See also: Men at Arms by Terry Pratchett. There is a part where Troll Detritus (normally slightly less than human intelligence) is being frozen to death, but his brain works better in the cold, and can make the leap mentioned above. Mathematics ensues.

"Equals what? Equals what??" - Lance-Constable Cuddy

Good answer, but it feels like moving the goalposts a bit. The premise is that this society only has two numbers, but this solution is essentially "invent more numbers". This society can represent a 2 as a series of 0s and 1s, but I don't see why they'd ever do that if they have no concept of 2 in the first place. I do think a society like this would in reality develop the ability to count, but it seems contrary to the OP's premise. This society can't count past 1, changing how they represent those numbers doesn't fix much - you can order 110101 bricks, but how does that order get filled?

Forget 178654 - I don't think humans can conceptualize 15.

@NuclearWang: The same way they conceptualised it in the first place. Maths. Pick up a brick. Add it to the pile. Use the rules of addition and subtraction to add '1' to the pile of many you know you already have. Write it down if you have to. When the number on your blackboard (or in your head, if you're super smart) is '110101' you know you have sufficient bricks and you can stop counting. Then you can be jealous of Jeff, who realised that he can arrange his bricks in piles of 10 and do half the maths.

I agree with @NuclearWang. You didn't have to make it this complicated, you could have just said that they could use hash marks instead of binary, i.e., they could represent the numbers as 1, 11, 111, 1111, 11111, 111111, etc... But that's inventing numbers they can't conceive, so perhaps the question needs to be clarified?

@JoeBloggs But that defies the OP's setting. These people can only count 0, 1, and many - in the scenario described here, someone has recognized that putting 1 brick next to another yields a distinct number of bricks, which will change when you add yet one more brick, and then another and another. They are counting, I don't know what else you could call it. The question is "How technologically advanced can a society without numbers become?" and this answer is "They will invent numbers", which doesn't really answer the question.

@NuclearWang: the very last sentence is ‘they will invent numbers’, because that’s a natural consequence of thinking about mathematics. The rest of it is just maths. If I know the rules for how one can be turned into many (which I must, or I wouldn’t be able to define either state), I can do maths. I don’t need to understand the numbers I’m inventing until later.

@ DavidLjungMadisonStellar: If the question had been ‘they understand none and some’ I’d have used unary as my example. The same idea still stands though: you can do maths (to add 1 to my many I write an extra mark) without understanding how many ‘many’ is.

Actually, @ DavidLjungMadisonStellar, if they can discern the difference between ‘none’ and ‘some’, but can’t tell if one ‘some’ is bigger than another ‘some’, then my answer becomes invalid. Otherwise maths is still go.

They don’t even need to use binary. We cope fine with base 10 even though we can’t intuitively see the difference between 8 and 9 — they’re both “many” to us.

"1001" is not in set "0, 1, or many". I won't "-1" as maybe they can do this without conceptualizing "1001" as a number but that is difficult to think about. Would they view any binary string longer than length 1 as "many"? Would they not really be able to intuit "10 is XX amount" but still be able to write 10 because some great genius among them noticed a pattern that allowed for the creation of binary but none of them comprehend what is going on, even with 10, and mindlessly push bits because the genius taught others a pattern? Like human math students not understanding a formula they use?

@Loduwijk: 100 is not in the set of numbers I can count intuitively in base 10. Nor is e. Or pi. Or i. Yet I am capable of using them because i know the right rules. Heck, if you try hard enough you can rederive all of modern mathematics using nothing but axioms of set theory which feature no numbers, just sets and the differences between them, and it still works.

@JoeBloggs I don't think those are good examples. At least as most people are reading OP (maybe you read it differently?). If you cannot fathom how many 99 is but count to 99 and read/write 99 and add/subtract to it, you're doing what I think OP forbade. If you really cannot count, yet you have some very complicated algorithm you follow "just because", like most do the chain rule or arc-length formula in class, and they find it equivalent, then your answer works. Either they are counting to 2 or they are not, and counting to binary 10 is counting to 2.

@Loduwijk: That’s sort of my point. I can’t count to the square root of minus one. I can take the square root of minus one to prove it exists. I can use the square root of minus one to prove all sorts of things, but I can’t count to the square root of minus one. Eventually I might say to myself ‘you know what’s easier than saying the square root of minus one every time? Saying i’, but to get to the concept of i I first have to do the maths. That’s exactly what these guys are doing. Follow the complex algorithm to prove numbers exist. Follow the complex algorithm to write them and use them.

And eventually, despite it being unnatural to think about numbers higher than one (just like humans find it unnatural to think much higher than the teens) you train yourself to understand the complex algorithms. Then you start giving the ones you use most names. Or you could not, but it seems weird to have a complex society that’s incapable of naming formulae.

