How inevitable is the concept of numbers? (2021) | Hacker News Reader

How inevitable is the concept of numbers? (2021)

writings.stephenwolfram.com

71 pointsmarcospgp65 comments

I think that if pattern recognition is essential to intelligence (it's got to be the best way to simplify the complexities of all the tons of data any intelligent being's senses will be giving it), then we will inherently group things by category.

There are ways to get along in common life while only making distinctions between "more" and "less", although you will still probably need counting, for example the classic case of shepards making tallies. For complicated behavior, numbers will inevitably arise. For example, the earliest writing we know comes from an sumerian accounting of a barley warehouse in Uruk making beer, which counts incoming and outgoing shipments. It miiight be possible to skip this sort of step some amount by only using comparative weights or things, although I would imagine to simplify things you'd want to move to a standardised system of weights, which it seems would inevitably progress towards some being multiples of others.

At the very least, a species advanced enough to develop ways to understand things like atoms or chemical bonds will have to start making distictions between hydrogen and helium all the way to lead etc, or HO and HO2. If they didnt think about numbers before, this will require them to in some regard.

"Solaris" is an interesting exploration of a situation in which this is maybe not the case.

If you mean the Book written by Stanislav Lem, i did read that but i don't understand how you connect the Ocean with "not having the concept of numbers".

Solaris is too vague to call it an exploration of the idea imo. Lem postulates that there is an intelligence so different to ours that it's incomprehensible. Naturally he can't describe it in any but the most nebulous terms. It remains a postulation without any supporting arguments that such a form of intelligence is realistic.

You can compare quantities geometrically: draw a line expressing incoming shipments and under it a line expressing outgoing shipments. You can also replace counting with an imperative drawing process, there's real imperative counting method using the 正 character, it has five strokes, they are drawn one by one until the character is complete, then next character is drawn.

Just curious, how would you represent discrete quantities geometrically?

Line is for continuous quantities, that's clear.

"there's real imperative counting method using the 正 character, it has five strokes, they are drawn one by one until the character is complete, then next character is drawn."

This is almost exactly what a tally is, like I mentioned its an early form of counting used in a lot of cultures to count animals incoming and outgoing of areas by shepards, often cut into a piece of wood. Suprisingly many cultures have almost the exact same system of making a pattern of 5 lines which then repeats

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

The problem with treating pattern recognition as fundamental, is you could describe almost any sort of analysis as being pattern recognition.

Inevitable because numbers are useful for survival. Even plants can count

https://www.iflscience.com/carnivorous-plants-know-how-count...

Likewise time. Being able to reliably model / predict cyclic phenomena is valuable to survival factors such as dealing with seasonal food sources.

Then there's prime numbers! [1]. I was blown away by the emergence of prime number from a very simple process that's optimised for survival.

https://www.livescience.com/14238-southern-cicadas-emerge-ex...

One thing we'd share with any alien species is our perception of the universe, and stars and galaxies are eminently countable, so the development of numbers seems inevitable.

There is the example of the sheepherder who cannot count but ensures all sheep are accounted for by with the bag of pebbles: as the sheep leave the enclosure in the morning, a pebble is placed in the (initially empty) bag; as the sheep return, a pebble is removed for each sheep, and if none are missing, the bag ends up empty. This quickly fails for many issues, such as two sheepherders wishing to discuss the relative size of their flocks, however.

Regardless, an alien species might have started from entirely different premises then numbers. A lot of the images in this article are graphs, node-and-edge constructions, but it's a bit odd that graphical algorithms are a relatively new feature of human mathematics - Dijkstra's algorithm was only described in 1956.

https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm

What if the ancient Greeks had instead started there, what would mathematics look like today? An alien species might have done so, leading to system in which number theory would be of much less importance than graph theory.

You can't do graphs without numbers, so the history makes sense. Finite graphs are relations over finite sets. You can't do much with finite sets before inventing numbers.

You are mixing up what something is with how it is conventionally formally modeled or defined. This is easy to do in modern mathematics, where students are trained from early on to develop every concept from basic axioms, imposing an ordered structure of definitions for convenience of exposition and proof without worrying as much about the inherent nature of the objects under study.

A concept of number is not required to draw or describe graphs. Indeed, if you wanted you could define numbers in terms of graphs. For example, you could start with the combinatorial game Hackenbush and build up numbers from there. https://en.wikipedia.org/wiki/Hackenbush

> One thing we'd share with any alien species is our perception of the universe, and stars and galaxies are eminently countable, so the development of numbers seems inevitable.

