Imagine a species of humanoid beings living on an Earth-like planet somewhere in the universe; they have developed complex spoken and written languages and they can study their own anatomy and the environment. Assuming they are capable to count using 0, 1 and many, how can they construct any kind of transportation and building infrastructure? How far can their technology progress?

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    – Monty Wild
    Mar 7 '20 at 17:15

As far as you want.

Humans have trouble conceptualising large numbers. We can’t count them, so above a certain point we see the number ‘178654’ and our brains turn it into ‘many’. Doesn’t change the value of the number, just how we intuit about it. For anything bigger than the number we can imagine (varies from person to person) we start doing maths rather than counting.

So how can we do maths?

We break the number down into smaller numbers. There’s one lot of 100000, 7 lots of 10000, etc. 100000 is just ten multiplied by itself lots of times. If I try to imagine 100 people what I actually imagine (again, this varies person to person) is a grid of 10x10 people, because I know that’s 100, even though I can’t count 100 people without my brain giving up and saying ‘many’. Neat.

But how does this help your species? They can’t count above 1!

They don’t need to. Introducing:

Base 2, AKA Binary!

The only numbers you need for mathematics in binary are 1 and 0. Everything else is simply a matter of placement. 0 is 0. 1 is 1. 10 is easy, it’s one lot of one more than one. 11 is one more than one plus one. 100 is one more than one lots of one more than one.

If you need to actually ‘count’ things, don’t do it in your head. Write it down. You know you can’t conceptualise past 1, so don’t try. Mathematics doesn’t require you to count the numbers, merely to trust that the symbols you write down and the rules you know work do, in fact, work. So you get a delivery of ‘1000 bricks’, then when you’ve moved one brick you write down ‘111’ bricks, because that’s the rule for subtracting 1. Doesn’t matter that you can’t conceive of what 111 bricks actually looks like. Maths doesn’t lie.

And we (as humans) know that maths in binary works. Our computers haven’t even got the concept of many. They work using nothing but 0 and 1, and somehow we’ve managed to use them to build some of the most complex buildings in the world.

The other answers cover what to do before you start up with the concepts of mathematics. After you’ve got basic maths down (even if it’s simply binary or, if you can grasp the idea of ‘the smallest many’ base 3) you can use that to do anything humans can do.

Calculated it will take 1111011011100111 bricks to build this house? Cool. Order them and get going. Need to measure a distance of 1000011 mm? Sure. Your tape measure has those markings.

And the weirdest thing is that once you have the methods to write and manipulate numbers you just might find some people start to think in terms of the maths instead of the numbers. And they might want a word for 10 that isn’t quite as clunky as ‘one lot of one more than one’. Say.. ‘Two’...


There have been quite a lot of comments along the lines of 'but this is just counting using a different number system'. That's not the point of this answer. The point of this answer is that this species is more than capable of doing the maths even if they can't wrap their heads around the actual numbers involved, much like I can use the concept of i (the square root of minus 1) even though it's impossible for me to conceptualise or even count to it.

To explain in a bit more detail, here's Professor Sneebleflarp with the first lecture of 'The theory of many' (AKA Rederiving mathematics when you can't count)

Good day. My name is Professor Sneebleflarp, head of advanced philosophy at the university of Gnurf.

Today you will be learning the beginnings of what is known as ‘The theory of Many’. You may all wish to shake off those hangovers and focus, because what I’m about to teach you is hard to wrap your head around, and it is examinable.

Now. Look around you. You may notice that although there are many seats in this hall, and many students here to listen to me drone, there are nonetheless many students still standing. By the way, if you each make sure you bring one extra seat from just down the hall next time we’ll have many empty seats instead, which I’m sure those of you with the hangovers will appreciate. The problem of ‘how can we make sure we have no standing students and no empty seats’ is the problem we will attempt to crack today, along with some notes on nomenclature and convention.

To solve this most complex problem, contemplate my desk. You may note a complete absence of stones. There are no stones on this desk.

Now, contemplate the basket next to my desk. It has an abundance of stones. A glut of stones. In short: The basket contains many stones.

If I take a stone from the basket and put it on the desk I now have one stone on the desk. This much is plain. I take another stone from the basket and place it on the desk I now have many stones. There has been a change. But if I take another stone and add it to my desk, I still have many stones. No change. The procedure of taking a stone and putting it on the pile is known as ‘adding one’. Adding one stone to one stone yields many stones. Adding one stone to many stones also equals many stones. This is natural and understandable.

Now, I shall put these stones to the leftmost end of my desk. At the other end of my desk I shall add a stone from the basket. And then another.

