The Mathematics of the Gods and the Algorithms of Men: A Cultural History | Hacker News Reader

The Mathematics of the Gods and the Algorithms of Men: A Cultural History

spectator.co.uk

74 pointsHooke21 comments

Any civilization, in any galaxy, would have the number one, no? They would have triangles and circles, too. There are certain universals that seem to transcend, even if we are living in a simulation, etc.

This idea, of the immaterial nature of numbers is the notion of platonic forms. That idea came from the pythagoreans who claimed that the universe is fundamentally made of numbers -- and that there are certain harmonies in numbers that result in order and beauty in the cosmos.

https://en.m.wikipedia.org/wiki/Pythagoreanism

> It does not make a wink of difference to your life whether the figures in your bank account or the digits on your clock are, in a philosophical sense, really real, so long as they work as expected.

Argh! It makes a crucial difference if you want not to be slaves to a dream! Beware: there's no such thing as "Tuesday". And Dollar is a small god.

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Anyhoo, operationally numbers come from counting, which comes from a cyclops matching up pebbles and sheep. Eventually someone gave names to the pebbles, and then someone (else?) realized that you could do without the actual pebbles, and there you are: numbers.

And yeah, math is very useful, culminating in the Standard Model and the Universal Machine and other fun stuff that I don't have to belabor here, eh?

Metaphysically (literally) numbers are each a kind of archetypal entity or quality (and see also "Sacred geometry") that are somehow "real" w/o being made out of any real stuff (protons, neutrons, electrons, pebbles, etc.)

It can be fun to reflect that e.g. the C language is of the same epistemological status: there are books on C, and C compilers, and C code, and so on, but the C language itself is no more (or less) real than the number two or Tuesdays.

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And yeah, Western history tends to ignore a lot of the world outside the West. This isn't confined to math. Ever try to read about African history other than Egypt?

There’s a history of Indian and African philosophy podcast that makes a good faith effort at it, but it’s limited by the fact that a lot of it wasn’t written down particularly outside of North Africa. That goes for African history in general, at least pre-colonial.

What is the podcast?

https://hopwag2.podbean.com/

https://archive.md/5hrEx

One of the worst translations I've ever seen.

Set theory already supposes that sets come before enumeration. I am not sure what the original Italian book is about then. Maybe it is rephrasing this with some added history.

Set theory didn't exist in ancient greece.

Can I ask how sets come before numbers?

And, does the concept of one/whole exist before sets?

Set theory constructs the natural/ordinal numbers from it's axioms, but set theory is a man-made formal language that didn't exist until the 19th century, so talking about what comes/before after is kind of moot.

https://en.wikipedia.org/wiki/Ordinal_number

It not about historical order, it's about the order in which you need things to build upon mathematically.

You don't need set theory for numbers, enumeration or arithmetic.

You don't need it, but sets are more primitive than numbers.

For example, the category of finite sets without an NNO [1] is simpler or more foundational than the set of natural numbers. At the same time, this category is actually a category of numbers.

My point is that numbers are complicated by nature, but they are more intuitive for humans (mostly I think) than sets are. Sets are simpler or more basic, but often less intuitive for humans.

[1] https://en.wikipedia.org/wiki/Natural_numbers_object

I think you can argue that the idea of finite sets is prior to numbers, though the machinery of set theory is obviously much newer. When you count a field of sheep, you need some idea of the set if sheep you are counting.

It was a kind of Peano arithmetic with pebbles: you kept a sack or pile of pebbles and moved them to another sack or pile one-by-one as your sheep went by one-by-one (Ye Olde Pigeon Hole Principle) and that way you knew you had all of your sheep. Two major innovations after that were the idea that you could e.g. add two piles of pebbles to "calculate" the size of combining two flocks of sheep into a larger flock, and so on, and the idea of naming the pebbles (perhaps from a "counting" song) and then letting the names themselves substitute for the actual pebbles.

So, yes, idea of finite sets is prior to numbers, but numbers were formalized before the idea of sets was formalized, IMO, FWIW.

One begins with nothing. Specifically, the set containing nothing: {}. From there, we can describe a sequence of sets, each containing the previous one: {{}}, {{{}}}, {{{{}}}}, etc. These sets can then be associated with natural numbers, though arithmetic in this context becomes somewhat perverse.

1 := {}

2 := {{}}

...

n+1 := {n}

This is the simplest set-based description of the natural numbers that I know, and it's very unwieldy -- one might like to construct negative numbers and zero; describe algorihms to perform arithmetic, etc. But it's all quite manageable and a satisfying exercise if one takes the time.

I believe a more standard approach is to say

0 := {}

1 := {{}} (= {0})

2 := {{}, {{}}} (= {0, 1})

3 := {{}, {{}}, {{}, {{}}}} (= {0, 1, 2})

n+1 := {0, 1, 2, ... n}

This has the advantage that the numbers are equal to the cardinality of the sets that define them. In the construction you gave (which I agree is conceptually easier), the cardinality of 1 is 0, and then the cardinality of every greater integer is 1.

A cool feature of this version is

x < y := x ∈ y

One begins with nothing... How nice! It seems quite ridiculous to ask which came first.

Cardinal numbers are sets. And after that comes ordinals, which are enumerated numbers.

[1] https://en.wikipedia.org/wiki/Cardinal_number [2] https://en.wikipedia.org/wiki/Ordinal_number

Von Neumann Construction https://en.wikipedia.org/wiki/Von_Neumann_universe

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