Posts filed under Logic

Overtaking Zeno

Shared Article from Slate Magazine

Zeno’s Paradox Is a Trick—But a Very Interesting Trick

The Greek philosopher Zeno wrote a book of paradoxes nearly 2,500 years ago. “Achilles and the Tortoise” is the easiest to understand, but it’s …

David Plotz @ slate.com


O.K., so, briefly: If you think that the point of Zeno’s Paradoxes of motion is to prove that the arrow never will reach its target, or that Achilles never does pass the tortoise, &c. — then I think that you are mistaken about the point of raising the paradox in the first place. Of course, it’s hard to be confident about the motives of dead philosophers who have no surviving books. But what we do know is that Zeno was a student of Parmenides; and Plato tells us that his books were written to defend Parmenides’s doctrines, by negative means,[1] showing that the views of his opponents led to contradictions.

So the most charitable understanding of Zeno’s aims is not that he’s trying to show you that Achilles can never catch the tortoise. Of course he does; just watch them race and you’ll see it happen. His point is to ask, given that Achilles passes the tortoise, well, how is that possible? And, for good or for ill, to argue from the paradox that you can only make sense of Achilles passing the tortoise if you reject presentism, and accept eternalist and Parmenidean conclusions about the nature of time and being.

Maybe he’s right about that, and maybe he’s wrong. (I’m inclined to think he’s wrong.) But note that if your solution is to try and settle the issue by introducing a lot of mathematical notation and conceptual apparatus from modern calculus — for example infinitesimal limit processes, convergent and divergent series, etc. — as is done in the Slate article here, and as is probably the overwhelmingly most common first response to Zeno’s paradoxes by mathematically-trained writers — then probably you are doing a better job than any pre-classical Greek philosopher could do in elaborating the precise nature of the problem.[2] But you’re not obviously refuting Zeno’s claims in any way, at least not yet. At the most you’re kicking the can down the road, and really you’re sort of strengthening Zeno’s own position. After all, naive formulations of mathematical notation are more or less always going to involve you in all kinds of specifically eternalist language, for example about moments in past and future time actually existing, instantiating the value of functions, etc. You cannot normally take the limit of ΔS(t) over values of t that don’t exist (no longer exist, do not yet exist).[3]

Or perhaps you can. But if you can, then doing so, and explaining what you’re doing when you do it, will take some very non-naive reinterpretation of ordinary mathematical language — and some nice metaphysics, too, to justify your reinterpretation. In any case the solution is going to have to be deeply philosophical, not just a matter of applying a technical innovation in maths.

  1. [1] In the Parmenides: I see, Parmenides, said Socrates, that Zeno would like to be not only one with you in friendship but your second self in his writings too; he puts what you say in another way . . . You affirm unity, he denies plurality. . . . Yes, Socrates, said Zeno… . The truth is, that these writings of mine were meant to protect the arguments of Parmenides against those who make fun of him and seek to show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one. My answer is addressed to the partisans of the many, whose attack I return with interest by retorting upon them that their hypothesis of the being of many, if carried out, appears to be still more ridiculous than the hypothesis of the being of one.
  2. [2] Since the 19th century, we’ve done a lot to really nicely rigorize the mathematics of infinites and infinitesimals, in ways that sometimes anticipated by but never fully available to ancient mathematicians.
  3. [3] If anything, this is even more true of late-modern mathematics than it was of classical mathematics. Contemporary mathematics constantly helps itself to a lot of the language of existence, actuality, etc., for mathematical objects, in areas where Euclid and other classical mathematicians were typically much more circumspect about making existence claims for mathematical objects that hadn’t yet been constructed.

Wartime Logic

Suppose that you have — somehow or another — conclusively proven that there is just no way to have a modern war without bombing cities and massacreing innocent people.[1] That leaves you with a hard incompatibility claim between moralism and militarism — so if you go around morally condemning military tactics (like the atomic bombing of Hiroshima and Nagasaki, say, or the firebombing of Tokyo) because they killed innocent people, then you’d end up having to condemn any modern war at all as immoral, no matter who fought it or how it was fought.

Many people, when they reach this point in the argument, want to shove it at you as if the incompatibility made for an obvious reductio ad absurdum of any kind of moralism about military tactics — Oh, well, if it’s always immoral to bomb cities then you couldn’t have any wars. That’s why it must not always be immoral to bomb cities. I honestly don’t know why so few of the people who give this argument ever even seem to have imagined that their conversation partner might take the incompatibility as an obvious reductio ad absurdum of any kind of militarismOh, well, if it’s always immoral to kill innocent people, you can’t bomb cities, and if you can’t bomb cities, you can’t have any wars. And that’s precisely why you shouldn’t have any wars.