I don't this is immediately obvious. There is an assumption here that stars and galaxies are these inherently well defined things, therefore they are countable. I don't think that's necessarily true.

Where does a galaxy really begin and end? Or a star for that matter? If there is a solar flare occurring and solar matter is being ejected into the surrounding area is that matter still part of the star?

We have this faculty which seems to be quite good at taking complex systems and approximating them to singular entities (planet, star, galaxy, table, audience, country), we can reason about. It seems to be a core aspect of human cognition but I'm not sure if this is a prerequisite for cognition in general.

The best conceptualization of the concept of number that I've come across is Von Staudt's construction of the rationals using the concept of harmonic tetrads. At first it seems weirdly over-complicated, but there's a moment where it clicks and then a shocking beauty. It's like a version of special relativity that's even more relative, like, what could observers agree on if they didn't even agree on the speed of light? Turns out they can still calculate cross ratios, but then even the notion of cross ratio can be turned inside out, and you get your fundamental notion of number out of a deeper concept of harmony.

> The best conceptualization of the concept of number that I've come across is Von Staudt's construction of the rationals using the concept of harmonic tetrads.

Do you have any more material for this? Google seems to give me no good results.

From what I understand, Von Staudt and some other geometers were trying to push back against algebra/arithmetic taking over geometry, and against projective geometry invariants like the cross ratio being defined in terms of algebraic relations between Euclidean distances. So he flipped things around and developed arithmetic in terms of basic projective geometry relations instead. (Projective geometry is geometry without circles, angle measures, distances, or parallelism, only straight lines.)

See Coolidge (1934) "The Rise and Fall of Projective Geometry". AMM 41(4), pp. 217-228 https://www.jstor.org/stable/2302023

Google scholar search: https://scholar.google.com/scholar?q=staudt+throws

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

https://en.wikipedia.org/wiki/Cross-ratio

Thanks!

This is kinda related to an idea I've been "struggling" with.

Some numbers represent real things. Like my little kid saw a pair of oven mitts and said 'two gloves'. Like, he was seeing a literal manifestation of two-ness in front of him - which is different than the logical idea of 1+1 or the symbol 2.

So you can physically see two of something. You can see ten of something. You can see a billion of something (sand on a beach). But at some point numbers get disconnected from concepts. We can easily say "ten to the power of million" as easily as "ten to the power of two" but the former is literally more than the number of particles in the universe. You can never encounter anything and say "oh, that's one with a million zeros of something."

Basically it seems like some numbers represent physical concepts (twoness, which can be manifested by a pair of mittens) and some numbers are purely theoretical.

And that is why even though Mathematics aptly represents the nature, it, at least to me, is a different thing entirely. Which again begs the question, why should Mathematics be so good at depicting natural phenomenons.

This really ties in to how brains and neurons work. A cluster of neurons will be primed with some sensory input. It might be the sound of the word that represents the number, or the visual impression of two items (and a sense of visual symmetry), etc...

Over time, neurons accumulate associations to other neurons associated to other stimuli that are related. Which trigger reinforcement as they are validated.

The number 10, might make you think of your fingers. The number 21 might make you think of Blackjack, 2 turtle doves, that 4 is related to "Four" written, 7 makes 8 when combined with a "counting" neuron impulse, etc...

Ultimately, we are a product of the inputs we are given, and a set of fundamental pre-wired associations (like visual symmetry detection).

This is similar to how AI neural nets in large language models can learn to create complex word associations just by inputing mountains of texts.

> Some numbers represent real things. Like my little kid saw a pair of oven mitts and said 'two gloves'. Like, he was seeing a literal manifestation of two-ness in front of him - which is different than the logical idea of 1+1 or the symbol 2.

Yes but even then you are using an abstraction. The two gloves forming a pair are a good exemple because they are not even actually identical: there is a left one and a right one. Yet, we can abstractly see them as a kind of glove.

Great point. I kinda snoozed on the article and didn't catch this idea but yes I see what you mean

But actually, does it stop being a pattern at some point? Is anything real. Like is h^2o really 3 atoms? Can we say that's a number. Can we count electrons?

> some numbers are purely theoretical.

