Find the Fallacy

Two mathematicians begin a discussion by assuming that A=B. Then they begin making changes. They add an equal term to both sides of the equation. A+A=A+B or 2A=A+B. Then they subtract an equal number from both sides. 2A -2B = A+B -2B. They look at this and agree that this all makes sense. Then they agree that this equation can be simplified by a simple operation. 2(A-B) = A+B-2B. This again simplifies to 2(A-B) = A-B. The result astounds them. Two times (A-B) equals (A-B)? Two equals one????

The answer is that the fallacy occurs only at the final step.

We get to 2(A-B) = A-B. But at this point we should recall that if you subtract something from itself you always get zero. So A-B = 0 and 2(A-B) also equals 0.

It is surely the case that 0 = 0, but thus stated the air of paradox has disappeared.

You obviously can’t get from there to the conclusion 2=1, only to the conclusion that

2(0) = 1(0).

The final step would be division by zero, which the basic rules prohibit. Precisely to avoid such paradoxes as this!

My point? Just that zero, and the rules governing its use are more exotic human conceptual inventions than one might think. The casualness with which we usually treat zero comes from familiarity, not from simplicity. This confirms the point I sought to make yesterday, that our most successful inventions are also discoveries, and vice versa.

This turns out to be, not especially Kantian, but certainly Jamesian, or well within the area of their overlap.

Interacting with the World, and Ciceronianus

This weekend, Ciceronianus, author of a wonderful neo-Stoic blog, posted on epistemology. He wrote: "I'm bemused from time to time by the view that the world (as in "reality" or the universe) is, in part at least, our creation, or perhaps is created by each of us for himself/herself."

He named first Kant and later Wilfred Sellars as examples of the sort of epistemologist he has in mind. Let us use the broad term "constructivism" for the broad PoV that Kant and Sellars share, and that bemuses our Stoic.

Ciceronianus, if I understand him, then proceeds to the assertion that such constructivism is either pointlessly obvious or wildly wrong. The obvious and uninteresting point is that "we are human beings, and as such interact with the world as human beings do." Yet those who are most serious about urging that "they shape the world" seem to want to go much further than this, and that furtherness is what bothers Ciceronianus.

I contributed a thought of my own to his comment section.
 

There is a tee shirt that bears upon some of the issues you raise. It shows somewhat anthropomorphized versions of the Greek letter pi (Π) and of the expression √-1.

Pi is saying to √-1, “Get real.” And√-1is replying, “Be rational!”

I’ll pause now while you slap your knees.

The joke, of course, is that pi is an example of an “irrational” but real number, while √-1 is the definition of i, the foundation of the imaginary numbers.

Philosophy: Some Terminology Changes

By the late 19th century, philosophers had formulated three theories about the nature of truth.

Before I should say any more, I should add: they were all using the word "truth" in a way that contemporary philosophers would not. They all meant, not "truth" specifically, as a property of belief -- but KNOWLEDGE of truth, as the reason why we know truths, when we do.

The three classical theories were these: correspondence, coherence, pragmatism.

According to the correspondence theory, a belief is true when it accurately copies a fact in the world. As Wittgenstein said, in his usual straightforward style, "In order to tell whether a picture is
true or false we must compare it with reality."

The correspondence theory carried with it some pretty heavy metaphysical baggage. Suppose I'm looking at a painted portrait of John Smith. If I want to decide whether that's a "true" portrait of
John Smith, I can go up to John himself, get a good like at him, then decide whether the likeness is an accurate one.

But even then, we should say that the portrait is either realistic or not. A surrealistic portrait of John Smith, painted say by Salvador Dali, might be as "true" in its own way as anything that can come out
of a camera in the noontime sun, but it would be true in a non-copyist way.

Furthermore, that is a very bad metaphor for my efforts to know the world. After all, both the portrait and the man who sat for it are facts outside of my skull -- I have the same kind of access to the
one as to the other. An analogy to that sort of comparison gets us nowhere if what is in question is the nature of truth/knowledge about people and paintings both.

The coherence and pragmatic theories of knowledge were both efforts to get rid of the metaphysical baggage of correspondence, or the copy theory. They both said that there is something about some of my experiences, my ideas, my beliefs, etc. that makes them true -- and that something isn't a transcendence of the whole body of my experience by the way that it relates to itself, the way the parts of it relate to one another.

