Can the Infinite Be Randomized?

Is the infinite always the same? Is the infinite tomorrow the same as is today? How many infinities are there? Only one? How about if there are infinite infinities? How about if all of them are not the same, one to another?

How about if one of those infinities is not linear, without any kind of order, but random? Why not?

We have a naive and simple idea of the infinite. In the best of the cases, we think of the infinite as the unending series 1, 2, 3, ... In the worst scenario, we think of the infinite as the quantity of the grains of sand in all the deserts and beaches on the Earth.

It was Archimedes of Syracuse, more than two millennia into the past, who proved that it was impossible that the grains of sand were infinite because he was able to give a fair estimate of the grains of sand that could be placed in a sphere the size of the orbit of Saturn.

He knew that it was impossible to count grain by grain all the deserts and beaches. So it is impossible to prove that the grains of sand are finite by enumerating them one by one.

So, his approach was to establish an upper limit to the number of grains that can hold the planet Earth. This is an indirect proof that the sand cannot be infinite.

The Sand Reckoner.

Download the free EBook The Sand Reckoner, the book where he developed his proof.

But what if we have an infinite book in our hands, how big can it be? Infinite in size? Infinite in weight?. Infinite in volume?

Have you ever heard of The Book of Sand? It’s a short story about an infinite book with no beginning and no end. But this book has a finite amount of pages; how come?

The infinite is incredible and surprising! What is going to limit the limitless?

Read the article Three unexpected behaviors of the infinite and see three unforeseen aspects of the infinite.

Do Random Numbers Really Exist?

One Million Random Digits.

While I was collecting information for this month's post, I was also looking for new material for a new free E-Book to compile for my readers. The idea of the so-called random numbers sprung into my mind, so I began to search for this topic.

I found an interesting book review titled: A Million Random Digits with 100,000 Normal Deviates. The book was originally published in 1955 by the RAND Corporation, so the "review" was a little late, but its OK; the author was "reviewing" one of the oldest books in his library. This book can be found and read at Google Book Search.

Doing a deeper search I also found another article and an E-Mail by Mr. Nathan Kennedy complaining to the RAND Corp about their stand that the One Million Random Digits table was of their property and that it cannot be redistributed on the Internet. By a great coincidence, my intention was the same as Mr. Kennedy's, and his intention of putting the table on the Internet was the same as mine. However, and in a great unselfish gesture, Mr. Kennedy generated his own table of random digits and proceeded to place it on the Internet for free as a text file on the same page.

Since he authorized the use of his million digits table, I reformatted the text file as a PDF file, designed a cover page, wrote a small introduction for it, and made an E-Book, to be distributed also for free.

I am not a statistician, so maybe I will never find a practical use for this kind of numerical table, however, random numbers are of interest for me, and probably for many others, for the degree of strangeness they bear.

Are there really random numbers, or there are random events?

Can we really speak of random "numbers"? Isn't it more appropriate to speak of random "events"? Can we make at least some arithmetic operations with them? Can we add two RNs and still say that the sum is also "random"? Can we multiply them to obtain -without a doubt- that the result is also random?

Note that the RNs are not obtained by any formula, or equation, or matrix, or any predefined mathematical operation; they are mainly obtained by algorithms fed by some "physical" events like atmospheric variations, radioactive decay, thermal processes, or the like. Hence, what we are doing is using unpredictable physical events, assign to each "event" a number, and say that this is an RN. Another nonphysical source of random sequences of digits (but sometimes questioned) is by selecting digits or portions of digits of the decimal expression of irrational or transcendental numbers.

But without physical events, can we still generate RNs? There are some rudimentary approaches, but they are mostly mere mathematical curiosities.

The interested reader can find more authoritative articles at Random.org where he/she can obtain instant (real-time) random numbers for the lottery, cards, passwords, etc.