Also.

  1. [1] Actually, I think this has been more or less conclusively proven. And that’s precisely why you shouldn’t have any wars.

Prooftexting

Show me an axiomatic approach to ethics, ideology or anything else in the marketplace of ideas, and I’ll show you a recipe designed to produce a specific result. . . . Besides, everyone since Gödel’s proof knows formal systems degenerate into mental masturbation at some point.[1]

Groundbreaking developments in the history of mathematics and logic: In 1931 Kurt Gödel published “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”[2] in the journal Monatshefte für Mathematik. The paper is famous among logicians and mathematicians for the two “Incompleteness Theorems” it contains,[3] logically demonstrating that no formal system rich enough to express truths of ordinary arithmetic can be both consistent and deductively complete while having a finite number of axioms.

The paper is famous among almost everyone else for containing a multi-page Rorschach inkblot, allowing a projection test in which the reader-subject can discern an easy dismissive response to whichever deductive argument they happen to like the least; or, if they prefer, to the exercise of deductive logic as a whole.

  1. [1] Lorraine Lee, Re: Julian Assange, the Left-Anarch. Comments at Social Memory Complex (21 April 2013). This is actually not even remotely what either of Gödel’s two major Incompleteness Theorem proofs says. —CJ.
  2. [2] A PDF blob of the article in its original German is available online thanks to Wilhelm K. Essler. An English translation of most of the paper is also available online thanks to Martin Hirzel.
  3. [3] Theorem VI and Theorem XI in the paper, specifically.

Clarity and clarifying

A couple of notes from a couple of different conversations on being clear and becoming clear. (It’s about philosophy, I promise, not about Scientology. . .)

Me, in reply to Andy Bass and Nemo during a conversation on Wittgenstein and philosophical method (Dec. 2011):

[Quoting A.B.:] Wittgenstein’s “end” to philosophy altogether would be some way of living with, and using, language in which linguistic inconsistencies and their resulting philosophical conundrums cannot arise at all. Wittgenstein doesn’t spend much time with this notion of a final treatment… .

I dunno, doesn’t he? It seems like this sort of end of analysis is importantly part of the goal of the Tractatus, and the struggle against that picture is part of the important shift in PI. To live with language in such a way as to end philosophical puzzling would be to become perfectly adept as a logical grammarian — to succeed in catching and keeping the will-o’-the-wisp of logical form. But if there is no such thing to catch, or no such thing as catching it … .

I’m rather inclined to think that if we take seriously what Cavell (for example) has to say about the projectability of concepts — and on the late Wittgensteinian themes that Cavell is drawing on here (on the urban geography of natural language, etc.) — then I think it has to be part of the nature of a certain sort of language-game — of any language-game of the sort you could reason or explain in, say — that there could not possibly be a way of living with language that does not raise the possibility of philosophical problems. To live with a language where concepts and linguistic structures can constantly be projected into novel forms is to live with the pervasiveness of risk, doubt, misfires, mistakes, confusion, — since to acknowledge the possibility of projection just is to acknowledge the risk of failing to cotton onto the novel uses, or to shift contexts appropriately, or to recognize the interplay between the old usage and the new, or . . . .

And often we should like to be perfectly adept at these things, but (1) it seems clear that we cannot do that with any set of ex ante rules about what good language ought to look like (as the positivists seem to have thought); (2) it also seems clear that we cannot do that with any set of ex ante principles about what good linguistic therapy ought to look like (as AoTLP[1] hinting); and (3) setting all that aside, it’s not clear that we possibly could count as being perfectly adept by any means within us (what if the conversational context is not something that’s always up to us, but depends on future contingents about what others will play or non-play? what if it involves external objects, like the meter-stick in Paris or the chemical structure of water, which may not be epistemically transparent to us? etc.). And it’s not even clear if this, were it possible, would always be desirable (what if projection serves a tentative or exploratory purpose, not just an analytical or declaratory one? not to allow a certain degree of risky or even confused behavior may simply be to close us off from some funky new neighborhoods that language might otherwise work itself into. . . .).

. . .

