1. This blog is about railroads, yet this is not a traditional railroad article. Many railfans (most? all?) will find what follows either silly, pointless or repugnant, perhaps all three. If only one person reads to the end, I will be pleased. If none does, I will be unbowed.
2. One of the most influential philosophers of the twentieth century, Ludwig Wittgenstein, wrote in numbered aphorisms.
3. Wittgenstein: "I don't know why we are here, but I'm pretty sure that it is not in order to enjoy ourselves."
4. Wittgenstein: "A nothing will serve just as well as a something about which nothing could be said."
5. Most of the time I feel like a dog who is standing in a fenced yard and wants to go forward past the fence. The gate to the fence is open, but the gate is behind the dog. Like the dog, I cannot comprehend that I must turn around and walk backward to the open gate in order to go forward past the fence.
6. The mathematician and philosopher Gotlob Frege spent a goodly portion of his professional life trying to establish a logical system from first premises that was complete and non-contradictory. He believed he had achieved his goal.
7. Bertrand Russell pointed out a paradox that undermined Frege's system.
8. Russell's Paradox: Consider the set of all sets that do not contain themselves. Should the set include itself? If so, then it is contradictory. If not, then it is incomplete.
9. Wittgenstein believed that he had solved Russell's Paradox. In his Tractitus Logico-Philosophicus, he stated:
"A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself. If, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition F(F(fx))”, and in this the outer function F and the inner function F must have different meanings; for the inner has the form φ(f x), the outer the form ψ(φ(f x)). Common to both functions is only the letter “F”, which by itself signifies nothing. This is at once clear, if instead of “F(F(u))” we write “(∃φ) : F(φu) . φu = Fu”. Herewith Russell’s paradox vanishes."
10. Wittgenstein's argument can be expressed in everyday language: I designate this bowl for cookies only; therefore, you cannot put another bowl in this bowl.
11. This is the same approach used by Russell and Whitehead in their formal mathematical system Principia Mathematica to solve the paradox: "Whatever involves all of a collection must not be involved in the collection."
12. In other words, Russell and Whitehead (as well as Wittgenstein)"solved" the paradox by prohibiting it.
13. This does not "solve" anything. It is like "solving" cancer by prohibiting it.
14. Godel's First Incompleteness Theorem demonstrated that formal systems expressing at least basic arithmetic cannot prohibit paradox.
15. The First Incompleteness Theorem: Any sufficiently powerful formal system (such as Principia Mathematica) that claims to derive only consistent statements can generate at least one paradoxical statement.
16. "Sufficiently powerful" means of sufficient complexity to express the basic arithmetic of the natural numbers -- i.e., addition, subtraction, multiplication, division.
17. "Consistent statements" means without paradox.
18. Showing that the formal system contains a paradox requires coding each abstract symbol in the formal system with a natural number. Example: "A" is coded as 1, "X" is coded as 2, "=" is coded as 3, "+" is coded as 4.
19. Each symbol is then separated by zero.
20. Thus, A+X = X+A would be coded as: "1040203020401."
21. This coding is called a Godel Number.
22. Each formula within a formal system will have a separate Godel Number.
23. Assume formula F(y) = (n not GF), where F is a specific formula in the formal system, y is a free variable, G stands for Godel Number, GF means that formula F uses its own Godel Number as its procedure, and n stands for a natural number. In plain English, this translates: "Formula F(y), as derived, shows that no natural number corresponds to the Gödel number of F(G(F))."
24. But the Godel Number of F(G(F)) is a natural number, albeit a very large one.
25. Thus (n not G(F)) is false.
26. Now consider F(G(F)) = (n not G(F)).
27. Because we have already shown that (n not G(F)) is false, we now have a self-referential statement that says F(G(F)) is false.
28. This is the equivalent of the Liar's Paradox: "This statement is false." If so, then it is true. But if true, it is false.
29. Godel expressly stated that his theorem was the equivalent of the Liar's Paradox.
30. Like formal mathematical systems, life is filled with paradoxes. There are several types:
31. Self-Referential Loop Statements: Like Russell's and Godel's, these statements loop back upon themselves and cannot be resolved:
A. "This statement is false." If true, then the statement is false. But if false, then the statement is true. But if true, then false. And on and on.
B. "I am a male barber, and I shave only men who do not shave themselves". If you shave yourself, the statement is false. But if you don't shave yourself, then the statement is also false. And on and on.
C. "Is the answer to this question no?" If yes, then the answer is no. But if no, then the answer is yes. And on and on.
D. This catalogue lists only catalogues that do not list themselves. If the catalogue lists itself, then it is in error. But if it does not list itself, then it is also in error. And on and on.
32. Self-Referential, Self-Contradictory Statements: These statements negate themselves by self-reference:
A. "I am certain that I am uncertain about everything." The statement is false, because you are not certain.
B. "All I know is that I know nothing." The statement is false, because you don't know this.
C. "I do not make absolute statements." The statement is false, because this is an absolute statement.
D. "I doubt that truth exists." The statement is false, because you don't doubt the truth that you doubt.
33: Self-Contradictory Statements: These statements negate themselves without self-reference:
A. N is "the smallest number not nameable in under ten words." The statement is false, because N is named in nine words.
B. "X is the only movie from Director Y that is not unique." The statement is false, because X is unique.
C. X beer is brewed from ingredients of beers that do not use the ingredients of other beers. The statement is false, because the other beers do use ingredients of another beer.