I now have a pile of many stones on my left and a pile of many stones on my right. I take a stone from my left and a stone from my right and place them back in the basket, a procedure known as ‘simultaneous reduction’. What do I find?

I now have one pile of many stones and one pile of only one stone. How can this be explained?

The answer is simple, though you may wish to write it down. One many is not necessarily the same as another many. If I remove another stone from my left and another from my right I now only have one stone left on the left of my desk, though I began with many piles of many stones.

We can see through the simple method of removing stones that the many stones on my right can be reduced to no stones before the many stones on my left. This is known as being a ‘larger’ many. The reverse is known as being a ‘smaller’ many. In the case I have just shown you the leftmost pile of many stones was ‘one larger’, as I was left with one stone there after I had reduced the rightmost pile to nothing.

Now. I will clear my desk again. Then I set up the same piles of many as before, and move the pile on my right into the centre of the desk. I shall add one more to each of these piles.

Then I add one to the right of my desk. Then I add another to the right of my desk.

I have many piles of many, as before. One to the right, one to the left, and one in the middle. I will state, and you can verify in your own time, that the pile on the left is ‘one larger’ than the pile in the middle, and the pile in the middle is ‘one larger’ than the one on the right. Remember the nomenclature from before? Good.

Now I will remove the pile in the middle altogether and place it back in the basket. If I simultaneously reduce these many piles as before, what would we expect to happen? We end up with one in the pile on the left, right? Well, let us do it.

But what is this? I have many on the left? This is the case known as being ‘many larger’. ‘One larger’ is actually a special case of ‘many larger’, though you may have to wait a few lectures for that to become apparent. It is also true, as proven by the great thinker Fleeblesnarp many years ago, that any case of ‘many larger’ can be broken down into intermediate steps, as we had with the pile in the centre, until it becomes nothing more than many cases of ‘one larger’.

Now, this offers a solution to our seating problem. If we simultaneously reduce the number of students and the number of seats in this hall it will become apparent that the number of students is many larger than the number of seats. Remember that I said any ‘many larger’ can be broken down into many cases of ‘one larger’? Could every student currently standing please come and take one stone and put it in a pile on the left of the hall.

Now. The important thing about ‘larger’ and ‘smaller’ sets of ‘many’ is that you can simultaneously reduce them to discern how ‘large’ or ‘small’ the difference between the sets is. In the case of my desk: the pile of many stones on the left of my desk represents this difference. It is many. In the case of the seats, the stones on the left of the hall represent the difference. It may be the same many. It may be a different many. That is immaterial for the purpose of this demonstration. Now. All of the sat students. Please come and take a stone and place it to the right of the hall.

Now we see we have many piles of many stones. One representing the many seats, and one representing the difference between the many students and the many seats.

Now for the tricky part. If I were to take one from the left of the hall and put it on the right I would subtract one from leftmost many and add it the rightmost many. If I do that many times? I am physically recreating Fleeblesnarp’s theorem on the divisibility of many larger. This in turn means that if I simply take the leftmost pile of many stones and add it to the rightmost pile of many stones, like so, I will end up with a pile of many stones that is many larger than the many I had before. This is known as ‘adding many’, and is the conceptual equal of ‘many larger’ just as ‘adding one’ is the equal of ‘one larger’. Specifically, I have added the difference between the many standing students and the many sat ones.

I know this is hard to wrap your head around. You’re all sat thinking ‘but now you just have a pile of many stones!’ and you’re right. But if every student could now come and pick up a stone…

You’ll see there are exactly as many stones as there are many students. Therefore, if we can get the students that have a seat to come put their stones back…

Then next lecture only the students that are holding stones will need to get a chair each from down the hall. We will have many chairs, and many students, but we will have no empty chairs and no standing students.

Next lecture I will teach you about many lots of many, and begin on the rudiments of recording the sizes of many , or ‘binary’, as well as the symbology of ‘larger’, ‘smaller’, ‘add’, ‘remove’ and so forth. Remember: This is the work of many great thinkers. You won’t get it in one day. So read your notes. There will be an exercise due many days from now.

P.S.: It's really hard to write from the perspective of this race. Two is too tempting!!

  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Monty Wild
    Mar 7 '20 at 17:16

Back in the day, you didn't specify how many bricks and logs it would require to build a house. You just baked bricks and made logs while building until the house was finished. Any leftovers were kept for repairs or the next time a house was built: "We need to build 1 house, and we will need many bricks and logs and buckets of mortar - until we need 0 more."; "Get me a log as long as this piece of string."

