It took a long while to reach division
and even longer to reach long division?
Comment on In this essay...
fushuan@lemmy.blahaj.zone 3 weeks agoIn logic class we kinda did prove most of the integer operations, but it was more like (extremely shortened and not properly written)
If 1+1=2 and 1+1+1=3 then prove that 1+2=3
2 was just a shortened representation of 1+1 so technically you were proving that 1+1 plus 1 equals 1+1+1.
Really fun stuff. It took a long while to reach division
MeThisGuy@feddit.nl 3 weeks ago
titanicx@lemmy.zip 3 weeks ago
None of that sounded fun…
JackbyDev@programming.dev 3 weeks ago
Lambda calculus be like
Taldan@lemmy.world 3 weeks ago
Presumably you were starting with a fundamental axiom such as 1 + 1 = 2, which is the difficult one to prove because it’s so fundamental
bleistift2@sopuli.xyz 3 weeks ago
I find this axiomatization of the naturals quite neat:
Now the neat part: If 0 is a constant, then s(0) is also a constant. So we can invent a name for that constant and call it “1.” Now s(s(0)) is a constant, too. Call it “2” and proceed to invent the natural numbers.
anton@lemmy.blahaj.zone 3 weeks ago
That axiomisation is incomplete as it doesn’t preclude stuff like loops, a predecessor to zero or a second number line.
TeddE@lemmy.world 3 weeks ago
Not sure what you mean by ‘loops’ - except perhaps modular arithmetic, but there are no natural numbers that are negative - you may be thinking of integers, which is constructed from the natural numbers. Similarly, rational numbers, real numbers, and complex numbers are also constructed from the naturals. Complex numbers are often expressed as though they’re two dimensional, since the imaginary part cannot be properly reduced, e.g. 3+2i.
I recommend this playlist by mathematician another roof: www.youtube.com/playlist?list=PLsdeQ7TnWVm_EQG1rm…
They build the whole modern number system ‘from scratch’
Eq0@literature.cafe 2 weeks ago
I think you are missing some properties of successors (uniqueness and s(n) different than any m<= n)
That would avoid “branching” of two different successors to n and loops in which a successor is a smaller number than n
kogasa@programming.dev 3 weeks ago
There are non-standard models of arithmetic. They follow the original first-order Peano axioms and any theorem about the naturals is true for them, but they have some wacky extra stuff in them like you mention.
unwarlikeExtortion@lemmy.ml 3 weeks ago
What’s missing here os the definition that we’re working in base 10. While it won’t be a proof, Fibbonaci has his nice little Liber Abbaci where he explains arabic numerals. A system of axioms for base 10, a definition of addition and your succession function would suffice. Probably what the originals were going for, but I can’t imagine how that would take 86 pages. Reading it’s been on my todo list, but I doubt I’ll manage 86 pages of modern math designed to be harder to read than egyptian hieroglyphs.
bleistift2@sopuli.xyz 2 weeks ago
That ‘86 pages’ factoid is misleading. They weren’t trying to prove that 1+1=2. They were trying to build a foundation for mathematics, and at some point along the way that prove fell out of the equations.
fushuan@lemmy.blahaj.zone 3 weeks ago
Yeah, that’s what meant with “2 is just the shortened representation of 1+1”.
Same with 1+1+1=3, really. We need to decide the value of 1,2,3,4… Before we can do anything. In hindsight if you think about it, for someone that doesn’t know the value of the symbols we use to represent numbers, any combination that mixes numbers requires the axiom of 1+1+1+1+… = X
I’d be surprised if someone proved that something equals 5 without any kind of axiom that already makes 5 equal to another thing.
Matriks404@lemmy.world 3 weeks ago
It’s only difficult to prove if you somehow aren’t able to observe objects in real world.
captainlezbian@lemmy.world 3 weeks ago
That’s just empirical data, not a mathematical axiom. I know it’s true, you know it’s true but this is math as philosophy not math as a tool