34. Non-Intuitive Statements: These statements describe actions that produce an opposite result:
A. "For every action, there is an equal and opposite reaction."
B. "The harder you try to untie a knot, the tighter it becomes."
C. "The more instructions you give a child, the fewer instructions the child will follow."
D. "The louder you shout, the fewer will listen."
E. "The softer you speak, the more will listen."
F. "You must spend money to make money."
G. "You must make money to spend money." ("F" and "G" together could be taken as Self-Referential Loop Statements.)
H. "If everyone drives on the same road at the same time, no one will drive anywhere."
I. "If you tell a child that you will give him a cookie if he will come out of the street, the child will run into the street."
35. Self-Creation Statements: Statements implying a condition that must create itself.
A. "Children can only learn to speak by hearing other people speak." How did the first person learn to speak?
B. "If a population drops below a certain critical mass, it will go extinct." How does any population reach the critical mass?
C. "Mammals are born alive from other mammals." How was the first mammal born?
D. "Chickens are hatched from eggs laid by other chickens." Where did the first chicken come from?
E. "Words are defined by other words." How was the first word defined?
36. Self-Referencing, Self-Contradictory Statements can be "fixed" by segregating themselves from the self-reference. (This type of paradox, in other words, is caused by imprecise use of language.)
A. "I am certain that I am uncertain about everything else."
B. "All I know is that I know nothing else."
C. "I do not make other absolute statements."
D. "I doubt that another truth exists."
37. Self-Contradictory Statements are also caused by imprecise use of language. They can be "fixed" simply by admitting error.
38. Non-Intuitive Statements are caused by the nature of the universe. They are caused by forces that humans understand and can learn to avoid. Therefore, they can also be "fixed."
39. Self-Creation Statements are also caused by the nature of the universe. Humans do not understand them, however, and can neither explain nor avoid them.
40. Self-Referential Loop Statements are not inherent in the universe. They are rather caused by the manner in which people think.
41. The dog that cannot understand how to go forward past the fence by turning around and walking backward through the open gate does not perceive the following Non-Intuitive Statement:
42. Sometimes the only way to move forward is to walk backward.
43. Lying in the sun on a pleasant day in April, my dog is happier than I will ever be.
44: The perception of paradoxes does not generate happiness.
45: Godel feared that someone was trying to poison him; thus, he would eat only food prepared by his wife. When she was hospitalized and could not cook, he refused to eat, eventually starving to death, weighing 65 pounds when he died.
46. Hereafter follow images of trains illustrating the Non-Intuitive Paradox "going backward to move forward." (I cannot see how trains might illustrate the other paradoxes discussed above. If you have any ideas, I would appreciate comments.)
|A loaded eastbound coal train is running west at the horseshoe near Folsom, New Mexico.|
|Another loaded eastbound coal train is running west at the same horseshoe.|
|The loaded coal train in the background is bound southeast to Texas but it currently heading northwest near Trinchera, Colorado.|
|The pusher on a Canadian Pacific potash train is running northeast, while the head is running southwest at Notch Hill, British Columbia.|
|A westbound empty UP coal train is headed north, south and east at Crater Loops, Colorado.|
|UP's westbound Craig local is running eastbound as it approaches Steamboat Springs, Colorado.|
|Another Craig local is running east at Crater Loops, Colorado.|
|A southbound (railroad east) Burlington Northern coal train is running north (railroad west) at Crawford Hill, Nebraska.|
|Another southbound (railroad east) BN coal train is running north (railroad west) at Crawford Hill.|
|A westbound grain train is headed east on Montana Rail Link in Lombard Canyon, Montana.|
|Another westbound running east in Lombard Canyon.|
|Westbound MRL running east in Lombard Canyon.|
|An eastbound Rio Grande mixed freight is traveling west on Big Ten Loop, Colorado.|
|An eastbound Rio Grande trailer-train is headed west in Big Ten Loop.|
|A short eastbound Rio Grande freight is turning west at Wondervu, Colorado.|
|A loaded eastbound Southern Pacific coal train running north, south and west at Crater Loops, Colorado.|
|Eastbound MRL running west beside the Missouri River in Lombard Canyon, Montana.|
|Loaded westbound coal running east beside the low water dam in Lombard Canyon.|
|Empty eastbound coal running west in Lombard Canyon.|
|Eastbound empty coal running southwest, turning back to the east, approaching Austin, Montana.|
|Westbound loaded coal turning southeast on MRL -- Mullan Pass, Montana.|
|Westbound loaded coal running northeast up Mullan Pass, Montana.|
|Pusher on westbound loaded coal is headed west as remainder of train has turned northeast at Austin, Montana.|
|An eastbound Rio Grande manifest is turning west at Gilluly, Utah.|
|A westbound Rio Grande mixed freight is running east at Gilluly, Utah.|
|An eastbound Santa Fe mixed freight is running west at Tehachapi Loop.|
|Eastbound Santa Fe Trailers running west at Tehachapi Loop.|
|Westbound Southern Pacific manifest running east at Tehachapi Loop.|
|Eastbound Southern Pacific manifest running west at Tehachapi Loop.|
|Eastbound BNSF trailers headed west at Tehachapi Loop.|
|Westbound Union Pacific manifest running east at Tehachapi Loop.|
|Westbound BNSF manifest running east at Caliente, California.|
|Eastbound UP manifest running west at Caliente, California.|
|A railroad east UP mixed freight is heading north at Arnold Loop, Nevada, before turning south to head east again.|
|A railroad west UP intermodal is headed north toward Arnold Loop, Nevada, before turning south to head west again.|
|Westbound BNSF stacks have turned east at Sullivan's Curve, Cajon Pass, California.|
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