Why the Number Line Freaks Me Out (2016)

It seems to be an article about all those "harmless" lies we tell students.

The vast majority of people think mathematics is about numbers, when it is actually about relations, and numbers are just some of the entities whose relations mathematics studies.

Nobody is born with this misconception; we teach it, and test it, and thereby ingrain it in the minds of every student, most of whom will never study mathematics at a level that makes them go "wait, what?". The overwhelming majority of people never get to this level.

I suspect this is also why statistics feels so counterintuitive to so many people, including me. The Monty Hall problem is only a problem to those who are naive about probability, which is most people, because most of us don't learn any of this stuff early enough to form long lasting, correct instincts.

It's not fair to students to bake "harmless" lies into their early education, as a way to simplify the topic such that it becomes more easily teachable. We've only done this because teaching is hard, and thus expensive. Education is expensive, at every step. It's not fair or productive to build a gate around proper education that makes it available only to those who can afford it at the level where the early misconceptions get corrected. Even those people end up spending a lot of cognitive capital on all those "wait, what?" moments, when their cognitive capital would be better spent elsewhere.

One thing not mentioned that still gets to me is that all the numbers we know, that we will ever know, is measure zero on the real number line. The lovecraftian nightmare numbers, the infinite maw of the unknowable, aren't some rare exception, they're everything.

This video parallels the article, going further. I love the tag line, "We know none of the numbers." https://www.youtube.com/watch?v=5TkIe60y2GI

Another number line mind-blown moment is that the complex plane is actually a half-plane since the distinction between i and -i is arbitrary, so any graph in the complex plane has to be symmetric about the real number line.

Just forget number line after high school. Number line is needed for engineering graphs and economic data

There are an infinite number of numbers between each number on the number line.

That's always something fun to think about.

There are more structures in non-computable numbers. As an example, a non-computable number could be "definable", i.e. you can describe it. For example, a real number the nth digit in the binary representation of which equals to whether the nth Turing machine halts.

Since the set of all English sentences is countable, whereas there are uncountably many real numbers, it follows that there must be numbers that are NOT even definable.

Think about that.

> Simon Gregg calls noncomputable numbers “the dark matter of the number world”...

Thanks but no, we don't need more of this kind of bs naming. "dark matter" already ruined physics because it implies something mysterious and magical is going on whereas it's quite the contrary. I hate it when people dumb down beautiful abstract concepts to the point that it's not only not intuitive, it actually makes the thing less accessible to those who are not in the know.

This helps explain the different kinds of numbers in that recent HN article about how hard it is to make an accurate calculator.

Related: https://math.stackexchange.com/questions/462790/are-there-an...

I find it hard to wrap my head around non computable numbers. How can I even “point to one” of them if I can’t express/describe it? And if I cannot communicate which number I’m referring to, does it really exist? In what way do they exist?

Its because its recursive in nature- every segment contains a new line and every segment of that line contains again - a infinite set of sub-segments.

Infinity in a box, right in front of your numeric microscope.

Which is why dividing by zero- is exactly the same operation. You take something finite- and you unpack the boxes- in parallel. Every time the operator hits something finite, it unpacks a new set of parallel boxes. The sum of all the boxes, is a infity with a signature.

And those parallel running overlapping infinityssquences, form the irrational numbers

Nothing so spooky deserves to be called a "Real" number.

From what I remember "things you can compute" splits into "things you can mathematically define" and "things for which you can construct a Turing machine which computes them".

If we define "x is computable" as "there exists a Turing machine T(x) which takes n as input and produces n-th digit of x" then there are numbers which are defineable but not computable.

I'm probably not very clever - but I don't get why calling the gap "fractions" is problematic.

The example the author gives of "fractions" is... rational numbers, and then proceeds to say "what about irrational numbers" - but in mymind (and this is probably where I'm a wrong?) an irrational number is still a fraction of a whole number, just we cannot express it "properly" (yet)

Even “e” and “pi” would have been noncomputable at one point in time.

But the noncomputable numbers make me wonder if our notion of mathematics is too general/powerful.

Math is the study of futility. Futility to calculate, to understand, to define, to rationalize. 1 is the only number. Everything else is a name.

The underlying problem is that infinity doesn't exist. It's a convenient illusion to make special cases go away. It's possible to have entirely constructive mathematics. In a true constructive model, everything can be constructed in a finite number of steps. There are only integers, no reals.

You can represent pi on a number line but it is absolutely completely impossible to randomly put a dot down on a number line and have it be pi. You can achieve endless measurable precision with decimal rational numbers. So randomly placing a dot on a line will always be a rational number.

Right???

I wonder if this was the inspiration for this numberphile video? https://www.youtube.com/watch?v=5TkIe60y2GI

The video does go further than the article.

When you say "most" of the numbers are non-computable, the word "most" is meaningless in this context. There are infinitely many of each kind of numbers you have listed there. You can't compare one infinity to another and say that one kind of infinite is bigger than the other. The concept of comparison (smaller/bigger) doesn't exist outside of finite numbers. Cantor was just what people thought he was - a crack, who did not consider the bounds of logical comparison.

And for the line itself, the line is not made up of numbers. Line is made up of continuity, while numbers are cuts in that continuum. Infinite number of cuts do not make up a continuous piece. Mathematical continuity (or extent or measure or span) is the essence of the imaginary spatial existence. It is not composed of cuts. A cut is a non-existence, completely opposite of the existence.

Searching for meaning is all good but sometimes its just the umbrella man - https://www.youtube.com/watch?v=yznRGS9f-jI

Recent and related is the discussion of "Dedekind's subtle knife": https://news.ycombinator.com/item?id=43084200

> It should be a timidating

Total nitpick, but i think the in in intimidating means "into a state of being timid" and not "in" in the sense of opposite of timidating.