Transportation: "This 1 cart needs 1 wheel in every corner, all of 1 size."; "The bus comes by whenever shadows on this sundial reaches a mark. If it is cloudy, guess."

Infrastructure: "We need 1 road from here to there - find many men and start working. You have 1 year to do it."

Also note that animals typically can't count more precisely than 0, 1, many, but they are still able to make housing (burrows, nests, termite mounds, beehives) and infrastructure (beaver dams, deer trails).

Science, however, will suffer. You can't develop mathematics, and without mathematics, you can't develop astronomy or much in the way of physics, beyond simple rules of thumbs. Medicine will be easier, as novices can learn by example from experienced medics, and medical drawings and charts usually don't need much in the way of numbers. Measures may not need to be finer than "1 thimble, 1 teaspoon, 1 soup spoon, 1 handful, 1 cup, 1 mug, 1 jug, 1 bucket", etc. to work for most things.

Overall, I believe that it'd be possible to develop something like an early industrial society, including railways, steamboats, and even simple aircraft, but probably not anything much more advanced than that, except in some fields like selective breeding.

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    $\begingroup$ Small point: if you can discern between ‘nothing’ and ‘something’, you can say that one ‘something’ is not the same as another ‘something’, and you can say that one ‘something’ is smaller/larger/the same as another ‘something’: you can use set mathematics to do pretty much everything. It’s a touch complex, but it’s doable. $\endgroup$
    – Joe Bloggs
    Mar 3 '20 at 9:33
  • $\begingroup$ You can also use geometry for some things but developing things like railways without actual mathematics will be a problem. You need straight railways and that means precise measurement over long distances and calculations. How much will it cost? $\endgroup$
    – Sulthan
    Mar 3 '20 at 18:58
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    $\begingroup$ @Sulthan: For straight railways, worker carry a rod indicating the space between rails. Rods can be copied with precision even though you don't have numbers. Just use the original as measure for any copy and refrain from too many generations of copies. $\endgroup$ Mar 5 '20 at 8:57
  • $\begingroup$ @KlausÆ.Mogensen Yes, that's the geometry. Tunnels, elevations etc. can become a serious problem. $\endgroup$
    – Sulthan
    Mar 5 '20 at 9:10
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    $\begingroup$ @Sulthan: Tunnels: Shine a narrow beam of light the way you want the tunnel to go. Carve where the light shines. Elevations: Build support where needed to prevent the slope from becoming too steep. I doubt early railmakers used much math. $\endgroup$ Mar 5 '20 at 9:35

They'd develop the same mathematics we did.

they have developed complex spoken and written languages

There is no rational reason a species capable of developing complex spoken and (particularly) written languages will not develop equally complex written mathematics. It's a natural progression.

Counting predates human written history. We have no idea when we replaced "ugg, ugg" with "two". It seems to be a function of developing a language to describe the world. The rest is just let try that rule, now can we extend that rule ? You cannot stop the develop of sophisticated number systems and mathematics unless you want them incapable of complex communication.

and they can study their own anatomy and the environment.

Then they'll ask questions like "how much can I lift using this lever ?" and so on. This is how numerical and later symbolic theory based physics and engineering develop.

It's going to happen.

Assuming they are capable to count using 0, 1 and many,

Zero is not a natural number - it's an invented number. We didn't start with a zero and one, two, three, we started with a ugg, ugg ugg, uggg ugg ugg ... - counting is a developing process and the extension of counting system is how we got from whole numbers (excluding zero - what's a zero of something anyway ?) to a system of numbers which includes complex numbers and non-computational numbers.

The curiosity which drives them to consider their own environment will drive them to develop mathematics to aid their exploration of that environment and the rules it works by. It's inevitable.

how can they construct any kind of transportation and building infrastructure? How far can their technology progress?

With only 0, 1 and many - not at all. You can get so far with empirical knowledge but it requires systematic study to develop proper industry. Most significantly the requirement to build anything large and expensive (as you must to develop a complex industry) also requires significant investment. We (and they) minimize the risk and reduce the potential for catastrophic error by using complex engineering based on a highly developed mathematics.

The Restaurant At the End of Your Fingers.

Let's say they miraculously develop a complex society which includes (naturally) restaurants and phones. They ring to book a table. Ann obvious and necessary question that's going to be asked is "how many of you will be coming ?". An answer restricted to 0,1 or many is not practically useful.

Your restaurant owner is going to want to be paid. Barter is great, but no society of Earth has failed to replace it with something better (or at least more practical) - money. But money and even the most basic form of business requires some kind of counting. "Many" is not going to cut it if you want to stay in business.

