been, our argument might be valid enough, because we would then be using words in a new external world, and ‘tree’ would usually refer to the new object. Our argument would only be invalidated if we were aware that we were in the Matrix, and if that aspect of our situation was the most apposite.

......But what about, since the sky is blue, and there’s little wind, I won’t need my umbrella? Technically, that’s invalid because it’s not impossible that it suddenly clouds over and rains. Some philosophers would therefore call it an induction. But I don’t see any generalisation over lots of observations there. And while such a generalisation may well be one of the argument’s many implicit premises, surely it is all the obvious implicit premises that make the argument sufficiently valid for human communication.

......Or, for a more philosophical example, consider Moore’s argument: Since that looks like a tree, therefore that is a tree. Now, we usually make such a deduction subconsciously, but nevertheless, surely such arguments are usually valid enough. And even in more rigorous contexts, how else are we to do science except by taking our readings to be as we read them? What would make such arguments invalid is something like bad lighting, not the mere possibility that we’ve just been taken into the Matrix.

......Indeed, even if we had been, our argument might be valid enough, because we would then be using words in a new external world, and ‘tree’ would usually refer to the new object. Our argument would only be invalidated if we were aware that we were in the Matrix, and if that aspect of our situation was the most apposite.

## Tuesday, October 19, 2010

### Definite Enough

Near enough is good enough in the real world; and in particular, it is good enough for reference, a concept that lies at the heart of philosophical logic. Reference occurs when someone refers someone else to something. Even prior to language, there is a primitive kind of reference that involves being seen to be looking at something, of some commonsensical kind. Now, analytical philosophers usually move quickly from reference to definite descriptions, before getting bogged down in the problem of vagueness. But I would like to suggest that vagueness is not so much a problem at the cutting edge of mathematical logic, as the best way to resolve most problems in philosophical logic. For a very simple example, consider the Lottery paradox: I believe, of each ticket, that it won’t win, so logically I ought to believe the conjunction, that none of the tickets will win, whereas I know that one will win.

......The paradox is resolved if I describe my belief that it very probably won’t win at least that precisely. And similarly, consider the Preface paradox: Each statement in this post is here because I believe it, but I also believe that I have probably made at least one mistake. Also similar is the fact that I believe that what I am now looking at, out my window, is a horse trotting past a tree, even though it might, just possibly, be a painted zebra about to be eaten by an alien stick insect. And what is common to all such paradoxes is that they are most satisfyingly resolved by our clarifying our terms. And quite generally, the power of natural language lies in its flexibility, which derives, I think, from the inherent vagueness of its terms. There are endless examples, so I challenge the reader to come up with a term that could not be replaced by more precise terms if necessary.

......Such subtle vagueness is ubiquitous precisely because our terms are almost always definite enough, so long as we speak carefully enough. And since logic is essentially the study of correct reasoning, it should not ignore the natural-linguistic clarification procedures—such as philosophical analysis itself—that aim at no more than an adequate bivalence. We have a strong bias towards bivalence because as we philosophize we clarify, aiming to maintain an adequate bivalence. But intuitions that logic ought to be bivalent are therefore quite compatible with logic not being perfectly bivalent. After all, questions of truth are essentially questions of how well our words describe the world, and so the logical primitive is not T (true), but True Enough—when we say “that’s true” we usually mean that it’s true enough—and it is implausible that statements are bound to be either true enough or else sufficiently false.

......The paradox is resolved if I describe my belief that it very probably won’t win at least that precisely. And similarly, consider the Preface paradox: Each statement in this post is here because I believe it, but I also believe that I have probably made at least one mistake. Also similar is the fact that I believe that what I am now looking at, out my window, is a horse trotting past a tree, even though it might, just possibly, be a painted zebra about to be eaten by an alien stick insect. And what is common to all such paradoxes is that they are most satisfyingly resolved by our clarifying our terms. And quite generally, the power of natural language lies in its flexibility, which derives, I think, from the inherent vagueness of its terms. There are endless examples, so I challenge the reader to come up with a term that could not be replaced by more precise terms if necessary.

......Such subtle vagueness is ubiquitous precisely because our terms are almost always definite enough, so long as we speak carefully enough. And since logic is essentially the study of correct reasoning, it should not ignore the natural-linguistic clarification procedures—such as philosophical analysis itself—that aim at no more than an adequate bivalence. We have a strong bias towards bivalence because as we philosophize we clarify, aiming to maintain an adequate bivalence. But intuitions that logic ought to be bivalent are therefore quite compatible with logic not being perfectly bivalent. After all, questions of truth are essentially questions of how well our words describe the world, and so the logical primitive is not T (true), but True Enough—when we say “that’s true” we usually mean that it’s true enough—and it is implausible that statements are bound to be either true enough or else sufficiently false.

