Inspired by Can there be democracy in a society that cannot count?, how advanced could that society get technologically without learning to count?
What important innovation on a hypothetical tech tree is the bottleneck that requires them to discover counting?
Or put another way - Can you do maths in any form without learning to count first?
You're going to have to try really hard to not count!
Counting is a very specific action in mathematics - integer incrementing. 1, 2, 3, 4, etc. From incrementing integers we can derive most of mathematics, however that need not be the source of it. I'm guessing perhaps alien biology here (no fingers to count on?)
If we start with Boolean logic: True / False. And from that equality and inequality (True != False) == True. And then Boolean operations (true and true == true, true and false == false etc). We could step into probabilities, giving us the constants Always and Never, and giving us a sense of ordering before giving us a way to represent numbers (Eg probability of rain this year > probability of earthquake this year).
From the definition of equality with regards to probabilities we could define "Fair" as equal probabilities of both outcomes happening. We could then subdivide the probability space from Never...Always into equal chunks, so we basically have recursive power of -2 constants, giving us fractions with power of 2 denominators, eg: 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1.0). You've basically discovered binary fractions.
From binary fractions and known logical operations you can discover addition, subtraction, and multiplication. The discovery of addition doesn't mean you've discovered counting yet, as we haven't discovered integer representation yet - 0.25 is just "halfway between Never and Fair.", lets say they represent it as FN for short (Start at Fair, then halfway to N). 0.5 is F. 0 is N. 1 is A. FAA is 7/8 (Start at fair, half to A, then half that again towards A). FAAN is 13/16. FAAAAAAAAAAAAAA is 65355/65356 (0.99995409755)
From multiplication you can discover division. You can discover square roots from self multiplication. From addition and subtraction you can discover numbers below "Never" and numbers above "Always". From division you can discover square roots, and from square roots you can propose "imaginary" numbers.
You can start applying this probability to the real world via geometry. Draw a circle that touches all 4 sides of a square, what's the probability of a random selected point in the square being also inside the circle? From that and other circle probabilities you can get an approximation of pi.
You could even get to algebra - by identifying an unknown value and recording it as such you could do all sorts of things.
I think you could do a lot of maths from this approach - basically all of it. You only need to "count" when your applying probabilities to collections of real world things. Where this kept entirely theoretically I see no reason why you couldn't get calculus, trigonometry, etc before discovering counting.
You can measure distance by making a ruler the length of a known thing. A is 1 ruler long. N is 0 rulers. F is half a ruler. If the ruler is 1m long, I'd be A + FANAN rulers tall. This is getting a bit of a grey area as determining that my farm is 431 rulers long is easiest done by incrementing an integer as I move the ruler. This is one of many ways in which you'll inadvertently discover counting.
Currency could work - especially if like bitcoin your just dealing it fractions of a big value, Eg 1.0 = "All the money in the kingdom", for example. But buying and selling goods is like ground zero for counting.
I think government will discover counting while trying to distribute resources, as in order to calculate the fraction of resources a group gets, one must count the groups size. Or some other Census. Trade and Taxation could theoretically work under this number system but opportunities to count would be everywhere.
No way to keep records.
Presumably by 'count' you mean they can say "one, two, three, four, . . . " and go on as long as they want. There are two parts to this:
Have names for all the numbers. They could be "un, deux, trois, cat, . . . " or "один, два, три, четыре, . . ." as long as the names are agreed upon
Can say the numbers in order when prompted, either with or without objects to help them count.
Some elements of democracy can be done without 1. For example suppose there are two candidates and everyone gets one vote. To manage this without counting everyone gets a bean. There are two bins, one for each candidate, and to vote for a candidate you put your bean in that bin. Once everyone has put their beans you take one bean out of each bin. Then repeat until one bin is empty. The owner of the non-empty bin is the winner.
Of course this assumes there are few enough candidates to make this possible logistically. If the country has two cities then you need two bins in each city, and to transport the bins to combine them. With counting you could simply count the beans and add the numbers. Without counting you need to transport the beans themselves.
Can you do maths in any form?
Yes! As described above you can do subtraction. You can also do addition. The problem is you have to do everything directly. In order to add 500 to 600 you need that many beans. After you have done the calculation, you need to keep those 1100 beans together since you cannot just write down "there are 1100 beans" as that would be counting.
One of the main cultural innovations that would require some sort of number system is a trade system- whether it's barter or currency-based. People need to count in order to know how much of a given good they're offering or obtaining in exchange. If this culture isn't trading, they'll have a much more limited range of materials to work with. (For comparison, real-world humans had trade routes for barter as early as the Neolithic/Chalcolithic.) Having a taxation system also requires some basic level of mathematics, so people who couldn't count would have a much harder time setting up a state. The same goes for establishing armies.