If they have fingers or even two legs, they're going to come up with the number two and probably as many basic numbers as they can count with their digits. If they're about to go to war with a neighboring tribe, no leader will be happy with the answer "many" from a scout sent to tell them how many enemy warriors are coming down the road.

The fact is we (and they) will naturally (and very early on) develop a need to produce a mathematics way beyond whatever basic "natural" counting system they start with.

So the idea is simply not possible.

How far could they get ?

As far as banging "many" rocks together and forget the languages.

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    $\begingroup$ We didn't even start with "ugg". Even the Romans with a vast empire lacked the concept of zero. $\endgroup$
    – March Ho
    Mar 2 '20 at 19:13
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    $\begingroup$ @MarchHo : that doesn't mean they were morons and they didn't have the concept of zero. They knew perfectly well what the complete lack of something meant. They just didn't feel the need to use mathematics to express it. Instead of saying "I have zero sheep", they just said "I don't have any sheep at all". $\endgroup$
    – vsz
    Mar 3 '20 at 5:32
  • $\begingroup$ Money is not a replacement of barter, it is a limited commodity which has its (admittedly rather arbitrary) value enforced by the government. On a base level we are still bartering; we just have one commodity which the government forces everybody to accept. $\endgroup$
    – The Daleks
    Mar 3 '20 at 5:37
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    $\begingroup$ Barter is not so great. There is no evidence that any pre-money society has made significant use of bartering. Mostly they just give one another presents. en.wikipedia.org/wiki/Gift_economy $\endgroup$ Mar 3 '20 at 9:22

Unary and artifacts

I'm going to assert that the aliens are human level intelligence, but for some reason cannot mentally conceptualize and create words for distinct numbers. As such, they can still understand relative sizes and so on, but cannot for the life of them keep a numerical value (aside from 0, 1 and many) in their head.

I am reminded of a story I heard about how ancient shepherds counted their sheep. I don't know if any of it is actually true, but it goes as follows: In the morning, the shepherd would round up his sheep and, for each one of them, put a pebble in a bag. In the evening, he would do the same, but remove a pebble for each. If, in the end, there were still pebbles in the bag, he had lost a sheep and had to go look for it.

If an alien is somehow incapable of naming and storing a numerical value mentally, they could still start of using basic unary arithmetic like above. Something like addition is a trivial development; pour one bag into the other. Subtraction isn't too far behind; remove one pebble from each bag at a time. When one bag is empty, the non-empty bag is the difference between them.

This method of storing numbers using artifacts could be further revolutionized by standardizing a weight for each individual "unit"; comparisons of large numbers can then be trivially performed by scales. This would allow for the next revolution, a simplistic base system to compactify the work of performing arithmetic.

They may decide to introduce a heavier pebble, one such that its weight equals some integer number of other pebbles. Since they have no innate concept of numbers, it'd probably be arbitrary, but let's say for the sake of simplicity that they pick 10. They make a new pebble that's as heavy as 10 units. They then make more and more copies of it such that they are all as heavy as the first one. Presumably, it would also have a different color or something so as to be more recognizable as special. When performing subtraction, they would make sure to first remove each pair of heavier pebbles. If there's an uneven amount, use another scale to measure out how many unit pebbles the heavier pebble correspond to and just pour those back into the bag, then continue as normal.

This notion of making progressively heavier pebbles may continue, creating fewer pebbles to be manually operated upon.

If pebbles in a bag are inconvenient (they roll all over the floor if dropped!) one could replace it with, for instance, disks on a rod (or rope?) for simpler long-term storage. For numbers that need to be stored very-long-term or transported far, one could smelt some metal and cast it into some artifact such that its weight exactly matches that of the corresponding number. (On the receiving end, you'd then just have to pour pebbles into the other side of the scale until they match to decode what number the artifact corresponds to and then perform arithmetic as normal to it.)

Eventually, some of the aliens may make an even bigger logical leap of storing this data on paper (or tablets, whatever). It could start out as simple as "one dot on the paper corresponds to one unit weight". One can just then add one unit in a bag for each dot on the paper, allowing numbers to be transported more easily (albeit at the cost of a lot of work to encode and decode the number).

The base system becomes even more useful here. They may yet again decide to use a different symbol for a larger amount. They may say "a circle means a heavy pebble instead of a unit one". Alternatively, they could just standardize a translation document. Everyone gets a tablet that says something like

  • O = ..........

...and so on, allowing the written numerical system to potentially diverge from the weight-based one.

Once you have numerals on paper, some people will probably make logical leaps that allow them to perform some operations on paper without pouring pebbles into a bag or whatever. Sure, it'll be way more cumbersome without an innate ability to keep numbers in your head, but it's perfectly doable.

