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Why real numbers are not.

October 5, 2015

Have you ever encountered mathematicians who do not believe in “real numbers”? Well there are some, mainly those who come form a computer science ideology. I am starting to understand why they do not think real numbers are real or useful as a concept. Firstly what do we mean by a real number? It comes from looking at the number line as a continuum. It is treating the number line consisting of infinite number of points. For example the numbers between 0 and 1 – there are infinite “real numbers” there.

Take an example of a so called real number \pi. It is written as 3.141592653589793… Now notice the ellipses in the number. They are there to say that the decimals after the last 3 as printed is infinitely long. So people think that a real number can be represented by those dot dot dot and so the real numbers have unending decimal series. In actuality the symbol \pi is the limit of that series of decimals once considered.

Now I can appreciate why A/Prof. N. Wildberger insists the need for something to be written down, and we will explain the reason why later. If you think for a moment, we are not capable of writing a real number down. Those dot dot dots are a semantic idea to signify to us that the digits following goes into an infinite series. That is not really writing a number down. Why do we need to be able to write something down with finality?

Well it is because we can put the process of writing into an algorithm. We can put it into a function. So imagine again \pi. The fact that we can not write the number down with completion means we can not put that generation of the numbers into an algorithm that will stop. It can not and we won’t let it stop precisely because the number of decimals in the tail end is infinite. So an algorithm that goes into an infinite loop, making it useless. Since  we can not locate precisely where \pi is in the continuum line we can not even have a function to compute it. In a sense the digits following are not decidable.

From a computer science point of view, the algorithm must terminate and if it does not, then the function is undefined at that point. The problem stems from the idea of infinity of points present in the number line. Yet in practice we can not really even locate real numbers in the number line. You can have the function stop at the 100th position of \pi but that is not \pi itself. This is the best we can do but that number is not exactly \pi rather it is “something like or close to \pi“.

So real numbers are unreal, man.

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3 Comments leave one →
  1. Jim Smith permalink
    October 7, 2015 10:18 pm

    Valuable obsevations. I’m not sure what to make of them, but I take care to point such things out to students — too many things are taught to them in a tone of cock-sure omniscience, so when the students learn otherwise, they tend to feel that they’ve been conned. And in a sense, they have. They deserve better, and your post will help me do a better job for them.

    • October 8, 2015 8:53 am

      Hi Jim,

      Some mathematicians think it is the idea that the number line is a continuum that is the source of the problem. It sure looks intuitive to rely on the number line for conceptual understanding but it fails in practical situations. Those who come from a computational background express skepticism on the viability of so called real numbers. I first heard about this skepticism in one of the talks by Prof. Donald Knuth (his video talk is found below), a well decorated soldier of computer science there in USA. For some, if something is not computable, it is of no concern. Some also come from a philosophical ideology too. Some mathematicians who are constructivist will also not believe in real numbers. For them you demonstrate existence by showing a proof of them but not by argument. The computability presupposition is indeed philosophical as well. However, at first I did not appreciate these people’s skepticism but I know understand where they are coming from when I caught that they were coming from an algorithmic/computational point of view.

      LPC

  2. Peter Osudar permalink
    November 26, 2017 5:27 am

    Thank you. Blogs like these are not easy to find. And even coming to this reflect is difficult as well, not well appreciated at all.

    Your ability to write so coherently about such a difficult to appreciate topic is mind blowing, to say the least. Try to keep in mind there are billions of people who are craving this type of rationality in their lives and you have demonstrated it in the public eye. Hats off my friend.

    The obvious place that this conversation needs to go next is to education of the young… how long will we let teachers talk to our students about “real numbers”. I imagine that the frustration and the undefinability of the real numbers are causing so much anger and strife among the population at large that, we could fund many hospitals with all the incoming patients if we just had the people who didn’t believe in real numbers and used things like finite fields to produce modelling that could be used to sell services to the public and industry at large. Whether you follow my logic or not, we need to all start appreciating the number theoretic possibilities of what the human mind derives and entails with respect to actual mathematics.

    We all need a way to bond over this.

    Please continue to write.

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