Silly details about numbers

The ancient Greeks found numbers fascinating in an unprecedented way (or so Morris Kline's Mathematics for the Nonmathematician informs me). Rather than adopting the more practical attitude of the Egyptians and Mesopotamians toward numbers, the Greeks recognized a conceptual beauty in the ability use the self-same methods on any possible collection of objects. That is to say, abstraction, and the universality thereof.

To assign a single, unique label to every quantity, and then to perform mental operations upon them which, amazingly, correctly modeled comparable situations in reality--astonishing. And I do mean that without irony.

Take a pie or any appropriately divisible object. Cut it into halves, then cut each half into thirds. We take it for granted today that the answer may be computed easily through a mechanical procedure: (1 * 1/2) * 1/3 = 1/6. This is to say that, without having made any cuts or further measurements, we already know with certainty what size the smaller pieces will be! (Or the size that they will approximate, because it is not an ideal world.)

But this is a fantastic discovery, for by beginning from only a few known facts, we discover what must be the case for physical objects after manipulating them--all without having left our armchairs.

Another Note Regarding Content

I've realized I still want to keep this blog alive, or at least in a feeble state of existence that approximates life. Unfortunately, for whatever reason, I don't have the interest or motivation to write out posts with the thought and research I'd originally planned. In fact, at this time (judging from my last post), I can't even assemble together posts that reach a definitive point backed by argument, or that are structured in a coherent manner.

But hey, then, why not just go with that? Basically, I am now relaxing the standards for my posts (wait, what? I had standards before?). Until further notice, Doubt Rests shall be a repository for half-formed thoughts, quotes, or whatever vaguely interesting factoids I come across. Don't expect explanation or justification for everything I write (although if you ask nicely I might elaborate), and certainly don't expect fully developed ideas. The contents will likely be of a more personal nature than previously. All will be fragmentary, but fragments put forth in the hope that failed attempts to progress are better than none at all.

And maybe someone, somewhere, some time might find a fragment here which fits well into his/her own thoughts.

Fallibility

Quantum mechanics tells us that the universe operates in very, very unusual ways. It is natural to want to dismiss some of the interpretations of QM (such as that natural laws are inherently probabilistic) as problems with our understanding, not as genuine features of reality. This is essentially a knee-jerk reaction to the oddness of QM and how it does not "mesh" with our everyday understanding of the world. However, science also informs us that we are the products of many years of evolution--and this evolution process equipped us to deal with one thing, and one thing only: survival. Our senses and reasoning faculties were not "designed" to help us humans apprehend truths about the world; rather, they were selected to enable humans qua systems to manipulate information in such a way that the systems preserve and replicate themselves. Luckily, knowing the truth--or approximating it--proved beneficial to the survival of these systems (these information processing patterns).

In simpler terms, more proto-humans who were able to correctly judge that there really is a savage tiger hiding in the grass over there successfully passed on their genes than those who hallucinated constantly. Or than those who had no knowledge whatsoever, who were not able to act on their knowledge, who had faulty reasoning processes, whatever.

Thus evolution tells us.

Unfortunately, to say "there really is a tiger over there" and leave it at that drastically oversimplifies the state of human existence. Because really, we don't know that there is or is not a tiger over there. All we know is that some collection of sights, sounds, scents, or inferences has caused us to believe that something over there can cause damage to us if we do not act accordingly. (Assume it's a hungry tiger and that we are defenseless in a savanna.) We may think to ourselves "There is a tiger," and we may believe "There is a tiger," and there is probably even a sense in which it is true that there is a tiger. But what we call a tiger is a convenient abstraction--a shorthand tag which bundles together a collection of concepts, memories, and/or feelings. So too, presumably, with our notion of existence--we have a certain understanding of what it means for something "to be" and "to be there." and we predicate this notion upon the abstraction "tiger."

[Edit as of 04-05-2009: I should mention that yes, Kant (and other philosophers) claim that existence is not a predicate. I do recognize that it's a contested notion and I disagree with the mainstream opinion; but I shan't defend it now.]

