When A Paradox Is Not A Paradox

“This blog is false.” ~ Theory Parker

In my free time, which is almost never these days, I like to think about paradoxes and the flaws that lead to their apparent existence. Paradoxes – statements that seems contradictory but may be or seem true – come in several types; they may be paradoxes of logic, of self-reference, of statistics, of probability, of vagueness, of mathematics, of geometry and even of physics. For the sake of brevity, I am going to focus on three paradoxes today that relate more to philosophy and demonstrate why they are in fact not paradoxes. I will start with one of the most famous of all, one of Zeno of Ela’s paradoxes, Achilles and the tortoise.

Achilles and the Tortoise

This “paradox of motion” basically states that in his race with a tortoise, the much faster Achilles must travel at least half the distance covered by his slower opponent an infinite number of times in order to even get close to the tortoise. Since the tortoise always moves ahead some distance, Achilles has to travel to the tortoise’s starting point of that distance before reaching the tortoise. So, as long as the tortoise is moving, Achilles can never catch up and pass the tortoise. Or, as Aristotle himself relates it in his work Physics, “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”

Solutions and rebuttals to this paradox have been offered by Aristotle, Archimedes, Thomas Aquinas, Bertrand Russell and philosopher Nick Huggett, with only the physicist Peter Lynds and mathematician Hermann Weyl coming close but still missing the mark. I’ve always found this puzzling as the solution to this paradox – thus not making it a paradox – is incredibly simple: The paradox is only a true paradox if we’re talking about Achilles and the tortoise as two-dimensional beings on a two-dimensional plane. But this is not the case in reality. In reality we live in three dimensions of space and one of time, to say nothing of the fact that the Earth is rotating while hurdling through space during which our solar system orbits the galactic center which is itself winding its way away from other galaxies as space is expanding. So, when Aristotle says something like, “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started,” it cannot be true that the faster runner ever occupies the same point in space (much less time) that the slower runner has started from, no matter the given starting point for the slower runner. Another way to put it would be to say that if we point to the starting point of the slower runner and tell the faster runner to go there, the faster runner may go to the same geographical location but the geographical location has changes its position in space-time by the time the faster runner gets there. And if the faster runner is fortunate to have the Earth rotate towards them as they run, they are in fact that much quicker (though this increase in speed is not detectable by human senses).

Zeno’s paradox here, like his other paradoxes of motion, winds up failing on account that he perhaps didn’t know about the four dimensions we live in. Furthermore, Zeno appears to make the mistake that time is static; that there is no flow to time. While time may occur in discreet packets at the quantum level (i.e. Plank time), there appears to be no interruption in the flow of time at the macro-level of the physical world. If Zeno has bothered to listen to his contemporary Heraclitus who said, “Everything is in flux,” Zeno would have readily seen the flaw in his paradox and never come up with his tale of Achilles and the tortoise.

“This Sentence is False.”

“This sentence is false,” is probably the most well-known version of the Liar Paradox and is categorized as a self-referential paradox. Quite simply, the apparent paradox is this: If "This sentence is false," is true then the sentence is false, but if "This sentence is false," is false, then the sentence is true.

I deny this is a paradox on two accounts. The first is that despite appearances, the sentence actually has no subject with which to refer to. The subject in question, “This sentence…” if taken by itself as a phrase, can have no truth value in the same way a phrase like ‘this book’

does, seeing how ‘this sentence’ is an abstract concept and ‘this book’ refers to an object in reality. That is to say, abstractions cannot have truth values, or perhaps it is more precise to say they have no truth values in reality, outside of individual minds. While a unicorn may exist as an abstract within our minds, unicorns don’t actually exist in reality. As such, I am of the position that abstractions have no truth or false values. If something has no truth value universally to all people everywhere, it simply has no truth value at all; the word, phrase, or sentence in question is ultimately devoid of truth or falsity.

Furthermore, I object that abstractions can reference themselves and maintain any truth value simply because referring to itself as being true or false cannot be verified. If a unicorn says, “This unicorn is true,” we don’t know what about the unicorn the unicorn is saying is true much less know whether or not the unicorn is lying, especially when unicorns don’t even exist in reality. 

“This sentence is false,” is merely a nonsensical utterance derived from the contrivances of language.