[Quoting Nemo:] After a conversation with Socrates, one would say to himself, I don’t know what t’m talking about! I don’t know what [the thing] really means. I’ve got a problem. With Wittgenstein, I know it now! Avoid logical fallacies and speak proper grammar, there is no problem at all.

Well, I think that the bit after I know it now! is for L.W. much easier said than done, but it’s the doing that he’s interested in. The AoTLP[1] seems to have some faith that there is a state you can be in where you will become perfectly adept in the avoiding and in the grammaticalizing — a state that can only be really understood by reaching it, but which will disclose itself to you, irresistibly when and to the extent that you reach it. (In many ways it ends up sounding something like what Socrates is portrayed as teaching Meno about the unforgetting of true knowledge in the second third of the dialogue.) Now, as I understand the later L.W., that faith in the End of Analysis is one of the things that really does change and come under the later L.W.’s criticism. In some ways this makes his project seem less Socratic (or Platonic, whichever), since it means a much less idealized picture of what logical understanding amounts to; in other ways, it makes it seem more Socratic, since it means that there is no end of philosophy to aim at — it’s not a matter of reaching some perfected state of clarity, only an ongoing process of recognizing confusion and clarifying. . . . (In PI, Wittgenstein says that the real discovery is the one that allows you to stop doing philosophy when you want to — but of course stopping it is rather different from finishing it.)

— Charles Johnson (Dec. 2011), comments re: Wittgenstein on Progress in Philosophy

Kelly Dean Jolley, on Clarity, Combative Clarity (Dec. 2011):

I am Wittgensteinian enough, or Kierkegaardian enough, or Marcelian enough to believe that what philosophy aims for is clarity. But one is always becoming clear; one is never finally clear.

Clarity. Clarity is internal to philosophical investigation: it is not a separable result, isolable from the activity that realizes it and such that it confers value onto the activity because of a value it has independent of that activity. If a result is separable, isolable and independent, then it has a career cut off in an important way from the process that realized it. Indeed, in one sense its history only begins after the process that realizes it is finished. The result can be seized and put to purposes quite different from anything that those involved in the process of realizing it intended or foresaw.

But clarity is valuable because of the process of philosophical investigation that realizes it. And there is no clarity in isolation from the philosophical investigation that realizes it. Philosophical investigation does not realize a clarity that someone could hope to enjoy who is no longer involved in philosophical investigation. (I got clear, you see; and now I am enjoying my clarity, although, thank God!, I am no longer involved in the travails of philosophical investigation.) –Kierkegaard’s Climacus talks about the true Christian, the subjective Christian, as combatively certain of Christianity, as certain in a way that requires that the certainty be daily won anew. Eternal certainty (his contrast-term) is not something that the subjective Christian can enjoy on this side of the blue. Similarly, the clarity realized by philosophical investigation is combative clarity, not eternal clarity.

— Kelly Dean Jolley (Dec. 2011), on Clarity, Combative Clarity, in Quantum Est In Rebus Inane

  1. [1] The Author of the Tractatus Logico-Philosophicus, the later Wittgenstein’s way of referring to his earlier views when he wished to criticize them.

Tertium Non Dant

From Tim Cavanaugh, Steven Chu, Oh Where Are You? (Solyndra Roundup), at Reason.com:

Meanwhile, the Wall Street Journal reports that a new poll indicates few Americans are paying attention to the Solyndra scandal, and most still support so-called clean energy initiatives … . More surprising than the continued support for solar power is the apparent support for spending taxpayer dollars on it, which the report from Public Opinion Strategies has at 62 percent, versus 31 percent opposed. However, I’m a little skeptical of the strongly leading questions:

There are thousands of successful and profitable clean energy and clean technology companies all across America; the failure of one California company should not stop us from continuing to make targeted public investments to help create American clean energy jobs. 62%
The collapse of Solyndra shows that investing taxpayer dollars in so-called green jobs is a waste of money; these businesses cannot compete or succeed on their own without government assistance, and we cannot afford to prop them up with government funding. 31%

I hope the remaining 7 percent answered, as I would have, Both of these options are stupid. I don’t want my taxes subsidizing private companies of any kind, and I’m aware that the amount of energy conventional solar power generates is modest. But how the hell should I know whether solar businesses can compete or succeed without government assistance?

The only way to find out whether these companies can work in the marketplace is to let them compete without government assistance.

— Tim Cavanaugh, Steven Chu, Oh Where Are You? (Solyndra Roundup), at Reason.com, 29 September 2011

See also.