"Well, what of it?" I hear you say. "You're not telling us anything that the pioneering philosophers of the 17th-18th century didn't when they first speculated about our psychological workings; and you're barely even consistent with today's psychological theories."

Fair enough. But, if we can suppose that the tenets of natural selection are true, then we should firmly keep in mind that our knowledge...

....Something...

This post is dissolving into incoherency. And as usual I don't have the patience to fix it. I shouldn't even post it. But whatever.

Fancy

[Please forgive this amateur flight of fanciful, mystical indulgence. This comes from a diary entry I wrote over the summer. I decided I needed to post it somewhere. I can't say I believe these sentiments, but I have sometimes taken comfort in them.]

Maybe I am starting to feel... that somewhere, somehow, in a realm outside of conceivability, our essences intermingle. And our essences love each other, they love each other so much that it almost tears them apart, yet simultaneously unites them all the more gloriously for it. Our essences love and understand why all this tragedy is necessary, why the struggle is necessary for growth, why all is truly joy and not sorrow. Our beings love each other because they created each other, because they dreamed themselves into being, and they chose each other out of all the other possible existences because this way was right. We love each other eternally, devoutly, devotedly, passionately, irrepressibly.

We love each other and are each other. We love each other because we are each other--because of that sympathetic resonance from one core to another--a unification in rhythm that reveals our inherent, underlying unity. Our oneness.

... You and I are one, just as everything else is one. Yet somehow you are special, and I am special, and our love is special.

So we play this game--because it is the unmasking, the revealing, the unraveling, the development, the exploration, the uncertainty, the discovery--these things give us more meaning than jumping straight toward the answer.

Because It's So True

A quote from www.friesian.com:

In the Twentieth Century, philosophy was like a confused and clumsy person who repeatedly tries to commit suicide, but keeps failing, though with the addition of debilitating damage at each attempt.

Mrrem

Yeah, so this blog is suffering from a definite lack of attention.

I don't have any particularly meaningful, worked-out content to post now, so I'll just mention some of the things I've been thinking about lately.

I recently started learning about Alonzo Church's lambda calculus (via Penrose's The Emperor's New Mind), and I'm very impressed. It's so cool that logicians were able to construct systems like this (and like Turing's work) before the advent of computers – indeed, these logical systems led directly to and facilitated the advent of computers. They start from such simplicity, but possess incredible power.

The lambda calculus reminds me of the programming language LISP (unsurprisingly, of course, since LISP was based on this very calculus), and reading about it makes me want to get back into programming again. I dabbled with Haskell a bit at the start of this summer, and I found it oddly fascinating. There is something quite elegant about these functional* languages that more practical languages (C/C++, Python, Perl) don't quite capture.

* Apparently only certain variants of LISP (like Scheme) are fully functional, but anyway.


Still, it is difficult for me to remain interested in these things for their own sake. I think I would need some kind of projects to work toward if I were to take up programming again seriously. Ah, motivation, that state which so often eludes me...

There are so many things I would love to learn, but somehow actually sitting down and doing the work required bores me terribly. Sometimes it doesn't; sometimes I go through brief periods where I feel a great deal of enthusiasm toward some subject (say, Wittgenstein's Tractatus, or some aspect of symbolic logic), but then I end up dropping it again, feeling utterly bored with it for months.

I would like to have a very extensive knowledge of mathematics, physics, philosophy, logic, computer science, linguistics, music, and (select aspects of) history. It would be swell to know Greek, Latin, German, and French. It is so difficult to care about, though; sometimes my mind just seems to shut off, and whatever I'm currently trying to study becomes excruciatingly dull; suddenly I can't remember why I wanted to learn about it in the first place. I value knowledge generally, but to actually feel something for it, to care – somehow that is different from merely saying "I value it"?

The Impossibility of Halting Philosophical Inquiry

[This post was largely inspired by a hypothetical scenario and ensuing discussion at Philosophy, et cetera.]