## Thursday, October 07, 2010

### Do Inconsistent Objects Exist?

Mark Colyvan (2009: ‘Applying Inconsistent Mathematics,’ in Otávio Bueno & Øystein Linnebo,

......His example was the infinitesimals of the early calculus, which are widely believed to have been inconsistent. E.g. they were equated to both zero and non-zero quantities (while Newton’s fluxions varied but were inconsistently equated to constants). Nevertheless, the early calculus was widely applied throughout the eighteenth century. So if, as many philosophers assert, ‘we should be committed to the existence of all and only the entities that are indispensible to our best scientific theories’ (p. 162), then it seems that Colyvan’s suggestion makes sense.

......But can we know what our best theories are, without hindsight? Epicycles, for example, were an arbitrarily effective way of coping with real-world ellipses, given a mathematical language of circles. So the astronomy of Copernicus, with his circular orbits and no epicycles, was originally less accurate than Ptolemaic astronomy. But of course, the former was a better theory. It was a step in the right direction. If we can’t, then, know what our best theories are without the benefit of hindsight, then insofar as we now have better theories without such inconsistencies in them, perhaps we should now say that we should not then have believed in such objects. After all, we could always take inconsistencies to be good indications that we need a better theory.

......It is perhaps easier to see that asking if inconsistent objects exist is a bit like asking if impossible things can happen. And of course, if something happens then it must have been possible. Nevertheless, things that seem impossible can happen. And similarly, existing objects may well have descriptions that, while true enough individually (for our usual purposes), are collectively inconsistent (at least on the surface). An apparent inconsistency usually means that our descriptions stand in need of more precision. But it doesn’t mean that they’re too bad to use most of the time. Nor does it mean that the objects so described don’t exist.

......For a ubiquitous example, light as we sense it can be bright (or dull), but photons are dense (or sparse). Our word ‘light’ equivocates between our sensations of light and the light itself. Furthermore, light itself is lots of photons, but it’s also electromagnetic waves, and waves aren’t particles. But it isn’t that light doesn’t exist, of course, and eventually quantum physics described such behaviour consistently enough. It certainly makes sense for us to believe that photons exist (and to be even surer that light exists). And although light also behaves according to relativity physics, which may well be inconsistent with quantum physics, such inconsistency may just be a reason to pursue an even better theory (of light).

......For a more apposite example, the infinitesimals of the early calculus, more precisely described, may be the so-called

......And since the standard axiom of infinity—which says that the natural numbers are collectively a standard set—goes well beyond the Peano axioms, and is rather prone to paradox (e.g. Levy’s paradox), let us further suppose that the natural numbers are collectively indefinitely extensible. If that’s indeed the case, then

......So if

......So if such continua as do exist are well enough described by such (informal) theories as mine, then surely we could take irreal infinitesimals to be the referents of ‘infinitesimal’ in the early calculus. (Points were not then the same as real numbers, and the natural numbers were widely regarded as indefinitely extensible, so that infinite space would contain infinite lengths and hence geometrical infinitesimals.) Furthermore, insofar as the natural numbers can be said to exist, we could then think of such infinitesimals as existing.

......Of course, if mathematical existence is equated with consistency, relative to some axioms, then inconsistent mathematical objects exist when, and only when, we have inconsistent axioms; and of course, such inconsistent objects shouldn’t exist because they would be too trivial.

*New Waves in Philosophy of Mathematics*, Palgrave Macmillan, pp. 160–172), while looking at Inconsistent Mathematics, tentatively suggested that (p. 163): ‘There are times when we ought to believe in inconsistent objects.’......His example was the infinitesimals of the early calculus, which are widely believed to have been inconsistent. E.g. they were equated to both zero and non-zero quantities (while Newton’s fluxions varied but were inconsistently equated to constants). Nevertheless, the early calculus was widely applied throughout the eighteenth century. So if, as many philosophers assert, ‘we should be committed to the existence of all and only the entities that are indispensible to our best scientific theories’ (p. 162), then it seems that Colyvan’s suggestion makes sense.