At this point, it seems to me like we have everything we need for mathematical progress to be made. Everything will be a gazillion times slower and some concepts (like, say, fractions) may be considerably harder for them to deal with, but it should in theory work.

If they then reach the point of building machines, maybe even electronic ones, then their problems are over. The machines can just do it all considerably easier (and faster!) than they can.

I'd hate to see what their programming languages would look like, though.

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    $\begingroup$ Your story about shepherds is not (only) some ancient thing. It (or minor variants such as marks on a stick or knots in a string) is still a common practice in primitive herding societies. $\endgroup$ Mar 2 '20 at 17:46
  • $\begingroup$ Good answer, though the part about having some units equal multiples of others may be essentially a number system, even more-so than roman numerals (I, II, III, IV, V, VI, VII, VIII, IX, X, XI, etc.). $\endgroup$
    – Loduwijk
    Mar 2 '20 at 22:04
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    $\begingroup$ @Loduwijk: Yes. I assert that a number system may develop, even for a species incapable of mentally retaining numbers, by externalizing the storage of and operations on numbers into physical systems and then optimizing those physical systems. $\endgroup$ Mar 3 '20 at 11:11
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    $\begingroup$ Also note en.wikipedia.org/wiki/Abacus . It's a simple mathematical tool that enables you to do complex calculations without really thinking. $\endgroup$
    – Sulthan
    Mar 3 '20 at 19:01
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    $\begingroup$ @Smallhacker That is an excellent way to explain it. If only we had been so elegant in the large discussion under Joe Bloggs' answer, as it seems he's trying to make a similar argument to justify his answer. $\endgroup$
    – Loduwijk
    Mar 6 '20 at 2:24

Expanding Misha and L.Dutch comments. As my grandfather said

when you're waiting for a bus on a winter night the only state it is, is "not here".

We would say 0.

When you're building a house you need many brick, logs and stuff. How many? Until the building will be 1. You don't need numbers to have dimensions. That's why Americans measure holes in dogs and washing machines. You have finger, palm, foot, pygmē (or forearm). Until industrialization brickmakers in England had their own, stamped by king, forms. Which right now help us identify the brickmaker just by size of bricks used to build a house.

I would say that maximum development is early industrial stage (mabe pre-industrial). A lot of waste during production but the supplies are so in excess that it don't stop the production. Almost everything can be changed in error/success method (bigger wheel, smaller wheel)

Note that you don't need numbers to count time. For transportation you just say that wheel size of pygmē is better for transportation that the size of foot because it goes from point A to B in finger lenght of a knot and not palm.

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    $\begingroup$ "How many? Until the building will be 1." This way of planning virtually guarantees that most of the time the building will stay at 0. $\endgroup$
    – Alexander
    Mar 2 '20 at 18:49
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    $\begingroup$ @Alexander I've seen plenty of people build buildings on YouTube without counting resources, instead just grabbing "many" of something and using it, then if the work is not done getting "many" again. Often results in having extra things left over after, though they can have other purposes. Edit: next answer down as-of-now, by Klaus, puts it nicely. $\endgroup$
    – Loduwijk
    Mar 2 '20 at 21:59
  • $\begingroup$ @Alexander Hardly. Sure, during construction the number of buildings is 0, but most buildings worthy of the name "building" continue to exist for far longer than it took to build them, so it will be 1 for much longer than it is 0. $\endgroup$ Mar 2 '20 at 22:00
  • $\begingroup$ To measure some dimension in "dogs" you'd need to have a precise number of dogs greater than one. The numbers do matter. (FWIW, a small number of Americans once measured a bridge in smoots, but I've never heard of measuring length or breadth in dogs.) $\endgroup$
    – jpaugh
    Mar 2 '20 at 22:28
  • $\begingroup$ @Loduwijk apart from possible lack of resources or waste, many times builders need to decide on the question "How many X do we need to support Y?" If we resort to trial and error here, only primitive buildings can be built. $\endgroup$
    – Alexander
    Mar 2 '20 at 22:38

The answer by Smallhacker gives yet another great example of how these "can't count" people could do just fine, but it brought an abacus to my mind. If we are permitted to allow for the concept that "this thing is equivalent to a certain quantity of something else" then we can also allow an abacus.

Even someone who could not count could still probably learn to use an abacus and do lots of math quickly and easily. They could produce complex mathematical answers without actually counting anything.

Even if you don't want to count an abacus and insist that's essentially using a number system, we could suggest that the race may invent some other device that has essentially the same qualities but which does not rely on the digital nature of our own abacus.

No numbers does not mean no math.


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