Ship of Theseus Paradox

A little more challenging is this paradox brought to us by the Greek historian, Plutarch. He writes in his Theseus, "The ship wherein Theseus and the youth of Athens returned from

Crete had thirty oars … they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same." What Plutarch is asking is if parts are taken away from the ship and replaced which new parts though of the same design, is it still the same ship? Furthermore, would it still be the same ship if all the pieces were eventually replaced? Further furthermore, if all the parts were eventually replaced with new ones and the old pieces were reassembled just as they had been on the ‘old’ ship, which ship is actually Theseus’ ship?!

Aristotle thought he’d solved this paradox by invoking four criteria (he called them ‘causes’) for the ‘identity’ of the ship. They include the Formal Cause (the ship’s design), the Efficient Cause (how the ship was built), the Material Cause (what the ship is made of), and the Final Cause (the purpose for which the ship is made). Aristotle was mostly concerned with a thing’s formal cause, so as far as he considered the problem, there was no paradox because despite the replacement of parts – no matter how extensive – resulted in the same design. This also satisfied Aristotle’s second most import cause, the Final Cause, since the design still allowed for sailing. So, if we were only concerned with the idea of Theseus’ ship, the ‘form’ of Theseus’ ship, then both ships are Theseus’ ship (which puts aside the question of ownership). But since Aristotle added criteria that to him indicated that only the ship that was still worthy of sailing was Theseus’ ship. Of course, if both ships are sailable, Aristotle’s argument sinks.

The solution that most philosophers ignore because it is just too obvious, is to say that Theseus’ ship is whichever one he owns and is at the helm of. If Theseus had paid for one single ship, there might be a question as to which ship he actually owns, but by adding that Theseus must be at the helm of the ship as well quite quickly resolves the apparent paradox. While the pieces of the ‘old’ ship may have been taken away and reassembled, it is not Theseus’ ship. It was Theseus’ ship but not anymore, just like there was a paradox but not anymore.

Have a paradox you need solved? Just send them to pi3.onefour@gmail.com and I’ll get back to just as soon as I return from time-travelling to the past to kill my grandfather.

The Asshats of Philosophy

Philosophy has a major problem, namely, the number of people who consider themselves capable of doing philosophy with any amount of integrity. I’ve got news for those of you who think you’re doing philosophy when you’re reading jokers like Aristotle or Kant: You’re not doing philosophy. Doing philosophy is reading Aristotle or Kant and thinking about why they are right or wrong (Aristotle or Kant are both wrong on several accounts, but what philosopher isn’t?). Instead, what appears to happen is that a person reads something from someone like Aristotle or Kant that sounds clever or that they agree with (because they already thought the same thing) and end their thinking at that point. In cases where they don’t end their thinking about a given philosopher’s assertion or general idea on the table, their thinking is routinely searching for arguments to strengthen what they already agree with, dispensing with the skepticism that is necessary to do good philosophy since any amount of skepticism will probably undermine what they are comfortable believing. Presumably this is due to the need to believe something at some point, which often results in sloppy thinking.

I routinely challenge the assertions of so-called rational atheists on atheist-related threads and what happens on a routine basis are ad hominem attacks against me simply because I choose to challenge what they believe or the reasons why they believe certain things. Basically, if you don’t kowtow to the party’s line, you’re evil in their view. As we all know, this is what theists do to outsiders and what many atheists proclaim to hate about theists. Meanwhile, many atheists do the exact same thing and don’t even seem to realize it. Not a socially uber-liberal femi-Nazi supporter? Evil. Support a single viewpoint with an opposing political party? Evil. Don’t think theists are necessarily worse people than atheists? Evil. But, people are arrogant and love nothing more than to proclaim their righteousness no matter how wrong they may be; atheists are no exception. And like the attitude of theists atheist claim to hate, so-called rational atheists are just as pig-headed and stubborn, unwilling to budge on their beliefs.

If you don’t believe me, ask a so-called rational atheist what it would take to change their mind about the existence of God. (Atheists often employ this kind of question against theists. What’s good for the goose is good for the gander, someone once said.) Their answer is always the same as a theist’s when such a question is posed to them: nothing. Why? One answer would be that they have no way of telling a god apart from a sufficiently powerful or knowledgeable alien. That is, if the god in question appeared to violate the laws of physics in some way, we as a species do not possess sufficient enough knowledge to know whether the laws of physics have actually been broken (which harkens back to not knowing how much a knowledgeable alien knows). Another answer would be that even if said being could prove they created us, creation does not entail that the entity in question is a god. (Do we program or even expect robots to view human beings as gods because we created them? Only in the movies.) The point is, you cannot call yourself rational if you are unwilling to change your mind.