It is naive (no offense indended) to think that only things that can be seen in our "human" reality exist (think of viruses etc.). But even if this was the case with numbers, what is the problem? Natural language allows us to entertain abstract thoughts with no immediate realization, and numbers are just one of these things.

No offense taken and I am kinda amused that I triggered you into being defensive about numbers (not saying you are, it's just a funny image in my head)

So to be clear, there's no problem but there's clearly a difference between things that are solely abstract and things that also have a physical manifestation.

I'm abstract, we can talk about "Napoleon's invasion of Russia," "Napoleon's invasion of Japan," and "Napoleon's invasion of the nucleus of an atom" as concepts on the same level, but it still matters is that one is a historical fact, one is a thing that didn't happen but logically could have existed, and one is a purely abstract idea without meaning.

Same pattern. "3 has definitely happened. There's definitely been 3 of something", "739163859115 may have happened. No guarantee that this exact number of anything was ever present but obviously could have", "10^1,000,000,000,000,000 isn't possible because it's more than countable things in the universe."

But I agree with you that this is just a human perspective. To the infinite G-d, no number is abstract :)

You might be interested in reading some mathematical philosophy.

Here's an excerpt from Bertrand Russell summarizing Gottlob Frege's answer to "What is a number" [1]:

> A trio of men, for example, is an instance of the number 3, and the number 3 is an instance of number; but the trio is not an instance of number. This point may seem elementary and scarcely worth mentioning; yet it has proved too subtle for the philosophers, with few exceptions.

> A particular number is not identical with any collection of terms having that number: the number 3 is not identical with [page 12] the trio consisting of Brown, Jones, and Robinson. The number 3 is something which all trios have in common, and which distinguishes them from other collections. A number is something that characterises certain collections, namely, those that have that number.

[1] https://people.umass.edu/klement/imp/imp.html#chapter2

The idea that "3" is not the same as a group of three things seems fairly straightforward, in the same way that a seeing a red ball and a red shirt next to each other and describing them as "red" is not the same as the concept of "red". The "weirdness" of numbers is that seeing 3 balls in one place and 2 balls somewhere else lets you say "3 + 2 is 5, so there are 5 balls"; we don't have a system of "composing" most descriptions of a shared category in a universal way. While we might be able to come up with a way of composing some of them, like "horizontal striped shirt plus vertical striped shirt equals plaid shirt", we don't (at least right now) have a system that makes it simple for two people to independently come up with consistent answers (does red shirt plus white shirt equal red and white striped shirt, or pink shirt, or something else entirely?)

The difference is the complexity of the concepts involved. There are predictable results when you e.g. combine and filter light spectra, stack several lenses and mirrors to make a telescope, mix two chemicals, put electric circuit components together, or perform a well-defined sequence of knitting stitches, but describing them is more difficult than whole-number arithmetic.

> a literal manifestation of two-ness in front of him

Did he though? Are there two gloves, or 100 wool threads, or a 10^23 atoms? Would a different mind see the same thing?

Rely good point!

All numbers are purely theoretical; an abstraction.

There are no "two gloves", just as there are no "two stones". There's energy in certain configuration and if this configuration matches a certain macroscopic criteria, we call it "a stone".

So, even though you see ten of something, the "ten" is only in your head.

There are numbers of energy configurations, e.g. the excitement states of the electron in the Hydrogen atom.

Sorry for the late response, I've been absent for a while.

I'm doubtful that the excitement states of the electron are more than an abstraction themselves; useful mathematical tricks.

But that's just my personal opinion.

This article made me think about how many things we (as humanity and as individuals) have not experienced enough times to realize their patterns, and to start identifying them as things. Without being able to symbolically reason about them as things, we can't make them, we can't manipulate them, and we can't begin to understand them. The pattern-seeking brain is amazing, but also dauntingly limited.

This seems to relate to the myth of Magellan's invisible ships. We probably overlook a million things in our lives just like that.

If we lived in a quantum world where everything is delocalized, where there are no things to count and the concept of time hardly exists, we wouldn't need numbers, but we would still need math to model reality, as that reality had distinct properties.

With the right brain and sensory abilities you don’t need numbers. You can make computations by interpreting quantities as sort of temperatures or waves. This however is not something a human can comprehend, nor is their mind and body optimized for it.