In the coherence view, error is inconsistency. If I could only get all my beliefs into a single coherent whole, I would know them as true -- or, strictly, I would know them as the whole single shining
body of Truth itself.

In the pragmatic view, error is failure. Beliefs guide me through experiences well or poorly, they are valuable when they lead me where I want to go, harmful when they get me lost. Truth, then, is success.

The tendency in Anglo-American philosophy since about the 1930s has been to distinguish sharply between the issue of truth and the issue of knowledge, then to render the notion of truth impoverished. To say "it is true that this computer is working" is just to say, with a little extra emphasis perhaps, that "this computer is working" -- the word "true" is the emphasis, but in terms of strict meaning its
redundant.

I've said all this before, but what I'd like to emphasize today is that it isn't only pragmatism that needs restatement in order to become comprehensible within contemporary philosophical jargon. It is each
of the original three positions. They all turn out, in 21st century terms, to have been more about knowledge -- and even more specifically one's WARRANT for a true belief -- than about truth as
such.

The correspondence theory looks backward for warrant, for evidence that the copy derives casually from the thing copied.

The coherence theory looks inward for warrant, and worries whether all parts of a world view cohere.

The pragmatic view would LIKE to look forward -- but since the success or failure of this moment's forward looks is necessarily still unsettled, it adapts by looking backward at earlier instances
of successful forward look, and imitating whatever made them succeed.

Schopenhauer on time

"Past and future (apart from the consequences of their content) are as empty and unreal as any dream; but present is only the boundary between the two, having neither extension nor duration. In just the same way, we shall also recognize the same emptiness in all the other forms of the principle of sufficient reason and shall see that, like time, space also, and like this, everything that exists simultaneously in space and time ... is only through and for another like itself, i.e. only just as enduring."

I understand the reasoning here in broad terms. Indeed, William James' notion of the 'specious present' was developed largely as a riposte to such reasoning.

Kant and the Austrian Economists I

As many of my readers presumably know, there are important differences among different schools of "free market economists."  If you are a Marxist, or for that matter a Keynesian, you'll likely think all free marketers sound pretty much the same.  This, though, is a trick of perspective.

One oft-noted contrast is between the "Chicago School" (associated with Milton Friedman and monetarism) on the one hand, and the "Austrian School" (of Friedrich Hayek and his close ideological kin) on the other.

Here's a full account of the difference, from an "Austrian." I put the label "Austrian" in quotation marks there because, although the school got its start there, by no means do all "Austrians" of today have names that sound, well ... Austrian.  It isn't all Hayek and von Mises anymore!  The author of the article to which I linked you above has the distinctly not-middle-Europe moniker, "Robert Murphy."

Anyway, many of the points of divergence are methodological -- even you might say epistemological.  Murphy quotes Friedman on the question of how economics should work as a science:

The relevant question to ask about the "assumptions" of a theory is not whether they are descriptively "realistic," for they never are, but whether they are sufficiently good approximations for the purpose in hand. And this question can be answered only by seeing whether the theory works, which means whether it yields sufficiently accurate predictions.

Testable predictions.  Newton tells us to assume objects are moving around in a frictionless environment.  They (almost) never are, but assuming that makes the math easy, and the easy math makes for testable predictions. One we test those predictions, we can note that they are slightly off (only "sufficiently" accurate as Friedman says) and then worry about correcting for friction!  So Friedman seeks to create an economic science that is just as genuinely a science as is physics.

Murphy, as an "Austrian," has a different view.  After all, a basic premise of economic theory, he says, is that "people respond to incentives."  Is that itself a testable prediction?  Murphy thinks that it is not, and he is unbothered by its untestability.

Think of falsifiability, the specific notion of "testability" developed by Karl Popper, an Austrian in the non-economists' sense.  How could anyone falsify the claim that people respond to incentives?  I ask you to cut off your big toe.  You ignore me.  I offer you $20 to cutr off your big toe.  You continue to ignore me.  I offer $1,000.  You continue to ignore me.  At what point has my little experiment falsified the view that people respond to incentives?  At none.  It is immune from falsification.

I don't know off hand what Friedman would say about the big toe experiment, but Murphy is unbothered by its lack of result.  This premise isn't supposed to be falsifiable, he says.  It is part of an internally coherent framework for interpreting data, not an empirical datum itself.

