How inevitable is the concept of numbers?

It is everything that has some relation to something that has that same relation to something that has that same relation to something that has that same relation to something that has that same relation to some unique thing that does NOT have that same relation to anything.

Ah, but we do, and it’s very simple: it is what five cows an five fingers have in common in the most obvious sense. There’s no mystery there.

Or, stated in another way, “fiveness is something you’ll be very sad not to see when you look at your hand.”

Using different bases, to me, is just about storing information. How many symbols do you want to create before you increase the length of your number and start re-using them.

I would say I knew what 5 was even before I learned any notation or written language whatsoever.

There is no Antimemetics Division

But for us humans, the tokens and concepts blend to some degree: maybe not so much with regards to numerals (although there seem to be exceptions ...). So even if 'Amour', 'Love' and 'Liebe' essentially point to the same concept, we might conceive of them differently, ever so slightly, depending on the language we chose to use.

Which gives an inkling of the vast difference between computers and humans, at the current stage.

The reason it seems mushy is because there are two related concepts at play: a numeral vs. a number.

A numeral is the symbol: 5 in Arabic numerals in Latin script or ٥ in Arabic script; V in Roman numerals etc.

A number is the quantity, the amount being expressed: five (English), ḫamsa (Arabic), cinq (French) etc.

Do we really know if adaptation is responding to discrete classifications of environmental pressure?

There could be other ways to describe how systems adapt. You are ascribing a lot to how you think it emerged.

I think your opinions are great, but I just have a hard time having beliefs about things that are as fundamental as numbers.

It's not that useful to count them. My personal interactions with trees have likely never involved counting them; not even when I was involved in woodland management.

Trees control their numbers according to whether or not they can reasonably live in a particular density of tree coverage.

Only humans want to count trees, perhaps in general to claim (abstract) ownership of a particular large number.

I really like this part of the explanation, although not all adaptive systems are intelligent. E.g if I add a grain of salt on a heap of salt and it suddenly collapses, that is an adaptive system but not as a result of intelligent modeling of how it should behave.

> Not because numbers are fundamental to the universe. Rather they're fundamental aspect of intelligently adapting to (i.e. thinking about and making predictions about) the universe.

Here the problem is assuming intelligent adaptivity and the agent that has it is something separate from the universe. Being able to have a model of the universe inherently requires internally resembling the structural functional organization of the universe, a mutual conformity if you will, and therefore if numbers, counting, modes, categories etc are fundamentally useful constructs, I think they also at least resemble a fundamental part of the universe.

Who would claim they are? Mathematics initially co-evolved with our ideas of numbers, but many other important objects have been thought about for a very long time now.

https://youtu.be/GAcUZ3my6E0?t=480

Actually such computer would look like a universe, moving particles around in a continous space.

Since to humans, explaining numbers with hydrogen would be harder, I think. The common ground between humans is larger, so we can rely on something less fundamental and abstract.

If i'm at a party and everybody wants 2 beers, then if there is 1 person I need 2 beers (1 * 2), 2 people need 4 beers (2 * 2), but 0 people need 0 beers (0 * 2).

Fish have 0 legs. So how many legs do N fish have? N*0 = 0.

Now division by zero, that is Devil's idea.

If I cover lunch for you, you would owe me $5.

In a way, you now have -$5. You won't have $0 until you pay me back the $5 you owe. We can also say you have $5 of debt. That makes the number positive while still representing the amount. But when reconciling your books, it'll be subtracted from your total.

The thesis, broadly, revolves around contemplation about whether an number is an object, but the larger question is whether it's possible to have any small part of mathematics, even a single integer, which doesn't imply the whole (or at least, a significant structure).

edit: another intuition is that ordering is a property of numbers. Positive numbers are ordered in one direction. Can numbers be ordered in other directions?

I recently found this to be helpful when investigation fixed and floating point numbers. If ”1011” means 2^3 + 2^1 + 2^0, which is 11 the “1011.101” means the same thing but with an additional 2^-1 + 2^-3 aka 5/8. Negative powers seem weird to begin with but it kind of just is there to just discover, due to the symmetry.

This kind of arithmetic is far less fundamental than what you and the article are talking about — it is just representation really, in computers — but I think it is a good example of how if you can tread a path in one direction then turning around and coming back to where you started then carrying on in the other direction is a useful tool for teaching and learning.

