Logical Reasoning: To Infinity and Beyond

“Logic is not a body of doctrine, but a mirror-image of the  world. Logic is transcendental.” — Ludwig Wittgenstein

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Some scholars argue that mathematics is the fundamental language: That it is both universal and scientific. “One does not understand anything until he or she quantifies it,” these scholars often say.

For the most part, I agree with them, and I hope you do as well. We can qualitatively describe and categorize an object to any extent we want. Therefore, grouping objects together in this fashion is a fairly effective and quick method of organization. However, there is always a chance in our qualitative analysis that two objects have almost the same characteristics, or moreover, they do have the same characteristics. It is herein that our qualitative analysis seems to decompose. We need a system of organization more concrete and more reliable. We need something that never lies. We need quantities.

How is it that quantites can never lie? (How is it that mathematics, in general, can never lie for that matter?) It is in the answer to this question that I argue that mathematics is not the fundamental language. The answer to my question is simple–logical reasoning. Mathematics is constructed on a beautifully intricate and delicate balance of axioms, postulates, constructions, lemmes, theorems, and corollaries. The soul of mathematics is to take old knowledge or knowledge that we know can never be false and, using these balanced logical constructions, discover new knowledge. (Of course, there is one other component to the soul of mathematics–imagination.) This is why mathematics never lies: All knowledge is true because it is based on true knowledge. To find new true knowledge requires logic, the true fundamental language.

If you cannot tell already, logic will be our main focus today, but rather than introduce the concept in the orthodox fashion, which would require you to sit down and read dozens of ancient essays on conditional statements and biconditional statements and contrapositive statements and counter-example statements, I am going to let you explore logic on your own. To do this, I am going to call upon paradoxes which arise from the perfectly logical-illlogical concept of infinity. It will be up to you to answer them using your logical intuition and critical thinking as a guide. (These paradoxes have really neat answers, and I really encourage you to try them–NOT skip over them.)

1. Hilbert’s Hotel

Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied–that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms. Suppose a new guest arrives and wishes to be accommodated in the hotel. The hotel is full. A countably infinite number of guests arrive at the same time. How can the hotel be arranged to fit the incoming guests?

2. Ross-Littlewood Paradox

Cards with numbers 1, 2, 3, … are put in a box in a particular order. One minute before noon numbers from 1 till 10 are put and a number 1 is taken back. 1/2 minute before noon numbers from 11 till 20 are put and a number 2 is taken back. 1/3 minute before noon numbers from 21 till 30 are put and a number 3 is taken back. This pattern continues infinitely many times. How many numbers are in the box at noon?

3. The Wizard and the Mermaid

An infinitely rich wizard has a mermaid in the pond of his garden. He likes to play with her and gold coins. Every minute he throws two coins in the pond and she throws him back one coin after a half minute. When this game is allowed to run forever, how will the money be distributed? We are allowed put a label on all the coins, numbering them 1,2,3,4, etc. This paradox actually has more than one solution.

I will post the solutions to these logic problems next week, and then, we will actually discuss some interesting properties of infinity, which I believe I alluded to a few weeks ago. (We are getting closer to the grand finale I have in store.)

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