4 ms·
Except that you’re assuming that 1 must necessarily correspond to that minimum distance. Keeping that situation, we can just say that 1 corresponds to two inche
by dhosek 3mo ago
Except that you’re assuming that 1 must necessarily correspond to that minimum distance. Keeping that situation, we can just say that 1 corresponds to two inches and then realize 1/2 as that number.
Classical geometry (a la Euclid) allows for constructing a lot of numbers. We get natural numbers pretty cheaply, negative integers through adding in a concept of directionality (zero is a bit of an imaginative leap which is why it was absent from Western mathematics for so long). Constructing arbitrary ratios is possible through similar triangles and square roots through right triangles, but some basic algebraic numbers like cube roots cannot be constructed with a ruler and straight edge (which raises the question of whether, in a quantized universe, whether irrational cube roots actually exist). Of course there’s no guarantee that the quantization is going to be uniform and we also have the ɣ factor of special relativity (1/sqrt(1-v²/c²)) which gives us a non-Euclidean space to complicate things, but it’s not clear that if you can find a value for 1 that allows you to get a measurement for every irrational number.
- dhosek 3mo agoOne could make the argument that the only numbers that actually “exist” are the natural numbers, but the question ultimately is can you model any real number in the physical universe. Modeling ½ is simply a question of picking a unit to be 1 and finding its midpoint (or for that matter, declaring two apples to be “1” and thus a single apple would be “½”, although it’s a bit of a challenge to use apples to model (2-√3)/5
- Dylan16807 3mo agoMake a line of 10 apples and declare it to be 2 units long, then make a square that's 15 applies diagonal, finally measure how much longer the 10 apples are than the side of the square. It's a pretty linear increase in complexity between the math and the apples.
- testaccount28 3mo agoin what way does drawing a diagram have anything to do with physically realizing a number?
- Dylan16807 3mo agoI didn't say a diagram? I'm talking primarily about physical objects. And I'm just explaining how to use the same methods as the comment I replied to.
- testaccount28 3mo ago"Make a line of 10 apples and declare it to be 2 units long" <- what number does this physically realize?
- Dylan16807 3mo agoBoth ten and two. And it lets you model fifths. But don't ask me about that. That part wasn't my idea at all. Ask dhosek about that way to do fractions. My contribution was the square root and the subtraction.