I have always been unsatisfied with the current foundations of math and their obtuse basis in Set Theory. Although I must admit, Category Theory has alleviated that quite a lot, especially with the relaxing of equivalence compared to equality.

1. https://en.wikipedia.org/wiki/Intuitionism

2. https://www.youtube.com/watch?v=HeQX2HjkcNo

3. https://www.maa.org/press/maa-reviews/the-philosophy-of-set-...

I don't have a good explanation of what zero is in this context... Would someone else have a way to explain it (i.e maybe related to the nullspace?).

So you start off with zero, and the successor to zero, and the successor to that number, etc, and they're the counting numbers...

> what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero.

...and there are all sorts of numbers that aren't counting numbers, but can be manipulated by the same rules of arithmetic. E.g.:

    h*2=1 -- fractions

    n+1=0 -- negative numbers

    r*r=2 -- irrational numbers

    i*i+1=0 -- imaginary numbers

So extend the concept of differences between natural numbers, by subtracting a large number from a smaller number and defining the result as belonging to a new class of number. Similarly, get fractions by defining non-integer ratios between integers, real numbers by defining non-fractional limits of infinite sums of fractions, imaginary numbers of defining non-real roots of polynomials with real co-efficients, and so on.

Each time, we have an operation that works for some subset of our numbers, so we look at what does not work, and see if we can make it work anyway by defining the result of such an operation as a new type of number. And then we repeat with a different operation.

But there are things we cannot consistently "extend", like division by zero, or 0^0, so we leave just those out.

Think about how electrical charge and currents work at the physical level. There's electrons, which have negative charge, or holes, which have positive charge. Holes are just an empty place an electron can go, rather than an extant particle.

Similarly, when we think of negative numbers in relation to counting numbers, we're just using a notational trick to keep track of a place holder or hole, a spot where a unary count can potentially go later to cancel it out.

There's a pretty fascinating book named Quantum Computing Since Democritus by Scott Aaronson, one of the leaders in that field. The idea of the book is "Could the ancient greeks have discovered quantum mechanics?"

Much of the higher level math in the book is past my familiarity, but the central theme is clear and intuitive: if you take ordinary probabilities, generalize them to allow negative probabilities, and then generalize those to allow complex numbers, out pops quantum mechanics quite naturally.

Why do these two generalization steps make sense? What the heck is a negative probability of an event? It's exactly just notational borrowing in the same sense as above. Why generalize to complex numbers? That's more tricky, but I think of it from two directions: 1. It allows you to model partial constructive and destructive interference of probabilities, rather than just simple union or intersection. 2. Complex numbers are algebraically closed, while more simple numbers are not. So it feels natural that our ultimate number system to model nature would need to extend all this way.

I realize some math heavy folks would find my way of thinking of this a bit hand wavy, but it really has helped me cut through the confusion and mystery. It also fits in very well with a bayesian perspective on probabilities.

This, the "fields are what's real" perspective on physics, and bayesian epistemology in general have greatly simplified the way I think of these big idea question topics.

At t1, you say one.

...

At t7, you say seven.

Positive seven is what you say while your clock says t7. Negative seven is what you plan and "will" say at t7 while your clock still says t0.

Positive operations and numbers move you to the right. Negative operations and numbers move you to the left.

So a negative number is simply a count of a change in direction.

This works very literally. Ten miles west followed by five miles east is five miles west. There's nothing mysterious or weird about the "absence of westness" when you've turned around and started off in the opposite direction.

It generalises neatly to complex numbers where i is a rotation in the complex plane - instead of being pointlessly-weird-for-the-sake-of-it: "the square root of -1 which we've spent years telling you can't exist, and now we're telling you it can."

Technically this is a form of basis vector. Beyond that things get complicated.

But the idea is still valid - you define your direction markers (even if they're functions instead of constants) and then you can work out where you are in the space you're exploring, and what "movement", "counting" and "position" mean in that space.

Multiplying by zero is always a null movement of no distance.

One of the things I did in my math for liberal arts majors class I used to teach was also to do a fictionalized version of the expansion of the concept of numbers from *N* to the whole numbers to integers to rationals to algebraic numbers to reals to complex numbers. I say fictionalized because whole numbers and negative integers historically come after (positive) rationals, algebraic numbers and transcendental reals, but for pedagogical purposes, its easier to go in order of increasing supersets.