How inevitable is the concept of numbers? - https://news.ycombinator.com/item?id=27279913 - May 2021 (202 comments)

I thought I knew about numbers but I didn't really have a solid grasp until a year or two into college.

tl;skimmed only - but my view is that all lifeforms are computers. The main thesis here seems to be that math evolves naturally as a way to reduce computational load. Makes sense to me.

This is a wonderful illustration of how dumb I am, I think.

Given Wolfram’s writing on computational essays, I don’t understand why he doesn’t publish essays like this as Mathematica notebooks, so we could play with the automata or fork the essay and explore or respond.

Publish the export, of course! But what a great chance to push the computational essay idea with a notebook export too.

I'm slowly starting a blog, and this is something I have struggled with. For now, I am writing the blog posts in markdown but then supplement the posts with notebooks so that there's clean and complete code without in-between versions.

I have often thought that it would be nice to just write the articles fully in the notebooks, but it's not ideal. One thing is that articles often build up ideas where providing code is often better to just show the complete pieces. Having everything in one notebook muddles these two things. Also, converting notebooks to markdown ot HTML/CSS/JavaScript is not all that straightforward when your notebook includes LaTeX and Mermaid diagrams.

So for now, I'm sticking with articles in markdown supplemented with notebooks. The downside is that the notebooks are not standalone.

[spoiler alert: emacs evangelism]

Org mode + babel allows exactly for this sort of literate programming notebooks.

That doesn't solve my problems though. F# notebooks with Polyglot Notebooks or Elixir notebooks with Livebook, which are what I use, are much more advanced than Jupyter notebooks and likely anything Emacs provides.

For example, in a Polyglot Notebook in VS Code, I can freely mix markdown, F#, C#, JavaScript, HTML, Python, LaTeX, Mermaid diagrams, images, and more.

I can only assume Emacs provides nothing close to this.

Your assumption is incorrect, I don't know how it works is VSCode, but what you described is exactly my flow in Emacs org-mode.

I can even annotate my code blocks to execute on remote machines, or to "tangle" my code blocks together to deploy standalone files my local or remote machines.

The meta language in tangle let's me reuse code snippets for machine specific configurations. I can almost replace my DevOps workflows with Emacs Org-mode.

It takes practice to use, and the documentation isn't very good though. Too terse for my preferences, but ChatGPT is really good at helping me understand.

You can display images, including interactive ones, rendered LaTeX and diagrams, and arbitrary HTML/CSS/JavaScript inside "notebooks" in Emacs?

Yes and read email, organize a file system, shell into the file system in Bash (or Lisp)…

…basically, “yes” is the most likely answer to “can eMacs…?”

(Though I am not sure about the current state of web browsing.)

I don't think the parent asks "what is available in Emacs in different places" or "what is theorically possible", but whether org mode has feature parity with what he uses, and the answer is not.

There might be a way to have plots inside org mode (e.g. org-plot) but you don't have "arbitrary html/js" plots (except as links to other in a browser, and I doubt you get even close to the same level of interactivity.

And even what you get regarding plotting is not turn-key that way VSCode Jupiter and Polyglot are.

Thanks for getting it. Steps to run my F# or Elixir notebooks are:

* Install F# or Elixir

* Install the Polyglot Notebook extension in VS Code (for F#) or install Livebook (for Elixir)

* Run notebook

The idea of what Emacs provides is irrelevant to my problems of how to efficiently write blog posts either in or support by notebooks. I gave up the path of Emacs long ago for the exact reasons of this chain. Yes, you can do anything or a collection of things, given enough time. For VS Code and these notebook technologies that I use, I do not spend time fussing about with the notebook technology.

I see, this makes sense, thank you for helping to clarify.

That being said, I feel like this kind of the entire point of emacs. Everything is pieced together, and very little is "out of the box".

At least, It's THE main reason I started learning Emacs, I was tired of the lack of interoperability of my different tools and I wanted an ecosystem that lets me piece things together as needed, like the Unix philosophy but with a tighter interactivity loop.

It's a double edged sword, in that it takes much more effort to build and maintain, but there's nothing really like it.

I still use VSCode and Jetbrains IDEs, but they don't drive my workflow like Emacs.

Anything that isn’t VS Code isn’t VS Code and won’t work just like VS Code.

That means it probably is not worth switching tools when the existing tool is good enough for something else that might be good enough.

But not-right-for-OP is not the same as eMacs-can’t.