Most living beings are only capable of counting up to 5. Apparently it's related to having a concept of numbers and ordinality in your mind. That's also the reason why young kids without a fundamental understanding of numbers have problems counting items.

A tally system is effectively "lots of ones". Each "one" is distinct. Being able to distinguish between different "many"s is important, even if you cant think of the exact amount. 1 gram vs 1 kilo. How many children do you have again?

I guess this just doesn't seem like a very Worldbuilding answer, as the only difference from our actual world is immediately brought back in line with how mathematics actually developed - this isn't worldbuilding, it's history. This seems like asking how advanced a society without language could become, and then saying they will develop language. I read the question as though this society does not have the capacity to understand numbers, but this answer essentially argues that they haven't developed numbers yet, but are perfectly capable of doing so. But then why haven't they?

@NuclearWang: I don’t have the capacity to understand i. It’s an impossible number, no more within my conceptual reach than an intuitive understanding of sneebleflarps (whatever they are). Yet: because someone wrote down basic rules like ‘1 is bigger than 0. Some is bigger than 1.’ and explored the logical outcome of the rules: I can use i to correctly orient a drone in 3D space without worrying about gimbal lock. The difference between ‘this is bigger than that’ and ‘lets go to the moon’ is curiosity, logic and time.

I will argue this is one of the worst answers. While binary numbers require only 0 and 1 to be represented, 0 and 1 in this case are symbols, not numbers. So the binary number 100 might consist of only 0 and 1, it is in reality the number 4. Hence representing all numbers as binary (for example saying we need 1000011 meters of distance) does not solve the OPs question at all, since 1000011 is still 67 as an absolute quantity (just written differently), but the alien species in the OP can only count to 1 and not 67.

@ultimA: You’ve missed the point of the answer. This isn’t about the representation of the numbers specifically. It’s about the maths used to deal with the numbers without the ability to directly contemplate the numbers themselves. 11 isn’t the number three. It’s just a handy way to denote ‘the many that’s one bigger than the many that’s one bigger than one’, in the same way that 100 in base 10 is a way of saying ‘a number that’s ten times bigger than the number that’s ten times bigger than 1’. You can use these numbers without being able to count or even conceptualise them, because Maths.

@Joe Bloggs: I don't think I've missed it. You cannot say "you cannot count up to X but you can get to it using math", because the very definition of counting to X is to keep adding 1. So if you have math and addition, you can count up to it. Your premise "Humans have trouble conceptualizing large numbers" is wrong as a demonstration, because sure we can count up to any specific large number. We might need a lot of time and patience, which we might not have, but the OP didn't imagine a species that is too slow to count, but one that is missing the whole concept of any number greater.

@ultimaA: If I ask you to add 10 to 100, do you count all the way to 110? No. You don't. You don't have the patience to count it and you don't have the conceptual framework to just 'know' what it is, so you use maths. You trust the rules you've been taught on how numbers work are true. So too for this species. They can discern the difference between 0, 1, and 'more than 1'. This is amply sufficient to come up with the idea of addition. Now, if they can't discern between different manies (because they can't count) it doesn't matter because they have mathematical rules for how addition works.

As an aside: I think this necessitates only that the people can conceptualize of a "smallest many", which is 1+1, and therefore distinct from many+many and many+1. This smallest many would just be 2, but we don't actually need the concept that it's a number, just that it's a many where subtracting an item from it yields 1. It's still a "many" but it's a many that follows some very specific rules with regards to addition and subtraction of ones. 10 is then a small many, and 11 is a small many and a one. 100 is a small many of small manies, and counting exponentiates upwards from there.

This is a fantastic answer, I would +many if I could. To all those that claim this is "just counting"... Just as it is difficult for the students to wrap their head around the ideas presented by Sneebleflarp, it can be difficult for those of us can count to limit our own mental models to put ourselves in their shoes. This is an abstraction, it is a tool for abstract measurement of many. To begin down the route of studying many, all one needs is to notice that some many are different from others. This pile is taller than that pile, I can carry this many, but I can't carry that many...

Point number 1 is extremely true. Here's an example. \$10 billion dollars is a lot of money. Most people have no idea just how much. At an annual salary of $50,000, assume a working life of 50 years, that's 4,000 LIFETIMES of working (50,000 x 50 x 4,000). Does the breakdown sound much larger than 10 billion? That's because your brain can't actually understand a billion to begin with, but you do understand annual salary, a lifetime, and 4,000.

@ckersch : The concept of ‘smallest many’ can be derived axiomatically as long as you can discern between different manies and the manies are discrete. You don’t even need the concept of 1 (though it cuts out a lot of deviration!)

@user253751 No, 15 can be done. You just picture 3 groups of 5. Both 3 and 5 easily accessible to us. The trouble comes when at least one factor of a number (or the number of factors themselves) is too large to picture.