Suppose that we have a magic device--a little black box with a screen and keyboard--which supplies us with true answers to every philosophical question we can possibly pose to it.

We can ask, e.g., "What is the true nature of the mind-body relation?" And the box might tell us, "Functionalism," (or whatever the "correct" answer is). Hurrah! In one fell swoop, we have settled a major dispute in the philosophy of mind. So, we proceed through the various fields: "Which understanding of ethics is correct?"; "What is the nature of ultimate reality?"; "What is the nature of knowledge?"; "Does God exist?". Ping! Ping! Ping! Perfectly formulated and accurate answers to each question spew out from the box. Academics everywhere can both rejoice that their long, bloody dialectical battles are over, but also mourn that now they're out of a job. Their passion--speculating about and debating abstruse philosophical topics--has lost its point, since there remains nothing further to be said.

Or does there?

The sagacious box in of itself would generate a slew of new philosophical questions to replace the old ones. In fact, it might not be adequate to answer even those traditional questions at all! Witness the following.

If there really existed such an oracular apparatus, before we even begin to think about taking it seriously, I would want to know one thing (and I think this is a very reasonable request): how can we trust the veracity of its answers in the first place? Now, this being a philosophical question, we could pose said question to the little black box itself in the hopes of receiving a satisfactory answer. But this presents a circularity problem, for no matter how the box answers, we would have no a priori reason to trust the accuracy of its answers (here a priori means less the traditional "knowable independent of experience," and more "given prior to the examination/subject at hand"). The box obviously cannot tell us, "My answers are true by virtue of their being outputted by a box that always outputs true answers." For indeed, now we are inclined to respond, "But how do we know you are one of those boxes? My dear black box, that is begging the question, plain and simple. You'll have to do better than that." In short, to answer this epistemological question, the box cannot appeal to its own authority as an infallible agent, for to do so would be circular.

In order to escape this trap, the box could try to appeal to an external, time-honored authority: reasoned arguments. Perhaps it could supply us with an airtight, incontrovertible argument to the effect that its answers are always correct. Indeed, perhaps all its answers might take this form of undeniably true premises leading irrefutably to a conclusion that no rational being could reject.

However, two things. First, is it really possible for there to be such irrefutable arguments? It seems that we can always doubt even the most basic of things, including the apparently incontestable. As Lewis Caroll pointed out in "What the Tortoise Said to Achilles," (Mind 4, No. 14 (April 1895): 278-280. Also freely available a number of places online, such as here, here, and here), nothing will force an interlocutor to accept a logical inference as valid. And we all know this, don't we? All reasoning must begin with axioms, and axioms by definition are unsupported or unjustified. So when we call something an irrefutable argument, it can only be irrefutable to some one person, or irrefutuable as considered from some one perspective (which means, starting from from a particular set of axioms and/or rules of inference). Is there such a thing as an "undeniably true premise"? Or can we reject an axiom that consists of something like, "For all A, A = A"? From what I understand, efforts have been made in the realm of dialetheism to investigate what happens if we reject such an obvious rule as the principle of non-contradiction and/or principle of excluded middle. Perhaps it is the case that there just is no such thing as an uncontroversial premise? (I realize that I am conflating premises and rules of inference a little bit in the above discussion, but the gist should be clear. I am too lazy at the moment to go back and fix things.)

Second, there might still be disagreement about understanding the logic employed by the black box. What if it spewed out proofs millions of lines long, with such convoluted chains of logic that it takes teams of experts just to have the barest idea of what is supposed to be going on, much less whether the steps are all valid or not?

To sum up, if we were unable to accept the black box's arguments at face value (and there is good reason to think that we would not be able to), we would have to resort to philosophical investigation in order to provide justification for why we should trust it in the first place. Hence, philosophical inquiry would continue, even with a magical device which, for all intents and purposes, should have ended it.


Curiously, we find ourselves in the same situation even when we replace the black box with the idea of a God.