This is where a couple of "aha" bulbs went off in my brain.  It seems that Murphy is saying (though without being explicit, for he is writing for a non-academic audience) that the Austrian school has a Kantian philosophical slant.  The idea that people respond to incentives might be what Kant called a synthetic a priori principle.  It tells us about any possible world, any experiencable world, and thus is a premise of science rather than a result.  A bit like the idea that every event has a cause.  That was the first aha bulb.

The second bulb had to do with the American novelist pop-philosopher Ayn Rand, by whom I have to be frank never been enamoured.

But I've gone on a bit long here and I think I'll finish this thought tomorrow.
 

James' thought for today

"Ordinary epistemology contents itself with the vague statement that the ideas must 'correspond' or 'agree' [with reality]; the pragmatist insists on being more concrete, and asks what such 'agreement' may mean in detail. He finds first that the ideas must point to or lead towards that reality and no other, and then that the pointings and leadings must yield satisfaction as their result. So far the pragmatist is hardly less abstract than the ordinary slouchy epistemologist; but as he defines himself farther, he grows more concrete."

"The Pragmatist Account of Truth and its Misunderstanders," PHILOSOPHICAL REVIEW (Jan. 1908).

Truth and knowledge

Recalling the title of this blog, let us do some old-school philosophizing.

By the late 19th century, philosophers had formulated three theories about the nature of truth.

Before I should say any more, I should add: they were all using the word "truth" in a way that contemporary philosophers would not. They all meant, not "truth" specifically, as a property of belief -- but KNOWLEDGE of truth, as the reason why we know truths, when we do.

The three classical theories were these: correspondence, coherence, pragmatism.

According to the correspondence theory, a belief is true when it accurately copies a fact in the world. As Wittgenstein said, in his usual straightforward style, "In order to tell whether a picture is true or false we must compare it with reality."

The correspondence theory carried with it some pretty heavy metaphysical baggage. Suppose I'm looking at a painted portrait of John Smith. If I want to decide whether that's a "true" portrait of John Smith, I can go up to John himself, get a good like at him, then decide whether the likeness is an accurate one.

But even then, we should say that the portrait is either realistic or not. A surrealistic portrait of John Smith, painted say by Salvador Dali, might be as "true" in its own way as anything that can come out of a camera in the noontime sun, but it would be true in a non-copyist way.

Furthermore, that is a very bad metaphor for my efforts to know the world. After all, both the portrait and the man who sat for it are facts outside of my skull -- I have the same kind of access to the one as to the other. An analogy to that sort of comparison gets us nowhere if what is in question is the nature of truth/knowledge about people and paintings both.

The coherence and pragmatic theories of knowledge were both efforts to get rid of the metaphysical baggage of correspondence, or the copy theory. They both said that there is something about some of my experiences, my ideas, my beliefs, etc. that makes them true -- and that something isn't a transcendence of the whole body of my
experience by the way that it relates to itself, the way the parts of it relate to one another.

In the coherence view, error is inconsistency. If I could only get all my beliefs into a single coherent whole, I would know them as true -- or, strictly, I would know them as the whole single shining body of Truth itself.

In the pragmatic view, error is failure. Beliefs guide me through experiences well or poorly, they are valuable when they lead me where I want to go, harmful when they get me lost. Truth, then, is success.

The tendency in Anglo-American philosophy since about the 1930s has been to distinguish sharply between the issue of truth and the issue of knowledge, then to render the notion of truth impoverished. To say "it is true that this computer is working" is just to say, with a little extra emphasis perhaps, that "this computer is working" -- the word "true" provides emphasis, but in terms of strict meaning it's
redundant.

It isn't only pragmatism that needs restatement in order to become comprehensible within contemporary philosophical jargon. It is each of the original three positions. They all turn out, in 21st century terms, to have been more about knowledge -- and even more specifically one's WARRANT for a true belief -- than about truth as such.

The correspondence theory looks backward for warrant, for evidence that the copy derives casually from the thing copied.

The coherence theory looks inward for warrant, and worries whether all parts of a world view cohere.

The pragmatic view would LIKE to look forward -- but since the success or failure of this moment's forward looks is necessarily still unsettled, it adapts by looking backward at earlier instances of successful forward looks, and imitating whatever made them succeed.

Analytical Philosophy

And they say the analytic tradition in philosophy is moribund!

In a recent book on the contending definitions of truth in philosophy today, a book with a distinctly Russellian/Wittgensteinian focus, I've found a passage that bears on that accusation. To me, its a very funny piece of writing too, though I'm sure it wasn't intended as such.