This is where any so-called rational atheist will roll their eyes because no one is willing to change their mind that one plus one equals two. It’s a stubborn belief because it appears so rational to the point it is intuitive, but one can easily think of counter-examples if one takes one plus one out of its prevailing mathematical context. Sure, one plus one equals two, but one has to rely on context to make that assertion true. Not changing one’s mind on the existence of God is much the same way as many atheists view God existence in the same context, as that of a (supposedly) physical being like you or me. Myself? I am willing to change my mind on God’s existence, which upon internal examination would have more to do with me wanting to believe in a god than the arguments for God’s existence, almost all of which I find specious. Perhaps as an exercise in critical thinking, a future blog will try to come up with arguments in favor of God’s existence. You will not see any so-called rational atheist taking on such an assignment, though. It’s because they are not the philosophers they think they are.

Here’s a quick test that strengthens my assertion. Ask a so-called rational atheist if they are a Humanist. (Some will answer “yes,” some will say “no.”) For those that answer yes, ask them if they subscribe to the Humanist notion that all people have inherent worth. (They must, if they identify as a Humanist.) Now ask them where that inherent worth comes from. They cannot say; their belief that all people are born with inherent worth is simply what they wish to believe. Or are they going to say that worth is inherent from the mere act of being born? Really, says who? Is that some kind of instinctual desire, not unlike the instinctual desire to beat stupid people about the head? It’s a nonsensical belief backed by no scientific evidence or rational reasoning. But, like the theist who will not admit they’re pretending to know something they do not, the so-called rational atheist does the same thing, just with a different belief.

This is what makes some people asshats. Take George W. Bush for example, when asked who his favorite philosopher is, answers, “Jesus.” Jesus was not a philosopher – he said such-and-such is the way it is and there is no room for doubt. That’s not philosophy and that’s why GWB is an asshat. So the next time you think you’re doing philosophy and you think you’ve come to some concrete answer, it’s no longer philosophy; you’ve either entered the realm of science or are completely wrong. My money is on the latter.

A New Paradigm for Philosophy?

“Man is a rational animal.” – Aristotle

“Man is not a rational animal, he is a rationalizing animal.” – Robert Heinlein

I recently read an article by Jamie Hale titled, “Are We Rational Animals?” To summarize the short article, Hale writes that there is growing evidence to support the idea that we are not the rational animals we think we are. This has been obvious to me for some time, but been on my mind a lot lately, especially after defending Abercrombie & Fitch’s Mike Jefferies in a blog a few months ago and the emotional vitriol against me that followed. Even in my last blog, I took so-called rational atheists to task for not being as rational as they pretend to be. And, if I take myself as an example, I’ll unabashedly tell you I can be as irrational as rational on any given day. Certainly, I don’t want to be irrational, beholden to the potentially negative effects of my capricious emotions, ever. It can’t be helped sometimes, though.

If it is true that we are not the rational animals we think we are, if we think and act more irrationally than rationally, why? Is our incredible capacity for reasoning not as incredible as we think we it is? We can tell ourselves that we indeed have an incredible capacity for reasoning, but is it rational to say so or are we rationalizing the assertion to inflate our egos? With these thoughts in the back of my rational mind, I re-read something that made me suspect that ever since Aristotle’s claim about man being a rational animal, we’ve all been going about philosophy the wrong way. That is, so few problems in philosophy have been solved due to the prevailing top-down approach, an approach that begins from the premise that man is indeed a rational animal and that any one person is egotistical enough to think they alone have figured something out. This approach is why no one can come up with a convincing argument for or against, say, abortion, that convinces everyone.

So, I was reading The Information: A History, A Theory, A Flood by James Gleick in which he begins taking about the early days of genetic theory and some innovative remarks Richard Dawkins made about genes at the time (which can be found in Dawkin’s book, The Selfish Gene). Dawkins basically said that we do not use our genes; our genes use us. Genes are essentially programmed to get themselves (or the genes in its group) into the next generation and it doesn’t matter how; genes aren’t conscious of what they’re doing, they are just trying to replicate. Dawkins said that if you consider this, it begins to explain many types of behavior that we observe. It explains tribalism, violence, rape, and even self-sacrifice. It explains why we tend to honor the unspoken Social Contract and trust people without any grounds for doing so. It explains why I, someone with no desire to have children, wants to contribute something lasting (an idea perhaps) to the human race. I really do try to ignore this feeling, but I am nonetheless compelled by something other than reason. The mission of genes explains why we – all of us – are more irrational than rational.