......But can we know what our best theories are, without hindsight? Epicycles, for example, were an arbitrarily effective way of coping with real-world ellipses, given a mathematical language of circles. So the astronomy of Copernicus, with his circular orbits and no epicycles, was originally less accurate than Ptolemaic astronomy. But of course, the former was a better theory. It was a step in the right direction. If we can’t, then, know what our best theories are without the benefit of hindsight, then insofar as we now have better theories without such inconsistencies in them, perhaps we should now say that we should not then have believed in such objects. After all, we could always take inconsistencies to be good indications that we need a better theory.

......It is perhaps easier to see that asking if inconsistent objects exist is a bit like asking if impossible things can happen. And of course, if something happens then it must have been possible. Nevertheless, things that seem impossible can happen. And similarly, existing objects may well have descriptions that, while true enough individually (for our usual purposes), are collectively inconsistent (at least on the surface). An apparent inconsistency usually means that our descriptions stand in need of more precision. But it doesn’t mean that they’re too bad to use most of the time. Nor does it mean that the objects so described don’t exist.

......For a ubiquitous example, light as we sense it can be bright (or dull), but photons are dense (or sparse). Our word ‘light’ equivocates between our sensations of light and the light itself. Furthermore, light itself is lots of photons, but it’s also electromagnetic waves, and waves aren’t particles. But it isn’t that light doesn’t exist, of course, and eventually quantum physics described such behaviour consistently enough. It certainly makes sense for us to believe that photons exist (and to be even surer that light exists). And although light also behaves according to relativity physics, which may well be inconsistent with quantum physics, such inconsistency may just be a reason to pursue an even better theory (of light).

......For a more apposite example, the infinitesimals of the early calculus, more precisely described, may be the so-called

*irreal*infinitesimals that were informally introduced in my To Continue with Continuity (pp. 105–107). Suppose there are such continua as I describe in that paper (e.g. space, perhaps). And let*x*be a real number (e.g. pi), in the sense of an integer (e.g. 3) plus, after the decimal point, an endless sequence of digits in all the decimal places: the first (e.g. 1), the second (4), the third (1) and so forth (59265...). While that isn’t the standard definition of a real number, it’s a definite enough concept, and highly applicable.......And since the standard axiom of infinity—which says that the natural numbers are collectively a standard set—goes well beyond the Peano axioms, and is rather prone to paradox (e.g. Levy’s paradox), let us further suppose that the natural numbers are collectively indefinitely extensible. If that’s indeed the case, then

*x*is an (infinitesimally) imprecise description of many (infinitesimally) different lengths.......So if

*l*is a non-zero irreal infinitesimal then*x*+*l*=*x*because, quite generally, infinitesimals are smaller than 1/*n*for any natural number*n*, so that adding them to*x*affects*x*in*none*of its decimal places. (And incidentally, the word ‘infinitesimal’ derives from a Latin word that originally meant the infiniteth term in a sequence.) There are, then, consistent (if informal) mathematical objects—irreal infinitesimals—that for the purposes of the early calculus could be adequately described by its descriptions of its infinitesimals.......So if such continua as do exist are well enough described by such (informal) theories as mine, then surely we could take irreal infinitesimals to be the referents of ‘infinitesimal’ in the early calculus. (Points were not then the same as real numbers, and the natural numbers were widely regarded as indefinitely extensible, so that infinite space would contain infinite lengths and hence geometrical infinitesimals.) Furthermore, insofar as the natural numbers can be said to exist, we could then think of such infinitesimals as existing.

......Of course, if mathematical existence is equated with consistency, relative to some axioms, then inconsistent mathematical objects exist when, and only when, we have inconsistent axioms; and of course, such inconsistent objects shouldn’t exist because they would be too trivial.

## Tuesday, October 05, 2010

### Monarchy And Democracy

Yesterday's Dispatches on Channel 4 was about "the illegal phone hacking carried out at the New of World during the six years Andy Coulson was either Deputy Editor and Editor," of current interest because "by hiring Andy Coulson David Cameron has sanctioned the News of the World culture of impunity." Rupert Murdoch certainly seems to have a worrying amount of influence over our police and politics, centre-left as well as centre-right. The origin of these revelations, in a successful prosecution for hacking a conversation between our two young princes, is of course less interesting. But the consequent importance of the fact that Murdoch could neither buy off nor intimidate the Palace does make me wonder whether the left, and other democrats, should question their traditional Republicanism, in these transitional days.

## Monday, October 04, 2010

### Ordinary Objects

Further to the question of whether or not Chairs Exist, I notice that Amie L. Thomasson’s 2007 book Ordinary Objects is out in paperback next month. Basically, it shows how “the claim that there are ordinary objects can form part of a coherent, reflective metaphysical view built up out of our common sense way of looking at the world, in a way that avoids the philosophical problems that have long been feared to plague a common sense ontology” (pp. 7–8).