>Your assumption is incorrect, I don't know how it works is VSCode, but what you described is exactly my flow in Emacs org-mode.

I doubt it. Care to post a GIF/video of a session with interactive html/js and plotting in org.mode?

I am not suggesting switching tools. Tools need to fit the ways we work and be good enough for the works we do.

If you want a better idea of why some people live in Emacs, maybe check out Literate Devops:

Video: https://howardism.org/Technical/Emacs/literate-devops.html

Text: https://howardism.org/Technical/Emacs/literate-devops.html

Fyi the two links are the same, and point to the text. (I was able to find the video though).

You might consider trying Observable notebooks, https://observablehq.com

These are more convenient both for the code creation and for the final written exposition than Mathematica notebooks. Among other convenient features, code can be split across notebooks, and cells can be arbitrarily re-ordered on the page without any inherent linear dependency.

Caveats: you need to write your code in Javascript, and your notebook is hosted on a commercial platform (though it’s not inordinately difficult to copy it out from there.)

You might be interested in this blog post, where the author had a similar problem to yours, which he solved by annotating his source files with XML. https://buttondown.email/hillelwayne/archive/i-finally-found...

For what it's worth, I've found it ok (not great, but just ok) to write in Colab notebooks (maybe because I use it at work and I'm familiar with it). Here are a couple of such "blog posts" (which are basically just explorations for myself, but still clean enough to share):

• https://colab.research.google.com/drive/1kEmqjgjudlmPwDReTH9...

• https://colab.research.google.com/drive/1wbktvb8XWISitThnpRA...

• https://colab.research.google.com/drive/1vsEE7XooIF1jHZKWZbE...

Pros: Can just type Markdown and LaTeX math; don't need a separate step to convert to HTML/CSS/JavaScript.

Cons: URL is ugly, code can only be Python (I think?), the UI is janky, can list lots more. But the friction was low enough that it got me to write at all (and move on without endlessly fiddling and never posting), and that's the most important thing for me.

I imagine if I were more comfortable doing everything in Javascript I could use Observable notebooks, as suggested in a sibling comment.

We know many useful maps from numbers to other things and from other things to numbers; my guess as to the lack of a CA export is that they're still trying to fill in the ???? between

> "Given the complete pattern of who wants to go where, we can dispatch specific vehicles to drive in whatever complicated arrangement is needed to optimally deliver people to their destinations. It won’t be like the trains, with their regular times. Instead, it’ll be something that looks more complex, and computationally irreducible."

and a notebook that actually does something both computationally irreducible yet useful.

(as far as I'm concerned, two naturals suffice for a turing tape — and the second is probably sugar)

The sheer number of scare quotes in this essay makes it very uncomfortable to read, and in the end, I don't think it says anything substantial about the concept of numbers. Even if you distill out the usual finite automata fillers

I think it was well written as a pop-science piece, albeit some things were too technical for me and unnecessary in my view. Wolfram tries to make everything about cellular automata, which I'm not sure why is uniquely relevant here.

The gist of it is pretty interesting to me: "things" only exist because we, as humans, observe the universe in a sequential manner through instants of time. If a collection of atoms stay the same through a few of those instants we consider that to have permanence, and be a distinct object or thing. By observing distinct things we can then count them, but that would be impossible if not for our experience of time.

It is indeed unclear if that is how every being would necessarily observe the universe.

tldr, visual symbolisation leads to numbers. Example being Inuit's (eskimos) have many different words for the different types of snow they experience, but Europeans have maybe a few words or even just one word to describe snow.

Vocabulary is defined by your environment, which will include concepts like numbers.

Look at Hungarian notation and decimal or imperial counting schemes. Different methods of counting numbers to suit the environment they evolved from.

> Europeans have maybe a few words or even just one word to describe snow

Really? In common use: snow, sleet, hail, flurry, frost, hoar, ice, rime, (snow)drift, powder, slush. Less commonly: firn, neve, sastrugi, suncups, ...

Well I understood they had lots of words, some suggestions have been as high as 52 words for snow, but other sources suggest otherwise. https://www.thecanadianencyclopedia.ca/en/article/inuktitut-... but it appears Scots have 421 words for snow. https://www.bbc.co.uk/news/uk-scotland-34323967

But another concept which an AI or aliens may not be familiar with is the concept of time and the way it is visualised.

https://www.bbc.com/future/article/20221103-how-language-war...

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