What Did You Do to Us Zeno?
Erasmo Calzadilla
I got myself into a big mess with the students at the faculty of Nuclear Sciences the day I explained to them that the paradoxes of reason were not resolved.
“The nonsense of sleepless philosophers!” they thought. “So Achilles can’t catch up with a turtle!” and they went on in search of their text book where it was explained how an infinitesimal calculation had solved many of these problems, especially the paradoxes of movement put forward by Zeno*.
Of course they didn’t believe me, and my reputation — which already wasn’t very good — was never the same. No one will believe that dogmatism is not plentiful among scientists and the projects of scientists.
Be reasonable, don’t go crazy!
Common sense opposes the reasoning of the insane, and it associates the first one with restraint in comparison to rage. “Sonny, think about things before doing them,” a grandmother would tell to her impulsive little grandson. Few people today know that reason has clay feet.
I’ll give a simple example to make what I’m talking about clearer: How could we judge if reason is guiding us along the right path? Would we have to look for an evaluator who was beyond reason, or are we going to let reason evaluate and congratulate itself?!
No, for real, that would be fraudulent, but beyond reason there is only unreasonableness. Can we guide ourselves by unreasonableness to judge the work of reason? Even the question sounds absurd.
These little things have scientific, human and civilizational consequences that are delicate and deep. How much cruelty has been committed and justified in the name of reason; and how much blood has been spilt in the name of its cousin, progress?
Figuring out the paradoxes of reason, that’s to say the limits of reason, is to lean out over an abyss (one that doesn’t give you vertigo because you still haven’t made it to the edge). They are able to undermine all of our certainties, to paralyze us and make us distressed… We would probably have to look for a way out.
One solution is to understand that thought itself, which leads to paradoxes, is paradoxical. This doesn’t solve a whole lot because this thought that allowed us to understand the paradoxical character of the paradoxes is also paradoxical, and so on ad infinitum.
Another way out could be to integrate reason and life, as was attempted by several Spanish philosophers of the last century. Let’s not trick ourselves with words, life here is synonymous with irrational; so to integrate reason and life is the same thing as integrating the rational with the irrational. It sounds as poetic as it is impossible.
My individual solution has been to forget the matter and to keep movin’ on without bogging myself down, but when I try to use truth and justice for guides, the problem returns.
If someone knows how to get me out of this mess, please don’t make me beg for your help.
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(*) Zeno: a Greek philosopher who put forward a group of paradoxes that derived from paying attention to the senses; for example, believing that movement is possible. If movement were not possible then Achilles, who was a fast runner, would never have been able to catch up with a turtle, no matter how fast he ran. For Zeno, paradoxes still did not come from reason; on the contrary, with the guidance of reason we could avoid them. Later we discover that paradoxes are innate to reason itself and everything is very confused.
just read Hegel’s Science of Logic, good therapy for paradoxes.
I hope some modern day Xenophon is collecting these dialogues! Erasmo poses the question; George and Julio present different responses. From these seemingly contradictory views can there be a synthesis? In any event, very entertaining–and expanding, too!
George,
Yes you can please point to me where is the contradiction?
and you are miss interpreting Godel’s incompleteness theorems.
The holly grail of deductive systems was from David Hilbert’s program to get a system of axioms that are complete (Means you can derive any theorem from it) and that is consistent (that means that is not self contradictory) in other words you can not deduce (A) and (not A) from the same set of axioms.
In the first of Godel’s incompleteness theorem he proved that there are true statements in arithmetic that can not be proof. That means that any axiom set you pick is always incomplete. You need to keep adding those statements that can not be proof as axioms themselves.
The second incompleteness theorem is a stronger version of the first.
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems
Julio, you cannot prove that there is no contradiction inherent in making use of symbols that represent infinities… this is Godel’s second incompleteness theorem… so I can safely disregard everything you have written… further you can prove that any complete description of the universe that includes symbols that represent infinities is contradictory… this is Godel’s first incompleteness theorem…
George, I am sorry to disagree with many of your statements above.