......The first problem addressed (pp. 9–24) was that if all such things are made of atoms then there’s

......The first problem addressed (pp. 9–24) was that if all such things are made of atoms then there’s

*causal redundancy*, e.g. if it’s really atoms arranged stone-wise that break a window (atoms arranged window-wise) then we shouldn’t also have a stone breaking it. But of course, the former just is the stone breaking it, if all such things are made of atoms. Thomasson’s analysis was more detailed, of course, but still readable, and seems to generalize nicely (e.g. the next section addressed the related issue of epiphenomenalism in the theory of mind).## Saturday, October 02, 2010

### Euler’s ‘2’

Incidentally, I’ve rewritten two of last month’s posts, and the rewrite—Did Euler’s ‘2’ refer to ZFC’s {Ø, {Ø}}?—will be appearing in next month’s issue of The Reasoner.

## Friday, October 01, 2010

### True Enough

What is truth? Clearly it includes (following Aristotle) saying, of what is, that it is, and saying of what isn’t that it isn’t. The obvious contrast is with falsity, with saying of what is that it isn’t, or of what isn’t that it is. Truth, then, is the fit of our words with the world.

......But words exist in a public language and so, given how we come to acquire our linguistic skills, vague meanings are inevitably ubiquitous. Still, we invariably speak within some context, wherein we need only say enough to make our meaning clear enough. Our words can describe the world well enough—they usually do—and then what we say is true enough. And when it isn’t, we can always be more precise.

......We can even introduce new terms into our language, if we have to (as scientists and philosophers). Indeed, there seems to be no logical limit to our ability to be ever more precise. And so to say that something is true is, more precisely, to say that it’s true enough. Bivalent propositional logics—in which each sentence is either true or else false—are just rough approximations to the truth.

......Consider some commonplace examples: The table at which I’m typing this is flat—it isn’t warped or lopsided—but in another sense it isn’t flat, not being perfectly smooth and horizontal. To say that it’s flat is to say something that’s

......And similarly, to return to the themes of previous posts, ‘grass is green’ is true because ordinary grass (such as fills lawns and pastures) reflects the green bits of daylight (fuzzily delineated bits) ordinarily (e.g. when there’s no drought).

......And it’s

......And do rainbows exist? Well, in a sense they do (e.g. we can refer each other to them), but there is clearly a sense in which they don’t (much as mirages are not oases).

......What do we mean by ‘grass’ or ‘chair’? Such things form obvious kinds, which is how we come to learn such words. What most of our words have, then, are meanings that are definite enough. Indeed, such vagueness may well be logically necessary, in any possible medium of communication. But even if a more definite language was possible, it’s the vagueness we have which means that our words can be given more definite meanings as required. So a less vague language would in any case be a less useful tool.

......But words exist in a public language and so, given how we come to acquire our linguistic skills, vague meanings are inevitably ubiquitous. Still, we invariably speak within some context, wherein we need only say enough to make our meaning clear enough. Our words can describe the world well enough—they usually do—and then what we say is true enough. And when it isn’t, we can always be more precise.

......We can even introduce new terms into our language, if we have to (as scientists and philosophers). Indeed, there seems to be no logical limit to our ability to be ever more precise. And so to say that something is true is, more precisely, to say that it’s true enough. Bivalent propositional logics—in which each sentence is either true or else false—are just rough approximations to the truth.

......Consider some commonplace examples: The table at which I’m typing this is flat—it isn’t warped or lopsided—but in another sense it isn’t flat, not being perfectly smooth and horizontal. To say that it’s flat is to say something that’s

*true enough*.......And similarly, to return to the themes of previous posts, ‘grass is green’ is true because ordinary grass (such as fills lawns and pastures) reflects the green bits of daylight (fuzzily delineated bits) ordinarily (e.g. when there’s no drought).

......And it’s

*insofar*as chairs exist that it’s true to say of them that they do.......And do rainbows exist? Well, in a sense they do (e.g. we can refer each other to them), but there is clearly a sense in which they don’t (much as mirages are not oases).

......What do we mean by ‘grass’ or ‘chair’? Such things form obvious kinds, which is how we come to learn such words. What most of our words have, then, are meanings that are definite enough. Indeed, such vagueness may well be logically necessary, in any possible medium of communication. But even if a more definite language was possible, it’s the vagueness we have which means that our words can be given more definite meanings as required. So a less vague language would in any case be a less useful tool.

Subscribe to:
Posts (Atom)