Left to nature alone with no nurturing to tame it, the mission of genes readily sees population growth go unchecked. When genes are allowed to run rampant, the effect in the animal kingdom is seen when an animal overpopulates its environment and exceeds the land’s capacity to sustain them. To take an example we as humans can identify with, we used to be haunted (now we’re just jaded) by pictures of starving children in Third World countries where there isn’t enough food being grown locally to feed them while there is little or no education to curb population growth. Starving children are considered a problem and the underlying cause is genes running amuck – it’s an irrational situation. Solving the problem hasn’t worked in the traditional ways; politics tend to obstruct aid for…irrational reasons, I’m sure. Meanwhile philosophers tell us what we should do and then don’t act on their convictions. My point is, in doing philosophy, no one (that I know of) is considering our genes as the actual starting point for solving such a problem. If it is the purpose of our genes to get themselves into the next generation and we are simply the vehicles to get genes where they are going, might this be a place from which to start doing philosophy? One reason to even do philosophy is to tame our genes so that we don’t have to see something like children starving. (We also do philosophy that leads to science, which makes products that placates our genes’ need for certain comforts.)

Let’s return to the issue of abortion and begin from the premise that our genes are trying to get into the next generation and we are compelled by our nature to get them there. On the face of it, considering the needs of our genes alone, abortion should not be permissible. Now other factors in the argument for or against abortion can be considered. Should anyone but a pregnant woman have a say in what happens to her potential baby? That may depend on the needs of the group or society that is otherwise going to support her baby, whether through her job, family support, or some other way, what have you. (Of course, how large a group or society should be to have any say, if they have any say, is open to debate; based upon the needs of our genes, of course.) What about a woman that becomes pregnant due to a rape; should she be made to carry a fetus to term? That may depend upon how far into the future a group or society wants a gene prone to such violence to propagate*. See how now we can ask which genes should and should not be allowed to propagate. If we allow all genes to go forward, anything becomes permissible and no philosophy or argument for or against something like abortion will ever do for everyone. Or we can select for some traits in an attempt to subvert others and construct some guidelines for what should be done (taking into account any number of auxiliary circumstances) when a woman becomes pregnant. I’m not saying an issue like abortion isn’t complex, but we cannot debate such an issue if our starting point is whatever position you’ve already taken on the issue. That’s the top-down approach which in any given group or society ultimately kowtows to mob mentality. Mob mentality should never do for a species that considers itself rational.

(* This is assuming there is a genetic explanation or component to rape. I’m sure that on occasion something as horrible as rape it is due to a prevailing culture, but I believe cultures are ultimately the expressions of genes. Culture have to be since without people, there are no cultures.)

We can't do anything to curb being irrational until we admit how irrational we are and why. It's sort of like admitting you're an alcoholic before you can stop yourself. Recognizing what role genes play in our lives - sorry, the other way 'round - can be as much a starting place for philosophy as any other, so why not try it out since all the other starting points have pretty much failed?

 

Since this idea is new to me (and probably even weirder to you) I haven’t yet taken the time to consider to just what extent approaching philosophy in this manner is useful, if at all. If it fails, it fails. No biggie. But, like rational atheists who think a world run by people like themselves would be better than it is now simply because they couldn’t do any worse than the people who have traditionally run the world, I don’t see how this possible new approach to philosophy could do worse than how Western philosophy has played out so far. Not that it’s been all bad…

Breaking the Law of Non-Contradiction

Although I bear little respect for theologians, I’ve become terribly annoyed by the insistence of analytic philosophers lately. My annoyance stems from their arrogance, an arrogance that was once the realm of those whom they seek now to overthrow, theologians. To ad-lib from the Law of Non-Contradiction before I break it, let’s say that two wrongs do not make a right.