Let me just pick on one of them.
“Zeno’s paradoxes arise because infinity is inherently contradictory to describe… the moment we write it down we have rendered it finite!”
There is no contradiction on de description or definition of infinity by writing it down.
We can write down infinity using a symbol that looks like horizontal 8. We can even defined different types of infinity from countably infinity (the cardinality of the natural numbers) to bigger infinities like aleph 0 the cardinality of the Real numbers.
Let me give you an even simpler example.
I can symbolically write 1/3 (that is with a finite number of symbols(only 3 in this case One,Three and /)) that actually express in itself the idea of infinite as you know 1/3 is equal to 0.333333333…. where the … means we continue writing 3 an infinite number of times. So you see there is multiple ways to write down infinite with a finite number of symbols but even is the writing is finite the concept that it inclosed is still infinite. The fact that we are able to write it with just a finite number of symbols does not make infinity finite.
That symbol or finite combination of symbols becomes the representation of infinity.
As you know 1/3 is not equal to just .3
nor is it equal to .33 or .333 and so on but to
.333… and as I mentioned the … dots mean that we continue writing 3 up to infinity.
The paradoxes of Zeno are what makes dialectics possible… I am truly amazed that this is not taught in Cuba… Erasmo, seriously, I would worry less about not being able to get a job because you step outside the box and more about not knowing the fundamental ideas that give rise to what’s in the box, namely dialectics… I hope that doesn’t sound too harsh… but really, sometimes I despair… what are they teaching in Cuba if not this?
Perhaps a good place for you to start is the Mathematical Manuscripts of Marx which can be found here: http://www.marxists.org/archive/marx/works/1881/mathematical-manuscripts/index.htm
Zeno’s paradoxes arise because infinity is inherently contradictory to describe… the moment we write it down we have rendered it finite!… however without such a “paradox”, infinity would be just another number and there would be no room for the contradictions that give rise to dialectics… I have written so many times on this that I am a bit tired of repeating myself, so I have attached below a similar comment I made some time ago elsewhere…
It is well known that the Pythagoreans were both mystics and mathematicians. Indeed when Platon wrote above the entrance of his school “let none who do not understand mathematics enter here” he was emphasising the necessity mathematics to the mystical path. The idea that the mystical becomes concrete is an essential law of progression towards the infinite, but the equally essential in the definition of the infinite is the idea that there will always be mystical experiences. As an example, for the ancient Pythagoreans, the square root of two, which arises from the right-angled triangle of unit base and height, and other similar numbers, were refered to as the “unspeakable numbers”. Due to their infinite nature, and inability to be expressed as fractions, they were held to be only understandable on a mystical level. Today we call them the irrationals, and our mathematical language has expanded to make them concrete. In other words we can express them finitely by expanding our language. However this is the essential paradox of the infinite. The moment we express it finitely, i.e. in words, we are no longer capturing it’s true essence. Thus the real infinite remains at the mystical level. Indeed in contemporary mathematics, we have “undefinable numbers”, the modern day equivalent of the “unspeakable numbers” of the past. Again one enters the realm of the mystical. Such a process will proceed indefinitely, for that is what the concept of the infinite means. One can expand ones vocabulary to make finite what was previously unexpressable and mystical, but then there is still more that is unexpressable and mystical on the horizon. It is essentially the process of counting. There is always a bigger number. What is more there are numbers whose existence is undecidable. In the contemporary setting this was proved formally by Kurt Godel, but it was known to the ancients. Indeed it is the presence of undecidable statements, which are a necessary consequence of the infinite, that gives rise to the logic of dialectics. This is the famous logic of yin and yang in the East. It is a logic precisely for dealing with infinity and the contradictions that such a concept involve. It is the logic of evolution recognised by both Marx and Lao Tsu. Indeed the process of going from describing root two as an “unspeakable number” to describing it as an “irrational number” is entirely dialectical, involving the resolution of an undecidable statement. Thus mathematics is an esoteric path, one in which the mystical is always present. As we follow this path, our concrete world expands, but the infinite neve diminishes.