IMO negative numbers, complex numbers for circles, etc can be thought of as practical overloads. If we agree that the symbol minus means a debt, we can use it like that. We could agree, on *5 as meaning something, and if enough find it useful in 200 years we will teach it in school as obvious.

I think the negative number is obviously useful (you can add up positive and negative numbers and get a balance, it has an easy way to be plotted, etc)

P.S. another example is the Orbifold notation, from Conway. https://en.wikipedia.org/wiki/Orbifold_notation

That's in a sense how every model works. All theoretical models of the world are human inventions that aid in making sense of the world in terms familiar to us and nothing more. If you get good results from thinking the world is made of atoms, then the world is made out of atoms. When someone comes along and explains the same thing with strings, then the world is made out of strings. Models are just 'manners of speaking'. It might very well be that you have a dozen entirely different, but equally accurate fundamental ways to talk about a thing.

It's as if Stephen Wolframs mental horizon starts getting more and more restricted over the years since he tries to frame everything he sees in terms of his physical theory.

I think if a lion could speak, we'd understand him, because we're much closer to "a speaking lion" ourselves than we realize. Our intelligence didn't just happen at the last mile between ape and humans. Our intelligence has been emerging from the simplest animal there is throughout our entire evolutionary history.

And even then we see birds, octopuses and so on use tools and solve problems very similarly to us, which have branched much earlier from us. There are many different types of minds that can evolve. But I believe the fundamentals will always be the same. They're fundamentals of information processing systems. And BTW, we know dolphins, parrots and so on can do basic math, like counting, and translating that count to another set of objects. So they understand numbers, that's what numbers are at their most basic.

We marvel at how enormous the universe is, and how varied life in it can be, how different the minds of aliens may be. But in that seemingly humble pondering, there underlies something very arrogant. We think our mind is special. We think it's unique. And other minds will think extremely differently.

I think we'll figure out sooner or later, that our mind is basically inevitable, the broad strokes will be replicated for every species throughout the universe. This doesn't mean it'll be immediately easy to figure out aliens and their culture. But those are the details, not the fundamentals.

In this analogy, the lion is our younger selves. Perhaps our brain changes so much that we cannot understand (or access) what we were thinking or experiencing anymore.

If we imagine that long term memories are encoded in a way that they can later be interpreted actively by the structures of the brain, we could expect that major developmental changes in brain function might impact the ability to interpret long term memory formed before the change.

Physical trauma, the major reorganisations of infancy and language acquisition all prompt changes in organisation and processing that could be what renders earlier memories inaccessible.

This (admittedly hand-wavey) idea seems to mesh well well with the observed details of the phenomenon. Frequently accessed memories may be re-encoded, leading to some continuity. Children themselves experience fairly continuous long-term memory. Types of memories where relevant processing might not have changed as much (smell, visual memory) might be more resilient. We would expect age of language acquisition to play a part. Etc.

You have pretty concisely described Wolfram's wheelhouse.

I understand that many of those well-developed fields which exist on their own terms, have standard methods, standard questions and standard approaches to moving towards answers.

The article jump between these fields to ask and grope for an answer to a simple question that in many ways can't be asked or answered in these fields.

One thing to consider is that present day computers can follow the mechanical production of mathematical propositions close to completely. But computers have a lot of trouble producing or following arguments like this, in "natural language", which have a definite logic to them but whose operation is not based on only explicit, codified rules.

Edit: To me, this sort of speculation is what philosophy actually should be doing. The questions that are "ill-defined but compelling" are the questions that have lead significant intellectual progress. How Zeno's paradox lead (or at least related) to the invention of calculus, how Einstein's thought experiments lead to relativity, etc.

We are Steven, Simon, and Sarah.

> How many berries do we need to collect for each.

A handful for each.

> How big a foundation is needed for this building.

What building? You haven't even built the foundation yet.

And so on. You can do without numbers. And in fact people did all of the above without measuring quantities in any meaningful way in the history of humanity.

Language or cave paintings or whatever, use discrete symbols to describe physical reality

But one two and three are also innate (immediately recognizable) quantities are they not? So does a person using such a number system actually count, or recognize and categorize only?

I mean numbers are fingers but they're also a kind of song "one two three.." you map on things.

https://news.ycombinator.com/newsguidelines.html