The author, Matthew McGrath, early on mentions sentences that contain the word "sake," as in "Lois Lane risked her life for Clark Kent's sake."

He then discusses the problems that we might run into if we regarded a "sake" as a thing, something Clark possesses. (Like a pet?) -- so that we might try to understand the sentence as analogous to, "Lois Lane risked her life for Clark Kent's collie." He calls this "sake realism."

"The opacity of sake-talk makes sake realism objectionable. But it is objectionable for reasons of a more general relevance. By accepting sakes, we would confront a panoply of seemingly unaswerable questions: Do all things have sakes? My socks? the number 7? Do sakes themselves have sakes? Why is it that the things that have sakes have only one sake?"

You might think that I'm being unfair, taking this out of context. Surely he was attempting a reductio? Some opponent's view strikes him as analogous to "sake realism" and, accordingly, absurd?

Not quite. He's playing defense in this passage. As near I can tell, he's trying to say that although his view of truth bears some relation to "sake realism," it is sufficiently distinquishable to survive such a reductio.

But never mind about that. The next time you want to give your mind the cleansing benefits of an Zen koan, just recall the question: "Do sakes themselves have sakes?"

But don't do it for my sake.

Mr. McGrath needs to get out more.

Epistemology

There continues to be life in the old debates about truth.

George Englebretsen, a logician who teaches at Bishop's University, Quebec, has just come out with a book entitled Bare Facts and Naked Truths: A New Correspondence Theory of Truth.

While researching that book, I discovered a similar one that came out a few years back (as long ago as 2000, but much more recent than most of the books and articles on the subject I've ever seen given my old-school habits of mind!) Between Deflationism and Correspondence Theory (2000).

Matthew McGrath at the University of Missouri.

What both books have in common is that they see the chief rivalry in the definition of truth as the one defined in Dr. McGrath's title. Either truth is merely a word of approval, (when you say "the sun is yellow" and I say "that's true," I could as well have nodded my head, or given you a thumb's up signal) OR it is an ontologically significant state of agreement between a proposition and a fact-in-the-world.

Englebretsen takes the side of correspondence. McGrath stays in the middle ground, which he also calls "weak deflationism." His theory is summarized on-line, in this article:

http://web.missouri.edu/~mcgrathma/pubs-papers/weakdeflationism.pdf

Something else both books have in common: they both accept and in fact turn on the idea of a "proposition." A proposition is the meaning of a sentence. Presumably, a proposition is the commonality we have in mind when we say that two sentences "mean the same thing."

The idea of correspondence inherent in both books is that a true PROPOSITION is one that corresponds to a fact.

But in practice (always my touchstone) the truth bearer is a sentence in some natural language, and if we slice propositions out of the account we can still (a) cope quite well with the fact that some pairs of sentences are equivalent to one another in meaning, while (b) leaving a ghostly pseudo-entity out of our ontology.

On Making Myself At Home

William James wrote somewhere (I'm feeling too lazy to look it up) that there is a class of cases where the word "it" has no clear antecedent, yet where this apparent ambiguity is perfectly acceptable as a matter of idiomatic English. "It is raining," after all, is a simple statement of fact. Nobody, hearing it, scratches his head and askes, "what is raining?" The phrase in question conveys the same meaning as "Rain is falling at this moment," yet conveys it in half as many syllables.

The lesson: we can't deduce a metaphysical fact from a grammatical subject.

So what of the pronoun "I" in the sentence "I think"? This is one crucial problem with Descartes' famous reasoning. The thoughts with which he tormented himself in the course of his methodical doubt, the thought that there might be a powerful evil demon, etc., and the thought "I think" itself, might all be rain, as it were, coming from no "substance" more specific than the atmosphere at large. Or much less specific than that. All we can really get from paraphrasing "cogito ergo sum" with such an understanding is this: "Thinking goes on, therefore a universe that includes thought, exists."

Okay, all of this is derivative. Not only have I stolen from James, I've stolen it from myself. I just posted the same thoughts on a listserve I run. But why should the first post on a new blog not be derivative? At least, in this way I don't set myself any impossibly high hurdle for future jumps.

I have been blogging for about a year and a half now at blog-city. Unfortunately, blog-city will cease offering us the cyber-real estate for free at the end of this year, and is already making various changes that seem designed to drive us out ahead of that. So I figured I'd begin my migration to blogspot today.