Analytic philosophers and logicians love the Law of Non-Contradiction (the LNC), maybe because they think Aristotle was so great. Or maybe they think it tells us something about the world, the Problem of Substance be damned. I’m not sure. I certainly don’t think Aristotle’s contributions to moral philosophy in particular were contributions at all and his formulation of physics certainly impeded progress in that field of study (seeing how he didn’t bother to test any of his hypotheses). Sorry, I'm already getting off track…What is this law analytic philosophers and logicians love? As blogger Bill Pratt puts it (via Leibniz via Aristotle), “There are at least three ways to state it:

1. A thing cannot both be A and not-A at the same time and in the same context.
2. A thing cannot both exist and not exist at the same time and in the same context.
3. A statement cannot both be true and not true at the same time and in the same context.” [Note: I exchanged his original word ‘sense’ for the more suitable word in this context, ‘context.’]

Bill Pratt goes on to say that he is amazed that people still try to deny this law. Never underestimate people, Bill; I am amazed how often analytic philosophers ignore underdetermination and other flaws in analytic and scientific reasoning. But I think that in part, my problem with The LNC stems from not liking being told what to do. So, if you tell me The LNC is a law, I’m going to find a way to refute it because, well, I can, at least some of the time. (And because I think Aristotle sucks.) What that means, kids, is that The LNC isn’t a law at all. Oooo, I can just feel the intense hatred of some analytic philosopher reading this right now…

My problem with the LNC comes via its linguistic analysis. Bill Pratt attempts to score one for his team by pointing out that you cannot negate the law without using the law itself to defeat it. He writes: “If you have any doubts about this fundamental law of rationality, try and deny it, but then write out your denial in a sentence – ‘The law of non-contradiction is false’ – and ask whether your statement is both true and false at the same time and in the same sense. If the law of non-contradiction is false, then your statement of denial must be both true and false.  But if your denial is false, then the law of non-contradiction is true! By denying the law of non-contradiction, you have just affirmed it.  The more you try to deny the law, the more you will affirm it.’

Typical; someone convinced they are right playing word games to prove their point. Pratt wants to put words in your mouth by claiming your denial of the law can only be structurally sentenced as ‘The law of non-contradiction is false.’ No, Bill, linguistically, all I have to do to deny the law is assert ‘A thing can be both A and not-A at the same time.’ (I do have to prove it, though.) Pratt continues to say that if you try to say the law is both true and false, the falseness of the statement makes the statement true, and if you say the law is true it’s both true and false and..that…doesn’t make much sense. Let us not forget that this word play just a mind-game, a paradox of language that is completely dependent upon what each word in the statement means. Aristotle engaged in this same word-play, insisting that a statement must have only one meaning if we’re to understand each other and gives the example of the word ‘man’ meaning ‘two-legged animal.’ Only, we do not have to consent to Aristotle’s definition and still understand what he means when he uses the word ‘man,’ though his definition may be more narrow than our own. I’d like to remark here that by way of his argument, Aristotle has proved the law to himself and only himself; the LNC doesn’t hold in the presence of two or more people.

By the way, Pratt suspiciously fails to point out that in order to prove the law, he has to use the law to prove itself. I’m not sure how he forgot such a minor detail.

To further illustrate the LNC is a contrivance of language, let me use this example: Pi is 3.14. Okay, so what are we saying here, that Pi is 3.14 (A)? Well, that can’t be since Pi and 3.14 are two different things, at least to my eyes. So, to say Pi is 3.14 would violate the LNC. Oh, or are we saying that Pi (A) is 3.14 (not-A)? Well, it can’t be A is not-A because that would violate the LNC, too! The way I see it, if Pi is 3.14, then two things are both A and not-A at the same time in the same context being that their meanings are synonymous. If we say Pi (A) is 3.14 (not-A), we’ve still refuted the LNC. If – in a desperate attempt to save the LNC – you’re thinking that ‘Pi’ is the written expression of 3.14 and ‘3.14’ is the numerical expression of Pi and this means the synonymous symbols make for two different contexts, I would strongly disagree; we’ve already agreed their meanings are synonymous and therefore the context is preserved. In fact, you can do this with almost any ‘is’ statement. This enigmatic vortex of the philosophy of language is academically referred to as a SNAFU.

I’m not the first to deny the LNC. The ancient Greek philosopher Heraclitus, Graham Priest and ‘inconsistency-tolerant’ logicians have also challenged the validity of the law, although in a manner unlike I have done here. And I’m not saying the LNC can always be refuted; in instances that are self-evident truths, of course you can’t refute the LNC. That is, when you see blue, of course you’re seeing blue. But ask a color-blind person what they see and you’ll get a different answer. Thus the LAW of Non-Contradiction isn’t a law at all. If it is, well, you know that old saying: Some laws were meant to be broken.